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Distributed Stochastic Gradient Tracking Methods

Shi Pu, Angelia Nedić

arXiv:1805.11454v5math.OCcs.DCcs.SIstat.ML

TL;DR

The paper addresses distributed optimization when agents have only noisy local gradients. It proposes DSGT and GSGT, showing exponential attraction to an optimal neighborhood under constant stepsizes, network-size-dependent DSGT error bounds, and lower GSGT communication cost on well-connected networks.

  • Problem

    The problem is to minimize the average of smooth, strongly convex local costs across networked agents using only stochastic gradient information.

  • Method

    The paper proposes DSGT with auxiliary gradient-tracking variables and a gossip-like GSGT variant using randomized local communication.

  • Results

    DSGT and GSGT reach optimal neighborhoods exponentially fast in expectation under sufficiently small constant stepsizes; DSGT’s error bounds decrease with n, while GSGT lowers communication on well-connected networks.

  • Takeaways & Limitations

    DSGT can achieve centralized-comparable stochastic optimization performance, while GSGT offers a lower-communication alternative on well-connected networks.

  • Takeaways & Limitations

    DSGT incurs two times the communication and storage costs per iteration compared with standard distributed subgradient methods.

Abstract

from arXiv · show

In this paper, we study the problem of distributed multi-agent optimization over a network, where each agent possesses a local cost function that is smooth and strongly convex. The global objective is to find a common solution that minimizes the average of all cost functions. Assuming agents only have access to unbiased estimates of the gradients of their local cost functions, we consider a distributed stochastic gradient tracking method (DSGT) and a gossip-like stochastic gradient tracking method (GSGT). We show that, in expectation, the iterates generated by each agent are attracted to a neighborhood of the optimal solution, where they accumulate exponentially fast (under a constant stepsize choice). Under DSGT, the limiting (expected) error bounds on the distance of the iterates from the optimal solution decrease with the network size $n$, which is a comparable performance to a centralized stochastic gradient algorithm. Moreover, we show that when the network is well-connected, GSGT incurs lower communication cost than DSGT while maintaining a similar computational cost. Numerical example further demonstrates the effectiveness of the proposed methods.

1 Introduction

The paper develops stochastic gradient-tracking methods for distributed optimization with noisy local gradients, establishing convergence and communication advantages under stated network and stepsize conditions.

  • Each agent minimizes a smooth, strongly convex local cost while collectively optimizing the average objective over a connected network.
  • Stochastic gradient estimates are assumed unbiased with bounded conditional second-moment error, covering online learning and simulation-based optimization settings.
  • DSGT tracks stochastic gradient averages and reaches an optimal neighborhood exponentially fast in expectation under a sufficiently small constant stepsize.
  • GSGT uses random wake-ups and neighbor communication, reducing communication cost by a factor of O(|E|/n) on well-connected networks while retaining similar computational cost.
  • DSGT’s limiting error bound decreases with network size n, making its performance comparable to centralized stochastic gradient methods.
  • With a diminishing stepsize, DSGT achieves the optimal O(1/k) convergence rate to the optimal point.

2 A Distributed Stochastic Gradient Tracking Method (DSGT)

DSGT combines consensus updates with auxiliary variables that track stochastic gradient averages. Under suitable stepsizes, it converges linearly in expectation to an error neighborhood whose network dependence worsens with poorer connectivity, while sufficiently small stepsizes yield centralized-comparable performance.

  • DSGT method: DSGT uses auxiliary variables y_i,k to track the average stochastic gradients while agents update through a doubly stochastic coupling matrix W.The coupling matrix is assumed nonnegative and doubly stochastic, with a positive self-weight for at least one agent.
  • Implementation cost: DSGT requires twice the communication and storage costs per iteration compared with standard distributed subgradient methods.This cost follows from maintaining and communicating the gradient-tracking variables.
  • Constant-stepsize convergence: Under a sufficiently small constant stepsize, DSGT achieves linear convergence in expectation at rate O(ρ(A)^k), where ρ(A) < 1.The rate is governed by the spectral radius of the matrix A.
  • Error bounds: The stochastic-gradient contribution to the ultimate error is independent of network topology, whereas additional error terms increase with ρ_w as connectivity worsens.Here, larger ρ_w indicates poorer network connectivity.
  • Error bounds: The average solution error across agents decreases with network size n when the constant stepsize is sufficiently small.The network-size dependence is tied to the spectral gap 1−ρ_w and the graph topology.
  • Stepsize selection: A larger early stepsize followed by smaller stepsizes can trade faster initial convergence for greater final accuracy.The paper presents this as a practical implementation strategy.
  • Comparison with centralized methods: With sufficiently small stepsizes, DSGT has ultimate error bounds comparable to centralized stochastic gradient, up to constant factors.Its convergence speed is also comparable to centralized schemes, whose linear rate is O((1 − 2αµ)^k).
  • Diminishing-stepsize convergence: With diminishing stepsizes α_k = θ/(m + k), DSGT attains an O(1/k) convergence rate matching centralized stochastic-gradient rates up to constant factors.The stated rate does not depend on the spectral norm ρ_w.

