Kiefer Wolfowitz Algorithm is Asymptotically Optimal for a Class of Non-Stationary Bandit Problems

We consider the problem of designing an allocation rule or an "online learning algorithm" for a class of bandit problems in which the set of control actions available at each time $s$ is a convex, compact subset of $\mathbb{R}^d$. Upon choosing an action $x$ at time $s$, the algorithm obtains a noisy value of the unknown and time-varying function $f_s$ evaluated at $x$. The "regret" of an algorithm is the gap between its expected reward, and the reward earned by a strategy which has the knowledge of the function $f_s$ at each time $s$ and hence chooses the action $x_s$ that maximizes $f_s$. For this non-stationary bandit problem set-up, we consider two variants of the Kiefer Wolfowitz (KW) algorithm i) KW with fixed step-size $\beta$, and ii) KW with sliding window of length $L$. We show that if the number of times that the function $f_s$ varies during time $T$ is $o(T)$, and if the learning rates of the proposed algorithms are chosen "optimally", then the regret of the proposed algorithms is $o(T)$, and hence the algorithms are asymptotically efficient.