Many learning algorithms, such as stochastic gradient descent, are affected by the order in which training examples are used. It is often observed that sampling the training examples without-replacement, also known as random reshuffling, causes learning algorithms to converge faster. We give a counterexample to the Operator Inequality of Noncommutative Arithmetic and Geometric Means, a longstanding conjecture that relates to the performance of random reshuffling in learning algorithms (Recht and Ré, "Toward a noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences," COLT 2012). We use this to give an example of a learning task and algorithm for which with-replacement random sampling actually outperforms random reshuffling.

Epoch-GD) proposed by (Hazan and Kale, 2011) was deemeda breakthrough for stochastic strongly convex minimization, which achieves theoptimal convergence rate of O(1/T) with T iterative updates for the objective gap. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whethera fast rate ofO(1/T)for theduality gapis achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some re-cent studies have proposed stochastic algorithms with fast convergence rates formin-max problems, they require additional assumptions about the problem, e.g.,smoothness, bi-linear structure, etc. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the optimal rate ofO(1/T)for the duality gapof general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can also be used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to a nearly optimal complexity without resorting to smoothness or other structural conditions.

A central issue in machine learning is how to train models on sensitive user data. Industry has widely adopted a simple algorithm: Stochastic Gradient Descent with noise (a.k.a. However, foundational theoretical questions about this algorithm's privacy loss remain open---even in the seemingly simple setting of smooth convex losses over a bounded domain. Our main result resolves these questions: for a large range of parameters, we characterize the differential privacy up to a constant. This result reveals that all previous analyses for this setting have the wrong qualitative behavior.

The use of momentum in stochastic gradient methods has become a widespread practice in machine learning. Different variants of momentum, including heavy-ball momentum, Nesterov's accelerated gradient (NAG), and quasi-hyperbolic momentum (QHM), have demonstrated success on various tasks. Despite these empirical successes, there is a lack of clear understanding of how the momentum parameters affect convergence and various performance measures of different algorithms. In this paper, we use the general formulation of QHM to give a unified analysis of several popular algorithms, covering their asymptotic convergence conditions, stability regions, and properties of their stationary distributions. In addition, by combining the results on convergence rates and stationary distributions, we obtain sometimes counter-intuitive practical guidelines for setting the learning rate and momentum parameters.

Classical parameter-space Bayesian inference for Bayesian neural networks (BNNs) suffers from several unresolved prior issues, such as knowledge encoding intractability and pathological behaviours in deep networks, which can lead to improper posterior inference. To address these issues, functional Bayesian inference has recently been proposed leveraging functional priors, such as the emerging functional variational inference. In addition to variational methods, stochastic gradient Markov Chain Monte Carlo (MCMC) is another scalable and effective inference method for BNNs to asymptotically generate samples from the true posterior by simulating continuous dynamics. However, existing MCMC methods perform solely in parameter space and inherit the unresolved prior issues, while extending these dynamics to function space is a non-trivial undertaking. In this paper, we introduce novel functional MCMC schemes, including stochastic gradient versions, based on newly designed diffusion dynamics that can incorporate more informative functional priors. Moreover, we prove that the stationary measure of these functional dynamics is the target posterior over functions. Our functional MCMC schemes demonstrate improved performance in both predictive accuracy and uncertainty quantification on several tasks compared to naive parameter-space MCMC and functional variational inference.

Minimax problems have achieved success in machine learning such as adversarial training, robust optimization, reinforcement learning. For theoretical analysis, current optimal excess risk bounds, which are composed by generalization error and optimization error, present 1/n-rates in strongly-convex-strongly-concave (SC-SC) settings. Existing studies mainly focus on minimax problems with specific algorithms for optimization error, with only a few studies on generalization performance, which limit better excess risk bounds. In this paper, we study the generalization bounds measured by the gradients of primal functions using uniform localized convergence. We obtain a sharper high probability generalization error bound for nonconvex-strongly-concave (NC-SC) stochastic minimax problems. Furthermore, we provide dimension-independent results under Polyak-Lojasiewicz condition for the outer layer. Based on our generalization error bound, we analyze some popular algorithms such as empirical saddle point (ESP), gradient descent ascent (GDA) and stochastic gradient descent ascent (SGDA). We derive better excess primal risk bounds with further reasonable assumptions, which, to the best of our knowledge, are n times faster than exist results in minimax problems.