3 Analysis

The analysis establishes convergence by bounding expected optimality, consensus, and gradient-tracking errors through linear systems. Under suitable stepsize conditions, these bounds decay geometrically toward a neighborhood, with rates governed by spectral radii.

  • Proof strategy: The proof tracks three expected error quantities through a linear system of inequalities.The stochastic-gradient variances are explicitly incorporated because they affect the resulting bounds.
  • Network dependence: The transient time needed for the normalized expected error to approach its asymptotic rate can depend significantly on network topology and n.Thus, asymptotic rate comparisons do not fully characterize finite-time behavior.
  • Convergence condition: ρ(A) < 1 implies geometric decay at rate O(ρ(A)^k) for the relevant expected errors.The analysis uses powers of the nonnegative matrix A to establish this linear convergence behavior.
  • Steady-state bounds: The limiting bounds are obtained from [(I − A)^−1B], linking steady-state errors to the noise vector B and system matrix A.The vector B contains terms involving stochastic-gradient noise and related constants.
  • Diminishing stepsize: With α_k = θ/(m + k), the analyzed quantities satisfy U_k ≤ Û/(m + k), X_k ≤ X̂/(m + k)^2, and Y_k ≤ Ŷ.These bounds are established inductively under the stated constraints on m, θ, and the constants.

4 A Gossip-Like Stochastic Gradient Tracking Method (GSGT)

GSGT replaces full-network communication with randomized pairwise or self-updates while retaining stochastic gradient tracking. Under sufficiently small constant stepsizes, it converges geometrically to an error neighborhood, with communication advantages on well-connected networks.

  • Algorithm: At each iteration, one agent wakes uniformly and either communicates with a neighbor or updates using only its own information.Nonparticipating agents retain their current variables, while communicating agents apply the pairwise update.
  • Algorithm: GSGT uses a doubled stepsize when one agent updates alone because each iteration represents two gradient updates within the network.This differs from standard gossip, where exactly two agents update at every round.
  • Network dependence: For complete networks and many regular graphs, 1 − ρ̄_w = O(1/n), supporting favorable communication scaling for GSGT.The spectral quantity ρ̄_w is bounded using the doubly stochastic interaction structure.
  • Convergence results: Under a sufficiently small constant stepsize, GSGT errors converge to a neighborhood of zero at rate O(ρ(A_g)^k), with ρ(A_g) < 1.The result is stated for the expected optimality, consensus, and tracking errors under the theorem’s assumptions.
  • Convergence results: GSGT converges more slowly than DSGT under the same stepsize because only two agents update their iterates per iteration.The comparison concerns convergence speed rather than the existence of geometric convergence.
  • Complexity comparison: On complete networks with |E| = O(n^2), GSGT requires O(n) fewer communications than DSGT while retaining comparable gradient-computation scaling.The communication comparison is for reaching an ε error level under constant stepsizes.

5 Numerical Example

The numerical study evaluates DSGT and GSGT on online Ridge regression across network sizes and stepsizes. DSGT matches CSG closely, while GSGT trades convergence speed for substantially lower communication requirements.

  • Setup: The experiment uses online Ridge regression with p = 20 and n ∈ {10, 25, 100} on random networks whose links appear with probability 0.4.Two constant stepsizes, α = 5 × 10^-3 and α = 5 × 10^-2, are tested across three instances.
  • Setup: The stochastic Ridge-regression gradient estimate is unbiased under continuously sampled feature-output pairs and Gaussian observation noise.The observed outputs satisfy v_i = u_i^T x̃_i + ε_i, with ε_i having mean zero and variance one.
  • Results: DSGT and CSG have nearly indistinguishable convergence speeds and ultimate error bounds, whereas EXTRA and DLM attain worse final error bounds.The performance gap relative to EXTRA and DLM increases with network size and stepsize.
  • Results: At α = 5 × 10^-3, GSGT is slower but reaches a comparable error level, while DSGT, GSGT, and CSG error bounds decrease with n.These observations match the theoretical dependence on network size.
  • Cost comparison: With equal stochastic-gradient evaluations, DSGT and GSGT have comparable convergence speeds; with equal communications, GSGT is much faster.The comparison uses the smaller stepsize α = 5 × 10^-3.

6 Conclusions and Future Work

The paper proposes DSGT and communication-efficient GSGT for distributed optimization with inexact gradients. Both achieve exponential approach to an error neighborhood under constant stepsizes, while DSGT has network-size-scaled error bounds and GSGT reduces communication on well-connected graphs.

