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LAG: Lazily Aggregated Gradient for Communication-Efficient Distributed Learning

Tianyi Chen, Georgios B. Giannakis, Tao Sun, Wotao Yin

arXiv:1805.09965v2stat.MLcs.DCcs.LGmath.OC

TL;DR

Distributed learning can be bottlenecked by costly communication between a server and workers, especially when latency dominates small-message transmission. LAG adaptively skips workers whose gradients change slowly and reuses lagged gradients. It retains GD-order convergence in smooth strongly-convex, convex, and nonconvex settings while reducing communication in heterogeneous learning problems, though broader theoretical conditions remain an open limitation.

  • Problem

    Distributed machine learning needs to reduce costly server-worker communication rounds while optimizing objectives whose data remain distributed across workers.

  • Method

    LAG uses simple adaptive rules to select workers for fresh gradient computation and reuses lagged gradients from the remaining workers.

  • Results

    LAG achieves the same convergence rates as batch GD in smooth strongly-convex, convex, and nonconvex cases and requires fewer communication rounds in heterogeneous settings.

  • Takeaways & Limitations

    LAG provides a communication-cognizant distributed gradient method whose empirical performance shows significant communication reduction compared with alternatives.

  • Takeaways & Limitations

    The current condition ensuring communication reduction is relatively restrictive, and broader settings matching the empirical performance remain future work.

Abstract

from arXiv · show

This paper presents a new class of gradient methods for distributed machine learning that adaptively skip the gradient calculations to learn with reduced communication and computation. Simple rules are designed to detect slowly-varying gradients and, therefore, trigger the reuse of outdated gradients. The resultant gradient-based algorithms are termed Lazily Aggregated Gradient --- justifying our acronym LAG used henceforth. Theoretically, the merits of this contribution are: i) the convergence rate is the same as batch gradient descent in strongly-convex, convex, and nonconvex smooth cases; and, ii) if the distributed datasets are heterogeneous (quantified by certain measurable constants), the communication rounds needed to achieve a targeted accuracy are reduced thanks to the adaptive reuse of lagged gradients. Numerical experiments on both synthetic and real data corroborate a significant communication reduction compared to alternatives.

1. Introduction

Distributed learning avoids single-server bottlenecks by aggregating worker gradients, but communication latency makes repeated server-worker exchanges costly. LAG addresses this by adaptively reusing lagged gradients while preserving GD-order convergence and reducing communication in heterogeneous settings.

  • Motivation: Massive datasets make sequential single-server gradient computation prohibitively slow, motivating distributed workers that keep data local and communicate gradients.Workers compute gradients on locally stored samples while a central server aggregates them into model updates.
  • Motivation: Communication latency, rather than message size, dominates when small models or aggregated gradients are exchanged, making fewer rounds especially valuable.The stated goal is minimizing the learning objective with as little communication overhead as possible.
  • LAG approach: LAG refines the previous aggregated gradient using only selected workers’ new gradients and reuses outdated gradients from the others.With lagged gradients stored at the server, communication per iteration falls from GD’s M rounds to LAG’s |M_k| rounds.
  • LAG approach: The method provides parameter-server and worker-side variants that select communicating workers using adaptive trigger rules.LAG-PS selects workers at the server, whereas LAG-WK has workers compute and check their own selection condition.
  • Contributions: LAG preserves the order of GD convergence rates in smooth strongly-convex, convex, and nonconvex cases while reducing communication under heterogeneous data.In certain learning settings, the paper states that LAG requires only O(1/M) of GD’s communication and empirically reports reductions by several orders of magnitude.

2. LAG: Lazily Aggregated Gradient Approach

LAG selects workers whose current gradients need refinement and reuses lagged gradients for the rest, targeting a principled communication-accuracy tradeoff. Its worker-side and server-side rules support simpler or more conservative operation, with worker-side selection typically saving more communication.

  • Trigger design: LAG skips a worker when its current gradient refinement is small relative to its stored gradient, thereby reusing the outdated value.The selection criterion is motivated by treating the difference between successive local gradients as the communication-required refinement.
  • Aggregation: The selected and nonselected worker sets determine whether each worker communicates, while the server updates using fresh gradients and lagged gradients together.Including more workers in the noncommunicating set saves more communication but produces a coarser gradient update.
  • Communication-accuracy tradeoff: The worker-selection criterion is designed to balance communication savings against update accuracy using the fundamental descent lemma.The analysis compares GD’s descent after communicating with all workers against LAG’s descent using a selected subset.
  • Trigger design: Directly checking the ideal refinement condition is expensive because it requires global-gradient information and workers do not know the selected set.The paper therefore develops implementable worker-side and server-side rules.
  • Variants: LAG-WK and LAG-PS have the same worst-case convergence guarantees, but LAG-PS is more conservative and empirically communicates with more workers.The worker-side rule generally yields a larger noncommunicating set and thus greater communication savings.
  • Implementation: Recursive updates make LAG as simple to implement as GD, and the method can be combined with other communication-efficient or accelerated techniques.The paper contrasts LAG’s communication reduction with approaches that reduce bandwidth or increase local computation.

