Source-linked AI summary
Distributed Subgradient Methods and Quantization Effects
Angelia Nedić, Alex Olshevsky, Asuman Ozdaglar, John N. Tsitsiklis
TL;DR
The paper studies distributed convex optimization when agents must cooperatively minimize summed local objectives using only local information over changing networks. It develops subgradient methods based on averaging, improves unquantized convergence-rate bounds, and analyzes quantized communication. The results characterize polynomial dependence on network size and quantization-dependent error in the quantized setting.
Problem
Distributed agents need to cooperatively optimize summed convex local objectives over time-varying networks when information is local and communication may be quantized.
Method
The paper combines local subgradient iterations with distributed averaging, then adapts the method so agents store and exchange quantized estimates.
Results
The paper derives improved convergence-rate upper bounds for unquantized subgradient methods and quantized bounds whose error depends on quantization levels and polynomially on n.
Takeaways & Limitations
The methods support convergence toward the optimal objective value in time-varying networks, with quantization introducing a bounded error characterized by the number of levels.
Abstract
from arXiv · showhide
We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a time-varying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels.
I. INTRODUCTION
The paper develops distributed optimization methods for networked agents with local information, time-varying connectivity, and quantized communication. It improves convergence-rate bounds for the unquantized method and characterizes quantization-dependent errors.
- Distributed optimization is motivated by resource allocation and sensor-network problems requiring local information, full distribution, and robustness to topology changes.
- The agents cooperatively solve an unconstrained problem that sums convex local objectives, each known only to its corresponding agent.
- The method combines local subgradient optimization with averaging so agents can incorporate information about other agents’ objectives.
- The paper establishes improved convergence-rate upper bounds with polynomial dependence on the number of agents n, rather than the exponential dependence in earlier bounds.
- A quantized-information variant provides convergence bounds containing additional error terms determined by the number of quantization levels.
II. DISTRIBUTED SUBGRADIENT METHOD
The distributed subgradient method updates local estimates through neighbor averaging and local subgradients over a time-varying directed topology. Its analysis uses transition matrices under weight and connectivity assumptions.
- Each agent maintains an estimate x_i(k) and updates it using estimates received from neighboring agents.
- The update combines nonnegative weighted neighbor estimates with a subgradient of the agent’s local cost function and a positive stepsize α.
- Transition matrices Φ(k, s) represent the evolution of estimates across time and relate later estimates to earlier agent estimates.
- The convergence analysis studies transition-matrix behavior under assumptions on agent interactions and weight matrices.
- The weight matrices are doubly stochastic with positive diagonal entries, and every positive weight is bounded below by η.
- Connectivity assumptions require sufficiently frequent influence among agents through strongly connected directed graphs.
A. Preliminary Results
The preliminary analysis tracks disagreement under doubly stochastic averaging and derives bounds on its decay and on transition-matrix mixing.
- A. Preliminary Results: The averaging update preserves the initial average when each update matrix is doubly stochastic.This makes the average a fixed reference for analyzing agent agreement.
- A. Preliminary Results: The decrease in V(kB) is linear in η and quadratic in n−1.This bound is used to quantify how quickly averaging reduces disagreement.
- A. Preliminary Results: The resulting transition-matrix estimates apply for all i, j and times k ≥ s under the stated assumptions.The corollary follows from the disagreement bound and its nonincreasing behavior.
- A. Preliminary Results: Products of the doubly stochastic transition matrices retain doubly stochasticity, so their average entry is 1/n.This property supports bounds on individual transition-matrix entries.
B. Convergence time
The convergence analysis assumes uniformly bounded subgradients and uses time-averaged iterates to establish convergence-time bounds. The resulting estimate scales polynomially rather than exponentially with the number of agents.
- B. Convergence time: Time-averaged iterates are introduced to bound objective-function improvement at every iteration.These averages are the quantities used in the convergence-rate theorem.
- B. Convergence time: The uniform subgradient-boundedness assumption can be relaxed according to prior work.The paper cites for this relaxation.
- B. Convergence time: The analysis assumes a nonempty optimal-solution set, bounded subgradients, and suitably bounded initial vectors.The initial-vector bound is used to present a compact convergence-time expression.
- B. Convergence time: Theorem 1 gives a convergence-rate bound for the time-averaged estimates generated by the distributed subgradient method.Its proof relies on the improved transition-matrix convergence bound.
- B. Convergence time: The new constant C1 increases polynomially with n, whereas the earlier bound had exponential dependence on n.The paper presents this as more favorable scaling with network size.
III. QUANTIZATION EFFECTS
The quantized distributed subgradient method models rounding as external noise and establishes convergence bounds whose error depends on quantization resolution. A stopped version converges to a common limit, and the bounds recover the unquantized results as Q increases.
- Quantized method: Quantized communication represents each estimate as a vector whose entries are integer multiples of 1/Q.The rounding operation is componentwise, and the initial estimates are also assumed quantized in the stated model.
- Quantized method: The quantized update can be rewritten as a subgradient method with external, possibly persistent, noise from the quantization error.The error arises from rounding to the nearest multiple of 1/Q, and its magnitude need not diminish over time.
- Stopped model: After subgradients and quantization stop, all agents’ iterates converge to the same limit vector y(k̃).The stopped model sets the subgradient and quantization errors to zero after the stopping time.
- Convergence bounds: As Q →∞, the quantized constants and error terms reduce to those of the unquantized convergence bound.Thus, increasingly fine quantization makes the results of Theorems 2 and 1 coincide.
IV. CONCLUSIONS
The paper studies distributed subgradient methods for convex optimization over time-varying networks, including both continuous-value and quantized communication settings.
- Conclusions: The continuous-value method has a convergence-rate bound for agents connected through a time-varying topology.The analysis covers agents that can exchange and store continuous values.
- Conclusions: The quantized method converges to the optimal objective value within an error whose bound depends on the number of quantization levels and polynomially on n.The paper identifies these dependencies as key features of the quantized performance guarantee.
- Conclusions: Future work includes studying other quantization schemes and noise in agents’ estimates.These are identified as directions for analyzing effects on algorithm performance.