Robustness of Accelerated First-Order Algorithms for Strongly Convex Optimization Problems

We study the robustness of accelerated first-order algorithms to stochastic uncertainties in gradient evaluation. Specifically, for unconstrained, smooth, strongly convex optimization problems, we examine the mean-squared error in the optimization variable when the iterates are perturbed by additive...

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Veröffentlicht in:IEEE transactions on automatic control Jg. 66; H. 6; S. 2480 - 2495
Hauptverfasser: Mohammadi, Hesameddin, Razaviyayn, Meisam, Jovanovic, Mihailo R.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.06.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Accelerated first-order algorithms
Acceleration
Algorithms
Computational geometry
consensus networks
control for optimization
Convergence
Convex analysis
convex optimization
Convexity
Eigenvalues
Evaluation
First order algorithms
Heuristic algorithms
integral quadratic constraints
linear matrix inequalities (LMIs)
Mathematical analysis
noise amplification
Noise measurement
Optimization
Robustness
second-order moments
semidefinite programming
Topology
Uncertainty
Upper bound
Upper bounds
White noise
ISSN:0018-9286, 1558-2523
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Zusammenfassung:We study the robustness of accelerated first-order algorithms to stochastic uncertainties in gradient evaluation. Specifically, for unconstrained, smooth, strongly convex optimization problems, we examine the mean-squared error in the optimization variable when the iterates are perturbed by additive white noise. This type of uncertainty may arise in situations where an approximation of the gradient is sought through measurements of a real system or in a distributed computation over a network. Even though the underlying dynamics of first-order algorithms for this class of problems are nonlinear, we establish upper bounds on the mean-squared deviation from the optimal solution that are tight up to constant factors. Our analysis quantifies fundamental tradeoffs between noise amplification and convergence rates obtained via any acceleration scheme similar to Nesterov's or heavy-ball methods. To gain additional analytical insight, for strongly convex quadratic problems, we explicitly evaluate the steady-state variance of the optimization variable in terms of the eigenvalues of the Hessian of the objective function. We demonstrate that the entire spectrum of the Hessian, rather than just the extreme eigenvalues, influences robustness of noisy algorithms. We specialize this result to the problem of distributed averaging over undirected networks and examine the role of network size and topology on the robustness of noisy accelerated algorithms.
Bibliographie:ObjectType-Article-1
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ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2020.3008297