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Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs
Angelia Nedic, Alex Olshevsky
TL;DR
The paper addresses convergence of distributed optimization over time-varying directed graphs when nodes have only local, noisy gradient information. It analyzes a subgradient-push variant under strong convexity and establishes an O((ln t)/t) convergence rate, while exact O(1/t) convergence remains open in this setting.
Problem
Distributed optimization must minimize aggregate local objectives over time-varying directed graphs without requiring global topology or network-size knowledge.
Method
The paper analyzes a subgradient-push variant in which nodes use noisy subgradients of their individual strongly convex functions and a weighted averaging strategy.
Results
O((ln t)/t) convergence is established for strongly convex functions with noisy gradients.
Takeaways & Limitations
The method improves over the previously known O((ln t)/√t) rate for general convex functions while remaining close to the centralized O(1/t) benchmark.
Abstract
from arXiv · showhide
We investigate the convergence rate of the recently proposed subgradient-push method for distributed optimization over time-varying directed graphs. The subgradient-push method can be implemented in a distributed way without requiring knowledge of either the number of agents or the graph sequence; each node is only required to know its out-degree at each time. Our main result is a convergence rate of $O \left((\ln t)/t \right)$ for strongly convex functions with Lipschitz gradients even if only stochastic gradient samples are available; this is asymptotically faster than the $O \left((\ln t)/\sqrt{t} \right)$ rate previously known for (general) convex functions.
I. INTRODUCTION
The paper studies distributed minimization of sums of local convex functions when communication is time-varying and directed. It extends subgradient-push to noisy gradients under strong convexity and obtains an improved convergence rate.
- Problem setting: Distributed optimization minimizes a sum of n local convex functions, with each function known only to its corresponding node.Such problems arise when networks must agree on parameters minimizing aggregate local costs.
- Problem setting: Time-varying directed topologies model mobile nodes, unpredictable interference, and unequal wireless transmission power.These settings produce changing communication links and unidirectional interactions.
- Method context: Subgradient-push requires no knowledge of the communication topology or total node count; each node only needs its out-degree at each time.The method is a distributed generalization of push-sum for directed graphs.
- Main contribution: O((ln t)/t) convergence is achieved for strongly convex functions even when nodes access only noisy gradients.The result improves on the previously established subgradient-push rate for the less restrictive setting.
- Main contribution: The rate is close to the centralized strongly convex noisy-gradient benchmark O(1/t), while achieving O(1/t) in this distributed setting remains open.The open problem concerns time-varying directed graphs of unknown size.
II. PROBLEM, ALGORITHM AND MAIN RESULT
The paper studies stochastic subgradient-push for minimizing distributed sums of strongly convex functions over time-varying directed graphs. With Lipschitz gradients and suitable stepsize and averaging, the method achieves an O((ln t)/t) convergence rate while requiring limited network knowledge.
- Problem and assumptions: Each node knows only its local function through noisy subgradient samples, while the network is modeled by a repeatedly strongly connected sequence of directed graphs.The graph sequence is B-strongly-connected, meaning connectivity recurs over sufficiently long time windows.
- Algorithm: The y variables cancel graph-induced scaling, allowing x_i(t)/y_i(t) to track the running average with increasing accuracy.This ratio mechanism removes the differing steady-state influence weights associated with column-stochastic dynamics.
- Algorithm: The subgradient-push method maintains vector and scalar variables whose push-sum ratios compensate for scaling effects caused by time-varying directed communication.Nodes broadcast x_j(t)/d_j(t) and y_j(t)/d_j(t), then use received messages to update their local variables.
- Assumptions and stability: Strong convexity implies a unique global minimizer, while differentiable Lipschitz gradients ensure the iterates remain bounded under a vanishing stepsize.The boundedness result holds with probability 1 under the stated assumptions.
- Scope and limitations: The result applies to directed and time-varying graphs, but its constants can grow exponentially with the number of nodes in the general directed case.For regular graphs, the bounds scale polynomially in n; obtaining polynomial-time average consensus for general directed graphs remains open.
III. PROOF OF THEOREM 1
The proof treats stochastic gradient-push as perturbed push-sum: consensus controls disagreement, while averaged iterates approximately follow subgradient descent. Strong convexity and weighted averaging then yield the stated near-O(1/t) rate.
- Consensus and averaging: Bounded subgradients and a decaying stepsize allow the protocol to achieve consensus.This consensus step is the basis for analyzing the evolution of the average iterate.
