Differentially Private Exploration in Reinforcement Learning with Linear Representation

This paper studies privacy-preserving exploration in Markov Decision Processes (MDPs) with linear representation. We first consider the setting of linear-mixture MDPs (Ayoub et al., 2020) (a.k.a.\ model-based setting) and provide an unified framework for analyzing joint and local differential private (DP) exploration. Through this framework, we prove a $\widetilde{O}(K^{3/4}/\sqrt{\epsilon})$ regret bound for $(\epsilon,\delta)$-local DP exploration and a $\widetilde{O}(\sqrt{K/\epsilon})$ regret bound for $(\epsilon,\delta)$-joint DP. We further study privacy-preserving exploration in linear MDPs (Jin et al., 2020) (a.k.a.\ model-free setting) where we provide a $\widetilde{O}(\sqrt{K/\epsilon})$ regret bound for $(\epsilon,\delta)$-joint DP, with a novel algorithm based on low-switching. Finally, we provide insights into the issues of designing local DP algorithms in this model-free setting.

Adaptive Multi-Goal Exploration

We introduce a generic strategy for provably efficient multi-goal exploration. It relies on AdaGoal, a novel goal selection scheme that is based on a simple constrained optimization problem, which adaptively targets goal states that are neither too difficult nor too easy to reach according to the agent's current knowledge. We show how AdaGoal can be used to tackle the objective of learning an $\epsilon$-optimal goal-conditioned policy for all the goal states that are reachable within $L$ steps in expectation from a reference state $s_0$ in a reward-free Markov decision process. In the tabular case with $S$ states and $A$ actions, our algorithm requires $\tilde{O}(L^3 S A \epsilon^{-2})$ exploration steps, which is nearly minimax optimal. We also readily instantiate AdaGoal in linear mixture Markov decision processes, which yields the first goal-oriented PAC guarantee with linear function approximation. Beyond its strong theoretical guarantees, AdaGoal is anchored in the high-level algorithmic structure of existing methods for goal-conditioned deep reinforcement learning.

Reinforcement Learning in Linear MDPs: Constant Regret and Representation Selection

We study the role of the representation of state-action value functions in regret minimization in finite-horizon Markov Decision Processes (MDPs) with linear structure. We first derive a necessary condition on the representation, called universally spanning optimal features (UNISOFT), to achieve constant regret in any MDP with linear reward function. This result encompasses the well-known settings of low-rank MDPs and, more generally, zero inherent Bellman error (also known as the Bellman closure assumption). We then demonstrate that this condition is also sufficient for these classes of problems by deriving a constant regret bound for two optimistic algorithms (LSVI-UCB and ELEANOR). Finally, we propose an algorithm for representation selection and we prove that it achieves constant regret when one of the given representations, or a suitable combination of them, satisfies the UNISOFT condition.

A Fully Problem-Dependent Regret Lower Bound for Finite-Horizon MDPs

We derive a novel asymptotic problem-dependent lower-bound for regret minimization in finite-horizon tabular Markov Decision Processes (MDPs). While, similar to prior work (e.g., for ergodic MDPs), the lower-bound is the solution to an optimization problem, our derivation reveals the need for an additional constraint on the visitation distribution over state-action pairs that explicitly accounts for the dynamics of the MDP. We provide a characterization of our lower-bound through a series of examples illustrating how different MDPs may have significantly different complexity. 1) We first consider a "difficult" MDP instance, where the novel constraint based on the dynamics leads to a larger lower-bound (i.e., a larger regret) compared to the classical analysis. 2) We then show that our lower-bound recovers results previously derived for specific MDP instances. 3) Finally, we show that, in certain "simple" MDPs, the lower bound is considerably smaller than in the general case and it does not scale with the minimum action gap at all. We show that this last result is attainable (up to $poly(H)$ terms, where $H$ is the horizon) by providing a regret upper-bound based on policy gaps for an optimistic algorithm.

A Unified Framework for Conservative Exploration

We study bandits and reinforcement learning (RL) subject to a conservative constraint where the agent is asked to perform at least as well as a given baseline policy. This setting is particular relevant in real-world domains including digital marketing, healthcare, production, finance, etc. For multi-armed bandits, linear bandits and tabular RL, specialized algorithms and theoretical analyses were proposed in previous work. In this paper, we present a unified framework for conservative bandits and RL, in which our core technique is to calculate the necessary and sufficient budget obtained from running the baseline policy. For lower bounds, our framework gives a black-box reduction that turns a certain lower bound in the nonconservative setting into a new lower bound in the conservative setting. We strengthen the existing lower bound for conservative multi-armed bandits and obtain new lower bounds for conservative linear bandits, tabular RL and low-rank MDP. For upper bounds, our framework turns a certain nonconservative upper-confidence-bound (UCB) algorithm into a conservative algorithm with a simple analysis. For multi-armed bandits, linear bandits and tabular RL, our new upper bounds tighten or match existing ones with significantly simpler analyses. We also obtain a new upper bound for conservative low-rank MDP.

Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to guarantee both optimism and convergence of the associated value iteration scheme. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i.e., it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$ which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e.g., positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first horizon-free regret bound beyond the finite-horizon MDP setting.

Leveraging Good Representations in Linear Contextual Bandits

The linear contextual bandit literature is mostly focused on the design of efficient learning algorithms for a given representation. However, a contextual bandit problem may admit multiple linear representations, each one with different characteristics that directly impact the regret of the learning algorithm. In particular, recent works showed that there exist "good" representations for which constant problem-dependent regret can be achieved. In this paper, we first provide a systematic analysis of the different definitions of "good" representations proposed in the literature. We then propose a novel selection algorithm able to adapt to the best representation in a set of $M$ candidates. We show that the regret is indeed never worse than the regret obtained by running LinUCB on the best representation (up to a $\ln M$ factor). As a result, our algorithm achieves constant regret whenever a "good" representation is available in the set. Furthermore, we show that the algorithm may still achieve constant regret by implicitly constructing a "good" representation, even when none of the initial representations is "good". Finally, we empirically validate our theoretical findings in a number of standard contextual bandit problems.

Homomorphically Encrypted Linear Contextual Bandit

Contextual bandit is a general framework for online learning in sequential decision-making problems that has found application in a large range of domains, including recommendation system, online advertising, clinical trials and many more. A critical aspect of bandit methods is that they require to observe the contexts -- i.e., individual or group-level data -- and the rewards in order to solve the sequential problem. The large deployment in industrial applications has increased interest in methods that preserve the privacy of the users. In this paper, we introduce a privacy-preserving bandit framework based on asymmetric encryption. The bandit algorithm only observes encrypted information (contexts and rewards) and has no ability to decrypt it. Leveraging homomorphic encryption, we show that despite the complexity of the setting, it is possible to learn over encrypted data. We introduce an algorithm that achieves a $\widetilde{O}(d\sqrt{T})$ regret bound in any linear contextual bandit problem, while keeping data encrypted.

Improved Sample Complexity for Incremental Autonomous Exploration in MDPs

We investigate the exploration of an unknown environment when no reward function is provided. Building on the incremental exploration setting introduced by Lim and Auer [1], we define the objective of learning the set of $\epsilon$-optimal goal-conditioned policies attaining all states that are incrementally reachable within $L$ steps (in expectation) from a reference state $s_0$. In this paper, we introduce a novel model-based approach that interleaves discovering new states from $s_0$ and improving the accuracy of a model estimate that is used to compute goal-conditioned policies to reach newly discovered states. The resulting algorithm, DisCo, achieves a sample complexity scaling as $\tilde{O}(L^5 S_{L+\epsilon} \Gamma_{L+\epsilon} A \epsilon^{-2})$, where $A$ is the number of actions, $S_{L+\epsilon}$ is the number of states that are incrementally reachable from $s_0$ in $L+\epsilon$ steps, and $\Gamma_{L+\epsilon}$ is the branching factor of the dynamics over such states. This improves over the algorithm proposed in [1] in both $\epsilon$ and $L$ at the cost of an extra $\Gamma_{L+\epsilon}$ factor, which is small in most environments of interest. Furthermore, DisCo is the first algorithm that can return an $\epsilon/c_{\min}$-optimal policy for any cost-sensitive shortest-path problem defined on the $L$-reachable states with minimum cost $c_{\min}$. Finally, we report preliminary empirical results confirming our theoretical findings.

An Asymptotically Optimal Primal-Dual Incremental Algorithm for Contextual Linear Bandits

In the contextual linear bandit setting, algorithms built on the optimism principle fail to exploit the structure of the problem and have been shown to be asymptotically suboptimal. In this paper, we follow recent approaches of deriving asymptotically optimal algorithms from problem-dependent regret lower bounds and we introduce a novel algorithm improving over the state-of-the-art along multiple dimensions. We build on a reformulation of the lower bound, where context distribution and exploration policy are decoupled, and we obtain an algorithm robust to unbalanced context distributions. Then, using an incremental primal-dual approach to solve the Lagrangian relaxation of the lower bound, we obtain a scalable and computationally efficient algorithm. Finally, we remove forced exploration and build on confidence intervals of the optimization problem to encourage a minimum level of exploration that is better adapted to the problem structure. We demonstrate the asymptotic optimality of our algorithm, while providing both problem-dependent and worst-case finite-time regret guarantees. Our bounds scale with the logarithm of the number of arms, thus avoiding the linear dependence common in all related prior works. Notably, we establish minimax optimality for any learning horizon in the special case of non-contextual linear bandits. Finally, we verify that our algorithm obtains better empirical performance than state-of-the-art baselines.