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[Introduction](Introduction) +2. [Markov Decision Process](Markov_Decision_Process) + 1. [States](States) + 2. [Actions](Actions) + 3. [Transition Model](Transition_Model) + 4. [Rewards](Rewards) + 5. [Formalization](Formalization) + 6. [Policy](Policy) +3. [MDP Search Trees](MDP_Search_Trees) +4. [Utilities of Sequences](Utilities_of_Sequences) + 1. [Discounting](Discounting) + 2. [Finite and infinite horizons](Finite_and_infinite_horizons) + 3. [Infinite Utilities](Infinite_Utilities) +5. [Bellman Equation](Bellman_Equation) +6. [Time Limited Values](Time_Limited_Values) +7. [Implementation](Implementation) +8. [Conclusion](Conclusion) +9. [References](References) + +## Introduction + +In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating reinforcement learning mathematically. In a typical Reinforcement Learning problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called environment. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with states and agent's possible decisions are modeled by actions. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with stochastic transition models are used. In reinforcement learning, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by reward function in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on Bellman equation, So in order to find the best guideline for the agent Bellman equation should be solved. + +In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. + +## Markov Decision Process + +Markov Decision Process is based on Markov chain. A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The chain holds the following independence assumption. + +$$ P(S_{t+1}|S_t, ..., S_0)=P(S_{t+1}|S_t) $$ + +$S_t$ represents the state at time $t$ . Intuitively, $S_t$ retains all of the information about the history that can affect the future states so *the future is conditionally independent of the past, given the present*. + +### States + +A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards, return, can be computed. + +As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Notice that overheated is a terminal state. + +
+ + + +
+ +### Actions + +Actions are set of operations an agent is allowed to do in the given environment. The set of actions is usually shown with $A$. In the robot car example, the machine can increase or decrease its speed. So the actions' set would be {slow, fast} + +
+ +
+ +### Transition Model + +At each state, the agent decides which action to perform. The resulting state ($s'$) depends on both the current state ($s$) and the action performed by the agent ($a$). The transition model $T(s, a, s’)$ gives probability $P(s’|s, a)$, that is, the probability of landing up in the new state $s’$ given that the agent takes an action, $a$, in the given state, $s$. + +Environments can be divided to two types according to their transition models. + +- Determined environment: In a determined environment, if a certain action ($a$) is taken, the resulting state is the expected state ($s'$) with probability 1. +- Stochastic environment: In a stochastic environment, if the same action (a) is taken, with a certain probability ,e.g. $0.8$, the resulting state will be the expected state and there is $0.2$ probability that the resulting state is not the expected state. Here, for the state $s$, the action $a$ and next state $s'$ the transition model is $T(s’, a, s) = P(s’| s, a) = 0.8$. + +In the machine robot example, we assume that the environment is stochastic. The probability of transitions is determined based on the current state, action, and next state. +One of the possible transitions is shown in the figure below. + +
+ +
+ +### Rewards + +The reward function, quantifies the usefulness of taking a specific action and entering a specific state. These rewards incorporate the action costs in addition to any prizes or penalties that may be given. Negative rewards are called punishments. The general form of the reward function is shown as $R(s, a, s')$. But in some MDPs the reward function is independent from the resulting state or both resulting state and action. In those models reward function is shown with $R(s, a)$ and $R(s)$ respectively. + +For a particular environment, the domain knowledge plays an important role in the assignment of rewards for different states as minor changes in the reward do matter for finding the optimal solution to an MDP problem. +Sum of the rewards is called *utility* or *return*. + +Usually the performance of the agent, which is also called *utility*, is measured by the sum of rewards on the visited path. But other utility functions are also possible. General form of utility is shown by $U_h[s_0,s_1, s_2, ..., s_N]$. + +In the robot car example, the robot must reach its destination as quickly as possible, so the high-speed reward must be considered more than the low-speed, e.g. +2 and +1 may be considered. But reward should be assigned in a way to prevent the robot from going to overheated state. So -10 is assigned to entering the overheated state. + +
+ +
+ +### Formalization + +By combining the concepts which were explained above the complete formulation for Markov Decision Process is achieved as below. + +- A set of possible states $S$. +- A set of possible actions $A$. +- A transition function $T(s, a, s’)$ + - Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s’| s, a)$ +- Also called the model or the dynamic. +- A reward function $R(s, a, s’)$ + - Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s’)$ can be replaced by $R(s’)$ or $R(s', a)$ +- A start state +- Maybe a terminal state + +### Policy + +The policy is a function that takes the state as an input and outputs the action to be taken. Policy $\pi : S \rightarrow A$ is a set of commands that the agent follows. + +The policy which maximizes the expected utility among all the possible policies is called optimal policy and shown with $\pi^*$. + +Finding $\pi^*$ is usually referred to as solving the MDP. There are different algorithms which attempt to find $\pi^*$ such as value iteration and policy evaluation. + +## MDP Search Trees + +In expectimax tree the path which leads to the most reward is desired. MDP problems could be demonstrated as a search tree. But there are two main differences. + +- In MDP Rewards are assigned to edges rather than nodes. +- In MDP utility is the sum of rewards gained in the path to the terminal state. But in expectimax utility is the the value assigned to the terminal state. + +In the robot car example, the search is as shown below. + +
+ +
+ +## Utilities of Sequences + +It's easy to choose between sequence of [1, 2, 2] and [2, 3, 4]. Obviously more utility as well as more reward in each step is gained . But choosing between [1, 0, 0] and [0, 0, 1] is not that easy. This motivates us to consider the time which the reward is being received. In most cases, it's preferred to gain the rewards as soon as possible. + +### Discounting + +Consider gaining reward $r_i$ in time step $t_i$. In discounting, $\sum \gamma^t r_i$ is considered instead of $\sum r_i$ as utility function such that $0\lt \gamma \lt 1$. $\gamma=1$ is just equivalent to simple summation. Apart from what we have said so far, this method has other advantages, which will be mentioned below briefly. + +### Finite and infinite horizons + +Decision making problems may be of two types. Some may be *finite horizon* and some *infinite horizon*. Finite horizon means that there is a fixed time N after which nothing matters. In other words what happens after the fixed time is not analysed. To be mathematically shown, $U_h([s_0, s_1,...,s_{N+k}]) = U_h([s_0, s_1,..., s_N ])$ +for all $k > 0$ . In finite horizon problems, the optimal action in a given state could change over time. That is, when there is little time left, the agent should take risk to gain reward before the deadline is reached. Because of this change in optimal action in a given state over time, the optimal policy for a finite horizon is non stationary. However, with no fixed time limit, there is no reason to behave differently in the same state at different times. Hence, the optimal policy depends only on the current state, and the optimal policy is stationary. Note that infinite horizon does not necessarily mean that all state sequences are infinite, it just means that there is no fixed deadline. + +#### Infinite Utilities + +In infinite horizon problems, calculating an upper bound on the gained utility is desired to distinguish different policies appropriately. But defining utility as $\sum r_i$ does not satisfy the mentioned condition. But if discounting is applied, according to the geometric sequence sum $\sum \gamma^t r_i < r_{max} \sum \gamma^t = \frac{r_{max}}{1 - \gamma}$ . + +#### Stationary Preference + +In infinite horizon only if utility is defined as $\sum r_i$ or $\sum \gamma^t r_i$ the agent’s preferences between state sequences are stationary. Stationarity for preferences means + +$[s, s_0, s_1, s_2, ... ] > [s, s_0', s_1', s_2', ... ] \iff [s_0, s_1, s_2, ... ] > [s_0', s_1', s_2', ... ].$ + +This means that if a sequence is preferred to another, it is also preferred after one state is passed. Explaining in simple words, this means that if one future is preferred to another starting tomorrow, then it is still preferred if it were to start today instead. + +## Bellman Equation + +Due to previous discussions, now objective is to find policy that maximizes expected discounted utility. +$$ max_\pi E \Bigg[ \sum_{i=0}^\infty \gamma^ir_i \Bigg] $$ + +Consider $V^*(s)$ as expected utility starting from state $s$ and act according to $\pi^*$, the optimal policy. + +$Q^*(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi^*$. + +By considering all forward steps and choosing the step which maximizes the utility, $V^*$ can be formulated as +$$ V^*(s) = max_a Q^*(s, a) . $$ +Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q^*$ will be written as +$$ Q^*(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^*(s') \big].$$ + +Since $V^*$ and $Q^*$ are jointly recursive, It is possible to replace $Q^*$ with $V^*$. +$$ V^*(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^*(s') \big] $$ + +Equation above is called as *bellman equation*. + +By solving bellman equation for all states, optimal policy will be obtained. + +## Time Limited Values + +Solving bellman equation via tree has two major problems: + +1. There are repeating sub problems, so there are a lots of overheads on the system. +2. The depth of the tree could be infinite. + +An approach to solve this problem is to try solving the problem in $k$ steps which converge to final answer instead of one-shot solving. + +Instead of $V^*(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi^*$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V^*$ so an approximate solution to the equations system can be obtained. + +Equations below are the limited form of what was discussed about $V^*$ and $Q^*$ above. +$$ V_{k+1}(s) = max_a Q_k(s, a) $$ +$$ Q_k(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big]$$ + +So $V_{k+1}$ can be calculated by +$$ V_{k+1}(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big] .$$ + +For each state $s$, $V_{k+1}(s)$ will be calculated by iteration over all states and all actions, which has time complexity $O(|S||A|)$. So the total time complexity for each iteration will be $O(|S|^2|A|)$. + +## Implementation + +States, terminal states, actions, transition and reward function should be implemented in order to define the MDP. +The robot car example explained above will be encoded as bellow + +```Python +car_states = ['Cool', 'Warm', 'Over'] +car_terminals = ['Over'] +car_actions = ['fast', 'slow'] + + +def car_transition(state, action, next_state): + if state == 'Cool': + if action == 'slow': + if next_state == 'Cool': + return 1.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 0.0 + if action == 'fast': + if next_state == 'Cool': + return 0.5 + if next_state == 'Warm': + return 0.5 + if next_state == 'Over': + return 0.0 + if state == 'Warm': + if action == 'slow': + if next_state == 'Cool': + return 0.5 + if next_state == 'Warm': + return 0.5 + if next_state == 'Over': + return 0.0 + if action == 'fast': + if next_state == 'Cool': + return 0.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 1.0 + if state == 'Over': + if action == 'slow': + if next_state == 'Cool': + return 0.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 1.0 + if action == 'fast': + if next_state == 'Cool': + return 0.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 1.0 + + +def car_reward(state, action, next_state): + if state == 'Cool': + if action == 'slow': + if next_state == 'Cool': + return 1 + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return 0 # Impossible + if action == 'fast': + if next_state == 'Cool': + return 2 + if next_state == 'Warm': + return 2 + if next_state == 'Over': + return 0 # Impossible + if state == 'Warm': + if action == 'slow': + if next_state == 'Cool': + return 1 + if next_state == 'Warm': + return 1 + if next_state == 'Over': + return 0 # Impossible + if action == 'fast': + if next_state == 'Cool': + return 0 # Impossible + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return -10 + if state == 'Over': + if action == 'slow': + if next_state == 'Cool': + return 0 # Impossible + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return 0 # Impossible + if action == 'fast': + if next_state == 'Cool': + return 0 # Impossible + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return 0 # Impossible +``` + +In order to find the optimal policy, while finding new values using value iteration, the action which maximizes the value should also be stored. + +```Python +from typing import List +from typing import Tuple + + +def argmax(l: List) -> int: + index_max = 0 + for i in range(len(l)): + if l[i] > l[index_max]: + index_max = i + return index_max + + +def mdp_iterate(transition_function, reward_function, gamma: float, states: List, terminals: List, actions: List, v: List): + new_v = [] + best_actions = [] + for i in range(len(states)): + state = states[i] + if state in terminals: + new_v.append(v[i]) + best_actions.append(None) + continue + values_action = [] + for action_index in range(len(actions)): + action = actions[action_index] + values_next = [] + for next_state_index in range(len(states)): + next_state = states[next_state_index] + expected = reward_function(state, action, next_state) + (gamma * v[i]) + values_next.append(transition_function(state, action, next_state) * expected) + value_next = sum(values_next) + values_action.append(value_next) + value_action = max(values_action) + new_v.append(value_action) + best_action_i = argmax(values_action) + best_actions.append(actions[best_action_i]) + return new_v, best_actions + + +def mdp_solve(transition_function, reward_function, gamma: float, states: List, terminals: List, actions: List, iter: int): + v = [0 for _ in range(len(states))] + p = [None for _ in range(len(states))] + for _ in range(iter): + v, p = mdp_iterate(transition_function, reward_function, gamma, states, terminals, actions, v) + return {states[i]: p[i] for i in range(len(states))} +``` + +By running the code above this result is generated. + +```Python +>>> mdp_solve(car_transition, car_reward, 0.9, car_states, car_terminals, car_actions, 10) +{'Cool': 'fast', 'Warm': 'slow', 'Over': None} +``` + +## Conclusion + +In conclusion Markov Decision Process is an appropriate tool to represent Reinforcement Learning problems. In order to represent a problem using MDP, states, actions, transition model and rewards should be determined properly. Afterwards finding the optimal policy, i.e. the one with the most utility, is desired. In order to find the optimal utility, bellman equation should be solved. Algorithms such as value iteration and policy iteration attempt to solve the bellman equation feasibly. In the end take into consideration that in real world Reinforcement Learning problems, the transition model and reward function are not known and must be learned. + +## References + +- Russell, Stuart, and Peter Norvig. "Artificial intelligence: a modern approach." (2002). +- +- +- +- +- +- diff --git a/notebooks/19_intro_to_mdp/metadata.yml b/notebooks/19_intro_to_mdp/metadata.yml new file mode 100644 index 00000000..f602bb3c --- /dev/null +++ b/notebooks/19_intro_to_mdp/metadata.yml @@ -0,0 +1,41 @@ +title: Introduction to Markov Decision Process + +header: + title: Introduction to Markov Decision Process + +authors: + label: + position: top + text: Authors + kind: people + content: + - name: Aryan Ahadinia + role: Author + contact: + - icon: fas fa-envelope + link: mailto:ahadinia@ce.sharif.edu + - icon: fab fa-github + link: https://github.com/AryanAhadinia + - icon: fab fa-linkedin + link: https://www.linkedin.com/in/ahadinia/ + - name: Rouzbeh Pirayadi + role: Author + contact: + - icon: fas fa-envelope + link: mailto:rouzbehpirayadi@gmail.com + - icon: fab fa-github + link: https://github.com/rpirayadi + - name: Arian Yazdanparast + role: Author + contact: + - icon: fas fa-envelope + link: mailto:arian.yazdan2001@gmail.com + - icon: fab fa-github + link: https://github.com/ArianYp + - name: Sepehr Amini Afshar + role: Supervisor + contact: + - icon: fas fa-envelope + link: mailto:sepehramini83@gmail.com + - icon: fab fa-linkedin + link: https://www.linkedin.com/in/sepehr-amini-afshar-2201381b8/ diff --git a/notebooks/19_intro_to_mdp/overheated.jpg b/notebooks/19_intro_to_mdp/overheated.jpg new file mode 100644 index 0000000000000000000000000000000000000000..cf31eb05642c2d9f7eef62d722727ed8a0f8fb30 GIT binary patch literal 5742 zcmb_gWmHsMxIQz(5K7EQgD4>}Fm$8zfHHIp5=wW6fS{DLv~&p4jdZJYBi)E}g9rkG zAb0fMwZ8A(KljJI?>>94XTRrJ&yMx1wf8yKGuKN13@xu94}d@*;5oJc*DIiT1zA}` zH4RmH1tmG`3jhEO0N4Oa0f4>3GZ&4AGW0sSdi21*!r0W+>0jOd39%ZdQvkLFfN@Us zXHb&=N%&v-1TNSCv7AF}CA0jCK{$V~$$w(1KiK1$(=+T`k$W0^k4M<*Ids7fdD=>2=EDR+#nz#CL|^zg~MQQ(%WPd*m9eL_AV{<;b7rnW@6@I zK?!nm3!+qHBxFp6fB0s_D=2n^dlT?Sy}01kAucr3tl#bAn!J{xMMh#uQ! zZVgh)ph|Hi8u^k64Z{lq-FuRpJ{cEjF|(BYHwnYT0Yo8yG%2Jq~y zzJn9LX&f`AVhs&>LJZZ^bwB;@aeM|BHJ($Ht(G=YZtg1=|6s$9K0E=O2lx#BY6f44ACj|8s4cl 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zg$sVCA^&bJuM$560HMTT5mxYq z3h7ZXBUlw9*+lfyO8(dixw7?))3VLL@XS0oZ8FA*36thjQQEmbmi;az;}RHnD>KV#m3q~4=n(o^ zN>YCm zpv60{EYH}Ycy>CKdDaHy2=;E6^$2tTYiY^DsIcC}-y%op$jSjtM|N;2UyF?OS=bPoHj;MOWUhBO)R05bJwe zETX)*J=kr)58eM#Ze>&|ZEtN{-+-mxx!{EW)yiV}D>6+lVll#STF!wjeT#F)SG3$J z$(pw#?<}mZuXl~u3}prgGSU&cd|+tY(&=;an5H*RRj<_)Nz&*3*|ec(n?29y9rL); hLe^_VL{{mXI^27iD literal 0 HcmV?d00001 diff --git a/notebooks/index.yml b/notebooks/index.yml index 1b2133c0..4fd07c0a 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -54,4 +54,8 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ - kind: S2021, LN, Notebook \ No newline at end of file + kind: S2021, LN, Notebook + - notebook: notebooks/19_intro_to_mdp/ + kind: F2021, LN, Markdown + metadata: notebooks/19_intro_to_mdp/metadata.yml + text: Introduction to Markov Decision Process \ No newline at end of file From 5f60ccd0e05c825e554171d59aa52d5e7ba46309 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:35:52 +0330 Subject: [PATCH 02/42] Modify images allignments --- notebooks/19_intro_to_mdp/index.md | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 27621af6..2453c757 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -41,19 +41,19 @@ A representation of the environment's features at a specific time is called stat As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Notice that overheated is a terminal state. -

