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Asynchronous Parallel Stochastic Gradient for Nonconvex Optimization
Xiangru Lian, Yijun Huang, Yuncheng Li, Ji Liu
TL;DR
Asynchronous stochastic-gradient methods are widely used, but existing theory provides limited explanation of their convergence and speedup in nonconvex optimization. This paper analyzes computer-network and shared-memory implementations, proving asymptotic convergence and linear-speedup guarantees. The results extend and improve earlier convex analyses and apply to broader shared-memory scenarios.
Problem
Existing theories do not explain the convergence and speedup properties of asynchronous stochastic gradient in nonconvex optimization, although the method is widely used in deep learning.
Method
The paper studies AsySG-con for computer networks and AsySG-incon for shared-memory systems, modeling their distinct read and write behavior.
Results
An asymptotic sublinear convergence rate is proven for both algorithms, with linear speedup achievable when the number of workers is bounded by O(√K).
Takeaways & Limitations
The analysis provides theoretical convergence and speedup guarantees for asynchronous stochastic-gradient methods used in deep learning and extends earlier convex results.
Takeaways & Limitations
Asynchronous parallelism uses old parameter values for stochastic-gradient evaluation, and communication-bandwidth ceilings can sharply worsen speedup beyond a machine-dependent threshold.
Abstract
from arXiv · showhide
Asynchronous parallel implementations of stochastic gradient (SG) have been broadly used in solving deep neural network and received many successes in practice recently. However, existing theories cannot explain their convergence and speedup properties, mainly due to the nonconvexity of most deep learning formulations and the asynchronous parallel mechanism. To fill the gaps in theory and provide theoretical supports, this paper studies two asynchronous parallel implementations of SG: one is on the computer network and the other is on the shared memory system. We establish an ergodic convergence rate $O(1/\sqrt{K})$ for both algorithms and prove that the linear speedup is achievable if the number of workers is bounded by $\sqrt{K}$ ($K$ is the total number of iterations). Our results generalize and improve existing analysis for convex minimization.
1 Introduction
This paper studies two asynchronous parallel stochastic-gradient implementations for nonconvex optimization, addressing limited theoretical understanding of their convergence and speedup. It establishes convergence and linear-speedup guarantees while identifying how delayed information affects asynchronous updates.
- Motivation: Existing theories do not explain AsySG convergence and speedup in nonconvex optimization, despite its use in deep neural networks and multiple computing platforms.The gap is attributed mainly to nonconvex formulations and asynchronous mechanisms.
- Approach: The paper analyzes two implementations: AsySG-con on computer networks and AsySG-incon on shared-memory systems.The architectures differ in whether they can atomically read and write the whole parameter vector or only individual coordinates.
- Results: O(1/√K) asymptotic convergence is established for both asynchronous algorithms on nonconvex smooth optimization.Here, K denotes the total iteration number.
- Results: Linear speedup is proven achievable when the number of workers is bounded by O(√K).The paper defines linear speedup as speedup with T workers greater than cT, for a constant c independent of T.
- Mechanism: Asynchronous updates trade reduced system overhead for stale information, whose additional deviation should remain minor relative to stochastic-gradient deviation.The analysis states that the negative effect of using old parameter values vanishes asymptotically.
2 Related Work
Related work covers stochastic-gradient convergence, synchronous and asynchronous parallel methods, and prior analyses of AsySG-con and Hogwild!. The paper positions its nonconvex analysis as an improvement over earlier convex results and as applicable to broader asynchronous settings.
- Stochastic Gradient: Stochastic gradient has established rates of O(1/√K) for convex problems and O(1/K) for strongly convex problems.The related-work discussion uses these rates as standard reference points for stochastic-gradient analysis.
- Synchronous Parallel Methods: Synchronous mini-batch stochastic gradient achieves O(1/√(KM)) for convex optimization, with workers synchronizing before each parameter update.A related variant is described as asymptotically robust to asynchrony when most processors remain synchronized most of the time.
- Related Asynchronous Methods: Prior asynchronous methods also studied coordinate descent, dual coordinate ascent, randomized Kaczmarz, and semi-asynchronous optimization.For parallel stochastic coordinate descent, linear speedup was reported under bounds depending on consistent versus inconsistent reads.
- Asynchronous SG: Agarwal and Duchi analyzed AsySG-con for convex smooth optimization and proved linear speedup when T is bounded by O(K^1/4/M^3/4).The present analysis extends this setting to nonconvex smooth optimization and improves the worker bound by K^1/4M^1/4.
