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A Dual Approach for Optimal Algorithms in Distributed Optimization over Networks
César A. Uribe, Soomin Lee, Alexander Gasnikov, Angelia Nedić
TL;DR
The paper asks how distributed convex optimization over communication networks can attain optimal centralized rates despite local information and consensus constraints. It formulates a dual problem using graph structure, develops algorithms for dual-friendly and non-dual-friendly functions, and obtains centralized-optimal rates up to logarithmic factors with network-dependent costs.
Problem
Distributed agents must minimize a sum of locally known convex functions while exchanging information only through a communication network.
Method
The paper uses a graph-constrained primal formulation and its dual to design distributed algorithms, extending from exact to approximate dual subproblem solutions.
Results
The algorithms match centralized optimal complexity bounds up to logarithmic factors, with an additional cost determined by network spectral properties.
Takeaways & Limitations
The results provide convergence-rate estimates for four convexity and smoothness regimes while quantifying the communication price imposed by the network.
Takeaways & Limitations
The proposed algorithms require some global knowledge of graph properties and the network function’s condition number; settings without such information require further study.
Abstract
from arXiv · showhide
We study dual-based algorithms for distributed convex optimization problems over networks, where the objective is to minimize a sum $\sum_{i=1}^{m}f_i(z)$ of functions over in a network. We provide complexity bounds for four different cases, namely: each function $f_i$ is strongly convex and smooth, each function is either strongly convex or smooth, and when it is convex but neither strongly convex nor smooth. Our approach is based on the dual of an appropriately formulated primal problem, which includes a graph that models the communication restrictions. We propose distributed algorithms that achieve the same optimal rates as their centralized counterparts (up to constant and logarithmic factors), with an additional optimal cost related to the spectral properties of the network. Initially, we focus on functions for which we can explicitly minimize its Legendre-Fenchel conjugate, i.e., admissible or dual friendly functions. Then, we study distributed optimization algorithms for non-dual friendly functions, as well as a method to improve the dependency on the parameters of the functions involved. Numerical analysis of the proposed algorithms is also provided.
1. Introduction
The paper develops dual-based distributed algorithms for minimizing sums of locally known convex functions over communication networks. It targets centralized-optimal convergence rates, up to logarithmic factors, while accounting for network spectral costs and dual-friendliness.
- Problem setting: The problem minimizes a sum of convex functions known separately by agents that communicate only through a static network.The network setup imposes local-information and neighbor-exchange constraints.
- Scope: The paper treats four combinations of strong convexity and smoothness, including nonsmooth and non-strongly-convex cases.The cases cover strongly convex and smooth, strongly convex and nonsmooth, smooth and convex, and merely convex Lipschitz functions.
- Scope and limitations: The rates require some common knowledge of graph quantities, whose preprocessing is outside the paper’s scope.The analysis also indicates how graph and objective-function properties affect algorithm performance.
- Contributions: The dual formulation yields distributed algorithms matching centralized optimal complexity bounds up to logarithmic factors, with an additional network-induced cost.The complexity is expressed in terms of communication rounds and network spectral properties.
- Method: The approach first handles dual-friendly functions with explicitly solvable conjugate minimizations, then extends to cases where the dual subproblem is solved approximately.The latter extension addresses non-dual-friendly functions.
2. Preliminaries
The preliminaries characterize aggregate function parameters and establish Nesterov’s fast gradient method as the paper’s centralized optimization foundation.
- Function properties: For separable sums, aggregate strong convexity is the minimum individual modulus, while aggregate smoothness is the maximum individual smoothness constant.These definitions determine the global condition number used in later rates.
- Optimization foundation: The proposed algorithms are built from Nesterov’s fast gradient method, which is stated to be optimal for unconstrained strongly convex smooth minimization.The preliminaries include a constant-step variant and its parameter condition.
3. Problem Statement
The distributed problem replicates local decision variables and enforces consensus through a graph Laplacian. The formulation supports rate analysis across four objective-function regimes.
- Network model: Each agent holds a local function and can exchange information only with immediate neighbors in a fixed connected undirected graph.No single agent is assumed to possess the complete distributed data in the regression example.
- Consensus constraints: The communication constraints are encoded by the graph Laplacian, lifted to the decision dimension as W = W̄ ⊗ I_n.The Laplacian uses negative edge entries, node degrees on the diagonal, and zeros elsewhere.
- Consensus constraints: For a connected undirected graph, W is symmetric positive semidefinite and its nullspace consists exactly of consensus vectors.Thus Wx = 0 is equivalent to all agents’ local variables being equal.
- Objective assumptions: The analysis covers four cases determined by the aggregate objective’s strong convexity, smoothness, and Lipschitz properties.The individual assumptions induce corresponding global parameters such as µ, L, and M.
- Example: The framework includes distributed regularized linear regression with data matrices and observations partitioned across agents.The example uses local blocks b_i and H_i whose concatenation forms the global problem.
4. Primal-Dual Analysis for Convex Problems with Linear Constraints
The paper reformulates distributed optimization as a convex problem with linear constraints encoding network communication, then analyzes its dual to derive primal solution guarantees.