+

-

+

### Actions Actions are set of operations an agent is allowed to do in the given environment. The set of actions is usually shown with $A$. In the robot car example, the machine can increase or decrease its speed. So the actions' set would be {slow, fast} -
+

-

+

### Transition Model @@ -67,9 +67,9 @@ Environments can be divided to two types according to their transition models. In the machine robot example, we assume that the environment is stochastic. The probability of transitions is determined based on the current state, action, and next state. One of the possible transitions is shown in the figure below. -
+

-

+

### Rewards @@ -82,9 +82,9 @@ Usually the performance of the agent, which is also called *utility*, is measure In the robot car example, the robot must reach its destination as quickly as possible, so the high-speed reward must be considered more than the low-speed, e.g. +2 and +1 may be considered. But reward should be assigned in a way to prevent the robot from going to overheated state. So -10 is assigned to entering the overheated state. -
+

-

+

### Formalization @@ -117,9 +117,9 @@ In expectimax tree the path which leads to the most reward is desired. MDP probl In the robot car example, the search is as shown below. -
+

-

+

## Utilities of Sequences From 4a392e4416c7c0c9dbdcc592c58d9ab354fd7ac5 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:54:29 +0330 Subject: [PATCH 03/42] aa --- notebooks/index.yml | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 4fd07c0a..d19b4031 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,7 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - notebook: notebooks/19_intro_to_mdp/ - kind: F2021, LN, Markdown + - notebook: notebooks/19_intro_to_mdp/index.md + kind: F2021, LN metadata: notebooks/19_intro_to_mdp/metadata.yml text: Introduction to Markov Decision Process \ No newline at end of file From 111959226dadd9ba57b49437f5ac2fb96fc84345 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:57:07 +0330 Subject: [PATCH 04/42] sdfasdadggd --- .github/workflows/main.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml index 7e8ef31a..70726084 100644 --- a/.github/workflows/main.yml +++ b/.github/workflows/main.yml @@ -24,5 +24,5 @@ jobs: github_token: ${{ secrets.GITHUB_TOKEN }} enable_jekyll: true allow_empty_commit: true - publish_dir: . + publish_dir: './webified/' exclude_assets: '.github' From 83d710f64a1b2b2dc21e99ee612de926e8dfd761 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:03:21 +0330 Subject: [PATCH 05/42] aaa --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index d19b4031..769750ba 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,7 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - notebook: notebooks/19_intro_to_mdp/index.md + - md: notebooks/19_intro_to_mdp/index.md kind: F2021, LN metadata: notebooks/19_intro_to_mdp/metadata.yml text: Introduction to Markov Decision Process \ No newline at end of file From 25fbb61b55fdc22be2cd6b1575ae8ffc4f332b93 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:04:05 +0330 Subject: [PATCH 06/42] sdaf --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 769750ba..5a132427 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,7 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - md: notebooks/19_intro_to_mdp/index.md + - md: notebooks/19_intro_to_mdp kind: F2021, LN metadata: notebooks/19_intro_to_mdp/metadata.yml text: Introduction to Markov Decision Process \ No newline at end of file From 25f6bd181a42cd8ccae016c0c9a111ae6a0c2df1 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:11:56 +0330 Subject: [PATCH 07/42] change --- notebooks/19_intro_to_mdp/index.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 2453c757..b1372f7b 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -42,9 +42,9 @@ A representation of the environment's features at a specific time is called stat As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Notice that overheated is a terminal state.

- - - + + +

### Actions @@ -52,7 +52,7 @@ As an example, consider a robot car that wants to reach its destination as quick Actions are set of operations an agent is allowed to do in the given environment. The set of actions is usually shown with $A$. In the robot car example, the machine can increase or decrease its speed. So the actions' set would be {slow, fast}

- +

### Transition Model @@ -68,7 +68,7 @@ In the machine robot example, we assume that the environment is stochastic. The One of the possible transitions is shown in the figure below.

- +

### Rewards @@ -83,7 +83,7 @@ Usually the performance of the agent, which is also called *utility*, is measure In the robot car example, the robot must reach its destination as quickly as possible, so the high-speed reward must be considered more than the low-speed, e.g. +2 and +1 may be considered. But reward should be assigned in a way to prevent the robot from going to overheated state. So -10 is assigned to entering the overheated state.

- +

### Formalization @@ -118,7 +118,7 @@ In expectimax tree the path which leads to the most reward is desired. MDP probl In the robot car example, the search is as shown below.

