Optimal regret bounds for Multi-Armed Bandit problems are now well documented. They can be classified into two categories based on the growth rate with respect to the time horizon T : (i) small, distribution-dependent, bounds of order of magnitude ln(T ) and (ii) robust, distribution-free, bounds of order of magnitude √ T . The Bandits with Knapsacks model, an extension to the framework allowing to model resource consumption, lacks this clear-cut distinction. While several algorithms have been shown to achieve asymptotically optimal distribution-free bounds on regret, there has been little progress toward the development of small distribution-dependent regret bounds. We partially bridge the gap by designing a general-purpose algorithm with distribution-dependent regret bounds that are logarithmic in the initial endowments of resources in several important cases that cover many practical applications, including dynamic pricing with limited supply, bid optimization in online advertisement auctions, and dynamic procurement.

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We consider the problem of finding an optimal history-dependent routing strategy on a directed graph weighted by stochastic arc costs when the objective is to minimize the risk of spending more than a prescribed budget. To help mitigate the impact of the lack of information on the arc cost probability distributions, we introduce a robust counterpart where the distributions are only known through confidence intervals on some statistics such as the mean, the mean absolute deviation, and any quantile. Leveraging recent results in distributionally robust optimization, we develop a general-purpose algorithm to compute an approximate optimal strategy. To illustrate the benefits of the robust approach, we run numerical experiments with field data from the Singapore road network.

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Quality Diversity (QD) has emerged as a powerful alternative optimization paradigm that aims at generating large and diverse collections of solutions, notably with its flagship algorithm MAP-ELITES (ME) which evolves solutions through mutations and crossovers. While very effective for some unstructured problems, early ME implementations relied exclusively on random search to evolve the population of solutions, rendering them notoriously sample-inefficient for highdimensional problems, such as when evolving neural networks. Follow-up works considered exploiting gradient information to guide the search in order to address these shortcomings through techniques borrowed from either Black-Box Optimization (BBO) or Reinforcement Learning (RL). While mixing RL techniques with ME unlocked state-of-the-art performance for robotics control problems that require a good amount of exploration, it also plagued these ME variants with limitations common among RL algorithms that ME was free of, such as hyperparameter sensitivity, high stochasticity as well as training instability, including when the population size increases as some components are shared across the population in recent approaches. Furthermore, existing approaches mixing ME with RL tend to be tied to a specific RL algorithm, which effectively prevents their use on problems where the corresponding RL algorithm fails. To address these shortcomings, we introduce a flexible framework that allows the use of any RL algorithm and alleviates the aforementioned limitations by evolving populations of agents (whose definition include hyperparameters and all learnable parameters) instead of just policies. We demonstrate the benefits brought about by our framework through extensive numerical experiments on a number of robotics control problems, some of which with deceptive rewards, taken from the QD-RL literature. We open source an efficient JAX-based implementation of our algorithm in the QDax library 1 .