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wsabiL.m
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function [ log_mu, log_Var, clktime, xxIter, hyp ] = wsabiL( ...
range, ... 1) 2 x D matrix, lower bnd top row.
priorMu, ... 2) Gaussian prior mean, D x 1.
priorVar, ... 3) Gaussian prior covariance, D x D.
kernelVar, ... 4) Initial input length scales, D x D.
lambda, ... 5) Initial output length scale.
alpha, ... 6) Alpha offset fraction, as in paper.
numSamples, ... 7) Number of BBQ samples to run.
loglikhandle, ... 8) Handle to log-likelihood function.
printing ) ... 9) If true, print intermediate output.
% Output structures:
% log_mu: log of the integral posterior mean.
% log_var: log of the integral posterior variance.
% clktime: vector of times per iteration, may want to cumulative sum.
% xxIter: numSamples x D array of sample locations used to build model.
% hyp: integral hyperparameters.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Relabel prior mean and covariance for brevity of code.
bb = priorMu;
BB = diag(priorVar)';
% Relabel input hyperparameters for brevity of code.
VV = diag(kernelVar)';
jitterNoise = 1e-6; % Jitter on the GP model over the log likelihood.
numEigs = inf; % If trying to use Nystrom ** NOT RECOMMENDED **
hypOptEvery = 1; % 1 => optimise hyperparameters every iteration.
dim = length(bb); % Dimensionality of integral.
% Limit absolute range of likelihood model hyperparameters for stability.
hypLims = 30*ones(1,dim+1);
% Allocate Storage
mu = zeros(numSamples-1,1);
logscaling = zeros(numSamples-1,1);
Var = zeros(numSamples-1,1);
clktime = zeros(numSamples-1,1);
lHatD_0_tmp = zeros(numSamples,1);
loglHatD_0_tmp = zeros(size(lHatD_0_tmp));
hyp = zeros(1,1+dim);
% Minimiser options (fmincon for hyperparameters)
options1 = optimset('fmincon');
options1.Display = 'none';
options1.GradObj = 'off';
options1.Algorithm = 'active-set';
options1.TolX = 1e-5;
options1.TolFun = 1e-5;
options1.MaxTime = 0.5;
options1.MaxFunEvals = 50;
%options1.UseParallel = 'always';
options1.AlwaysHonorConstraints = 'true';
% Minimiser options (fmincon if desired for active sampling)
options2 = optimset('fmincon');
options2.Display = 'none';
options2.GradObj = 'on';
%options2.DerivativeCheck = 'on';
options2.TolX = 1e-5;
options2.TolFun = 1e-5;
%options2.MaxTime = 0.5;
%options2.MaxFunEvals = 75;
options2.UseParallel = 'always';
options2.AlwaysHonorConstraints = 'true';
% Minimiser options (CMAES - advised for active sampling)
opts = cmaes('defaults');
opts.LBounds = range(1,:)';
opts.UBounds = range(2,:)';
opts.DispModulo = Inf;
opts.DispFinal = 'off';
opts.SaveVariables = 'off';
%opts.EvalParallel = 'on';
%opts.PopSize = 100;
%opts.Restarts = 1;
% Initial Sample set to priorMean:
xx = zeros(numSamples,dim);
xx(end,:) = bb+1e-6;
currNumSamples = 1;
if printing
fprintf('Iter: ');
end
for t = 1:numSamples - 1
if printing
if ~mod(t,100)
prstr = sprintf('Log Current Mean Integral: %g', ...
mu(t-1)*exp(logscaling(t-1)));
fprintf(prstr);
pause(1);
fprintf(repmat('\b',1,length(prstr)))
end
if t > 1
fprintf(repmat('\b',1,length(num2str(t-1))));
end
fprintf('%i',t);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Pre-process new samples -- i.e. convert to log space etc.
