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Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming
Saeed Ghadimi, Guanghui Lan
TL;DR
The paper asks whether Nesterov’s accelerated gradient method can extend beyond smooth convex optimization to nonconvex, composite, and stochastic problems. It modifies stepsize policies and develops stochastic AG variants, obtaining best-known, optimal, or improved convergence rates within the studied settings. The analysis also identifies computational trade-offs in mini-batch choices for stochastic composite problems.
Problem
The paper addresses the limited applicability of accelerated gradient methods to nonlinear programming that may be nonconvex, composite, or stochastic.
Method
The paper modifies AG stepsize policies for smooth and composite nonlinear programs and develops randomized stochastic AG methods, including mini-batching for stochastic composite problems.
Results
The methods attain the best-known rate for general smooth nonconvex NLP, optimal rates in convex settings, and improved rates for several nonconvex stochastic problems.
Takeaways & Limitations
AG can be analyzed across convex, nonconvex, composite, and stochastic optimization settings while retaining aggressive stepsize policies in the studied composite class.
Takeaways & Limitations
For stochastic composite problems, constant mini-batches may prevent bounds from vanishing, whereas very large mini-batches make each iteration expensive.
Abstract
from arXiv · showhide
In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature.
1 Introduction
The paper extends Nesterov’s accelerated gradient method from smooth convex optimization to nonconvex, composite, and stochastic nonlinear programming. It establishes best-known or improved convergence guarantees through tailored stepsize policies and stochastic variants.
- Nesterov’s accelerated gradient method improves convex smooth optimization complexity from O(1/ε) for gradient descent to O(1/√ε).
- The paper aims to provide a unified theoretical treatment of convex, nonconvex, and stochastic optimization while exploiting locally convex structure in general nonlinear objectives.The motivation includes understanding whether AG’s aggressive convex stepsize policy remains applicable beyond globally convex problems.
- With a suitable stepsize policy, AG finds an ε-solution of general smooth nonconvex NLP in at most O(1/ε) iterations.The criterion is a point x̄ satisfying ∥∇Ψ(x̄)∥2 ≤ ε.
- For composite problems, AG uses the same aggressive convex-case stepsize policy whether the objective is convex or nonconvex.The nonsmooth component requires a gradient-mapping termination criterion rather than a gradient norm.
- The composite AG method achieves an optimal functional-gap rate when the objective is convex and improves an existing projected-gradient bound for nonconvex problems.The improvement is especially significant when L_f is sufficiently small, and the bound improves dependence on L_h.
- For stochastic NLP, the paper introduces randomized stochastic AG methods that extend convergence analysis to noisy first-order oracles and nonconvex stochastic composite problems.A mini-batch approach is used for stochastic composite optimization, improving several existing complexity results.
2 The accelerated gradient algorithm
The paper modifies accelerated gradient methods to converge on smooth nonconvex problems and develops a uniform treatment for convex and nonconvex composite optimization. Proper stepsize policies retain optimal convex rates while achieving best-known or improved nonconvex rates.
- 2 The accelerated gradient algorithm: The modified AG method converges for general nonconvex smooth optimization after incorporating an appropriate stepsize policy.The section presents separate AG methods for general nonlinear problems and nonconvex composite problems.
- 2 The accelerated gradient algorithm: For convex problems, a more aggressive stepsize improves the complexity to O(1/ε^1/3), whereas matching β_k and λ_k reduces AG to gradient descent.The convex rate is contrasted with the O(1/ε) bound for general nonconvex problems.
- 2 The accelerated gradient algorithm: The composite AG method uses the same aggressive stepsize policies for convex and possibly nonconvex composite objectives.The composite objective includes a smooth component and a convex, possibly nonsmooth component with bounded domain.
- 2 The accelerated gradient algorithm: The composite analysis assumes an optimal solution exists and uses bounded-domain conditions to ensure the required proximal-search-point assumption.A proper closed convex function with bounded domain satisfies Assumption 2.
- 2 The accelerated gradient algorithm: For composite problems, the method can improve projected-gradient dependence on L_h, while projected gradient applies more broadly because it does not require a bounded domain for X.The comparison is stated for the convergence bounds discussed in the section.
3 The stochastic accelerated gradient method
The paper develops stochastic accelerated-gradient methods for smooth and composite optimization, including nonconvex problems, with stepsize policies tailored to convexity and stochasticity. The resulting RSAG methods match known nonconvex rates, attain optimal convex rates, and provide unified treatment of stochastic composite problems.
- 3.1 Minimization of stochastic smooth functions: The stochastic analysis permits dependent random vectors, requiring weaker conditions than the standard independent-sampling setting.The paper assumes stochastic-oracle conditions without requiring the random vectors to be independent.
- 3.1 Minimization of stochastic smooth functions: RSAG replaces exact gradients with stochastic ones and adds a randomized termination criterion for general stochastic nonlinear programming.The method is presented as a stochastic counterpart of the accelerated-gradient algorithm.
- 3.1 Minimization of stochastic smooth functions: For general smooth nonconvex stochastic problems, RSAG achieves the same convergence-rate order as RSG.The paper explicitly compares the nonconvex RSAG rate with the RSG rate.
- 3.1 Minimization of stochastic smooth functions: For convex stochastic problems, RSAG achieves an optimal convergence rate, improving over RSG's nearly optimal rate.This result follows from an appropriate specification of the stepsize policies.
- 3.2 Minimization of nonconvex stochastic composite functions: For stochastic composite problems, RSAG provides unified convergence analysis whether the objective is convex or nonconvex.The composite method uses gradient mappings and the same aggressive stepsize policy associated with the convex case.
- 3.2 Minimization of nonconvex stochastic composite functions: Composite stochastic bounds depend on minibatch sizes: constant minibatches may prevent convergence, whereas very large minibatches make each iteration expensive.The paper therefore develops specific minibatch-size selections and also notes dependence on the iteration limit in one choice.
4 Concluding remarks
The paper generalizes Nesterov’s accelerated gradient method to nonconvex and stochastic optimization, establishing strong convergence results across general and composite problems. It also introduces a randomized stochastic variant and claims the first analysis of AG for nonconvex optimization.
- The generalized AG method achieves the best known convergence rate for smooth nonconvex problems under a proper stepsize policy.The result is comparable to the gradient descent method.
- AG provides a uniform treatment for a class of composite problems whether they are convex or nonconvex.The same aggressive stepsize policy used in the convex case can be applied even when the composite problem is nonconvex.
- For convex composite problems, AG achieves an optimal convergence rate, while for nonconvex composite problems it improves the best known rate.
- A randomized stochastic AG method improves several existing convergence-rate results for nonconvex stochastic optimization.
- The paper presents the first literature analysis of Nesterov’s method for nonconvex optimization problems, to the authors’ knowledge.