Source-linked AI summary
Communication-Efficient Algorithms for Decentralized and Stochastic Optimization
Guanghui Lan, Soomin Lee, Yi Zhou
TL;DR
Decentralized optimization is hindered by costly communication, difficult local subproblems, and settings where only noisy first-order information is available. The paper develops primal-dual communication-sliding methods, including stochastic SDCS, and establishes low communication complexity while preserving subgradient-evaluation bounds. For general convex and strongly convex objectives, DCS and SDCS achieve O(1/ε) and O(1/√ε) communication rounds, respectively, with corresponding O(1/ε^2) and O(1/ε) evaluation bounds.
Problem
Decentralized optimization requires communication among agents, while dual methods may leave local subproblems inefficient to solve and stochastic settings provide only noisy first-order information.
Method
The paper develops decentralized primal-dual communication-sliding methods that skip communications during iterative local subproblem solution, plus a stochastic counterpart called SDCS.
Results
DCS and SDCS achieve O(1/ε) or O(1/√ε) communication rounds for general convex or strongly convex objectives, with O(1/ε^2) or O(1/ε) subgradient evaluations.
Takeaways & Limitations
The methods reduce inter-node communication while retaining subgradient-evaluation bounds comparable to centralized nonsmooth and stochastic optimization.
Takeaways & Limitations
The approach assumes local objective and constraint information is private to each agent, and dual methods remain difficult when local subproblems cannot be solved efficiently.
Abstract
from arXiv · showhide
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 $ε$-solution both in terms of functional optimality gap and feasibility residual in $O(1/ε)$ 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 $ε$-solution in $O(1/ε)$ (resp., $O(1/\sqrtε)$) communication rounds for general convex functions (resp., strongly convex functions), while maintaining the $O(1/ε^2)$ (resp., $O(1/ε)$) 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.
1 Introduction
The paper targets decentralized convex and stochastic optimization where communication is costly, local subproblems may be difficult, and agents have only partial or noisy information. It develops primal-dual, DCS, and SDCS methods with separate communication and subgradient-computation guarantees.
- Problem setting: Decentralized optimization requires agents to communicate neighboring information while cooperatively minimizing a sum of private local objectives.The network enforces agreement across agents despite each agent lacking full knowledge of the global problem.
- Communication bottleneck: Dual methods can use fewer communication rounds, but their local Lagrangian subproblems cannot be solved efficiently in many cases.This motivates methods that retain dual-style communication efficiency while using first-order information locally.
- Communication bottleneck: O(1/ε^2) inter-node communications and intra-node gradient computations are required by decentralized subgradient methods for an ε-optimal solution.Existing methods can be slow because each iteration entails communication and diminishing stepsizes.
- Primal-dual framework: The proposed primal-dual framework reaches an ε-optimal solution in both primal optimality gap and feasibility residual using O(1/ε) communication rounds when local subproblems are solved exactly.The framework uses only two inter-node communications per iteration and serves as a communication benchmark.
- DCS methods: DCS skips communications during iterative local subproblem solution through successive linearizations, achieving O(1/ε) or O(1/√ε) communication rounds for general or strongly convex objectives.It maintains O(1/ε^2) or O(1/ε) intra-node subgradient evaluations, respectively.
- Stochastic extension: SDCS extends communication sliding to stochastic optimization, with O(1/ε) or O(1/√ε) communication rounds and up to O(1/ε^2) or O(1/ε) stochastic subgradient evaluations.For finite-sum objectives, each iteration uses one randomly selected component, giving up to an O(l) savings over DCS.
- Contribution: The paper presents separate communication-round and subgradient-evaluation complexity bounds for decentralized communication sliding algorithms.The authors characterize this as a first presentation of these algorithms and separate bounds in the literature.
2 Preliminaries
The preliminaries establish prox-functions and gap-based criteria for measuring approximate decentralized solutions, including separate optimality and feasibility residuals despite an unbounded dual feasible set.
- Distance Generating Functions and Prox-functions: Prox-functions generalize Euclidean projections by allowing the geometry of a constraint set to determine proximity.They are induced by distance-generating functions that are continuously differentiable and strongly convex with respect to the chosen norm.
- Distance Generating Functions and Prox-functions: The network-level prox-function aggregates the agents’ individual prox-functions, optionally weighted by coefficients p_i.The weights can be selected using information about the individual constraint sets.
- Gap Functions and Termination Criteria: The primal-dual gap function measures the accuracy of an approximate solution to the saddle-point formulation.At a saddle point, the gap has the expected nonpositive or nonnegative orientation depending on its argument order.
- Gap Functions and Termination Criteria: Because the saddle-point feasible set may be unbounded, the paper uses a perturbation-based gap function for termination.The perturbed gap function is defined over a closed dual set and is designed to support separate objective and feasibility bounds.
- Gap Functions and Termination Criteria: An (ε, δ)-solution has primal residual ε and feasibility residual δ, while the stochastic version bounds their expectations.For the decentralized formulation, feasibility residual measures disagreement among agents’ local copies.
3 Decentralized Primal-Dual
The decentralized primal-dual framework adapts saddle-point methods to networked agents and establishes communication and subgradient complexity for approximate solutions.
- Framework and Goals: The decentralized primal-dual framework adapts a saddle-point method to networked optimization and separately analyzes optimality and consistency violation.The basic method also provides a benchmark for the communication cost of the later communication-sliding algorithms.
