├── LICENSE.txt ├── README.md ├── basis_cst.m ├── basis_none.m ├── bfgs_search_prior.m ├── chaining_tree.m ├── chaining_ucb.m ├── cholpsd.m ├── cummax.m ├── enet_greedy.m ├── gp_dist.m ├── gp_downdate.m ├── gp_inf.m ├── gp_inf_update.m ├── gp_lik.m ├── gp_loolik.m ├── gp_pred.m ├── gp_sample.m ├── gpopt.m ├── gpucb.m ├── kernel_matern.m ├── kernel_se.m ├── kernel_se_normiso.m ├── solve_chol.m └── sq_dist.m /LICENSE.txt: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Gp Optimization 2 | 3 | The MATLAB code provided here includes several optimization algorithms (purely sequential or batch) using Gaussian processes. Available algorithms include GP-UCB, EI, Chaining-UCB for sequential optimization, and GP-UCB-PE, GP-B-UCB for batch optimization. The implementation uses efficient updating formulae in order to remain scalable to a large number of training points. 4 | 5 | ## Documentation 6 | 7 | Please go to http://econtal.perso.math.cnrs.fr/software/ for example of use and documentation. 8 | 9 | ## Relevant publications 10 | 11 | **Statistical Learning Approaches for Global Optimization** 12 | Emile Contal 13 | PhD 14 | http://econtal.perso.math.cnrs.fr/publications/phd.pdf 15 | 16 | 17 | **Stochastic Process Bandits: Upper Confidence Bounds Algorithms via Generic Chaining** 18 | Emile Contal, Nicolas Vayatis 19 | http://arxiv.org/pdf/1602.04976v1.pdf 20 | 21 | 22 | **A Ranking Approach to Global Optimization** 23 | Cédric Malherbe, Emile Contal, Nicolas Vayatis 24 | ICML 2016 25 | http://arxiv.org/pdf/1603.04381v1.pdf 26 | 27 | 28 | **Optimization for Gaussian Processes via Chaining** 29 | Emile Contal, Cédric Malherbe, Nicolas Vayatis 30 | NIPS Workshop on Bayesian Optimization 31 | http://arxiv.org/pdf/1510.05576v1.pdf 32 | 33 | 34 | **Gaussian Process Optimization with Mutual Information** 35 | Emile Contal, Vianney Perchet, Nicolas Vayatis 36 | ICML 2014 37 | http://jmlr.csail.mit.edu/proceedings/papers/v32/contal14.pdf 38 | 39 | 40 | **Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration** 41 | ECML 2013 42 | Emile Contal, David Buffoni, Alexandre Robicquet, Nicolas Vayatis 43 | http://arxiv.org/pdf/1304.5350v3.pdf 44 | -------------------------------------------------------------------------------- /basis_cst.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [B] = basis_cst(X) 18 | %% 19 | % Constant mean function 20 | %% Syntax 21 | % B = basis_cst(X) 22 | %% Arguments 23 | % * _X_ matrix _(n, d)_ where _n_ is the number of data points and _d_ is the dimension 24 | %% Outputs 25 | % * _B_ vector _(n, 1)_ of ones 26 | %% See also 27 | % 28 | 29 | B = [ones(size(X,1),1)]; 30 | end 31 | -------------------------------------------------------------------------------- /basis_none.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [B] = basis_none(X) 18 | %% 19 | % Zero mean GP inferance 20 | %% Syntax 21 | % B = basis_none(X) 22 | %% Arguments 23 | % * _X_ matrix _(n, d)_ where _n_ is the number of data points and _d_ is the dimension 24 | %% See also 25 | % 26 | 27 | B = []; 28 | end 29 | -------------------------------------------------------------------------------- /bfgs_search_prior.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [HP, kernel, noise, fval] = bfgs_search_prior(Xt, Yt, HPini, kfun, basis) 18 | %% 19 | % Optimize prior hyper-parameter with respect to the pseudo-likelihood, using the BFGS algorithm 20 | %% Syntax 21 | % HP = bfgs_search_prior(Xt, Yt, HPini, kfun, basis) 22 | %% Arguments 23 | % * _Xt_ matrix _(n, d)_ where _n_ is the number of data points and _d_ is the dimension 24 | % * _Yt_ vector _(n, 1)_ of noisy observations 25 | % * _HPini_ vector _(h, 1)_ of initial hyper-parameters formatted as : _[log(sf2) log(sn) log(w1) log(w2)]_ 26 | % * _kfun_ kernel function such as __ 27 | % * _basis_ basis function such as __ 28 | %% Outputs 29 | % * _HP_ vector _(h, 1)_ of locally optimal log hyper-parameters 30 | % * _kernel_ found kernel function 31 | % * _noise_ found noise standard deviation 32 | %% See also 33 | % 34 | 35 | NelderMeadIters = 50; 36 | 37 | Ht = basis(Xt); 38 | f = @(hp) nllcost(Xt, Yt, hp(1), hp(2), hp(3:end), kfun, Ht); 39 | 40 | % Starts with few Nelder-Mead iterations 41 | [hpopt, fvalNM] = fminsearch(f, HPini, optimset('maxFunEvals',NelderMeadIters,'display','off')); 42 | % BGFS search 43 | [hpopt, fval] = fminunc(f, hpopt,optimset('display','off','LargeScale','off')); 44 | 45 | HP = hpopt; 46 | kernel = @(x,y) kfun(x,y, exp(HP(1)), exp(HP(3:end))); 47 | noise = exp(HP(1)+HP(2)); 48 | 49 | end 50 | 51 | 52 | function [nll] = nllcost(Xt, Yt, sf2, rsn, W, kfun, Ht) 53 | 54 | Ktt = kfun(Xt, Xt, exp(sf2), exp(W)); 55 | noise = exp(sf2+rsn); 56 | BayesInv = gp_inf(Ktt, Yt, noise, Ht); 57 | nll = gp_loolik(Ktt, Yt, BayesInv, Ht); 58 | 59 | end -------------------------------------------------------------------------------- /chaining_tree.