arXiv subjects

Alberto Maria Metelli

Publications and source records attributed to Alberto Maria Metelli.

At least 55 records · Page 3Linked to original sources

arXiv2024

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.

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arXiv2024

Best Arm Identification for Stochastic Rising Bandits

Stochastic Rising Bandits (SRBs) model sequential decision-making problems in which the expected reward of the available options increases every time they are selected. This setting captures a wide range of scenarios in which the available options are learning entities whose performance improves (in expectation) over time (e.g., online best model selection). While previous works addressed the regret minimization problem, this paper focuses on the fixed-budget Best Arm Identification (BAI) problem for SRBs. In this scenario, given a fixed budget of rounds, we are asked to provide a recommendation about the best option at the end of the identification process. We propose two algorithms to tackle the above-mentioned setting, namely R-UCBE, which resorts to a UCB-like approach, and R-SR, which employs a successive reject procedure. Then, we prove that, with a sufficiently large budget, they provide guarantees on the probability of properly identifying the optimal option at the end of the learning process and on the simple regret. Furthermore, we derive a lower bound on the error probability, matched by our R-SR (up to constants), and illustrate how the need for a sufficiently large budget is unavoidable in the SRB setting. Finally, we numerically validate the proposed algorithms in both synthetic and realistic environments.

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arXiv2024

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~$\widetilde{\mathcal{O}}(ε^{-2-d/(ν+1)})$ sample complexity, where $d$ is the dimension of the state-action space and $ν$ the order of smoothness, recovers the state-of-the-art result of discretization approaches for the special case of Lipschitz MDPs $(ν=0)$. At the same time, for $ν\to\infty$, it recovers and greatly generalizes the $\mathcal{O}(ε^{-2})$ 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.

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arXiv2024

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 $O(ε^{-4})$ 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.

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arXiv2024

Parameterized Projected Bellman Operator

Approximate value iteration (AVI) is a family of algorithms for reinforcement learning (RL) that aims to obtain an approximation of the optimal value function. Generally, AVI algorithms implement an iterated procedure where each step consists of (i) an application of the Bellman operator and (ii) a projection step into a considered function space. Notoriously, the Bellman operator leverages transition samples, which strongly determine its behavior, as uninformative samples can result in negligible updates or long detours, whose detrimental effects are further exacerbated by the computationally intensive projection step. To address these issues, we propose a novel alternative approach based on learning an approximate version of the Bellman operator rather than estimating it through samples as in AVI approaches. This way, we are able to (i) generalize across transition samples and (ii) avoid the computationally intensive projection step. For this reason, we call our novel operator projected Bellman operator (PBO). We formulate an optimization problem to learn PBO for generic sequential decision-making problems, and we theoretically analyze its properties in two representative classes of RL problems. Furthermore, we theoretically study our approach under the lens of AVI and devise algorithmic implementations to learn PBO in offline and online settings by leveraging neural network parameterizations. Finally, we empirically showcase the benefits of PBO w.r.t. the regular Bellman operator on several RL problems.

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arXiv2024

Performance Improvement Bounds for Lipschitz Configurable Markov Decision Processes

Configurable Markov Decision Processes (Conf-MDPs) have recently been introduced as an extension of the traditional Markov Decision Processes (MDPs) to model the real-world scenarios in which there is the possibility to intervene in the environment in order to configure some of its parameters. In this paper, we focus on a particular subclass of Conf-MDP that satisfies regularity conditions, namely Lipschitz continuity. We start by providing a bound on the Wasserstein distance between $γ$-discounted stationary distributions induced by changing policy and configuration. This result generalizes the already existing bounds both for Conf-MDPs and traditional MDPs. Then, we derive a novel performance improvement lower bound.

