We study continual learning on multiple linear classification tasks by sequentially running gradient descent (GD) for a fixed budget of iterations per task. When all tasks are jointly linearly separable and are presented in a cyclic/random order, we show the directional convergence of the trained linear classifier to the joint (offline) max-margin solution. This is surprising because GD training on a single task is implicitly biased towards the individual max-margin solution for the task, and the direction of the joint max-margin solution can be largely different from these individual solutions. Additionally, when tasks are given in a cyclic order, we present a non-asymptotic analysis on cycle-averaged forgetting, revealing that (1) alignment between tasks is indeed closely tied to catastrophic forgetting and backward knowledge transfer and (2) the amount of forgetting vanishes to zero as the cycle repeats. Lastly, we analyze the case where the tasks are no longer jointly separable and show that the model trained in a cyclic order converges to the unique minimum of the joint loss function.

We study the complexity of online stochastic gradient descent (SGD) for learning a two-layer neural network with $P$ neurons on isotropic Gaussian data: $f_*(\boldsymbol{x}) = \sum_{p=1}^P a_p\cdot \sigma(\langle\boldsymbol{x},\boldsymbol{v}_p^*\rangle)$, $\boldsymbol{x} \sim \mathcal{N}(0,\boldsymbol{I}_d)$, where the activation $\sigma:\mathbb{R}\to\mathbb{R}$ is an even function with information exponent $k_*>2$ (defined as the lowest degree in the Hermite expansion), $\{\boldsymbol{v}^*_p\}_{p\in[P]}\subset \mathbb{R}^d$ are orthonormal signal directions, and the non-negative second-layer coefficients satisfy $\sum_{p} a_p^2=1$. We focus on the challenging ``extensive-width'' regime $P\gg 1$ and permit diverging condition number in the second-layer, covering as a special case the power-law scaling $a_p\asymp p^{-\beta}$ where $\beta\in\mathbb{R}_{\ge 0}$. We provide a precise analysis of SGD dynamics for the training of a student two-layer network to minimize the mean squared error (MSE) objective, and explicitly identify sharp transition times to recover each signal direction. In the power-law setting, we characterize scaling law exponents for the MSE loss with respect to the number of training samples and SGD steps, as well as the number of parameters in the student neural network. Our analysis entails that while the learning of individual teacher neurons exhibits abrupt transitions, the juxtaposition of $P\gg 1$ emergent learning curves at different timescales leads to a smooth scaling law in the cumulative objective.

This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization, called the Asymmetric Projected Gradient Descent (APGD), for reconstructing the point set configuration from partial Euclidean distance measurements, known as the Euclidean Distance Matrix Completion (EDMC) problem. By paralleling the incoherence matrix completion framework, we show for the first time that global convergence guarantee with exact recovery of this routine can be established given $\mathcal{O}(\mu^2 r^3 \kappa^2 n \log n)$ Bernoulli random observations without any sample splitting. Unlike leveraging the tangent space Restricted Isometry Property (RIP) and local curvature of the low-rank embedding manifold in some very recent works, our proof provides new upper bounds to replace the random graph lemma under EDMC setting. The APGD works surprisingly well and numerical experiments demonstrate exact linear convergence behavior in rich-sample regions yet deteriorates fast when compared with the performance obtained by optimizing the s-stress function, i.e., the standard but unexplained non-convex approach for EDMC, if the sample size is limited. While virtually matching our theoretical prediction, this unusual phenomenon might indicate that: (i) the power of implicit regularization is weakened when specified in the APGD case; (ii) the stabilization of such new gradient direction requires substantially more samples than the information-theoretic limit would suggest.

Imitation learning holds the promise of equipping robots with versatile skills by learning from expert demonstrations. However, policies trained on finite datasets often struggle to generalize beyond the training distribution. In this work, we present a unified perspective on the generalization capability of imitation learning, grounded in both information theorey and data distribution property. We first show that the generalization gap can be upper bounded by (i) the conditional information bottleneck on intermediate representations and (ii) the mutual information between the model parameters and the training dataset. This characterization provides theoretical guidance for designing effective training strategies in imitation learning, particularly in determining whether to freeze, fine-tune, or train large pretrained encoders (e.g., vision-language models or vision foundation models) from scratch to achieve better generalization. Furthermore, we demonstrate that high conditional entropy from input to output induces a flatter likelihood landscape, thereby reducing the upper bound on the generalization gap. In addition, it shortens the stochastic gradient descent (SGD) escape time from sharp local minima, which may increase the likelihood of reaching global optima under fixed optimization budgets. These insights explain why imitation learning often exhibits limited generalization and underscore the importance of not only scaling the diversity of input data but also enriching the variability of output labels conditioned on the same input.

W e generalize the connection between activation function and spline regression/smoothing and characterize how this choice may influence spectral bias within a 1D shallow network. W e then demonstrate how gradient descent (GD) can be reinterpreted as a shrinkage operator that masks the singular values of a neural network's Jacobian. Viewed this way, GD implicitly selects the number of frequency components to retain, thereby controlling the spectral bias. An explicit relationship is proposed between the choice of GD hyperparameters (learning rate & number of iterations) and bandwidth (the number of active components). GD regularization is shown to be effective only with monotonic activation functions. Finally, we highlight the utility of non-monotonic activation functions (sinc, Gaussian) as iteration-efficient surrogates for spectral bias.

