Dichotomy of Feature Learning and Unlearning: Fast-Slow Analysis on Neural Networks with Stochastic Gradient Descent

The dynamics of gradient-based training in neural networks often exhibit nontrivial structures; hence, understanding them remains a central challenge in theoretical machine learning. In particular, a concept of feature unlearning, in which a neural network progressively loses previously learned features over long training, has gained attention. In this study, we consider the infinite-width limit of a two-layer neural network updated with a large-batch stochastic gradient, then derive differential equations with different time scales, revealing the mechanism and conditions for feature unlearning to occur. Specifically, we utilize the fast-slow dynamics: while an alignment of first-layer weights develops rapidly, the second-layer weights develop slowly. The direction of a flow on a critical manifold, determined by the slow dynamics, decides whether feature unlearning occurs. We give numerical validation of the result, and derive theoretical grounding and scaling laws of the feature unlearning. Our results yield the following insights: (i) the strength of the primary nonlinear term in data induces the feature unlearning, and (ii) an initial scale of the second-layer weights mitigates the feature unlearning. Technically, our analysis utilizes Tensor Programs and the singular perturbation theory.

High-Dimensional Limit of Stochastic Gradient Flow via Dynamical Mean-Field Theory

Modern machine learning models are typically trained via multi-pass stochastic gradient descent (SGD) with small batch sizes, and understanding their dynamics in high dimensions is of great interest. However, an analytical framework for describing the high-dimensional asymptotic behavior of multi-pass SGD with small batch sizes for nonlinear models is currently missing. In this study, we address this gap by analyzing the high-dimensional dynamics of a stochastic differential equation called a \emph{stochastic gradient flow} (SGF), which approximates multi-pass SGD in this regime. In the limit where the number of data samples $n$ and the dimension $d$ grow proportionally, we derive a closed system of low-dimensional and continuous-time equations and prove that it characterizes the asymptotic distribution of the SGF parameters. Our theory is based on the dynamical mean-field theory (DMFT) and is applicable to a wide range of models encompassing generalized linear models and two-layer neural networks. We further show that the resulting DMFT equations recover several existing high-dimensional descriptions of SGD dynamics as special cases, thereby providing a unifying perspective on prior frameworks such as online SGD and high-dimensional linear regression. Our proof builds on the existing DMFT technique for gradient flow and extends it to handle the stochasticity in SGF using tools from stochastic calculus.

Neuron Block Dynamics for XOR Classification with Zero-Margin

The ability of neural networks to learn useful features through stochastic gradient descent (SGD) is a cornerstone of their success. Most theoretical analyses focus on regression or on classification tasks with a positive margin, where worst-case gradient bounds suffice. In contrast, we study zero-margin nonlinear classification by analyzing the Gaussian XOR problem, where inputs are Gaussian and the XOR decision boundary determines labels. In this setting, a non-negligible fraction of data lies arbitrarily close to the boundary, breaking standard margin-based arguments. Building on Glasgow's (2024) analysis, we extend the study of training dynamics from discrete to Gaussian inputs and develop a framework for the dynamics of neuron blocks. We show that neurons cluster into four directions and that block-level signals evolve coherently, a phenomenon essential in the Gaussian setting where individual neuron signals vary significantly. Leveraging this block perspective, we analyze generalization without relying on margin assumptions, adopting an average-case view that distinguishes regions of reliable prediction from regions of persistent error. Numerical experiments confirm the predicted two-phase block dynamics and demonstrate their robustness beyond the Gaussian setting.

Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval

Spectral gradient methods, such as the Muon optimizer, modify gradient updates by preserving directional information while discarding scale, and have shown strong empirical performance in deep learning. We investigate the mechanisms underlying these gains through a dynamical analysis of a nonlinear phase retrieval model with anisotropic Gaussian inputs, equivalent to training a two-layer neural network with the quadratic activation and fixed second-layer weights. Focusing on a spiked covariance setting where the dominant variance direction is orthogonal to the signal, we show that gradient descent (GD) suffers from a variance-induced misalignment: during the early escaping stage, the high-variance but uninformative spike direction is multiplicatively amplified, degrading alignment with the true signal under strong anisotropy. In contrast, spectral gradient descent (SpecGD) removes this spike amplification effect, leading to stable alignment and accelerated noise contraction. Numerical experiments confirm the theory and show that these phenomena persist under broader anisotropic covariances.

Finite-Sample Inference for Sparsely Permuted Linear Regression

We study a noisy linear observation model with an unknown permutation called permuted/shuffled linear regression, where responses and covariates are mismatched and the permutation forms a discrete, factorial-size parameter. This unknown permutation is a key component of the data-generating process, yet its statistical investigation remains challenging due to its discrete nature. In this study, we develop a general statistical inference framework on the permutation and regression coefficients. First, we introduce a localization step that reduces the permutation space to a small candidate set building on recent advances in the repro samples method, whose miscoverage decays polynomially with the number of Monte Carlo samples. Then, based on this localized set, we provide statistical inference procedures: a conditional Monte Carlo test of permutation structures with valid finite-sample Type-I error control. We also develop coefficient inference that remains valid under alignment uncertainty of permutations. For computational purposes, we develop a linear assignment problem computable in polynomial time complexity and demonstrate that its solution asymptotically converges to that of the conventional least squares problem with large computational cost. Extensions to partially permuted designs and ridge regularization are also discussed. Extensive simulations and an application to Beijing air-quality data corroborate finite-sample validity, strong power to detect mismatches, and practical scalability.

