Learning-to-Defer (L2D) methods route each query either to a predictive model or to external experts. While existing work studies this problem in batch settings, real-world deployments require handling streaming data, changing expert availability, and shifting expert distribution. We introduce the first online L2D algorithm for multiclass classification with bandit feedback and a dynamically varying pool of experts. Our method achieves regret guarantees of $O((n+n_e)T^{2/3})$ in general and $O((n+n_e)\sqrt{T})$ under a low-noise condition, where $T$ is the time horizon, $n$ is the number of labels, and $n_e$ is the number of distinct experts observed across rounds. The analysis builds on novel $\mathcal{H}$-consistency bounds for the online framework, combined with first-order methods for online convex optimization. Experiments on synthetic and real-world datasets demonstrate that our approach effectively extends standard Learning-to-Defer to settings with varying expert availability and reliability.

Wide neural networks in the feature-learning regime drive modern deep learning, and yet they remain far less studied than their kernel-regime counterparts. We consider a critical yet under-explored difference between these two regimes: the regulariser and prior implied by gradient flow training. This canonical regularisation property is well-studied in kernel regime networks -- of all the infinite global minima, gradient flow selects exactly the vanishing ridge solution -- and underpins the celebrated NN-GP correspondence, precisely allowing the modelling of noise during training. However, we prove ridge regularisation biases gradient flow in feature-learning regime networks, even in the infinitesimal limit of vanishing regularisation. Over training, ridge distorts the inductive bias of the network, with a particular damage done to pretrained networks where the implicit prior is informative. We resolve this by axiomatising the canonical regulariser as a regime-agnostic function-space energy and lift, which uniquely identifies ridge in the kernel regime, and crucially generalises to the feature-learning regime. By studying the Riemannian geometry of feature-learning networks, we derive geodesic ridge from our framework, generalising ridge to the feature-learning regime. Correspondingly, we prove the canonical function-space prior is a Riemannian Gibbs Process, generalising the more familiar Gaussian Process. As a practical contribution, we propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge, revealing a deep relationship between early stopping and canonical regularisation across learning regimes. Finally, we demonstrate the consequences of our theory empirically on both image processing and NLP transfer-learning problems.

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.

Graph neural networks (GNNs) are the de facto standard deep learning architectures for machine learning on graphs. This has led to a large body of work analyzing the capabilities and limitations of these models, particularly pertaining to their representation and extrapolation capacity. We offer a novel theoretical perspective on the representation and extrapolation capacity of GNNs, by answering the question: how do GNNs behave as the number of graph nodes become very large? Under mild assumptions, we show that when we draw graphs of increasing size from the Erd os-Rényi model, the probability that such graphs are mapped to a particular output by a class of GNN classifiers tends to either zero or to one. This class includes the popular graph convolutional network architecture. The result establishes'zero-one laws' for these GNNs, and analogously to other convergence laws, entails theoretical limitations on their capacity. We empirically verify our results, observing that the theoretical asymptotic limits are evident already on relatively small graphs.

There is growing interest in improving our algorithmic understanding of fundamental statistical problems such as mean estimation, driven by the goal of understanding the fundamental limits of what we can extract from limited and valuable data. The state of the art results for mean estimation in R are 1) the optimal sub-Gaussian mean estimator by [Lee and Valiant, 2022], attaining the optimal sub-Gaussian error constant for all distributions with finite but unknown variance, and 2) the analysis of the median-of-means algorithm by [Bubeck, Cesa-Bianchi and Lugosi, 2013] and a matching lower bound by [Devroye, Lerasle, Lugosi, and Oliveira, 2016], characterizing the big-O optimal errors for distributions that have tails heavy enough that only a 1 + α moment exists for some α (0,1). Both of these results, however, are optimal only in the worst case. Motivated by the recent effort in the community to go "beyond the worst-case analysis" of algorithms, we initiate the fine-grained study of the mean estimation problem: Is it possible for algorithms to leverage beneficial features/quirks of their input distribution to beat the sub-Gaussian rate, without explicit knowledge of these features? We resolve this question, finding an unexpectedly nuanced answer: "Yes in limited regimes, but in general no". Given a distribution p, assuming only that it has a finite mean and absent any additional assumptions, we show how to construct a distribution qn,δ such that the means of p and q are well-separated, yet p and q are impossible to distinguish with n samples with probability 1 δ, and q further preserves the finiteness of moments of p.

Location estimation is one of the most basic questions in parametric statistics. Suppose we have a known distribution density f, and we get ni.i.d.

We consider computationally-efficient estimation of population parameters when observations are subject to missing data. In particular, we consider estimation under the realizable contamination model of missing data in which an $ε$ fraction of the observations are subject to an arbitrary (and unknown) missing not at random (MNAR) mechanism. When the true data is Gaussian, we provide evidence towards statistical-computational gaps in several problems. For mean estimation in $\ell_2$ norm, we show that in order to obtain error at most $ρ$, for any constant contamination $ε\in (0, 1)$, (roughly) $n \gtrsim d e^{1/ρ^2}$ samples are necessary and that there is a computationally-inefficient algorithm which achieves this error. On the other hand, we show that any computationally-efficient method within certain popular families of algorithms requires a much larger sample complexity of (roughly) $n \gtrsim d^{1/ρ^2}$ and that there exists a polynomial time algorithm based on sum-of-squares which (nearly) achieves this lower bound. For covariance estimation in relative operator norm, we show that a parallel development holds. Finally, we turn to linear regression with missing observations and show that such a gap does not persist. Indeed, in this setting we show that minimizing a simple, strongly convex empirical risk nearly achieves the information-theoretic lower bound in polynomial time.

This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a $\mathcal{O}(1/k)$ best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the $n$-finite-sum and expectation settings are $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, respectively, when employing loopless-SVRG or SAGA, where $ε$ is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ for the $n$-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.