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Online Distributed Optimization on Dynamic Networks
Saghar Hosseini, Airlie Chapman, Mehran Mesbahi
TL;DR
The paper addresses distributed optimization under uncertain costs and switching communication topologies. It proposes adaptive dual-averaging methods with dynamic communication weights, and reports convergence rate O(1/T), sub-linear online regret O(√T), and hindsight-comparable weight selection.
Problem
The problem is distributed optimization with disturbances in cost functions and network structure, including switching communication topologies.
Method
The paper develops distributed weighted dual averaging with dynamic weight selection and extends it to online distributed optimization.
Results
The analysis provides convergence rate O(1/T) and sub-linear online regret O(√T), while dynamic weight selection performs on average as well as the best strategy in hindsight.
Takeaways & Limitations
The framework enables distributed optimization using only local information and supports network design through regret bounds linked to adaptive weight selection.
Abstract
from arXiv · showhide
This paper presents a distributed optimization scheme over a network of agents in the presence of cost uncertainties and over switching communication topologies. Inspired by recent advances in distributed convex optimization, we propose a distributed algorithm based on a dual sub-gradient averaging. The objective of this algorithm is to minimize a cost function cooperatively. Furthermore, the algorithm changes the weights on the communication links in the network to adapt to varying reliability of neighboring agents. A convergence rate analysis as a function of the underlying network topology is then presented, followed by simulation results for representative classes of sensor networks.
I. INTRODUCTION
The paper addresses distributed optimization when both local cost functions and communication networks are uncertain or dynamically changing. It proposes adaptive dual-averaging methods for fixed and switching networks, extending them to online settings with regret analysis.
- Gap: Existing distributed online optimization work largely uses fixed communication structures and does not exploit dynamic weight selection.The paper identifies adaptive edge weighting as an unaddressed feature of prior distributed algorithms.
- Motivation: The paper considers uncertainties in both cost functions and network structure, including arbitrarily varying costs without probabilistic assumptions.Online learning treats the cost function as unknown when decisions are made, and good performance corresponds to sub-linear regret.
- Contributions: DWDA extends distributed dual averaging with a distributed dynamic weight-selection procedure based on an online weighted-majority approach.The procedure adaptively changes edge weights to optimize information diffusion.
- Contributions: The proposed framework applies DWDA to switching networks and extends it to Online-DWDA for uncertain costs and unavailable noise statistics.The paper derives regret bounds linking adaptive weight selection with online performance.
- Analysis and evaluation: The paper studies convergence and regret over network topologies and evaluates the online approach in distributed sensor-network estimation.The analysis and simulations cover switching topologies and representative sensor-network settings.
II. BACKGROUND AND PRELIMINARIES
This section introduces notation for vectors, matrices, stochastic communication processes, weighted directed graphs, and time-varying network topologies. It also identifies path and random graphs as simulation models.
- Notation: Vectors, matrices, inner products, norms, probability vectors, and row-stochastic matrices establish the notation used for network updates.Time-varying matrices are represented through backward products of communication matrices.
- Graphs: A weighted directed graph represents agents as vertices, interactions as edges, and communication influence through edge weights.The graph also defines in-neighborhoods, distances, adjacency matrices, and time-varying topologies.
- Graphs: The graph Laplacian is defined as L(G) = Δ(G) − A(G), and every graph has the all-ones right eigenvector associated with eigenvalue 0.The weighted in-degree determines the diagonal matrix Δ(G).
- Simulation graph models: The paper uses path graphs and random graphs, including Erdős-Rényi, random-tree, and random k-regular constructions, in simulations.A path graph connects vertices exactly when their indices differ by one.
B. Regret
The paper defines online regret by comparing incurred costs with the best fixed decision in hindsight and formulates distributed convex optimization over a network. It then introduces dual averaging as the algorithmic basis for distributed updates.
- Regret: At each iteration, an online algorithm chooses a decision before a previously unknown convex cost function is revealed and incurs the resulting loss.The objective is to make the average difference from the best fixed decision small.
- Regret: An algorithm performs well when regret is sub-linear, satisfying lim_T→∞ R_T/T = 0 and matching the best fixed hindsight strategy on average.This criterion is independent of the adversary’s moves.
- Problem formulation: The distributed problem has agents cooperatively minimizing a global convex objective over a network using local decision variables x_i ∈ χ.Each agent has a convex local cost function and solves locally over the closed convex set χ.
- Distributed dual averaging: The distributed algorithm adapts dual averaging so agents combine neighboring sub-gradients through a stochastic matrix preserving the graph’s zero structure.Strong connectivity is required for every agent to access all cost functions.