- +

## Utilities of Sequences From 59ce75370857485f4e48f4289b23e86e5da8f912 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:27:37 +0330 Subject: [PATCH 08/42] segardg --- notebooks/19_intro_to_mdp/index.md | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index b1372f7b..5e5ea01d 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -104,9 +104,9 @@ By combining the concepts which were explained above the complete formulation fo The policy is a function that takes the state as an input and outputs the action to be taken. Policy $\pi : S \rightarrow A$ is a set of commands that the agent follows. -The policy which maximizes the expected utility among all the possible policies is called optimal policy and shown with $\pi^*$. +The policy which maximizes the expected utility among all the possible policies is called optimal policy and shown with $\pi\star$. -Finding $\pi^*$ is usually referred to as solving the MDP. There are different algorithms which attempt to find $\pi^*$ such as value iteration and policy evaluation. +Finding $\pi\star$ is usually referred to as solving the MDP. There are different algorithms which attempt to find $\pi\star$ such as value iteration and policy evaluation. ## MDP Search Trees @@ -151,17 +151,17 @@ This means that if a sequence is preferred to another, it is also preferred afte Due to previous discussions, now objective is to find policy that maximizes expected discounted utility. $$ max_\pi E \Bigg[ \sum_{i=0}^\infty \gamma^ir_i \Bigg] $$ -Consider $V^*(s)$ as expected utility starting from state $s$ and act according to $\pi^*$, the optimal policy. +Consider $V\star(s)$ as expected utility starting from state $s$ and act according to $\pi\star$, the optimal policy. -$Q^*(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi^*$. +$Q\star(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi\star$. -By considering all forward steps and choosing the step which maximizes the utility, $V^*$ can be formulated as -$$ V^*(s) = max_a Q^*(s, a) . $$ -Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q^*$ will be written as -$$ Q^*(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^*(s') \big].$$ +By considering all forward steps and choosing the step which maximizes the utility, $V\star$ can be formulated as +$$ V\star(s) = max_a Q\star(s, a) . $$ +Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q\star$ will be written as +$$ Q^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V\star(s') \big].$$ -Since $V^*$ and $Q^*$ are jointly recursive, It is possible to replace $Q^*$ with $V^*$. -$$ V^*(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^*(s') \big] $$ +Since $V\star$ and $Q\star$ are jointly recursive, It is possible to replace $Q\star$ with $V\star$. +$$ V\star(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V\star(s') \big] $$ Equation above is called as *bellman equation*. @@ -176,9 +176,9 @@ Solving bellman equation via tree has two major problems: An approach to solve this problem is to try solving the problem in $k$ steps which converge to final answer instead of one-shot solving. -Instead of $V^*(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi^*$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V^*$ so an approximate solution to the equations system can be obtained. +Instead of $V\star(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi\star$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V\star$ so an approximate solution to the equations system can be obtained. -Equations below are the limited form of what was discussed about $V^*$ and $Q^*$ above. +Equations below are the limited form of what was discussed about $V\star$ and $Q\star$ above. $$ V_{k+1}(s) = max_a Q_k(s, a) $$ $$ Q_k(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big]$$ From 145d4de3a5cb00140a48aadfb6c37775ca5860e6 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:30:47 +0330 Subject: [PATCH 09/42] =?UTF-8?q?=D9=87=D8=A7=D8=B9=D9=84=D8=BA?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- notebooks/19_intro_to_mdp/index.md | 24 ++++++++++++------------ 1 file changed, 12 insertions(+), 12 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 5e5ea01d..d5655b1b 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -104,9 +104,9 @@ By combining the concepts which were explained above the complete formulation fo The policy is a function that takes the state as an input and outputs the action to be taken. Policy $\pi : S \rightarrow A$ is a set of commands that the agent follows. -The policy which maximizes the expected utility among all the possible policies is called optimal policy and shown with $\pi\star$. +The policy which maximizes the expected utility among all the possible policies is called optimal policy and shown with $\pi^\star$. -Finding $\pi\star$ is usually referred to as solving the MDP. There are different algorithms which attempt to find $\pi\star$ such as value iteration and policy evaluation. +Finding $\pi^\star$ is usually referred to as solving the MDP. There are different algorithms which attempt to find $\pi^\star$ such as value iteration and policy evaluation. ## MDP Search Trees @@ -151,17 +151,17 @@ This means that if a sequence is preferred to another, it is also preferred afte Due to previous discussions, now objective is to find policy that maximizes expected discounted utility. $$ max_\pi E \Bigg[ \sum_{i=0}^\infty \gamma^ir_i \Bigg] $$ -Consider $V\star(s)$ as expected utility starting from state $s$ and act according to $\pi\star$, the optimal policy. +Consider $V^\star(s)$ as expected utility starting from state $s$ and act according to $\pi^\star$, the optimal policy. -$Q\star(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi\star$. +$Q^\star(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi^\star$. -By considering all forward steps and choosing the step which maximizes the utility, $V\star$ can be formulated as -$$ V\star(s) = max_a Q\star(s, a) . $$ -Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q\star$ will be written as -$$ Q^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V\star(s') \big].$$ +By considering all forward steps and choosing the step which maximizes the utility, $V^\star$ can be formulated as +$$ V^\star(s) = max_a Q^\star(s, a) . $$ +Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q^\star$ will be written as +$$ Q^^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big].$$ -Since $V\star$ and $Q\star$ are jointly recursive, It is possible to replace $Q\star$ with $V\star$. -$$ V\star(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V\star(s') \big] $$ +Since $V^\star$ and $Q^\star$ are jointly recursive, It is possible to replace $Q^\star$ with $V^\star$. +$$ V^\star(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big] $$ Equation above is called as *bellman equation*. @@ -176,9 +176,9 @@ Solving bellman equation via tree has two major problems: An approach to solve this problem is to try solving the problem in $k$ steps which converge to final answer instead of one-shot solving. -Instead of $V\star(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi\star$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V\star$ so an approximate solution to the equations system can be obtained. +Instead of $V^\star(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi^\star$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V^\star$ so an approximate solution to the equations system can be obtained. -Equations below are the limited form of what was discussed about $V\star$ and $Q\star$ above. +Equations below are the limited form of what was discussed about $V^\star$ and $Q^\star$ above. $$ V_{k+1}(s) = max_a Q_k(s, a) $$ $$ Q_k(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big]$$ From b06786c73304643d88c39baa83cf612d419adedf Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:37:02 +0330 Subject: [PATCH 10/42] efsge --- notebooks/19_intro_to_mdp/index.md | 36 +++++++++++++++--------------- 1 file changed, 18 insertions(+), 18 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index d5655b1b..deb7c443 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -2,24 +2,24 @@ ## Table of Contents -1. [Introduction](Introduction) -2. [Markov Decision Process](Markov_Decision_Process) - 1. [States](States) - 2. [Actions](Actions) - 3. [Transition Model](Transition_Model) - 4. [Rewards](Rewards) - 5. [Formalization](Formalization) - 6. [Policy](Policy) -3. [MDP Search Trees](MDP_Search_Trees) -4. [Utilities of Sequences](Utilities_of_Sequences) - 1. [Discounting](Discounting) - 2. [Finite and infinite horizons](Finite_and_infinite_horizons) - 3. [Infinite Utilities](Infinite_Utilities) -5. [Bellman Equation](Bellman_Equation) -6. [Time Limited Values](Time_Limited_Values) -7. [Implementation](Implementation) -8. [Conclusion](Conclusion) -9. [References](References) +1. [Introduction](#Introduction) +2. [Markov Decision Process](#Markov_Decision_Process) + 1. [States](#States) + 2. [Actions](#Actions) + 3. [Transition Model](#Transition_Model) + 4. [Rewards](#Rewards) + 5. [Formalization](#Formalization) + 6. [Policy](#Policy) +3. [MDP Search Trees](#MDP_Search_Trees) +4. [Utilities of Sequences](#Utilities_of_Sequences) + 1. [Discounting](#Discounting) + 2. [Finite and infinite horizons](#Finite_and_infinite_horizons) + 3. [Infinite Utilities](#Infinite_Utilities) +5. [Bellman Equation](#Bellman_Equation) +6. [Time Limited Values](#Time_Limited_Values) +7. [Implementation](#Implementation) +8. [Conclusion](#Conclusion) +9. [References](#References) ## Introduction From 0432eba06f45cfa0ffb834231e97ede67b1c54d8 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:42:33 +0330 Subject: [PATCH 11/42] afZD --- notebooks/19_intro_to_mdp/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index deb7c443..ee382279 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -2,16 +2,16 @@ ## Table of Contents -1. [Introduction](#Introduction) -2. [Markov Decision Process](#Markov_Decision_Process) +1. [Introduction](.#Introduction) +2. [Markov Decision Process](..#Markov_Decision_Process) 1. [States](#States) 2. [Actions](#Actions) 3. [Transition Model](#Transition_Model) 4. [Rewards](#Rewards) 5. [Formalization](#Formalization) 6. [Policy](#Policy) -3. [MDP Search Trees](#MDP_Search_Trees) -4. [Utilities of Sequences](#Utilities_of_Sequences) +3. [MDP Search Trees](./#MDP_Search_Trees) +4. [Utilities of Sequences](../#Utilities_of_Sequences) 1. [Discounting](#Discounting) 2. [Finite and infinite horizons](#Finite_and_infinite_horizons) 3. [Infinite Utilities](#Infinite_Utilities) From 3599a053ac9808db387cda293e4d1d2890978cb3 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:46:36 +0330 Subject: [PATCH 12/42] SASD --- notebooks/19_intro_to_mdp/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index ee382279..deb7c443 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -2,16 +2,16 @@ ## Table of Contents -1. [Introduction](.#Introduction) -2. [Markov Decision Process](..#Markov_Decision_Process) +1. [Introduction](#Introduction) +2. [Markov Decision Process](#Markov_Decision_Process) 1. [States](#States) 2. [Actions](#Actions) 3. [Transition Model](#Transition_Model) 4. [Rewards](#Rewards) 5. [Formalization](#Formalization) 6. [Policy](#Policy) -3. [MDP Search Trees](./#MDP_Search_Trees) -4. [Utilities of Sequences](../#Utilities_of_Sequences) +3. [MDP Search Trees](#MDP_Search_Trees) +4. [Utilities of Sequences](#Utilities_of_Sequences) 1. [Discounting](#Discounting) 2. [Finite and infinite horizons](#Finite_and_infinite_horizons) 3. [Infinite Utilities](#Infinite_Utilities) From 2ac5976475caa073fd7663007678657df0f97fd5 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 01:59:21 +0330 Subject: [PATCH 13/42] sadsd --- notebooks/19_intro_to_mdp/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index deb7c443..e77f53e1 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -158,7 +158,7 @@ $Q^\star(s, a)$ is the expected utility starting from state $s$ then performing By considering all forward steps and choosing the step which maximizes the utility, $V^\star$ can be formulated as $$ V^\star(s) = max_a Q^\star(s, a) . $$ Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q^\star$ will be written as -$$ Q^^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big].$$ +$$ Q^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big].$$ Since $V^\star$ and $Q^\star$ are jointly recursive, It is possible to replace $Q^\star$ with $V^\star$. $$ V^\star(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big] $$ From 31ee8ae5c481d86f6b9df6a852dff224e6444784 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:04:50 +0330 Subject: [PATCH 14/42] Add MDP_Intro --- .github/workflows/main.yml | 2 +- notebooks/19_intro_to_mdp/action.jpg | Bin 0 -> 7766 bytes notebooks/19_intro_to_mdp/cool.jpg | Bin 0 -> 5247 bytes notebooks/19_intro_to_mdp/index.md | 365 +++++++++++++++++++++++ notebooks/19_intro_to_mdp/metadata.yaml | 41 +++ notebooks/19_intro_to_mdp/overheated.jpg | Bin 0 -> 5742 bytes notebooks/19_intro_to_mdp/reward.jpg | Bin 0 -> 88373 bytes notebooks/19_intro_to_mdp/transition.jpg | Bin 0 -> 87699 bytes notebooks/19_intro_to_mdp/tree.png | Bin 0 -> 57505 bytes notebooks/19_intro_to_mdp/warm.jpg | Bin 0 -> 5333 bytes notebooks/index.yml | 6 +- 11 files changed, 412 insertions(+), 2 deletions(-) create mode 100644 notebooks/19_intro_to_mdp/action.jpg create mode 100644 notebooks/19_intro_to_mdp/cool.jpg create mode 100644 notebooks/19_intro_to_mdp/index.md create mode 100644 notebooks/19_intro_to_mdp/metadata.yaml create mode 100644 notebooks/19_intro_to_mdp/overheated.jpg create mode 100644 notebooks/19_intro_to_mdp/reward.jpg create mode 100644 notebooks/19_intro_to_mdp/transition.jpg create mode 100644 notebooks/19_intro_to_mdp/tree.png create mode 100644 notebooks/19_intro_to_mdp/warm.jpg diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml index 7e8ef31a..70726084 100644 --- a/.github/workflows/main.yml +++ b/.github/workflows/main.yml @@ -24,5 +24,5 @@ jobs: github_token: ${{ secrets.GITHUB_TOKEN }} enable_jekyll: true allow_empty_commit: true 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[Introduction](#Introduction) +2. [Markov Decision Process](#Markov_Decision_Process) + 1. [States](#States) + 2. [Actions](#Actions) + 3. [Transition Model](#Transition_Model) + 4. [Rewards](#Rewards) + 5. [Formalization](#Formalization) + 6. [Policy](#Policy) +3. [MDP Search Trees](#MDP_Search_Trees) +4. [Utilities of Sequences](#Utilities_of_Sequences) + 1. [Discounting](#Discounting) + 2. [Finite and infinite horizons](#Finite_and_infinite_horizons) + 3. [Infinite Utilities](#Infinite_Utilities) +5. [Bellman Equation](#Bellman_Equation) +6. [Time Limited Values](#Time_Limited_Values) +7. [Implementation](#Implementation) +8. [Conclusion](#Conclusion) +9. [References](#References) + +## Introduction + +In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating reinforcement learning mathematically. In a typical Reinforcement Learning problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called environment. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with states and agent's possible decisions are modeled by actions. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with stochastic transition models are used. In reinforcement learning, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by reward function in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on Bellman equation, So in order to find the best guideline for the agent Bellman equation should be solved. + +In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. + +## Markov Decision Process + +Markov Decision Process is based on Markov chain. A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The chain holds the following independence assumption. + +$$ P(S_{t+1}|S_t, ..., S_0)=P(S_{t+1}|S_t) $$ + +$S_t$ represents the state at time $t$ . Intuitively, $S_t$ retains all of the information about the history that can affect the future states so *the future is conditionally independent of the past, given the present*. + +### States + +A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards, return, can be computed. + +As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Notice that overheated is a terminal state. + +