% Get batch of samples & variables from stack.
tmpT = cputime;
xxIter = xx(numSamples-currNumSamples+1:end,:); % Curr samples
% Call loglik handle for latest sample.
loglHatD_0_tmp(numSamples-currNumSamples+1) = ...
loglikhandle( xxIter(1,:) );
% Find the max in log space.
logscaling(t) = max(loglHatD_0_tmp(numSamples-currNumSamples+1:end));
% Scale batch by max, and exponentiate.
lHatD_0_tmp(numSamples-currNumSamples+1:end) = ...
exp(loglHatD_0_tmp(numSamples-currNumSamples+1:end) - logscaling(t));
% Evaluate the offset, alpha fraction of minimum value seen.
aa = alpha * min( lHatD_0_tmp(numSamples-currNumSamples+1:end) );
% Transform into sqrt space.
lHatD = sqrt(abs(lHatD_0_tmp(numSamples-currNumSamples+1:end)- aa)*2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ML-II On GP Likelihood model hyperparameters
hyp(1) = log( lambda );
hyp(2:end) = log(VV);
if currNumSamples > 3 && ~mod(currNumSamples,hypOptEvery)
if currNumSamples < numEigs + 1
hypLogLik = @(x) logLikGPDim(xxIter, lHatD, x);
else
hypLogLik = @(x) logLikGPDimNystrom(xxIter, lHatD, x, numEigs);
end
[hyp] = fmincon(hypLogLik, ...
hyp,[],[],[],[],-hypLims,hypLims,[],options1);
end
lambda = exp(hyp(1));
VV = exp(hyp(2:end));
% Scale samples by input length scales.
xxIterScaled = xxIter .* repmat(sqrt(1./VV),currNumSamples,1);
% Squared distance matrix
dist2 = pdist2_squared_fast(xxIterScaled, xxIterScaled);
% Evaluate Gram matrix
Kxx = lambda.^2 * (1/(prod(2*pi*VV).^0.5)) * exp(-0.5*dist2);
Kxx = Kxx + ...
lambda.^2*(1/(prod(2*pi*VV).^0.5))*jitterNoise*eye(size(Kxx));
Kxx = Kxx/2 + Kxx'/2; % Make sure symmetric for stability.
% Invert Gram matrix.
if currNumSamples < numEigs + 1
invKxx = Kxx \ eye(size(Kxx));
else % If using nystrom
idx = randperm( length(xxIter(:,1)) );
xxuScaled = xxIterScaled( idx(1:numEigs), : );
xxsScaled = xxIterScaled;
AA1 = pdist2_squared_fast(xxuScaled,xxuScaled);
AA2 = pdist2_squared_fast(xxsScaled,xxuScaled);
Kuu = lambda^2 * 1/sqrt(det(2*pi*VV)) * (exp( -0.5 * AA1 ) + ...
jitterNoise*eye(size(AA1)));
Ksu = lambda^2 * 1/sqrt(det(2*pi*VV)) * exp( -0.5 * AA2 );
[eVec, eVal] = eig(Kuu);
eVec = Ksu * ...
repmat(sqrt(numEigs)./diag(eVal)',length(eVal(:,1)),1) ...
.* eVec;
eVal = eVal / numEigs;
Z = jitterNoise * diag(1./diag(eVal)) + eVec'*eVec;
invKxx = (1./jitterNoise)*(eye(currNumSamples)-eVec*(Z \ eVec'));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Expected value of integral:
ww = invKxx * lHatD;
Yvar = (VV.*VV + 2*VV.*BB)./VV;
postProd = ww*ww';
xx2sq = xxIter .* repmat(sqrt(1./Yvar),currNumSamples,1);
bbch = bb .* sqrt(1./Yvar);
xx2sqFin = pdist2_squared_fast(xx2sq,bbch);
xxIterScaled2 = xxIter .* repmat(sqrt(BB./(VV.*Yvar)),currNumSamples,1);
dist4 = pdist2_squared_fast(xxIterScaled2,xxIterScaled2);
YY = lambda^4 * ...