- Distributed dual averaging: DWDA updates local decisions and working variables using local sub-gradients, regularized projection, a non-increasing step sequence, and a proximal function.The proximal function is assumed strongly convex to control projection behavior.
V. DISTRIBUTED DYNAMIC WEIGHT SELECTION
The dynamic weight-selection procedure treats neighboring agents and the agent itself as experts whose weights are updated from local losses. These weights form a stochastic communication matrix and adapt information diffusion to local confidence.
- Dynamic weight selection: The paper adapts network weight selection to improve information diffusion within distributed optimization.The procedure is embedded in the DWDA algorithm.
- Expert-based allocation: Each agent allocates weights across neighboring links and its self-loop using an online allocation formulation with bounded expert losses.The expert set contains the agent’s neighbors and the agent itself.
- Performance guarantee: The distributed online allocation regret is sub-linear, so weight allocation performs as well as the best strategy in hindsight.The bound includes the loss upper bound M and the number of neighboring experts.
- Weight adaptation: The procedure places more weight on neighbors with higher confidence and updates self-confidence from the agent’s local loss.The resulting communication matrix has a positive diagonal.
- Communication structure: Because each agent’s allocation is a probability distribution, the time-varying communication matrix is row stochastic at every time step.For strongly connected graphs with positive diagonal elements, the matrix is irreducible, indecomposable, and aperiodic.
A. Switching Topologies
The paper extends dynamic weight selection to switching topologies whose unions are strongly connected over fixed intervals. The resulting communication matrices remain row stochastic, supporting subsequent convergence analysis.
- Switching-topology model: Switching topologies are modeled under the assumption that the union of graphs over each interval of length δ is strongly connected.The interval length satisfies δ ≥1.
- Communication construction: The communication matrix for switching topologies is constructed from the sum of the interval’s communication matrices.This construction is given by P 0 + P 1 + · · · + P δ−1.
- Communication properties: Each row of P(t) is a probability distribution, so P(t) is row stochastic at every time step.The associated weighted graph Laplacian therefore retains the form used in the analysis.
- Role in analysis: These communication-matrix properties are used in the convergence analysis of the distributed optimization algorithm.
VI. CONVERGENCE ANALYSIS
The convergence analysis separates standard dual-averaging error from network-induced disagreement. It represents network-level quantities through weighted averages and bounds the resulting network effect for switching communication.
- Assumptions: The analysis assumes each local convex function is positive and L-Lipschitz.
- Network averages: The sequences ȳ(t) and ġ(t) denote network-level weighted averages of dual variables and subgradients.
- Algorithmic structure: The weighted-average dynamics are analogous to the standard dual-averaging update, motivating the distributed update rule.
- Error decomposition: After T iterations, each agent’s total-cost evaluation error is bounded by the dual-averaging error together with network effects.
- Network disagreement: The network error is defined by the deviation of each local dual variable from the network’s weighted dual-variable average.
- Convergence result: The convergence analysis extends the distributed optimization algorithm to switching graphs and establishes a sub-linear convergence rate for DWDA.
A. Switching Topologies
The switching-network analysis uses weak ergodicity of inhomogeneous Markov chains to characterize communication-product convergence. Its rate depends on an ergodicity factor and a connectivity measure tied to graph diameter.
- Matrix convergence: Products of stochastic SIA matrices converge exponentially to a rank-one matrix of the form 1πT as t →∞.
- Switching connectivity: For strongly connected switching topologies, some product of at most n−1 matrices is scrambling.The bound follows from a connectivity argument over successive graph products.
- Switching connectivity: The scrambling bound is obtained by propagating positive entries along paths across successive graph products.
- Connectivity dependence: The parameter ν is bounded by n−1, with the worst case attained by a directed cycle graph.
- Rate bound: Theorem 4 gives a sub-linear convergence rate whose parameters include γ, ν, δ, and an arbitrary constant k > 0.Here γ < 1 captures communication-matrix ergodicity, while ν measures network connectivity and is bounded by graph diameter.
- Spectral interpretation: The ergodic coefficient bounds the second-largest eigenvalue, and 1−λ2(Pt) equals the second-smallest weighted-Laplacian eigenvalue.
- Connectivity dependence: Higher network connectivity promotes better performance of the proposed algorithm.
B. Adaptive Weight Selection
The adaptive weight-selection procedure is embedded within DWDA to improve information flow. Its analysis bounds the communication ergodicity coefficient using communication history and agent performance.
- Adaptive weighting: Theorem 5 analyzes communication matrices constructed from the adaptive probability sequence generated by Algorithm 2.
- Network dependence: The connectivity parameter ν remains bounded by the diameter of the graph, while δ is a positive integer.
