[Paper Review] Gradient and gradient-free methods for stochastic convex optimization with inexact oracle

Alexander Gasnikov, Pavel Dvurechensky+1 more|arXiv (Cornell University)|Feb 22, 2015

Stochastic Gradient Optimization Techniques7 references3 citations

TL;DR

This paper extends universal gradient and intermediate gradient methods to stochastic convex optimization under an inexact oracle model, where gradient information is corrupted by noise. It establishes convergence rates for both gradient-based and gradient-free methods (e.g., random direction search), proving that optimal iteration complexity is achievable even with noisy oracles, thus generalizing Nesterov's framework to practical, noisy settings.

ABSTRACT

In the paper we generalize universal gradient method (Yu. Nesterov) to strongly convex case and to Intermediate gradient method (Devolder-Glineur-Nesterov). We also consider possible generalizations to stochastic and online context. We show how these results can be generalized to gradient-free method and method of random direction search. But the main ingridient of this paper is assumption about the oracle. We considered the oracle to be inexact.

Motivation & Objective

• To generalize Nesterov's universal gradient method and the intermediate gradient method to the strongly convex and stochastic settings.
• To extend these methods to gradient-free optimization techniques such as random direction search.
• To analyze convergence properties under an inexact oracle model, where gradient evaluations are corrupted by noise.
• To establish optimal iteration complexity bounds for both gradient-based and derivative-free methods in the presence of inexact oracles.
• To unify theoretical analysis of first-order methods under uncertainty, bridging deterministic and stochastic optimization frameworks.

Proposed method

• Adapts Nesterov's universal gradient method to the strongly convex case and extends it to stochastic and online optimization settings.
• Introduces a modified update rule that accounts for inexact gradient information, using a recursive error compensation mechanism.
• Applies the intermediate gradient method framework to stochastic problems with inexact oracles, ensuring convergence under weaker assumptions.
• Derives convergence rates for both gradient-based and gradient-free methods under the inexact oracle model, using a unified analysis framework.
• Employs a stochastic approximation approach with diminishing step-sizes to handle noisy oracle queries in the optimization process.
• Generalizes the analysis to random direction search by modeling directional derivatives as noisy estimates of the true gradient.

Experimental results

Research questions

• RQ1 Can universal and intermediate gradient methods be extended to the stochastic and strongly convex settings with inexact oracles?
• RQ2 What convergence rates can be achieved for gradient-free methods such as random direction search under inexact oracle assumptions?
• RQ3 How does the presence of inexact gradient information affect the iteration complexity of first-order methods in convex optimization?
• RQ4 Can optimal convergence rates be preserved when using noisy or approximate gradient information in stochastic optimization?
• RQ5 What modifications to standard first-order methods are required to maintain convergence and optimality under inexact oracle conditions?

Key findings

• The paper establishes optimal iteration complexity bounds for both gradient-based and gradient-free methods under an inexact oracle model.
• Convergence rates for the universal and intermediate gradient methods are derived and shown to be optimal in the stochastic setting with inexact gradients.
• The analysis confirms that gradient-free methods like random direction search can achieve the same convergence rates as gradient-based methods when the oracle is inexact.
• The proposed framework maintains optimal convergence under noise, with the error in gradient estimates controlled via recursive compensation.
• The results generalize Nesterov's framework to inexact oracles, extending its applicability to real-world problems with measurement or computational noise.
• The theoretical bounds are tight and match known lower bounds for stochastic convex optimization, confirming optimality.