On ergodic two-armed bandits

TL;DR

This paper proves that the Narendra algorithm reliably identifies the best arm in ergodic two-armed bandit problems under weak assumptions, extending previous results to deterministic payoffs with convergence rates.

Contribution

It generalizes the convergence guarantees of the Narendra algorithm from i.i.d. payoffs to deterministic ergodic payoffs with explicit convergence rates.

Findings

Almost sure convergence to the best arm under ergodic assumptions

Extension of previous i.i.d. results to deterministic payoffs

Existence of positive probability of choosing the wrong arm

Abstract

A device has two arms with unknown deterministic payoffs and the aim is to asymptotically identify the best one without spending too much time on the other. The Narendra algorithm offers a stochastic procedure to this end. We show under weak ergodic assumptions on these deterministic payoffs that the procedure eventually chooses the best arm (i.e., with greatest Cesaro limit) with probability one for appropriate step sequences of the algorithm. In the case of i.i.d. payoffs, this implies a "quenched" version of the "annealed" result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] by the law of iterated logarithm, thus generalizing it. More precisely, if $(\eta_{\ell,i})_{i\in \mathbb {N}}\in\{0,1\}^{\mathbb {N}}$, $\ell\in\{A,B\}$, are the deterministic reward sequences we would get if we played at time $i$, we obtain infallibility with the same assumption on nonincreasing step sequences on the payoffs as in Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454], replacing the i.i.d. assumption by the hypothesis that the empirical averages $\sum_{i=1}^n\eta_{A,i}/n$ and $\sum_{i=1}^n\eta_{B,i}/n$ converge, as $n$ tends to infinity, respectively, to $\theta_A$ and $\theta_B$, with rate at least $1/(\log n)^{1+\varepsilon}$, for some $\varepsilon >0$. We also show a fallibility result, that is, convergence with positive probability to the choice of the wrong arm, which implies the corresponding result of Lamberton, Pag\`{e}s and Tarr\`{e}s [Ann. Appl. Probab. 14 (2004) 1424--1454] in the i.i.d. case.