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

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 $1\mathrm{/}2$-Tsallis entropy regularizer $\mathrm{\Psi }(p)=-4{\sum }_{i=1}^d\sqrt{p_i}$ (the $1\mathrm{/}2$-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 $1\mathrm{/}2$-Tsallis-INF algorithm and partially confirms this intuition through theoretical analysis, showing that the Bregman divergence, defined by $\mathrm{\Psi }(p)$, 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 $t^{-1\mathrm{/}2}$.