Last-Iterate Analyses of FTRL with the 1/2-Tsallis Entropy in Stochastic Bandits

TL;DR

This paper analyzes the last-iterate convergence of the FTRL algorithm with 1/2-Tsallis entropy in stochastic bandits, showing a decay rate of t^{-1/2} in Bregman divergence, linking regret and convergence.

Contribution

It provides the first theoretical analysis of the last-iterate convergence rate for FTRL with 1/2-Tsallis entropy in stochastic bandits.

Findings

Bregman divergence decays at a rate of t^{-1/2}.

Logarithmic regret implies a t^{-1} last-iterate convergence rate.

Partially confirms the intuition connecting regret and convergence rate.

Abstract

The convergence analysis of online learning algorithms is central to machine learning theory, where the last-iterate convergence is particularly important, as it captures the learner's actual decisions and describes the evolution of the learning process over time. However, in multi-armed bandits, most existing algorithmic analyses mainly focus on the order of regret, while the last-iterate (simple regret) convergence rate remains less explored -- especially for the widely studied Follow-the-Regularized-Leader (FTRL) algorithms. Recently, FTRL with the $1/2$-Tsallis entropy regularizer $\Psi(p) = -4\sum_{i=1}^d \sqrt{p_i}$ (the $1/2$-Tsallis-INF algorithm, by arXiv:1807.07623) was shown to achieve logarithmic regret in stochastic bandits. Nevertheless, its last-iterate convergence rate has not yet been studied. Intuitively, logarithmic regret should correspond to a $t^{-1}$ last-iterate convergence rate. This paper studies the $1/2$-Tsallis-INF algorithm and partially confirms this intuition through theoretical analysis, showing that the Bregman divergence, defined by $\Psi(p)$, between the point mass on the optimal arm and the probability distribution over the arm set obtained at iteration $t$, decays at a rate of $t^{-1/2}$.