VITS : Variational Inference Thomson Sampling for contextual bandits

In this paper, we introduce and analyze a variant of the Thompson sampling (TS) algorithm for contextual bandits. At each round, traditional TS requires samples from the current posterior distribution, which is usually intractable. To circumvent this issue, approximate inference techniques can be used and provide samples with distribution close to the posteriors. However, current approximate techniques yield to either poor estimation (Laplace approximation) or can be computationally expensive (MCMC methods, Ensemble sampling...). In this paper, we propose a new algorithm, Varational Inference Thompson sampling VITS, based on Gaussian Variational Inference. This scheme provides powerful posterior approximations which are easy to sample from, and is computationally efficient, making it an ideal choice for TS. In addition, we show that VITS achieves a sub-linear regret bound of the same order in the dimension and number of round as traditional TS for linear contextual bandit. Finally, we demonstrate experimentally the effectiveness of VITS on both synthetic and real world datasets.

Towards Minimax Optimality of Model-based Robust Reinforcement Learning

We study the sample complexity of obtaining an $\epsilon$-optimal policy in \emph{Robust} discounted Markov Decision Processes (RMDPs), given only access to a generative model of the nominal kernel. This problem is widely studied in the non-robust case, and it is known that any planning approach applied to an empirical MDP estimated with $\tilde{\mathcal{O}}(\frac{H^3 \mid S \mid\mid A \mid}{\epsilon^2})$ samples provides an $\epsilon$-optimal policy, which is minimax optimal. Results in the robust case are much more scarce. For $sa$- (resp $s$-)rectangular uncertainty sets, the best known sample complexity is $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid^2\mid A \mid}{\epsilon^2})$ (resp. $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid^2\mid A \mid^2}{\epsilon^2})$), for specific algorithms and when the uncertainty set is based on the total variation (TV), the KL or the Chi-square divergences. In this paper, we consider uncertainty sets defined with an $L_p$-ball (recovering the TV case), and study the sample complexity of \emph{any} planning algorithm (with high accuracy guarantee on the solution) applied to an empirical RMDP estimated using the generative model. In the general case, we prove a sample complexity of $\tilde{\mathcal{O}}(\frac{H^4 \mid S \mid\mid A \mid}{\epsilon^2})$ for both the $sa$- and $s$-rectangular cases (improvements of $\mid S \mid$ and $\mid S \mid\mid A \mid$ respectively). When the size of the uncertainty is small enough, we improve the sample complexity to $\tilde{\mathcal{O}}(\frac{H^3 \mid S \mid\mid A \mid }{\epsilon^2})$, recovering the lower-bound for the non-robust case for the first time and a robust lower-bound when the size of the uncertainty is small enough.

Robust Reinforcement Learning with Distributional Risk-averse formulation

Robust Reinforcement Learning tries to make predictions more robust to changes in the dynamics or rewards of the system. This problem is particularly important when the dynamics and rewards of the environment are estimated from the data. In this paper, we approximate the Robust Reinforcement Learning constrained with a $\Phi$-divergence using an approximate Risk-Averse formulation. We show that the classical Reinforcement Learning formulation can be robustified using standard deviation penalization of the objective. Two algorithms based on Distributional Reinforcement Learning, one for discrete and one for continuous action spaces are proposed and tested in a classical Gym environment to demonstrate the robustness of the algorithms.