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.

Truncating Trajectories in Monte Carlo Policy Evaluation: an Adaptive Approach

Policy evaluation via Monte Carlo (MC) simulation is at the core of many MC Reinforcement Learning (RL) algorithms (e.g., policy gradient methods). In this context, the designer of the learning system specifies an interaction budget that the agent usually spends by collecting trajectories of fixed length within a simulator. However, is this data collection strategy the best option? To answer this question, in this paper, we propose as a quality index a surrogate of the mean squared error of a return estimator that uses trajectories of different lengths, i.e., \emph{truncated}. Specifically, this surrogate shows the sub-optimality of the fixed-length trajectory schedule. Furthermore, it suggests that adaptive data collection strategies that spend the available budget sequentially can allocate a larger portion of transitions in timesteps in which more accurate sampling is required to reduce the error of the final estimate. Building on these findings, we present an adaptive algorithm called Robust and Iterative Data collection strategy Optimization (RIDO). The main intuition behind RIDO is to split the available interaction budget into mini-batches. At each round, the agent determines the most convenient schedule of trajectories that minimizes an empirical and robust version of the surrogate of the estimator's error. After discussing the theoretical properties of our method, we conclude by assessing its performance across multiple domains. Our results show that RIDO can adapt its trajectory schedule toward timesteps where more sampling is required to increase the quality of the final estimation.

Efficient Learning of POMDPs with Known Observation Model in Average-Reward Setting

Dealing with Partially Observable Markov Decision Processes is notably a challenging task. We face an average-reward infinite-horizon POMDP setting with an unknown transition model, where we assume the knowledge of the observation model. Under this assumption, we propose the Observation-Aware Spectral (OAS) estimation technique, which enables the POMDP parameters to be learned from samples collected using a belief-based policy. Then, we propose the OAS-UCRL algorithm that implicitly balances the exploration-exploitation trade-off following the $\textit{optimism in the face of uncertainty}$ principle. The algorithm runs through episodes of increasing length. For each episode, the optimal belief-based policy of the estimated POMDP interacts with the environment and collects samples that will be used in the next episode by the OAS estimation procedure to compute a new estimate of the POMDP parameters. Given the estimated model, an optimization oracle computes the new optimal policy. We show the consistency of the OAS procedure, and we prove a regret guarantee of order $\mathcal{O}(\sqrt{T \log(T)})$ for the proposed OAS-UCRL algorithm. We compare against the oracle playing the optimal stochastic belief-based policy and show the efficient scaling of our approach with respect to the dimensionality of the state, action, and observation space. We finally conduct numerical simulations to validate and compare the proposed technique with other baseline approaches.

Learning Utilities from Demonstrations in Markov Decision Processes

Our goal is to extract useful knowledge from demonstrations of behavior in sequential decision-making problems. Although it is well-known that humans commonly engage in risk-sensitive behaviors in the presence of stochasticity, most Inverse Reinforcement Learning (IRL) models assume a risk-neutral agent. Beyond introducing model misspecification, these models do not directly capture the risk attitude of the observed agent, which can be crucial in many applications. In this paper, we propose a novel model of behavior in Markov Decision Processes (MDPs) that explicitly represents the agent's risk attitude through a utility function. We then define the Utility Learning (UL) problem as the task of inferring the observed agent's risk attitude, encoded via a utility function, from demonstrations in MDPs, and we analyze the partial identifiability of the agent's utility. Furthermore, we devise two provably efficient algorithms for UL in a finite-data regime, and we analyze their sample complexity. We conclude with proof-of-concept experiments that empirically validate both our model and our algorithms.

