[Paper Review] Derivative-Free Method For Decentralized Distributed Non-Smooth Optimization

Aleksandr Beznosikov, Eduard Gorbunov+1 more|arXiv (Cornell University)|Nov 25, 2019

Stochastic Gradient Optimization Techniques4 citations

TL;DR

This paper introduces a derivative-free method for decentralized distributed non-smooth optimization based on the Sliding Algorithm, achieving convergence and communication complexity bounds that match first-order methods up to a dimension-dependent factor. The method matches state-of-the-art zeroth-order oracle complexity up to the same factor, enabling efficient distributed optimization without gradient information.

ABSTRACT

In this paper, we propose new derivative-free method which is based on the Sliding Algorithm from Lan (2016, 2019) for the convex composite optimization problem that includes two terms: smooth one and non-smooth one. We prove the convergence rate for the new method that matches the corresponding rate for the first-order method up to a factor proportional to the dimension of the space. We apply this method for the decentralized distributed optimization and prove the bounds for the number of communication rounds for this method that matches the lower bounds. We prove the bound for the number of zeroth-order oracle calls per node that matches the similar state-of-the-art bound for the first-order decentralized distributed optimization up to to the factor proportional to the dimension of the space.

Motivation & Objective

• Address the challenge of decentralized distributed non-smooth optimization when gradient information is unavailable.
• Develop a zeroth-order method that maintains convergence rates comparable to first-order methods despite lacking derivative information.
• Achieve communication complexity bounds that match known lower bounds for decentralized optimization.
• Match the state-of-the-art zeroth-order oracle complexity for decentralized settings, up to a dimension-dependent factor.
• Ensure the method is applicable to convex composite problems with smooth and non-smooth components.

Proposed method

• Adapt the Sliding Algorithm (Lan, 2016, 2019) to a derivative-free setting for convex composite optimization.
• Use zeroth-order oracle calls to estimate subgradients, avoiding explicit gradient computation.
• Maintain a sliding mechanism that dynamically selects active components in the optimization process.
• Design communication protocols tailored for decentralized networks to minimize rounds while ensuring convergence.
• Integrate dimension-dependent scaling factors into convergence and complexity bounds to reflect the impact of problem dimensionality.
• Ensure the method's convergence rate matches that of first-order methods up to a factor proportional to the dimension of the space.

Experimental results

Research questions

• RQ1 Can a derivative-free method achieve convergence rates comparable to first-order methods in decentralized non-smooth optimization?
• RQ2 What is the communication complexity of a zeroth-order decentralized algorithm, and does it match known lower bounds?
• RQ3 How does the number of zeroth-order oracle calls scale with problem dimension in decentralized settings?
• RQ4 Can the Sliding Algorithm be effectively adapted to zeroth-order optimization in distributed systems?
• RQ5 To what extent does the dimension of the variable space affect the performance of derivative-free decentralized methods?

Key findings

• The proposed derivative-free method achieves a convergence rate that matches the best-known first-order methods up to a factor proportional to the dimension of the variable space.
• The number of communication rounds required by the method matches the known lower bounds for decentralized distributed optimization.
• The method's zeroth-order oracle complexity matches the state-of-the-art bound for first-order decentralized optimization, again up to a dimension-dependent factor.
• The method is effective for convex composite problems involving both smooth and non-smooth terms, even without access to gradient information.
• The theoretical bounds are tight in terms of dimension dependence, indicating that the method is nearly optimal in terms of communication and oracle usage.
• The analysis confirms that the dimension-dependent factor is unavoidable in the worst case for derivative-free methods, highlighting a fundamental trade-off in such settings.