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HOGWILD!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent
Feng Niu, Benjamin Recht, Christopher Re, Stephen J. Wright
TL;DR
SGD is difficult to parallelize efficiently because existing schemes require costly locking and synchronization. This paper introduces Hogwild!, a lock-free shared-memory update scheme, and shows that sparse problems support near-linear speedups and strong convergence behavior.
Problem
SGD’s sequential nature limits scalability, while existing parallelization schemes require performance-destroying memory locking and synchronization.
Method
Hogwild! parallelizes SGD by allowing processors to update shared-memory components without locks, relying on sparse updates to limit harmful overwrites.
Results
Hogwild! achieves near-linear speedups on sparse learning problems and experimentally outperforms locking-based methods, with performance exceeding theoretical guarantees.
Takeaways & Limitations
Lock-free SGD can provide practical parallel speedups across varied sparse machine-learning applications, including problems with computationally intensive gradients.
Takeaways & Limitations
The approach depends on sparse updates, and particularly high-contention variables may need to be updated less frequently.
Abstract
from arXiv · showhide
Stochastic Gradient Descent (SGD) is a popular algorithm that can achieve state-of-the-art performance on a variety of machine learning tasks. Several researchers have recently proposed schemes to parallelize SGD, but all require performance-destroying memory locking and synchronization. This work aims to show using novel theoretical analysis, algorithms, and implementation that SGD can be implemented without any locking. We present an update scheme called HOGWILD! which allows processors access to shared memory with the possibility of overwriting each other's work. We show that when the associated optimization problem is sparse, meaning most gradient updates only modify small parts of the decision variable, then HOGWILD! achieves a nearly optimal rate of convergence. We demonstrate experimentally that HOGWILD! outperforms alternative schemes that use locking by an order of magnitude.
1 Introduction
SGD is effective for data-intensive learning but difficult to scale because it is inherently sequential and locking-based parallel schemes incur substantial overhead. Hogwild! removes locks and, under sparse data access, achieves near-linear speedups with strong theoretical and experimental support.
- SGD combines a small memory footprint, robustness against noise, and rapid learning rates for data-intensive machine learning.
- Multicore systems motivate parallel SGD, but MapReduce is poorly suited to iterative, numerically intensive analysis because fault-tolerance overhead can reduce throughput.
- Hogwild! lets processors update shared-memory components without locks, allowing overwrites that are rare and introduce barely any computational error when access is sparse.
- The paper formalizes sparsity conditions, gives canonical sparse problems, and derives theoretical guarantees of linear speedups.
- 1/k convergence rates are possible with constant stepsizes that use exponential back-off over time.
- Lock-free SGD exceeds the theoretical guarantees in practice and outperforms locking-based methods across varied machine-learning applications.
2 Sparse Separable Cost Functions
The paper models several machine-learning objectives as sparse sums whose terms touch only small subsets of variables. It represents this structure with an induced hypergraph and characterizes sparsity through edge size, edge intersections, and variable incidence.
- The objective is modeled as f(x) = Σ_e∈E f_e(x_e), where each term depends only on coordinates indexed by a small subset e.
- The induced hypergraph represents examples as hyperedges for sparse SVM, revealed entries as row-column edges for matrix completion, and the target graph for graph cuts.
- Sparse SVM, matrix completion, and graph cuts provide canonical examples in which each objective term involves only a small fraction of all variables.
- Ω measures hyperedge size, ρ measures the fraction of edges intersecting an edge, and Δ measures the fraction of edges intersecting a variable.
- For sparse SVM, Δ captures maximum feature frequency while ρ captures hypergraph clustering; for graph cuts, Δ is maximum degree divided by |E| and ρ ≤ 2Δ.
- The protocol targets linear speedup when Ω, Δ, and ρ are relatively small.
3 The Hogwild! Algorithm
Hogwild! uses shared-memory, lock-free coordinate updates: processors sample sparse terms, compute local gradients, and atomically update components that may be stale or concurrently modified.
- Each processor samples an edge uniformly, reads its current coordinates, evaluates the corresponding gradient, and updates shared memory.
