Communication-Efficient Algorithms for Decentralized and Stochastic Optimization (1701.03961v2)

Abstract: We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. We first propose a decentralized primal-dual method which can find an $\epsilon$-solution both in terms of functional optimality gap and feasibility residual in $O(1/\epsilon)$ inter-node communication rounds when the objective functions are convex and the local primal subproblems are solved exactly. Our major contribution is to present a new class of decentralized primal-dual type algorithms, namely the decentralized communication sliding (DCS) methods, which can skip the inter-node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions. By employing DCS, agents can still find an $\epsilon$-solution in $O(1/\epsilon)$ (resp., $O(1/\sqrt{\epsilon})$) communication rounds for general convex functions (resp., strongly convex functions), while maintaining the $O(1/\epsilon^2)$ (resp., $O(1/\epsilon)$) bound on the total number of intra-node subgradient evaluations. We also present a stochastic counterpart for these algorithms, denoted by SDCS, for solving stochastic optimization problems whose objective function cannot be evaluated exactly. In comparison with existing results for decentralized nonsmooth and stochastic optimization, we can reduce the total number of inter-node communication rounds by orders of magnitude while still maintaining the optimal complexity bounds on intra-node stochastic subgradient evaluations. The bounds on the subgradient evaluations are actually comparable to those required for centralized nonsmooth and stochastic optimization.

• The paper introduces the DCS and SDCS methods that achieve ε-optimal solutions with significantly fewer communication rounds.
• It presents decentralized primal-dual frameworks with convergence rates of O(1/ε) for convex and O(1/√ε) for strongly convex problems.
• These methods are pivotal for distributed machine learning and sensor networks, effectively reducing communication overhead in decentralized systems.

Communication-Efficient Algorithms for Decentralized and Stochastic Optimization: An Academic Overview

Decentralized optimization is an area of growing interest, particularly in applications related to signal processing, decentralized machine learning, and a variety of networked systems. The paper by Lan, Lee, and Zhou investigates this domain by presenting a novel series of decentralized primal-dual algorithms aimed at overcoming a critical bottleneck: the communication overhead among network nodes. This paper specifically addresses optimization problems characterized by nonsmooth, stochastic objective functions over multiagent network structures.

The core contribution of this work is the development of the decentralized communication sliding (DCS) method and its stochastic counterpart, SDCS. These algorithms effectively reduce the number of inter-node communication rounds without compromising the overall computational complexity associated with intra-node subgradient evaluations. This is significant given the growing disparity between intra-node processing speeds and inter-node communication bandwidths.

Algorithmic Insights

The paper delineates a new class of decentralized primal-dual frameworks for solving optimization problems involving convex and strongly convex objective functions. It extends this approach to stochastic settings where the objective functions can only be estimated through noisy observations. The primary components of these methodologies include:

1. Decentralized Primal-Dual Method: Achieves an $O(1/\epsilon)$ communication complexity for convex functions and $O(1/\sqrt{\epsilon})$ complexity for strongly convex functions. This establishes a baseline in terms of communication efficiency.
2. DCS and SDCS Methods: By leveraging successive linearizations and skipping communication rounds, these methods find $\epsilon$-optimal solutions with only $O(1/\epsilon)$ (and $O(1/\sqrt{\epsilon})$ for strongly convex problems) total communication rounds. This is in conjunction with subgradient evaluations comparable to those in centralized approaches, particularly under certain accuracy constraints.

These innovative schemes are pivotal in reducing communication costs—showing potential reductions by orders of magnitude—while preserving theoretical bounds on the number of subgradient computations.

Theoretical and Practical Implications

This research holds several implications:

• For Decentralized Optimization Theory: It advances fundamental understanding of primal-dual methods in decentralized settings, shedding light on how such methods can be refined to achieve communication efficiency.
• For Practical Applications: Ensuring low communication overheads while maintaining computational rigor is crucial for practical applications in distributed machine learning and real-time data analytics over sensor networks.

The paper also explores theoretical aspects, presenting assumptions and proving convergence rates under given conditions. Assumptions about the network's connectivity and the accessible noisy first-order information are rigorously handled to establish a strong theoretical foundation for the proposed algorithms.

Conclusion and Future Directions

Lan, Lee, and Zhou's work on communication-efficient algorithms provides a robust foundation for further research in decentralized and stochastic optimization. Future investigations might explore extending these methods to handle dynamic networks with changing connectivity or adapting them to optimize other function classes like nonconvex objectives. Moreover, practical implementation in real-world distributed systems can provide empirical insights, potentially leading to further refinements and optimizations.

In conclusion, this paper advances the field by addressing a critical challenge in decentralized optimization, leveraging primal-dual methodologies to achieve significant reductions in communication overhead, a critical constraint in real-world networked systems.