Luca Viano

Fanghui Liu, Luca Viano, Volkan Cevher

This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $ε$-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e.g., width and depth) beyond the ``linear'' regime. To be specific, we focus on the value based algorithm with the $ε$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces, respectively, which aims at approximating an $α$-smooth Q-function in a $d$-dimensional feature space. We prove that, with $T$ episodes, scaling the width $m = \widetilde{\mathcal{O}}(T^{\frac{d}{2α+ d}})$ and the depth $L=\mathcal{O}(\log T)$ of the neural network for deep RL is sufficient for learning with sublinear regret in Besov spaces. Moreover, for a two layer neural network endowed by the Barron space, scaling the width $Ω(\sqrt{T})$ is sufficient. To achieve this, the key issue in our analysis is how to estimate the temporal difference error under deep neural function approximation as the $ε$-greedy exploration is not enough to ensure ``optimism''. Our analysis reformulates the temporal difference error in an $L^2(\mathrm{d}μ)$-integrable space over a certain averaged measure $μ$, and transforms it to a generalization problem under the non-iid setting. This might have its own interest in RL theory for better understanding $ε$-greedy exploration in deep RL.

Fanghui Liu, Luca Viano, Volkan Cevher

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the $L^p$ variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond $\widetilde{O}(\sqrt{T})$ and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

Stefano Viel, Luca Viano, Volkan Cevher

This paper introduces the SOAR framework for imitation learning. SOAR is an algorithmic template that learns a policy from expert demonstrations with a primal dual style algorithm that alternates cost and policy updates. Within the policy updates, the SOAR framework uses an actor critic method with multiple critics to estimate the critic uncertainty and build an optimistic critic fundamental to drive exploration. When instantiated in the tabular setting, we get a provable algorithm with guarantees that matches the best known results in $ε$. Practically, the SOAR template is shown to boost consistently the performance of imitation learning algorithms based on Soft Actor Critic such as f-IRL, ML-IRL and CSIL in several MuJoCo environments. Overall, thanks to SOAR, the required number of episodes to achieve the same performance is reduced by half.