arXiv subjects

Francesco Emanuele Stradi

Publications and source records attributed to Francesco Emanuele Stradi.

At least 19 recordsLinked to original sources

arXiv2026

Toward Optimal Regret in Adversarial MDPs with Stochastic Hard Constraints

We study episodic constrained Markov decision processes with adversarial losses under stochastic hard constraints. Specifically, starting from a known strictly feasible policy with margin $d$, we seek to obtain optimal regret while satisfying the expected cost constraints in every episode. In this setting, Stradi et al. (2025) show that a carefully designed mixing rule attains regret of order $\widetilde{\mathcal{O}}(\sqrt{T}/\min\{d,d^2\})$. Interestingly, they also provide a lower bound of order $Ω(\sqrt{T}/ρ)$ for the same setting, where $ρ$ is the Slater margin of the offline problem and can be much larger than $d$. In this work, we build on their approach to obtain optimal regret dependence on these margins. Specifically, we propose MA-OPS, an algorithm that combines an optimistic search for the Slater margin with a pessimistic evaluation of the selected policies to safely learn a policy with a large feasibility margin. This policy is then used to minimize regret while satisfying the constraints at every episode. In particular, we show that MA-OPS attains regret $\widetilde{\mathcal{O}}(\sqrt{T}/ρ+ 1/(dρ))$. Finally, we provide a matching lower bound, showing that the dependence on $T$, $d$, $ρ$ in the regret bound is optimal up to logarithmic factors.

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arXiv2026

Optimal Regret for Online Market Making with Limit Order Book

We study online learning in market making, where, at each round, a market maker posts bid and ask prices before observing the market price and the private valuation of an incoming trader. In this setting, Maran et al. 2026 introduce a feedback model motivated by limit order books, in which the trader's valuation is revealed only if no transaction occurs. Assuming that trader valuations are drawn i.i.d. from an unknown distribution while market prices are chosen adversarially, they establish an expected regret bound of $\widetilde{\mathcal{O}}(T^{2/3})$. In this work, we improve upon this guarantee by establishing a high-probability regret bound of $\widetilde{\mathcal{O}}(\sqrt{T})$. As a warm-up, we first consider the full-feedback setting. We introduce a discretization of the bid-ask space based on two coupled grids and combine it with Hedge to achieve the desired regret rate. Building on these ideas, we then address the substantially weaker feedback induced by a limit order book and develop an algorithm that achieves the same guarantee. Finally, we investigate the limits of learnability in fully adversarial environments, where the valuations may vary arbitrarily as well. Perhaps surprisingly, we show that when both market prices and trader valuations are chosen adversarially, sublinear regret is impossible even under full feedback, thereby motivating our stochastic assumption on the valuations.

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arXiv2026

Marrying Pricing and Advertising with LLMs

We study a sequential pricing problem in which a seller jointly posts a price and an advertisement generated by a large language model (LLM). The seller aims to maximize revenue under an unknown product demand that depends on both decisions, while observing only whether each offer leads to a purchase. We propose an online actor-critic algorithm that combines low-rank adaptation (LoRA) of a pretrained LLM with a demand model fitted to available data. At each round, the actor generates an advertisement, and the critic estimates purchase probabilities to guide price selection. Then, the resulting feedback is used to update both the actor and the critic, with the critic's revenue estimates providing a baseline for policy gradient updates of the actor. To evaluate our approach, we develop an evaluation framework with three synthetic demand models and a demand simulator built from real-world marketplace data. Finally, we compare our algorithm with benchmarks that do not jointly optimize price selection and advertisement generation, achieving expected revenue gains over the reference policy of 5.69%, 5.18% and 55.96% under the three synthetic demand models and 5.81% under the marketplace simulator.

