Streaming kernel regression with provably adaptive mean, variance, and regularization

Abstract

We consider the problem of streaming kernel regression, when the observations arrive sequentially and the goal is to recover the underlying mean function, assumed to belong to an RKHS. The variance of the noise is not assumed to be known. In this context, we tackle the problem of tuning the regularization parameter adaptively at each time step, while maintaining tight confidence bounds estimates on the value of the mean function at each point. To this end, we first generalize existing results for finite-dimensional linear regression with fixed regularization and known variance to the kernel setup with a regularization parameter allowed to be a measurable function of past observations. Then, using appropriate self-normalized inequalities we build upper and lower bound estimates for the variance, leading to Bersntein-like concentration bounds. The later is used in order to define the adaptive regularization. The bounds resulting from our technique are valid uniformly over all observation points and all time steps, and are compared against the literature with numerical experiments. Finally, the potential of these tools is illustrated by an application to kernelized bandits, where we revisit the Kernel UCB and Kernel Thompson Sampling procedures, and show the benefits of the novel adaptive kernel tuning strategy.

References in corpus (5)

• A Contextual-Bandit Approach to Personalized News Article Recommendation
• Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design
• Thompson Sampling for Contextual Bandits with Linear Payoffs
• Finite-Time Analysis of Kernelised Contextual Bandits
• Theoretical Analysis of Bayesian Optimisation with Unknown Gaussian Process Hyper-Parameters

Cited by in corpus (12)

• Stochastic Bandits with Context Distributions
• On Function Approximation in Reinforcement Learning: Optimism in the Face of Large State Spaces
• Simple Regret Minimization for Contextual Bandits
• Local Differential Privacy for Bayesian Optimization
• No-regret Algorithms for Multi-task Bayesian Optimization
• Online Learning in Kernelized Markov Decision Processes
• No-Regret Bayesian Optimization with Unknown Hyperparameters
• On Batch Bayesian Optimization
• Combining Pessimism with Optimism for Robust and Efficient Model-Based Deep Reinforcement Learning
• Bregman Deviations of Generic Exponential Families
• Improved Algorithms for Stochastic Linear Bandits Using Tail Bounds for Martingale Mixtures
• Indexed Minimum Empirical Divergence for Unimodal Bandits