The capacity of deep learning models is often large enough to both learn the underlying statistical signal and overfit to noise in the training set. This noise memorization can be harmful especially for data with a low signal-to-noise ratio (SNR), leading to poor generalization. Inspired by prior observations that label noise provides implicit regularization that improves generalization, in this work, we investigate whether introducing label noise to the gradient updates can enhance the test performance of neural network (NN) in the low SNR regime. Specifically, we consider training a two-layer NN with a simple label noise gradient descent (GD) algorithm, in an idealized signal-noise data setting. We prove that adding label noise during training suppresses noise memorization, preventing it from dominating the learning process; consequently, label noise GD enjoys rapid signal growth while the overfitting remains controlled, thereby achieving good generalization despite the low SNR. In contrast, we also show that NN trained with standard GD tends to overfit to noise in the same low SNR setting and establish a non-vanishing lower bound on its test error, thus demonstrating the benefit of introducing label noise in gradient-based training.

Task arithmetic has emerged as a simple yet powerful technique for model merging, enabling the combination of multiple finetuned models into one. Despite its empirical success, a clear theoretical explanation of why and when it works is lacking. This paper provides a rigorous theoretical foundation for task arithmetic by establishing a connection between task vectors and gradients of the task losses. We show that under standard gradient descent, a task vector generated from one epoch of finetuning is exactly equivalent to the negative gradient of the loss, scaled by the learning rate. For the practical multi-epoch setting, we prove that this equivalence holds approximately, with a second-order error term that we explicitly bound for feed-forward networks. Our empirical analysis across seven vision benchmarks corroborates our theory, demonstrating that the first-epoch gradient dominates the finetuning trajectory in both norm and direction. A key implication is that merging models finetuned for only a single epoch often yields performance comparable to merging fully converged models. These findings reframe task arithmetic as a form of approximate multitask learning, providing a clear rationale for its effectiveness and highlighting the critical role of early training dynamics in model merging.

Despite growing empirical evidence of bias amplification in machine learning, its theoretical foundations remain poorly understood. We develop a formal framework for majority-minority learning tasks, showing how standard training can favor majority groups and produce stereotypical predictors that neglect minority-specific features. Assuming population and variance imbalance, our analysis reveals three key findings: (i) the close proximity between ``full-data'' and stereotypical predictors, (ii) the dominance of a region where training the entire model tends to merely learn the majority traits, and (iii) a lower bound on the additional training required. Our results are illustrated through experiments in deep learning for tabular and image classification tasks.

The interplay between optimization and privacy has become a central theme in privacy-preserving machine learning. Noisy stochastic gradient descent (SGD) has emerged as a cornerstone algorithm, particularly in large-scale settings. These variants of gradient methods inject carefully calibrated noise into each update to achieve differential privacy, the gold standard notion of rigorous privacy guarantees. Prior work primarily provides various bounds on statistical risk and privacy loss for noisy SGD, yet the \textit{exact} behavior of the process remains unclear, particularly in high-dimensional settings. This work leverages a diffusion approach to analyze noisy SGD precisely, providing a continuous-time perspective that captures both statistical risk evolution and privacy loss dynamics in high dimensions. Moreover, we study a variant of noisy SGD that does not require explicit knowledge of gradient sensitivity, unlike existing work that assumes or enforces sensitivity through gradient clipping. Specifically, we focus on the least squares problem with $\ell_2$ regularization.

Vision-Language-Action (VLA) models trained on large robot datasets promise general-purpose, robust control across diverse domains and embodiments. However, existing approaches often fail out-of-the-box when deployed in novel environments, embodiments, or tasks. We introduce Mixture of Skills VLA (MoS-VLA), a framework that represents robot manipulation policies as linear combinations of a finite set of learned basis functions. During pretraining, MoS-VLA jointly learns these basis functions across datasets from the Open X-Embodiment project, producing a structured skill space. At test time, adapting to a new task requires only a single expert demonstration. The corresponding skill representation is then inferred via a lightweight convex optimization problem that minimizes the L1 action error, without requiring gradient updates. Empirically, MoS-VLA achieves lower action-prediction error on five out of five unseen datasets and succeeds in both simulation and real-robot tasks where a pretrained VLA model fails outright. Inspired by the success of large language models, modern robotics aims to achieve generalization and human-like performance through the use of internet-scale data and large, attention-based architectures. To this end, researchers have collected enormous datasets of robotic arm trajectories (Open X-Embodiment Collaboration et al., 2023) and trained so-called vision-language-action foundation models to map natural language task descriptions and state observations to robot actions (Kim et al., 2024; Octo Model Team et al., 2024; Brohan et al., 2023b;a; Ma et al., 2024).