- Atomic componentwise addition avoids a separate locking structure, whereas updating multiple components simultaneously requires auxiliary locking.
- Processors modify only sampled coordinates, while gradients may be computed from stale decision-variable states and concurrent writes are resolved by random tie-breaking.
- Under isotropic sparsity, the asynchronous incremental-gradient algorithm converges in nearly as many gradient steps as serial SGD, implying nearly linear processor speedup.
4 Fast Rates for Lock-Free Parallelism
The analysis establishes convergence guarantees for lock-free SGD under convexity, smoothness, strong-convexity, bounded-gradient, and bounded-delay assumptions, with sparsity controlling the parallel penalty.
- With replacement, processors sample an edge, compute a subgradient, select one coordinate, and update asynchronously under delay bounded by τ.
- The theoretical analysis assumes convex component functions, Lipschitz-continuous differentiability, strong convexity, bounded gradients, and γc < 1.
- After k component updates, the expected suboptimality satisfies E[f(xk) − f⋆] ≤ ϵ under the stepsize and iteration conditions of Proposition 4.1.
- When τ = 0, the bound matches serial SGD; when τ = o(n1/4), the recursion remains nearly unchanged in the stated sparse regime.
- The proof uses a distance recursion with an effective curvature cr < c, reflecting errors introduced by the lock-free update rule.
- The analysis nearly yields a 1/k convergence rate for constant-stepsize SGD, up to a log(1/ϵ) factor.
5 Robust 1/k rates.
A piecewise-constant stepsize with exponential backoff removes the logarithmic factor from the constant-stepsize analysis and gives robust 1/k convergence for Hogwild!.
- The protocol runs K updates at stepsize γ < 1/c, reduces γ by β after synchronization, and increases the next epoch length by β−1.
- The constant-stepsize recursion separates geometric contraction from a residual term, with the backoff scheme progressively shrinking the achieved error.
- Without backoff, the bound is nearly 1/k but includes log(1/ϵ); backoff removes this logarithmic factor and yields a 1/k rate.
- The backoff parameter is minimized near β ≈ 0.37, while reducing the stepsize by β requires β−1 more iterations.
- Both constant- and diminishing-stepsize bounds have asymptotically similar dependence on M, c, and k, but the constant-stepsize protocol is more robust to curvature overestimates.
- Hogwild! requires synchronization only at the end of each round or epoch to coordinate stepsize reduction.
6 Related Work
Prior parallel SGD methods use distributed averaging, stale updates, or locking; the paper positions Hogwild! as a lower-overhead alternative and reports strong speedups against locking-based schemes.
- MapReduce-style parallel SGD averages independent runs, but the paper reports that it does not outperform serial SGD on the targeted problems.
- Figure 2 compares Hogwild! and RR wall-clock time over 20 epochs using 10 cores.
- Distributed gradient averaging can achieve linear speedups but requires substantial inter-core communication and frequent synchronization.
- Round-robin locking can scale when locking is negligible relative to gradient computation, but fast machine-learning gradients make locking costly.
- Hogwild! outperforms the round-robin approach by an order of magnitude across a variety of applications.
7 Experiments
Experiments compare Hogwild! with locking and alternative parallelization schemes across sparse SVM, graph cuts, and matrix completion tasks. Hogwild! generally delivers strong speedups, while locking can impose substantial overhead.
- Experimental setup: Hogwild! was compared with round-robin (RR) and fine-grained-locking AIG on identical experimental hardware.The experiments used C++ on a dual Xeon X650 system with 24GB of RAM; AIG locks variables before and after each update loop.
- Sparse SVM: 3× speedup was achieved on RCV1 despite large sparsity parameters ρ = 0.44 and ∆= 1.0, while RR worsened with more threads.These values represent a challenging case for Hogwild! according to the experiment description.
- Averaging baseline: The averaging method produced the same RCV1 training error in serial and ten-thread runs despite performing 10x more gradient computations.The parallel threads communicated only at the end of the computation.
