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How can a scientist use a Reinforcement Learning (RL) algorithm to design experiments over a dynamical system's state space? In the case of finite and Markovian systems, an area called Active Exploration (AE) relaxes the optimization problem of experiments design into Convex RL, a generalization of RL admitting a wider notion of reward. Unfortunately, this framework is currently not scalable and the potential of AE is hindered by the vastness of experiment spaces typical of scientific discovery applications. However, these spaces are often endowed with natural geometries, e.g., permutation invariance in molecular design, that an agent could leverage to improve the statistical and computational efficiency of AE. To achieve this, we bridge AE and MDP homomorphisms, which offer a way to exploit known geometric structures via abstraction. Towards this goal, we make two fundamental contributions: we extend MDP homomorphisms formalism to Convex RL, and we present, to the best of our knowledge, the first analysis that formally captures the benefit of abstraction via homomorphisms on sample efficiency. Ultimately, we propose the Geometric Active Exploration (GAE) algorithm, which we analyse theoretically and experimentally in environments motivated by problems in scientific discovery., Comment: ICML 2024

The problem of pure exploration in Markov decision processes has been cast as maximizing the entropy over the state distribution induced by the agent's policy, an objective that has been extensively studied. However, little attention has been dedicated to state entropy maximization under partial observability, despite the latter being ubiquitous in applications, e.g., finance and robotics, in which the agent only receives noisy observations of the true state governing the system's dynamics. How can we address state entropy maximization in those domains? In this paper, we study the simple approach of maximizing the entropy over observations in place of true latent states. First, we provide lower and upper bounds to the approximation of the true state entropy that only depends on some properties of the observation function. Then, we show how knowledge of the latter can be exploited to compute a principled regularization of the observation entropy to improve performance. With this work, we provide both a flexible approach to bring advances in state entropy maximization to the POMDP setting and a theoretical characterization of its intrinsic limits.

Building on the one-to-one relationship between generalized FGM copulas and multivariate Bernoulli distributions, we prove that the class of multivariate distributions with generalized FGM copulas is a convex polytope. Therefore, we find sharp bounds in this class for many aggregate risk measures, such as value-at-risk, expected shortfall, and entropic risk measure, by enumerating their values on the extremal points of the convex polytope. This is infeasible in high dimensions. We overcome this limitation by considering the aggregation of identically distributed risks with generalized FGM copula specified by a common parameter $p$. In this case, the analogy with the geometrical structure of the class of Bernoulli distribution allows us to provide sharp analytical bounds for convex risk measures., Comment: 31 pages, 1 figure

In online Inverse Reinforcement Learning (IRL), the learner can collect samples about the dynamics of the environment to improve its estimate of the reward function. Since IRL suffers from identifiability issues, many theoretical works on online IRL focus on estimating the entire set of rewards that explain the demonstrations, named the feasible reward set. However, none of the algorithms available in the literature can scale to problems with large state spaces. In this paper, we focus on the online IRL problem in Linear Markov Decision Processes (MDPs). We show that the structure offered by Linear MDPs is not sufficient for efficiently estimating the feasible set when the state space is large. As a consequence, we introduce the novel framework of rewards compatibility, which generalizes the notion of feasible set, and we develop CATY-IRL, a sample efficient algorithm whose complexity is independent of the cardinality of the state space in Linear MDPs. When restricted to the tabular setting, we demonstrate that CATY-IRL is minimax optimal up to logarithmic factors. As a by-product, we show that Reward-Free Exploration (RFE) enjoys the same worst-case rate, improving over the state-of-the-art lower bound. Finally, we devise a unifying framework for IRL and RFE that may be of independent interest.

Recent works have studied *state entropy maximization* in reinforcement learning, in which the agent's objective is to learn a policy inducing high entropy over states visitation (Hazan et al., 2019). They typically assume full observability of the state of the system, so that the entropy of the observations is maximized. In practice, the agent may only get *partial* observations, e.g., a robot perceiving the state of a physical space through proximity sensors and cameras. A significant mismatch between the entropy over observations and true states of the system can arise in those settings. In this paper, we address the problem of entropy maximization over the *true states* with a decision policy conditioned on partial observations *only*. The latter is a generalization of POMDPs, which is intractable in general. We develop a memory and computationally efficient *policy gradient* method to address a first-order relaxation of the objective defined on *belief* states, providing various formal characterizations of approximation gaps, the optimization landscape, and the *hallucination* problem. This paper aims to generalize state entropy maximization to more realistic domains that meet the challenges of applications.

