In real-world decision-making problems, one needs to pick among multiple policies the one that performs best while respecting economic constraints. This motivates the problem of constrained best-arm identification for bandit problems where every arm is a joint distribution of reward and cost. We investigate the general case where reward and cost are dependent. The goal is to accurately identify the arm with the highest mean reward among all arms whose mean cost is below a given threshold. We prove information-theoretic lower bounds on the sample complexity for three models: Gaussian with fixed covariance, Gaussian with unknown covariance, and non-parametric distributions of rectangular support. We propose a combination of a sampling and a stopping rule that correctly identifies the constrained best arm and matches the optimal sample complexities for each of the three models. Simulations demonstrate the performance of our algorithms.

In online inverse linear optimization, a learner observes time-varying sets of feasible actions and an agent's optimal actions, selected by solving linear optimization over the feasible actions. The learner sequentially makes predictions of the agent's true linear objective function, and their quality is measured by the regret, the cumulative gap between optimal objective values and those achieved by following the learner's predictions. A seminal work by Bärmann et al. (2017) obtained a regret bound of O( T), where T is the time horizon. Subsequently, the regret bound has been improved to O(n4 lnT) by Besbes et al. (2021, 2025) and to O(nlnT) by Gollapudi et al. (2021), where nis the dimension of the ambient space of objective vectors. However, these logarithmic-regret methods are highly inefficient when T is large, as they need to maintain regions specified by O(T) constraints, which represent possible locations of the true objective vector.

Conformal prediction provides a powerful framework for constructing prediction intervals with finite-sample guarantees, yet its robustness under distribution shifts remains a significant challenge. This paper addresses this limitation by modeling distribution shifts using Lévy-Prokhorov (LP) ambiguity sets, which capture both local and global perturbations. We provide a self-contained overview of LP ambiguity sets and their connections to popular metrics such as Wasserstein and Total Variation. We show that the link between conformal prediction and LP ambiguity sets is a natural one: by propagating the LP ambiguity set through the scoring function, we reduce complex high-dimensional distribution shifts to manageable onedimensional distribution shifts, enabling exact quantification of worst-case quantiles and coverage. Building on this analysis, we construct robust conformal prediction intervals that remain valid under distribution shifts, explicitly linking LP parameters to interval width and confidence levels. Experimental results on real-world datasets demonstrate the effectiveness of the proposed approach.

The signature kernel [10] is a recent state-of-the-art tool for analyzing highdimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via adaptively truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE [17], our approach employs tilewise Neumann-series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent Goursat PDEs via adaptively truncated local power series expansions and recursive propagation of boundary conditions along a directed graph in a topological ordering.

Flow-based generative models achieve state-of-the-art sample quality, but require the expensive solution of a differential equation at inference time. Flow map models, commonly known as consistency models, encompass many recent efforts to improve inference-time efficiency by learning the solution operator of this differential equation. Yet despite their promise, these models lack a unified description that clearly explains how to learn them efficiently in practice. Here, building on the methodology proposed in Boffi et al. (2024), we present a systematic algorithmic framework for directly learning the flow map associated with a flow or diffusion model. By exploiting a relationship between the velocity field underlying a continuous-time flow and the instantaneous rate of change of the flow map, we show how to convert any distillation scheme into a direct training algorithm via self-distillation, eliminating the need for pre-trained teachers. We introduce three algorithmic families based on different mathematical characterizations of the flow map: Eulerian, Lagrangian, and Progressive methods, which we show encompass and extend all known distillation and direct training schemes for consistency models. We find that the novel class of Lagrangian methods, which avoid both spatial derivatives and bootstrapping from small steps by design, achieve significantly more stable training and higher performance than more standard Eulerian and Progressive schemes. Our methodology unifies existing training schemes under a single common framework and reveals new design principles for accelerated generative modeling.

The problem of optimally covering a given compact subset of RN with a preassigned number n of Euclidean metric balls has a long-standing history and it is well-recognized to be computationally hard. This article establishes a numerically viable algorithm for obtaining optimal covers of compact sets via two key contributions. The first is a foundational result establishing Lipschitz continuity of the marginal function of a certain parametric non-convex maximization problem in the optimal covering problem, and it provides the substrate for numerical gradient algorithms to be employed in this context. The second is an adaptation of a stochastically smoothed numerical gradient-based (zeroth-order) algorithm for a non-convex minimization problem, that, equipped with randomized restarts, spurs global convergence to an optimal cover. Several numerical experiments with complicated nonconvex compact sets demonstrate the excellent performance of our techniques.

We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on n data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order log(n)neurons suffices for global convergence with high probability. Our analysis uses a Polyak-Łojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of 1n. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders 1n and 1 n, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order 1log(n).

Fairness-accuracy trade-offs are a central concern in the deployment of fairness-aware machine learning methods. When sensitive attributes are unavailable at inference time-the so called unawareness setting, principled methods for obtaining accurate predictions under relaxed fairness constraints are largely missing. In this work, we address this gap by formulating regression under a demographic parity penalty as an optimal transport problem. Our framework unifies both the \emph{aware} and \emph{unaware} settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the choice of penalty reflects fundamentally different fairness philosophies: the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals. Building on these theoretical characterizations, we propose an algorithm that is simple to implement, computationally efficient, and consistently matches or outperforms state-of-the-art baselines on real-world benchmarks.

The ability of deep neural networks to learn hierarchical features is widely regarded as a key mechanism underlying their success in high-dimensional learning. Existing theory partially supports this view by establishing approximation rates based on parameter counts and sample complexity guarantees for compositional models without incurring the curse of dimensionality (CoD). To study overparameterized regimes, where the number of parameters exceeds the sample size, we develop a framework that measures complexity via the parameter norm. Within this approach, we establish approximation rates and excess risk bounds for learning sparse compositional functions whose compositional structure is represented by directed acyclic graphs (DAGs), using Frobenius norm-constrained deep neural networks. Our results have broad applicability since every function that is efficiently Turing computable admits sparse compositional representations. In particular, we cover a range of representative models, including multi-index models, binary tree structures, and general compositional architectures. The rates we derive show that deep networks can exploit the compositional structure of the target functions, effectively avoiding the CoD through hierarchical representations.

We investigate the global optimization of the objective function arising in continuous sparse regression, specifically the Beurling LASSO (BLASSO), over the space of measures. While Conic Particle Gradient Descent (CPGD) methods are computationally efficient, they may become trapped in local minima due to the non-convexity of the parameterization. To overcome this limitation, we introduce Fast Spawn\&Prune (FS\&P), a stochastic algorithm that extends FastPart introduced in De Castro et al. (2025) and combines CPGD with a birth-death process. The birth mechanism ensures asymptotic global exploration by introducing particles in regions where first-order optimality conditions are violated, while the death process preserves computational efficiency by pruning non-informative particles. We provide the first theoretical guarantee of global convergence for this class of discrete-time stochastic algorithms, without requiring exponentially large initializations. Furthermore, we derive explicit convergence rates for the excess risk, which scale as $\mathcal{O}\big(\left(\log K / K\right)^{\frac{1}{2(2+d)}}\big)$, where $K$ denotes the number of iterations and d the dimension of the domain, thereby quantifying the trade-off between global exploration and local refinement. Moreover, the sample complexity is $\mathcal{O}\big(N^{-\frac{1}{4(2+d)}}\big)$ (up to logarithmic factors). We also propose a horizon-free variant that does not require prior knowledge of the iteration budget.