3. Iteration and communication complexity

LAG preserves GD-order convergence under strongly convex, convex, and nonconvex smooth objectives while reducing communication under suitable data heterogeneity. Its guarantees rely on Lyapunov descent and worker-specific trigger rules that reuse gradients when local losses vary slowly.

  • Convergence analysis: The Lyapunov function decreases under properly chosen stepsize and trigger parameters, generalizing GD’s descent lemma.The admissible stepsize region includes α ∈ (0, 2/L), including α = 1/L.
  • Iteration complexity: LAG’s iteration complexity has the same order as GD’s, κ log(ϵ−1), although the analyzed bound has a worse constant.The larger constant follows from using a smaller stepsize in the stated parameter choice; empirically, α = 1/L performs nearly like GD.
  • Communication complexity: Workers satisfying the lazy-communication condition communicate at most k/(d + 1) rounds by iteration k, versus k rounds per worker under GD.The condition favors workers with small smoothness constants, corresponding to close-to-linear local losses.
  • Communication complexity: The heterogeneity score h(γ) lower-bounds the fraction of workers communicating at most k/(d + 1) rounds, enabling lower communication than GD under favorable heterogeneity.For strongly convex objectives, a sufficient condition is Δ̄C(h; ξ) > √Dξ; analogous conditions apply in convex and nonconvex cases.
  • Limitations: The current theoretical communication-reduction condition on h(γ) is relatively restrictive, leaving broader guarantees as future work.This limitation is attributed to technical issues in the convergence analysis.
  • Convex and nonconvex cases: For general convex and nonconvex smooth objectives, LAG achieves GD-identical convergence order and provably lower communication under certain conditions on h(γ).The authors state that empirical gains extend beyond the settings covered by the worst-case theory.

4. Numerical tests

The experiments evaluate LAG on synthetic and real linear and logistic regression tasks, comparing iteration and communication complexity with distributed baselines. LAG consistently reduces communication, while matching GD’s iteration complexity on Gisette.

  • Experimental setup: The evaluation covers linear and logistic regression on synthetic and real datasets, using Batch-GD, Cyc-IAG, Num-IAG, LAG-WK, and LAG-PS as benchmarks.The default setup uses one server and nine workers, with objective optimality error as the figure of merit.
  • Synthetic data: LAG-WK and LAG-PS require fewer communication rounds than GD on synthetic data with increasing smoothness constants.LAG-WK also improves communication under uniform smoothness by exploiting hidden local curvature.
  • Real datasets: LAG-WK consistently achieves the lowest communication complexity across tested real-dataset settings and worker counts.Its communication rounds can be smaller than the iteration count when no worker violates the trigger condition.
  • Gisette: On Gisette, LAG-WK and LAG-PS match GD’s iteration complexity and reduce communication rounds by several orders of magnitude versus the alternatives.The dataset contains 2000 samples with 4837 features and is split across nine workers.

5. Conclusions

The paper concludes that LAG adaptively skips communication by reusing lagged gradients while retaining GD-order convergence. Under heterogeneous data, this yields lower communication requirements, supported by synthetic and real-data experiments.

  • Conclusions: LAG is a communication-cognizant distributed batch-gradient method that skips communication at selected rounds by reusing stored gradients.The method is designed to remain as simple as GD while reducing communication rounds.
  • Convergence: LAG achieves the same convergence-rate orders as batch GD in smooth strongly-convex, convex, and nonconvex settings.The conclusion states this result across the three objective-function regimes.
  • Communication: For heterogeneous distributed datasets, LAG requires fewer communication rounds than GD to reach a targeted accuracy.The reported reduction depends on measurable heterogeneity conditions.
  • Empirical evidence: Experiments on synthetic and real datasets corroborate substantial communication reductions compared with alternative distributed methods.The empirical evidence is summarized as confirming the communication advantage of LAG.

Appendix B. Proof of Lemma 3

The proof establishes a sufficient-descent property for a Lyapunov function under parameter conditions tied to the LAG triggers. It then derives admissible parameter choices and shows the Lyapunov function is non-increasing.

