Thompson Sampling Guided Stochastic Searching on the Line for Deceptive Environments with Applications to Root-Finding Problems

The multi-armed bandit problem forms the foundation for solving a wide range of on-line stochastic optimization problems through a simple, yet effective mechanism. One simply casts the problem as a gambler that repeatedly pulls one out of N slot machine arms, eliciting random rewards. Learning of reward probabilities is then combined with reward maximization, by carefully balancing reward exploration against reward exploitation. In this paper, we address a particularly intriguing variant of the multi-armed bandit problem, referred to as the {\it Stochastic Point Location (SPL) Problem}. The gambler is here only told whether the optimal arm (point) lies to the "left" or to the "right" of the arm pulled, with the feedback being erroneous with probability $1-π$. This formulation thus captures optimization in continuous action spaces with both {\it informative} and {\it deceptive} feedback. To tackle this class of problems, we formulate a compact and scalable Bayesian representation of the solution space that simultaneously captures both the location of the optimal arm as well as the probability of receiving correct feedback. We further introduce the accompanying Thompson Sampling guided Stochastic Point Location (TS-SPL) scheme for balancing exploration against exploitation. By learning $π$, TS-SPL also supports {\it deceptive} environments that are lying about the direction of the optimal arm. This, in turn, allows us to solve the fundamental Stochastic Root Finding (SRF) Problem. Empirical results demonstrate that our scheme deals with both deceptive and informative environments, significantly outperforming competing algorithms both for SRF and SPL.