arXiv subjects

Matteo Castiglioni

Publications and source records attributed to Matteo Castiglioni.

At least 19 recordsLinked to original sources

arXiv2026

A Subsampling Theorem for Constraint Satisfaction Problems with Large Arity

Subsampling theorems for constraint satisfaction problems (CSPs) guarantee that the value of the CSP is approximately preserved after restricting it to small random subsets of variables. We provide the first subsampling theorem for CSPs, which requires a sample size that is polynomial in the arity $k$ and error $\varepsilon$, and polylogarithmic in the alphabet size $q$. This improves upon the subsampling theorem of Barak, Hardt, Holenstein, and Steurer (SODA '11), which achieves a polynomial dependency on $\varepsilon$ and polylogarithmic in $q$ only in the constant-arity regime. Our subsampling theorem has applications in interactive proofs and property testing. In interactive proofs, it provides a key missing ingredient for the proof of Aaronson, Impagliazzo, and Moshkovitz (CCC '14) that $\textsf{AM}(\textsf{poly})=\textsf{AM}$ (where $\textsf{AM}(k)$ is the class of languages decidable by Arthur-Merlin protocols with $k$ non-communicating Merlins with independent questions). In property testing, it yields the first one-sided tester for satisfiability with sample size polynomial in the arity $k$ and the error $\varepsilon^{-1}$, and polylogarithmic in the alphabet size $q$.

cs.DS↗

arXiv2026

A Unifying Framework for Quasi-Polynomial Optimization of Fixed-degree Polynomials

We study the simultaneous approximation of constant-degree polynomials over convex sets. For any family of $m$ degree-$d$ polynomials and any convex set ${H} \subseteq \mathbb{R}_{\ge0}^n$, we construct an $\epsilon$-Cover of the joint value set $\{(f_1(x), \dots, f_m(x)) : x \in {H}\}$ in the $\ell_\infty$-norm. This cover is of size $n^{O(\log(mn)/\epsilon^2)}$, provided the polynomials have constant range over the smallest $\ell_1$-ball inscribing ${H}$. Our approach extends classical net-based sparsifications for linear functions (e.g., Lipton, Markakis, and Mehta [2003]) to arbitrary families of constant-degree polynomials over general convex sets. We use a two-step scheme: first, we construct a quasi-polynomial pre-cover of the family on the smallest $\ell_1$-ball containing ${H}$ by using a concentration argument and leveraging a connection between Bernstein approximation and multinomial distributions; we then compress the pre-cover to ${H}$ by using a recursive degree reduction and feasibility programs anchored at points of the pre-cover. The existence of these covers immediately yields a unified framework for Quasi-Polynomial Time Approximation Schemes (QPTAS) across a wide range of a problems, including fixed-degree polynomial minimization over polyhedral sets, Constraint Satisfaction Problems (CSPs), Free Games, variational inequalities with polynomial operators (which implies guarantees for local Nash equilibria in polynomial games), and additive approximation for normalized densest $k$-subhypergraph on $O(1)$-uniform hypergraphs.

cs.DS↗

arXiv2026

Breaking the $T^{3/4}$ Barrier for Regret Minimization With Bi-Dimensional CDFs

We study regret minimization for learning CDF-related objectives of the form \[ g(x)\cdot\mathbb{P}_{X\sim\mathcal{D}}(X\le x), \] over $[0,1]^2$, where $g$ is a known Lipschitz function and $\mathcal{D}$ is an unknown distribution. At each round $t$, the learner selects a point $x_t$ and observes the binary feedback $\mathbb{I}(X_t\le x_t)$, where $X_t\sim\mathcal{D}$. We design an algorithm achieving regret $\widetilde{\mathcal{O}}(T^{7/10})$, improving over the previous best-known bound of $\widetilde{\mathcal{O}}(T^{3/4})$ and showing that the curse of dimensionality can be at least partially lifted for this class of objectives, though a gap remains with the $\Omega(T^{2/3})$ lower bound. As an application, our techniques yield the same $\widetilde{\mathcal{O}}(T^{7/10})$ regret bound for profit maximization in repeated bilateral trade with fixed prices.

cs.LG↗

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).

cs.GT↗

arXiv2026

Strict Fairness at What Cost? Envy-Free Contracts with Subsidies

We study algorithmic fair contract design, where a principal designs task-level contracts and fairly delegates a set of tasks to a set of agents. Prior work reveals a fairness-revenue dilemma: exact envy-free (EF) contracts may have an unbounded price of fairness (PoF), while approximate notions avoid this unboundedness only by weakening strict fairness. To address this dilemma, we propose a novel scheme, called {\it Envy-free Contracts with Subsidies} (EFS), in which the principal may additionally offer agent-specific subsidies to restore strict fairness. Our main technical result is a tight characterization of the price of fairness for EFS contracts. In sharp contrast to EF contracts, whose PoF can be unbounded, we show that the PoF of EFS contracts is $n^{n+O(1)}$, where $n$ is the number of agents. Moreover, EFS contracts can outperform EF contracts by an arbitrarily large factor in terms of the principal's revenue. Finally, we present the complexity landscape: computing optimal EFS contracts is NP-hard in general, whereas a polynomial-time algorithm exists when the number of tasks is constant.

cs.GT↗

arXiv2026

The Complexity of Min-Max Optimization for Quadratic Polynomials

We prove that computing approximate stationary points of min-max optimization over the hypercube is PPAD-hard for quadratic polynomials. This holds even when the polynomials are multilinear, each variable appears in at most three monomials, and the approximation factor is inverse polynomial. As a direct consequence, we obtain the first PPAD-hardness results for two-team zero-sum polymatrix games.

cs.CC↗

arXiv2026

Min-Max Optimization Requires Exponentially Many Queries

We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an $\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\varepsilon$ or $d$.

