Adafactor is an early memory-efficient optimization algorithm proposed as an alternative to Adam. By eliminating first-order momentum and employing a rank-1 matrix factorization to approximate the second-moment matrix, Adafactor achieves near-zero memory overhead compared to traditional gradient descent methods. Despite its practical suitability for large-scale training tasks where memory efficiency is critical, its theoretical convergence analysis remains unexplored, largely due to the challenges posed by its matrix factorization and update clipping mechanisms. In this work, we provide a convergence analysis of Adafactor for non-convex smooth optimization. We establish optimal convergence rates (up to logarithmic factors) for finding stationary points in both deterministic and stochastic settings, the latter under sub-Gaussian noise. Central to our analysis is viewing Adafactor as an approximation of Adam, and the use of a new proxy step-size to approximate the unique adaptive step-size induced by Adafactor's matrix factorization and update clipping, along with an induction argument to control the gradient magnitude. Our findings may theoretically suggest that involving rank-1 matrix approximation of the second-moment matrix in Adam does not fundamentally hinder the convergence.

Large-scale neural language models exhibit remarkable performance in in-context learning: the ability to learn and reason about the input context on the fly. This work studies in-context counterfactual reasoning in language models, that is, the ability to predict consequences of a hypothetical scenario. We focus on a well-defined, synthetic linear regression task that requires noise abduction. Accurate prediction is based on (1) inferring an unobserved latent concept and (2) copying contextual noise from factual observations. We show that language models are capable of counterfactual reasoning. Further, we enhance existing identifiability results and reduce counterfactual reasoning for a broad class of functions to a transformation on in-context observations.

PU (Positive Unlabeled) learning is a variant of supervised classification learning in which the only labels revealed to the learner are of positively labeled instances. PU learning arises in many real-world applications. Most existing work relies on the simplifying assumptions that the positively labeled training data is drawn from the restriction of the data generating distribution to positively labeled instances and/or that the proportion of positively labeled points (a.k.a. the class prior) is known apriori to the learner. This paper provides a theoretical analysis of the statistical complexity of PU learning under a wider range of setups. Unlike most prior work, our study does not assume that the class prior is known to the learner. We prove upper and lower bounds on the required sample sizes (of both the positively labeled and the unlabeled samples).

Large Language Models (LLMs) have demonstrated the ability to tackle increasingly complex tasks through advanced reasoning, long-form content generation, and tool use. Solving these tasks often involves long inference-time computations. In human problem solving, a common strategy to expedite work is collaboration: by dividing the problem into sub-tasks, exploring different strategies concurrently, etc. Recent research has shown that LLMs can also operate in parallel by implementing explicit cooperation frameworks, such as voting mechanisms or the explicit creation of independent sub-tasks that can be executed in parallel. However, each of these frameworks may not be suitable for all types of tasks, which can hinder their applicability.

Instruction-Tuning (IT) was recently found the impressive data efficiency in posttraining large language models (LLMs). While the pursuit of efficiency predominantly focuses on sequence-level curation, often overlooking the nuanced impact of critical tokens and the inherent risks of token noise and biases. Drawing inspiration from bi-level coreset selection, our work provides the principled view of the motivation behind selecting instructions' responses. It leads to our approach Quadratic Coreset Selection (QCS) that reconciles sequence-level and token-level influence contributions, deriving more expressive LLMs with established theoretical result. Despite the original QCS framework challenged by prohibitive computation from inverted LLM-scale Hessian matrices, we overcome this barrier by proposing a novel QCS probabilistic variant, which relaxes the original formulation through re-parameterized densities. This innovative solver is efficiently learned using hierarchical policy gradients without requiring back-propagation, achieving provable convergence and certified asymptotic equivalence to the original objective. Our experiments demonstrate QCS's superior sequence-level data efficiency and reveal how strategically leveraging token-level influence elevates the performance ceiling of data-efficient IT. Furthermore, QCS's adaptability is showcased through its successes in regular IT and challenging targeted IT scenarios, particularly in the cases of free-form complex instruction-following and CoT reasoning. They underscore QCS's potential for a wide array of versatile post-training applications.