This paper focuses on finite sample analysis for stochastic gradient algorithms. The motivation stems from a vast varieties of applications. In particular, the recent advances on stochastic optimization in conjunction with machine learning have opened up new domains. A particular emphasis of the learning community requires us taking a careful look at of the finite sample analysis. Well, it is well known that stochastic gradient algorithms or stochastic approximation algorithms are normally concentrated on dealing with asymptotic properties of the recursive algorithms. However, the learning community placed more effort for carrying out analysis of finite sample properties of the recursive algorithms; see for example,... and references therein.

While momentum-based optimization algorithms are commonly used in the notoriously non-convex optimization problems of deep learning, their analysis has historically been restricted to the convex and strongly convex setting. In this article, we partially close this gap between theory and practice and demonstrate that virtually identical guarantees can be obtained in optimization problems with a `benign' non-convexity. We show that these weaker geometric assumptions are well justified in overparametrized deep learning, at least locally. Variations of this result are obtained for a continuous time model of Nesterov's accelerated gradient descent algorithm (NAG), the classical discrete time version of NAG, and versions of NAG with stochastic gradient estimates with purely additive noise and with noise that exhibits both additive and multiplicative scaling.

Fine-tuning has become a popular approach to adapting large foundational models to specific tasks. As the size of models and datasets grows, parameter-efficient fine-tuning techniques are increasingly important. One of the most widely used methods is Low-Rank Adaptation (LoRA), with adaptation update expressed as the product of two low-rank matrices. While LoRA was shown to possess strong performance in fine-tuning, it often under-performs when compared to full-parameter fine-tuning (FPFT). Although many variants of LoRA have been extensively studied empirically, their theoretical optimization analysis is heavily under-explored. The starting point of our work is a demonstration that LoRA and its two extensions, Asymmetric LoRA and Chain of LoRA, indeed encounter convergence issues. To address these issues, we propose Randomized Asymmetric Chain of LoRA (RAC-LoRA) -- a general optimization framework that rigorously analyzes the convergence rates of LoRA-based methods. Our approach inherits the empirical benefits of LoRA-style heuristics, but introduces several small but important algorithmic modifications which turn it into a provably convergent method. Our framework serves as a bridge between FPFT and low-rank adaptation. We provide provable guarantees of convergence to the same solution as FPFT, along with the rate of convergence. Additionally, we present a convergence analysis for smooth, non-convex loss functions, covering gradient descent, stochastic gradient descent, and federated learning settings. Our theoretical findings are supported by experimental results.

Neural networks are trained by choosing an architecture and training the parameters. The choice of architecture is often by trial and error or with Neural Architecture Search (NAS) methods. While NAS provides some automation, it often relies on discrete steps that optimize the architecture and then train the parameters. We introduce a novel neural network training framework that fundamentally transforms the process by learning architecture and parameters simultaneously with gradient descent. With the appropriate setting of the loss function, it can discover sparse and compact neural networks for given datasets. Central to our approach is a multi-scale encoder-decoder, in which the encoder embeds pairs of neural networks with similar functionalities close to each other (irrespective of their architectures and weights). To train a neural network with a given dataset, we randomly sample a neural network embedding in the embedding space and then perform gradient descent using our custom loss function, which incorporates a sparsity penalty to encourage compactness. The decoder generates a neural network corresponding to the embedding. Experiments demonstrate that our framework can discover sparse and compact neural networks maintaining a high performance.