arXiv subjects

Gianmarco Genalti

Publications and source records attributed to Gianmarco Genalti.

10 recordsLinked to original sources

arXiv2026

Parameter-Free Heavy-Tailed Bandits

Heavy-tailed distributions arise naturally in sequential decision-making problems such as financial investment, online advertising, and network management, where rare but extreme outcomes can dominate performance. Heavy-tailed bandits model online decision-making in these settings by assuming only that rewards $X$ satisfy $\mathbb{E}[|X|^{1+ε}]\leq u$, for some tail exponent $ε\in(0,1]$ and moment bound $u<+\infty$. However, most existing regret minimization algorithms require these parameters to be known. This assumption is particularly restrictive in practice: $ε$ and $u$ govern the frequency and magnitude of rare events and are therefore precisely the quantities that are hardest to infer reliably from limited observations. Motivated by an open problem posed by Genalti and Metelli at COLT 2025, we resolve the assumption-free adaptation problem for heavy-tailed bandits and characterize the price in the regret of not knowing the tail parameters. We first study adaptation to the moment bound $u$ for a fixed tail exponent $ε$. We prove that every algorithm unaware of $u$, or of any upper bound on it, must obey a sharp trade-off between its distribution-dependent and distribution-free regret guarantees. We then introduce a scheduled-exploration algorithm that requires no knowledge of $u$ and matches the resulting adaptation frontier up to logarithmic factors. Finally, we show that the same algorithm can be instanced without knowing $ε$ by calibrating its exploration schedule to the endpoint $ε=1$. It achieves sublinear regret for every fixed $ε>0$, while no algorithm can guarantee sublinear regret uniformly over all $ε\in(0,1]$. Altogether, our results resolve the COLT open problem without additional distributional assumptions and provide a sharp characterization of the statistical cost of adapting to unknown heavy tails.

cs.LG↗

arXiv2026

Bridging Rested and Restless Bandits with Graph-Triggering: Rising and Rotting

Rested and Restless Bandits are two well-known bandit settings that are useful to model real-world sequential decision-making problems in which the expected reward of an arm evolves over time due to the actions we perform or due to the nature. In this work, we propose Graph-Triggered Bandits (GTBs), a unifying framework to generalize and extend rested and restless bandits. In this setting, the evolution of the arms' expected rewards is governed by a graph defined over the arms. An edge connecting a pair of arms $(i,j)$ represents the fact that a pull of arm $i$ triggers the evolution of arm $j$, and vice versa. Interestingly, rested and restless bandits are both special cases of our model for some suitable (degenerated) graph. As relevant case studies for this setting, we focus on two specific types of monotonic bandits: rising, where the expected reward of an arm grows as the number of triggers increases, and rotting, where the opposite behavior occurs. For these cases, we study the optimal policies. We provide suitable algorithms for all scenarios and discuss their theoretical guarantees, highlighting the complexity of the learning problem concerning instance-dependent terms that encode specific properties of the underlying graph structure.

stat.ML↗

arXiv2026

Online Packet Scheduling with Deadlines and Learning

Network routers that enforce Quality-of-Service (QoS) guarantees must decide, at every clock cycle, which expiring packet of information to transmit, even when the value of the packet is unknown until it is processed. We frame this problem as the Online Packet Scheduling with Deadlines (OPSD) problem under Partial Feedback: packets arrive at every clock cycle, with different deadlines, but the weights are only observed after execution. Under a stochastic assumption on the unknown weights, we explore different variants of the OPSD problem with bandit feedback. We establish a connection between our setting and the sleeping bandits problem, and set our learning goal to $α$-regret minimization. We provide algorithms with provable $α$-regret guarantees under different spans of slackness, distinguishing systems allowing for randomization and systems that do not. In every scenario, our algorithms achieve an $α$-regret upper bound of $\widetilde{\mathcal{O}}\left(\sqrt{KT}\right)$, matching the lower bound for the standard bandit setting. In the practically relevant case of $2$-bounded deadline instances, where the deadline is set at most one clock cycle away from the arrival, our deterministic algorithm achieves the provably tightest possible competitive ratio. Remarkably, when the number of distinct packet types $K\ge 2$ is finite, it is possible to break the well-established $Φ= \frac{1+\sqrt{5}}{2}$ competitive ratio barrier and attain a tighter competitive ratio $θ_K$ ranging in $[\sqrt{2}, Φ)$.