Demystifying MaskGIT Sampler and Beyond: Adaptive Order Selection in Masked Diffusion

Masked diffusion models have shown promising performance in generating high-quality samples in a wide range of domains, but accelerating their sampling process remains relatively underexplored. To investigate efficient samplers for masked diffusion, this paper theoretically analyzes the MaskGIT sampler for image modeling, revealing its implicit temperature sampling mechanism. Through this analysis, we introduce the "moment sampler," an asymptotically equivalent but more tractable and interpretable alternative to MaskGIT, which employs a "choose-then-sample" approach by selecting unmasking positions before sampling tokens. In addition, we improve the efficiency of choose-then-sample algorithms through two key innovations: a partial caching technique for transformers that approximates longer sampling trajectories without proportional computational cost, and a hybrid approach formalizing the exploration-exploitation trade-off in adaptive unmasking. Experiments in image and text domains demonstrate our theory as well as the efficiency of our proposed methods, advancing both theoretical understanding and practical implementation of masked diffusion samplers.

SONA: Learning Conditional, Unconditional, and Mismatching-Aware Discriminator

Deep generative models have made significant advances in generating complex content, yet conditional generation remains a fundamental challenge. Existing conditional generative adversarial networks often struggle to balance the dual objectives of assessing authenticity and conditional alignment of input samples within their conditional discriminators. To address this, we propose a novel discriminator design that integrates three key capabilities: unconditional discrimination, matching-aware supervision to enhance alignment sensitivity, and adaptive weighting to dynamically balance all objectives. Specifically, we introduce Sum of Naturalness and Alignment (SONA), which employs separate projections for naturalness (authenticity) and alignment in the final layer with an inductive bias, supported by dedicated objective functions and an adaptive weighting mechanism. Extensive experiments on class-conditional generation tasks show that \ours achieves superior sample quality and conditional alignment compared to state-of-the-art methods. Furthermore, we demonstrate its effectiveness in text-to-image generation, confirming the versatility and robustness of our approach.

Precise Dynamics of Diagonal Linear Networks: A Unifying Analysis by Dynamical Mean-Field Theory

Diagonal linear networks (DLNs) are a tractable model that captures several nontrivial behaviors in neural network training, such as initialization-dependent solutions and incremental learning. These phenomena are typically studied in isolation, leaving the overall dynamics insufficiently understood. In this work, we present a unified analysis of various phenomena in the gradient flow dynamics of DLNs. Using Dynamical Mean-Field Theory (DMFT), we derive a low-dimensional effective process that captures the asymptotic gradient flow dynamics in high dimensions. Analyzing this effective process yields new insights into DLN dynamics, including loss convergence rates and their trade-off with generalization, and systematically reproduces many of the previously observed phenomena. These findings deepen our understanding of DLNs and demonstrate the effectiveness of the DMFT approach in analyzing high-dimensional learning dynamics of neural networks.

High-dimensional Nonparametric Contextual Bandit Problem

We consider the kernelized contextual bandit problem with a large feature space. This problem involves $K$ arms, and the goal of the forecaster is to maximize the cumulative rewards through learning the relationship between the contexts and the rewards. It serves as a general framework for various decision-making scenarios, such as personalized online advertising and recommendation systems. Kernelized contextual bandits generalize the linear contextual bandit problem and offers a greater modeling flexibility. Existing methods, when applied to Gaussian kernels, yield a trivial bound of $O(T)$ when we consider $\Omega(\log T)$ feature dimensions. To address this, we introduce stochastic assumptions on the context distribution and show that no-regret learning is achievable even when the number of dimensions grows up to the number of samples. Furthermore, we analyze lenient regret, which allows a per-round regret of at most $\Delta > 0$. We derive the rate of lenient regret in terms of $\Delta$.

Minimax Rates of Estimation for Optimal Transport Map between Infinite-Dimensional Spaces

We investigate the estimation of an optimal transport map between probability measures on an infinite-dimensional space and reveal its minimax optimal rate. Optimal transport theory defines distances within a space of probability measures, utilizing an optimal transport map as its key component. Estimating the optimal transport map from samples finds several applications, such as simulating dynamics between probability measures and functional data analysis. However, some transport maps on infinite-dimensional spaces require exponential-order data for estimation, which undermines their applicability. In this paper, we investigate the estimation of an optimal transport map between infinite-dimensional spaces, focusing on optimal transport maps characterized by the notion of $\gamma$-smoothness. Consequently, we show that the order of the minimax risk is polynomial rate in the sample size even in the infinite-dimensional setup. We also develop an estimator whose estimation error matches the minimax optimal rate. With these results, we obtain a class of reasonably estimable optimal transport maps on infinite-dimensional spaces and a method for their estimation. Our experiments validate the theory and practical utility of our approach with application to functional data analysis.