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Gaussian Process Optimization with Adaptive Sketching: Scalable and No Regret
Michal Valko

Fri Jun 14 05:20 PM -- 05:40 PM (PDT) @
Gaussian processes (GP) are a popular Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to complex high-dimensional functions, as their per-iteration time and space cost is at least quadratic in the number of dimensions $d$ and iterations~$t$. Given a set of $A$ alternative to choose from, the overall runtime $O(t^3A)$ quickly becomes prohibitive. In this paper, we introduce BKB (budgeted kernelized bandit), an approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and no assumption on the input space or the GP's covariance. Combining a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching technique (i.e., leverage score sampling), we prove that selecting inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most $\widetilde{O}(d_{eff})$ points, where $d_{eff}$ is the \emph{effective} dimension of the explored space, which is typically much smaller than both $d$ and $t$. This greatly reduces the dimensionality of the problem, thus leading to a $\widetilde{O}(TAd_{eff}^2)$ runtime and $\widetilde{O}(A d_{eff})$ space complexity.

Author Information

Michal Valko (DeepMind)

Michal is a research scientist in DeepMind Paris and SequeL team at Inria Lille - Nord Europe, France, lead by Philippe Preux and Rémi Munos. He also teaches the course Graphs in Machine Learning at l'ENS Cachan. Michal is primarily interested in designing algorithms that would require as little human supervision as possible. This means 1) reducing the “intelligence” that humans need to input into the system and 2) minimising the data that humans need spend inspecting, classifying, or “tuning” the algorithms. Another important feature of machine learning algorithms should be the ability to adapt to changing environments. That is why he is working in domains that are able to deal with minimal feedback, such as bandit algorithms, semi-supervised learning, and anomaly detection. Most recently he has worked on sequential algorithms with structured decisions where exploiting the structure can lead to provably faster learning. In the past the common thread of Michal's work has been adaptive graph-based learning and its application to the real world applications such as recommender systems, medical error detection, and face recognition. His industrial collaborators include Adobe, Intel, Technicolor, and Microsoft Research. He received his PhD in 2011 from University of Pittsburgh under the supervision of Miloš Hauskrecht and after was a postdoc of Rémi Munos.