- The Algorithm: Each agent maintains local primal estimates and dual variables associated with its consistency constraints with neighboring agents.The primal estimates represent local decision-variable estimates, while the dual variables enforce network consistency.
- The Algorithm: Each iteration uses two communication rounds: one for primal estimates and one for updated dual variables, followed by a local proximal projection subproblem.Agents first broadcast primal predictions, compute neighborhood disagreement and dual updates, then communicate the updated dual variables.
- Assumptions and Analysis: The presented analysis initially treats general convex local objectives, postponing strongly convex cases to the communication-sliding sections.The method’s parameters are later instantiated to establish convergence properties.
- Complexity: O(1/ε) communication rounds achieve an (ε, δ)-solution’s primal functional optimality, while feasibility requires O(1/δ) rounds.These orders follow because each iteration uses a constant number of communication rounds.
4 Decentralized Communication Sliding
DCS approximately solves difficult local primal subproblems through communication sliding, preserving communication complexity while shifting work into intra-node subgradient evaluations.
- Algorithmic framework: DCS iteratively solves otherwise difficult local primal subproblems through linearizations and a communication-sliding inner procedure.The CS inner loop repeatedly computes local subgradients and solves relatively easy projection subproblems.
- Communication reduction: DCS maintains the same inter-node communication count as the exact method despite inexactly solving the local subproblems.The outer-loop design is modified to attain the best possible convergence rate with inexact subproblem solutions.
- Communication reduction: The same dual variable is reused across Tk inner iterations, so these iterations require no additional dual-variable communications.This distinguishes DCS from accelerated gradient methods whose communication counts can grow with the outer iteration index.
- General convex case: For general convex objectives, the method establishes convergence and complexity bounds for computing an (ε, δ)-solution.The analysis uses weighted primal-dual iterates and parameter policies specified for the general convex case.
- Complexity: The total subgradient evaluations are comparable to classic mirror descent under the stated parameter conditions and are not improvable in general.This comparison is stated for both the general convex and nonsmooth strongly convex settings.
- Strongly convex case: DCS uses variable stepsizes to accelerate convergence for strongly convex objectives while retaining comparable subgradient complexity to centralized mirror descent.The strongly convex results also provide bounds on communication rounds and intra-node subgradient evaluations.
5 Stochastic Decentralized Communication Sliding
SDCS extends communication sliding to stochastic first-order information, preserving communication savings while matching centralized stochastic subgradient complexity in supported regimes.
- Stochastic framework: SDCS addresses stochastic objectives by replacing exact subgradients in DCS’s communication-sliding procedure with stochastic-oracle subgradients.The oracle returns a random vector at a queried point, modeling noisy first-order information.
- Communication reduction: SDCS performs Tk stochastic inner iterations without additional communications of the dual variables.The same dual variable is used throughout the inner loop, as in deterministic DCS.
- General convex case: For general convex objectives, the paper establishes expected convergence and complexity bounds for stochastic (ε, δ)-solutions.The result is derived under the stated stochastic assumptions and parameter choices.
- Complexity: SDCS requires a total number of stochastic subgradients comparable to mirror-descent stochastic approximation, even after skipping many communication rounds.The paper characterizes this sample complexity as optimal relative to centralized stochastic optimization.
- Strongly convex case: For strongly convex objectives, SDCS uses variable stepsizes and achieves stated stochastic communication and subgradient complexity bounds.The strongly convex analysis assumes quadratically growing prox-functions and compares stochastic subgradient computations with centralized optimal bounds.
- High-probability results: Under compactness and additional stochastic assumptions, SDCS provides high-probability convergence results with both primal and feasibility residuals of order O(1/N).The corollary explicitly states this rate for nonsmooth convex objectives.
6 Convergence Analysis
The convergence analysis builds recursive bounds for the primal-dual method, DCS, and SDCS by analyzing their projection steps and aggregating inner- and outer-loop inequalities.
- Projection analysis: The analysis characterizes primal, dual, and inner-loop projection steps through a common lemma for prox-based subproblems.This lemma underlies the deterministic and stochastic convergence proofs.
- Outer-loop recursion: The outer-loop recursion combines local projection inequalities with the decentralized gap-function definition and parameter conditions.The proof repeatedly telescopes Bregman-distance terms across iterations.
- Deterministic and stochastic analysis: DCS and SDCS convergence proofs use weighted summation of inner-loop inequalities and convexity of the gap function.The resulting bounds control approximate primal-dual iterates and saddle-point evaluations.
- Relationship between methods: SDCS contains DCS as the special case in which the stochastic error δt is zero for every inner iteration.The paper therefore proves the stochastic lemmas first and specializes them to the deterministic setting.
7 Concluding Remarks
The paper’s concluding result is that communication sliding substantially reduces inter-node communication while retaining centralized-comparable subgradient complexity for deterministic and stochastic decentralized optimization.
- Communication complexity: The proposed decentralized primal-dual methods reduce communication rounds to O(1/ε) for convex, potentially nonsmooth objectives.This communication target applies to the distributed optimization problem considered in the paper.
- Local computation: The corresponding intra-node subgradient bounds are O(1/ε^2) for convex objectives and O(1/ε) for strongly convex objectives.The paper states these bounds are comparable to centralized nonsmooth optimization and not improvable in general.
- Stochastic extension: Stochastic communication-sliding methods provide analogous complexity guarantees for streaming data and decentralized statistical inference.The conclusion frames these methods as communication-efficient approaches for stochastic decentralized optimization.