Source-linked AI summary
Distributed optimization over time-varying directed graphs
Angelia Nedic, Alex Olshevsky
TL;DR
The paper studies distributed minimization of a sum of convex functions when each node knows only its local function and communication is directed and time-varying. It develops subgradient-push, combining subgradient optimization with push-sum under uniformly strong connectivity. All nodes converge to a common optimal solution at rate O(ln t/√t), with the constant depending on initial vectors, subgradient bounds, consensus speed, and influence imbalance.
Problem
The problem is to minimize a sum of convex functions distributed across nodes, with each function available only to its owning node.
Method
The paper develops subgradient-push, a broadcast-based combination of the subgradient method and push-sum for directed, time-varying networks.
Results
All nodes converge to the same point in the optimal-solution set at rate O(ln t/√t), with the constant depending on initial vectors, subgradient bounds, consensus speed, and influence imbalance.
Takeaways & Limitations
Under uniformly strong connectivity, distributed optimization succeeds without requiring knowledge of the graph sequence or the number of agents.
Abstract
from arXiv · showhide
We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of $O(\ln(t)/\sqrt{t})$, where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes.
I. INTRODUCTION
The paper addresses distributed minimization when convex-function knowledge is scattered across nodes communicating through directed, time-varying graphs. It introduces subgradient-push, a broadcast protocol that operates without knowing the graph sequence or agent count.
- Distributed optimization minimizes a sum of convex functions when each function is known only to one node.
- The setting focuses on directed, time-varying communication between nodes.
- Subgradient-push steers every node toward an optimal value under bounded subgradients.
- The protocol requires each node to know its out-degree but not the graph sequence or number of agents.
- The convergence-rate constant depends on information-diffusion speed and influence imbalance among nodes.
- Unlike prior approaches for fixed directed topologies, the method handles time-varying networks without requiring balanced communications or doubly stochastic matrices.
II. PROBLEM, ALGORITHM AND MAIN RESULTS
The paper formulates distributed minimization of a sum of locally known convex functions over uniformly strongly connected, time-varying directed graphs. Its subgradient-push algorithm combines subgradient optimization with push-sum and requires only local out-degree knowledge.
- Each node knows one convex function, and all nodes must converge to a common optimal solution.
- Communication uses a sequence of directed graphs that is uniformly strongly connected over time.
- Uniform strong connectivity allows connectivity edges to appear over time while still supporting bounds on information propagation.
- Subgradient-push combines the subgradient method with push-sum for distributed minimization.
- Implementation requires node i to know only its local out-degree d_i(t).
A. The subgradient-push method
The subgradient-push method combines push-sum-style aggregation with subgradient updates for distributed optimization. Its broadcast implementation maintains vector and scalar states, using out-degree-normalized messages and local subgradients.
- Each node maintains vector variables x_i(t), w_i(t) and a scalar variable y_i(t).
- At each step, nodes update x_i(t+1) by subtracting a stepsize-scaled local subgradient from w_i(t+1).The subgradient is evaluated at z_i(t+1), and the method starts with y_i(0)=1.
- Nodes broadcast x_j(t)/d_j(t) and y_j(t)/d_j(t), then sum received messages to obtain w_i(t+1) and y_i(t+1).This requires each node to know its out-degree at the current time.
- The protocol uses push-sum aggregation so the ratios z_i(t)=w_i(t)/y_i(t) compensate for unequal influence in directed communication.The subgradient term perturbs the underlying push-sum averaging process.
B. Our results
Under uniformly strongly connected directed networks, convex local objectives, nonempty optimal solutions, and bounded subgradients, subgradient-push converges to an optimizer. With α(t)=1/√t, its objective convergence is O(ln t/√t), with constants reflecting both optimization and network properties.
- Under uniform strong connectivity, convexity, a nonempty optimizer set, and bounded subgradients, subgradient-push converges to an optimal point at every node.
- α(t)=1/√t yields objective convergence at rate O(ln t/√t).The theorem uses running time-averages of the node iterates.
- The theorem establishes convergence of objective values along time-averages but does not establish convergence of the time-average sequences themselves.
- The convergence constant depends on subgradient bounds, initial conditions, information-diffusion speed, and influence imbalance.The parameter λ measures diffusion speed, while δ measures long-term uniformity of influence.
- For time-varying regular directed networks, δ=1 and 1/(δ(1−λ)) can scale polynomially in n.
- The proof treats subgradient-push as a perturbed push-sum protocol whose perturbations are controlled using consensus properties.
III. PERTURBED PUSH-SUM PROTOCOL
The perturbed push-sum protocol extends directed-network averaging to updates with decaying perturbations. Its analysis shows how uniform strong connectivity, stochastic matrix products, and influence lower bounds support consensus-related guarantees.