Meta reinforcement learning sets a distribution over a set of tasks on which the agent can train at will, then is asked to learn an optimal policy for any test task efficiently. In this paper, we consider a finite set of tasks modeled through Markov decision processes with various dynamics. We assume to have endured a long training phase, from which the set of tasks is perfectly recovered, and we focus on regret minimization against the optimal policy in the unknown test task. Under a separation condition that states the existence of a state-action pair revealing a task against another, Chen et al. (2022) show that $O(M^2 \log(H))$ regret can be achieved, where $M, H$ are the number of tasks in the set and test episodes, respectively. In our first contribution, we demonstrate that the latter rate is nearly optimal by developing a novel lower bound for test-time regret minimization under separation, showing that a linear dependence with $M$ is unavoidable. Then, we present a family of stronger yet reasonable assumptions beyond separation, which we call strong identifiability, enabling algorithms achieving fast rates $\log (H)$ and sublinear dependence with $M$ simultaneously. Our paper provides a new understanding of the statistical barriers of test-time regret minimization and when fast rates can be achieved.

Inverse reinforcement learning (IRL) aims to recover the reward function of an expert agent from demonstrations of behavior. It is well-known that the IRL problem is fundamentally ill-posed, i.e., many reward functions can explain the demonstrations. For this reason, IRL has been recently reframed in terms of estimating the feasible reward set (Metelli et al., 2021), thus, postponing the selection of a single reward. However, so far, the available formulations and algorithmic solutions have been proposed and analyzed mainly for the online setting, where the learner can interact with the environment and query the expert at will. This is clearly unrealistic in most practical applications, where the availability of an offline dataset is a much more common scenario. In this paper, we introduce a novel notion of feasible reward set capturing the opportunities and limitations of the offline setting and we analyze the complexity of its estimation. This requires the introduction an original learning framework that copes with the intrinsic difficulty of the setting, for which the data coverage is not under control. Then, we propose two computationally and statistically efficient algorithms, IRLO and PIRLO, for addressing the problem. In particular, the latter adopts a specific form of pessimism to enforce the novel desirable property of inclusion monotonicity of the delivered feasible set. With this work, we aim to provide a panorama of the challenges of the offline IRL problem and how they can be fruitfully addressed., Comment: International Conference on Machine Learning 41 (ICML 2024)

The growing deployment of reinforcement learning from human feedback (RLHF) calls for a deeper theoretical investigation of its underlying models. The prevalent models of RLHF do not account for neuroscience-backed, partially-observed "internal states" that can affect human feedback, nor do they accommodate intermediate feedback during an interaction. Both of these can be instrumental in speeding up learning and improving alignment. To address these limitations, we model RLHF as reinforcement learning with partially observed reward-states (PORRL). We accommodate two kinds of feedback $-$ cardinal and dueling feedback. We first demonstrate that PORRL subsumes a wide class of RL problems, including traditional RL, RLHF, and reward machines. For cardinal feedback, we present two model-based methods (POR-UCRL, POR-UCBVI). We give both cardinal regret and sample complexity guarantees for the methods, showing that they improve over naive history-summarization. We then discuss the benefits of a model-free method like GOLF with naive history-summarization in settings with recursive internal states and dense intermediate feedback. For this purpose, we define a new history aware version of the Bellman-eluder dimension and give a new guarantee for GOLF in our setting, which can be exponentially sharper in illustrative examples. For dueling feedback, we show that a naive reduction to cardinal feedback fails to achieve sublinear dueling regret. We then present the first explicit reduction that converts guarantees for cardinal regret to dueling regret. In both feedback settings, we show that our models and guarantees generalize and extend existing ones., Comment: 58 pages. 12 pages formain paper, 46 pages for references + appendix

In 2019, the President of the European Commission announced the creation of the European Child Guarantee, an initiative that aimed to combat child poverty by making sure that every child who lives in poverty has access to free healthcare, free education, free childcare, decent housing and adequate nutrition. With "Delivering the EU Child Guarantee: Practical lessons for effective interventions," UNICEF Innocenti provides a synthesis of findings of operational research on 15 pilot models of intervention for children and families in Bulgaria, Croatia, Greece and Italy. The goal of the report is to learn from the experiences in these four countries to inform the rollout of the Child Guarantee programme throughout Europe. The study provides lessons learned in three themes: laying the foundations for the Child Guarantee work at the national level; developing interventions at the local level; supporting the Child Guarantee at the EU level.