- Asynchronous SG: Hogwild! introduced a lock-free asynchronous stochastic-gradient implementation for shared memory and established a sublinear convergence rate O(1/K) for strongly convex settings.The paper distinguishes its shared-memory analysis from Hogwild! because the problem settings differ.
3 Asynchronous parallel stochastic gradient for computer network
The computer-network implementation AsySG-con uses asynchronous worker updates with delayed but consistent parameter reads. Under stated independence and bounded-delay assumptions, it achieves an ergodic rate matching serial and synchronous stochastic-gradient cases, with linear speedup for sufficiently few workers.
- Algorithm description: AsySG-con models asynchronous stochastic gradient on a star-shaped computer network, where a master maintains x and workers independently pull parameters and push stochastic gradients.The master aggregates a predefined number of gradients and performs atomic updates.
- Algorithm description: The main cost of asynchronous execution is that stochastic gradients may use old parameter values, although the paper states this negative effect vanishes asymptotically.This delay is the trade-off for substantially reduced system overhead and tolerance of large network delays.
- Algorithm description: The algorithm’s consistent-read property means each stochastic gradient uses a real state of x, ensured by atomicity of the master’s update.The network architecture can atomically read and write the whole parameter vector, unlike the shared-memory setting discussed elsewhere.
- Convergence analysis: Under the paper’s assumptions, AsySG-con has an ergodic convergence rate for nonconvex optimization based on expected gradient norms.The analysis uses a weighted average of gradient norms and expectation over the algorithm’s random variables.
- Convergence analysis: O(1/√(MK)) is attained when K is sufficiently larger than O(T^2), making the delay parameter’s asymptotic effect disappear.Here K is the total iteration number, M is minibatch size, and T bounds delay and is roughly proportional to worker count.
- Speedup and comparison: O(K^1/2 M^-1/2) workers suffice for linear speedup, improving the earlier O(K^1/4 M^-3/4) worker bound by a factor O(K^1/4 M^1/4).The resulting convergence rate is consistent with several serial and synchronous stochastic-gradient special cases.
4 Asynchronous parallel stochastic gradient for shared memory architecture
The paper analyzes a lock-free asynchronous stochastic-gradient implementation for shared-memory systems under inconsistent reads, bounded delays, and explicit assumptions. It establishes ergodic convergence results and compares their scope with prior analyses.
- Algorithm Description: AsySG-incon: The AsySG-incon algorithm models shared-memory workers that independently read, compute stochastic gradients, and update parameters without software locks.Its lock-free design permits simultaneous memory access, unlike the lock-based network implementation.
- Algorithm Description: AsySG-incon: Inconsistent reads can combine components from different parameter states, so a worker’s read may not equal any real state previously held in shared memory.This distinguishes AsySG-incon from AsySG-con, where workers obtain a consistent parameter value.
- Analysis for AsySG-incon: The analysis represents each read through prior-iteration subsets and assumes independent iteration variables together with a global bounded delay T.The bounded-age condition restricts the age of missing components, while T is proportional to the number of workers and independent of minibatch size M.
- Analysis for AsySG-incon: O(1/√(MK)) is achieved asymptotically when K exceeds a threshold of order O(T^2), with n treated as a constant.The same rate and threshold are reported as consistent with the AsySG-con result.
- Analysis for AsySG-incon: The analysis provides a more precise inconsistent-read treatment than Hogwild!, applies to smooth nonconvex optimization, and extends to broader scenarios despite not strictly dominating prior results.A sparse-gradient extension slightly improves the convergence rate by exploiting sparsity.
5 Experiments
The experiments evaluate AsySG-con on LENET and CIFAR10-FULL and AsySG-incon on synthetic neural-network data, comparing iteration and running-time speedups across workers or machines. Results show near-linear iteration scaling, with running-time speedup generally lower and communication bandwidth limiting cluster performance at higher machine counts.
- Experimental measures: The experiments measure both iteration speedup and running-time speedup, with the former less affected by hardware.Iteration speedup compares iterations needed by serial SG and T workers at the same precision; running-time speedup compares their elapsed times.
- 5.1 AsySG-con: AsySG-con is evaluated on LENET and CIFAR10-FULL using up to 8 machines.The study implements AsySG-con with Caffe and reports objective loss against iterations and running time.
- 5.1 AsySG-con: Iteration speedup is always better than running-time speedup in the AsySG-con experiments.The paper attributes the difference to hardware effects on elapsed time.
- 5.1 AsySG-con: AsySG-con speedups are comparable overall across LENET and CIFAR10-FULL, while CIFAR10-FULL time speedup is slightly more stable.LENET time speedup drops from 5.29 with mpi-7 to 2.88 with mpi-8.