- Push-sum enables nodes with directed communication links to compute averages and other network aggregates.
- When perturbations decay to zero, node iterates converge toward one another, although they need not converge to a common point.
- Perturbed push-sum updates add a perturbation ε_i(t+1) to each node’s aggregated state.Without this term, the update reduces to ordinary push-sum.
- The push-sum ratios z_i(t)=w_i(t)/y_i(t) compensate for influence imbalances among nodes.
- The analysis represents the dynamics with column-stochastic matrices that need not be row-stochastic.
- Uniform strong connectivity provides stochastic limiting vectors and bounds on products of the communication matrices.
- For regular graphs, the matrices are doubly stochastic, the influence lower bound satisfies δ′=1, and unperturbed push-sum computes the initial average.
- The central lemma shows that each z_i(t+1) increasingly tracks the network average x̄(t).
IV. CONVERGENCE RESULTS FOR SUBGRADIENT-PUSH METHOD
The convergence proof combines perturbed push-sum analysis with subgradient arguments: node states approach a common average process, which converges to an optimal solution. With α(t)=1/√(t+1), the resulting bounds yield the stated subgradient-push convergence rate.
- Proof strategy: The proof treats subgradient-push as a perturbed push-sum protocol, with bounded subgradient terms acting as diminishing perturbations.The perturbations use α(t+1) times the stacked subgradients, and α(t) tends to zero.
- Consensus: The entries of z_i(t) approach one another relative to the average process, satisfying lim t→∞∥z_i(t+1)−x̄(t)∥=0 for every node.This establishes asymptotic agreement between each node state and the average sequence.
- Optimality: The average sequence x̄(t) converges to a solution in the optimal set, and consensus then implies that every node sequence converges to the same optimal solution.The proof first establishes convergence of the average process and then transfers it to the individual node states.
- Rate analysis: α(t)=1/√(t+1) produces bounds containing 1/√t and logarithmic terms, which establish the convergence-rate theorem.The displayed bounds include terms proportional to 2∥x̄(0)−z*∥_1/√(t+1), L^2(1+ln(t+1)), and 16dL^2(1+ln t).
- Rate analysis: The proof bounds disagreement terms using subgradient-norm bounds and perturbed push-sum estimates before combining them with a subgradient-like recursion.The resulting argument invokes Lemmas 8 and 9 and the convergence lemmas for scalar recursions.
V. SIMULATIONS
Simulations study distributed quadratic estimation on random time-varying graphs. They show fast initial error decay, sub-geometric convergence, and approximately linear threshold time as the number of nodes increases.
- Experimental setup: The experiments optimize F(θ)=∑_i p_i(θ−u_i)^2, a distributed estimation problem in which each node knows only its own measurement.Half the nodes receive zero measurement weights, while the remaining weights are uniformly random between 0 and 1.
- Figure measurements: The top plots track the number of iterations against ∥z(t)−θ*1∥ for a 1000-node graph instance.The bottom plots track node count against average time to reach error at most 0.1 over 30 simulations.
- Experimental setup: The simulations use random graphs in which each node has two out-neighbors: one on a fixed cycle and one selected uniformly at random at each step.Initial points are independent standard Gaussian random variables.
- Observed behavior: The error initially decays quickly, requiring only a small number of iterations to bring all nodes reasonably close to the optimum.This behavior is reported for the simulated random-graph cases.
- Observed behavior: The decay is sub-geometric, while the time to reduce error below 0.1 appears to scale linearly with the number of nodes.The scaling statement concerns runs starting from random initial points.
VI. CONCLUSIONS
The paper introduces subgradient-push for distributed optimization over uniformly strongly connected directed graphs and proves convergence of all nodes to a common optimum. Its objective convergence rate is O(ln t/√t), with constants shaped by initialization, subgradient bounds, consensus speed, and influence imbalance.
- Contribution: The subgradient-push is a broadcast-based protocol for minimizing a sum of convex functions over directed graphs.It succeeds when the communication graph sequence is uniformly strongly connected.
- Guarantee: All nodes converge to the same point in the set of optimal solutions under the stated connectivity condition.The conclusion concerns the node states generated by subgradient-push.
- Rate: O(ln t/√t) is the objective-function convergence rate, with constants depending on initial vectors, subgradient-norm bounds, consensus speed λ, and influence imbalance δ.The rate constant reflects both information diffusion and unequal influence among nodes.
- Open problems: Determining how consensus speed λ depends on graph-sequence properties remains an open problem.The paper also identifies the dependence of influence imbalance δ on graph combinatorial properties as an open question.