  • Contributions: DSGT handles distributed optimization when agents possess only inexact gradients of their local cost functions.Each local objective is smooth and strongly convex, and the global objective averages the agents’ costs.
  • DSGT: With a constant stepsize, DSGT iterates approach a neighborhood of the optimum exponentially fast in expectation.With a diminishing stepsize, DSGT achieves the O(1/k) convergence rate.
  • DSGT: DSGT’s limiting error bounds decrease with network size n and are comparable to centralized stochastic-gradient performance.The reported convergence rate can match centralized stochastic-gradient rates up to constant factors.
  • GSGT: GSGT is designed to reduce communication and requires fewer communications than DSGT on well-connected interaction graphs.The paper identifies complete networks and almost all regular graphs as examples.
  • Comparison: DSGT still has an early-stage advantage over DSG because it can achieve similar convergence speed with lower communication and storage costs in that comparison.This comparison concerns the early stage of convergence.

7 APPENDIX

This appendix develops intermediate bounds for stochastic gradient-tracking quantities using conditional expectations, smoothness, and variance assumptions. The derivation combines auxiliary lemmas to establish the desired relations.

  • Proof setup: The proof introduces shorthand Gk := G(xk, ξk) and ∇k := ∇F(xk) before analyzing successive gradient-estimator differences.The analysis conditions expectations on the filtration Fk.
  • Conditional bounds: Conditional expansion separates E[∥Gk+1 − Gk∥2 | Fk] into gradient-difference and cross-term contributions.The cross term involves the estimator errors at iterations k and k+1.
  • Proof completion: The appendix invokes additional lemmas and prior relations to complete the desired inequalities.Several steps explicitly defer to Lemma 1 and earlier relations before concluding the proof.
  • Conditional bounds: Under Assumption 2, the key cross term is bounded by αLσ2.The bound uses the variance estimate E[∥gi(xi,k, ξi,k) − ∇fi(xi,k)∥2 | Fk] ≤ σ2.

7.2 Proof of Lemma 5

The proof analyzes the characteristic polynomial of S and derives a spectral-radius condition, then bounds stochastic tracking terms using the update rules and assumptions. These bounds support the stated inequalities for the lemma.

  • Spectral analysis: The characteristic function of S is defined as g(λ) := det(λI − S) and expanded using diagonal entries and paired off-diagonal products.The expansion is used to study the real roots associated with S.
  • Spectral analysis: The proof shows that det(λ*I − S) > 0 is sufficient for locating all real roots of g(λ) within (−λ*, λ*).It establishes monotonicity outside the interval and evaluates g(−λ*) < 0.
  • Spectral conclusion: ρ(S) < λ* follows because the Perron-Frobenius theorem identifies ρ(S) as a real eigenvalue of S.The preceding root-location argument places this eigenvalue inside the required interval.
  • Inequality assembly: The resulting inequalities are assembled from relations (84), (89), and (90) together with the stated assumptions and lemmas.The proof proceeds term by term before concluding the target relations.
  • Stochastic bounds: The argument further uses conditional expectations over the random selections of ik and jk and the stepsize condition α < n/(µ + L).These ingredients are applied to bound expressions involving yik,k + yjk,k.

Lemma 10

Lemma 10 supplies an additional bound used in the analysis of selected gradient-tracking differences. Its application combines update-rule terms, smoothness, and stochastic-noise control.

  • Lemma bound: The combined estimate in (95) contains weighted iterate-difference terms and the noise contribution 4(αL + 1)σ2.The proof states that this inequality follows from equation (92), equation (93), equation (94), and Lemma 10.
  • Role in proof: The proof introduces Lemma 10 specifically to further bound the sum of the selected tracking-difference terms.The lemma is then incorporated into the subsequent inequality chain.

Lemma 11

The proof of Lemma 11 is referenced rather than reproduced, and subsequent relations invoke it together with earlier bounds to continue the analysis.

  • Proof reference: The proof of Lemma 11 is deferred to Appendix 7.4.No derivation of the lemma appears in the supplied passage.
  • Use in analysis: Relation (97) and Lemma 11 are combined in the subsequent argument.The passage indicates that Lemma 11 is used as an ingredient rather than restating its conclusion.
  • Use in analysis: The later bounds also invoke relations (37a), (38a), and Assumption 2 before applying relation (98).These steps contribute to the inequality assembled from relations (91), (96), and (99).

7.4 Proof of Lemma 11

The proof introduces a relation and connects it to the updating rules (37b) and (38b), but the supplied passage text does not state the resulting lemma conclusion.

  • The proof begins by stating that a relation holds.
  • The displayed expression involves gradient-related terms associated with agents i and j.
  • The proof derives the next step from updating rules (37b) and (38b).
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