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Taming the Wild: A Unified Analysis of Hogwild!-Style Algorithms

Christopher De Sa, Ce Zhang, Kunle Olukotun, Christopher Ré

arXiv:1506.06438v2cs.LGmath.OCstat.ML

TL;DR

Asynchronous execution and reduced precision complicate SGD convergence analysis through stale updates and additional noise. This paper develops a unified martingale-based framework and applies it across convex, non-convex, and low-precision settings, with BUCKWILD! achieving speedups of up to 2.3× over HOGWILD!-based algorithms for logistic regression.

  • Problem

    Convergence analysis for asynchronous SGD is difficult because stale updates create race conditions, especially in non-convex problems.

  • Method

    The paper develops a martingale-based framework that models sampling, initialization, and asynchronous delays as unified noise and adapts sequential convergence proofs to asynchronous algorithms.

  • Results

    The framework derives rates for relaxed-sparsity convex HOGWILD!, asynchronous non-convex matrix completion, and quantization errors, while BUCKWILD! achieves speedups of up to 2.3× over HOGWILD!-based logistic-regression algorithms.

  • Takeaways & Limitations

    The results provide a common convergence framework for asynchronous and low-precision SGD variants, with BUCKWILD! achieving near-linear parallel speedup over sequential algorithms.

  • Takeaways & Limitations

    The analysis assumes asynchronous delays are bounded above by a random variable, rather than assuming independent delays.

Abstract

from arXiv · show

Stochastic gradient descent (SGD) is a ubiquitous algorithm for a variety of machine learning problems. Researchers and industry have developed several techniques to optimize SGD's runtime performance, including asynchronous execution and reduced precision. Our main result is a martingale-based analysis that enables us to capture the rich noise models that may arise from such techniques. Specifically, we use our new analysis in three ways: (1) we derive convergence rates for the convex case (Hogwild!) with relaxed assumptions on the sparsity of the problem; (2) we analyze asynchronous SGD algorithms for non-convex matrix problems including matrix completion; and (3) we design and analyze an asynchronous SGD algorithm, called Buckwild!, that uses lower-precision arithmetic. We show experimentally that our algorithms run efficiently for a variety of problems on modern hardware.

1 Introduction

The paper develops a martingale-based framework that models asynchronous execution and reduced precision as noise within a unified analysis of SGD. It applies this framework to relaxed-sparsity HOGWILD!, asynchronous non-convex matrix completion, and quantized BUCKWILD!, which achieves speedups up to 2.3× over HOGWILD!-based logistic-regression algorithms.

  • Motivation: SGD is a ubiquitous machine-learning algorithm [1], used in applications including neural-network backpropagation, recommendation systems, and optimization [20].Its theoretical success on non-convex problems, particularly deep learning, remains poorly understood.
  • Motivation: Asynchronous lock-free execution, including HOGWILD! [17], speeds SGD by allowing multiple threads to update in parallel without locks.Related methods have been applied to PageRank approximations [16], deep learning, and recommender systems.
  • Unified framework: A martingale-based result unifies diverse extensions by treating asynchrony and precision changes as different forms of noise.This avoids requiring an entirely new analytical model for each type of asynchrony or precision.
  • Applications: For convex HOGWILD!, the analysis relaxes strict sparsity assumptions while deriving convergence rates and recovers prior rates under HOGWILD!’s stricter assumptions.The result addresses the convex case without losing the established guarantees under the original conditions.
  • Applications: The framework derives rates for asynchronous SGD on non-convex matrix completion, gives the first such rates following synchronous work by De Sa et al. [4], and analyzes quantization errors in fixed-point arithmetic.Experimentally, BUCKWILD! achieves speedups up to 2.3× over HOGWILD!-based algorithms for logistic regression.

2 Main Result

The paper develops a martingale-based technique for bounding convergence rates under asynchronous updates, including some non-convex problems. It models stochastic updates and delays together, yielding rates controlled by the worst-case expected delay.

