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Distributed Delayed Stochastic Optimization
Alekh Agarwal, John C. Duchi
TL;DR
The paper addresses how asynchronous delays affect stochastic optimization when distributed workers compute gradients in parallel. It extends dual averaging and mirror descent to delayed gradients and shows that smooth stochastic problems retain asymptotically optimal distributed convergence, while the cyclic architecture has a communication-related limitation.
Problem
Distributed stochastic optimization needs methods that handle delayed gradient information without sacrificing the benefits of parallel computation or requiring synchronization.
Method
The paper extends dual averaging and mirror descent by updating with gradients computed at earlier iterates under bounded expected delays.
Results
O(1/(nε^2)) is the asymptotic time to reach an ε-optimal solution, a factor n better than a centralized procedure despite delay.
Takeaways & Limitations
Smooth stochastic distributed optimization can exploit asynchronous processing without incurring an asymptotic penalty from delays.
Takeaways & Limitations
The cyclic architecture can suffer a large n^2m/T term because information from a worker may take O(n) time to reach the master.
Abstract
from arXiv · showhide
We analyze the convergence of gradient-based optimization algorithms that base their updates on delayed stochastic gradient information. The main application of our results is to the development of gradient-based distributed optimization algorithms where a master node performs parameter updates while worker nodes compute stochastic gradients based on local information in parallel, which may give rise to delays due to asynchrony. We take motivation from statistical problems where the size of the data is so large that it cannot fit on one computer; with the advent of huge datasets in biology, astronomy, and the internet, such problems are now common. Our main contribution is to show that for smooth stochastic problems, the delays are asymptotically negligible and we can achieve order-optimal convergence results. In application to distributed optimization, we develop procedures that overcome communication bottlenecks and synchronization requirements. We show $n$-node architectures whose optimization error in stochastic problems---in spite of asynchronous delays---scales asymptotically as $\order(1 / \sqrt{nT})$ after $T$ iterations. This rate is known to be optimal for a distributed system with $n$ nodes even in the absence of delays. We additionally complement our theoretical results with numerical experiments on a statistical machine learning task.
1 Introduction
The paper studies stochastic convex optimization with delayed gradients in distributed systems, where asynchronous updates can avoid synchronization constraints. It shows that smooth stochastic problems can retain parallelization benefits despite delays, with convergence rates ranging from O(n^3/T + 1/√(nT)) to O(1/T^2/3 + 1/√(nT)).
- Problem setting: The optimization problem minimizes a convex expected objective over a closed convex set using stochastic gradients.The objective is f(x) = E_P[F(x; ξ)], with convex sample functions F(·; ξ).
- Distributed motivation: Distributed workers compute stochastic gradients in parallel while a master updates parameters, creating potentially asynchronous delays.The architecture is motivated by large datasets and is intended to avoid synchronization issues in distributed computation.
- Main contribution: For smooth stochastic problems, delays are asymptotically negligible, allowing delayed updates to preserve parallelization benefits.The paper contrasts this with prior asynchronous methods whose rates incur an asymptotic delay penalty.
- Results: O(n^3/T + 1/√(nT)) to O(n/T + 1/√(nT)) are the reported convergence-rate range under different network and delay assumptions.The rates depend on the network and stochastic delay process.
- Results: O(1/(nε^2)) is the asymptotic time needed to reach an ε-optimal solution, a factor n better than a centralized procedure despite delays.The improvement is stated for the distributed setting analyzed in the paper.
2 Setup and Algorithms
The paper extends dual averaging and mirror descent to delayed stochastic gradients under convexity, smoothness, variance, and compactness assumptions. The delayed methods replace current gradients with gradients computed at earlier iterates while retaining established convergence guarantees under bounded expected delays.
- Delay-free algorithms: Dual averaging and mirror descent are the two first-order algorithms used as the delay-free foundations.Both use a proximal function to regularize iterates while approximating the objective linearly.
- Delay-free algorithms: The stochastic oracle samples ξ(t) independently and returns a subgradient or gradient evaluated at the current iterate.Dual averaging updates a dual vector and a primal vector, while mirror descent performs a proximal update.
- Assumptions: The analysis assumes convex sample functions, bounded expected subgradient norms, smoothness with a variance bound, and a compact optimization set.These are collected in Assumptions A–C and support the stated convergence rates.