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arXiv2024

Autoregressive Bandits

Autoregressive processes naturally arise in a large variety of real-world scenarios, including stock markets, sales forecasting, weather prediction, advertising, and pricing. When facing a sequential decision-making problem in such a context, the temporal dependence between consecutive observations should be properly accounted for guaranteeing convergence to the optimal policy. In this work, we propose a novel online learning setting, namely, Autoregressive Bandits (ARBs), in which the observed reward is governed by an autoregressive process of order $k$, whose parameters depend on the chosen action. We show that, under mild assumptions on the reward process, the optimal policy can be conveniently computed. Then, we devise a new optimistic regret minimization algorithm, namely, AutoRegressive Upper Confidence Bound (AR-UCB), that suffers sublinear regret of order $\widetilde{\mathcal{O}} \left( \frac{(k+1)^{3/2}\sqrt{nT}}{(1-Γ)^2}\right)$, where $T$ is the optimization horizon, $n$ is the number of actions, and $Γ< 1$ is a stability index of the process. Finally, we empirically validate our algorithm, illustrating its advantages w.r.t. bandit baselines and its robustness to misspecification of key parameters.

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arXiv2024

Information Capacity Regret Bounds for Bandits with Mediator Feedback

This work addresses the mediator feedback problem, a bandit game where the decision set consists of a number of policies, each associated with a probability distribution over a common space of outcomes. Upon choosing a policy, the learner observes an outcome sampled from its distribution and incurs the loss assigned to this outcome in the present round. We introduce the policy set capacity as an information-theoretic measure for the complexity of the policy set. Adopting the classical EXP4 algorithm, we provide new regret bounds depending on the policy set capacity in both the adversarial and the stochastic settings. For a selection of policy set families, we prove nearly-matching lower bounds, scaling similarly with the capacity. We also consider the case when the policies' distributions can vary between rounds, thus addressing the related bandits with expert advice problem, which we improve upon its prior results. Additionally, we prove a lower bound showing that exploiting the similarity between the policies is not possible in general under linear bandit feedback. Finally, for a full-information variant, we provide a regret bound scaling with the information radius of the policy set.

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arXiv2024

$(ε, u)$-Adaptive Regret Minimization in Heavy-Tailed Bandits

Heavy-tailed distributions naturally arise in several settings, from finance to telecommunications. While regret minimization under subgaussian or bounded rewards has been widely studied, learning with heavy-tailed distributions only gained popularity over the last decade. In this paper, we consider the setting in which the reward distributions have finite absolute raw moments of maximum order $1+ε$, uniformly bounded by a constant $u<+\infty$, for some $ε\in (0,1]$. In this setting, we study the regret minimization problem when $ε$ and $u$ are unknown to the learner and it has to adapt. First, we show that adaptation comes at a cost and derive two negative results proving that the same regret guarantees of the non-adaptive case cannot be achieved with no further assumptions. Then, we devise and analyze a fully data-driven trimmed mean estimator and propose a novel adaptive regret minimization algorithm, AdaR-UCB, that leverages such an estimator. Finally, we show that AdaR-UCB is the first algorithm that, under a known distributional assumption, enjoys regret guarantees nearly matching those of the non-adaptive heavy-tailed case.

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arXiv2024

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 $ν-$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 $ν-$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.

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arXiv2024

Pure Exploration under Mediators' Feedback

Stochastic multi-armed bandits are a sequential-decision-making framework, where, at each interaction step, the learner selects an arm and observes a stochastic reward. Within the context of best-arm identification (BAI) problems, the goal of the agent lies in finding the optimal arm, i.e., the one with highest expected reward, as accurately and efficiently as possible. Nevertheless, the sequential interaction protocol of classical BAI problems, where the agent has complete control over the arm being pulled at each round, does not effectively model several decision-making problems of interest (e.g., off-policy learning, partially controllable environments, and human feedback). For this reason, in this work, we propose a novel strict generalization of the classical BAI problem that we refer to as best-arm identification under mediators' feedback (BAI-MF). More specifically, we consider the scenario in which the learner has access to a set of mediators, each of which selects the arms on the agent's behalf according to a stochastic and possibly unknown policy. The mediator, then, communicates back to the agent the pulled arm together with the observed reward. In this setting, the agent's goal lies in sequentially choosing which mediator to query to identify with high probability the optimal arm while minimizing the identification time, i.e., the sample complexity. To this end, we first derive and analyze a statistical lower bound on the sample complexity specific to our general mediator feedback scenario. Then, we propose a sequential decision-making strategy for discovering the best arm under the assumption that the mediators' policies are known to the learner. As our theory verifies, this algorithm matches the lower bound both almost surely and in expectation. Finally, we extend these results to cases where the mediators' policies are unknown to the learner obtaining comparable results.

