In this paper, we study a Markov chain-based stochastic gradient algorithm in general Hilbert spaces, aiming to approximate the optimal solution of a quadratic loss function. We establish probabilistic upper bounds on its convergence. We further extend these results to an online regularized learning algorithm in reproducing kernel Hilbert spaces, where the samples are drawn along a Markov chain trajectory hence the samples are of the non i.i.d.

Simple ideas of thermodynamics and kinetic theory allow us to explain better generalization observed for learning by the stochastic gradient optimization procedure, see also [7] (where also overfitting control for GAN model was discussed). We also have explained the grokking(delayed generalization) phenomenon and some properties of grokking observed in [8].

In minimax optimization, the extragradient (EG) method has been extensively studied because it outperforms the gradient descent-ascent method in convex-concave (C-C) problems. Yet, stochastic EG (SEG) has seen limited success in C-C problems, especially for unconstrained cases. Motivated by the recent progress of shuffling-based stochastic methods, we investigate the convergence of shuffling-based SEG in unconstrained finite-sum minimax problems, in search of convergent shuffling-based SEG. Our analysis reveals that both random reshuffling and the recently proposed flip-flop shuffling alone can suffer divergence in C-C problems. However, with an additional simple trick called anchoring, we develop the SEG with flip-flop anchoring (SEG-FFA) method which successfully converges in C-C problems. We also show upper and lower bounds in the strongly-convex-strongly-concave setting, demonstrating that SEG-FFA has a provably faster convergence rate compared to other shuffling-based methods.

Feature selection is essential in the analysis of molecular systems and many other fields, but several uncertainties remain: What is the optimal number of features for a simplified, interpretable model that retains essential information? How should features with different units be aligned, and how should their relative importance be weighted? Here, we introduce the Differentiable Information Imbalance (DII), an automated method to rank information content between sets of features. Using distances in a ground truth feature space, DII identifies a low-dimensional subset of features that best preserves these relationships. Each feature is scaled by a weight, which is optimized by minimizing the DII through gradient descent. This allows simultaneously performing unit alignment and relative importance scaling, while preserving interpretability. DII can also produce sparse solutions and determine the optimal size of the reduced feature space. We demonstrate the usefulness of this approach on two benchmark molecular problems: (1) identifying collective variables that describe conformations of a biomolecule, and (2) selecting features for training a machine-learning force field. These results show the potential of DII in addressing feature selection challenges and optimizing dimensionality in various applications. The method is available in the Python library DADApy.

Federated Learning (FL) enables clients to share learning parameters instead of local data, reducing communication overhead. Traditional wireless networks face latency challenges with FL. In contrast, Cell-Free Massive MIMO (CFmMIMO) can serve multiple clients on shared resources, boosting spectral efficiency and reducing latency for large-scale FL. However, clients' communication resource limitations can hinder the completion of the FL training. To address this challenge, we propose an energy-efficient, low-latency FL framework featuring optimized uplink power allocation for seamless client-server collaboration. Our framework employs an adaptive quantization scheme, dynamically adjusting bit allocation for local gradient updates to reduce communication costs. We formulate a joint optimization problem covering FL model updates, local iterations, and power allocation, solved using sequential quadratic programming (SQP) to balance energy and latency. Additionally, clients use the AdaDelta method for local FL model updates, enhancing local model convergence compared to standard SGD, and we provide a comprehensive analysis of FL convergence with AdaDelta local updates. Numerical results show that, within the same energy and latency budgets, our power allocation scheme outperforms the Dinkelbach and max-sum rate methods by increasing the test accuracy up to $7$\% and $19$\%, respectively. Moreover, for the three power allocation methods, our proposed quantization scheme outperforms AQUILA and LAQ by increasing test accuracy by up to $36$\% and $35$\%, respectively.

Training and fine-tuning large language models (LLMs) come with challenges related to memory and computational requirements due to the increasing size of the model weights and the optimizer states. Various techniques have been developed to tackle these challenges, such as low-rank adaptation (LoRA), which involves introducing a parallel trainable low-rank matrix to the fixed pre-trained weights at each layer. However, these methods often fall short compared to the full-rank weight training approach, as they restrict the parameter search to a low-rank subspace. This limitation can disrupt training dynamics and require a full-rank warm start to mitigate the impact. In this paper, we introduce a new method inspired by a phenomenon we formally prove: as training progresses, the rank of the estimated layer gradients gradually decreases, and asymptotically approaches rank one. Leveraging this, our approach involves adaptively reducing the rank of the gradients during Adam optimization steps, using an efficient online-updating low-rank projections rule. We further present a randomized SVD scheme for efficiently finding the projection matrix. Our technique enables full-parameter fine-tuning with adaptive low-rank gradient updates, significantly reducing overall memory requirements during training compared to state-of-the-art methods while improving model performance in both pretraining and fine-tuning. Finally, we provide a convergence analysis of our method and demonstrate its merits for training and fine-tuning language and biological foundation models.