- Delayed algorithms: The delayed algorithms replace g(t) with g(t − τ(t)) in the dual averaging and mirror descent updates.The delay process may be non-i.i.d., provided E[τ(t)] ≤ B < ∞ for every t.
- Convergence guarantees: Under bounded expected delays, delayed dual averaging and mirror descent inherit convergence guarantees through suitable stepsizes.The paper presents a generalization of prior asynchronous subgradient results under Assumptions A and C.
3 Convergence rates for delayed optimization of smooth functions
The section establishes convergence results for asynchronous stochastic optimization under smoothness, including single and multiple delayed stochastic gradients. Its central conclusion is that delay penalties are asymptotically negligible, with implications for distributed stochastic optimization.
- Single delayed gradients: Theorems 1 and 2 analyze asynchronous updates using one stochastic subgradient, including dual averaging and mirror descent.The two update rules receive separate convergence theorems under the stated assumptions.
- Smooth delayed optimization: Smoothness makes stale gradients only slightly perturbed when the algorithm constrains variability in x(t).This addresses the severe delay penalty that can arise for non-smooth optimization.
- Random delays: Random delays are permitted when their second moments satisfy E[τ(t)^2] ≤ B^2 for all t.Corollary 1 gives convergence guarantees for either update rule under this bounded-second-moment condition.
- Single delayed gradients: Delay penalties in the convergence rate are asymptotically negligible for the analyzed asynchronous methods.The result also supports favorable implications for robust distributed stochastic optimization.
- Combinations of delays: Theorems 3 and 4 extend the analysis to updates combining n stochastic gradients with potentially different delays.The gradients are weighted by λ in the probability simplex, with separate dual-averaging and mirror-descent updates.
- Distributed implications: The multi-gradient results have consequences for distributed architectures, and their rates can be adapted when communication costs are known.The paper highlights settings with large n or expensive communication to the master node.
4 Distributed Optimization
The paper develops asynchronous master-worker architectures for distributed stochastic optimization, using delayed and locally averaged gradients to reduce communication and synchronization burdens. These methods achieve faster convergence, with locally communicated schemes benefiting from smaller graph-diameter-dependent delays.
- Motivation: Distributed stochastic optimization targets datasets too large for one computer by splitting computation across worker nodes and a master.Workers compute stochastic gradients from local data or streams, while the master updates the shared parameter.
- Distributed architectures: The proposed protocols allow asynchronous parallel gradient computation without imposing full synchronization requirements.The paper considers independent master-worker communication and distance-based communication through a spanning tree.
- Convergence rates: O(1/√(nT)) convergence is achieved asymptotically by both distributed protocols, with optimization error scaling optimally in the number of nodes and iterations.The locally averaged result is stated as an optimal dependence on the number of stochastic samples.
- Locally averaged architecture: The locally averaged architecture gives the master a convex combination of gradients with delays bounded by the network diameter D.Its delays depend on spanning-tree height rather than the full number of workers.
- Communication-aware rates: For communication-aware cyclic optimization, convergence follows two regimes: O(n2/3/T2/3) when T ≤ n7/C3 and O(1/√(TCn)) when T > n7/C3.The latter rate is order-optimal given the total number of computed stochastic gradients.
- Topology and running time: Locally averaged methods always improve on cyclic methods when D ≤ n, while topology determines the size of the improvement.Cycles or paths yield equal rates; balanced trees and expanders provide an n2/3 improvement over cyclic architectures, ignoring logarithmic terms.
5 Numerical Results
The experiments evaluate a cyclic delayed optimization method for logistic regression on RCV1, measuring time to ε-accuracy across worker counts and per-worker sample sizes. Delayed updates achieve nearly linear speedup for moderate worker counts, while communication and delay costs reduce gains at larger scales.
- RCV1 contains approximately 800,000 news articles represented by roughly 10^5-dimensional word-feature vectors for logistic regression.
- The experiments vary the number of workers n and samples m computed at each worker, measuring time to reach ε = .05 accuracy over ten trials.
- Running time is estimated by multiplying iterations to ε-optimality by max{m/n, 1} for the distributed method and m for the centralized method.
- The delayed update achieves speedup nearly linear in n until approximately n ≥ 15 workers.