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arXiv2026

Instance-Dependent Regret for CMDPs with Step-Wise Constraints

We study online learning in episodic tabular constrained Markov decision processes with step-wise safety constraints. In such a setting, the constraints induce a safe subgraph that shapes the variance of cumulative rewards under feasible policies and, consequently, the difficulty of learning. Exploiting this structure, however, requires learning which actions are safe while controlling constraint violations. We propose Safe Variance-Adaptive Exploration (SVAE), an efficient algorithm that learns candidate safe subgraphs and performs variance-adaptive optimistic planning within them. With high probability, SVAE achieves cumulative regret of order $\widetilde{\mathcal{O}}(\sqrt{SAH\min\{\mathbb{V}_Σ,K\mathrm{Var}^{\star}\}}+S\sqrt{AH^3\min\{K,\mathcal{C}\}}+S^2AH^2)$ over $K$ episodes, where $H$ is the horizon of a single episode, while $S$ and $A$ are the numbers of states and actions, respectively. Here, $\mathrm{Var}^{\star}$ is the maximum return variance among safe policies, $\mathbb{V}_Σ$ is the variance accumulated before the first unsafe action is encountered, and $\mathcal{C}$ captures the statistical complexity of eliminating actions incorrectly considered potentially safe. SVAE additionally attains $\widetilde{\mathcal{O}}(H\sqrt{SAK}+S^2AH^2)$ step-wise constraint violation and a gap-dependent violation bound that is polylogarithmic in $K$. Finally, we establish a lower bound showing that dependence on these instance-specific quantities is unavoidable.

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arXiv2026

Online Resource Allocation with Replenishable Budgets

Online Resource Allocation (ORA) is a fundamental framework for sequential decision-making problems under budget constraints. Classical ORA models typically assume that resources are monotonic, meaning that selecting actions can only decrease the available budget. In this work, we study a more general setting with replenishable budgets, in which actions may either consume or replenish resources over time. This extension is necessary to capture scenarios such as inventory systems or energy markets in which capacity can be actively recovered. We develop a dual-based algorithm that recovers best-of-both-worlds guarantees for standard ORA when the replenishment factor $β= 0$, and improves them when $β> 0$. In particular, our algorithm attains $\widetilde{\mathcal O}(\sqrt{T})$ regret in the stochastic setting and $\widetilde{\mathcal O}(\sqrt{T})$ $α$-regret in the adversarial setting, where $α$ depends on the per-round budget and on the replenishment factor of the void action. Moreover, the algorithm ensures strict satisfaction of the budget constraints.

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arXiv2026

Beyond Slater's Condition in Online CMDPs with Stochastic and Adversarial Constraints

We study \emph{online episodic Constrained Markov Decision Processes} (CMDPs) under both stochastic and adversarial constraints. We provide a novel algorithm whose guarantees greatly improve those of the state-of-the-art best-of-both-worlds algorithm introduced by Stradi et al. (2025). In the stochastic regime, \emph{i.e.}, when the constraints are sampled from fixed but unknown distributions, our method achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation without relying on Slater's condition, thereby handling settings where no strictly feasible solution exists. Moreover, we provide guarantees on the stronger notion of \emph{positive} constraint violation, which does not allow to recover from large violation in the early episodes by playing strictly safe policies. In the adversarial regime, \emph{i.e.}, when the constraints may change arbitrarily between episodes, our algorithm ensures sublinear constraint violation without Slater's condition, and achieves sublinear $α$-regret with respect to the \emph{unconstrained} optimum, where $α$ is a suitably defined multiplicative approximation factor. We further validate our results through synthetic experiments, showing the practical effectiveness of our algorithm.

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arXiv2026

Decoupling Corruption and Horizon in Robust Contextual Pricing

We study robust repeated contextual pricing, where valuations depends linearly on the features. At each round $t\in[T]$, a seller observes a context, posts a price, and receives only a possibly corrupted binary sale feedback. The seller knows an upper bound $C$ on the number of corrupted rounds. We design an algorithm with regret $\mathcal O(Cd+d^2\log T)$, where $d$ is the context dimension. This is the first guarantee for robust contextual pricing that separates the dependence on the corruption budget $C$ from the horizon $T$, closing the problem left open by Gupta, Guruganesh, Paes Leme, and Schneider (2025).