  • Problem formulation: Asynchronous execution creates stale reads and race conditions because cores maintain separate cached copies of the iterate.The algorithm is formalized through update samples, a success region, and a failure event indicating that no iterate reaches that region by time T.
  • Martingale framework: Martingale methods combine stochastic sampling, random initialization, and asynchronous delays in one model for analyzing asynchronous SGD and related algorithms.The technique extends convergence analysis beyond sequential settings and applies to convex problems and some non-convex algorithms.
  • Asynchronous hardware model: The hardware model serializes writes in a central store, uses atomic single-coordinate read-add-write updates, and represents asynchronous effects as coordinate-wise delays.Update samples are assumed i.i.d., while delays may be correlated but are bounded by a random variable with expected value τ.
  • Asynchronous convergence theorem: Theorem 1 shows that an asynchronous stochastic algorithm with a bounded rate supermartingale converges when HRξτ < 1.The required boundedness conditions control the supermartingale’s iterate sensitivity, expected update sensitivity, and expected update magnitude.
  • Asynchronous convergence theorem: The asynchronous failure probability depends only on the worst-case expected delay τ and increases over the sequential bound by a factor of 1 − HRξτ.The paper states that this increase is negligible in practical cases where HRξτ ≪ 1; the proof constructs a corrected supermartingale V_t from W_t.

3 Applications

The paper applies its martingale-based analysis to convex asynchronous SGD, low-precision BUCKWILD!, and non-convex HOGWILD! Alecton, deriving failure-rate bounds under relaxed assumptions and demonstrating the technique’s generality.

  • Convex HOGWILD!: The convex HOGWILD! analysis derives a failure-rate bound without assuming gradient sparsity, depending instead on the gradient-sample second moment and expected delay; under stricter assumptions, it recovers Niu et al.’s rate [17].The result follows by constructing a rate supermartingale and applying Theorem 1.
  • Low-Precision BUCKWILD!: BUCKWILD! models low-precision arithmetic as unbiased, bounded round-off noise and applies the same supermartingale approach to obtain a convex convergence bound.When κ ≪ 1, the additional error is negligible, so BUCKWILD! reaches comparable output quality in less wall-clock time than HOGWILD!.
  • Non-Convex Alecton: For non-convex problems, the paper specializes its analysis to rank-1 low-rank least-squares matrix completion, analyzing asynchronous HOGWILD! Alecton under eigengap, incoherence, and bounded-space conditions [4].The method tracks recovery of the dominant eigenvector’s direction rather than its eigenvalue and uses entrywise matrix sampling.
  • Non-Convex Alecton: The resulting supermartingale yields a failure-probability bound for HOGWILD! Alecton, showing that the framework extends beyond convex SGD.The analysis parameter γ trades off the allowable runtime horizon against the supermartingale’s initial value.

4 Experiments

Experiments validate the theoretical results for asynchronous non-convex matrix completion and BUCKWILD!, showing that low-precision arithmetic can substantially speed training while asynchronous matrix completion converges comparably to sequential training.

  • BUCKWILD! implementation: BUCKWILD! rounds input data to 8-bit or 16-bit integers instead of HOGWILD!’s 32-bit floating point representation.The reduced precision decreases memory usage and enables single-instruction-multiple-data (SIMD) instructions while threads update in parallel without locking.
  • Precision experiments: Table 1 evaluates how logistic-regression training loss changes with arithmetic precision across four DimmWitted-favored datasets [25].The experiments use step size α = 0.0001, with similar results reported across a range of step sizes; the datasets include Reuters, RCV1, and Forest.
  • BUCKWILD! speedup: BUCKWILD! achieves significant speedups over both sequential full-precision SGD and best-case HOGWILD! on dense RCV1.Figure 1(a) compares both baselines on a machine with two Xeon X650 CPUs, each with six hyperthreaded cores, and 24GB of RAM.
  • Asynchronous matrix completion: Asynchronous and sequential Alecton matrix completion show qualitatively similar convergence and reach the same noise floor on a synthetic matrix.Each plotted series is a different randomized run, so trajectories vary somewhat.

5 Conclusion

The paper presents a unified martingale-based framework for convergence rates of asynchronous and low-precision random algorithms, including a hardware-oriented SGD strategy called BUCKWILD!.

  • The paper develops a unified theoretical framework for convergence rates of asynchronous and low-precision random algorithms such as SGD.
  • A martingale-based convergence rate for sequential, full-precision algorithms can be leveraged to obtain rates for asynchronous, low-precision versions.
  • The paper introduces BUCKWILD!, an SGD strategy designed to exploit modern hardware resources for task and dat...

