Optimization for Gaussian Processes via Chaining

In this paper, we consider the problem of stochastic optimization under a bandit feedback model. We generalize the GP-UCB algorithm [Srinivas and al., 2012] to arbitrary kernels and search spaces. To do so, we use a notion of localized chaining to control the supremum of a Gaussian process, and provide a novel optimization scheme based on the computation of covering numbers. The theoretical bounds we obtain on the cumulative regret are more generic and present the same convergence rates as the GP-UCB algorithm. Finally, the algorithm is shown to be empirically more efficient than its natural competitors on simple and complex input spaces.

[1]  R. Dudley.  The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes , 1967 .

[2]  David S. Johnson.  Approximation algorithms for combinatorial problems , 1973, STOC '73.

[3]  Jonas Mockus,et al.  Bayesian Approach to Global Optimization , 1989 .

[4]  D. Pollard.  Empirical Processes: Theory and Applications , 1990 .

[5]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[6]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[7]  Hans-Peter Kriegel,et al.  Shortest-path kernels on graphs , 2005, Fifth IEEE International Conference on Data Mining (ICDM'05).

[8]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[9]  Michael A. Osborne.  Bayesian Gaussian processes for sequential prediction, optimisation and quadrature , 2010 .

[10]  John Shawe-Taylor,et al.  Regret Bounds for Gaussian Process Bandit Problems , 2010, AISTATS 2010.

[11]  Andreas Krause,et al.  Contextual Gaussian Process Bandit Optimization , 2011, NIPS.

[12]  Jasper Snoek,et al.  Practical Bayesian Optimization of Machine Learning Algorithms , 2012, NIPS.

[13]  Philipp Hennig,et al.  Entropy Search for Information-Efficient Global Optimization , 2011, J. Mach. Learn. Res..

[14]  S. Kakade,et al.  Information-Theoretic Regret Bounds for Gaussian Process Optimization in the Bandit Setting , 2012, IEEE Transactions on Information Theory.

[15]  Alexander J. Smola,et al.  Regret Bounds for Deterministic Gaussian Process Bandits , 2012, ArXiv.

[16]  Nicolas Vayatis,et al.  Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration , 2013, ECML/PKDD.

[17]  Andreas Krause,et al.  High-Dimensional Gaussian Process Bandits , 2013, NIPS.

[18]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[19]  Vianney Perchet,et al.  Gaussian Process Optimization with Mutual Information , 2013, ICML.

[20]  R. Nickl,et al.  Mathematical Foundations of Infinite-Dimensional Statistical Models , 2015 .

[21]  Pierre Gaillard,et al.  A Chaining Algorithm for Online Nonparametric Regression , 2015, COLT.