The problem of benign overfitting asks whether it is possible for a model to perfectly fit noisy training data and still generalize well. We study benign overfitting in two-layer leaky ReLU networks trained with the hinge loss on a binary classification task. We consider input data which can be decomposed into the sum of a common signal and a random noise component, which lie on subspaces orthogonal to one another. We characterize conditions on the signal to noise ratio (SNR) of the model parameters giving rise to benign versus non-benign, or harmful, overfitting: in particular, if the SNR is high then benign overfitting occurs, conversely if the SNR is low then harmful overfitting occurs. We attribute both benign and non-benign overfitting to an approximate margin maximization property and show that leaky ReLU networks trained on hinge loss with Gradient Descent (GD) satisfy this property. In contrast to prior work we do not require near orthogonality conditions on the training data: notably, for input dimension $d$ and training sample size $n$, while prior work shows asymptotically optimal error when $d = \Omega(n^2 \log n)$, here we require only $d = \Omega\left(n \log \frac{1}{\epsilon}\right)$ to obtain error within $\epsilon$ of optimal.

Maximum entropy (Maxent) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, Maxent models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for Maxent models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical Maxent models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth Maxent models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth Maxent models efficiently. For Maxent models with discrete probability distribution of $n$ elements built from samples, each containing $m$ features, the stepsize parameters estimation and iterations in our algorithms scale on the order of $O(mn)$ operations and can be trivially parallelized. Moreover, the strong $\ell_{1}$ convexity of the Kullback--Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the arts by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.

Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to infer response variables at points, where no explanatory data were observed. The data considered here are located in compact sets in higher dimensions and the paper addresses approximations of the kernel itself. The new approach considers Taylor series approximations of radial kernel functions. For the Gauss kernel on the unit cube, the paper establishes an upper bound of the associated eigenfunctions, which grows only polynomially with respect to the index. The novel approach substantiates smaller regularization parameters than considered in the literature, overall leading to better approximations. This improvement confirms low rank approximation methods such as the Nystr\"om method.

Cross-silo federated learning (FL) allows data owners to train accurate machine learning models by benefiting from each others private datasets. Unfortunately, the model accuracy benefits of collaboration are often undermined by privacy defenses. Therefore, to incentivize client participation in privacy-sensitive domains, a FL protocol should strike a delicate balance between privacy guarantees and end-model accuracy. In this paper, we study the question of when and how a server could design a FL protocol provably beneficial for all participants. First, we provide necessary and sufficient conditions for the existence of mutually beneficial protocols in the context of mean estimation and convex stochastic optimization. We also derive protocols that maximize the total clients' utility, given symmetric privacy preferences. Finally, we design protocols maximizing end-model accuracy and demonstrate their benefits in synthetic experiments.

While many Machine Learning methods were developed or transposed on Riemannian manifolds to tackle data with known non Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on these spaces is the Wasserstein distance which suffers from a heavy computational burden. On Euclidean spaces, a popular alternative is the Sliced-Wasserstein distance, which leverages a closed-form solution of the Wasserstein distance in one dimension, but which is not readily available on manifolds. In this work, we derive general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices. Then, we propose different applications. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows.

Multi-source domain adaptation aims to reduce performance degradation when applying machine learning models to unseen domains. A fundamental challenge is devising the optimal strategy for feature selection. Existing literature is somewhat paradoxical: some advocate for learning invariant features from source domains, while others favor more diverse features. To address the challenge, we propose a statistical framework that distinguishes the utilities of features based on the variance of their correlation to label $y$ across domains. Under our framework, we design and analyze a learning procedure consisting of learning approximately shared feature representation from source tasks and fine-tuning it on the target task. Our theoretical analysis necessitates the importance of learning approximately shared features instead of only the strictly invariant features and yields an improved population risk compared to previous results on both source and target tasks, thus partly resolving the paradox mentioned above. Inspired by our theory, we proposed a more practical way to isolate the content (invariant+approximately shared) from environmental features and further consolidate our theoretical findings.

