Statistical Analysis of Policy Space Compression Problem

Policy search methods are crucial in reinforcement learning, offering a framework to address continuous state-action and partially observable problems. However, the complexity of exploring vast policy spaces can lead to significant inefficiencies. Reducing the policy space through policy compression emerges as a powerful, reward-free approach to accelerate the learning process. This technique condenses the policy space into a smaller, representative set while maintaining most of the original effectiveness. Our research focuses on determining the necessary sample size to learn this compressed set accurately. We employ R\'enyi divergence to measure the similarity between true and estimated policy distributions, establishing error bounds for good approximations. To simplify the analysis, we employ the $l_1$ norm, determining sample size requirements for both model-based and model-free settings. Finally, we correlate the error bounds from the $l_1$ norm with those from R\'enyi divergence, distinguishing between policies near the vertices and those in the middle of the policy space, to determine the lower and upper bounds for the required sample sizes.

Local Linearity: the Key for No-regret Reinforcement Learning in Continuous MDPs

Achieving the no-regret property for Reinforcement Learning (RL) problems in continuous state and action-space environments is one of the major open problems in the field. Existing solutions either work under very specific assumptions or achieve bounds that are vacuous in some regimes. Furthermore, many structural assumptions are known to suffer from a provably unavoidable exponential dependence on the time horizon $H$ in the regret, which makes any possible solution unfeasible in practice. In this paper, we identify local linearity as the feature that makes Markov Decision Processes (MDPs) both learnable (sublinear regret) and feasible (regret that is polynomial in $H$). We define a novel MDP representation class, namely Locally Linearizable MDPs, generalizing other representation classes like Linear MDPs and MDPS with low inherent Belmman error. Then, i) we introduce Cinderella, a no-regret algorithm for this general representation class, and ii) we show that all known learnable and feasible MDP families are representable in this class. We first show that all known feasible MDPs belong to a family that we call Mildly Smooth MDPs. Then, we show how any mildly smooth MDP can be represented as a Locally Linearizable MDP by an appropriate choice of representation. This way, Cinderella is shown to achieve state-of-the-art regret bounds for all previously known (and some new) continuous MDPs for which RL is learnable and feasible.

Last-Iterate Global Convergence of Policy Gradients for Constrained Reinforcement Learning

Constrained Reinforcement Learning (CRL) tackles sequential decision-making problems where agents are required to achieve goals by maximizing the expected return while meeting domain-specific constraints, which are often formulated as expected costs. In this setting, policy-based methods are widely used since they come with several advantages when dealing with continuous-control problems. These methods search in the policy space with an action-based or parameter-based exploration strategy, depending on whether they learn directly the parameters of a stochastic policy or those of a stochastic hyperpolicy. In this paper, we propose a general framework for addressing CRL problems via gradient-based primal-dual algorithms, relying on an alternate ascent/descent scheme with dual-variable regularization. We introduce an exploration-agnostic algorithm, called C-PG, which exhibits global last-iterate convergence guarantees under (weak) gradient domination assumptions, improving and generalizing existing results. Then, we design C-PGAE and C-PGPE, the action-based and the parameter-based versions of C-PG, respectively, and we illustrate how they naturally extend to constraints defined in terms of risk measures over the costs, as it is often requested in safety-critical scenarios. Finally, we numerically validate our algorithms on constrained control problems, and compare them with state-of-the-art baselines, demonstrating their effectiveness.

Projection by Convolution: Optimal Sample Complexity for Reinforcement Learning in Continuous-Space MDPs

We consider the problem of learning an $\varepsilon$-optimal policy in a general class of continuous-space Markov decision processes (MDPs) having smooth Bellman operators. Given access to a generative model, we achieve rate-optimal sample complexity by performing a simple, \emph{perturbed} version of least-squares value iteration with orthogonal trigonometric polynomials as features. Key to our solution is a novel projection technique based on ideas from harmonic analysis. Our~$\widetilde{\mathcal{O}}(\epsilon^{-2-d/(\nu+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $\nu$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(\nu=0)$. At the same time, for $\nu\to\infty$, it recovers and greatly generalizes the $\mathcal{O}(\epsilon^{-2})$ rate of low-rank MDPs, which are more amenable to regression approaches. In this sense, our result bridges the gap between two popular but conflicting perspectives on continuous-space MDPs.

Policy Gradient with Active Importance Sampling

Importance sampling (IS) represents a fundamental technique for a large surge of off-policy reinforcement learning approaches. Policy gradient (PG) methods, in particular, significantly benefit from IS, enabling the effective reuse of previously collected samples, thus increasing sample efficiency. However, classically, IS is employed in RL as a passive tool for re-weighting historical samples. However, the statistical community employs IS as an active tool combined with the use of behavioral distributions that allow the reduction of the estimate variance even below the sample mean one. In this paper, we focus on this second setting by addressing the behavioral policy optimization (BPO) problem. We look for the best behavioral policy from which to collect samples to reduce the policy gradient variance as much as possible. We provide an iterative algorithm that alternates between the cross-entropy estimation of the minimum-variance behavioral policy and the actual policy optimization, leveraging on defensive IS. We theoretically analyze such an algorithm, showing that it enjoys a convergence rate of order $O(\epsilon^{-4})$ to a stationary point, but depending on a more convenient variance term w.r.t. standard PG methods. We then provide a practical version that is numerically validated, showing the advantages in the policy gradient estimation variance and on the learning speed.