- Network communication is represented by linear constraints Ax = 0, enabling distributed optimization to be analyzed through constrained primal-dual methods.The framework uses a graph-dependent operator to encode consensus constraints.
- An (ε, ˜ε)-solution simultaneously requires objective suboptimality at most ε and constraint violation at most ˜ε.
- The dual problem has no duality gap, and primal solutions can be recovered from inner maximizers x∗(A^Ty∗).When the dual solution is nonunique, y∗ denotes the minimum-norm solution with norm R.
- The method’s parameterization depends on quantities such as the minimum-norm dual solution and graph spectral values, some of which may require distributed estimation.The paper notes that R and Rx are typically unknown, while certain graph eigenvalue parameters can be computed distributively.
- The analysis uses dual smoothing and accelerated gradient iterations while avoiding the previously required bounded-variation assumption on the admissible set.The dual is not globally strongly convex, but the analysis exploits strong convexity on a restricted gradient-generated subspace.
- Dual-friendly functions permit efficient computation of the inner maximizer, often through an explicit Legendre-Fenchel conjugate gradient.Examples include ridge regression and entropy-based optimization problems.
5. Dual Friendly Functions: Algorithms and Iteration Complexity Analysis
For dual-friendly objectives, the paper develops distributed accelerated methods across four convexity and smoothness classes, with communication complexities stated for approximate optimality and consensus.
- 5.1 Sums of Strongly Convex and Smooth Functions: A communication round is one iteration in which every node shares local estimates with neighbors and updates its local variables.The iteration count N therefore directly measures communication rounds.
- 5.1 Sums of Strongly Convex and Smooth Functions: Strongly convex and smooth dual-friendly functions are handled by applying fast gradient descent to the dual and changing variables for distributed execution.The resulting local updates require only neighbor information because the communication matrix follows the graph sparsity pattern.
- 5.2 Sums of Strongly Convex and M-Lipschitz Functions on a Bounded set: The strongly convex Lipschitz case uses regularization to induce strong convexity in the dual before applying the distributed accelerated method.The analysis assumes a bound on ∥∇F(x∗)∥2 and derives a communication guarantee for an (ε, ε/R)-solution.
- 5.3 Sums of Smooth and Convex Functions: For smooth convex objectives, regularization is instead applied to the primal problem to induce strong convexity and enable accelerated optimization.The section states a communication complexity bound for the resulting distributed algorithm.
- 5.4 Sums of Convex and M-Lipschitz Functions: When objectives are merely convex and Lipschitz, the method regularizes both primal and dual problems and builds on the preceding strongly convex cases.Theorem 5.5 provides the corresponding approximate-solution guarantee and communication bound.
6. Non-dual Friendly Functions: Algorithms and Iteration Complexity Analysis
For non-dual-friendly functions, the paper approximates the inner dual maximization with an inexact oracle and develops distributed accelerated methods with corresponding communication and oracle-complexity bounds.
- Approximate solutions of the inner dual maximization can serve as inexact first-order oracles for the outer dual problem.The paper uses fast gradient methods with inexact oracles to remove the dual-friendliness requirement.
- For non-dual-friendly strongly convex smooth objectives, Algorithm 7 uses an inner loop for the auxiliary problem and an outer loop applying fast gradient descent to the dual.The resulting method guarantees an (ε, ε/R)-solution with stated communication and local-oracle bounds.
- The total local-oracle complexity can exceed centralized fast gradient descent, although local computations are performed in parallel across agents.The paper argues the estimates can nevertheless be optimal when the number of agents is sufficiently large relative to the condition-number scale.
- For non-dual-friendly smooth convex objectives, Algorithm 8 similarly uses approximate inner solutions and obtains an inexact-oracle-based communication guarantee.The analysis regularizes the inner problem so it becomes strongly convex and smooth, then applies the strongly convex result.
- For nonsmooth cases, the paper applies accelerated gradient sliding methods and reports communication and oracle complexities that are optimal up to logarithmic factors.The same estimates can extend to stochastic optimization problems.
7. Improving on the Dependence of the Strong Convexity Parameter: Computation-Communication Trade-Off
The section reformulates the problem to reduce dependence on the smallest individual strong-convexity constant, while trading this improvement for additional communication rounds.
- Reformulation: The reformulation preserves correctness because any solution of the augmented problem is also a solution of the original problem.The network-structured gradient term enables decentralized computation even though the inner problem itself is not directly locally computable.
- Strongly Convex and Smooth Problems: The augmented strongly convex and smooth method combines outer accelerated optimization with an auxiliary dual problem solved approximately by fast gradient iterations.The additional term has network structure and can be computed using information shared among neighboring agents.
- Complexity and Applicability: The total communication and oracle complexity replaces the smallest strong-convexity constant with the aggregate parameter, at the price of graph-conditioning and extra inner communication.The approach is identified as useful for large, well-connected networks with m ≫ 1 and χ(W) = O(1).
- Computation-Communication Trade-Off: Replacing the smallest individual strong-convexity constant improves the function-condition-number dependence but adds graph-condition-number and inner-loop communication costs.The added inner communication rounds are proportional to the number of local oracle calls for f.