+ + + +

+ +### Actions + +Actions are set of operations an agent is allowed to do in the given environment. The set of actions is usually shown with $A$. In the robot car example, the machine can increase or decrease its speed. So the actions' set would be {slow, fast} + +

+ +

+ +### Transition Model + +At each state, the agent decides which action to perform. The resulting state ($s'$) depends on both the current state ($s$) and the action performed by the agent ($a$). The transition model $T(s, a, s’)$ gives probability $P(s’|s, a)$, that is, the probability of landing up in the new state $s’$ given that the agent takes an action, $a$, in the given state, $s$. + +Environments can be divided to two types according to their transition models. + +- Determined environment: In a determined environment, if a certain action ($a$) is taken, the resulting state is the expected state ($s'$) with probability 1. +- Stochastic environment: In a stochastic environment, if the same action (a) is taken, with a certain probability ,e.g. $0.8$, the resulting state will be the expected state and there is $0.2$ probability that the resulting state is not the expected state. Here, for the state $s$, the action $a$ and next state $s'$ the transition model is $T(s’, a, s) = P(s’| s, a) = 0.8$. + +In the machine robot example, we assume that the environment is stochastic. The probability of transitions is determined based on the current state, action, and next state. +One of the possible transitions is shown in the figure below. + +

+ +

+ +### Rewards + +The reward function, quantifies the usefulness of taking a specific action and entering a specific state. These rewards incorporate the action costs in addition to any prizes or penalties that may be given. Negative rewards are called punishments. The general form of the reward function is shown as $R(s, a, s')$. But in some MDPs the reward function is independent from the resulting state or both resulting state and action. In those models reward function is shown with $R(s, a)$ and $R(s)$ respectively. + +For a particular environment, the domain knowledge plays an important role in the assignment of rewards for different states as minor changes in the reward do matter for finding the optimal solution to an MDP problem. +Sum of the rewards is called *utility* or *return*. + +Usually the performance of the agent, which is also called *utility*, is measured by the sum of rewards on the visited path. But other utility functions are also possible. General form of utility is shown by $U_h[s_0,s_1, s_2, ..., s_N]$. + +In the robot car example, the robot must reach its destination as quickly as possible, so the high-speed reward must be considered more than the low-speed, e.g. +2 and +1 may be considered. But reward should be assigned in a way to prevent the robot from going to overheated state. So -10 is assigned to entering the overheated state. + +

+ +

+ +### Formalization + +By combining the concepts which were explained above the complete formulation for Markov Decision Process is achieved as below. + +- A set of possible states $S$. +- A set of possible actions $A$. +- A transition function $T(s, a, s’)$ + - Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s’| s, a)$ +- Also called the model or the dynamic. +- A reward function $R(s, a, s’)$ + - Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s’)$ can be replaced by $R(s’)$ or $R(s', a)$ +- A start state +- Maybe a terminal state + +### Policy + +The policy is a function that takes the state as an input and outputs the action to be taken. Policy $\pi : S \rightarrow A$ is a set of commands that the agent follows. + +The policy which maximizes the expected utility among all the possible policies is called optimal policy and shown with $\pi^\star$. + +Finding $\pi^\star$ is usually referred to as solving the MDP. There are different algorithms which attempt to find $\pi^\star$ such as value iteration and policy evaluation. + +## MDP Search Trees + +In expectimax tree the path which leads to the most reward is desired. MDP problems could be demonstrated as a search tree. But there are two main differences. + +- In MDP Rewards are assigned to edges rather than nodes. +- In MDP utility is the sum of rewards gained in the path to the terminal state. But in expectimax utility is the the value assigned to the terminal state. + +In the robot car example, the search is as shown below. + +