- Weight construction: The adaptive weights depend on communication-round counts through directed edges.
- Performance signal: The quantity Ck(t) measures an agent’s weighted cost difference relative to the minimum neighboring cost.
- Adaptive effect: The DOA algorithm yields a tighter upper bound than the conservative bound and mitigates network-topology effects when agents perform poorly.
- Graph classes: DWDA performs well for graph classes including k-regular and expander graphs, where the maximum neighbor count can be bounded.
VII. ONLINE DISTRIBUTED OPTIMIZATION
Online distributed optimization addresses time-varying local costs and environmental uncertainties using a distributed scheme whose regret is analyzed under changing communication conditions. The resulting sub-linear regret depends explicitly on network connectivity and communication-matrix properties.
- Problem and setup: The online-DWDA scheme handles local convex costs that may change unpredictably after each agent commits to a decision.Each local cost is revealed only after the decision, while its sub-gradient is available to the agent and neighbors.
- Algorithm: The algorithm extends DWDA to online settings and provides a regret analysis quantifying its performance.
- Regret analysis: The regret bound depends on γ, describing communication-matrix ergodicity, and ν, measuring network connectivity.The bound also uses a positive integer δ associated with unions of directed topologies.
- Regret analysis: Theorem 6 establishes sub-linear regret for online-DWDA under bounded decision-domain and cost-regularity conditions.The theorem applies to sequences generated by Algorithm 1 and assumes ψ(x∗) ≤ R2.
- Running averages: Running-average estimates exhibit a similar dependence on network connectivity and achieve a corresponding bounded running-average regret.
VIII. ONLINE DISTRIBUTED ESTIMATION
Online distributed estimation applies the adaptive optimization framework to sensor networks with uncertain, time-varying measurements and unknown noise characteristics. Simulations show sub-linear regret across considered noise types and improved regret for adaptive topologies in jamming scenarios.
- Problem setup: The estimation problem seeks a common parameter estimate from distributed sensors when measurements and environmental conditions are uncertain.Local costs can change unpredictably, and the optimization variable lies in a closed convex set.
- Adaptive network: Dynamic weight selection can eliminate information from jammed sensors when noise resolution and characteristics are unknown in advance.
- Online estimation: Each sensor computes an estimate from local information before an oracle reveals its time-varying local cost, without requiring prior noise statistics.
- Simulation results: Adaptive sensor networks achieve better regret performance than fixed topologies in the reported jamming scenario.
- Simulation results: R_T(θ∗, ˆθ_1) = O(√T) for all considered noise types without a prior assumption on noise characteristics.
- Topology effects: Performance varies with network connectivity γ, suggesting topology-design metrics such as λ_2(P(G_0)) for scalable graph classes.
IX. CONCLUSION
The paper presents a distributed online algorithm that uses local information and dynamic weight selection for optimization on uncertain, changing networks. Its analysis establishes convergence and sub-linear regret, while simulations examine sensor estimation under errors, jamming, and varying connectivity.
- The algorithm evolves distributively using only local information available to network agents.
- The analysis provides a convergence rate of O(1/T) and a sub-linear regret of O(√T) in the online setting.
- A distributed dynamic weight selection procedure performs on average as well as the best information-diffusion strategy in hindsight.
- The approach improves convergence rate by mitigating network effects.
- In sensor estimation, real-time simulations address unknown errors and jamming, with average performance as good as the best fixed solution in hindsight.
- The simulations examine sensor-network classes and highlight network connectivity’s role in network-level regret.
X. APPENDIX
The appendix develops supporting bounds for distributed optimization over fixed-topology networks and describes simulation settings used to examine regret and connectivity effects.
- Supporting results: Lemmas 8 and 9 are included as reference results for the proximal function, step-size sequence, and associated inequalities.The text attributes both lemmas to Duchi et al.
- Simulation settings: Figure 4 evaluates regret under three observation-noise characteristics on a 100-node random 4-regular graph.The noise signals are generated from distributions with mean −bmax and standard deviation bmax.
- Supporting results: Lemma 10 bounds the deviation between the network average dual variable and each decentralized iterate by the error from the decentralized update.The bound applies to sequences generated by Algorithm 1 and equation (18).
- Fixed-topology analysis: Proposition 11 gives an upper bound on ν in the convergence rate for fixed-topology networks represented by stochastic matrices.The matrices are assumed to have positive diagonal elements and represent arbitrarily strongly connected topologies.
- Fixed-topology analysis: The proof establishes that a product of ν matrices is scrambling when every row of the relevant product has positive entries.The argument builds positivity across graph paths and concludes that Q is scrambling.