  • Descent: The proof uses the definition of V_k to establish that the Lyapunov function is non-increasing.Specifically, it concludes V_k+1 ≤ V_k.
  • Proof steps: The sufficient-descent argument combines the LAG update, square-distance decomposition, Young’s inequality, smoothness, and trigger conditions.These ingredients connect the intermediate inequalities to the descent claim.
  • Parameter conditions: The descent conditions constrain the stepsize, trigger constants, and β_d parameters through inequalities indexed by d.One stated condition is α^2|M|^2 − β_d + β_d+1 ≤ 0 for d = 1, …, D − 1.
  • Parameter choices: When ρ → 0 and β_1 → 0, the admissible stepsize region becomes 0 ≤ α ≤ 2/L, matching GD’s stepsize region.The proof also discusses the special choice α = 1/L and selects a simpler parameterization for subsequent analysis.

Appendix C. Proof of Theorem 1

The proof derives Q-linear convergence of the Lyapunov function under the stated assumptions and converts it into R-linear convergence of the objective gap. A specific parameter choice then yields an iteration-complexity bound.

  • Convergence proof: Under the strong-convexity or PL condition, the proof derives a contraction for the Lyapunov function.The intermediate argument uses the strong convexity or PL condition from Assumption 3.
  • Convergence proof: Q-linear convergence of V_k implies R-linear convergence of L(θ_k) − L(θ*).This is the proof’s stated objective-convergence consequence.
  • Iteration complexity: The iteration-complexity analysis selects specific parameters because the generic rate constant depends on several design variables.The chosen parameters are not necessarily optimal.
  • Iteration complexity: The resulting iteration complexity is expressed using the condition number κ and a logarithmic accuracy term.The displayed derivation concludes with the bound denoted I_LAG(ϵ).

Appendix D. Proof of Lemma 4

Lemma 4 shows that under condition (21), a worker that recently communicated does not trigger either communication condition, so it can reuse its gradient for up to d iterations.

  • Condition (21) prevents trigger condition (15b) from activating for a worker that communicated within the previous d iterations.The argument considers the most recent communication at iteration k−d′, where 1 ≤ d′ ≤ d.
  • An additional bound prevents trigger condition (15a) from activating at the current iteration.
  • Consequently, the worker communicates with the server at most every other d iterations.The conclusion holds for any 1 ≤ d′ ≤ d.

Appendix E. Proof of Proposition 1

The proposition bounds LAG’s communication complexity by grouping workers according to heterogeneity and quantifying how often each group communicates. Under sufficiently favorable heterogeneity, LAG requires fewer communications than GD.

  • Communication-complexity bound: The quantity h(γd) lower-bounds the percentage of workers communicating at most every other d iterations, or at most K/(d + 1) times by iteration K.
  • Communication-complexity bound: Workers are partitioned into D + 1 nonoverlapping subgroups according to thresholds on their heterogeneity scores H2(m).Subgroup Md contains workers satisfying the corresponding threshold interval, with MD containing workers satisfying H2(m) < γD.
  • Communication-complexity bound: For workers in subgroup Md, the analysis upper-bounds communication rounds through the current iteration and aggregates these bounds across groups.The subgroup construction and h(γ) determine the resulting total communication-complexity upper bound.
  • Comparison with GD: With the selected parameters, the communication reduction is represented through Δ̄C(h; {γd}), which determines the fraction of GD’s average communication rounds avoided by LAG.
  • Comparison with GD: If ξ ≤ D/(D + 1)^2, there exists a distributed learning setting in which LAG’s communication complexity is lower than GD’s.This follows because Δ̄C(h; ξ) can attain values up to D/(D + 1) for suitable h.

Appendix G. Proof of Theorem 3

The proof establishes Theorem 3 by summing a Lyapunov-function inequality whose nonnegative terms are summable under the specified stepsize and trigger parameters.

  • Summing inequality (70) and using the Lyapunov function’s nonnegativity and finite initial value bounds the accumulated terms.
  • The parameter choice makes the coefficient c(α; {ξd}) positive, so the two left-hand-side terms are summable.
  • Summability implications complete the theorem’s proof.

Appendix H. Proof of Proposition 2

The proposition compares LAG and GD communication complexity, showing that adaptive communication can reduce total communications under a condition on the heterogeneity function. Experiments evaluate both LAG algorithms on regression tasks with synthetic and real datasets.

  • Communication comparison: LAG’s average communication rounds per iteration are (1 − Δ̄C(h; {γd})) times those of GD.
  • Communication comparison: LAG’s total communications are reduced when the stated condition on h(γ′) holds.The condition is equivalent to h(γ′) > (D + 1)ξ and requires the corresponding threshold relation involving γD and γ′.
  • Experiments: The evaluation tests two LAG algorithms on linear and logistic regression using square and binary logistic losses.
  • Experiments: Real-data experiments use Housing, Body fat, Ionosphere, Adult, and Derm datasets.
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