+ +

+ +## Utilities of Sequences + +It's easy to choose between sequence of [1, 2, 2] and [2, 3, 4]. Obviously more utility as well as more reward in each step is gained . But choosing between [1, 0, 0] and [0, 0, 1] is not that easy. This motivates us to consider the time which the reward is being received. In most cases, it's preferred to gain the rewards as soon as possible. + +### Discounting + +Consider gaining reward $r_i$ in time step $t_i$. In discounting, $\sum \gamma^t r_i$ is considered instead of $\sum r_i$ as utility function such that $0\lt \gamma \lt 1$. $\gamma=1$ is just equivalent to simple summation. Apart from what we have said so far, this method has other advantages, which will be mentioned below briefly. + +### Finite and infinite horizons + +Decision making problems may be of two types. Some may be *finite horizon* and some *infinite horizon*. Finite horizon means that there is a fixed time N after which nothing matters. In other words what happens after the fixed time is not analysed. To be mathematically shown, $U_h([s_0, s_1,...,s_{N+k}]) = U_h([s_0, s_1,..., s_N ])$ +for all $k > 0$ . In finite horizon problems, the optimal action in a given state could change over time. That is, when there is little time left, the agent should take risk to gain reward before the deadline is reached. Because of this change in optimal action in a given state over time, the optimal policy for a finite horizon is non stationary. However, with no fixed time limit, there is no reason to behave differently in the same state at different times. Hence, the optimal policy depends only on the current state, and the optimal policy is stationary. Note that infinite horizon does not necessarily mean that all state sequences are infinite, it just means that there is no fixed deadline. + +#### Infinite Utilities + +In infinite horizon problems, calculating an upper bound on the gained utility is desired to distinguish different policies appropriately. But defining utility as $\sum r_i$ does not satisfy the mentioned condition. But if discounting is applied, according to the geometric sequence sum $\sum \gamma^t r_i < r_{max} \sum \gamma^t = \frac{r_{max}}{1 - \gamma}$ . + +#### Stationary Preference + +In infinite horizon only if utility is defined as $\sum r_i$ or $\sum \gamma^t r_i$ the agent’s preferences between state sequences are stationary. Stationarity for preferences means + +$[s, s_0, s_1, s_2, ... ] > [s, s_0', s_1', s_2', ... ] \iff [s_0, s_1, s_2, ... ] > [s_0', s_1', s_2', ... ].$ + +This means that if a sequence is preferred to another, it is also preferred after one state is passed. Explaining in simple words, this means that if one future is preferred to another starting tomorrow, then it is still preferred if it were to start today instead. + +## Bellman Equation + +Due to previous discussions, now objective is to find policy that maximizes expected discounted utility. +$$ max_\pi E \Bigg[ \sum_{i=0}^\infty \gamma^ir_i \Bigg] $$ + +Consider $V^\star(s)$ as expected utility starting from state $s$ and act according to $\pi^\star$, the optimal policy. + +$Q^\star(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi^\star$. + +By considering all forward steps and choosing the step which maximizes the utility, $V^\star$ can be formulated as +$$ V^\star(s) = max_a Q^\star(s, a) . $$ +Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q^\star$ will be written as +$$ Q^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big].$$ + +Since $V^\star$ and $Q^\star$ are jointly recursive, It is possible to replace $Q^\star$ with $V^\star$. +$$ V^\star(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big] $$ + +Equation above is called as *bellman equation*. + +By solving bellman equation for all states, optimal policy will be obtained. + +## Time Limited Values + +Solving bellman equation via tree has two major problems: + +1. There are repeating sub problems, so there are a lots of overheads on the system. +2. The depth of the tree could be infinite. + +An approach to solve this problem is to try solving the problem in $k$ steps which converge to final answer instead of one-shot solving. + +Instead of $V^\star(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi^\star$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V^\star$ so an approximate solution to the equations system can be obtained. + +Equations below are the limited form of what was discussed about $V^\star$ and $Q^\star$ above. +$$ V_{k+1}(s) = max_a Q_k(s, a) $$ +$$ Q_k(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big]$$ + +So $V_{k+1}$ can be calculated by +$$ V_{k+1}(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big] .$$ + +For each state $s$, $V_{k+1}(s)$ will be calculated by iteration over all states and all actions, which has time complexity $O(|S||A|)$. So the total time complexity for each iteration will be $O(|S|^2|A|)$. + +## Implementation + +States, terminal states, actions, transition and reward function should be implemented in order to define the MDP. +The robot car example explained above will be encoded as bellow + +```Python +car_states = ['Cool', 'Warm', 'Over'] +car_terminals = ['Over'] +car_actions = ['fast', 'slow'] + + +def car_transition(state, action, next_state): + if state == 'Cool': + if action == 'slow': + if next_state == 'Cool': + return 1.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 0.0 + if action == 'fast': + if next_state == 'Cool': + return 0.5 + if next_state == 'Warm': + return 0.5 + if next_state == 'Over': + return 0.0 + if state == 'Warm': + if action == 'slow': + if next_state == 'Cool': + return 0.5 + if next_state == 'Warm': + return 0.5 + if next_state == 'Over': + return 0.0 + if action == 'fast': + if next_state == 'Cool': + return 0.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 1.0 + if state == 'Over': + if action == 'slow': + if next_state == 'Cool': + return 0.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 1.0 + if action == 'fast': + if next_state == 'Cool': + return 0.0 + if next_state == 'Warm': + return 0.0 + if next_state == 'Over': + return 1.0 + + +def car_reward(state, action, next_state): + if state == 'Cool': + if action == 'slow': + if next_state == 'Cool': + return 1 + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return 0 # Impossible + if action == 'fast': + if next_state == 'Cool': + return 2 + if next_state == 'Warm': + return 2 + if next_state == 'Over': + return 0 # Impossible + if state == 'Warm': + if action == 'slow': + if next_state == 'Cool': + return 1 + if next_state == 'Warm': + return 1 + if next_state == 'Over': + return 0 # Impossible + if action == 'fast': + if next_state == 'Cool': + return 0 # Impossible + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return -10 + if state == 'Over': + if action == 'slow': + if next_state == 'Cool': + return 0 # Impossible + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return 0 # Impossible + if action == 'fast': + if next_state == 'Cool': + return 0 # Impossible + if next_state == 'Warm': + return 0 # Impossible + if next_state == 'Over': + return 0 # Impossible +``` + +In order to find the optimal policy, while finding new values using value iteration, the action which maximizes the value should also be stored. + +```Python +from typing import List +from typing import Tuple + + +def argmax(l: List) -> int: + index_max = 0 + for i in range(len(l)): + if l[i] > l[index_max]: + index_max = i + return index_max + + +def mdp_iterate(transition_function, reward_function, gamma: float, states: List, terminals: List, actions: List, v: List): + new_v = [] + best_actions = [] + for i in range(len(states)): + state = states[i] + if state in terminals: + new_v.append(v[i]) + best_actions.append(None) + continue + values_action = [] + for action_index in range(len(actions)): + action = actions[action_index] + values_next = [] + for next_state_index in range(len(states)): + next_state = states[next_state_index] + expected = reward_function(state, action, next_state) + (gamma * v[i]) + values_next.append(transition_function(state, action, next_state) * expected) + value_next = sum(values_next) + values_action.append(value_next) + value_action = max(values_action) + new_v.append(value_action) + best_action_i = argmax(values_action) + best_actions.append(actions[best_action_i]) + return new_v, best_actions + + +def mdp_solve(transition_function, reward_function, gamma: float, states: List, terminals: List, actions: List, iter: int): + v = [0 for _ in range(len(states))] + p = [None for _ in range(len(states))] + for _ in range(iter): + v, p = mdp_iterate(transition_function, reward_function, gamma, states, terminals, actions, v) + return {states[i]: p[i] for i in range(len(states))} +``` + +By running the code above this result is generated. + +```Python +>>> mdp_solve(car_transition, car_reward, 0.9, car_states, car_terminals, car_actions, 10) +{'Cool': 'fast', 'Warm': 'slow', 'Over': None} +``` + +## Conclusion + +In conclusion Markov Decision Process is an appropriate tool to represent Reinforcement Learning problems. In order to represent a problem using MDP, states, actions, transition model and rewards should be determined properly. Afterwards finding the optimal policy, i.e. the one with the most utility, is desired. In order to find the optimal utility, bellman equation should be solved. Algorithms such as value iteration and policy iteration attempt to solve the bellman equation feasibly. In the end take into consideration that in real world Reinforcement Learning problems, the transition model and reward function are not known and must be learned. + +## References + +- Russell, Stuart, and Peter Norvig. "Artificial intelligence: a modern approach." (2002). +- +- +- +- +- +- diff --git a/notebooks/19_intro_to_mdp/metadata.yaml b/notebooks/19_intro_to_mdp/metadata.yaml new file mode 100644 index 00000000..fe086037 --- /dev/null +++ b/notebooks/19_intro_to_mdp/metadata.yaml @@ -0,0 +1,41 @@ +title: Introduction to Markov Decision Process + +header: + title: Introduction to Markov Decision Process + +authors: + label: + position: top + text: Authors + kind: people + content: + - name: Aryan Ahadinia + role: Author + contact: + - icon: fas fa-envelope + link: mailto:ahadinia@ce.sharif.edu + - icon: fab fa-github + link: https://github.com/AryanAhadinia + - icon: fab fa-linkedin + link: https://www.linkedin.com/in/ahadinia/ + - name: Rouzbeh Pirayadi + role: Author + contact: + - icon: fas fa-envelope + link: mailto:rouzbehpirayadi@gmail.com + - icon: fab fa-github + link: https://github.com/rpirayadi + - name: Arian Yazdanparast + role: Author + contact: + - icon: fas fa-envelope + link: mailto:arian.yazdan2001@gmail.com + - icon: fab 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zg$sVCA^&bJuM$560HMTT5mxYq z3h7ZXBUlw9*+lfyO8(dixw7?))3VLL@XS0oZ8FA*36thjQQEmbmi;az;}RHnD>KV#m3q~4=n(o^ zN>YCm zpv60{EYH}Ycy>CKdDaHy2=;E6^$2tTYiY^DsIcC}-y%op$jSjtM|N;2UyF?OS=bPoHj;MOWUhBO)R05bJwe zETX)*J=kr)58eM#Ze>&|ZEtN{-+-mxx!{EW)yiV}D>6+lVll#STF!wjeT#F)SG3$J z$(pw#?<}mZuXl~u3}prgGSU&cd|+tY(&=;an5H*RRj<_)Nz&*3*|ec(n?29y9rL); hLe^_VL{{mXI^27iD literal 0 HcmV?d00001 diff --git a/notebooks/index.yml b/notebooks/index.yml index 1b2133c0..5a132427 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -54,4 +54,8 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ - kind: S2021, LN, Notebook \ No newline at end of file + kind: S2021, LN, Notebook + - md: notebooks/19_intro_to_mdp + kind: F2021, LN + metadata: notebooks/19_intro_to_mdp/metadata.yml + text: Introduction to Markov Decision Process \ No newline at end of file From f0539e3859192421ebbd61ac77989f8d8399ff78 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:12:02 +0330 Subject: [PATCH 15/42] Fix metadata.yaml --- notebooks/19_intro_to_mdp/metadata.yaml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/metadata.yaml b/notebooks/19_intro_to_mdp/metadata.yaml index fe086037..7c0683b7 100644 --- a/notebooks/19_intro_to_mdp/metadata.yaml +++ b/notebooks/19_intro_to_mdp/metadata.yaml @@ -1,4 +1,4 @@ -title: Introduction to Markov Decision Process +title: LN | Introduction to Markov Decision Process header: title: Introduction to Markov Decision Process From 71fadecb65cc5da0b68ba86fc32760f72186dd28 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:19:11 +0330 Subject: [PATCH 16/42] Fix metadata.yaml --- notebooks/19_intro_to_mdp/metadata.yaml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/metadata.yaml b/notebooks/19_intro_to_mdp/metadata.yaml index 7c0683b7..f8fb61ac 100644 --- a/notebooks/19_intro_to_mdp/metadata.yaml +++ b/notebooks/19_intro_to_mdp/metadata.yaml @@ -1,7 +1,7 @@ title: LN | Introduction to Markov Decision Process header: - title: Introduction to Markov Decision Process + title: LN | Introduction to Markov Decision Process authors: label: From 84585d2922514205910da11bd55f2276365de543 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:24:10 +0330 Subject: [PATCH 17/42] Fix Display --- notebooks/19_intro_to_mdp/metadata.yaml | 2 +- notebooks/index.yml | 2 -- 2 files changed, 1 insertion(+), 3 deletions(-) diff --git a/notebooks/19_intro_to_mdp/metadata.yaml b/notebooks/19_intro_to_mdp/metadata.yaml index f8fb61ac..7c0683b7 100644 --- a/notebooks/19_intro_to_mdp/metadata.yaml +++ b/notebooks/19_intro_to_mdp/metadata.yaml @@ -1,7 +1,7 @@ title: LN | Introduction to Markov Decision Process header: - title: LN | Introduction to Markov Decision Process + title: Introduction to Markov Decision Process authors: label: diff --git a/notebooks/index.yml b/notebooks/index.yml index 5a132427..8d6a2276 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -57,5 +57,3 @@ notebooks: kind: S2021, LN, Notebook - md: notebooks/19_intro_to_mdp kind: F2021, LN - metadata: notebooks/19_intro_to_mdp/metadata.yml - text: Introduction to Markov Decision Process \ No newline at end of file From 1fee2a528651a74ed1fdbf8455c8ed4a2d197eb5 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:28:11 +0330 Subject: [PATCH 18/42] Fix Title --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 8d6a2276..8013ed84 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,5 +55,5 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - md: notebooks/19_intro_to_mdp + - md: notebooks/19_intro_to_mdp/ kind: F2021, LN From 754183080d0e0fa6b680503f99a5335b63cc9f5f Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:32:23 +0330 Subject: [PATCH 19/42] Fix --- notebooks/index.yml | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 8013ed84..5a132427 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,5 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - md: notebooks/19_intro_to_mdp/ + - md: notebooks/19_intro_to_mdp kind: F2021, LN + metadata: notebooks/19_intro_to_mdp/metadata.yml + text: Introduction to Markov Decision Process \ No newline at end of file From bd2fc05fe2d5fe8480a8f930baed577237174e85 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:37:21 +0330 Subject: [PATCH 20/42] Fix --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 5a132427..31ca6c50 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,7 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - md: notebooks/19_intro_to_mdp + - md: notebooks/19_intro_to_mdp/ kind: F2021, LN metadata: notebooks/19_intro_to_mdp/metadata.yml text: Introduction to Markov Decision Process \ No newline at end of file From 5335c071d19921ebe8d9d142ebac9b1e4f05953b Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:45:12 +0330 Subject: [PATCH 21/42] Fix --- notebooks/19_intro_to_mdp/{metadata.yaml => metadata.yml} | 1 + 1 file changed, 1 insertion(+) rename notebooks/19_intro_to_mdp/{metadata.yaml => metadata.yml} (95%) diff --git a/notebooks/19_intro_to_mdp/metadata.yaml b/notebooks/19_intro_to_mdp/metadata.yml similarity index 95% rename from notebooks/19_intro_to_mdp/metadata.yaml rename to notebooks/19_intro_to_mdp/metadata.yml index 7c0683b7..42690fe1 100644 --- a/notebooks/19_intro_to_mdp/metadata.yaml +++ b/notebooks/19_intro_to_mdp/metadata.yml @@ -2,6 +2,7 @@ title: LN | Introduction to Markov Decision Process header: title: Introduction to Markov Decision Process + description: LN | Introduction to Markov Decision Process authors: label: From 73529e183d3d57c59b6c01e81f5dd91efc2b5a2c Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:53:16 +0330 Subject: [PATCH 22/42] Delete metadata --- notebooks/19_intro_to_mdp/metadata.yml | 42 -------------------------- notebooks/index.yml | 2 +- 2 files changed, 1 insertion(+), 43 deletions(-) delete mode 100644 notebooks/19_intro_to_mdp/metadata.yml diff --git a/notebooks/19_intro_to_mdp/metadata.yml b/notebooks/19_intro_to_mdp/metadata.yml deleted file mode 100644 index 42690fe1..00000000 --- a/notebooks/19_intro_to_mdp/metadata.yml +++ /dev/null @@ -1,42 +0,0 @@ -title: LN | Introduction to Markov Decision Process - -header: - title: Introduction to Markov Decision Process - description: LN | Introduction to Markov Decision Process - -authors: - label: - position: top - text: Authors - kind: people - content: - - name: Aryan Ahadinia - role: Author - contact: - - icon: fas fa-envelope - link: mailto:ahadinia@ce.sharif.edu - - icon: fab fa-github - link: https://github.com/AryanAhadinia - - icon: fab fa-linkedin - link: https://www.linkedin.com/in/ahadinia/ - - name: Rouzbeh Pirayadi - role: Author - contact: - - icon: fas fa-envelope - link: mailto:rouzbehpirayadi@gmail.com - - icon: fab fa-github - link: https://github.com/rpirayadi - - name: Arian Yazdanparast - role: Author - contact: - - icon: fas fa-envelope - link: mailto:arian.yazdan2001@gmail.com - - icon: fab fa-github - link: https://github.com/ArianYp - - name: Sepehr Amini Afshar - role: Supervisor - contact: - - icon: fas fa-envelope - link: mailto:sepehramini83@gmail.com - - icon: fab fa-linkedin - link: https://www.linkedin.com/in/sepehr-amini-afshar-2201381b8/ \ No newline at end of file diff --git a/notebooks/index.yml b/notebooks/index.yml index 31ca6c50..5a132427 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,7 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - md: notebooks/19_intro_to_mdp/ + - md: notebooks/19_intro_to_mdp kind: F2021, LN metadata: notebooks/19_intro_to_mdp/metadata.yml text: Introduction to Markov Decision Process \ No newline at end of file From 730ddf8d4770d1ccf184ca12584f96206fe6d1ae Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 8 Jan 2022 02:54:00 +0330 Subject: [PATCH 23/42] Recreate --- notebooks/19_intro_to_mdp/metadata.yml | 41 ++++++++++++++++++++++++++ 1 file changed, 41 insertions(+) create mode 100644 notebooks/19_intro_to_mdp/metadata.yml diff --git a/notebooks/19_intro_to_mdp/metadata.yml b/notebooks/19_intro_to_mdp/metadata.yml new file mode 100644 index 00000000..fe086037 --- /dev/null +++ b/notebooks/19_intro_to_mdp/metadata.yml @@ -0,0 +1,41 @@ +title: Introduction to Markov Decision Process + +header: + title: Introduction to Markov Decision Process + +authors: + label: + position: top + text: Authors + kind: people + content: + - name: Aryan Ahadinia + role: Author + contact: + - icon: fas fa-envelope + link: mailto:ahadinia@ce.sharif.edu + - icon: fab fa-github + link: https://github.com/AryanAhadinia + - icon: fab fa-linkedin + link: https://www.linkedin.com/in/ahadinia/ + - name: Rouzbeh Pirayadi + role: Author + contact: + - icon: fas fa-envelope + link: mailto:rouzbehpirayadi@gmail.com + - icon: fab fa-github + link: https://github.com/rpirayadi + - name: Arian Yazdanparast + role: Author + contact: + - icon: fas fa-envelope + link: mailto:arian.yazdan2001@gmail.com + - icon: fab fa-github + link: https://github.com/ArianYp + - name: Sepehr Amini Afshar + role: Supervisor + contact: + - icon: fas fa-envelope + link: mailto:sepehramini83@gmail.com + - icon: fab fa-linkedin + link: https://www.linkedin.com/in/sepehr-amini-afshar-2201381b8/ \ No newline at end of file From cce01e8efe9f7aca7b68db9173b3ab8ce1595b0b Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 20:21:23 +0330 Subject: [PATCH 24/42] Simplify Code --- notebooks/19_intro_to_mdp/index.md | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index e77f53e1..d6a4762f 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -319,11 +319,9 @@ def mdp_iterate(transition_function, reward_function, gamma: float, states: List best_actions.append(None) continue values_action = [] - for action_index in range(len(actions)): - action = actions[action_index] + for action in actions: values_next = [] - for next_state_index in range(len(states)): - next_state = states[next_state_index] + for next_state in states: expected = reward_function(state, action, next_state) + (gamma * v[i]) values_next.append(transition_function(state, action, next_state) * expected) value_next = sum(values_next) From b801b954e47cabbb5d1540eefa1c0d08a481f69c Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 20:47:05 +0330 Subject: [PATCH 25/42] Change Images --- 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zpv60{EYH}Ycy>CKdDaHy2=;E6^$2tTYiY^DsIcC}-y%op$jSjtM|N;2UyF?OS=bPoHj;MOWUhBO)R05bJwe zETX)*J=kr)58eM#Ze>&|ZEtN{-+-mxx!{EW)yiV}D>6+lVll#STF!wjeT#F)SG3$J z$(pw#?<}mZuXl~u3}prgGSU&cd|+tY(&=;an5H*RRj<_)Nz&*3*|ec(n?29y9rL); hLe^_VL{{mXI^27iD From e224ec7c27d799c6982ffe7bf1a5f9bdba9822aa Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 20:52:29 +0330 Subject: [PATCH 26/42] Make state images inrow --- notebooks/19_intro_to_mdp/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index d6a4762f..3bb2bdb6 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -42,9 +42,9 @@ A representation of the environment's features at a specific time is called stat As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Notice that overheated is a terminal state.