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arXiv2026

Online Resource Allocation With General Constraints

Online resource allocation (ORA) is a fundamental framework for sequential decision-making problems under budget constraints, with applications ranging from online advertising to revenue management. In this work, we study a broader setting that includes both budget constraints and general constraints, extending the classical budget-only model. This extension is essential for modeling critical economic requirements, such as Return-on-Investment (ROI) constraints. We develop an algorithm that achieves best-of-both-world guarantees within this generalized framework. In particular, against a dynamic benchmark, our algorithm achieves $\widetilde{\mathcal O}(\sqrt{T})$ regret in the \emph{stochastic} regime and $α$-regret of order $\widetilde{\mathcal O}(\sqrt{T})$ in the \emph{adversarial} regime, where $α$ depends on the feasibility margin of the corresponding offline problem. At the same time, our algorithm guarantees strict satisfaction of the budget constraints and $\widetilde{\mathcal O}(\sqrt{T})$ cumulative violation for the general ones. From a technical perspective, introducing general constraints alongside budgets precludes the use of standard budget-focus methods. While budget methods rely on a zero-consumption ``safe'' action to ensure feasibility, general constraints are much less ``aligned'' towards feasibility. We overcome these difficulties with a new analysis that exploits \emph{weak adaptivity} to get boundedness of the Lagrangian multipliers and best-of-both-world guarantees.

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arXiv2026

Multi-Armed Bandits With Best-Action Queries

We study \emph{multi-armed bandits} (MABs) augmented with \emph{best-action queries}, in which the learner may additionally query an oracle that reveals the best arm in the current round. This setting was recently characterized by Russo et al. [2024] in the \emph{full-feedback} model, where the learner observes the rewards of all arms after each round. They show that, in both \emph{stochastic} and \emph{adversarial} environments, $k$ best-action queries reduce the optimal $\widetilde{\mathcal{O}}(\sqrt{T})$ regret to $\widetilde{\mathcal{O}}(\min\{T/k,\sqrt{T}\})$. Whether this improvement extends to the more realistic \emph{bandit-feedback} model -- where the learner observes only the reward of the played arm -- was left as an open problem. We fully resolve this question. When rewards are stochastic but correlated among arms, we show that the full-feedback result does not carry over: any algorithm must incur regret at least $Ω(\sqrt{T-k})$. This lower bound directly extends to adversarial environments. On the positive side, we show that $\widetilde{\mathcal{O}}(\min\{T/k,\sqrt{T-k}\})$ regret is still achievable when rewards are stochastic and i.i.d., and establish a matching lower bound, up to logarithmic factors. Together, these results provide a complete characterization of the benefits of \emph{best-action queries} in the \emph{bandit-feedback} model.

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arXiv2026

Toward Optimal Regret in Robust Pricing: Decoupling Corruption and Time

We design the first regret guarantees for robust dynamic pricing that decouple the dependence on the corruption $C$ and the time horizon $T$. In dynamic pricing, a seller with unlimited supply of a good interacts with a stream of buyers over $ T $ rounds, with the goal of maximizing revenue. At each round $t$, the seller posts a price $p_t$, and the buyer purchases the good only if their unknown valuation $v^\star$ exceeds this price. The seller observes only the binary feedback $\mathbb{I} \left\{ p_t \leq v^\star \right\}$, indicating whether a sale occurred. In the \emph{robust} pricing setting, a malicious adversary is allowed to corrupt this feedback in at most $C$ rounds. Even if the learner knows the corruption $C$, the best known regret bound is $\mathcal{O}(C\log\log T)$ by Gupta et al. [2025]. This leaves as an open problem to ``decouple'' the dependence on $C$ and $T$. In this work, we resolve this open problem. In particular, we develop a robust variant of binary search that achieves regret $\mathcal{O}(C+\log T)$ when the corruption $C$ is known and $\mathcal{O}(C+\log^2 T)$ when the corruption is unknown.

