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.

We derive a differential equation that governs the evolution of the generalization gap when a deep network is trained by gradient descent. This differential equation is controlled by two quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. We analyze this differential equation to compute an ``effective Gram matrix'' that characterizes the generalization gap after training in terms of the alignment between this Gram matrix and a certain initial ``residual''. Empirical evaluations on image classification datasets indicate that this analysis can predict the test loss accurately. Further, at any point during training, the residual predominantly lies in the subspace of the effective Gram matrix with the smallest eigenvalues. This indicates that the training process is benign, i.e., it does not lead to significant deterioration of the generalization gap (which is zero at initialization). The alignment between the effective Gram matrix and the residual is different for different datasets and architectures. The match/mismatch of the data and the architecture is primarily responsible for good/bad generalization.

Wireless Sensor Networks (WSNs) are a cutting-edge domain in the field of intelligent sensing. Due to sensor failures and energy-saving strategies, the collected data often have massive missing data, hindering subsequent analysis and decision-making. Although Latent Factor Learning (LFL) has been proven effective in recovering missing data, it fails to sufficiently consider data privacy protection. To address this issue, this paper innovatively proposes a federated latent factor learning (FLFL) based spatial signal recovery (SSR) model, named FLFL-SSR. Its main idea is two-fold: 1) it designs a sensor-level federated learning framework, where each sensor uploads only gradient updates instead of raw data to optimize the global model, and 2) it proposes a local spatial sharing strategy, allowing sensors within the same spatial region to share their latent feature vectors, capturing spatial correlations and enhancing recovery accuracy. Experimental results on two real-world WSNs datasets demonstrate that the proposed model outperforms existing federated methods in terms of recovery performance.

Counterfactuals are a popular framework for interpreting machine learning predictions. These what if explanations are notoriously challenging to create for computer vision models: standard gradient-based methods are prone to produce adversarial examples, in which imperceptible modifications to image pixels provoke large changes in predictions. We introduce a new, easy-to-implement framework for counterfactual images that can flexibly adapt to contemporary advances in generative modeling. Our method, Counterfactual Attacks, resembles an adversarial attack on the representation of the image along a low-dimensional manifold. In addition, given an auxiliary dataset of image descriptors, we show how to accompany counterfactuals with feature attribution that quantify the changes between the original and counterfactual images. These importance scores can be aggregated into global counterfactual explanations that highlight the overall features driving model predictions. While this unification is possible for any counterfactual method, it has particular computational efficiency for ours. We demonstrate the efficacy of our approach with the MNIST and CelebA datasets.

While test-time fine-tuning is beneficial in few-shot learning, the need for multiple backpropagation steps can be prohibitively expensive in real-time or low-resource scenarios. To address this limitation, we propose an approach that emulates gradient descent without computing gradients, enabling efficient test-time adaptation. Specifically, we formulate gradient descent as an Euler discretization of an ordinary differential equation (ODE) and train an auxiliary network to predict the task-conditional drift using only the few-shot support set. The adaptation then reduces to a simple numerical integration (e.g., via the Euler method), which requires only a few forward passes of the auxiliary network -- no gradients or forward passes of the target model are needed. In experiments on cross-domain few-shot classification using the Meta-Dataset and CDFSL benchmarks, our method significantly improves out-of-domain performance over the non-fine-tuned baseline while incurring only 6\% of the memory cost and 0.02\% of the computation time of standard fine-tuning, thus establishing a practical middle ground between direct transfer and fully fine-tuned approaches.

The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to solve this problem, leveraging an alternating optimization approach that decomposes the global non-convex problem into a series of convex sub-problems. To improve performance, we explore advanced initialization strategies inspired by the singular value decomposition (SVD) and incorporate acceleration techniques by introducing momentum-based updates. Beyond optimizing the two-matrix case, we also extend the Hadamard decomposition framework to support more than two low-rank matrices, enabling approximations with higher effective ranks while preserving computational efficiency. Finally, we conduct extensive experiments to compare our method with the existing gradient descent-based approaches for the Hadamard decomposition and with traditional low-rank approximation techniques. The results highlight the effectiveness of our proposed method across diverse datasets.

--The main approach to hybrid quantum-classical neural networks (QNN) is employing quantum computing to build a neural network (NN) that has quantum features, which is then optimized classically. Here, we propose a different strategy: to use quantum computing in order to optimize the weights of a classical NN. As such, we design an instance of Grover's quantum search algorithm to accelerate the search for the optimal parameters of an NN during the training process, a task traditionally performed using the backpropagation algorithm with the gradient descent method. Indeed, gradient descent has issues such as exploding gradient, vanishing gradient, or convexity problem. Other methods tried to address such issues with strategies like genetic searches, but they carry additional problems like convergence consistency. Our original method avoids these issues--because it does not calculate gradients--and capitalizes on classical architectures' robustness and Grover's quadratic speedup in high-dimensional search spaces to significantly reduce test loss (58.75%) and improve test accuracy (35.25%), compared to classical NN weight optimization, on small datasets. Unlike most QNNs that are trained on small datasets only, our method is also scalable, as it allows the optimization of deep networks; for an NN with 3 hidden layers, trained on the Digits dataset from scikit-learn, we obtained a mean accuracy of 97.7%. Moreover, our method requires a much smaller number of qubits compared to other QNN approaches, making it very practical for near-future quantum computers that will still deliver a limited number of logical qubits. Since the dawn of Deep Learning (DL) in 2012, one of its most investigated topics has been the convergence of NN [1]. Traditionally, we may achieve convergence after minimizing the loss function by calculating the loss's derivative for the network's initial weights. This way, the gradient descent method determines the optimal parameter values [2]. There are several variants of gradient descent in the literature: stochastic gradient descent, batch gradient descent, and mini-batch gradient descent [3], [4].