Quantized Distributed Gradient Tracking Algorithm with Linear Convergence in Directed Networks

Abstract

Communication efficiency is a major bottleneck in the applications of distributed networks. To address the problem, the problem of quantized distributed optimization has attracted a lot of attention. However, most of the existing quantized distributed optimization algorithms can only converge sublinearly. To achieve linear convergence, this paper proposes a novel quantized distributed gradient tracking algorithm (Q-DGT) to minimize a finite sum of local objective functions over directed networks. Moreover, we explicitly derive the update rule for the number of quantization levels, and prove that Q-DGT can converge linearly even when the exchanged variables are respectively one bit. Numerical results also confirm the efficiency of the proposed algorithm.

Communication Compression for Distributed Nonconvex Optimization

“…the second inequality holds due to the Cauchy-Schwarz inequality; and the last inequality holds due to (28). Denote C r (•) = C(•)/r, then we have…”

“…algorithms; [15] employed biased but contractive compressors to design a decentralized SGD algorithm; [16] and [17], [18] utilized unbiased compressors to respectively design decentralized gradient descent and primal-dual algorithms; [19] and [20] made use of the standard uniform quantizer to respectively design decentralized subgradient methods and alternating direction method of multipliers appraoches; [21], [22] and [23] respectively adopted the unbiased random quantization and the adaptive quantization to design decentralized projected subgradient algorithms; [24] and [25]- [28] exploited the standard uniform quantizer with dynamic quantization level to respectively design decentralized subgradient and primal-dual algorithms; and [29] applied the standard uniform quantizer with a fixed quantization level to design a decentralized gradient descent algorithm. The compressors mentioned above can be unified into three general classes.…”

“…Especially, some of them showed that the achieved convergence rates under compressed communication are comparable to and even match those under accurate communication. For instance, linear convergence was achieved in [17], [18], [25]- [28], [30], [32] under the standard strong convexity assumption.…”

“…The same class of compressors satisfying Assumption 3 has also been used in [32], which covers the standard uniform quantizer with dynamic and fixed quantization levels respectively used in [24]- [28] and [29], and the Moniqua used in [11]. Moreover, as pointed out in [32],…”

“…Moreover, [32] required an unpractical condition that the unique optimal point needs to be known a priori to design algorithm parameters, which is a drawback. Thirdly, compared to [25]- [28] which used the standard uniform quantizer with dynamic quantization level and established linear convergence under the condition that each local cost function is strongly convex, we use the more general compressors and the weaker P-Ł condition to show linear convergence. Finally, compared to [29] which used the standard uniform quantizer with fixed quantization level and established linear convergence under the assumption that each local cost function is quadratic and the global cost function is strongly convex, we not only use the more general compressors but also consider the more general nonconvex functions satisfying the weaker P-Ł condition.…”