- Consensus and averaging: The average iterate approximately satisfies the recursion of the ordinary subgradient method after consensus is established.The proof uses column-stochasticity and the preserved sum of the y variables to relate node variables to averaged quantities.
- Rate conclusion: Weighting the t’th iterate proportionally to t yields a nearly O((ln t)/t) decay for the perturbed stochastic subgradient method.The resulting theorem bound has τ in the denominator and logarithmic or constant numerators.
- Perturbed push-sum: The proof starts from perturbed push-sum, where perturbations create disagreement and Lemma 1 bounds that disagreement using network and external-force effects.The push-sum variables track running averages geometrically, while perturbations push node values apart.
- Optimization recursion: Strong convexity controls the optimization error through inequalities involving function values and subgradients at the averaged and local iterates.The argument compares F(¯x(t)) and F(z*) while bounding disagreement-dependent cross-terms.
IV. PROOF OF THEOREM 2
The proof establishes boundedness of the iterates under strong convexity, Lipschitz gradients, and vanishing stepsizes. It combines a one-step stability lemma with a ratio representation of push-sum updates.
- Stability of gradient steps: The proof argues that sufficiently large iterates decrease, using level-set boundedness of strongly convex functions with Lipschitz gradients.The key auxiliary result provides a compact set controlling a gradient step perturbed by bounded noise.
- Stability of gradient steps: A strongly convex function with Lipschitz gradients admits a compact level set that contains the relevant perturbed updates under suitable stepsize conditions.The construction uses bounds on the gradient and perturbation mapping.
- Push-sum ratios: The push-sum ratio transformation converts column-stochastic updates into a convex-combination or row-stochastic consensus representation.Applying the ratio lemma to coordinate vectors shows each new z_i(t+1) is a convex combination of prior x_i(t)/y_i(t) values.
- Push-sum ratios: Because the y variables remain positive with preserved total mass, α(t)/y_i(t) eventually becomes small enough for the stability lemma to apply to each ratio x_i(t)/y_i(t).The proof uses y_i(t)≤n and α(t)→0 to obtain the required eventual stepsize bound.
- Boundedness conclusion: Induction then propagates a uniform bound on all z_i(t), with an initial finite-time bound converted into a deterministic global bound.The recursion uses constants γ1, γ2, and γ3 and deterministic initial points.
- Boundedness conclusion: The analysis notes that more general noise models may be handled when their perturbations are small compared with the strong-convexity parameters.The details are omitted because the proofs are described as essentially identical to the zero-perturbation case.
V. SIMULATIONS
Simulations evaluate stochastic subgradient-push on a distributed quadratic estimation problem over large sparse, time-varying graph sequences. The reported trajectories show rapid error decay in both random and alternating-star settings.
- Experimental setup: The experiments use a scalar quadratic objective whose local observations define a distributed maximum-likelihood estimation problem.Each node knows its own observation, while the global minimizer is the unique estimate when at least one precision weight is positive.
- Experimental setup: The method runs for 200 iterations with α(t)=p/t on a 1000-node graph sequence with random connectivity.The parameter is set using the sum of the randomly sampled precision weights.
- Random graphs: Figure 1 compares average and single-run log absolute errors for five selected nodes on random graphs with two out-neighbors per node.One out-neighbor belongs to a fixed cycle and the other is chosen uniformly at random at each step.
- Alternating-star graphs: Figure 2 reports the same quantities for graph sequences alternating between two undirected star graphs.The comparison uses the same error quantities as Figure 1.
- Observed behavior: The simulations show fairly speedy error decay despite 1000 nodes and sparse per-stage connectivity.The authors describe the results as suggesting potential effectiveness for network optimization problems.
VI. CONCLUSION
The paper establishes an improved convergence rate for subgradient-push with strongly convex, Lipschitz-smooth objectives and noisy subgradients, while identifying several open problems and broader evaluation considerations.
- O(ln t/t) convergence is established for strongly convex functions with Lipschitz gradients and noisy subgradients.
- The result improves on the previously known O(ln t/√t) rate for the noiseless subgradient-push method.
- Polynomially scaling directed average-consensus algorithms remain unavailable for arbitrary time-varying graphs, limiting convergence-time scaling guarantees.
- Geometric convergence under directed, time-varying graphs remains of unclear attainability under additional assumptions.
- Iteration count may not capture application-relevant convergence speed, which can instead involve communication bits or physical distance traveled.
- Protocols for convex optimization evaluated by these alternative metrics remain an open problem.