The last decade has seen tremendous progress in our ability to generate realistic-looking data, be it images, text, audio, or video. Here, we discuss the closely related problem of quantifying realism, that is, designing functions that can reliably tell realistic data from unrealistic data. This problem turns out to be significantly harder to solve and remains poorly understood, despite its prevalence in machine learning and recent breakthroughs in generative AI. Drawing on insights from algorithmic information theory, we discuss why this problem is challenging, why a good generative model alone is insufficient to solve it, and what a good solution would look like. In particular, we introduce the notion of a universal critic, which unlike adversarial critics does not require adversarial training. While universal critics are not immediately practical, they can serve both as a North Star for guiding practical implementations and as a tool for analyzing existing attempts to capture realism.

Data sources in contemporary machine learning applications are often heterogeneous, leading to potential distribution shifts Sugiyama and Kawanabe [2012], Shen et al. [2021]. This is a particularly relevant problem in computer vision Csurka [2017], healthcare Zhang et al. [2021], finance, Earth and climate sciences [Tuia et al., 2016, Kellenberger et al., 2021] and social sciences, as variations in data patterns can significantly impact model performance and generalisation in the out-of-distribution (OOD) setting, also referred to as domain generalisation [Shen et al., 2021, Zhou et al., 2023]. Various frameworks have been proposed to formally address the emergence of distribution shifts during the testing phase Peters et al. [2016], Arjovsky et al. [2020]. In cases where the data distribution is entailed by a Structural Causal Model (SCM) Peters et al. [2017], one can consider distribution shifts arising from intervention on specific variables of the SCM. Notably, the Instrumental Variable (IV) regression exhibits robustness to arbitrarily strong interventions [Bowden and Turkington, 1990]. However, pursuing algorithms robust to strong interventions may be overly conservative, especially when prior knowledge is available regarding the intervention strength that generates the distribution shift. Anchor Regression (AR) addresses this challenge by explicitly considering interventions on exogenous variables up to a specified strength [Rothenhäusler et al., 2018].

Recently, neural networks (NN) have made great strides in combinatorial optimization. However, they face challenges when solving the capacitated arc routing problem (CARP) which is to find the minimum-cost tour covering all required edges on a graph, while within capacity constraints. In tackling CARP, NN-based approaches tend to lag behind advanced metaheuristics, since they lack directed arc modeling and efficient learning methods tailored for complex CARP. In this paper, we introduce an NN-based solver to significantly narrow the gap with advanced metaheuristics while exhibiting superior efficiency. First, we propose the direction-aware attention model (DaAM) to incorporate directionality into the embedding process, facilitating more effective one-stage decision-making. Second, we design a supervised reinforcement learning scheme that involves supervised pre-training to establish a robust initial policy for subsequent reinforcement fine-tuning. It proves particularly valuable for solving CARP that has a higher complexity than the node routing problems (NRPs). Finally, a path optimization method is proposed to adjust the depot return positions within the path generated by DaAM. Experiments illustrate that our approach surpasses heuristics and achieves decision quality comparable to state-of-the-art metaheuristics for the first time while maintaining superior efficiency.

While exposure to diverse viewpoints may reduce polarization, it can also have a backfire effect and exacerbate polarization when the discussion is adversarial. Here, we examine the question whether intergroup interactions around important events affect polarization between majority and minority groups in social networks. We compile data on the religious identity of nearly 700,000 Indian Twitter users engaging in COVID-19-related discourse during 2020. We introduce a new measure for an individual's group conformity based on contextualized embeddings of tweet text, which helps us assess polarization between religious groups. We then use a meta-learning framework to examine heterogeneous treatment effects of intergroup interactions on an individual's group conformity in the light of communal, political, and socio-economic events. We find that for political and social events, intergroup interactions reduce polarization. This decline is weaker for individuals at the extreme who already exhibit high conformity to their group. In contrast, during communal events, intergroup interactions can increase group conformity. Finally, we decompose the differential effects across religious groups in terms of emotions and topics of discussion. The results show that the dynamics of religious polarization are sensitive to the context and have important implications for understanding the role of intergroup interactions.