- 5.1 AsySG-con: Communication bandwidth can sharply reduce LENET performance when machine count reaches a threshold.LENET has more parameters than CIFAR10-FULL and therefore requires greater communication cost; the reported threshold case is 8 machines.
- 5.2 AsySG-incon: AsySG-incon shows almost linear iteration speedup and slightly lower running-time speedup on a shared-memory system with 1 to 32 cores.The experiment uses synthetic five-layer neural-network data, minibatch size M = 32, and step length γ = 1.1 × 10^-7.
6 Conclusion
The conclusion reports sublinear convergence for both asynchronous SG implementations in nonconvex smooth optimization and establishes linear speedup under a worker-count bound. It also presents AsySG-incon as a more precise model of lock-free shared-memory implementation than Hogwild!.
- 6 Conclusion: Both AsySG-con and AsySG-incon have an asymptotic sublinear convergence rate for nonconvex smooth optimization.The paper states that this rate is consistent with SG for convex optimization.
- 6 Conclusion: Linear speedup is achievable when the number of workers is bounded by the stated condition involving K.The conclusion presents this result as an improvement over earlier AsySG-con analysis for convex optimization.
- 6 Conclusion: AsySG-incon provides a more precise description of lock-free shared-memory implementation than Hogwild!.The conclusion also states that its result applies to more scenarios.
Proofs to Theorem 1
The proof of Theorem 1 combines stochastic-gradient expectations with bounds for auxiliary terms, using smoothness, delay limits, and full expectation to derive the theorem’s inequality.
- Proof strategy: The proof uses the unbiased stochastic-gradient assumption when taking expectation with respect to the stochastic sample.This establishes the expectation step used in the theorem’s derivation.
- Proof strategy: The proof separately estimates auxiliary terms T1 and T2 using the stated assumptions and Lipschitzian gradient smoothness.The bound for T1 also applies a norm inequality to combine vector terms.
- Proof strategy: The delayed-information term is bounded using the delay-age condition τk,µ ≤ T.The proof substitutes upper bounds for additional expectation terms before aggregating the inequality.
- Proof strategy: Summing the derived inequality from k = 1 to k = K and using the stated condition completes the convergence proof.The proof then invokes the global optimization point x∗.
Proofs to Corollary 2
The proof of Corollary 2 verifies that the theorem’s condition holds globally and then applies the theorem’s bound to obtain the corollary.
- Proof strategy: The proof derives a global implication of conditions (9) and (10) for condition (7) in Theorem 1.This verification permits direct use of the theorem’s result.
- Proof strategy: The proof completes Corollary 2 by applying equation (8) after establishing the theorem condition.The final step uses the preceding inequality and the equality associated with condition (9).
Proofs to Theorem 3
The proof derives bounds for the stochastic and asynchronous cross terms by taking expectations over worker indices and stochastic samples. These bounds are then combined to establish the required inequality and complete the proof of Theorem 3.
- Proofs to Theorem 3: The proof decomposes the analysis into terms T1 through T4 and treats them separately under expectation.The cited passages explicitly introduce T1 and T2, then evaluate T3 and T4 before substitution.
- Proofs to Theorem 3: Expectation identities transform cross-worker inner products using the random-worker variables E_k,m′ and the stochastic samples ξ_k,∗.The proof takes expectations over i_k and ξ_k,∗ and rewrites the cross terms accordingly.
- Proofs to Theorem 3: The bounds for T3 and T4 are substituted into the bound for T1, followed by substitution of T1 and T2 into the preceding inequality.This sequential substitution produces the final bound used in the theorem’s proof.
- Proofs to Theorem 3: The final inequality uses assumption (15) together with E(f(x_k+1)) ≥ f(x∗) to complete the proof.The argument concludes immediately after applying the lower bound on the objective value.
Proofs to Corollary 4
The proof verifies that the steplength satisfies the condition required by Theorem 3. It then applies the theorem’s bound after controlling the steplength using the lower bound on K.
- Proofs to Corollary 4: The steplength definition and the lower bound on K yield an upper bound for γ.The proof further relaxes this upper bound before verifying the theorem condition.
- Proofs to Corollary 4: The relaxed bound ensures that the steplength satisfies condition (15) globally.The proof invokes L_max ≤ L_T and an inequality involving √nT to establish the condition.
- Proofs to Corollary 4: After verifying condition (15), the proof applies (16) from Theorem 3 and substitutes γ using its definition in (17).The argument concludes after using the upper bound of γ and the steplength definition.