- - - + + +

### Actions From 757e97390b74f1b44827fbe1aab7c5f07be56f7d Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 20:53:44 +0330 Subject: [PATCH 27/42] Modify images widths --- notebooks/19_intro_to_mdp/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 3bb2bdb6..25797752 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -52,7 +52,7 @@ As an example, consider a robot car that wants to reach its destination as quick Actions are set of operations an agent is allowed to do in the given environment. The set of actions is usually shown with $A$. In the robot car example, the machine can increase or decrease its speed. So the actions' set would be {slow, fast}

- +

### Transition Model From 6461dc7f2cfdfc7f01e7183a9c50b674e400a0a6 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 21:07:55 +0330 Subject: [PATCH 28/42] Modify text body --- notebooks/19_intro_to_mdp/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 25797752..365c4a6c 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -39,7 +39,7 @@ $S_t$ represents the state at time $t$ . Intuitively, $S_t$ retains all of the i A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards, return, can be computed. -As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Notice that overheated is a terminal state. +As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Note that overheated is a terminal state.

From d822189546205a2f0a480351396016e19bea6ace Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 22:55:41 +0330 Subject: [PATCH 29/42] Modify equations --- notebooks/19_intro_to_mdp/index.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 365c4a6c..b9a178f6 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -149,19 +149,19 @@ This means that if a sequence is preferred to another, it is also preferred afte ## Bellman Equation Due to previous discussions, now objective is to find policy that maximizes expected discounted utility. -$$ max_\pi E \Bigg[ \sum_{i=0}^\infty \gamma^ir_i \Bigg] $$ +$$ \max_{\pi} E \Bigg[ \sum_{i=0}^\infty \gamma^ir_i \Bigg] $$ Consider $V^\star(s)$ as expected utility starting from state $s$ and act according to $\pi^\star$, the optimal policy. $Q^\star(s, a)$ is the expected utility starting from state $s$ then performing action $a$ and after that acting according to $\pi^\star$. By considering all forward steps and choosing the step which maximizes the utility, $V^\star$ can be formulated as -$$ V^\star(s) = max_a Q^\star(s, a) . $$ +$$ V^\star(s) = \max_{a} Q^\star(s, a) . $$ Then by calculating expectation on earned utility by marginalizing on all possible next states which can be succeeded after action $a$ in state $s$, $Q^\star$ will be written as $$ Q^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big].$$ Since $V^\star$ and $Q^\star$ are jointly recursive, It is possible to replace $Q^\star$ with $V^\star$. -$$ V^\star(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big] $$ +$$ V^\star(s) = \max_{a} \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big] $$ Equation above is called as *bellman equation*. @@ -179,11 +179,11 @@ An approach to solve this problem is to try solving the problem in $k$ steps whi Instead of $V^\star(s)$, Consider $V_k(s)$ which is the expected utility starting from state $s$ and acting according to $\pi^\star$ with consideration that the process will be terminated after $k$ steps. The idea is that with the growth of $k$, $V_k$ will converge to $V^\star$ so an approximate solution to the equations system can be obtained. Equations below are the limited form of what was discussed about $V^\star$ and $Q^\star$ above. -$$ V_{k+1}(s) = max_a Q_k(s, a) $$ +$$ V_{k+1}(s) = \max_{a} Q_k(s, a) $$ $$ Q_k(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big]$$ So $V_{k+1}$ can be calculated by -$$ V_{k+1}(s) = max_a \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big] .$$ +$$ V_{k+1}(s) = \max_{a} \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V_k(s') \big] .$$ For each state $s$, $V_{k+1}(s)$ will be calculated by iteration over all states and all actions, which has time complexity $O(|S||A|)$. So the total time complexity for each iteration will be $O(|S|^2|A|)$. From 6351fcfb516c7e67cfa24d843f0d8fbfb2a5913d Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 22:57:37 +0330 Subject: [PATCH 30/42] Modify text body --- notebooks/19_intro_to_mdp/index.md | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index b9a178f6..ecb000d8 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -92,11 +92,9 @@ By combining the concepts which were explained above the complete formulation fo - A set of possible states $S$. - A set of possible actions $A$. -- A transition function $T(s, a, s’)$ - - Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s’| s, a)$ +- A transition function $T(s, a, s’)$: Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s’| s, a)$ - Also called the model or the dynamic. -- A reward function $R(s, a, s’)$ - - Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s’)$ can be replaced by $R(s’)$ or $R(s', a)$ +- A reward function $R(s, a, s’)$: Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s’)$ can be replaced by $R(s’)$ or $R(s', a)$ - A start state - Maybe a terminal state From a0ae0212e0970c995cfb2a43b571ac9b77e493bf Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Sat, 15 Jan 2022 22:58:33 +0330 Subject: [PATCH 31/42] Modify text body --- notebooks/19_intro_to_mdp/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index ecb000d8..03275238 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -59,7 +59,7 @@ Actions are set of operations an agent is allowed to do in the given environment At each state, the agent decides which action to perform. The resulting state ($s'$) depends on both the current state ($s$) and the action performed by the agent ($a$). The transition model $T(s, a, s’)$ gives probability $P(s’|s, a)$, that is, the probability of landing up in the new state $s’$ given that the agent takes an action, $a$, in the given state, $s$. -Environments can be divided to two types according to their transition models. +Environments can be divided into two types according to their transition models. - Determined environment: In a determined environment, if a certain action ($a$) is taken, the resulting state is the expected state ($s'$) with probability 1. - Stochastic environment: In a stochastic environment, if the same action (a) is taken, with a certain probability ,e.g. $0.8$, the resulting state will be the expected state and there is $0.2$ probability that the resulting state is not the expected state. Here, for the state $s$, the action $a$ and next state $s'$ the transition model is $T(s’, a, s) = P(s’| s, a) = 0.8$. From fb1293e551a03f0d99fca91083ef0a21e124b5f9 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Wed, 26 Jan 2022 23:13:41 +0330 Subject: [PATCH 32/42] Modify --- notebooks/19_intro_to_mdp/index.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 03275238..57181b93 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -37,7 +37,7 @@ $S_t$ represents the state at time $t$ . Intuitively, $S_t$ retains all of the i ### States -A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards, return, can be computed. +A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards {which can be also called return} can be computed. As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Note that overheated is a terminal state. @@ -353,6 +353,7 @@ In conclusion Markov Decision Process is an appropriate tool to represent Reinfo ## References - Russell, Stuart, and Peter Norvig. "Artificial intelligence: a modern approach." (2002). +- https://inst.eecs.berkeley.edu/~cs188/fa18/assets/notes/n4.pdf - - - From 5512416b536717ed86f2ed4a3bb5aa7b9600641c Mon Sep 17 00:00:00 2001 From: Arian Yazdanparast Date: Wed, 26 Jan 2022 23:59:56 +0330 Subject: [PATCH 33/42] Fix abbreviations --- notebooks/19_intro_to_mdp/index.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 03275238..7c339485 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -23,13 +23,13 @@ ## Introduction -In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating reinforcement learning mathematically. In a typical Reinforcement Learning problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called environment. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with states and agent's possible decisions are modeled by actions. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with stochastic transition models are used. In reinforcement learning, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by reward function in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on Bellman equation, So in order to find the best guideline for the agent Bellman equation should be solved. +In nature, learning by trial and error is the most common way of learning. Learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating RL mathematically. In a typical RL problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called environment. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with states and agent's possible decisions are modeled by actions. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with stochastic transition models are used. In RL, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by reward function in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on Bellman equation, So in order to find the best guideline for the agent Bellman equation should be solved. In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. ## Markov Decision Process -Markov Decision Process is based on Markov chain. A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The chain holds the following independence assumption. +MDP is based on Markov chain. A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules. The chain holds the following independence assumption. $$ P(S_{t+1}|S_t, ..., S_0)=P(S_{t+1}|S_t) $$ @@ -39,7 +39,7 @@ $S_t$ represents the state at time $t$ . Intuitively, $S_t$ retains all of the i A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards, return, can be computed. -As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated} .Note that overheated is a terminal state. +As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated}. Note that overheated is a terminal state.