Learning Optimal Deterministic Policies with Stochastic Policy Gradients

Policy gradient (PG) methods are successful approaches to deal with continuous reinforcement learning (RL) problems. They learn stochastic parametric (hyper)policies by either exploring in the space of actions or in the space of parameters. Stochastic controllers, however, are often undesirable from a practical perspective because of their lack of robustness, safety, and traceability. In common practice, stochastic (hyper)policies are learned only to deploy their deterministic version. In this paper, we make a step towards the theoretical understanding of this practice. After introducing a novel framework for modeling this scenario, we study the global convergence to the best deterministic policy, under (weak) gradient domination assumptions. Then, we illustrate how to tune the exploration level used for learning to optimize the trade-off between the sample complexity and the performance of the deployed deterministic policy. Finally, we quantitatively compare action-based and parameter-based exploration, giving a formal guise to intuitive results.

Optimisic Information Directed Sampling

We study the problem of online learning in contextual bandit problems where the loss function is assumed to belong to a known parametric function class. We propose a new analytic framework for this setting that bridges the Bayesian theory of information-directed sampling due to Russo and Van Roy (2018) and the worst-case theory of Foster, Kakade, Qian, and Rakhlin (2021) based on the decision-estimation coefficient. Drawing from both lines of work, we propose a algorithmic template called Optimistic Information-Directed Sampling and show that it can achieve instance-dependent regret guarantees similar to the ones achievable by the classic Bayesian IDS method, but with the major advantage of not requiring any Bayesian assumptions. The key technical innovation of our analysis is introducing an optimistic surrogate model for the regret and using it to define a frequentist version of the Information Ratio of Russo and Van Roy (2018), and a less conservative version of the Decision Estimation Coefficient of Foster et al. (2021). Keywords: Contextual bandits, information-directed sampling, decision estimation coefficient, first-order regret bounds.

No-Regret Reinforcement Learning in Smooth MDPs

Obtaining no-regret guarantees for reinforcement learning (RL) in the case of problems with continuous state and/or action spaces is still one of the major open challenges in the field. Recently, a variety of solutions have been proposed, but besides very specific settings, the general problem remains unsolved. In this paper, we introduce a novel structural assumption on the Markov decision processes (MDPs), namely $\nu-$smoothness, that generalizes most of the settings proposed so far (e.g., linear MDPs and Lipschitz MDPs). To face this challenging scenario, we propose two algorithms for regret minimization in $\nu-$smooth MDPs. Both algorithms build upon the idea of constructing an MDP representation through an orthogonal feature map based on Legendre polynomials. The first algorithm, \textsc{Legendre-Eleanor}, archives the no-regret property under weaker assumptions but is computationally inefficient, whereas the second one, \textsc{Legendre-LSVI}, runs in polynomial time, although for a smaller class of problems. After analyzing their regret properties, we compare our results with state-of-the-art ones from RL theory, showing that our algorithms achieve the best guarantees.

Importance-Weighted Offline Learning Done Right

We study the problem of offline policy optimization in stochastic contextual bandit problems, where the goal is to learn a near-optimal policy based on a dataset of decision data collected by a suboptimal behavior policy. Rather than making any structural assumptions on the reward function, we assume access to a given policy class and aim to compete with the best comparator policy within this class. In this setting, a standard approach is to compute importance-weighted estimators of the value of each policy, and select a policy that minimizes the estimated value up to a "pessimistic" adjustment subtracted from the estimates to reduce their random fluctuations. In this paper, we show that a simple alternative approach based on the "implicit exploration" estimator of \citet{Neu2015} yields performance guarantees that are superior in nearly all possible terms to all previous results. Most notably, we remove an extremely restrictive "uniform coverage" assumption made in all previous works. These improvements are made possible by the observation that the upper and lower tails importance-weighted estimators behave very differently from each other, and their careful control can massively improve on previous results that were all based on symmetric two-sided concentration inequalities. We also extend our results to infinite policy classes in a PAC-Bayesian fashion, and showcase the robustness of our algorithm to the choice of hyper-parameters by means of numerical simulations.

Offline Primal-Dual Reinforcement Learning for Linear MDPs

Offline Reinforcement Learning (RL) aims to learn a near-optimal policy from a fixed dataset of transitions collected by another policy. This problem has attracted a lot of attention recently, but most existing methods with strong theoretical guarantees are restricted to finite-horizon or tabular settings. In constrast, few algorithms for infinite-horizon settings with function approximation and minimal assumptions on the dataset are both sample and computationally efficient. Another gap in the current literature is the lack of theoretical analysis for the average-reward setting, which is more challenging than the discounted setting. In this paper, we address both of these issues by proposing a primal-dual optimization method based on the linear programming formulation of RL. Our key contribution is a new reparametrization that allows us to derive low-variance gradient estimators that can be used in a stochastic optimization scheme using only samples from the behavior policy. Our method finds an $\varepsilon$-optimal policy with $O(\varepsilon^{-4})$ samples, improving on the previous $O(\varepsilon^{-5})$, while being computationally efficient for infinite-horizon discounted and average-reward MDPs with realizable linear function approximation and partial coverage. Moreover, to the best of our knowledge, this is the first theoretical result for average-reward offline RL.