cs.LG↗

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.

cs.LG↗

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.

cs.LG↗

arXiv2025

Catoni-Style Change Point Detection for Regret Minimization in Non-Stationary Heavy-Tailed Bandits

Regret minimization in stochastic non-stationary bandits gained popularity over the last decade, as it can model a broad class of real-world problems, from advertising to recommendation systems. Existing literature relies on various assumptions about the reward-generating process, such as Bernoulli or subgaussian rewards. However, in settings such as finance and telecommunications, heavy-tailed distributions naturally arise. In this work, we tackle the heavy-tailed piecewise-stationary bandit problem. Heavy-tailed bandits, introduced by Bubeck et al., 2013, operate on the minimal assumption that the finite absolute centered moments of maximum order $1+ε$ are uniformly bounded by a constant $v<+\infty$, for some $ε\in (0,1]$. We focus on the most popular non-stationary bandit setting, i.e., the piecewise-stationary setting, in which the mean of reward-generating distributions may change at unknown time steps. We provide a novel Catoni-style change-point detection strategy tailored for heavy-tailed distributions that relies on recent advancements in the theory of sequential estimation, which is of independent interest. We introduce Robust-CPD-UCB, which combines this change-point detection strategy with optimistic algorithms for bandits, providing its regret upper bound and an impossibility result on the minimum attainable regret for any policy. Finally, we validate our approach through numerical experiments on synthetic and real-world datasets.

cs.LG↗

arXiv2024

A Best-of-Both-Worlds Algorithm for Constrained MDPs with Long-Term Constraints

We study online learning in episodic constrained Markov decision processes (CMDPs), where the learner aims at collecting as much reward as possible over the episodes, while satisfying some long-term constraints during the learning process. Rewards and constraints can be selected either stochastically or adversarially, and the transition function is not known to the learner. While online learning in classical (unconstrained) MDPs has received considerable attention over the last years, the setting of CMDPs is still largely unexplored. This is surprising, since in real-world applications, such as, e.g., autonomous driving, automated bidding, and recommender systems, there are usually additional constraints and specifications that an agent has to obey during the learning process. In this paper, we provide the first best-of-both-worlds algorithm for CMDPs with long-term constraints, in the flavor of Balseiro et al. (2023). Our algorithm is capable of handling settings in which rewards and constraints are selected either stochastically or adversarially, without requiring any knowledge of the underling process. Moreover, our algorithm matches state-of-the-art regret and constraint violation bounds for settings in which constraints are selected stochastically, while it is the first to provide guarantees in the case in which they are chosen adversarially.

cs.LG↗

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.

cs.LG↗

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.

cs.LG↗

arXiv2022

Dynamic Pricing with Volume Discounts in Online Settings

According to the main international reports, more pervasive industrial and business-process automation, thanks to machine learning and advanced analytic tools, will unlock more than 14 trillion USD worldwide annually by 2030. In the specific case of pricing problems-which constitute the class of problems we investigate in this paper-, the estimated unlocked value will be about 0.5 trillion USD per year. In particular, this paper focuses on pricing in e-commerce when the objective function is profit maximization and only transaction data are available. This setting is one of the most common in real-world applications. Our work aims to find a pricing strategy that allows defining optimal prices at different volume thresholds to serve different classes of users. Furthermore, we face the major challenge, common in real-world settings, of dealing with limited data available. We design a two-phase online learning algorithm, namely PVD-B, capable of exploiting the data incrementally in an online fashion. The algorithm first estimates the demand curve and retrieves the optimal average price, and subsequently it offers discounts to differentiate the prices for each volume threshold. We ran a real-world 4-month-long A/B testing experiment in collaboration with an Italian e-commerce company, in which our algorithm PVD-B-corresponding to A configuration-has been compared with human pricing specialists-corresponding to B configuration. At the end of the experiment, our algorithm produced a total turnover of about 300 KEuros, outperforming the B configuration performance by about 55%. The Italian company we collaborated with decided to adopt our algorithm for more than 1,200 products since January 2022.

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