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arXiv2026

Truly Adapting to Adversarial Constraints in Constrained MABs

We study the constrained variant of the \emph{multi-armed bandit} (MAB) problem, in which the learner aims not only at minimizing the total loss incurred during the learning dynamic, but also at controlling the violation of multiple \emph{unknown} constraints, under both \emph{full} and \emph{bandit feedback}. We consider a non-stationary environment that subsumes both stochastic and adversarial models and where, at each round, both losses and constraints are drawn from distributions that may change arbitrarily over time. In such a setting, it is provably not possible to guarantee both sublinear regret and sublinear violation. Accordingly, prior work has mainly focused either on settings with stochastic constraints or on relaxing the benchmark with fully adversarial constraints (\emph{e.g.}, via competitive ratios with respect to the optimum). We provide the first algorithms that achieve optimal rates of regret and \emph{positive} constraint violation when the constraints are stochastic while the losses may vary arbitrarily, and that simultaneously yield guarantees that degrade smoothly with the degree of adversariality of the constraints. Specifically, under \emph{full feedback} we propose an algorithm attaining $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ regret and $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ {positive} violation, where $C$ quantifies the amount of non-stationarity in the constraints. We then show how to extend these guarantees when only bandit feedback is available for the losses. Finally, when \emph{bandit feedback} is available for the constraints, we design an algorithm achieving $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ {positive} violation and $\widetilde{\mathcal{O}}(\sqrt{T}+C\sqrt{T})$ regret.

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arXiv2026

Replicable Constrained Bandits

Algorithmic \emph{replicability} has recently been introduced to address the need for reproducible experiments in machine learning. A \emph{replicable online learning} algorithm is one that takes the same sequence of decisions across different executions in the same environment, with high probability. We initiate the study of algorithmic replicability in \emph{constrained} MAB problems, where a learner interacts with an unknown stochastic environment for $T$ rounds, seeking not only to maximize reward but also to satisfy multiple constraints. Our main result is that replicability can be achieved in constrained MABs. Specifically, we design replicable algorithms whose regret and constraint violation match those of non-replicable ones in terms of $T$. As a key step toward these guarantees, we develop the first replicable UCB-like algorithm for \emph{unconstrained} MABs, showing that algorithms that employ the optimism in-the-face-of-uncertainty principle can be replicable, a result that we believe is of independent interest.

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arXiv2025

Data-Dependent Regret Bounds for Constrained MABs

This paper initiates the study of data-dependent regret bounds in constrained MAB settings. These bounds depend on the sequence of losses that characterize the problem instance. Thus, they can be much smaller than classical $\widetilde{\mathcal{O}}(\sqrt{T})$ regret bounds, while being equivalent to them in the worst case. Despite this, data-dependent regret bounds have been completely overlooked in constrained MAB settings. The goal of this paper is to answer the following question: Can data-dependent regret bounds be derived in the presence of constraints? We answer this question affirmatively in constrained MABs with adversarial losses and stochastic constraints. Specifically, our main focus is on the most challenging and natural settings with hard constraints, where the learner must ensure that the constraints are always satisfied with high probability. We design an algorithm with a regret bound consisting of two data-dependent terms. The first term captures the difficulty of satisfying the constraints, while the second one encodes the complexity of learning independently of the presence of constraints. We also prove a lower bound showing that these two terms are not artifacts of our specific approach and analysis, but rather the fundamental components that inherently characterize the complexities of the problem. Finally, in designing our algorithm, we also derive some novel results in the related (and easier) soft constraints settings, which may be of independent interest.

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arXiv2025

No-Regret Learning Under Adversarial Resource Constraints: A Spending Plan Is All You Need!

We study online decision making problems under resource constraints, where both reward and cost functions are drawn from distributions that may change adversarially over time. We focus on two canonical settings: $(i)$ online resource allocation where rewards and costs are observed before action selection, and $(ii)$ online learning with resource constraints where they are observed after action selection, under full feedback or bandit feedback. It is well known that achieving sublinear regret in these settings is impossible when reward and cost distributions may change arbitrarily over time. To address this challenge, we analyze a framework in which the learner is guided by a spending plan--a sequence prescribing expected resource usage across rounds. We design general (primal-)dual methods that achieve sublinear regret with respect to baselines that follow the spending plan. Crucially, the performance of our algorithms improves when the spending plan ensures a well-balanced distribution of the budget across rounds. We additionally provide a robust variant of our methods to handle worst-case scenarios where the spending plan is highly imbalanced. To conclude, we study the regret of our algorithms when competing against benchmarks that deviate from the prescribed spending plan.

