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

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

Stochastic Gradient Optimization Techniques4 citations

TL;DR

This paper proposes adaptive quantization for distributed subgradient methods to achieve convergence rates matching unquantized methods under finite communication bandwidth. By dynamically adjusting quantizer resolution based on algorithm progress, the method maintains optimal convergence rates—within a constant factor dependent on quantizer resolution—under both convex and strongly convex objectives.

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. Recent advances using the distributed gradient algorithm with a quantization scheme at a fixed resolution have established convergence, but at rates significantly slower than when the communications are unquantized. In this paper, we introduce a novel quantization method, which we refer to as adaptive quantization, that allows us to match the convergence rates under perfect communications. Our approach adjusts the quantization scheme used by each node as the algorithm progresses: as we approach the solution, we become more certain about where the state variables are localized, and adapt the quantizer codebook accordingly. We bound the convergence rates of the proposed method as a function of the communication bandwidth, the underlying network topology, and structural properties of the constituent objective functions. In particular, we show that if the objective functions are convex or strongly convex, then using adaptive quantization does not affect the rate of convergence of the distributed subgradient methods when the communications are quantized, except for a constant that depends on the resolution of the quantizer. To the best of our knowledge, the rates achieved in this paper are better than any existing work in the literature for distributed gradient methods under finite communication bandwidths. We also provide numerical simulations that compare convergence properties of the distributed gradient methods with and without quantization for solving distributed regression problems for both quadratic and absolute loss functions.

Motivation & Objective

• Address the performance degradation in distributed subgradient methods due to quantized communication in bandwidth-constrained networks.
• Overcome the slow convergence of fixed-resolution quantization schemes in prior work.
• Design a quantization strategy that adapts to the algorithm's progress to preserve convergence rates.
• Establish theoretical convergence bounds for both convex and strongly convex objective functions under adaptive quantization.
• Demonstrate via simulations that 8-bit adaptive quantization achieves performance nearly identical to unquantized communication.

Proposed method

• Introduce a modified distributed subgradient algorithm that explicitly accounts for quantization error at each iteration.
• Design an adaptive quantization scheme where the quantizer codebook is adjusted at each iteration based on the current stepsize and estimated solution localization.
• Use a lazy Metropolis matrix as the mixing matrix to ensure network connectivity and satisfy the spectral gap condition.
• Apply a quantization rule that reduces the quantization interval as the algorithm approaches the solution, improving resolution in regions of interest.
• Theoretical analysis bounds convergence rates using the spectral gap $1 - ho_2$ and quantizer resolution $ riangle$.
• Leverage subgradient descent with diminishing stepsize and average consensus over a connected network to ensure convergence.

Experimental results

Research questions

• RQ1 Can adaptive quantization eliminate the performance gap between quantized and unquantized distributed subgradient methods?
• RQ2 What convergence rate can be achieved with adaptive quantization under finite communication bandwidth?
• RQ3 How does the choice of quantizer resolution affect convergence speed and final accuracy?
• RQ4 Can the convergence rate of distributed subgradient methods be preserved under quantization, matching the unquantized case?
• RQ5 How does adaptive quantization compare to fixed-resolution or random quantization in terms of convergence speed and stability?

Key findings

• For convex objectives, the convergence rate is bounded by $ f( extbf{z}_i(k)) - f^* riangleq rac{ riangle^2}{(1 - ho_2)^2} rac{ ext{ln } k}{ ext{sqrt}(k)} $, matching unquantized DSG up to a constant factor.
• For strongly convex objectives, the convergence rate is $ ext{||} extbf{z}_i(k) - extbf{x}^* ext{||}^2 riangleq rac{ riangle^2}{(1 - ho_2)^2} rac{ ext{ln } k}{k} $, again matching unquantized DSG within a constant.
• Numerical simulations show that 8-bit adaptive quantization achieves convergence performance nearly indistinguishable from unquantized communication.
• With only 4 bits, the algorithm still converges effectively, and the number of iterations scales approximately as $ 1/(2^b - 1)^2 $, matching theoretical upper bounds.
• Adaptive quantization outperforms both time-varying quantization (which requires infinite bits) and random quantization (which suffers from slow convergence).
• The proposed method achieves the best of both worlds: optimal convergence rates and finite-bit communication, unlike prior methods that sacrifice one for the other.