(1 / prod(4*pi^2*((VV.*VV + 2*VV.*BB)))^0.5) * ...
exp(-0.5 * (pdist2(xx2sqFin,-xx2sqFin) + dist4)) .* ...
postProd;
% Mean of the integral at iteration 't', before scaling back up:
mu(t) = (aa + 0.5*sum(YY(:)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Variance of the integral, before scaling back up:
GG_coeff = lambda^6 * 1/prod(16*pi^4*(VV.*VV + 2*VV.*BB).*VV.*BB.* ...
(((VV.*VV + 2*VV.*BB)./BB) + ...
((VV.*VV + 2*VV.*BB)./VV) + VV + BB))^0.5 * ...
prod(2*pi*(VV.*VV + 2*VV.*BB))^0.5;
tmpVar = ((VV.*VV + 2*VV.*BB).*(Yvar + Yvar.*(VV./BB + VV + BB)));
xx3sq = xxIter .* ...
repmat(sqrt((Yvar.*VV)./tmpVar),currNumSamples,1);
bb3ch = bb .* sqrt((Yvar.*VV)./tmpVar);
xx2sqGGlh = xx2sqFin + pdist2_squared_fast(xx3sq,bb3ch);
xx4sq = xxIter .* ...
repmat(sqrt((VV.*Yvar)./tmpVar),currNumSamples,1);
bb4ch = bb .* sqrt((VV.*Yvar)./tmpVar);
xx5sq = xxIter .* ...
repmat(sqrt((VV.*Yvar.^2)./(tmpVar.*BB)),currNumSamples,1);
bb5ch = bb .* sqrt((VV.*Yvar.^2)./(tmpVar.*BB));
xx2sqGGrh = pdist2_squared_fast(xx4sq,bb4ch) + ...
pdist2_squared_fast(xx5sq,bb5ch);
xxIterScaled3 = xxIter .* ...
repmat(sqrt(Yvar.*BB./tmpVar),currNumSamples,1);
dist4 = pdist2_squared_fast(xxIterScaled3,xxIterScaled3);
xxIterScaled4 = xxIter .* ...
repmat(sqrt(BB./(Yvar.*VV)),currNumSamples,1);
dist5 = pdist2_squared_fast(xxIterScaled4,xxIterScaled4);
GG = GG_coeff * postProd .* ...
exp(-0.5*(pdist2(xx2sqGGlh, -xx2sqGGrh) + dist4));
YY2 = lambda^4 * (1 / prod(4*pi^2*((VV.*VV + 2*VV.*BB)))^0.5) * ...
exp(-0.5 * (pdist2(xx2sqFin,-xx2sqFin) + dist5)) .* ...
repmat(ww',length(ww),1);
Var(t) = (sum(sum(GG)) - sum(YY2,2)'*(invKxx * sum(YY2,2)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Actively select next sample point:
if rand < 1.1 % Sample starting location for search from prior.
strtSamp = mvnrnd(bb,diag(BB),1);
else
strtSamp = 2*range(2,:).*rand(1,dim) - 50;
end
% If using local optimiser (fast):
%EV = @(x) expectedVarL( transp(x), lambda, VV, ...
% lHatD, xxIter, invKxx, jitterNoise, bb, BB ); %Utility function
%newX = fmincon( EV, strtSamp,[],[],[],[], ...
% range(1,:),range(2,:),[],options2 );
% If using global optimiser (cmaes):
newX = cmaes( 'expectedVarL', strtSamp', [],opts, lambda, VV, ...
lHatD, xxIter, invKxx, jitterNoise, bb, BB);
newX = newX';
xx(numSamples-currNumSamples,:) = newX;
clktime(t) = cputime - tmpT;
currNumSamples = currNumSamples + 1;
end
fprintf('\n done. \n');
log_mu = log(mu) + logscaling;
log_Var = log(Var) + 2*logscaling;
end