@@ -88,7 +88,7 @@ In the robot car example, the robot must reach its destination as quickly as pos ### Formalization -By combining the concepts which were explained above the complete formulation for Markov Decision Process is achieved as below. +By combining the concepts which were explained above the complete formulation for MDP is achieved as below. - A set of possible states $S$. - A set of possible actions $A$. @@ -348,7 +348,7 @@ By running the code above this result is generated. ## Conclusion -In conclusion Markov Decision Process is an appropriate tool to represent Reinforcement Learning problems. In order to represent a problem using MDP, states, actions, transition model and rewards should be determined properly. Afterwards finding the optimal policy, i.e. the one with the most utility, is desired. In order to find the optimal utility, bellman equation should be solved. Algorithms such as value iteration and policy iteration attempt to solve the bellman equation feasibly. In the end take into consideration that in real world Reinforcement Learning problems, the transition model and reward function are not known and must be learned. +In conclusion MDP is an appropriate tool to represent RL problems. In order to represent a problem using MDP, states, actions, transition model and rewards should be determined properly. Afterwards finding the optimal policy, i.e. the one with the most utility, is desired. In order to find the optimal utility, bellman equation should be solved. Algorithms such as value iteration and policy iteration attempt to solve the bellman equation feasibly. In the end take into consideration that in real world RL problems, the transition model and reward function are not known and must be learned. ## References From 212bbf174f3cc9e0128598f48b268eea7bf45bb5 Mon Sep 17 00:00:00 2001 From: "r.pirayadi" Date: Thu, 27 Jan 2022 00:00:15 +0330 Subject: [PATCH 34/42] make keywords bold --- notebooks/19_intro_to_mdp/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 03275238..79558887 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -23,7 +23,7 @@ ## Introduction -In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating reinforcement learning mathematically. In a typical Reinforcement Learning problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called environment. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with states and agent's possible decisions are modeled by actions. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with stochastic transition models are used. In reinforcement learning, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by reward function in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on Bellman equation, So in order to find the best guideline for the agent Bellman equation should be solved. +In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating reinforcement learning mathematically. In a typical Reinforcement Learning problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called __environment__. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with __states__ and agent's possible decisions are modeled by __actions__. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with __stochastic transition models__ are used. In reinforcement learning, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by __reward function__ in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on __Bellman equation__, So in order to find the best guideline for the agent Bellman equation should be solved. In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. @@ -129,7 +129,7 @@ Consider gaining reward $r_i$ in time step $t_i$. In discounting, $\sum \gamma^t ### Finite and infinite horizons -Decision making problems may be of two types. Some may be *finite horizon* and some *infinite horizon*. Finite horizon means that there is a fixed time N after which nothing matters. In other words what happens after the fixed time is not analysed. To be mathematically shown, $U_h([s_0, s_1,...,s_{N+k}]) = U_h([s_0, s_1,..., s_N ])$ +Decision making problems may be of two types. Some may be __finite horizon__ and some __infinite horizon__. Finite horizon means that there is a fixed time N after which nothing matters. In other words what happens after the fixed time is not analysed. To be mathematically shown, $U_h([s_0, s_1,...,s_{N+k}]) = U_h([s_0, s_1,..., s_N ])$ for all $k > 0$ . In finite horizon problems, the optimal action in a given state could change over time. That is, when there is little time left, the agent should take risk to gain reward before the deadline is reached. Because of this change in optimal action in a given state over time, the optimal policy for a finite horizon is non stationary. However, with no fixed time limit, there is no reason to behave differently in the same state at different times. Hence, the optimal policy depends only on the current state, and the optimal policy is stationary. Note that infinite horizon does not necessarily mean that all state sequences are infinite, it just means that there is no fixed deadline. #### Infinite Utilities @@ -161,7 +161,7 @@ $$ Q^\star(s, a) = \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \bi Since $V^\star$ and $Q^\star$ are jointly recursive, It is possible to replace $Q^\star$ with $V^\star$. $$ V^\star(s) = \max_{a} \sum_{s'}T(s, a, s')\big[R(s, a, s') + \gamma V^\star(s') \big] $$ -Equation above is called as *bellman equation*. +Equation above is called as __bellman equation__. By solving bellman equation for all states, optimal policy will be obtained. From 4ade6786c2d6c652ca8293b7f9923ca64b3a5f2c Mon Sep 17 00:00:00 2001 From: "r.pirayadi" Date: Thu, 27 Jan 2022 00:32:24 +0330 Subject: [PATCH 35/42] Fix decision maker and learner ambiguity --- notebooks/19_intro_to_mdp/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 395449f6..b9876b78 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -27,7 +27,7 @@ ## Introduction -In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating RL mathematically. In a typical RL problem, there is a learner and a decision maker called agent. The surrounding which agent interacts with is called __environment__. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with __states__ and agent's possible decisions are modeled by __actions__. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with __stochastic transition models__ are used. In RL, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by __reward function__ in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on __Bellman equation__, So in order to find the best guideline for the agent Bellman equation should be solved. +In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating RL mathematically. In a typical RL problem, there is a learner called agent, which makes decisions. The surrounding which agent interacts with is called __environment__. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with __states__ and agent's possible decisions are modeled by __actions__. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with __stochastic transition models__ are used. In RL, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by __reward function__ in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on __Bellman equation__, So in order to find the best guideline for the agent Bellman equation should be solved. In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. From aca42a12f689380ee90db414e239ca193d9ec3c1 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Thu, 27 Jan 2022 08:11:25 +0330 Subject: [PATCH 36/42] Fix Warnings --- notebooks/19_intro_to_mdp/index.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index b9876b78..7ab961b3 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -26,7 +26,6 @@ ## Introduction - In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating RL mathematically. In a typical RL problem, there is a learner called agent, which makes decisions. The surrounding which agent interacts with is called __environment__. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with __states__ and agent's possible decisions are modeled by __actions__. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with __stochastic transition models__ are used. In RL, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by __reward function__ in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on __Bellman equation__, So in order to find the best guideline for the agent Bellman equation should be solved. In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. @@ -357,7 +356,7 @@ In conclusion MDP is an appropriate tool to represent RL problems. In order to r ## References - Russell, Stuart, and Peter Norvig. "Artificial intelligence: a modern approach." (2002). -- https://inst.eecs.berkeley.edu/~cs188/fa18/assets/notes/n4.pdf +- - - - From d5c45e0cd0279c812589c9c03e77e8473dfca6de Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Thu, 27 Jan 2022 08:21:20 +0330 Subject: [PATCH 37/42] Fix --- notebooks/19_intro_to_mdp/index.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 7ab961b3..8e579360 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -40,7 +40,7 @@ $S_t$ represents the state at time $t$ . Intuitively, $S_t$ retains all of the i ### States -A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent’s sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards {which can be also called return} can be computed. +A representation of the environment's features at a specific time is called state and shown by $s$. Thus, any input from the agent's sensors can play an important role in state formation. The $S$ state set is a set of different states, represented as $s$ which shows the status of the environment. Each MDP consists of some states, which some of them are stopping state or so called terminal state. Terminal state is a state in which no action could be taken. When agent enters a stopping state, sum of rewards {which can be also called return} can be computed. As an example, consider a robot car that wants to reach its destination as quickly as possible. The engine condition of this car can be cool, warm, or overheated depending on its speed.So in the above example, set of states would be {cool, warm, overheated}. Note that overheated is a terminal state. @@ -60,12 +60,12 @@ Actions are set of operations an agent is allowed to do in the given environment ### Transition Model -At each state, the agent decides which action to perform. The resulting state ($s'$) depends on both the current state ($s$) and the action performed by the agent ($a$). The transition model $T(s, a, s’)$ gives probability $P(s’|s, a)$, that is, the probability of landing up in the new state $s’$ given that the agent takes an action, $a$, in the given state, $s$. +At each state, the agent decides which action to perform. The resulting state ($s'$) depends on both the current state ($s$) and the action performed by the agent ($a$). The transition model $T(s, a, s')$ gives probability $P(s'|s, a)$, that is, the probability of landing up in the new state $s'$ given that the agent takes an action, $a$, in the given state, $s$. Environments can be divided into two types according to their transition models. - Determined environment: In a determined environment, if a certain action ($a$) is taken, the resulting state is the expected state ($s'$) with probability 1. -- Stochastic environment: In a stochastic environment, if the same action (a) is taken, with a certain probability ,e.g. $0.8$, the resulting state will be the expected state and there is $0.2$ probability that the resulting state is not the expected state. Here, for the state $s$, the action $a$ and next state $s'$ the transition model is $T(s’, a, s) = P(s’| s, a) = 0.8$. +- Stochastic environment: In a stochastic environment, if the same action (a) is taken, with a certain probability ,e.g. $0.8$, the resulting state will be the expected state and there is $0.2$ probability that the resulting state is not the expected state. Here, for the state $s$, the action $a$ and next state $s'$ the transition model is $T(s', a, s) = P(s'| s, a) = 0.8$. In the machine robot example, we assume that the environment is stochastic. The probability of transitions is determined based on the current state, action, and next state. One of the possible transitions is shown in the figure below. @@ -95,9 +95,9 @@ By combining the concepts which were explained above the complete formulation fo - A set of possible states $S$. - A set of possible actions $A$. -- A transition function $T(s, a, s’)$: Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s’| s, a)$ +- A transition function $T(s, a, s')$: Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s'| s, a)$ - Also called the model or the dynamic. -- A reward function $R(s, a, s’)$: Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s’)$ can be replaced by $R(s’)$ or $R(s', a)$ +- A reward function $R(s, a, s')$: Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s')$ can be replaced by $R(s')$ or $R(s', a)$ - A start state - Maybe a terminal state @@ -141,7 +141,7 @@ In infinite horizon problems, calculating an upper bound on the gained utility i #### Stationary Preference -In infinite horizon only if utility is defined as $\sum r_i$ or $\sum \gamma^t r_i$ the agent’s preferences between state sequences are stationary. Stationarity for preferences means +In infinite horizon only if utility is defined as $\sum r_i$ or $\sum \gamma^t r_i$ the agent's preferences between state sequences are stationary. Stationarity for preferences means $[s, s_0, s_1, s_2, ... ] > [s, s_0', s_1', s_2', ... ] \iff [s_0, s_1, s_2, ... ] > [s_0', s_1', s_2', ... ].$ From bb91e91963280ed071236c40b7cea4055e73066c Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Thu, 27 Jan 2022 11:44:58 +0330 Subject: [PATCH 38/42] Modify --- notebooks/19_intro_to_mdp/index.md | 15 ++++++++++++++- 1 file changed, 14 insertions(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 8e579360..ed364c89 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -21,6 +21,7 @@ - [Bellman Equation](#bellman-equation) - [Time Limited Values](#time-limited-values) - [Implementation](#implementation) + - [Real Life Example](#real-life-example) - [Conclusion](#conclusion) - [References](#references) @@ -295,6 +296,18 @@ def car_reward(state, action, next_state): return 0 # Impossible ``` +If we ignore the impossible cases, we will find that in this problem the reward is only a function of the action and next state. So we can rewrite reward function as below. + +```Python +def car_reward(state, action, next_state): + if action == 'slow': + return 1 + if action == 'fast': + if next_state == 'Over': + return -10 + return 2 +``` + In order to find the optimal policy, while finding new values using value iteration, the action which maximizes the value should also be stored. ```Python @@ -342,7 +355,7 @@ def mdp_solve(transition_function, reward_function, gamma: float, states: List, return {states[i]: p[i] for i in range(len(states))} ``` -By running the code above this result is generated. +By running the script below, this result will be generated which is the optimal policy. ```Python >>> mdp_solve(car_transition, car_reward, 0.9, car_states, car_terminals, car_actions, 10) From 8c90a06ba9ee4bacf7a39fa10654ffd7fd7d40c5 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Thu, 27 Jan 2022 18:27:28 +0330 Subject: [PATCH 39/42] Add new example --- notebooks/19_intro_to_mdp/index.md | 135 ++++++++++++++++++++--------- notebooks/19_intro_to_mdp/quiz.png | Bin 0 -> 33744 bytes 2 files changed, 96 insertions(+), 39 deletions(-) create mode 100644 notebooks/19_intro_to_mdp/quiz.png diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index ed364c89..0a6540f9 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -205,47 +205,21 @@ def car_transition(state, action, next_state): if action == 'slow': if next_state == 'Cool': return 1.0 - if next_state == 'Warm': - return 0.0 - if next_state == 'Over': - return 0.0 if action == 'fast': if next_state == 'Cool': return 0.5 if next_state == 'Warm': return 0.5 - if next_state == 'Over': - return 0.0 if state == 'Warm': if action == 'slow': if next_state == 'Cool': return 0.5 if next_state == 'Warm': return 0.5 - if next_state == 'Over': - return 0.0 if action == 'fast': - if next_state == 'Cool': - return 0.0 - if next_state == 'Warm': - return 0.0 - if next_state == 'Over': - return 1.0 - if state == 'Over': - if action == 'slow': - if next_state == 'Cool': - return 0.0 - if next_state == 'Warm': - return 0.0 - if next_state == 'Over': - return 1.0 - if action == 'fast': - if next_state == 'Cool': - return 0.0 - if next_state == 'Warm': - return 0.0 if next_state == 'Over': return 1.0 + return 0 def car_reward(state, action, next_state): @@ -323,7 +297,7 @@ def argmax(l: List) -> int: return index_max -def mdp_iterate(transition_function, reward_function, gamma: float, states: List, terminals: List, actions: List, v: List): +def mdp_iterate(transition_function, reward_function, gamma: float, states: List, terminals: List, actions: List, v: List): new_v = [] best_actions = [] for i in range(len(states)): @@ -332,18 +306,11 @@ def mdp_iterate(transition_function, reward_function, gamma: float, states: List new_v.append(v[i]) best_actions.append(None) continue - values_action = [] + values_actions = [] for action in actions: - values_next = [] - for next_state in states: - expected = reward_function(state, action, next_state) + (gamma * v[i]) - values_next.append(transition_function(state, action, next_state) * expected) - value_next = sum(values_next) - values_action.append(value_next) - value_action = max(values_action) - new_v.append(value_action) - best_action_i = argmax(values_action) - best_actions.append(actions[best_action_i]) + values_actions.append(sum([transition_function(state, action, states[j]) * (reward_function(state, action, states[j]) + gamma * v[j]) for j in range(len(states))])) + new_v.append(max(values_actions)) + best_actions.append(actions[argmax(values_actions)]) return new_v, best_actions @@ -362,6 +329,96 @@ By running the script below, this result will be generated which is the optimal {'Cool': 'fast', 'Warm': 'slow', 'Over': None} ``` +## Real Life Example + +In a TV quiz show, there are several levels. At each level, if the participant answers the question correctly, he / she will receive some prize. If the participant's answer is wrong, the participant leaves the competition empty-handed. Before each stage begins, the participant can decide whether to continue or withdraw and leave with reward that he / she already earned. + +Beside states defined for each level, There are three terminal states of *Win*, *Lost* and *Quit* in the game. Actions in each state are quit and play. The player will go to the quit state with probability of 1 if he / she decide to take action quit. otherwise he / she will pass the level i by probability of `win_ratio[i]` and go to the state which is for the next level. + +So we can model the problem of play / quit decision as a MDP as below. + +