cs.DS↗

arXiv2026

Regret Minimization in Bilateral Trade With Perturbed Markets

We address the problem of maximizing Gain from Trade (GFT) in repeated buyer-seller exchanges subject to global budget balance constraints. While this problem is well-understood in purely adversarial and stochastic settings, these environments exhibit a sharp dichotomy: adversarial environments allow for no-regret learning against the best fixed-price mechanism, whereas stochastic environments allow for no-regret learning against the best distribution over prices that is budget balanced in expectation. This gap is significant, as policies balanced in expectation can increase the GFT by a multiplicative factor of two. In this work, we bridge these extremes by studying perturbed markets, where an underlying stochastic distribution is subject to an adversarial corruption $C$. We design an algorithm that adaptively scales with the level of corruption, achieving an $\tilde{\mathcal{O}}(T^{3/4}) + \mathcal{O}(C\log(T))$ regret bound against the best budget-balanced distribution over prices. Simultaneously, our algorithm maintains the worst-case $\tilde{\mathcal{O}}(T^{3/4})$ regret bound relative to a per-round budget-balanced baseline, ensuring optimality even in fully adversarial environments.

cs.GT↗

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 $\alpha$-regret of order $\widetilde{\mathcal O}(\sqrt{T})$ in the \emph{adversarial} regime, where $\alpha$ 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.

cs.GT↗

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 $\Omega(\sqrt{T-k})$. 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.

cs.LG↗

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.

cs.LG↗

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.

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↗

arXiv2026

The Sample Complexity of Uniform Approximation for Multi-Dimensional CDFs and Fixed-Price Mechanisms

We study the sample complexity of learning a uniform approximation of an $n$-dimensional cumulative distribution function (CDF) within an error $\epsilon > 0$, when observations are restricted to a minimal one-bit feedback. This serves as a counterpart to the multivariate DKW inequality under ''full feedback'', extending it to the setting of ''bandit feedback''. Our main result shows a near-dimensional-invariance in the sample complexity: we get a uniform $\epsilon$-approximation with a sample complexity $\frac{1}{\epsilon^3}{\log\left(\frac 1 \epsilon \right)^{\mathcal{O}(n)}}$ over a arbitrary fine grid, where the dimensionality $n$ only affects logarithmic terms. As direct corollaries, we provide tight sample complexity bounds and novel regret guarantees for learning fixed-price mechanisms in small markets, such as bilateral trade settings.

cs.LG↗

arXiv2026

A Stronger Benchmark for Online Bilateral Trade: From Fixed Prices to Distributions

We study online bilateral trade, where a learner facilitates repeated exchanges between a buyer and a seller to maximize the Gain From Trade (GFT), i.e., the social welfare. In doing so, the learner must guarantee not to subsidize the market. This constraint is usually imposed per round through Weak Budget Balance (WBB). Despite that, Bernasconi et al. [2024] show that a Global Budget Balance (GBB) constraint on the profit -- enforced over the entire time horizon -- can improve the GFT by a multiplicative factor of two. While this might appear to be a marginal relaxation, this implies that all existing WBB-focused algorithms suffer linear regret when measured against the GBB optimum. In this work, we provide the first algorithm to achieve sublinear regret against the GBB benchmark in stochastic environments under one-bit feedback. In particular, we show that when the joint distribution of valuations has a bounded density, our algorithm achieves $\widetilde{\mathcal{O}}(T^{3/4})$ regret. Our result shows that there is no separation between the one-dimensional problem of learning the optimal WBB price and the two-dimensional problem of learning the optimal GBB distribution over pairs of prices.

cs.GT↗

arXiv2026

The Complexity of Min-Max Optimization with Product Constraints

We study the computational complexity of the problem of computing local min-max equilibria of games with a nonconvex-nonconcave utility function $f$. From the work of Daskalakis, Skoulakis, and Zampetakis [DSZ21], this problem was known to be hard in the restrictive case in which players are required to play strategies that are jointly constrained, leaving open the question of its complexity under more natural constraints. In this paper, we settle the question and show that the problem is PPAD-hard even under product constraints and, in particular, over the hypercube.

cs.CC↗

arXiv2026

Learning in Bayesian Stackelberg Games With Unknown Follower's Types

We study online learning in Bayesian Stackelberg games, where a leader repeatedly interacts with a follower whose unknown private type is independently drawn at each round from an unknown probability distribution. The goal is to design algorithms that minimize the leader's regret with respect to always playing an optimal commitment computed with knowledge of the game. We consider, for the first time to the best of our knowledge, the most realistic case in which the leader does not know anything about the follower's types, i.e., the possible follower payoffs. This raises considerable additional challenges compared to the commonly studied case in which the payoffs of follower types are known. First, we prove a strong negative result: no-regret is unattainable under action feedback, i.e., when the leader only observes the follower's best response at the end of each round. Thus, we focus on the easier type feedback model, where the follower's type is also revealed. In such a setting, we propose a no-regret algorithm that achieves a regret of $\widetilde{O}(\sqrt{T})$, when ignoring the dependence on other parameters.

cs.GT↗

arXiv2025

Fair Team Contracts

A principal selects a team of agents for collaborating on a joint project. The principal aims to design a revenue-optimal contract that incentivizes the team of agents to exert costly effort while satisfying fairness constraints. We show that the optimal fair contract ensures that there is a minimum share, and every agent receives a linear contract weakly higher than the minimum share that is sufficient to incentivize them to exert costly effort. Leveraging this structural characterization, we design an FPTAS for additive success functions and a constant approximation algorithm for submodular success functions. Moreover, we show that the optimal fair contract can outperform the non-discriminatory one by a factor $1.445$, and this bound is tight for both additive and submodular success functions.

cs.GT↗