- Smooth Problems: For smooth convex objectives, regularization induces strong convexity so the augmented method can obtain an (ε, ε/R)-solution with bounded inner and outer costs.The smooth-case construction uses Algorithm 10 and a regularized function with Lipschitz-gradient parameter ˆLα.
8. Discussion
The discussion frames the communication complexity through the graph’s spectral condition number and compares the resulting estimates with centralized accelerated optimization limits.
- Complexity Guarantees: The estimates in Table 2 are optimal up to logarithmic factors for smooth objectives and up to constant factors otherwise, in communication rounds.The table summarizes communication rounds needed for an (ε, ε/R)-solution across the stated function classes.
- Consensus Over Networks: Each distributed optimization iteration requires O(χ(W) log(1/ε)) additional consensus steps to enforce network agreement.These steps correspond to the iterations needed to solve the consensus problem.
- Consensus Over Networks: The consensus-round estimate cannot be improved up to constant factors under the stated smoothness setting.The discussion connects this lower-bound behavior to the graph-dependent communication overhead.
- Graph Dependence: For fixed undirected graphs, the worst-case graph condition number scales as χ(W) = O(m^2), matching the best known upper bound for consensus and distributed optimization.The dependence of χ(W) on network size has been extensively studied for broad graph classes.
- Norm Generalization: The analysis extends convex and strongly convex cases to p-norms, with the condition-number definitions adjusted to the chosen norm.The discussion assumes strong convexity and Lipschitz-continuous gradients in the new norm when applying this generalization.
- Parameter Knowledge: The method typically requires estimates of R, Rx, µ, and L, although some network and function parameters can be computed or shared distributively.Restarting methods for unknown strong-convexity or smoothness parameters are noted, but their distributed effects remain outside the paper’s scope.
9. Experimental results
Experiments evaluate the proposed distributed algorithms on cycle and Erdős-Rényi graphs across strongly convex, smooth, and logistic-regression problems. The reported results show fast convergence and graph-dependent scalability, with Algorithm 7 outperforming comparison methods in the logistic-regression experiments.
- Experimental setup: The experiments compare distance to optimality, distance to consensus, and network scalability across the evaluated graph topologies and problem classes.Cycle graphs and Erdős-Rényi random graphs are used to examine scalability properties.
- Ridge regression: The proposed algorithm shows linear convergence and faster performance than compared methods, with linear scalability as cycle-graph size increases.The comparison includes centralized FGM, Acc-DNGD, EXTRA, and DIGing for ridge regression.
- KL barycenter: The KL-barycenter experiment measures optimality, consensus, and scalability for Algorithm 4 under varied regularization parameters on a 100-agent cycle graph.The problem is strongly convex and M-Lipschitz, with n = 10.
- Smooth problems: For smooth but non-strongly-convex problems, smaller regularization parameters increase Algorithm 5’s precision in experiments on cycle and Erdős-Rényi graphs.This setting assumes c = 0 and wide data matrices with n much larger than l.
- Logistic regression: Algorithm 7 shows faster geometric convergence than Acc-DNGD, EXTRA, and DIGing on synthetic logistic-regression problems.Experiments use 10,000 data points over cycle and Erdős-Rényi graphs with 10, 100, and 1,000 agents.
- Logistic regression: Algorithm 7 outperforms the compared algorithms on four LibSVM datasets, while Acc-DNGD has the second-best performance.The experiments use Erdős-Rényi graphs with 100 agents; EXTRA and DIGing exhibit worse scaling as network size increases.
10. Conclusions
The conclusions state that the proposed distributed methods attain near-optimal convergence estimates whose dependence on graph properties is explicit. They also identify global graph information, smoothing requirements, and changing-network assumptions as boundaries for practical use and further study.
- Convergence guarantees: For dual-friendly functions, the estimates are optimal up to logarithmic factors when F(x) is smooth and up to constant factors otherwise.The bounds are expressed in communication rounds and depend explicitly on the optimized function’s properties.
- Network dependence: The network condition measure χ(W) represents the additional cost of distributed optimization relative to centralized optimal estimates.The convergence analysis also indicates how graph and objective-function quantities affect performance.
- Limitations: The algorithms require, to some extent, global knowledge of graph properties and the network function’s condition number; cases without such information remain open.The paper identifies unavailable global information as requiring additional study.
- Limitations: Nesterov’s smoothing requires explicit norm knowledge and introduces an accuracy-dependent strong-convexity term that may slow convergence.Distributed adaptive methods are identified as requiring further work.
- Extensions: Time-varying graphs can be handled with restarting techniques only under additional assumptions, including infrequent changes and detectable change events.The graph-sequence condition number becomes the worst condition number among graphs encountered.
12. Funding
The paper acknowledges financial support from the National Science Foundation, RFBR, and the Yahoo! Research Faculty Engagement Program.
- Funding: Support came from the National Science Foundation, RFBR, and the Yahoo! Research Faculty Engagement Program.The listed grants and affiliations are specified in the funding statement.