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arXiv2025

Learning Adversarial MDPs with Stochastic Hard Constraints

We study online learning in constrained Markov decision processes (CMDPs) with adversarial losses and stochastic hard constraints, under bandit feedback. We consider three scenarios. In the first one, we address general CMDPs, where we design an algorithm attaining sublinear regret and cumulative positive constraints violation. In the second scenario, under the mild assumption that a policy strictly satisfying the constraints exists and is known to the learner, we design an algorithm that achieves sublinear regret while ensuring that constraints are satisfied at every episode with high probability. In the last scenario, we only assume the existence of a strictly feasible policy, which is not known to the learner, and we design an algorithm attaining sublinear regret and constant cumulative positive constraints violation. Finally, we show that in the last two scenarios, a dependence on the Slater's parameter is unavoidable. To the best of our knowledge, our work is the first to study CMDPs involving both adversarial losses and hard constraints. Thus, our algorithms can deal with general non-stationary environments subject to requirements much stricter than those manageable with existing ones, enabling their adoption in a much wider range of applications.

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arXiv2025

Best-of-Both-Worlds Policy Optimization for CMDPs with Bandit Feedback

We study online learning in constrained Markov decision processes (CMDPs) in which rewards and constraints may be either stochastic or adversarial. In such settings, Stradi et al.(2024) proposed the first best-of-both-worlds algorithm able to seamlessly handle stochastic and adversarial constraints, achieving optimal regret and constraint violation bounds in both cases. This algorithm suffers from two major drawbacks. First, it only works under full feedback, which severely limits its applicability in practice. Moreover, it relies on optimizing over the space of occupancy measures, which requires solving convex optimization problems, an highly inefficient task. In this paper, we provide the first best-of-both-worlds algorithm for CMDPs with bandit feedback. Specifically, when the constraints are stochastic, the algorithm achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation, while, when they are adversarial, it attains $\widetilde{\mathcal{O}}(\sqrt{T})$ constraint violation and a tight fraction of the optimal reward. Moreover, our algorithm is based on a policy optimization approach, which is much more efficient than occupancy-measure-based methods.

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arXiv2024

Optimal Strong Regret and Violation in Constrained MDPs via Policy Optimization

We study online learning in \emph{constrained MDPs} (CMDPs), focusing on the goal of attaining sublinear strong regret and strong cumulative constraint violation. Differently from their standard (weak) counterparts, these metrics do not allow negative terms to compensate positive ones, raising considerable additional challenges. Efroni et al. (2020) were the first to propose an algorithm with sublinear strong regret and strong violation, by exploiting linear programming. Thus, their algorithm is highly inefficient, leaving as an open problem achieving sublinear bounds by means of policy optimization methods, which are much more efficient in practice. Very recently, Muller et al. (2024) have partially addressed this problem by proposing a policy optimization method that allows to attain $\widetilde{\mathcal{O}}(T^{0.93})$ strong regret/violation. This still leaves open the question of whether optimal bounds are achievable by using an approach of this kind. We answer such a question affirmatively, by providing an efficient policy optimization algorithm with $\widetilde{\mathcal{O}}(\sqrt{T})$ strong regret/violation. Our algorithm implements a primal-dual scheme that employs a state-of-the-art policy optimization approach for adversarial (unconstrained) MDPs as primal algorithm, and a UCB-like update for dual variables.

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arXiv2024

Learning Constrained Markov Decision Processes With Non-stationary Rewards and Constraints

In constrained Markov decision processes (CMDPs) with adversarial rewards and constraints, a well-known impossibility result prevents any algorithm from attaining both sublinear regret and sublinear constraint violation, when competing against a best-in-hindsight policy that satisfies constraints on average. In this paper, we show that this negative result can be eased in CMDPs with non-stationary rewards and constraints, by providing algorithms whose performances smoothly degrade as non-stationarity increases. Specifically, we propose algorithms attaining $\tilde{\mathcal{O}} (\sqrt{T} + C)$ regret and positive constraint violation under bandit feedback, where $C$ is a corruption value measuring the environment non-stationarity. This can be $Θ(T)$ in the worst case, coherently with the impossibility result for adversarial CMDPs. First, we design an algorithm with the desired guarantees when $C$ is known. Then, in the case $C$ is unknown, we show how to obtain the same results by embedding such an algorithm in a general meta-procedure. This is of independent interest, as it can be applied to any non-stationary constrained online learning setting.

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