We introduce Gradient Agreement Filtering (GAF) to improve on gradient averaging in distributed deep learning optimization. Traditional distributed data-parallel stochastic gradient descent involves averaging gradients of microbatches to calculate a macrobatch gradient that is then used to update model parameters. We find that gradients across microbatches are often orthogonal or negatively correlated, especially in late stages of training, which leads to memorization of the training set, reducing generalization. In this paper, we introduce a simple, computationally effective way to reduce gradient variance by computing the cosine distance between micro-gradients during training and filtering out conflicting updates prior to averaging. We improve validation accuracy with significantly smaller microbatch sizes. We also show this reduces memorizing noisy labels. We demonstrate the effectiveness of this technique on standard image classification benchmarks including CIFAR-100 and CIFAR-100N-Fine. We show this technique consistently outperforms validation accuracy, in some cases by up to 18.2\% compared to traditional training approaches while reducing the computation required nearly an order of magnitude because we can now rely on smaller microbatch sizes without destabilizing training.

Machine learning (ML) and artificial intelligence (AI) can provide powerful tools for the scientific community, as demonstrated by the recent Nobel Prize in Chemistry. Reversely, insights from traditional physics theories also contribute to a deeper understanding of the mechanism of learning. Ref. [1] contains a broad overview of the successful cross-fertilisation between ML and the physical sciences, covering a number of domains. One way to mitigate against possible scepticism with regard to using ML as a "black box" is by unveiling the dynamics of training (or learning) and explaining how the relevant information is engraved in the model during the training stage. To further develop this programme, we study here the dynamics of first-order stochastic gradient descent as applied to weight matrices, reporting and expanding on the work presented in Ref. [2]. When training ML models, weight matrices are commonly updated by one of the variants of the stochastic gradient descent algorithm. The dynamics can then be decomposed into a drift and a fluctuating term, and such a system can be described by a discrete Langevin equation. The dynamics of stochastic matrix updates is richer than the dynamics for vector or scalar quantities, as captured by Dyson Brownian motion and random matrix theory (RMT), with the appearance of universal features for the eigenvalues [3-9]. Earlier descriptions of the statistical properties of weight matrices in terms of RMT can be found in e.g.

Recent findings by Cohen et al., 2021, demonstrate that when training neural networks with full-batch gradient descent at a step size of $\eta$, the sharpness--defined as the largest eigenvalue of the full batch Hessian--consistently stabilizes at $2/\eta$. These results have significant implications for convergence and generalization. Unfortunately, this was observed not to be the case for mini-batch stochastic gradient descent (SGD), thus limiting the broader applicability of these findings. We show that SGD trains in a different regime we call Edge of Stochastic Stability. In this regime, what hovers at $2/\eta$ is, instead, the average over the batches of the largest eigenvalue of the Hessian of the mini batch (MiniBS) loss--which is always bigger than the sharpness. This implies that the sharpness is generally lower when training with smaller batches or bigger learning rate, providing a basis for the observed implicit regularization effect of SGD towards flatter minima and a number of well established empirical phenomena. Additionally, we quantify the gap between the MiniBS and the sharpness, further characterizing this distinct training regime.

Single-cell RNA sequencing allows the quantitation of gene expression at the individual cell level, enabling the study of cellular heterogeneity and gene expression dynamics. Dimensionality reduction is a common preprocessing step to simplify the visualization, clustering, and phenotypic characterization of samples. This step, often performed using principal component analysis or closely related methods, is challenging because of the size and complexity of the data. In this work, we present a generalized matrix factorization model assuming a general exponential dispersion family distribution and we show that many of the proposed approaches in the single-cell dimensionality reduction literature can be seen as special cases of this model. Furthermore, we propose a scalable adaptive stochastic gradient descent algorithm that allows us to estimate the model efficiently, enabling the analysis of millions of cells. Our contribution extends to introducing a novel warm start initialization method, designed to accelerate algorithm convergence and increase the precision of final estimates. Moreover, we discuss strategies for dealing with missing values and model selection. We benchmark the proposed algorithm through extensive numerical experiments against state-of-the-art methods and showcase its use in real-world biological applications. The proposed method systematically outperforms existing methods of both generalized and non-negative matrix factorization, demonstrating faster execution times while maintaining, or even enhancing, matrix reconstruction fidelity and accuracy in biological signal extraction. Finally, all the methods discussed here are implemented in an efficient open-source R package, sgdGMF, available at github/CristianCastiglione/sgdGMF