In recent works on the theory of machine learning, it has been observed that heavy tail properties of Stochastic Gradient Descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, G\"{u}rb\"{u}zbalaban et al. (arXiv:2006.04740) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_k=A_k X_{k-1}+B_k$, for independent and identically distributed pairs $(A_k, B_k)$, where $A_k$ is a random symmetric matrix and $B_k$ is a random vector. In this work, we will answer several open questions of the quoted paper and extend their results by applying the theory of irreducible-proximal (i-p) matrices.

Yet, releasing R(x) might reveal sensitive information on x. Instead, the The Gaussian Mechanism (GM), which consists curator may use a private algorithm A to release a in adding Gaussian noise to a vectorvalued sanitized approximation A(R)(x) of R(x). To guarantee query before releasing it, is a standard that the amount of information leaked by releasing privacy protection mechanism. In particular, A(R)(x) is limited, DP ensures that the distributions given that the query respects some of A(R)(x) and A(R)(y) are close for any y x, i.e., L2 sensitivity property (the L2 distance between that is close to x according to a neighboring relation outputs on any two neighboring inputs (databases that only differ in one row for instance). Several is bounded), GM guarantees Rényi Differential divergences have been considered to measure the Privacy (RDP). Unfortunately, precisely closeness between these two distributions, leading to different bounding the L2 sensitivity can be hard, thus variants of DP. Among them, Rényi-Differential leading to loose privacy bounds. In this work, Privacy (RDP), which is based on the Rényi divergence, we consider a Relative L2 sensitivity assumption, has become popular for its mathematical properties in which the bound on the distance between [Mironov, 2017].

Training a deep neural network using gradient-based methods necessitates the calculation of gradients at each level. However, using backpropagation or reverse mode differentiation, to calculate the gradients necessities significant memory consumption, rendering backpropagation an inefficient method for computing gradients. This paper focuses on analyzing the performance of the well-known Frank-Wolfe algorithm, a.k.a. conditional gradient algorithm by having access to the forward mode of automatic differentiation to compute gradients. We provide in-depth technical details that show the proposed Algorithm does converge to the optimal solution with a sub-linear rate of convergence by having access to the noisy estimate of the true gradient obtained in the forward mode of automated differentiation, referred to as the Projected Forward Gradient. In contrast, the standard Frank-Wolfe algorithm, when provided with access to the Projected Forward Gradient, fails to converge to the optimal solution. We demonstrate the convergence attributes of our proposed algorithms using a numerical example.

The asymptotically precise estimation of the generalization of kernel methods has recently received attention due to the parallels between neural networks and their associated kernels. However, prior works derive such estimates for training by kernel ridge regression (KRR), whereas neural networks are typically trained with gradient descent (GD). In the present work, we consider the training of kernels with a family of $\textit{spectral algorithms}$ specified by profile $h(\lambda)$, and including KRR and GD as special cases. Then, we derive the generalization error as a functional of learning profile $h(\lambda)$ for two data models: high-dimensional Gaussian and low-dimensional translation-invariant model. Under power-law assumptions on the spectrum of the kernel and target, we use our framework to (i) give full loss asymptotics for both noisy and noiseless observations (ii) show that the loss localizes on certain spectral scales, giving a new perspective on the KRR saturation phenomenon (iii) conjecture, and demonstrate for the considered data models, the universality of the loss w.r.t. non-spectral details of the problem, but only in case of noisy observation.

It is challenging for autonomous control systems to perform complex tasks in the presence of latent risks. Motivated by this challenge, this paper proposes an integrated framework that involves Large Language Models (LLMs), stochastic gradient descent (SGD), and optimization-based control. In the first phrase, the proposed framework breaks down complex tasks into a sequence of smaller subtasks, whose specifications account for contextual information and latent risks. In the second phase, these subtasks and their parameters are refined through a dual process involving LLMs and SGD. LLMs are used to generate rough guesses and failure explanations, and SGD is used to fine-tune parameters. The proposed framework is tested using simulated case studies of robots and vehicles. The experiments demonstrate that the proposed framework can mediate actions based on the context and latent risks and learn complex behaviors efficiently.