Posterior sampling allows exploitation of prior knowledge on the environment's transition dynamics to improve the sample efficiency of reinforcement learning. The prior is typically specified as a class of parametric distributions, the design of which can be cumbersome in practice, often resulting in the choice of uninformative priors. In this work, we propose a novel posterior sampling approach in which the prior is given as a (partial) causal graph over the environment's variables. The latter is often more natural to design, such as listing known causal dependencies between biometric features in a medical treatment study. Specifically, we propose a hierarchical Bayesian procedure, called C-PSRL, simultaneously learning the full causal graph at the higher level and the parameters of the resulting factored dynamics at the lower level. We provide an analysis of the Bayesian regret of C-PSRL that explicitly connects the regret rate with the degree of prior knowledge. Our numerical evaluation conducted in illustrative domains confirms that C-PSRL strongly improves the efficiency of posterior sampling with an uninformative prior while performing close to posterior sampling with the full causal graph.

The key result of this paper is to find all the joint distributions of random vectors whose sums $S=X_1+\ldots+X_d$ are minimal in convex order in the class of symmetric Bernoulli distributions. The minimal convex sums distributions are known to be strongly negatively dependent. Beyond their interest per se, these results enable us to explore negative dependence within the class of copulas. In fact, there are two classes of copulas that can be built from multivariate symmetric Bernoulli distributions: the extremal mixture copulas, and the FGM copulas. We study the extremal negative dependence structure of the copulas corresponding to symmetric Bernoulli vectors with minimal convex sums and we explicitly find a class of minimal dependence copulas. Our main results stem from the geometric and algebraic representations of multivariate symmetric Bernoulli distributions, which effectively encode several of their statistical properties.

Bayesian persuasion studies the problem faced by an informed sender who strategically discloses information to influence the behavior of an uninformed receiver. Recently, a growing attention has been devoted to settings where the sender and the receiver interact sequentially, in which the receiver's decision-making problem is usually modeled as a Markov decision process (MDP). However, previous works focused on computing optimal information-revelation policies (a.k.a. signaling schemes) under the restrictive assumption that the receiver acts myopically, selecting actions to maximize the one-step utility and disregarding future rewards. This is justified by the fact that, when the receiver is farsighted and thus considers future rewards, finding an optimal Markovian signaling scheme is NP-hard. In this paper, we show that Markovian signaling schemes do not constitute the "right" class of policies. Indeed, differently from most of the MDPs settings, we prove that Markovian signaling schemes are not optimal, and general history-dependent signaling schemes should be considered. Moreover, we also show that history-dependent signaling schemes circumvent the negative complexity results affecting Markovian signaling schemes. Formally, we design an algorithm that computes an optimal and {\epsilon}-persuasive history-dependent signaling scheme in time polynomial in 1/{\epsilon} and in the instance size. The crucial challenge is that general history-dependent signaling schemes cannot be represented in polynomial space. Nevertheless, we introduce a convenient subclass of history-dependent signaling schemes, called promise-form, which are as powerful as general history-dependent ones and efficiently representable. Intuitively, promise-form signaling schemes compactly encode histories in the form of honest promises on future receiver's rewards.

The most relevant problems in discounted reinforcement learning involve estimating the mean of a function under the stationary distribution of a Markov reward process, such as the expected return in policy evaluation, or the policy gradient in policy optimization. In practice, these estimates are produced through a finite-horizon episodic sampling, which neglects the mixing properties of the Markov process. It is mostly unclear how this mismatch between the practical and the ideal setting affects the estimation, and the literature lacks a formal study on the pitfalls of episodic sampling, and how to do it optimally. In this paper, we present a minimax lower bound on the discounted mean estimation problem that explicitly connects the estimation error with the mixing properties of the Markov process and the discount factor. Then, we provide a statistical analysis on a set of notable estimators and the corresponding sampling procedures, which includes the finite-horizon estimators often used in practice. Crucially, we show that estimating the mean by directly sampling from the discounted kernel of the Markov process brings compelling statistical properties w.r.t. the alternative estimators, as it matches the lower bound without requiring a careful tuning of the episode horizon., Comment: AISTATS 2023

In reinforcement learning, we encode the potential behaviors of an agent interacting with an environment into an infinite set of policies, the policy space, typically represented by a family of parametric functions. Dealing with such a policy space is a hefty challenge, which often causes sample and computation inefficiencies. However, we argue that a limited number of policies are actually relevant when we also account for the structure of the environment and of the policy parameterization, as many of them would induce very similar interactions, i.e., state-action distributions. In this paper, we seek for a reward-free compression of the policy space into a finite set of representative policies, such that, given any policy $\pi$, the minimum R\'enyi divergence between the state-action distributions of the representative policies and the state-action distribution of $\pi$ is bounded. We show that this compression of the policy space can be formulated as a set cover problem, and it is inherently NP-hard. Nonetheless, we propose a game-theoretic reformulation for which a locally optimal solution can be efficiently found by iteratively stretching the compressed space to cover an adversarial policy. Finally, we provide an empirical evaluation to illustrate the compression procedure in simple domains, and its ripple effects in reinforcement learning., Comment: In 25th International Conference on Artificial Intelligence and Statistics