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [eNet, PiX, DeltaT, U] = chaining_tree(D2, dmax, step, iMin, iMax, varargin) 18 | %% 19 | % Compute the chaining tree given the canonical distance matrix between point of _X_ 20 | %% Syntax 21 | % [eNet, PiX, DeltaT, U] = chaining_tree(D2, dmax, step, iMin, iMax) 22 | % [eNet, PiX, DeltaT, U] = chaining_tree(..., 'Name',Value) 23 | %% Arguments 24 | % * _D2_ matrix _(n, n)_ of canonical squared distance 25 | % * _dmax_ scalar of initial diameter of _X_ 26 | % * _step_ scalar > 1 for the geometric decay of $\epsilon_i$ 27 | % * _iMin_ integer for the first level to consider 28 | % * _iMax_ integer for the last level to consider 29 | %% Name-Value Pair Arguments 30 | % * _a_ scalar > 1 power to use for the geometric decay in the union bound, e.g. 2 31 | % * _lza_ scalar of logarithm of the Riemann zeta of a, e.g. _log(pi^2/6)_ 32 | %% Outputs 33 | % * _eNet_ vector _(1, N)_ of indices of points in the final $\epsilon$-net 34 | % * _PiX_ matrix _(h, n)_ of indices of the closest element of the net for all _h=iMax-iMin_ levels 35 | % * _DeltaT_ matrix _(h, N)_ of diameters of the cells of the net for all _h_ levels 36 | % * _U_ vector _(h, 1)_ of negative log probabilities w.r.t the union bounds for all _h_ levels 37 | %% See also 38 | % | 39 | 40 | ip = inputParser; 41 | ip.addOptional('a', 2.2); 42 | ip.addOptional('lza', 0.399141); 43 | ip.parse(varargin{:}); 44 | opt = ip.Results; 45 | 46 | n = size(D2,1); 47 | 48 | Ti = []; 49 | nSteps = iMax-iMin; 50 | PiX = zeros(nSteps,n,'uint32'); 51 | DeltaT = inf(nSteps,n); 52 | U = inf(nSteps,1); 53 | 54 | for ind=1:nSteps 55 | i = ind+iMin-1; 56 | ei = dmax*step^-i; 57 | [ni,Ti] = enet_greedy(D2,ei,Ti); 58 | U(ind) = log(ni+1) + opt.a*log(i) + opt.lza; 59 | 60 | [TiDist2, TiInd] = min(D2(Ti,:), [], 1); 61 | PiX(ind,:) = TiInd; 62 | for j=1:length(Ti) 63 | DeltaTi_j = sqrt(max(TiDist2(TiInd==j))); 64 | if ~isempty(DeltaTi_j) 65 | DeltaT(ind,j) = DeltaTi_j; 66 | end 67 | end 68 | end 69 | 70 | eNet = Ti; 71 | 72 | end -------------------------------------------------------------------------------- /chaining_ucb.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [ucb] = chaining_ucb(D2, s2, u, varargin) 18 | %% 19 | % Compute the chaining UCB given the canonical distance matrix between point of _X_ 20 | %% Syntax 21 | % ucb = chaining_ucb(D2, s2, u) 22 | % ucb = chaining_ucb(..., 'Name',Value) 23 | %% Arguments 24 | % * _D2_ matrix _(n, n)_ of canonical squared distance 25 | % * _s2_ matrix _(n, 1)_ of posterior variance 26 | % * _u_ scalar for negative log probability 27 | %% Name-Value Pair Arguments 28 | % * _step_ scalar > 1 power to use for the geometric decay of $\epsilon$ 29 | % * _cdfinv_ function to upper bound the tail probabilities e.g. _@(u,d) d.*sqrt(2*u)_ 30 | %% Ouputs 31 | % * _ucb_ vector _(n, 1)_ such that $P[\sup_{x^\star}f(x^\star)-f(x)-\mu(x^\star)+\mu(x^\star) > ucb(x)+cst] < \exp(-u)$ 32 | %% See also 33 | % | 34 | 35 | ip = inputParser; 36 | ip.addOptional('step', 2); 37 | ip.addOptional('cdfinv', @(u,d) sqrt(2)*d.^2*erfcinv(2*exp(-u))); 38 | ip.parse(varargin{:}); 39 | opt = ip.Results; 40 | 41 | n = size(D2,1); 42 | 43 | % chaining tree 44 | dmax = max(sqrt(2*s2)); 45 | deltaInd = ceil(-log(sqrt(s2)./dmax)/log(opt.step)); % indices such that an upper bound on Delta(X) is < s(X) 46 | [Tree,PiX,DeltaT,U] = chaining_tree(D2,dmax,opt.step,min(deltaInd),max(deltaInd)); 47 | 48 | % compute ucb(X) 49 | ucb = zeros(n,1); 50 | L1 = false(n,1); % i > min{ i: Delta_i(X) < s(X) } 51 | DeltaXi1 = inf(n,1); 52 | for i=1:size(PiX,1) 53 | ui = U(i)+u; 54 | DeltaXi = DeltaT(i,PiX(i,:))'; 55 | LT = DeltaXi < sqrt(s2); 56 | L0 = LT & (~L1); 57 | ucb(L0) = opt.cdfinv(ui, sqrt(s2(Tree(PiX(i,L0))))); % i = min{ i: Delta_i(X) < s(X) } 58 | ucb(L1) = ucb(L1) + opt.cdfinv(ui, DeltaXi1(L1)); % i > min{ } 59 | DeltaXi1 = DeltaXi; 60 | L1(LT) = true; 61 | end 62 | 63 | end -------------------------------------------------------------------------------- /cholpsd.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [R] = cholpsd(X) 18 | %% 19 | % Upper Cholesky decomposition of psd matrix 20 | % Tries to approach numerically $\lim_{\epsilon \to 0} chol(X+\epsilon I)$ 21 | %% Syntax 22 | % [R] = cholpsd(X) 23 | %% Arguments 24 | % * _X_ psd matrix _(n, n)_ 25 | %% Outputs 26 | % * _R_ upper triangular matrix _(n, n)_ such that _X=R'R_ 27 | %% See also 28 | % 29 | 30 | m = min(min(X)); 31 | if m<0; fprintf('Error, X is negative in cholpsd\n'); error(''); end 32 | m = max(eps, m*1e-14); 33 | e = m; 34 | I = eye(size(X)); 35 | ok = false; 36 | while ~ok 37 | try 38 | R = chol(X); 39 | ok = true; 40 | catch 41 | % if the Cholesky decomposition failed, try to add a small epsilon on the diagonal 42 | X = X+e*I; 43 | if e > 1e6 * m 44 | fprintf('Warning, adding %f for cholpsd\n', e); 45 | end 46 | e = 10*e; 47 | end 48 | end 49 | 50 | end -------------------------------------------------------------------------------- /cummax.