Source-linked AI summary
On the Convergence Rate of Incremental Aggregated Gradient Algorithms
Mert Gurbuzbalaban, Asuman Ozdaglar, Pablo Parrilo
TL;DR
The paper addresses the lack of explicit deterministic convergence rates for incremental aggregated gradient methods in strongly convex finite-sum optimization. It analyzes IAG with a Lyapunov function and extends the method with momentum, establishing global linear convergence and a corresponding rate analysis.
Problem
The paper addresses the lack of explicit convergence-rate analysis for deterministic IAG methods in strongly convex finite-sum optimization.
Method
The analysis treats IAG as gradient descent with delayed-gradient errors and uses a carefully chosen Lyapunov function, also studying an IAG variant with momentum.
Results
IAG is globally linearly convergent with an explicit rate result, and the momentum variant also has a linear convergence and rate analysis.
Takeaways & Limitations
The analysis explains delayed gradient errors as finite-duration shocks and provides a simpler global convergence framework for deterministic IAG and its momentum extension.
Takeaways & Limitations
Extending the linear-rate results to objectives with an additional possibly nonsmooth convex term remains future work.
Abstract
from arXiv · showhide
Motivated by applications to distributed optimization over networks and large-scale data processing in machine learning, we analyze the deterministic incremental aggregated gradient method for minimizing a finite sum of smooth functions where the sum is strongly convex. This method processes the functions one at a time in a deterministic order and incorporates a memory of previous gradient values to accelerate convergence. Empirically it performs well in practice; however, no theoretical analysis with explicit rate results was previously given in the literature to our knowledge, in particular most of the recent efforts concentrated on the randomized versions. In this paper, we show that this deterministic algorithm has global linear convergence and characterize the convergence rate. We also consider an aggregated method with momentum and demonstrate its linear convergence. Our proofs rely on a careful choice of a Lyapunov function that offers insight into the algorithm's behavior and simplifies the proofs considerably.
1 Introduction
The paper studies deterministic incremental methods for finite-sum optimization, motivated by distributed and machine-learning applications. It develops explicit global linear convergence results for IAG and extends the analysis to momentum.
- The objective is a sum of convex, continuously differentiable component functions arising in applications including least squares, parameter estimation, sensor networks, and machine learning.
- Incremental gradient methods reduce per-step computation by processing one component at a time, but generally require diminishing stepsizes and therefore have typical sublinear convergence.
- IAG and related methods retain recent gradients for all components while processing one function at a time, supporting applications with structured deterministic processing orders.
- IAG analysis provides explicit global linear convergence rates for deterministic finite-sum optimization, addressing a gap left by earlier work.Earlier results established global convergence or restricted linear convergence without explicit rate or stepsize bounds.
- The paper uses a Lyapunov-based analysis and introduces IAG with momentum, proving global and linear convergence for the aggregated-gradient method with memory.
- Unlike randomized SAG analyses, the paper targets deterministic convergence for any order that visits every function within a finite number of steps, while accounting for potentially worse order-dependent rates.
2 IAG method
IAG updates iterates using an aggregate of recently computed component gradients rather than a fully current gradient. Bounded delays characterize the method, with cyclic processing yielding the original IAG scheme.
- IAG updates x_k using an aggregate gradient g_k with constant stepsize γ, following x_{k+1} = x_k − γg_k.
- Each component gradient must be sampled at least once during the preceding K iterations, so K bounds the delay of stored gradients.
- The aggregate direction approximates steepest descent at x_k using gradients computed at earlier iterates, making IAG a perturbed gradient-descent method.
- Under cyclic processing, the original IAG method has K = m − 1 because every component is revisited within one cycle.
- With K = 0, all gradients are current and IAG reduces to classical gradient descent; the analysis focuses on the more interesting case K > 0.
- The analysis initializes stored iterates simply, while allowing extension to arbitrary initial points and more general sampling times satisfying the delay condition.
3 Convergence analysis
Under strong convexity and Lipschitz-gradient assumptions, the analysis bounds delayed-gradient errors and proves globally linear IAG convergence for an explicit constant-stepsize range. The resulting rate is explicit, while gradient errors behave as finite-duration shocks that degrade convergence.
- Gradient-error analysis: The Lyapunov-style recursion applies when the contraction and perturbation terms satisfy s(γ) = p(γ) + q(γ) < 1, with a stricter condition when K > 0.For delayed gradients, the condition becomes fourth order in γ, unlike the K = 0 case.
