We present an algorithm for fast stochastic gradient descent that uses a nonlinear adaptive momentum scheme to optimize the late time convergence rate. The algorithm makes effective use of cur(cid:173) vature information, requires only O(n) storage and computation, and delivers convergence rates close to the theoretical optimum. We demonstrate the technique on linear and large nonlinear back(cid:173) prop networks. Learning algorithms that perform gradient descent on a cost function can be for(cid:173) mulated in either stochastic (on-line) or batch form. Stochastic learning provides several advantages over batch learning.

The choice of batch sizes in stochastic gradient optimizers is critical for model training. However, the practice of varying batch sizes throughout the training process is less explored compared to other hyperparameters. We investigate adaptive batch size strategies derived from adaptive sampling methods, traditionally applied only in stochastic gradient descent. Given the significant interplay between learning rates and batch sizes, and considering the prevalence of adaptive gradient methods in deep learning, we emphasize the need for adaptive batch size strategies in these contexts. We introduce AdAdaGrad and its scalar variant AdAdaGradNorm, which incrementally increase batch sizes during training, while model updates are performed using AdaGrad and AdaGradNorm. We prove that AdaGradNorm converges with high probability at a rate of $\mathscr{O}(1/K)$ for finding a first-order stationary point of smooth nonconvex functions within $K$ iterations. AdaGrad also demonstrates similar convergence properties when integrated with a novel coordinate-wise variant of our adaptive batch size strategies. Our theoretical claims are supported by numerical experiments on various image classification tasks, highlighting the enhanced adaptability of progressive batching protocols in deep learning and the potential of such adaptive batch size strategies with adaptive gradient optimizers in large-scale model training.

We introduce the first temporal-difference learning algorithm that is stable with linear function approximation and off-policy training, for any finite Markov decision process, target policy, and exciting behavior policy, and whose complexity scales linearly in the number of parameters. We consider an i.i.d.\ policy-evaluation setting in which the data need not come from on-policy experience. The gradient temporal-difference (GTD) algorithm estimates the expected update vector of the TD(0) algorithm and performs stochastic gradient descent on its L_2 norm. Our analysis proves that its expected update is in the direction of the gradient, assuring convergence under the usual stochastic approximation conditions to the same least-squares solution as found by the LSTD, but without its quadratic computational complexity. GTD is online and incremental, and does not involve multiplying by products of likelihood ratios as in importance-sampling methods.

Markov Random Fields (MRFs) have proven very powerful both as density estimators and feature extractors for classification. However, their use is often limited by an inability to estimate the partition function Z . In this paper, we exploit the gradient descent training procedure of restricted Boltzmann machines (a type of MRF) to {\bf track} the log partition function during learning. Our method relies on two distinct sources of information: (1) estimating the change \Delta Z incurred by each gradient update, (2) estimating the difference in Z over a small set of tempered distributions using bridge sampling. The two sources of information are then combined using an inference procedure similar to Kalman filtering.

Although many variants of stochastic gradient descent have been proposed for large-scale convex optimization, most of them require projecting the solution at {\it each} iteration to ensure that the obtained solution stays within the feasible domain. For complex domains (e.g., positive semidefinite cone), the projection step can be computationally expensive, making stochastic gradient descent unattractive for large-scale optimization problems. We address this limitation by developing a novel stochastic gradient descent algorithm that does not need intermediate projections. Instead, only one projection at the last iteration is needed to obtain a feasible solution in the given domain. Our theoretical analysis shows that with a high probability, the proposed algorithms achieve an O(1/\sqrt{T}) convergence rate for general convex optimization, and an O(\ln T/T) rate for strongly convex optimization under mild conditions about the domain and the objective function.

Bayesian neural networks (BNNs) offer uncertainty quantification but come with the downside of substantially increased training and inference costs. Sparse BNNs have been investigated for efficient inference, typically by either slowly introducing sparsity throughout the training or by post-training compression of dense BNNs. The dilemma of how to cut down massive training costs remains, particularly given the requirement to learn about the uncertainty. To solve this challenge, we introduce Sparse Subspace Variational Inference (SSVI), the first fully sparse BNN framework that maintains a consistently highly sparse Bayesian model throughout the training and inference phases. Starting from a randomly initialized low-dimensional sparse subspace, our approach alternately optimizes the sparse subspace basis selection and its associated parameters. While basis selection is characterized as a non-differentiable problem, we approximate the optimal solution with a removal-and-addition strategy, guided by novel criteria based on weight distribution statistics. Our extensive experiments show that SSVI sets new benchmarks in crafting sparse BNNs, achieving, for instance, a 10-20x compression in model size with under 3\% performance drop, and up to 20x FLOPs reduction during training compared with dense VI training. Remarkably, SSVI also demonstrates enhanced robustness to hyperparameters, reducing the need for intricate tuning in VI and occasionally even surpassing VI-trained dense BNNs on both accuracy and uncertainty metrics.