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function M = cummax(R) 18 | %% 19 | % Cumulative maximum as used in simple regrets 20 | %% Syntax 21 | % M = cummax(R) 22 | %% Arguments 23 | % * _R_ matrix _(n,t)_ for _n_ run of length _t_ 24 | %% Outputs 25 | % * _M_ matrix _(n,t)_ of maximum along the column _(:,1:i)_ for each row 26 | 27 | [d,n] = size(R); 28 | M = zeros(d,n); 29 | m = R(:,1); 30 | M(:,1) = m; 31 | 32 | for i=2:n 33 | m = max(m, R(:,i)); 34 | M(:,i) = m; 35 | end 36 | 37 | end -------------------------------------------------------------------------------- /enet_greedy.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [N, eNet] = enet_greedy (D2, e, eNetInit) 18 | %% 19 | % Compute an $\epsilon$-net of _Xt_ given the distance matrix 20 | %% Syntax 21 | % [N, eNet] = enet_greedy(D2, e, eNetInit) 22 | %% Arguments 23 | % * _D2_ matrix _(n, n)_ of canonical squared distance 24 | % * _e_ scalar for $\epsilon$ 25 | % * _eNetInit_ vector _(1, N)_ of indices of points in the initial $\epsilon$-net 26 | %% Outputs 27 | % * _N_ scalar for the size of the final $\epsilon$-net 28 | % * _eNet_ vector _(1, N)_ of indices of points in the final $\epsilon$-net 29 | %% See also 30 | % | 31 | 32 | e2 = e^2; 33 | [n,~] = size(D2); 34 | L = sparse(D2<=e2); 35 | eNet = eNetInit; 36 | if eNet 37 | nc = ~any(L(:,eNet),2); % not covered 38 | else 39 | nc = true(n,1); 40 | end 41 | while any(nc) 42 | nc_ind = find(nc)'; 43 | P = sum(L(nc,nc), 2); % number of points in the ball of center X_i and radius e 44 | [~,maxP] = max(P); % we choose the maximum 45 | isolated = find(P==1)'; % and all the isolated points 46 | eNet = [eNet nc_ind([maxP isolated])]; 47 | nc(nc) = ~any(L(nc,eNet),2); 48 | end 49 | N = length(eNet); 50 | 51 | end 52 | -------------------------------------------------------------------------------- /gp_dist.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [D2, Kuv_post] = gp_dist(Kuv, Ktu, Ktv, dKuu, dKvv, BayesInv, Ht, Hu, Hv) 18 | %% 19 | % Canonical GP distance $d^2(u,v) = Var[f(u)-f(v) \mid Xt] = \sigma_t^2(u)+\sigma_t^2(v)-2k(u,v)$ 20 | %% Syntax 21 | % D2 = gp_dist(Kuv, Ktu, Ktv, dKuu, dKvv, BayesInv) 22 | % D2 = gp_dist(Kuv, Ktu, Ktv, dKuu, dKvv, BayesInv, Ht, Hu, Hv) 23 | %% Arguments 24 | % * _Kuv_ kernel matrix _(nu, nv)_ between the points of _U_ and _V_ 25 | % * _Ktu_ kernel matrix _(nt, nu)_ between the points of _Xt_ and _U_ 26 | % * _Ktv_ kernel matrix _(nt, nv)_ between the points of _Xt_ and _V_ 27 | % * _dKuu_ vector _(nu, 1)_ of the diagonal kernel between the points of _U_ 28 | % * _dKvv_ vector _(nv, 1)_ of the diagonal kernel between the points of _V_ 29 | % * _BayesInv_ struct array as returned by _(Ht, Ktt, Yt, noise)_ 30 | % * _Ht_ matrix _(nt, b)_ basis for the points of _Xt_ 31 | % * _Hu_ matrix _(nu, b)_ basis for the points of _U_ 32 | % * _Hv_ matrix _(nv, b)_ basis for the points of _V_ 33 | %% Outputs 34 | % * _D2_ matrix _(nu, nv)_ of squared distance between _U_ and _V_ 35 | %% See also 36 | % 37 | 38 | if nargin<7; Ht = []; end 39 | if nargin<8; Hu = []; end 40 | if nargin<9; Hv = []; end 41 | 42 | [~,s2u] = gp_pred(Ktu, dKuu, BayesInv, Ht, Hu); 43 | [~,s2v] = gp_pred(Ktv, dKvv, BayesInv, Ht, Hv); 44 | 45 | % TODO update with basis 46 | if Ht 47 | error('gp_dist is not implemented with basis functions.') 48 | end 49 | Kuv_post = Kuv - Ktu'*solve_chol(BayesInv.RC, Ktv); 50 | D2 = bsxfun(@plus,s2u,bsxfun(@minus,s2v',2*Kuv_post)); 51 | D2(D2. 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [mu, sigma2] = gp_downdate(Ktt, Yt, i, BayesInv, Ht) 18 | %% 19 | % Posterior $\mu(x_i)$ and $\sigma^2(x_i)$ given $X_t \setminus \{x_i\}$ 20 | %% Syntax 21 | % [mu, sigma2] = gp_downdate(Ktt, Yt, i, BayesInv) 22 | % [mu, sigma2] = gp_downdate(Ktt, Yt, i, BayesInv, Ht) 23 | %% Arguments 24 | % * _Ktt_ kernel matrix _(nt, nt)_ between the points of _Xt_ 25 | % * _Yt_ vector _(nt, 1)_ of observations 26 | % * _i_ indice of removed observation 27 | % * _BayesInv_ struct array returned by _(Ht, Ktt, Yt, noise)_ 28 | % * _Ht_ matrix _(nt, b)_ of basis data as returned by _(Xt)_ 29 | %% Outputs 30 | % * _mu_ scalar _(1,1)_ posterior mean $E[f(x_i) \mid X_t\setminus x_i, Y_t \setminus y_i]$ 31 | % * _sigma2_ scalar _(1,1)_ posterior variance $V[f(x_i) \mid X_t\setminus x_i, Y_t \setminus y_i]$ 32 | %% See also 33 | % | 34 | 35 | n = size(Ktt,1); 36 | T = [1:i-1 i+1:n]; 37 | Yt1 = Yt(T); 38 | 39 | % Covariance 40 | Kti = Ktt(T,i); 41 | Kii = Ktt(i,i); 42 | 43 | % Cholsky downdates (cf Osborne2010 p216) 44 | RC = BayesInv.RC; 45 | RC11 = RC(1:i-1,1:i-1); 46 | RC13 = RC(1:i-1,i+1:end); 47 | S23 = RC(i,i+1:end); 48 | S33 = RC(i+1:end,i+1:end); 49 | 50 | RC33 = cholupdate(S33, S23'); 51 | RC1 = [RC11 RC13; zeros(size(RC13')) RC33]; 52 | 53 | if Ht 54 | Ht1 = Ht(T,:); 55 | Hi = Ht(i,:); 56 | 57 | RHCH1 = cholpsd(Ht1'*solve_chol(RC1,Ht1)); 58 | 59 | % System resolution (cf RasmussenWilliams2006 Ch2 p28 Eq2.42) 60 | Ri = Hi - Ht1'*solve_chol(RC1,Kti); 61 | bet = solve_chol(RHCH1, (Ht1'*solve_chol(RC1, Yt1))); 62 | invCY = solve_chol(RC1, (Yt1 - Ht1*bet)); 63 | mu = Hi'*bet + Kti'*invCY; 64 | else 65 | invCY = solve_chol(RC1, Yt1); 66 | mu = Kti'*invCY; 67 | bet = []; 68 | end 69 | 70 | % sigma2 71 | if nargout > 1 72 | Vf = RC1'\Kti; 73 | covf = Kii - sum(Vf.*Vf, 1)'; 74 | 75 | if Ht 76 | Vb = RHCH1'\Ri; 77 | covb = sum(Vb.*Vb, 1)'; 78 | sigma2 = covb + covf; 79 | else 80 | sigma2 = covf; 81 | end 82 | end 83 | 84 | end 85 | -------------------------------------------------------------------------------- /gp_inf.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [BayesInv] = gp_inf(Ktt, Yt, noise, Ht) 18 | %% 19 | % Bayesian system resolution for computing posterior of GP given the observations _Yt_ at _Xt_ 20 | %% Syntax 21 | % BayesInv = gp_inf(Ktt, Yt, noise) 22 | % BayesInv = gp_inf(Ktt, Yt, noise, Ht) 23 | %% Arguments 24 | % * _Ktt_ matrix _(nt, nt)_ of kernel between the points of _Xt_ 25 | % * _Yt_ vector _(nt, 1)_ of observations 26 | % * _noise_ noise standard deviation $\eta^2$ 27 | % * _Ht_ matrix _(nt, b)_ of basis data as returned by _(Xt)_ 28 | %% Outputs 29 | % struct array containing: 30 | % 31 | % * _RC_ upper triangular matrix _(nt,nt)_ of Cholesky decomposition of _Ktt+noise*I_ 32 | % * _invCY_ vector _(nt,1)_ solution of $(K + \eta^2 I)^{-1} Y$ 33 | % * _beta_ vector _(b,1)_ solution of the basis system 34 | %% See also 35 | % | 36 | 37 | if nargin<4; Ht = []; end 38 | 39 | C = Ktt + noise*eye(size(Ktt)); % (nt x nt) 40 | 41 | % Cholesky decomposition 42 | RC = cholpsd(C); % (nt x nt) 43 | if Ht 44 | RHCH = cholpsd(Ht' * (solve_chol(RC, Ht))); % (b x b) 45 | 46 | % system resolution 47 | bet = solve_chol(RHCH, (Ht' * solve_chol(RC, Yt))); % (b x 1) 48 | invCY = solve_chol(RC, (Yt - Ht*bet)); 49 | else 50 | invCY = solve_chol(RC, Yt); 51 | bet = []; 52 | RHCH = []; 53 | end 54 | 55 | BayesInv = struct('RC',RC, 'invCY',invCY, 'bet',bet, 'RHCH',RHCH); 56 | 57 | end 58 | -------------------------------------------------------------------------------- /gp_inf_update.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function BayesInv = gp_inf_update(Ktt12, Ktt22, Yt, noise, BayesInv1, Ht) 18 | %% 19 | % Bayesian system update with new observations at _Xt2_ 20 | %% Syntax 21 | % BayesInv = gp_inf_update(Ktt12, Ktt22, Yt, noise, BayInv1) 22 | % BayesInv = gp_inf_update(Ktt12, Ktt22, Yt, noise, BayInv1, Ht) 23 | %% Arguments 24 | % * _Ktt12_ matrix _(nt1, nt2)_ of kernel between the points of _Xt1_ and _Xt2_ 25 | % * _Ktt22_ matrix _(nt2, nt2)_ of kernel between the points of _Xt2_ 26 | % * _Yt_ vector _(nt1+nt2, 1)_ of observations 27 | % * _noise_ noise standard deviation $\eta^2$ 28 | % * _BayInv1_ old system resolution for _Xt1_ 29 | % * _Ht_ matrix _(nt1+nt2, b)_ of basis data 30 | %% Outputs 31 | % struct array containing: 32 | % 33 | % * _RC_ upper triangular matrix _(nt1+nt2,nt1+nt2)_ of Cholesky decomposition of _Ktt+noise*I_ 34 | % * _invCY_ vector _(nt1+nt2,1)_ solution of $(K + \eta^2 I)^{-1} Y$ 35 | % * _beta_ vector _(b,1)_ solution of the basis system 36 | %% See also 37 | % 38 | 39 | [nt1,nt2] = size(Ktt12); 40 | if size(Ktt22,1)~=nt2 | size(Ktt22,2)~=nt2 | size(Yt,1)~=nt1+nt2 41 | error('Wrong arguments size: Ktt12:(%d,%d), Ktt22:(%d,%d), Yt:(%d,%d)',size(Ktt12),size(Ktt22),size(Yt)); 42 | end 43 | if size(BayesInv1.RC,1)~=nt1 44 | error('Arguments size incompatible with previous BayesInv: Ktt12:(%d,%d), RC1:(%d,%d)',size(Ktt12),size(BayesInv1.RC)); 45 | end 46 | 47 | C12 = Ktt12; 48 | if length(noise)==1 49 | C22 = Ktt22 + noise*eye(size(Ktt22)); % (nt x nt) 50 | else 51 | C22 = Ktt22 + diag(noise); % (nt x nt) 52 | end 53 | 54 | % Cholsky updates (cf Osborne2010 p214) 55 | RC1 = BayesInv1.RC; 56 | RC12 = RC1'\C12; 57 | RC22 = chol(C22 - RC12'*RC12); 58 | RC = [RC1 RC12; zeros(size(RC12')) RC22]; % (nt x nt) 59 | 60 | if Ht 61 | RHCH = cholpsd(Ht'*(solve_chol(RC, Ht))); % (b x b) 62 | 63 | % system resolution (cf RasmussenWilliams2006 Ch2 p28 Eq2.42) 64 | bet = solve_chol(RHCH, (Ht' * solve_chol(RC, Yt))); % (b x 1) 65 | invCY = solve_chol(RC, (Yt-Ht*bet)); 66 | else 67 | invCY = solve_chol(RC, Yt); 68 | bet = []; 69 | RHCH = []; 70 | end 71 | 72 | BayesInv = struct('RC',RC, 'invCY',invCY, 'bet',bet, 'RHCH',RHCH); 73 | 74 | end 75 | -------------------------------------------------------------------------------- /gp_lik.