skip to main content

Explore Research Products in the NSF-PAR

Title: Performance of noisy Nesterov's accelerated method for strongly convex optimization problems
We study the performance of noisy gradient descent and Nesterov's accelerated methods for strongly convex objective functions with Lipschitz continuous gradients. The steady-state second-order moment of the error in the iterates is analyzed when the gradient is perturbed by an additive white noise with zero mean and identity covariance. For any given condition number κ, we derive explicit upper bounds on noise amplification that only depend on κ and the problem size. We use quadratic objective functions to derive lower bounds and to demonstrate that the upper bounds are tight up to a constant factor. The established upper bound for Nesterov's accelerated method is larger than the upper bound for gradient descent by a factor of √κ. This gap identifies a fundamental tradeoff that comes with acceleration in the presence of stochastic uncertainties in the gradient evaluation.  more » « less
Award ID(s):
1809833
NSF-PAR ID:
10128666
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
2019 American Control Conference (ACC)
Page Range / eLocation ID:
3426 to 3431
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ( , Proceedings of Machine Learning Research)
    Momentum methods such as Polyak's heavy ball (HB) method, Nesterov's accelerated gradient (AG) as well as accelerated projected gradient (APG) method have been commonly used in machine learning practice, but their performance is quite sensitive to noise in the gradients. We study these methods under a first-order stochastic oracle model where noisy estimates of the gradients are available. For strongly convex problems, we show that the distribution of the iterates of AG converges with the accelerated linear rate to a ball of radius " centered at a unique invariant distribution in the 1-Wasserstein metric where  is the condition number as long as the noise variance is smaller than an explicit upper bound we can provide. Our analysis also certifies linear convergence rates as a function of the stepsize, momentum parameter and the noise variance; recovering the accelerated rates in the noiseless case and quantifying the level of noise that can be tolerated to achieve a given performance. To the best of our knowledge, these are the first linear convergence results for stochastic momentum methods under the stochastic oracle model. We also develop finer results for the special case of quadratic objectives, extend our results to the APG method and weakly convex functions showing accelerated rates when the noise magnitude is sufficiently small. 
    more » « less
  2. ; ; ; ( , Advances in Neural Information Processing Systems)
    In this work, we study the convergence \emph{in high probability} of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1 more » « less
  3. ; ; ; ( , Advances in neural information processing systems)
    In this work, we study the convergence in high probability of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1< p \leq 2$. Prior works in this setting follow the same recipe of using concentration inequalities and an inductive argument with union bound to bound the iterates across all iterations. This method results in an increase in the failure probability by a factor of $T$, where $T$ is the number of iterations. We instead propose a new analysis approach based on bounding the moment generating function of a well chosen supermartingale sequence. We improve the dependency on $T$ in the convergence guarantee for a wide range of algorithms with clipped gradients, including stochastic (accelerated) mirror descent for convex objectives and stochastic gradient descent for nonconvex objectives. Our high probability bounds achieve the optimal convergence rates and match the best currently known in-expectation bounds. Our approach naturally allows the algorithms to use time-varying step sizes and clipping parameters when the time horizon is unknown, which appears difficult or even impossible using existing techniques from prior works. Furthermore, we show that in the case of clipped stochastic mirror descent, several problem constants, including the initial distance to the optimum, are not required when setting step sizes and clipping parameters. 
    more » « less
  4. ; ; ; ( , American Control Conference)
    This work proposes an accelerated first-order algorithm we call the Robust Momentum Method for optimizing smooth strongly convex functions. The algorithm has a single scalar parameter that can be tuned to trade off robustness to gradient noise versus worst-case convergence rate. At one extreme, the algorithm is faster than Nesterov's Fast Gradient Method by a constant factor but more fragile to noise. At the other extreme, the algorithm reduces to the Gradient Method and is very robust to noise. The algorithm design technique is inspired by methods from classical control theory and the resulting algorithm has a simple analytical form. Algorithm performance is verified on a series of numerical simulations in both noise-free and relative gradient noise cases. 
    more » « less
  5. ; ; ; ( , Journal of machine learning research)
    The theory of integral quadratic constraints (IQCs) allows the certification of exponential convergence of interconnected systems containing nonlinear or uncertain elements. In this work, we adapt the IQC theory to study first-order methods for smooth and strongly-monotone games and show how to design tailored quadratic constraints to get tight upper bounds of convergence rates. Using this framework, we recover the existing bound for the gradient method~(GD), derive sharper bounds for the proximal point method~(PPM) and optimistic gradient method~(OG), and provide for the first time a global convergence rate for the negative momentum method~(NM) with an iteration complexity O(κ1.5), which matches its known lower bound. In addition, for time-varying systems, we prove that the gradient method with optimal step size achieves the fastest provable worst-case convergence rate with quadratic Lyapunov functions. Finally, we further extend our analysis to stochastic games and study the impact of multiplicative noise on different algorithms. We show that it is impossible for an algorithm with one step of memory to achieve acceleration if it only queries the gradient once per batch (in contrast with the stochastic strongly-convex optimization setting, where such acceleration has been demonstrated). However, we exhibit an algorithm which achieves acceleration with two gradient queries per batch. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email