[Paper Review] Accelerating the Convergence Rates of Distributed Subgradient Methods with Adaptive Quantization

Thinh T. Doan, Siva Theja Maguluri+1 more|arXiv (Cornell University)|Oct 30, 2018

Distributed Control Multi-Agent Systems10 citations

TL;DR

This paper proposes an adaptive quantization method for distributed subgradient algorithms that eliminates quantization errors by dynamically adjusting quantization levels based on algorithm progress. With this method, convergence rates match those of unquantized methods, differing only by a factor dependent on the number of quantization bits, for both convex and strongly convex functions.

ABSTRACT

We study distributed optimization problems over a network when the communication between the nodes is constrained, and so information that is exchanged between the nodes must be quantized. This imperfect communication poses a fundamental challenge, and this imperfect communication, if not properly accounted for, prevents the convergence of these algorithms. In this paper, our main contribution is to propose a novel quantization method, which we refer to as an adaptive quantization. The main idea of our approach is to quantize the nodes' estimates based on the progress of the algorithm, which helps to eliminate the quantized errors. Under the adaptive quantization, we then derive the bounds on the convergence rates of the proposed method as a function of the bandwidths and the underlying network topology, for both convex and strongly convex objective functions. Our results shows that using the adaptive quantization, the rate of convergence of distributed consensus-based subgradient methods with and without quantization are the same, except for a factor which captures the number of quantization bits. Finally, we provide numerical simulations to compare the convergence properties of the distributed gradient methods with and without quantization for solving the well-known regression problems over networks, for both quadratic and absolute loss functions.

Motivation & Objective

• Address the challenge of communication constraints in distributed optimization, where node-to-node information exchange is limited by bandwidth.
• Overcome the fundamental issue that standard quantization degrades convergence performance in distributed subgradient methods.
• Design a quantization strategy that adapts to algorithm progress to eliminate persistent quantization errors.
• Analyze convergence rates under adaptive quantization for both convex and strongly convex objective functions.
• Demonstrate that the convergence rate with adaptive quantization matches that of unquantized methods, up to a factor depending on quantization bits.

Proposed method

• Propose an adaptive quantization scheme that adjusts quantization levels based on the current progress of the distributed subgradient algorithm.
• Introduce a dynamic quantization rule that reduces quantization error over time by leveraging the decreasing subgradient norm as the algorithm converges.
• Formulate convergence bounds for the distributed subgradient method under adaptive quantization, explicitly depending on network topology and bandwidth (number of bits).
• Derive theoretical convergence rates for both convex and strongly convex functions under the proposed quantization framework.
• Use a consensus-based update rule where each node updates its estimate using quantized information from neighbors, with quantization levels adjusted iteratively.
• Validate the method through numerical simulations on regression problems using quadratic and absolute loss functions under various quantization levels.

Experimental results

Research questions

• RQ1 Can adaptive quantization eliminate the convergence degradation typically caused by fixed quantization in distributed subgradient methods?
• RQ2 How does the convergence rate of the distributed subgradient method with adaptive quantization compare to that of the unquantized method?
• RQ3 What is the dependence of the convergence rate on the number of quantization bits and network topology?
• RQ4 Does the adaptive quantization strategy maintain convergence for both convex and strongly convex objective functions?
• RQ5 How do the convergence properties of the method vary across different loss functions, such as quadratic and absolute loss, in regression problems?

Key findings

• The proposed adaptive quantization method achieves the same convergence rate as the unquantized distributed subgradient method, differing only by a factor that depends on the number of quantization bits.
• For convex functions, the convergence rate under adaptive quantization is O(1/√t) up to a logarithmic factor in the number of bits, matching the unquantized rate.
• For strongly convex functions, the convergence rate is O(1/t) under adaptive quantization, again matching the unquantized rate up to a bit-dependent factor.
• The convergence bounds explicitly depend on the network's algebraic connectivity and the number of quantization bits, showing the impact of topology and bandwidth.
• Numerical simulations confirm that the adaptive quantization method converges faster and more stably than fixed quantization methods in regression problems with both quadratic and absolute loss functions.
• The method effectively eliminates persistent quantization errors that otherwise hinder convergence in standard quantized distributed algorithms.