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [nll] = gp_lik(Ktt, Yt, BayesInv, Ht) 18 | %% 19 | % Negative log likelihood given oberservations _Yt_ at _Xt_ 20 | %% Syntax 21 | % nll = gp_lik(Ktt, Yt, BayesInv) 22 | % nll = gp_lik(Ktt, Yt, BayesInv, Ht) 23 | %% Arguments 24 | % * _Ktt_ matrix _(nt, nt)_ of kernel between the points of _Xt_ 25 | % * _Yt_ vector _(nt, 1)_ of observations 26 | % * _BayesInv_ structure array returned by _(Ht, Ktt, Yt, noise)_ 27 | % * _Ht_ matrix _(nt, b)_ of basis data as returned by _(Xt)_ 28 | %% Outputs 29 | % * _nll_ float, negative of the logarithm of the likelihood 30 | %% See also 31 | % | 32 | 33 | if nargin<4; Ht = []; end 34 | 35 | n = size(Yt,1); 36 | m = rank(Ht); 37 | if m 38 | RK = cholpsd(Ktt); 39 | A = Ht'*(solve_chol(RK, Ht)); 40 | RA = cholpsd(A); 41 | KHAHK = solve_chol(RK, Ht*solve_chol(RA, Ht'*solve_chol(RK, eye(n)))); 42 | 43 | nll = .5*Yt'*BayesInv.invCY - .5*Yt'*KHAHK*Yt ... 44 | + sum(log(diag(RK))) + sum(log(diag(A))) + .5*(n-m)*log(2*pi); 45 | else 46 | nll = .5*Yt'*(BayesInv.invCY) + .5*sum(log(diag(BayesInv.RC))) + .5*n*log(2*pi); 47 | end 48 | 49 | end 50 | -------------------------------------------------------------------------------- /gp_loolik.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [nll] = gp_loolik(Ktt, Yt, BayesInv, Ht) 18 | %% 19 | % Negative log leave-one-out likelihood also called Pseudo-likelihood given oberservations _Yt_ at _Xt_ 20 | %% Syntax 21 | % nll = gp_loolik(Ktt, Yt, BayesInv) 22 | % nll = gp_loolik(Ktt, Yt, BayesInv, Ht) 23 | %% Arguments 24 | % * _Ktt_ matrix _(nt, nt)_ of kernel between the points of _Xt_ 25 | % * _Yt_ vector _(nt, 1)_ of observations 26 | % * _BayesInv_ structure array returned by _(Ht, Ktt, Yt, noise)_ 27 | % * _Ht_ matrix _(nt, b)_ of basis data as returned by _(Xt)_ 28 | %% Output 29 | % * _nll_ float, negative of the logarithm of the pseudo-likelihood 30 | %% See also 31 | % | 32 | 33 | if nargin<5; Ht = []; end 34 | 35 | n = size(Ktt, 1); 36 | nll = 0; 37 | 38 | for i=1:n 39 | [mui, s2i] = gp_downdate(Ktt, Yt, i, BayesInv, Ht); 40 | nll = nll + .5*log(s2i) + (Yt(i)-mui)^2/(2*s2i) + .5*log(2*pi); 41 | end 42 | 43 | if ~isreal(nll) 44 | nll = inf; 45 | end 46 | 47 | end 48 | -------------------------------------------------------------------------------- /gp_pred.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [mu, sigma2] = gp_pred(Kts, dKss, BayesInv, Ht, Hs) 18 | %% 19 | % Posterior mean and variance of GP given the kernel matrices and the Bayesian inferance 20 | %% Syntax 21 | % [mu, sigma2] = gp_pred(Kts, dKss, BayesInv) 22 | % [mu, sigma2] = gp_pred(Kts, dKss, BayesInv, Ht, Hs) 23 | %% Arguments 24 | % * _Kts_ matrix _(nt, ns)_ of kernel between the points of _Xt_ and _Xs_ 25 | % * _dKss_ matrix _(ns, 1)_ of diagonal kernel between the points of _Xs_ 26 | % * _BayesInv_ structure array returned by _(Ht, Ktt, Yt, noise)_ 27 | % * _Ht_ matrix _(nt, b)_ of basis data for _Xt_ 28 | % * _Hs_ matrix _(ns, b)_ of basis data for _Xs_ 29 | %% Outputs 30 | % * _mu_ matrix _(ns, 1)_ of posterior mean $E[f(X_s) \mid X_t, Y_t]$ 31 | % * _sigma2_ matrix _(ns, 1)_ of posterior variance $V[f(X_s) \mid X_t, Y_t]$ 32 | %% See also 33 | % | 34 | 35 | if nargin<5; Hs = []; end 36 | if nargin<4; Ht = []; end 37 | 38 | % mu 39 | if Ht 40 | mu = Hs*BayesInv.bet + Kts'*BayesInv.invCY; % (ns x 1) 41 | else 42 | mu = Kts'*BayesInv.invCY; 43 | end 44 | 45 | % sigma2 46 | if nargout > 1 47 | Vf = BayesInv.RC'\Kts; % (nt x ns) 48 | covf = dKss - sum(Vf.*Vf, 1)'; 49 | 50 | if Ht 51 | Rs = Hs' - Ht'*solve_chol(BayesInv.RC, Kts); % (b x ns) 52 | LHCH = cholpsd(Ht'*(solve_chol(BayesInv.RC, Ht))); % (b x b) 53 | Vb = LHCH'\Rs; % (b x ns) 54 | covb = sum(Vb.*Vb, 1)'; 55 | sigma2 = covb + covf; 56 | else 57 | sigma2 = covf; 58 | end 59 | end 60 | 61 | end 62 | -------------------------------------------------------------------------------- /gp_sample.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [f, Xs, Fs, Xt, Yt, Kss] = gp_sample(varargin) 18 | %% 19 | % Sample a Gaussian process in a hypercube and perform Bayesian inferance 20 | %% Syntax 21 | % [f, Xs, Fs, Xt, Yt, Kss] = gp_sample(..., 'Name',Value) 22 | %% Name-Value Pair Arguments 23 | % * _d_ integer for the dimension (default: 1) 24 | % * _size_ scalar for the length of the hypercube (default: 40) 25 | % * _ns_ integer for the number of sampled points (default: 1000) 26 | % * _nt_ integer for the number of training points (default: 10) 27 | % * _kernel_ kernel function (default: ) 28 | % * _basis_ basis function (default: ) 29 | % * _noise_ scalar for the noise standard deviation (default: 0.01) 30 | % * _plot_ boolean to visualize the optimization (default: true) 31 | % * _posterior_ boolean to compute the Bayesian inferance (default: true) 32 | % * _verbose_ boolean to monitor the process (default: true) 33 | %% Outputs 34 | % * _f_ function which returns noisy observations 35 | % * _Xs_ matrix _(ns, d)_ of sampled data 36 | % * _Fs_ matrix _(ns, 1)_ of sampled values 37 | % * _Xt_ matrix _(nt, d)_ of training data 38 | % * _Yt_ vector _(nt, 1)_ of training observations 39 | % * _Kss_ matrix _(ns, ns)_ of kernel values 40 | 41 | 42 | ip = inputParser; 43 | ip.addOptional('d', 1); 44 | ip.addOptional('size', 40); 45 | ip.addOptional('ns', 1000); 46 | ip.addOptional('nt', 10); 47 | ip.addOptional('kernel', @kernel_se_normiso); 48 | ip.addOptional('basis', @basis_none); 49 | ip.addOptional('noise', 1e-2); 50 | ip.addOptional('plot', true); 51 | ip.addOptional('posterior', true); 52 | ip.addOptional('verbose', true); 53 | ip.parse(varargin{:}); 54 | opt = ip.Results; 55 | 56 | if opt.verbose; fprintf('generating fn...\n'); end 57 | Xs = opt.size * rand(opt.ns, opt.d); 58 | Xt = Xs(1:opt.nt,:); 59 | Kss = opt.kernel(Xs,Xs); 60 | B = opt.basis(Xs); 61 | Fs = cholpsd(Kss)' * randn(opt.ns, 1); 62 | if B 63 | Fs = Fs + B*randn(size(B,2),1); 64 | end 65 | f = @(X) Fs(X) + opt.noise * randn(size(X,1), 1); 66 | Yt = f(1:opt.nt); 67 | 68 | if opt.posterior 69 | if opt.plot; plot(Xt, Yt, 'r+'); hold on; end 70 | 71 | if opt.verbose; fprintf('Bayesian inferance...\n'); end 72 | Ktt = Kss(1:opt.nt,1:opt.nt); 73 | Ht = opt.basis(Xt); 74 | BayesInv = gp_inf(Ktt, Yt, opt.noise, Ht); 75 | 76 | if opt.verbose; fprintf('Prediction...\n'); end 77 | Kts = Kss(1:opt.nt,:); 78 | Hs = opt.basis(Xs); 79 | dKss = diag(Kss); 80 | [mu, s2] = gp_pred(Kts, dKss, BayesInv, Ht, Hs); 81 | 82 | if opt.plot 83 | [Z,IZ] = sort(Xs(:,1)); 84 | plot(Z, mu(IZ), 'k'); 85 | fill([Z; flipdim(Z,1)], [mu(IZ)+2*sqrt(s2(IZ)); flipdim(mu(IZ)-2*sqrt(s2(IZ)),1)], ... 86 | [7 8 7]/8, 'EdgeColor','None', 'FaceAlpha',.8); 87 | end 88 | else 89 | if opt.plot 90 | if opt.d == 1 91 | [Z,IZ] = sort(Xs(:,1)); 92 | plot(Z, Fs(IZ), 'k'); 93 | elseif opt.d == 2 94 | scatter(Xs(:,1),Xs(:,2),30,Fs,'fill'); 95 | end 96 | end 97 | end 98 | 99 | end -------------------------------------------------------------------------------- /gpopt.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [queries, Yt] = gpopt(f, Xt, Yt, Xs, T, varargin) 18 | %% 19 | % Sequential optimization using GP algorithms 20 | %% Syntax 21 | % [queries, Yt] = gpopt(f, Xt, Yt, Xs, T) 22 | % [queries, Yt] = gpopt(..., 'Name',Value) 23 | %% Arguments 24 | % * _f_ function which returns observations 25 | % * _Xt_ matrix _(nt, d)_ of initial data 26 | % * _Yt_ vector _(nt, 1)_ of initial observations 27 | % * _Xs_ matrix _(ns, d)_ of search data 28 | % * _T_ integer for the number of iterations 29 | %% Name-Value Pair Arguments 30 | % * _algo_ string code for the query algorithm 31 | % possible values are 'gpucb' (default), 'chaining', 'ei' for purely sequential optimization 32 | % and 'gpucbpe', 'greedyucb' for batch sequential optimization 33 | % * _noise_ scalar for the noise standard deviation (default: 1e-2) 34 | % * _u_ scalar for the negative logarithm of the upper bound probability (default: 3) 35 | % * _B_ integer for the size of the batch in batch sequential optimization (default: 1) 36 | % * _kernel_ kernel function for the inferance (default: ) 37 | % * _basis_ basis function for the inferance (default: ) 38 | % * _Kss_ matrix _(ns,ns)_ of the kernel between points of _Xs_, used by the 'chaining' algorithm 39 | % * _plot_ boolean to visualize the optimization (default: false) 40 | % * _verbose_ boolean to monitor the optimization (default: true) 41 | %% Outputs 42 | % * _queries_ vector _(1,T)_ of queries indices 43 | % * _Yt_ vector _(T,1)_ of observations 44 | %% See also 45 | % | 46 | 47 | ip = inputParser; 48 | ip.addOptional('algo', 'gpucb'); 49 | ip.addOptional('u', 3); 50 | ip.addOptional('B', 1); 51 | ip.addOptional('kernel', @kernel_se_normiso); 52 | ip.addOptional('basis', @basis_none); 53 | ip.addOptional('noise', 1e-2); 54 | ip.addOptional('plot', false); 55 | ip.addOptional('verbose', true); 56 | ip.addOptional('Kss', []); 57 | ip.parse(varargin{:}); 58 | opt = ip.Results; 59 | 60 | % check input 61 | if strcmp(opt.algo, 'chaining') & isempty(opt.Kss) 62 | error('You must provide Kss in the optional arguments for using the chaining algorithm'); 63 | end 64 | if opt.B>1 & ~(strcmp(opt.algo,'gpucbpe') | strcmp(opt.algo,'gpbucb') | strcmp(opt.algo,'greedyucb')) 65 | error('The algorithm is not adapted for batch optimization'); 66 | end 67 | target_plot = []; 68 | 69 | [ns,d] = size(Xs); 70 | 71 | queries = []; 72 | Ht = opt.basis(Xt); 73 | Hs = opt.basis(Xs); 74 | Ktt = opt.kernel(Xt,Xt); 75 | Kts = opt.kernel(Xt,Xs); 76 | dKss = opt.kernel(Xs,'diag'); 77 | 78 | BayesInv = gp_inf(Ktt, Yt, opt.noise, Ht); 79 | 80 | for iter=1:T 81 | % GP pred 82 | [mu, s2] = gp_pred(Kts, dKss, BayesInv, Ht, Hs); 83 | 84 | % compute ucb 85 | u = opt.u + log(iter^2*pi^2/6); 86 | switch opt.algo 87 | case 'gpucb' 88 | % GP-UCB - Srinivas et al. (2012) 89 | ucb = gpucb(s2, u, ns); 90 | target = mu + ucb; 91 | [target_val,xt] = max(target); 92 | case 'chaining' 93 | % Chainig-UCB - Contal et al. (2015) 94 | D2 = gp_dist(opt.Kss, Kts, Kts, dKss, dKss, BayesInv); 95 | ucb = chaining_ucb(D2, s2, opt.u + log(iter^2*pi^2/6)); 96 | target = mu + ucb; 97 | [target_val,xt] = max(target); 98 | case 'ei' 99 | % Expected Improvement 100 | fmax = max(Yt); 101 | ni = (mu-fmax) ./ sqrt(s2); 102 | target = (mu-fmax).