Contrastive learning involves learning representations via a loss function that encourages each (unlabeled) sample to be far from other samples, but close to its own . In this paper, we aim to understand why this simple idea performs remarkably well, by theoretically analyzing it for a simple, natural problem setting: dimensionality reduction in Gaussian Mixture Models (GMMs). Note that the standard GMM setup lacks the concept of augmentations. We study an intuitive extension: we define the pair of data sample and its augmentation as a coupled random draw from the GMM such that the marginal over the noisy augmentation is towards the component of the data sample. For this setup, we show that vanilla contrastive loss, e.g., InfoNCE, is able to find the lower-dimensional subspace even when the Gaussian components are non-isotropic. In particular, we show that InfoNCE can match the performance of a fully supervised algorithm, e.g., LDA, (where each data point is labeled with the mixture component it comes from) even when the augmentations are noisy. We further extend our setup to the multi-modal case, and develop a GMM-like setting to study the contrastive CLIP loss. We corroborate our theoretical with real-data experiments on CIFAR100; representations learned by InfoNCE loss match the performance of LDA on clustering metrics.

We introduce, a novel method for reconstructing relightable, high-resolution head avatars that can be rendered from novel viewpoints at interactive rates. Therefore, we propose a new low-cost light stage capture setup, tailored specifically towards capturing faces. Using this setup, we collect a novel dataset consisting of diverse multi-view sequences of numerous subjects under varying illumination conditions and facial expressions. By leveraging our new dataset, we introduce a new relightable avatar representation based on 3D Gaussian primitives that we animate with a parametric head model and an expression-dependent dynamics module. We propose a new hybrid neural shading approach, combining a neural diffuse BRDF with an analytical specular term. Our method reconstructs disentangled materials from our dynamic light stage recordings and enables all-frequency relighting of our avatars with both point lights and environment maps. In addition, our avatars can easily be animated and controlled from monocular videos. We validate our approach in extensive experiments on our dataset, where we consistently outperform existing state-of-the-art methods in relighting and reenactment by a significant margin.

Soccer officials already rely on cameras to see who's offside and who sent the ball out of bounds. But during this World Cup, refs will use digital twins of each player to view plays from every angle. At the 2026 World Cup, the refs on the field and the officials on the sidelines will be able to use an abundance of tech to help call penalties, spot offside violations, and make other consequential decisions. The video assistant referee system, known as VAR, and the semi-automated offside technology (SAOT) have been used in soccer for years. But the setup at this summer's World Cup represents some of the most advanced uses of adjudication tech to date--not just in soccer, but across all high-level sports.

We consider a decentralized setup in which the participants collaboratively train and serve a large neural network, and where each participant only processes a subset of the model. In this setup, we explore the possibility of unmaterializable weights, where a full weight set is never available to any one participant. We introduce Unextractable Protocol Models (UPMs): a training and inference framework that leverages the sharded model setup to ensure model shards (i.e.,, subsets) held by participants are incompatible at different time steps. UPMs periodically inject time-varying, random, invertible transforms at participant boundaries; preserving the overall network function yet rendering cross-time assemblies incoherent. On Qwen-2.5-0.5B and Llama-3.2-1B, 10 000 transforms leave FP32 perplexity unchanged ($\Delta$PPL$ < 0.01$; Jensen-Shannon drift $

We introduce FLASH-MAX, a shallow, exact-by-construction neural network architecture for predicting homogeneous electromagnetic fields from sparse pointwise observations. Each hidden neuron represents a separate exact solution to Maxwell's equations, so that the network satisfies the governing equations symbolically by construction and can be trained end-to-end from sparse data within seconds. We prove a universal approximation result showing that this exact model class remains universal on arbitrary domains. FLASH-MAX reaches sub-1% relative validation error from about 1K sparse pointwise observations in seconds, all while maintaining a zero PDE residual, and keeps single-digit errors even for only 100 observations sampled from 3D space. These results suggest that moving governing structure from the loss into the hypothesis class can dramatically improve the trade-off between precision and optimization speed in scientific machine learning.