Parallel data collection has redefined Reinforcement Learning (RL), unlocking unprecedented efficiency and powering breakthroughs in large-scale real-world applications. In this paradigm, $N$ identical agents operate in $N$ replicas of an environment simulator, accelerating data collection by a factor of $N$. A critical question arises: Does specializing the policies of the parallel agents hold the key to surpass the $N$ factor acceleration? In this paper, we introduce a novel learning framework that maximizes the entropy of collected data in a parallel setting. Our approach carefully balances the entropy of individual agents with inter-agent diversity, effectively minimizing redundancies. The latter idea is implemented with a centralized policy gradient method, which shows promise when evaluated empirically against systems of identical agents, as well as synergy with batch RL techniques that can exploit data diversity. Finally, we provide an original concentration analysis that shows faster rates for specialized parallel sampling distributions, which supports our methodology and may be of independent interest.

In reinforcement learning, we typically refer to *unsupervised* pre-training when we aim to pre-train a policy without a priori access to the task specification, i.e., rewards, to be later employed for efficient learning of downstream tasks. In single-agent settings, the problem has been extensively studied and mostly understood. A popular approach casts the unsupervised objective as maximizing the *entropy* of the state distribution induced by the agent's policy, from which principles and methods follow. In contrast, little is known about state entropy maximization in multi-agent settings, which are ubiquitous in the real world. What are the pros and cons of alternative problem formulations in this setting? How hard is the problem in theory, how can we solve it in practice? In this paper, we address these questions by first characterizing those alternative formulations and highlighting how the problem, even when tractable in theory, is non-trivial in practice. Then, we present a scalable, decentralized, trust-region policy search algorithm to address the problem in practical settings. Finally, we provide numerical validations to both corroborate the theoretical findings and pave the way for unsupervised multi-agent reinforcement learning via state entropy maximization in challenging domains, showing that optimizing for a specific objective, namely *mixture entropy*, provides an excellent trade-off between tractability and performances.

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.

The Maximum Entropy (Max-Ent) framework has been effectively employed in a variety of Reinforcement Learning (RL) tasks. In this paper, we first propose a novel Max-Ent framework for policy evaluation in a distributional RL setting, named *Distributional Maximum Entropy Policy Evaluation* (D-Max-Ent PE). We derive a generalization-error bound that depends on the complexity of the representation employed, showing that this framework can explicitly take into account the features used to represent the state space while evaluating a policy. Then, we exploit these favorable properties to drive the representation learning of the state space in a Structural Risk Minimization fashion. We employ state-aggregation functions as feature functions and we specialize the D-Max-Ent approach into an algorithm, named *D-Max-Ent Progressive Factorization*, which constructs a progressively finer-grained representation of the state space by balancing the trade-off between preserving information (bias) and reducing the effective number of states, i.e., the complexity of the representation space (variance). Finally, we report the results of some illustrative numerical simulations, showing that the proposed algorithm matches the expected theoretical behavior and highlighting the relationship between aggregations and sample regimes.