*normcdf(ni) + sqrt(s2) .* normpdf(ni); 103 | [target_val,xt] = max(target); 104 | case 'greedyucb' 105 | % Batch-greedy ucb policy, similar to GP-BUCB without the initialization phase 106 | xt = []; 107 | for b=1:opt.B 108 | ucb = gpucb(s2, u, ns); 109 | target = mu + ucb; 110 | [target_val,xtb] = max(target); 111 | xt = [xt; xtb]; 112 | if b= ydot; 131 | for b=2:opt.B 132 | % update 133 | Ht = [Ht; opt.basis(Xs(xtb,:))]; 134 | Ktt12 = opt.kernel(Xt,Xs(xtb,:)); 135 | Ktt22 = opt.kernel(Xs(xtb,:),Xs(xtb,:)); 136 | BayesInv = gp_inf_update(Ktt12, Ktt22, [Yt; mu(xt)], opt.noise, BayesInv, Ht); 137 | Kts = [Kts; opt.kernel(Xs(xtb,:),Xs)]; 138 | [~, s2] = gp_pred(Kts, dKss, BayesInv, Ht, Hs); 139 | Xt = [Xt; Xs(xtb,:)]; 140 | % next query 141 | s2_in_Rt = s2 .* is_in_Rt; 142 | [~, xtb] = max(s2_in_Rt); 143 | xt = [xt; xtb]; 144 | end 145 | end 146 | 147 | % query point 148 | queries = [queries xt]; 149 | yt = f(xt); 150 | Xt = [Xt; Xs(xt(end),:)]; % if batch, Xt is already partially updated 151 | Yt = [Yt; yt]; 152 | 153 | % GP update 154 | Ht = [Ht; opt.basis(Xt(end,:))]; 155 | Ktt12 = opt.kernel(Xt(1:end-1,:),Xt(end,:)); 156 | Ktt22 = opt.kernel(Xt(end,:),Xt(end,:)); 157 | BayesInv = gp_inf_update(Ktt12, Ktt22, Yt, opt.noise, BayesInv, Ht); 158 | Kts = [Kts; opt.kernel(Xt(end,:),Xs)]; 159 | 160 | % monitoring 161 | switch opt.verbose 162 | case 1 163 | fprintf('%d\tmax: %f\ttarget:%f\tobserved %f\r', iter, max(Yt), target_val, yt(end)); 164 | end 165 | if opt.plot 166 | clf 167 | hold on 168 | [Z, IZ] = sort(Xs(:,1)); 169 | plot(Z,mu(IZ),'k'); 170 | target_plot = min(mu) + (target-min(target))*(max(mu)-min(mu))/(max(target)-min(target)); 171 | plot(Z,target_plot(IZ),'g'); 172 | plot(Xt(:,1),Yt,'+r'); 173 | drawnow 174 | end 175 | end 176 | 177 | switch opt.verbose 178 | case 1 179 | fprintf('\n'); 180 | end 181 | 182 | end -------------------------------------------------------------------------------- /gpucb.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [ucb] = gpucb(s2, u, n) 18 | %% 19 | % Compute the ucb as in the GP-UCB algorithm 20 | %% Syntax 21 | % ucb = chaining_ucb(s2, u, n) 22 | %% Arguments 23 | % * _s2_ matrix _(n, 1)_ of posterior variance 24 | % * _u_ scalar for negative log probability 25 | % * _n_ number of test points 26 | %% Outputs 27 | % * _ucb_ vector _(n, 1)_ such that $P[\forall x,~f(x)-\mu(x)>ucb(x)] < \exp(-u)$ 28 | %% See also 29 | % 30 | 31 | ucb = sqrt(2*(u+log(n))*s2); 32 | 33 | end -------------------------------------------------------------------------------- /kernel_matern.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [K] = kernel_matern(X, Y, s, ells, nu) 18 | %% 19 | % Matern covariance function for $\nu \in \{ \frac 1 2, \frac 3 2, \frac 5 2\}$ 20 | %% Syntax 21 | % K = kernel_matern(X,'diag',s,ells,nu) 22 | % K = kernel_matern(X,Y,s,ells,nu) 23 | %% Arguments 24 | % * _X_ matrix _(nx, d)_ where _nx_ is the number of data points and _d_ is the dimension 25 | % * _Y_ matrix _(ny, d)_ or 'diag' for diagonal self covariance 26 | % * _s_ scalar _(1,1)_ for covariance scale 27 | % * _ells_ vector _(1,d)_ of covariance length-scales 28 | % * _nu_ scalar _(1,1)_ for Matern parameter $2\nu$, available values are _1_,_3_ and _5_ 29 | %% Outputs 30 | % * _K_ matrix _(nx, ny)_ or diagonal _(nx, 1)_ 31 | %% See also 32 | % 33 | 34 | ARD = diag(1./ells); 35 | 36 | if isa(Y, 'char') && strcmp(Y, 'diag') 37 | D = zeros(size(X,1),1); 38 | else 39 | D = sqrt(sq_dist(sqrt(nu)*ARD*X', sqrt(nu)*ARD*Y')); 40 | end 41 | 42 | switch nu 43 | case 1, K = s * exp(-D); 44 | case 3, K = s * (1+D) .* exp(-D); 45 | case 5, K = s * (1+D.*(1+D/3)) .* exp(-D); 46 | otherwise 47 | error('kernel_matern is only defined for n=1,3 or 5') 48 | end 49 | 50 | end -------------------------------------------------------------------------------- /kernel_se.