- Gradient-error analysis: Gradient errors are bounded through recent iterate distances and modeled as bounded-duration shocks, preserving linear convergence but worsening the rate.The degradation depends on shock amplitude and duration, controlled by q and d(k).
- Theorem 3.3: IAG iterates converge globally and linearly for constant stepsizes 0 < γ < γ̄ under the stated strong-convexity and smoothness assumptions.The theorem applies when the gradient-delay parameter satisfies K > 0.
- Theorem 3.3: The optimized stepsize γ* = γ̄/2 yields an explicit linear convergence rate for the iterates.The rate is obtained by first analyzing squared distances and then taking square roots.
- Comparison with previous results: Compared with earlier analyses, the paper provides transparent explicit constants and a computable stepsize interval rather than only asymptotic existence results.Earlier results used implicit constants or established convergence only for sufficiently small stepsizes.
- IAG versus IG: Unlike incremental gradient with a constant stepsize, IAG converges to the optimum because its gradient error is controlled by distances to the solution.Incremental gradient generally converges only to a neighborhood under a constant stepsize.
- Local strong convexity: If the objective is only locally strongly convex around a stationary point, the same reasoning supports local linear convergence rather than a global guarantee.This extends the result beyond global strong convexity only in a local sense.
4 IAG with momentum
The paper introduces incremental aggregated gradient with momentum (IAG-M) to address the limited convergence guarantees of deterministic incremental methods with momentum. It argues that IAG-M can converge globally and linearly for suitable momentum and stepsize parameters.
- 4 IAG with momentum: Deterministic incremental methods with momentum had fewer favorable convergence results than non-incremental momentum methods.The paper positions IAG-M against heavy-ball and accelerated-gradient approaches, while noting limitations of IG with momentum.
- 4 IAG with momentum: IAG-M is expected to outperform IAG on problems with oscillatory individual gradients because momentum provides additional smoothing or averaging of iterates.This is presented as an expected advantage rather than an established empirical result.
- 4 IAG with momentum: The paper develops a global linear convergence analysis for IAG-M over a suitable range of momentum values.The proof modifies the IAG argument to account for the additional momentum terms in iterate differences.
5 Discussion
The discussion interprets IAG as gradient descent with fading, finite-duration gradient-error shocks and reports an explicit linear-rate analysis. It also presents IAG-M and identifies extension to nonsmooth convex terms as future work.
- 5 Discussion: The analysis views IAG as gradient descent with errors whose finite-duration shocks fade as iterates approach the optimum.Strong convexity and a sufficiently small stepsize yield a contraction in distance to the optimal solution.
- 5 Discussion: The paper claims its analysis provides the first explicit linear rate result for IAG under strongly convex component functions.It contrasts the approach with earlier analyses based on more technical or asymptotic arguments.
- 5 Discussion: The paper also develops IAG with momentum and provides a corresponding linear convergence and rate analysis.Its expected benefit is additional smoothing when individual gradients oscillate.
- 5 Discussion: Extending the linear-rate results to a convex, possibly nonsmooth additional term remains future work.A proximal step can extend the algorithmic framework to such terms, but the paper does not establish the corresponding rate result.
A Proof sketch of the global linear convergence of the IAG-M method
The proof sketch extends the IAG contraction argument to momentum by bounding iterate changes and gradient errors. It concludes that IAG-M is globally linearly convergent when momentum and stepsize satisfy suitable bounds.
- A Proof sketch of the global linear convergence of the IAG-M method: The analysis explicitly reduces to the IAG inequality when β = 0.This connects the momentum proof to the previously established no-momentum case.
- A Proof sketch of the global linear convergence of the IAG-M method: The proof bounds each iterate displacement using the previous gradient, gradient error, and distances to the optimum.The momentum term contributes the additional distance terms involving consecutive iterates.
- A Proof sketch of the global linear convergence of the IAG-M method: Combining the bounds for the Lyapunov-related quantities produces an IAG-M analogue of the key IAG inequality.The resulting inequality supports the same contraction-style reasoning used for Theorem 3.3.
- A Proof sketch of the global linear convergence of the IAG-M method: IAG-M is globally linearly convergent when β ≤ bγ^p with p ≥ 1/2 and γ sufficiently small.The constants b and the admissible stepsize bounds arise from an argument analogous to the IAG proof.