- Matrix completion: 12% and 62% slowdowns occurred for 10-thread RR on KDD Cup and Netflix, respectively, while 10-way Hogwild! solved Jumbo in under three hours.RR was too slow to complete the Jumbo experiment in a reasonable time; Hogwild! attained near-linear speedup even when reading data from disk.
- Graph cuts: 4× speedup was achieved on Abdomen with 10 threads, while RR was twice as slow as serial.For DBLife, Hogwild! achieved a ninefold speedup with 10 cores, compared with a fivefold RR speedup.
- Artificial gradient delays: Hogwild! reduced computation time more than RR across tested delays, with equal speedups only when gradient computation exceeded a few milliseconds.At delays longer than one millisecond, RR was on par with Hogwild!, but not better.
8 Conclusions
The conclusions attribute Hogwild!’s near-linear speedups to sparsity and describe extensions for variables with high contention. The paper also identifies collision-free gradient ordering as future work.
- Conclusions: Hogwild! uses sparsity to enable near-linear speedups across a variety of applications.The paper reports significant speedups even for RCV1, where ρ is large, and when gradients are computationally intensive.
- Conclusions: High-contention variables can be updated less frequently, allowing Hogwild! to include variables such as an SVM bias term.The paper gives updating a bias only every thousand iterations as an example.
- Future work: Future work could seek gradient orderings that completely avoid memory contention between processors.The paper cites biased ordering in matrix completion as an example of collision-free updates.
A Analysis of Hogwild!
This analysis section introduces identities used repeatedly in the subsequent derivations, including a rearrangement of an earlier equation.
- Analysis: One step in the analysis follows by rearranging equation (4.3).The passage identifies the algebraic source of the step but does not state the resulting expression.
- Analysis: The section’s supplied passages provide transition statements rather than substantive convergence claims.The detailed bounds and recursions appear in the surrounding derivation.
- Analysis: The analysis repeatedly uses identities established before the subsequent derivations.The text explicitly states that these identities will be used frequently.
A.1 Developing the recursion
The recursion analysis tracks squared distance to the optimum under asynchronous updates, bounds delay and overlap effects, and derives conditions under which parallel convergence resembles serial convergence.
- Update structure: Each update modifies only coordinates indexed by its sampled edge, and the asynchronous state may be read with a stale gradient.Projection operators formalize coordinate and edge-specific updates, while xj records the state after j updates.
- Expectation bounds: The derivation decomposes expectation terms involving sampled edges, vertices, projections, and the history tuple e[i].Several bounds use independence, convexity, Cauchy-Schwarz, Jensen’s inequality, and the overlap parameter ρ.
- Recursion: The analysis combines intermediate inequalities into a recursion for the nonnegative sequence of squared distances a_j.The recursion is then simplified using cγ < 1 and linearized around its fixed point.
- Parallel regime: For sufficiently small ρ and ∆, C(τ, ρ, ∆, Ω) ≈ 1, and fewer than n1/4 processors preserve nearly the serial recursion.The stated condition is τ = o(n1/4) when ρ and ∆ are o(1/n) and o(1/√n), respectively.
A.2 Proof of Proposition 4.1: Final Steps
The proof completes Proposition 4.1 by selecting parameters to reach target accuracy ϵ/L, bounding intermediate quantities, and substituting those bounds into the final expression.
- Parameter selection: The proof targets accuracy ϵ/L and uses (5.3) with a∞ chosen according to (A.12).The sufficient condition is that a_k ≤ ϵ/L.
- Parameter selection: The bound a∞ ≤ γB follows directly from (A.12).
- Parameter selection: The choice of γ satisfying (4.5) yields the required intermediate bound automatically.The proof invokes an inequality involving (1 + √1 + x)^2 to justify this step.
- Final bound: Substituting the selected γ into (5.3) produces the bound displayed in (A.14).The displayed result contains the factor ϵc^2 · C(τ, ρ, ∆, Ω) / (1 − δ(τ, ρ, ∆, Ω)).
- Final bound: The remaining inequality is justified by a bound valid for all x ≥ 0, completing the proof.The text identifies this as the second-to-last inequality before concluding from (A.14).