Mean-field variational inference (MFVI) is a widely used method for approximating high-dimensional probability distributions by product measures. It has been empirically observed that MFVI optimizers often suffer from mode collapse. Specifically, when the target measure $π$ is a mixture $π= w P_0 + (1 - w) P_1$, the MFVI optimizer tends to place most of its mass near a single component of the mixture. This work provides the first theoretical explanation of mode collapse in MFVI. We introduce the notion to capture the separatedness of the two mixture components -- called $\varepsilon$-separateness -- and derive explicit bounds on the fraction of mass that any MFVI optimizer assigns to each component when $P_0$ and $P_1$ are $\varepsilon$-separated for sufficiently small $\varepsilon$. Our results suggest that the occurrence of mode collapse crucially depends on the relative position of the components. To address this issue, we propose the rotational variational inference (RoVI), which augments MFVI with a rotation matrix. The numerical studies support our theoretical findings and demonstrate the benefits of RoVI.

Dataset condensation (DC) enables the creation of compact, privacy-preserving synthetic datasets that can match the utility of real patient records, supporting democratised access to highly regulated clinical data for developing downstream clinical models. State-of-the-art DC methods supervise synthetic data by aligning the training dynamics of models trained on real and those trained on synthetic data, typically using full stochastic gradient descent (SGD) trajectories as alignment targets; however, these trajectories are often noisy, high-curvature, and storage-intensive, leading to unstable gradients, slow convergence, and substantial memory overhead. We address these limitations by replacing full SGD trajectories with smooth, low-loss parametric surrogates, specifically quadratic Bézier curves that connect the initial and final model states from real training trajectories. These mode-connected paths provide noise-free, low-curvature supervision signals that stabilise gradients, accelerate convergence, and eliminate the need for dense trajectory storage. We theoretically justify Bézier-mode connections as effective surrogates for SGD paths and empirically show that the proposed method outperforms state-of-the-art condensation approaches across five clinical datasets, yielding condensed datasets that enable clinically effective model development.

Contrary to common belief, we show that gradient ascent-based unconstrained optimization methods frequently fail to perform machine unlearning, a phenomenon we attribute to the inherent statistical dependence between the forget and retain data sets. This dependence, which can manifest itself even as simple correlations, undermines the misconception that these sets can be independently manipulated during unlearning. We provide empirical and theoretical evidence showing these methods often fail precisely due to this overlooked relationship. For random forget sets, this dependence means that degrading forget set metrics (which, for a retrained model, should mirror test set metrics) inevitably harms overall test performance. Going beyond random sets, we consider logistic regression as an instructive example where a critical failure mode emerges: inter-set dependence causes gradient descent-ascent iterations to progressively diverge from the ideal retrained model. Strikingly, these methods can converge to solutions that are not only far from the retrained ideal but are potentially even further from it than the original model itself, rendering the unlearning process actively detrimental. A toy example further illustrates how this dependence can trap models in inferior local minima, inescapable via finetuning. Our findings highlight that the presence of such statistical dependencies, even when manifest only as correlations, can be sufficient for ascent-based unlearning to fail. Our theoretical insights are corroborated by experiments on complex neural networks, demonstrating that these methods do not perform as expected in practice due to this unaddressed statistical interplay.

We develop a framework for analyzing the training and learning rate dynamics on a variety of high- dimensional optimization problems trained using one-pass stochastic gradient descent (SGD) with data generated from multiple anisotropic classes. We give exact expressions for a large class of functions of the limiting dynamics, including the risk and the overlap with the true signal, in terms of a deterministic solution to a system of ODEs. We extend the existing theory of high-dimensional SGD dynamics to Gaussian-mixture data and a large (growing with the parameter size) number of classes. We then investigate in detail the effect of the anisotropic structure of the covariance of the data in the problems of binary logistic regression and least square loss. We study three cases: isotropic covariances, data covariance matrices with a large fraction of zero eigenvalues (denoted as the zero-one model), and covariance matrices with spectra following a power-law distribution. We show that there exists a structural phase transition. In particular, we demonstrate that, for the zero-one model and the power-law model with sufficiently large power, SGD tends to align more closely with values of the class mean that are projected onto the "clean directions" (i.e., directions of smaller variance). This is supported by both numerical simulations and analytical studies, which show the exact asymptotic behavior of the loss in the high-dimensional limit.

We compare Ising ({-1,+1}) and QUBO ({0,1}) encodings for Boltzmann machine learning under a controlled protocol that fixes the model, sampler, and step size. Exploiting the identity that the Fisher information matrix (FIM) equals the covariance of sufficient statistics, we visualize empirical moments from model samples and reveal systematic, representation-dependent differences. QUBO induces larger cross terms between first- and second-order statistics, creating more small-eigenvalue directions in the FIM and lowering spectral entropy. This ill-conditioning explains slower convergence under stochastic gradient descent (SGD). In contrast, natural gradient descent (NGD)-which rescales updates by the FIM metric-achieves similar convergence across encodings due to reparameterization invariance. Practically, for SGD-based training, the Ising encoding provides more isotropic curvature and faster convergence; for QUBO, centering/scaling or NGD-style preconditioning mitigates curvature pathologies. These results clarify how representation shapes information geometry and finite-time learning dynamics in Boltzmann machines and yield actionable guidelines for variable encoding and preprocessing.