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arXiv2024

Inverse Reinforcement Learning with Sub-optimal Experts

Inverse Reinforcement Learning (IRL) techniques deal with the problem of deducing a reward function that explains the behavior of an expert agent who is assumed to act optimally in an underlying unknown task. In several problems of interest, however, it is possible to observe the behavior of multiple experts with different degree of optimality (e.g., racing drivers whose skills ranges from amateurs to professionals). For this reason, in this work, we extend the IRL formulation to problems where, in addition to demonstrations from the optimal agent, we can observe the behavior of multiple sub-optimal experts. Given this problem, we first study the theoretical properties of the class of reward functions that are compatible with a given set of experts, i.e., the feasible reward set. Our results show that the presence of multiple sub-optimal experts can significantly shrink the set of compatible rewards. Furthermore, we study the statistical complexity of estimating the feasible reward set with a generative model. To this end, we analyze a uniform sampling algorithm that results in being minimax optimal whenever the sub-optimal experts' performance level is sufficiently close to the one of the optimal agent.

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arXiv2023

Causal Feature Selection via Transfer Entropy

Machine learning algorithms are designed to capture complex relationships between features. In this context, the high dimensionality of data often results in poor model performance, with the risk of overfitting. Feature selection, the process of selecting a subset of relevant and non-redundant features, is, therefore, an essential step to mitigate these issues. However, classical feature selection approaches do not inspect the causal relationship between selected features and target, which can lead to misleading results in real-world applications. Causal discovery, instead, aims to identify causal relationships between features with observational data. In this paper, we propose a novel methodology at the intersection between feature selection and causal discovery, focusing on time series. We introduce a new causal feature selection approach that relies on the forward and backward feature selection procedures and leverages transfer entropy to estimate the causal flow of information from the features to the target in time series. Our approach enables the selection of features not only in terms of mere model performance but also captures the causal information flow. In this context, we provide theoretical guarantees on the regression and classification errors for both the exact and the finite-sample cases. Finally, we present numerical validations on synthetic and real-world regression problems, showing results competitive w.r.t. the considered baselines.

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arXiv2023

Nonlinear Feature Aggregation: Two Algorithms driven by Theory

Many real-world machine learning applications are characterized by a huge number of features, leading to computational and memory issues, as well as the risk of overfitting. Ideally, only relevant and non-redundant features should be considered to preserve the complete information of the original data and limit the dimensionality. Dimensionality reduction and feature selection are common preprocessing techniques addressing the challenge of efficiently dealing with high-dimensional data. Dimensionality reduction methods control the number of features in the dataset while preserving its structure and minimizing information loss. Feature selection aims to identify the most relevant features for a task, discarding the less informative ones. Previous works have proposed approaches that aggregate features depending on their correlation without discarding any of them and preserving their interpretability through aggregation with the mean. A limitation of methods based on correlation is the assumption of linearity in the relationship between features and target. In this paper, we relax such an assumption in two ways. First, we propose a bias-variance analysis for general models with additive Gaussian noise, leading to a dimensionality reduction algorithm (NonLinCFA) which aggregates non-linear transformations of features with a generic aggregation function. Then, we extend the approach assuming that a generalized linear model regulates the relationship between features and target. A deviance analysis leads to a second dimensionality reduction algorithm (GenLinCFA), applicable to a larger class of regression problems and classification settings. Finally, we test the algorithms on synthetic and real-world datasets, performing regression and classification tasks, showing competitive performances.

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arXiv2023

Dynamical Linear Bandits

In many real-world sequential decision-making problems, an action does not immediately reflect on the feedback and spreads its effects over a long time frame. For instance, in online advertising, investing in a platform produces an instantaneous increase of awareness, but the actual reward, i.e., a conversion, might occur far in the future. Furthermore, whether a conversion takes place depends on: how fast the awareness grows, its vanishing effects, and the synergy or interference with other advertising platforms. Previous work has investigated the Multi-Armed Bandit framework with the possibility of delayed and aggregated feedback, without a particular structure on how an action propagates in the future, disregarding possible dynamical effects. In this paper, we introduce a novel setting, the Dynamical Linear Bandits (DLB), an extension of the linear bandits characterized by a hidden state. When an action is performed, the learner observes a noisy reward whose mean is a linear function of the hidden state and of the action. Then, the hidden state evolves according to linear dynamics, affected by the performed action too. We start by introducing the setting, discussing the notion of optimal policy, and deriving an expected regret lower bound. Then, we provide an optimistic regret minimization algorithm, Dynamical Linear Upper Confidence Bound (DynLin-UCB), that suffers an expected regret of order $\widetilde{\mathcal{O}} \Big( \frac{d \sqrt{T}}{(1-\overlineρ)^{3/2}} \Big)$, where $\overlineρ$ is a measure of the stability of the system, and $d$ is the dimension of the action vector. Finally, we conduct a numerical validation on a synthetic environment and on real-world data to show the effectiveness of DynLin-UCB in comparison with several baselines.