In the sequential decision making setting, an agent aims to achieve systematic generalization over a large, possibly infinite, set of environments. Such environments are modeled as discrete Markov decision processes with both states and actions represented through a feature vector. The underlying structure of the environments allows the transition dynamics to be factored into two components: one that is environment-specific and another that is shared. Consider a set of environments that share the laws of motion as an example. In this setting, the agent can take a finite amount of reward-free interactions from a subset of these environments. The agent then must be able to approximately solve any planning task defined over any environment in the original set, relying on the above interactions only. Can we design a provably efficient algorithm that achieves this ambitious goal of systematic generalization? In this paper, we give a partially positive answer to this question. First, we provide a tractable formulation of systematic generalization by employing a causal viewpoint. Then, under specific structural assumptions, we provide a simple learning algorithm that guarantees any desired planning error up to an unavoidable sub-optimality term, while showcasing a polynomial sample complexity., Comment: AAAI 2023

In the maximum state entropy exploration framework, an agent interacts with a reward-free environment to learn a policy that maximizes the entropy of the expected state visitations it is inducing. Hazan et al. (2019) noted that the class of Markovian stochastic policies is sufficient for the maximum state entropy objective, and exploiting non-Markovianity is generally considered pointless in this setting. In this paper, we argue that non-Markovianity is instead paramount for maximum state entropy exploration in a finite-sample regime. Especially, we recast the objective to target the expected entropy of the induced state visitations in a single trial. Then, we show that the class of non-Markovian deterministic policies is sufficient for the introduced objective, while Markovian policies suffer non-zero regret in general. However, we prove that the problem of finding an optimal non-Markovian policy is NP-hard. Despite this negative result, we discuss avenues to address the problem in a tractable way and how non-Markovian exploration could benefit the sample efficiency of online reinforcement learning in future works., Comment: ICML 2022

The classic Reinforcement Learning (RL) formulation concerns the maximization of a scalar reward function. More recently, convex RL has been introduced to extend the RL formulation to all the objectives that are convex functions of the state distribution induced by a policy. Notably, convex RL covers several relevant applications that do not fall into the scalar formulation, including imitation learning, risk-averse RL, and pure exploration. In classic RL, it is common to optimize an infinite trials objective, which accounts for the state distribution instead of the empirical state visitation frequencies, even though the actual number of trajectories is always finite in practice. This is theoretically sound since the infinite trials and finite trials objectives can be proved to coincide and thus lead to the same optimal policy. In this paper, we show that this hidden assumption does not hold in the convex RL setting. In particular, we show that erroneously optimizing the infinite trials objective in place of the actual finite trials one, as it is usually done, can lead to a significant approximation error. Since the finite trials setting is the default in both simulated and real-world RL, we believe shedding light on this issue will lead to better approaches and methodologies for convex RL, impacting relevant research areas such as imitation learning, risk-averse RL, and pure exploration among others., Comment: NeurIPS 2022

Several recent works have been dedicated to unsupervised reinforcement learning in a single environment, in which a policy is first pre-trained with unsupervised interactions, and then fine-tuned towards the optimal policy for several downstream supervised tasks defined over the same environment. Along this line, we address the problem of unsupervised reinforcement learning in a class of multiple environments, in which the policy is pre-trained with interactions from the whole class, and then fine-tuned for several tasks in any environment of the class. Notably, the problem is inherently multi-objective as we can trade off the pre-training objective between environments in many ways. In this work, we foster an exploration strategy that is sensitive to the most adverse cases within the class. Hence, we cast the exploration problem as the maximization of the mean of a critical percentile of the state visitation entropy induced by the exploration strategy over the class of environments. Then, we present a policy gradient algorithm, $\alpha$MEPOL, to optimize the introduced objective through mediated interactions with the class. Finally, we empirically demonstrate the ability of the algorithm in learning to explore challenging classes of continuous environments and we show that reinforcement learning greatly benefits from the pre-trained exploration strategy w.r.t. learning from scratch., Comment: In 36th AAAI Conference on Artificial Intelligence (AAAI 2022)