Change Detection-Based Procedures for Piecewise Stationary MABs: A Modular Approach

Conventional Multi-Armed Bandit (MAB) algorithms are designed for stationary environments, where the reward distributions associated with the arms do not change with time. In many applications, however, the environment is more accurately modeled as being nonstationary. In this work, piecewise stationary MAB (PS-MAB) environments are investigated, in which the reward distributions associated with a subset of the arms change at some change-points and remain stationary between change-points. Our focus is on the asymptotic analysis of PS-MABs, for which practical algorithms based on change detection (CD) have been previously proposed. Our goal is to modularize the design and analysis of such CD-based Bandit (CDB) procedures. To this end, we identify the requirements for stationary bandit algorithms and change detectors in a CDB procedure that are needed for the modularization. We assume that the rewards are sub-Gaussian. Under this assumption and a condition on the separation of the change-points, we show that the analysis of CDB procedures can indeed be modularized, so that regret bounds can be obtained in a unified manner for various combinations of change detectors and bandit algorithms. Through this analysis, we develop new modular CDB procedures that are order-optimal. We compare the performance of our modular CDB procedures with various other methods in simulations.

Tangential Randomization in Linear Bandits (TRAiL): Guaranteed Inference and Regret Bounds

We propose and analyze TRAiL (Tangential Randomization in Linear Bandits), a computationally efficient regret-optimal forced exploration algorithm for linear bandits on action sets that are sublevel sets of strongly convex functions. TRAiL estimates the governing parameter of the linear bandit problem through a standard regularized least squares and perturbs the reward-maximizing action corresponding to said point estimate along the tangent plane of the convex compact action set before projecting back to it. Exploiting concentration results for matrix martingales, we prove that TRAiL ensures a $\Omega(\sqrt{T})$ growth in the inference quality, measured via the minimum eigenvalue of the design (regressor) matrix with high-probability over a $T$-length period. We build on this result to obtain an $\mathcal{O}(\sqrt{T} \log(T))$ upper bound on cumulative regret with probability at least $ 1 - 1/T$ over $T$ periods, and compare TRAiL to other popular algorithms for linear bandits. Then, we characterize an $\Omega(\sqrt{T})$ minimax lower bound for any algorithm on the expected regret that covers a wide variety of action/parameter sets and noise processes. Our analysis not only expands the realm of lower-bounds in linear bandits significantly, but as a byproduct, yields a trade-off between regret and inference quality. Specifically, we prove that any algorithm with an $\mathcal{O}(T^\alpha)$ expected regret growth must have an $\Omega(T^{1-\alpha})$ asymptotic growth in expected inference quality. Our experiments on the $L^p$ unit ball as action sets reveal how this relation can be violated, but only in the short-run, before returning to respect the bound asymptotically. In effect, regret-minimizing algorithms must have just the right rate of inference -- too fast or too slow inference will incur sub-optimal regret growth.

Compressed Online Learning of Conditional Mean Embedding

The conditional mean embedding (CME) encodes Markovian stochastic kernels through their actions on probability distributions embedded within the reproducing kernel Hilbert spaces (RKHS). The CME plays a key role in several well-known machine learning tasks such as reinforcement learning, analysis of dynamical systems, etc. We present an algorithm to learn the CME incrementally from data via an operator-valued stochastic gradient descent. As is well-known, function learning in RKHS suffers from scalability challenges from large data. We utilize a compression mechanism to counter the scalability challenge. The core contribution of this paper is a finite-sample performance guarantee on the last iterate of the online compressed operator learning algorithm with fast-mixing Markovian samples, when the target CME may not be contained in the hypothesis space. We illustrate the efficacy of our algorithm by applying it to the analysis of an example dynamical system.