Error Feedback (EF) is a highly popular and immensely effective mechanism for fixing convergence issues which arise in distributed training methods (such as distributed GD or SGD) when these are enhanced with greedy communication compression techniques such as TopK. While EF was proposed almost a decade ago (Seide et al., 2014), and despite concentrated effort by the community to advance the theoretical understanding of this mechanism, there is still a lot to explore. In this work we study a modern form of error feedback called EF21 (Richtarik et al., 2021) which offers the currently best-known theoretical guarantees, under the weakest assumptions, and also works well in practice. In particular, while the theoretical communication complexity of EF21 depends on the quadratic mean of certain smoothness parameters, we improve this dependence to their arithmetic mean, which is always smaller, and can be substantially smaller, especially in heterogeneous data regimes. We take the reader on a journey of our discovery process. Starting with the idea of applying EF21 to an equivalent reformulation of the underlying problem which (unfortunately) requires (often impractical) machine cloning, we continue to the discovery of a new weighted version of EF21 which can (fortunately) be executed without any cloning, and finally circle back to an improved analysis of the original EF21 method. While this development applies to the simplest form of EF21, our approach naturally extends to more elaborate variants involving stochastic gradients and partial participation. Further, our technique improves the best-known theory of EF21 in the rare features regime (Richtarik et al., 2023). Finally, we validate our theoretical findings with suitable experiments.

Traditional language models, adept at next-token prediction in text sequences, often struggle with transduction tasks between distinct symbolic systems, particularly when parallel data is scarce. Addressing this issue, we introduce \textit{symbolic autoencoding} ($\Sigma$AE), a self-supervised framework that harnesses the power of abundant unparallel data alongside limited parallel data. $\Sigma$AE connects two generative models via a discrete bottleneck layer and is optimized end-to-end by minimizing reconstruction loss (simultaneously with supervised loss for the parallel data), such that the sequence generated by the discrete bottleneck can be read out as the transduced input sequence. We also develop gradient-based methods allowing for efficient self-supervised sequence learning despite the discreteness of the bottleneck. Our results demonstrate that $\Sigma$AE significantly enhances performance on transduction tasks, even with minimal parallel data, offering a promising solution for weakly supervised learning scenarios.

The Gradient Descent-Ascent (GDA) algorithm, designed to solve minimax optimization problems, takes the descent and ascent steps either simultaneously (Sim-GDA) or alternately (Alt-GDA). While Alt-GDA is commonly observed to converge faster, the performance gap between the two is not yet well understood theoretically, especially in terms of global convergence rates. To address this theory-practice gap, we present fine-grained convergence analyses of both algorithms for strongly-convex-strongly-concave and Lipschitz-gradient objectives. Our new iteration complexity upper bound of Alt-GDA is strictly smaller than the lower bound of Sim-GDA; i.e., Alt-GDA is provably faster. Moreover, we propose Alternating-Extrapolation GDA (Alex-GDA), a general algorithmic framework that subsumes Sim-GDA and Alt-GDA, for which the main idea is to alternately take gradients from extrapolations of the iterates. We show that Alex-GDA satisfies a smaller iteration complexity bound, identical to that of the Extra-gradient method, while requiring less gradient computations. We also prove that Alex-GDA enjoys linear convergence for bilinear problems, for which both Sim-GDA and Alt-GDA fail to converge at all.

It is desirable in many multi-objective machine learning applications, such as multi-task learning with conflicting objectives and multi-objective reinforcement learning, to find a Pareto solution that can match a given preference of a decision maker. These problems are often large-scale with available gradient information but cannot be handled very well by the existing algorithms. To tackle this critical issue, this paper proposes a novel predict-and-correct framework for locating a Pareto solution that fits the preference of a decision maker. In the proposed framework, a constraint function is introduced in the search progress to align the solution with a user-specific preference, which can be optimized simultaneously with multiple objective functions. Experimental results show that our proposed method can efficiently find a particular Pareto solution under the demand of a decision maker for standard multiobjective benchmark, multi-task learning, and multi-objective reinforcement learning problems with more than thousands of decision variables. Code is available at: https://github.com/xzhang2523/pmgda. Our code is current provided in the pgmda.rar attached file and will be open-sourced after publication.}