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [K] = kernel_se(X, Y, s, ells) 18 | %% 19 | % Squared exponential covariance function 20 | %% Syntax 21 | % K = kernel_se(X,'diag',s,ells) 22 | % K = kernel_se(X,Y,s,ells) 23 | %% Arguments 24 | % * _X_ matrix _(nx, d)_ where _nx_ is the number of data points and _d_ is the dimension 25 | % * _Y_ matrix _(ny, d)_ or 'diag' for diagonal self covariance 26 | % * _s_ scalar _(1,1)_ for covariance scale 27 | % * _ell_ vector _(1,d)_ for covariance length-scales 28 | %% Outputs 29 | % * _K_ matrix _(nx, ny)_ or diagonal _(nx, 1)_ 30 | %% See also 31 | % | 32 | 33 | ARD = diag(1./ells); 34 | if isa(Y, 'char') && strcmp(Y, 'diag') 35 | D = zeros(size(X,1),1); 36 | else 37 | D = sq_dist(ARD*X', ARD*Y'); 38 | end 39 | 40 | K = s * exp(-D/2); 41 | 42 | end -------------------------------------------------------------------------------- /kernel_se_normiso.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [K] = kernel_se_normiso(X, Y) 18 | %% 19 | % Isotropic squared exponential covariance function 20 | %% Syntax 21 | % K = kernel_se_normiso(X,'diag') 22 | % K = kernel_se_normiso(X,Y) 23 | %% Arguments 24 | % * _X_ matrix _(nx, d)_ where _nx_ is the number of data points and _d_ is the dimension 25 | % * _Y_ matrix _(ny, d)_ or 'diag' for diagonal self covariance 26 | %% Outputs 27 | % * _K_ matrix _(nx, ny)_ or diagonal _(nx, 1)_ 28 | %% See also 29 | % 30 | 31 | ells = ones(size(X,2),1); 32 | s = 1; 33 | 34 | K = kernel_se(X, Y, s, ells); 35 | 36 | end -------------------------------------------------------------------------------- /solve_chol.m: -------------------------------------------------------------------------------- 1 | % This file is part of GpOptimization. 2 | % 3 | % GpOptimization is free software: you can redistribute it and/or modify 4 | % it under the terms of the GNU General Public License as published by 5 | % the Free Software Foundation, version 3 of the License. 6 | % 7 | % GpOptimization is distributed in the hope that it will be useful, 8 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 9 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 10 | % GNU General Public License for more details. 11 | % 12 | % You should have received a copy of the GNU General Public License 13 | % along with GpOptimization. If not, see . 14 | % 15 | % Copyright (c) by Emile Contal, 2015 16 | 17 | function [X] = solve_chol(R, Y) 18 | %% 19 | % Linear system resolution $MX=Y$ given upper Cholesky decomposition of $M=R^\top R$ 20 | %% Syntax 21 | % [X] = solve_chol(R, Y) 22 | %% Arguments 23 | % * _R_ upper triangular matrix _(n, n)_ of Cholesky decomposition of _M_ 24 | % * _Y_ matrix _(n, 1)_ 25 | %% Outputs 26 | % * _X_ matrix _(n, 1)_ such that _M*X=Y_ 27 | %% See also 28 | % 29 | 30 | X = R\(R'\Y); -------------------------------------------------------------------------------- /sq_dist.m: -------------------------------------------------------------------------------- 1 | % sq_dist - a function to compute a matrix of all pairwise squared distances 2 | % between two sets of vectors, stored in the columns of the two matrices, a 3 | % (of size D by n) and b (of size D by m). If only a single argument is given 4 | % or the second matrix is empty, the missing matrix is taken to be identical 5 | % to the first. 6 | % 7 | % Usage: C = sq_dist(a, b) 8 | % or: C = sq_dist(a) or equiv.: C = sq_dist(a, []) 9 | % 10 | % Where a is of size Dxn, b is of size Dxm (or empty), C is of size nxm. 11 | % 12 | % Copyright (c) by Carl Edward Rasmussen and Hannes Nickisch, 2010-12-13. 13 | 14 | function C = sq_dist(a, b) 15 | 16 | if nargin<1 || nargin>3 || nargout>1, error('Wrong number of arguments.'); end 17 | bsx = exist('bsxfun','builtin'); % since Matlab R2007a 7.4.0 and Octave 3.0 18 | if ~bsx, bsx = exist('bsxfun'); end % bsxfun is not yes "builtin" in Octave 19 | [D, n] = size(a); 20 | 21 | % Computation of a^2 - 2*a*b + b^2 is less stable than (a-b)^2 because numerical 22 | % precision can be lost when both a and b have very large absolute value and the 23 | % same sign. For that reason, we subtract the mean from the data beforehand to 24 | % stabilise the computations. This is OK because the squared error is 25 | % independent of the mean. 26 | if nargin==1 % subtract mean 27 | mu = mean(a,2); 28 | if bsx 29 | a = bsxfun(@minus,a,mu); 30 | else 31 | a = a - repmat(mu,1,size(a,2)); 32 | end 33 | b = a; m = n; 34 | else 35 | [d, m] = size(b); 36 | if d ~= D, error('Error: column lengths must agree.'); end 37 | mu = (m/(n+m))*mean(b,2) + (n/(n+m))*mean(a,2); 38 | if bsx 39 | a = bsxfun(@minus,a,mu); b = bsxfun(@minus,b,mu); 40 | else 41 | a = a - repmat(mu,1,n); b = b - repmat(mu,1,m); 42 | end 43 | end 44 | 45 | if bsx % compute squared distances 46 | C = bsxfun(@plus,sum(a.*a,1)',bsxfun(@minus,sum(b.*b,1),2*a'*b)); 47 | else 48 | C = repmat(sum(a.*a,1)',1,m) + repmat(sum(b.*b,1),n,1) - 2*a'*b; 49 | end 50 | C = max(C,0); % numerical noise can cause C to negative i.e. C > -1e-14 51 | --------------------------------------------------------------------------------