- For larger n, the n^3/T term becomes non-negligible, so communication and delay costs mitigate parallelization benefits in nonasymptotic regimes.
6 Delayed Updates for Smooth Optimization
The paper analyzes delayed stochastic gradient updates for smooth optimization using dual averaging and mirror descent. Smoothness lets the analysis control stale-gradient errors, making delay effects asymptotically negligible under suitable stepsize and delay conditions.
- Lemma 4 uses Bregman-divergence identities and iterate-distance bounds to show that gradient-differencing terms are essentially second order.
- Decreasing stepsizes control stochastic variance and combine with telescoping Bregman-divergence terms in the proofs of dual averaging and mirror descent.
- Delayed updates replace current stochastic gradients with out-of-date gradients, creating gradient-differencing error terms in the convergence analysis.
- Random delays are accommodated when their second moments are bounded, using expected iterate-distance and delayed-gradient error bounds.
7 Proof of Theorem 3
Theorem 3 extends the delayed stochastic analysis to distributed gradients with worker-specific delays. Its proof controls deterministic delay errors through smoothness and stochastic errors through conditional independence and variance bounds.
- The distributed proof defines worker-specific gradient errors using gradients evaluated at delayed iterates and combines them in the master update analysis.
- The analysis applies the same Bregman-divergence and telescoping techniques as the earlier theorems to bound the resulting update terms.
- Conditional independence makes one stochastic error term have zero expectation, while the remaining terms are controlled using Fenchel-Young inequalities and prior lemmas.
8 Conclusion and Discussion
The paper concludes that delayed dual averaging and mirror descent can preserve parallelization benefits for smooth stochastic optimization without an asymptotic delay penalty. The methods are also described as robust to failures, communication latency, and node slowdowns.
- Smooth delayed optimization preserves the performance benefits of parallelization over centralized stochastic optimization while relaxing synchronization requirements.
- The presented methods use distributed computation and are robust to node failures, communication latency, and node slowdowns.
- The expected convergence results can also be extended to high-probability guarantees.
A Technical Results about Proximal Functions
This section collects proximal-function and continuity results used to analyze dual-averaging and mirror-descent updates. It introduces properties of proximal dual functions and bounds changes in their iterates.
- The section develops results for the dual-averaging updates (4) and (9).
- The proximal dual function is introduced, with strong convexity of ψ supporting continuity properties of its solution map.
- α(z) is α-Lipschitz continuous under the norm associated with ψ’s strong convexity and its dual norm.
- A lemma characterizes the minimizer x+ of ⟨z, x⟩ + Aψ(x) over X and provides a result used for dual-averaging updates.
- Additional lemmas bound changes between consecutive iterates and between x(t) and x(t + τ) for dual-averaging and mirror-descent families.
- Lemma 7 applies to specified dual-averaging and mirror-descent updates with α(t)^−1 = L + η(t + t0)^c, where c ∈ [0, 1].
B Proofs of Proximal Operator Properties
The proofs establish continuity and delayed-iterate bounds using strong convexity, Hölder’s inequality, triangle inequalities, and expectation bounds. The arguments cover both dual-averaging and mirror-descent updates.
- The proof of Lemma 6 derives the relevant inequality from strong convexity and Hölder’s inequality.
- Lemma 7’s dual-averaging proof bounds iterate differences using changes in α(t), z(t), and z(t + τ).
- 4G^2τ^2/(η^2t^(2c)) bounds the final term when c ≤ 1.
- A tighter first-moment bound follows by applying the triangle inequality to bound (34).
- The mirror-descent proof focuses on the non-delayed update (5), while the other updates modify the indexing of g(t + s).
- The argument squares the iterate-difference bound, takes expectations, and uses that α(t) is non-increasing.
C Error in [LSZ09]
This section identifies a constrained-optimization error in a lemma from Langford et al. The stated relation holds as an equality only in the unconstrained case and can fail even as an upper bound under constraints.
- Langford et al.’s Lemma 1 is essential to their proofs but is valid as an equality only for unconstrained optimization, X = R^d.
- With constraints, the lemma can fail even as an upper bound, as shown by the example X = [−1, 1], f(x) = |x|, η ≡ 2, τ = 1, and t = 2.