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arXiv2023

An Option-Dependent Analysis of Regret Minimization Algorithms in Finite-Horizon Semi-Markov Decision Processes

A large variety of real-world Reinforcement Learning (RL) tasks is characterized by a complex and heterogeneous structure that makes end-to-end (or flat) approaches hardly applicable or even infeasible. Hierarchical Reinforcement Learning (HRL) provides general solutions to address these problems thanks to a convenient multi-level decomposition of the tasks, making their solution accessible. Although often used in practice, few works provide theoretical guarantees to justify this outcome effectively. Thus, it is not yet clear when to prefer such approaches compared to standard flat ones. In this work, we provide an option-dependent upper bound to the regret suffered by regret minimization algorithms in finite-horizon problems. We illustrate that the performance improvement derives from the planning horizon reduction induced by the temporal abstraction enforced by the hierarchical structure. Then, focusing on a sub-setting of HRL approaches, the options framework, we highlight how the average duration of the available options affects the planning horizon and, consequently, the regret itself. Finally, we relax the assumption of having pre-trained options to show how in particular situations, learning hierarchically from scratch could be preferable to using a standard approach.

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arXiv2023

Truncating Trajectories in Monte Carlo Reinforcement Learning

In Reinforcement Learning (RL), an agent acts in an unknown environment to maximize the expected cumulative discounted sum of an external reward signal, i.e., the expected return. In practice, in many tasks of interest, such as policy optimization, the agent usually spends its interaction budget by collecting episodes of fixed length within a simulator (i.e., Monte Carlo simulation). However, given the discounted nature of the RL objective, this data collection strategy might not be the best option. Indeed, the rewards taken in early simulation steps weigh exponentially more than future rewards. Taking a cue from this intuition, in this paper, we design an a-priori budget allocation strategy that leads to the collection of trajectories of different lengths, i.e., truncated. The proposed approach provably minimizes the width of the confidence intervals around the empirical estimates of the expected return of a policy. After discussing the theoretical properties of our method, we make use of our trajectory truncation mechanism to extend Policy Optimization via Importance Sampling (POIS, Metelli et al., 2018) algorithm. Finally, we conduct a numerical comparison between our algorithm and POIS: the results are consistent with our theory and show that an appropriate truncation of the trajectories can succeed in improving performance.

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arXiv2023

Towards Theoretical Understanding of Inverse Reinforcement Learning

Inverse reinforcement learning (IRL) denotes a powerful family of algorithms for recovering a reward function justifying the behavior demonstrated by an expert agent. A well-known limitation of IRL is the ambiguity in the choice of the reward function, due to the existence of multiple rewards that explain the observed behavior. This limitation has been recently circumvented by formulating IRL as the problem of estimating the feasible reward set, i.e., the region of the rewards compatible with the expert's behavior. In this paper, we make a step towards closing the theory gap of IRL in the case of finite-horizon problems with a generative model. We start by formally introducing the problem of estimating the feasible reward set, the corresponding PAC requirement, and discussing the properties of particular classes of rewards. Then, we provide the first minimax lower bound on the sample complexity for the problem of estimating the feasible reward set of order $Ω\Bigl( \frac{H^3SA}{ε^2} \bigl( \log \bigl(\frac{1}δ\bigl) + S \bigl)\Bigl)$, being $S$ and $A$ the number of states and actions respectively, $H$ the horizon, $ε$ the desired accuracy, and $δ$ the confidence. We analyze the sample complexity of a uniform sampling strategy (US-IRL), proving a matching upper bound up to logarithmic factors. Finally, we outline several open questions in IRL and propose future research directions.

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