A Proof of Theorem 1

The proof constructs a corrected process V_t and shows that its expected value does not increase under the asynchronous update. Recursively combining this supermartingale property with the initial condition and total expectation yields a bound on success probability.

  • A Proof of Theorem 1: The proof defines V_t by correcting W_t with a delay-dependent term, then analyzes its evolution after the update x_t+1 = x_t + ˜G(˜v_t).The correction uses H_R, ξ, and the stopping-time quantity τ_t.
  • A Proof of Theorem 1: Lipschitz continuity, the rate supermartingale property of W, and the update-distance bound establish the one-step inequality for V_t.The argument uses the equality of the 1-norm and 2-norm because each step updates only a single entry of x.
  • A Proof of Theorem 1: For every T < B, recursive application gives E[V_T] ≤ E[V_0], while the hardware initialization implies E[V_0] = E[W_0(x_0)].The inequality remains valid after success because V_t does not change once the algorithm succeeds.
  • A Proof of Theorem 1: The law of total expectation separates successful and unsuccessful executions, and the rate-supermartingale bound is then used to solve for P(F_T).The success event is lower-bounded through the conditional value of W_T and the correction term H_Rξτ_T.

B Proofs for Convex Case

The section proves the low-precision convex SGD rate-supermartingale lemma by combining a piecewise-logarithm argument with strong convexity, rounding-error bounds, and explicit boundedness verification. It then derives the associated corollary by applying Theorem 1.

  • B Proofs for Convex Case: The lemma establishes boundedness parameters R = αL and ξ2 = α2(1 + κ2)M 2 for the low-precision convex SGD rate supermartingale.The proof verifies the required continuity bounds using the logarithm derivative, mean value theorem, sparsity structure, rounding-error bounds, and unbiased quantization.
  • B Proofs for Convex Case: The piecewise logarithm is differentiable and concave, enabling a Jensen-based inequality used in the rate-supermartingale proof.Its supporting inequality follows from first-order overestimation of concave functions.
  • B Proofs for Convex Case: The proof constructs W_t so it satisfies the rate-supermartingale properties before success and remains constant after the iterates enter the success set.Nonnegativity of the logarithm establishes the required post-horizon bound.
  • B Proofs for Convex Case: The convex-case corollary follows by applying Theorem 1 directly to the lemma’s result.This completes the convergence consequence after proving the rate-supermartingale conditions and parameter bounds.

C Proofs for Non-Convex Case

The proof of the non-convex case specializes lemmas from De Sa et al. [4] to rank-1 entrywise sampling and establishes convergence for asynchronous Alecton via a rate supermartingale. It then bounds the supermartingale parameters and derives the corresponding failure-probability guarantee.

  • τ-bound: The proof uses a specialized τ-bound for Alecton under entrywise sampling, incoherence, and the Section 3.3 step-size conditions.The bound is a rank-1, entrywise-sampling specialization of results from De Sa et al. [4].
  • Rate supermartingale: A rate supermartingale W_t is constructed for asynchronous Alecton and is frozen once the iterate enters S ∪ Z.The proof verifies the supermartingale properties before and after stopping, including the required horizon bound.
  • Parameter bounds: The proof establishes boundedness parameters H, R, and ξ, with R = ηµ ∥A∥F and ξ = ηµ ∥A∥F C.The H bound follows by differentiating τ(x), while the R bound uses incoherence and the ξ bound controls update magnitudes.
  • Failure probability: Applying Theorem 1 with B = T and the established constants yields the stated failure-probability bound for asynchronous Alecton.The corollary substitutes the chosen horizon and parameter values into Lemma 2 to obtain the desired result.

D Simplified Convex Result

This section gives an elementary proof of a convex-case result similar to the main result, using a constant step size under the conditions of Section 3.1. The proof bounds expected squared distance to the optimum but does not extend to the non-convex case.

  • The simplified convex proof does not use martingale results, making it more elementary than the preceding proofs but limiting it to the convex case.
  • Theorem 2: Theorem 2 establishes the convex-case result under the conditions of Section 3.1 using a constant step size selected for any ǫ > 0 and ϑ ∈ (0, 1).
  • Proof: The proof controls squared distance to the optimum by showing that it generally decreases at each timestep.
  • Proof: The one-step bound separates current distance, first-order change, delayed-update noise, and random-sampling noise before taking conditional and full expectations.
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