+ +

+ +Considering $100, $200, $300, $400 and $500 as rewards and 0.9, 0.7, 0.6, 0.3, 0.1 as win ratio for levels 0 to 4 respectively, The model can be implemented as below + +```Python +quiz_levels = [f'{i}' for i in range(5)] +quiz_terminals = ['Win', 'Lost', 'Quit'] +quiz_states = quiz_levels + quiz_terminals +quiz_actions = ['play', 'quit'] + +quiz_win_ratio = [0.9, 0.7, 0.6, 0.3, 0.1] +quiz_win_amount = [100, 200, 300, 400, 500] + + +def quiz_transition(state, action, next_state): + if state in quiz_terminals: + return 0 + else: # quiz levels + state_level = int(state) + if action == 'quit': + if next_state == 'Quit': + return 1 + else: + return 0 + else: # play + if next_state == 'Win': + if state_level == 4: + return quiz_win_ratio[4] + else: + return 0 + elif next_state == 'Lost': + return 1 - quiz_win_ratio[state_level] + elif next_state == 'Quit': + return 0 + else: + next_state_level = int(next_state) + if next_state_level == state_level + 1: + return quiz_win_ratio[state_level] + else: + return 0 + + +def quiz_reward(state, action, next_state): + state_level = int(state) + if action == 'quit': + return 0 + else: # play + if next_state == 'Win': + return quiz_win_amount[4] + elif next_state == 'Lost': + return -1 * sum(quiz_win_amount[:state_level]) + elif next_state == 'Quit': + return 0 + else: + return quiz_win_amount[state_level] +``` + +By running the script below, this result will be generated which is the optimal policy. Consider that is this specific problem, we don't want to discount the reward. So we set gamma equal to one. + +```Python +>>> mdp_solve(quiz_transition, quiz_reward, 1, quiz_states, quiz_terminals, quiz_actions, 1000) +{'0': 'play', + '1': 'play', + '2': 'play', + '3': 'quit', + '4': 'quit', + 'Win': None, + 'Lost': None, + 'Quit': None} +``` + +By Calculating Expectation of reward earned in each step, It's obvious that in level 0, 1, and 2 the expection is greater than zero so continue playing is optimal policy, but in level 3 and 4, the expectation is less than zero so it's better to quit the game. + +$ +E_0 = 0.9 \times 100 + 0.1 \times 0 = 90 \\ +E_1 = 0.7 \times 200 + 0.3 \times -100 = 110 \\ +E_2 = 0.6 \times 300 + 0.4 \times -300 = 60 \\ +E_3 = 0.3 \times 400 + 0.7 \times -600 = -300 \\ +E_4 = 0.1 \times 500 + 0.9 \times -1000 = -850 +$ ## Conclusion In conclusion MDP is an appropriate tool to represent RL problems. In order to represent a problem using MDP, states, actions, transition model and rewards should be determined properly. Afterwards finding the optimal policy, i.e. the one with the most utility, is desired. In order to find the optimal utility, bellman equation should be solved. Algorithms such as value iteration and policy iteration attempt to solve the bellman equation feasibly. 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zqh8r)oac^}Y@{0Oz$_*2vsxI6X?-65BMo=`t30fE$Pg~W*o@$jU8PM;yeTFu$>FRF zoyGbE*!atD5gs2+5`%TUa0>)tRNAJ_lf*c`KPS+AF*eiHtl`!uofY2QDNn0xtpAPND2+rb$ft5`Sa?~&>xLWe@T%Ks&q6o(O+ob>u8!Ej5IW(J2grt zA9PVD?zgA8Rgu3QMWO5tj;jSuKIo~c@=9yK*Dwk0{U67~L&6GjJUoJ{dYWpM{~P2q|C_1)H{D|XZ>||Qya%NCq_`K`qj>P| Iznz8u4^hWpqyPW_ literal 0 HcmV?d00001 From 2b6c9b72a916dbe0572dd8510d4a222e8cc37886 Mon Sep 17 00:00:00 2001 From: "r.pirayadi" Date: Thu, 27 Jan 2022 19:20:43 +0330 Subject: [PATCH 40/42] Fix real life example explanations --- notebooks/19_intro_to_mdp/index.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 0a6540f9..5182adb6 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -331,17 +331,17 @@ By running the script below, this result will be generated which is the optimal ## Real Life Example -In a TV quiz show, there are several levels. At each level, if the participant answers the question correctly, he / she will receive some prize. If the participant's answer is wrong, the participant leaves the competition empty-handed. Before each stage begins, the participant can decide whether to continue or withdraw and leave with reward that he / she already earned. +In a TV quiz show, there are several levels. At each level, if the participant answers the question correctly, they will receive some prize. If the participant's answer is wrong, they leave the competition empty-handed. Before each stage begins, the participant can decide whether to continue or withdraw and leave the game with the reward which they already earned. -Beside states defined for each level, There are three terminal states of *Win*, *Lost* and *Quit* in the game. Actions in each state are quit and play. The player will go to the quit state with probability of 1 if he / she decide to take action quit. otherwise he / she will pass the level i by probability of `win_ratio[i]` and go to the state which is for the next level. +Beside states representing each level, There are three terminal states of *Win*, *Lost* and *Quit* in the game. Actions in each state are quit and play. The player will go to the quit state with probability of 1 if they decide to take action quit. otherwise they will pass the level i by probability of `win_ratio[i]` and go to the state which represents the next level. -So we can model the problem of play / quit decision as a MDP as below. +So the play / quit decision problem can be modeled as an MDP as below.

-Considering $100, $200, $300, $400 and $500 as rewards and 0.9, 0.7, 0.6, 0.3, 0.1 as win ratio for levels 0 to 4 respectively, The model can be implemented as below +Considering $100, $200, $300, $400 and $500 as rewards and 0.9, 0.7, 0.6, 0.3, 0.1 as win ratio for levels 0 to 4 respectively, The model can be implemented as below. ```Python quiz_levels = [f'{i}' for i in range(5)] @@ -396,7 +396,7 @@ def quiz_reward(state, action, next_state): return quiz_win_amount[state_level] ``` -By running the script below, this result will be generated which is the optimal policy. Consider that is this specific problem, we don't want to discount the reward. So we set gamma equal to one. +By running the script below, this result will be generated which is the optimal policy. Consider that in this specific problem, we don't want to discount the reward, So we set gamma to one. ```Python >>> mdp_solve(quiz_transition, quiz_reward, 1, quiz_states, quiz_terminals, quiz_actions, 1000) @@ -410,7 +410,7 @@ By running the script below, this result will be generated which is the optimal 'Quit': None} ``` -By Calculating Expectation of reward earned in each step, It's obvious that in level 0, 1, and 2 the expection is greater than zero so continue playing is optimal policy, but in level 3 and 4, the expectation is less than zero so it's better to quit the game. +In oreder to check the result acheived by the script above, expectation of reward earned in each step, is calculated. It can be seen that in level 0, 1, and 2 the expection is greater than zero so continuing the game is the optimal action in these states, but in level 3 and 4, the expectation is less than zero so it's better to quit the game. $ E_0 = 0.9 \times 100 + 0.1 \times 0 = 90 \\ From 7e64cc4068b6dfaf6164602cd735d3f086b1f727 Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Fri, 28 Jan 2022 01:09:49 +0330 Subject: [PATCH 41/42] Modify --- notebooks/19_intro_to_mdp/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index 5182adb6..ebfd6ca6 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -29,7 +29,7 @@ In nature, learning by trial and error is the most common way of learning. learning by trial and error is called Reinforcement Learning (RL) in computer literature. Markov Decision Process (MDP) is a foundational element of formulating RL mathematically. In a typical RL problem, there is a learner called agent, which makes decisions. The surrounding which agent interacts with is called __environment__. The environment provides rewards and a new state based on the actions the agent perform. All these could be modeled in a single MDP. The environment is modeled with __states__ and agent's possible decisions are modeled by __actions__. Since In real problems unexpected events may happen, agent's actions may not lead to expected results. This is why MDPs with __stochastic transition models__ are used. In RL, agent is not told how it should act but it is presented with rewards whether positive or negative based on its actions. These rewards are modeled by __reward function__ in an MDP. Solving an MDP means to find the best way that the agent can act. There are different ways to solve an MDP which all of them are based on __Bellman equation__, So in order to find the best guideline for the agent Bellman equation should be solved. -In this note, first MDP formalization is introduced. Then MDP search trees as a tool for presenting MDPs are explained. At the end Bellman equation is discussed as the way of solving an MDP. +In this note, first MDP formalization is introduced. A way to represent MDP is MDP search trees. MDP search trees are explained in the next part. At the end Bellman equation is discussed as the way of solving an MDP. ## Markov Decision Process From 3a7da504ad7187c22d8acaf55d4e097bef1a8def Mon Sep 17 00:00:00 2001 From: Aryan Ahadinia <62179268+AryanAhadinia@users.noreply.github.com> Date: Fri, 28 Jan 2022 21:50:11 +0330 Subject: [PATCH 42/42] Final Change --- notebooks/19_intro_to_mdp/index.md | 1 - 1 file changed, 1 deletion(-) diff --git a/notebooks/19_intro_to_mdp/index.md b/notebooks/19_intro_to_mdp/index.md index ebfd6ca6..07aecb60 100644 --- a/notebooks/19_intro_to_mdp/index.md +++ b/notebooks/19_intro_to_mdp/index.md @@ -97,7 +97,6 @@ By combining the concepts which were explained above the complete formulation fo - A set of possible states $S$. - A set of possible actions $A$. - A transition function $T(s, a, s')$: Probability that taking action $a$ in state $s$ leads to $s'$, i.e., $P(s'| s, a)$ -- Also called the model or the dynamic. - A reward function $R(s, a, s')$: Sometimes reward only depends on the resulting state or on the resulting state and action. So $R(s, a, s')$ can be replaced by $R(s')$ or $R(s', a)$ - A start state - Maybe a terminal state

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