Bridging Rested and Restless Bandits with Graph-Triggering: Rising and Rotting

Rested and Restless Bandits are two well-known bandit settings that are useful to model real-world sequential decision-making problems in which the expected reward of an arm evolves over time due to the actions we perform or due to the nature. In this work, we propose Graph-Triggered Bandits (GTBs), a unifying framework to generalize and extend rested and restless bandits. In this setting, the evolution of the arms' expected rewards is governed by a graph defined over the arms. An edge connecting a pair of arms $(i,j)$ represents the fact that a pull of arm $i$ triggers the evolution of arm $j$, and vice versa. Interestingly, rested and restless bandits are both special cases of our model for some suitable (degenerated) graph. As relevant case studies for this setting, we focus on two specific types of monotonic bandits: rising, where the expected reward of an arm grows as the number of triggers increases, and rotting, where the opposite behavior occurs. For these cases, we study the optimal policies. We provide suitable algorithms for all scenarios and discuss their theoretical guarantees, highlighting the complexity of the learning problem concerning instance-dependent terms that encode specific properties of the underlying graph structure.

Sliding-Window Thompson Sampling for Non-Stationary Settings

$\textit{Restless Bandits}$ describe sequential decision-making problems in which the rewards evolve with time independently from the actions taken by the policy-maker. It has been shown that classical Bandit algorithms fail when the underlying environment is changing, making clear that in order to tackle more challenging scenarios specifically crafted algorithms are needed. In this paper, extending and correcting the work by \cite{trovo2020sliding}, we analyze two Thompson-Sampling inspired algorithms, namely $\texttt{BETA-SWTS}$ and $\texttt{$\gamma$-SWGTS}$, introduced to face the additional complexity given by the non-stationary nature of the settings; in particular we derive a general formulation for the regret in $\textit{any}$ arbitrary restless environment for both Bernoulli and Subgaussian rewards, and, through the introduction of new quantities, we delve in what contribution lays the deeper foundations of the error made by the algorithms. Finally, we infer from the general formulation the regret for two of the most common non-stationary settings: the $\textit{Abruptly Changing}$ and the $\textit{Smoothly Changing}$ environments.

State and Action Factorization in Power Grids

The increase of renewable energy generation towards the zero-emission target is making the problem of controlling power grids more and more challenging. The recent series of competitions Learning To Run a Power Network (L2RPN) have encouraged the use of Reinforcement Learning (RL) for the assistance of human dispatchers in operating power grids. All the solutions proposed so far severely restrict the action space and are based on a single agent acting on the entire grid or multiple independent agents acting at the substations level. In this work, we propose a domain-agnostic algorithm that estimates correlations between state and action components entirely based on data. Highly correlated state-action pairs are grouped together to create simpler, possibly independent subproblems that can lead to distinct learning processes with less computational and data requirements. The algorithm is validated on a power grid benchmark obtained with the Grid2Op simulator that has been used throughout the aforementioned competitions, showing that our algorithm is in line with domain-expert analysis. Based on these results, we lay a theoretically-grounded foundation for using distributed reinforcement learning in order to improve the existing solutions.

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.

Open Problem: Tight Bounds for Kernelized Multi-Armed Bandits with Bernoulli Rewards

We consider Kernelized Bandits (KBs) to optimize a function $f : \mathcal{X} \rightarrow [0,1]$ belonging to the Reproducing Kernel Hilbert Space (RKHS) $\mathcal{H}_k$. Mainstream works on kernelized bandits focus on a subgaussian noise model in which observations of the form $f(\mathbf{x}_t)+\epsilon_t$, being $\epsilon_t$ a subgaussian noise, are available (Chowdhury and Gopalan, 2017). Differently, we focus on the case in which we observe realizations $y_t \sim \text{Ber}(f(\mathbf{x}_t))$ sampled from a Bernoulli distribution with parameter $f(\mathbf{x}_t)$. While the Bernoulli model has been investigated successfully in multi-armed bandits (Garivier and Capp\'e, 2011), logistic bandits (Faury et al., 2022), bandits in metric spaces (Magureanu et al., 2014), it remains an open question whether tight results can be obtained for KBs. This paper aims to draw the attention of the online learning community to this open problem.