More Like this

1. Low-Precision Stochastic Gradient Langevin Dynamics

   Zhang, Ruqi; Wilson, Andrew Gordon; De Sa, Christopher (July 2022, Proceedings of Machine Learning Research)

   Kamalika Chaudhuri, Stefanie Jegelka (Ed.)

   While low-precision optimization has been widely used to accelerate deep learning, low-precision sampling remains largely unexplored. As a consequence, sampling is simply infeasible in many large-scale scenarios, despite providing remarkable benefits to generalization and uncertainty estimation for neural networks. In this paper, we provide the first study of low-precision Stochastic Gradient Langevin Dynamics (SGLD), showing that its costs can be significantly reduced without sacrificing performance, due to its intrinsic ability to handle system noise. We prove that the convergence of low-precision SGLD with full-precision gradient accumulators is less affected by the quantization error than its SGD counterpart in the strongly convex setting. To further enable low-precision gradient accumulators, we develop a new quantization function for SGLD that preserves the variance in each update step. We demonstrate that low-precision SGLD achieves comparable performance to full-precision SGLD with only 8 bits on a variety of deep learning tasks. 

   more » « less
2. Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo

   Gürbüzbalaban, M.; Gao, X.; Hu, Y.; Zhu, L. (January 2021, Journal of machine learning research)

   Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian Monte Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC) algorithms for Bayesian inference that can scale to large datasets, allowing to sample from the posterior distribution of the parameters of a statistical model given the input data and the prior distribution over the model parameters. However, these algorithms do not apply to the decentralized learning setting, when a network of agents are working collaboratively to learn the parameters of a statistical model without sharing their individual data due to privacy reasons or communication constraints. We study two algorithms: Decentralized SGLD (DE-SGLD) and Decentralized SGHMC (DE-SGHMC) which are adaptations of SGLD and SGHMC methods that allow scaleable Bayesian inference in the decentralized setting for large datasets. We show that when the posterior distribution is strongly log-concave and smooth, the iterates of these algorithms converge linearly to a neighborhood of the target distribution in the 2-Wasserstein distance if their parameters are selected appropriately. We illustrate the efficiency of our algorithms on decentralized Bayesian linear regression and Bayesian logistic regression problems 

   more » « less
3. Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning

   Lamperski, A. (August 2021, Proceedings of Thirty Fourth Conference on Learning Theory)

   Belkin, M.; Kpotufe, S. (Ed.)

   Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov Chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with compact convex constraint sets and IID external data variables. We term the resulting method the projected stochastic gradient Langevin algorithm (PSGLA). We show the algorithm achieves a deviation of 𝑂(𝑇−1/4(𝑙𝑜𝑔𝑇)1/2) from its target distribution in 1-Wasserstein distance. For optimization and learning, we show that the algorithm achieves 𝜖-suboptimal solutions, on average, provided that it is run for a time that is polynomial in 𝜖 and slightly super-exponential in the problem dimension. 

   more » « less
4. A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization

   https://doi.org/10.1287/stsy.2021.0083  

   Liu, Tianyi; Chen, Zhehui; Zhou, Enlu; Zhao, Tuo (December 2021, Stochastic Systems)

   Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks. 

   more » « less
5. Efficient Privacy-Preserving Stochastic Nonconvex Optimization

   Lingxiao Wang, Bargav Jayaraman (January 2023, Proceedings of Machine Learning Research)

   Robin J. Evans, Ilya Shpitser (Ed.)

   In Proceedings of Uncertainty in Artificial Intelligence, 31-4 August 2023, Pittsburgh, PA, USA While many solutions for privacy-preserving convex empirical risk minimization (ERM) have been developed, privacy-preserving nonconvex ERM remains a challenge. We study nonconvex ERM, which takes the form of minimizing a finite-sum of nonconvex loss functions over a training set. We propose a new differentially private stochastic gradient descent algorithm for nonconvex ERM that achieves strong privacy guarantees efficiently, and provide a tight analysis of its privacy and utility guarantees, as well as its gradient complexity. Our algorithm reduces gradient complexity while matching the best-known utility guarantee. Our experiments on benchmark nonconvex ERM problems demonstrate superior performance in terms of both training cost and utility gains compared with previous differentially private methods using the same privacy budgets. 

   more » « less