Two-time-scale stochastic approximation is an algorithm with coupled iterations which has found broad applications in reinforcement learning, optimization and game control. While several prior works have obtained a mean square error bound of $O(1/k)$ for linear two-time-scale iterations, the best known bound in the non-linear contractive setting has been $O(1/k^{2/3})$. In this work, we obtain an improved bound of $O(1/k)$ for non-linear two-time-scale stochastic approximation. Our result applies to algorithms such as gradient descent-ascent and two-time-scale Lagrangian optimization. The key step in our analysis involves rewriting the original iteration in terms of an averaged noise sequence which decays sufficiently fast. Additionally, we use an induction-based approach to show that the iterates are bounded in expectation.

This paper investigates regularized stochastic gradient descent (SGD) algorithms for estimating nonlinear operators from a Polish space to a separable Hilbert space. We assume that the regression operator lies in a vector-valued reproducing kernel Hilbert space induced by an operator-valued kernel. Two significant settings are considered: an online setting with polynomially decaying step sizes and regularization parameters, and a finite-horizon setting with constant step sizes and regularization parameters. We introduce regularity conditions on the structure and smoothness of the target operator and the input random variables. Under these conditions, we provide a dimension-free convergence analysis for the prediction and estimation errors, deriving both expectation and high-probability error bounds. Our analysis demonstrates that these convergence rates are nearly optimal. Furthermore, we present a new technique for deriving bounds with high probability for general SGD schemes, which also ensures almost-sure convergence. Finally, we discuss potential extensions to more general operator-valued kernels and the encoder-decoder framework.

In social learning, a network of agents assigns probability scores (beliefs) to some hypotheses of interest, which rule the generation of local streaming data observed by each agent. Belief formation takes place by means of an iterative two-step procedure where: i) the agents update locally their beliefs by using some likelihood model; and ii) the updated beliefs are combined with the beliefs of the neighboring agents, using a pooling rule. This procedure can fail to perform well in the presence of dynamic drifts, leading the agents to incorrect decision making. Here, we focus on the fully online setting where both the true hypothesis and the likelihood models can change over time. This goal is achieved by exploiting two adaptation stages: i) a stochastic gradient descent update to learn and track the drifts in the decision model; ii) and an adaptive belief update to track the true hypothesis changing over time. These stages are controlled by two adaptation parameters that govern the evolution of the error probability for each agent. We show that all agents learn consistently for sufficiently small adaptation parameters, in the sense that they ultimately place all their belief mass on the true hypothesis. Index T erms Social learning, belief formation, decision making, distributed optimization, online leaerning, opinion diffusion over graphs. Marco Carpentiero and Vincenzo Matta are with the Department of Information and Electrical Engineering and Applied Mathematics (DIEM), University of Salerno, via Giovanni Paolo II, I-84084, Fisciano (SA), Italy, and Vincenzo Matta is also with the National Inter-University Consortium for Telecommunications (CNIT), Italy (e-mails: { mcarpentiero, vmatta }@unisa.it). Matta was partially supported by the European Union under the Italian National Recovery and Resilience Plan (NRRP) of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 - program "REST ART"). This work was produced while Virginia Bordignon was a post-doc with the Ecole Polytechnique F ed erale de Lausanne EPFL, School of Engineering, CH-1015 Lausanne, Switzerland (e-mail: virginia.bordignon@alumni.epfl.ch).

It often happens that some sensitive personal information, such as credit card numbers or passwords, are mistakenly incorporated in the training of machine learning models and need to be removed afterwards. The removal of such information from a trained model is a complex task that needs to partially reverse the training process. There have been various machine unlearning techniques proposed in the literature to address this problem. Most of the proposed methods revolve around removing individual data samples from a trained model. Another less explored direction is when features/labels of a group of data samples need to be reverted. While the existing methods for these tasks do the unlearning task by updating the whole set of model parameters or only the last layer of the model, we show that there are a subset of model parameters that have the largest contribution in the unlearning target features. More precisely, the model parameters with the largest corresponding diagonal value in the Hessian matrix (computed at the learned model parameter) have the most contribution in the unlearning task. By selecting these parameters and updating them during the unlearning stage, we can have the most progress in unlearning. We provide theoretical justifications for the proposed strategy by connecting it to sharpness-aware minimization and robust unlearning. We empirically show the effectiveness of the proposed strategy in improving the efficacy of unlearning with a low computational cost.

We present a design called Proof of Gradient Optimization (PoGO) for blockchain consensus, where miners produce veri fiable evidence of training large-scale machine-learning models. Bu ilding on previous work [1,2,3], we incorporate quantized gradients (4-bit precision [7] [8][9]) to reduce storage and computation requirements, wh ile still preserving the ability of verifiers to check that real progress h as been made on lowering the model's loss. Additionally, we employ Merkl e proofs over the full 32-bit model to handle large parameter sets and to enable random leaf checks with minimal on-chain data. We illustrate these ideas using GPT-3 (175B parameters) [5] as a reference example and also r efer to smaller but high-performance models (e.g., Gemma 3 with 27B parameters). We provide an empirical cost analysis showing that ve rification is significantly cheaper than training, thanks in part to quant ization and sampling. We also discuss the necessity of longer block time s (potentially hours) when incorporating meaningful training steps, the t rade-offs when using specialized GPU hardware, and how binary diffs may incr ementally optimize updates. Finally, we note that fine-tuning can be ha ndled in a similar manner, merely changing the dataset and the manner o f sampling but preserving the overall verification flow. Our protocol al lows verifiers to issue either positive or negative attestations; these are aggregated at finalization to either confirm the update or slash the miner.