The stochastic proximal gradient method is a powerful generalization of the widely used stochastic gradient descent (SGD) method and has found numerous applications in Machine Learning. However, it is notoriously known that this method fails to converge in non-convex settings where the stochastic noise is significant (i.e. when only small or bounded batch sizes are used). In this paper, we focus on the stochastic proximal gradient method with Polyak momentum. We prove this method attains an optimal convergence rate for non-convex composite optimization problems, regardless of batch size. Additionally, we rigorously analyze the variance reduction effect of the Polyak momentum in the composite optimization setting and we show the method also converges when the proximal step can only be solved inexactly. Finally, we provide numerical experiments to validate our theoretical results.

Sharpness-Aware Minimization (SAM) has been instrumental in improving deep neural network training by minimizing both training loss and loss sharpness. Despite the practical success, the mechanisms behind SAM's generalization enhancements remain elusive, limiting its progress in deep learning optimization. In this work, we investigate SAM's core components for generalization improvement and introduce "Friendly-SAM" (F-SAM) to further enhance SAM's generalization. Our investigation reveals the key role of batch-specific stochastic gradient noise within the adversarial perturbation, i.e., the current minibatch gradient, which significantly influences SAM's generalization performance. By decomposing the adversarial perturbation in SAM into full gradient and stochastic gradient noise components, we discover that relying solely on the full gradient component degrades generalization while excluding it leads to improved performance. The possible reason lies in the full gradient component's increase in sharpness loss for the entire dataset, creating inconsistencies with the subsequent sharpness minimization step solely on the current minibatch data. Inspired by these insights, F-SAM aims to mitigate the negative effects of the full gradient component. It removes the full gradient estimated by an exponentially moving average (EMA) of historical stochastic gradients, and then leverages stochastic gradient noise for improved generalization. Moreover, we provide theoretical validation for the EMA approximation and prove the convergence of F-SAM on non-convex problems. Extensive experiments demonstrate the superior generalization performance and robustness of F-SAM over vanilla SAM. Code is available at https://github.com/nblt/F-SAM.

A common approach for handling the complexity and inherent ambiguities of 3D human pose estimation is to use pose priors learned from training data. Existing approaches however, are either too simplistic (linear), too complex to learn, or can only learn latent spaces from "simple data", i.e., single activities such as walking or running. In this paper, we present an efficient stochastic gradient descent algorithm that is able to learn probabilistic non-linear latent spaces composed of multiple activities. Furthermore, we derive an incremental algorithm for the online setting which can update the latent space without extensive relearning. We demonstrate the effectiveness of our approach on the task of monocular and multi-view tracking and show that our approach outperforms the state-of-the-art.

We analyze the convergence of gradient-based optimization algorithms whose updates depend on delayed stochastic gradient information. The main application of our results is to the development of distributed minimization algorithms where a master node performs parameter updates while worker nodes compute stochastic gradients based on local information in parallel, which may give rise to delays due to asynchrony. Our main contribution is to show that for smooth stochastic problems, the delays are asymptotically negligible. In application to distributed optimization, we show n-node architectures whose optimization error in stochastic problems--in spite of asynchronous delays--scales asymptotically as O(1/ nT), which is known to be optimal even in the absence of delays.

Many real-world networks are described by both connectivity information and features for every node. To better model and understand these networks, we present structure preserving metric learning (SPML), an algorithm for learning a Mahalanobis distance metric from a network such that the learned distances are tied to the inherent connectivity structure of the network. Like the graph embedding algorithm structure preserving embedding, SPML learns a metric which is structure preserving, meaning a connectivity algorithm such as k-nearest neighbors will yield the correct connectivity when applied using the distances from the learned metric. We show a variety of synthetic and real-world experiments where SPML predicts link patterns from node features more accurately than standard techniques. We further demonstrate a method for optimizing SPML based on stochastic gradient descent which removes the running-time dependency on the size of the network and allows the method to easily scale to networks of thousands of nodes and millions of edges.