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Better Mini-Batch Algorithms via Accelerated Gradient Methods
Andrew Cotter, Ohad Shamir, Nathan Srebro, Karthik Sridharan
TL;DR
The paper asks whether mini-batch stochastic convex optimization can gain more from acceleration than standard gradient methods provide. It refines convergence analyses using L(w⋆), proposes a mini-batch-optimized accelerated method, and reports significant speed-ups with empirical validation. The tightness of one accelerated-bound term remains open.
Problem
The paper addresses whether standard gradient methods are insufficient for significant mini-batch speed-ups in stochastic convex optimization.
Method
The paper develops L(w⋆)-dependent convergence analyses and a mini-batch-optimized stochastic accelerated gradient method with adaptive step-size behavior.
Results
The accelerated method allows significant mini-batch speed-ups, has uniformly superior theoretical upper bounds, and is empirically validated.
Takeaways & Limitations
Appropriate acceleration can provide stronger mini-batch stochastic optimization guarantees than standard gradient methods in the regimes studied.
Takeaways & Limitations
Whether the accelerated bound’s 1/(bn) dependence is necessary remains an open practical and theoretical question.
Abstract
from arXiv · showhide
Mini-batch algorithms have been proposed as a way to speed-up stochastic convex optimization problems. We study how such algorithms can be improved using accelerated gradient methods. We provide a novel analysis, which shows how standard gradient methods may sometimes be insufficient to obtain a significant speed-up and propose a novel accelerated gradient algorithm, which deals with this deficiency, enjoys a uniformly superior guarantee and works well in practice.
TTIC Chicago, IL 60637 USA
The passage identifies Microsoft Research in Cambridge, Massachusetts, as an affiliation.
- The listed affiliation is Microsoft Research, Cambridge, MA 02142, USA.
- The location is Cambridge, Massachusetts, United States.
- The passage provides an institutional address rather than research content.
1 Introduction
The paper examines whether accelerated methods can improve mini-batch stochastic optimization. It argues that standard gradient methods can lack significant speed-ups in some regimes, while the proposed accelerated approach has stronger theoretical and empirical support.
- Mini-batching averages several instance gradients per update, enabling parallel gradient computation in distributed settings.
- In stochastic optimization, accelerated and standard methods generally share an O(1/√n) dominant term, limiting acceleration’s apparent benefit.
- The refined standard-gradient analysis depends on L(w⋆) and finds no significant mini-batch speed-up when desired suboptimality is comparable to or larger than L(w⋆).
- The paper proposes a mini-batch-optimized stochastic accelerated gradient variant that is implicitly adaptive to L(w⋆).
- The accelerated method is reported to allow significant mini-batch speed-ups with uniformly superior theoretical upper bounds and empirical validation.
2 Preliminaries
The preliminaries define the stochastic convex optimization setting and introduce SGD and accelerated gradient methods as alternative mini-batch update strategies. Existing bounds motivate a more refined comparison because both methods can share the same leading dependence on sample size.
- The optimization domain W is convex, with Euclidean norms used in the main setting and a Banach-space generalization in the Appendix.
- The loss is assumed convex, non-negative, and H-smooth in its predictor argument.
- Guarantees compare L(w) with L(w⋆), where w⋆ may be optimal or a lower-norm nonoptimal comparator.
- SGD and accelerated gradient methods average subgradients over mini-batches but use different update rules.
- When b = o(√m), both methods tolerate large mini-batches without significant degradation, but their shared 1/√m term does not establish an acceleration advantage.
3 Convergence Guarantees
The paper develops refined convergence guarantees for SGD and accelerated gradient methods that depend explicitly on the best predictor’s expected loss. Its analysis yields a new adaptive accelerated strategy, while exposing projection and bound-tightness qualifications.
- The refined convergence guarantees for SGD and accelerated gradient methods explicitly depend on L(w⋆), the best predictor’s expected loss.
- Theorem 2 specifies an accelerated strategy whose step sizes scale polynomially with iteration index and depend on mini-batch size and L(w⋆).
- Unlike SGD, the accelerated bound depends on the radius D, and the analysis requires a projection step.
- The analysis uses the self-bounding property of non-negative smooth convex losses, linking small loss to a small gradient norm.
4 Optimizing with Mini-Batches
The analysis compares SGD and the proposed accelerated gradient method by translating convergence bounds into runtime as functions of loss, accuracy, and mini-batch size. In parallel settings, acceleration supports larger useful mini-batches and uniformly matches or improves SGD's runtime bound; in serial settings, it permits larger batches without more than constant-factor degradation, and can be necessary for significant speed-ups.
- Runtime comparison: The comparison fixes target suboptimality ϵ and measures the samples m and iterations n required by each method as functions of L(w⋆), ϵ, and b.Parallel runtime is approximated by n, while serial runtime is measured by m.
- Parallel setting: The parallel analysis separates SGD and AG into regimes determined by which terms dominate their runtime bounds and by the relationship between L(w⋆) and ϵ.The regime summaries ignore constants and logarithmic factors.
- Parallel setting: AG retains linear mini-batch speedup for larger b than SGD and remains equal or significantly faster in runtime bounds across all regimes.Both methods initially improve linearly with b, but AG extends that regime and preserves a b speedup when ϵ ≥ L(w⋆)^2.
- Serial setting: In serial execution, increasing b requires more samples for both methods, so mini-batching offers no large serial gains despite possible implementation-level constant-factor improvements.The stated implementation benefits include reduced overhead, loop-control savings, pipelining, and concurrent memory access.
- Serial setting: The serial mini-batch size can scale as L/ϵ for SGD versus L/ϵ^3/2 for AG without degrading the runtime guarantee beyond a constant factor.Thus AG tolerates a larger mini-batch under the serial guarantee considered.
- When acceleration is necessary: When ϵ = Ω(L(w⋆)), including the separable case L(w⋆) = 0, AG can be necessary for significant mini-batch speed-ups because SGD may have no non-constant parallel speedup.The bounds give SGD L(w⋆)/ϵ = O(1) in this regime, while AG still achieves a speedup up to b = m^2/3.
5 Experiments
Experiments compared SGD and AG on astro-physics and CCAT while varying mini-batch size, including modified separable datasets with L(w⋆)=0. AG generally retained speedups over larger batches and often outperformed SGD, especially near suboptimality L(w⋆).
- Step-size exponent: Neither p = 0 nor p = 1 was best in all circumstances; performance depended on the data and showed a weak tendency toward higher p for larger b.The parameter p controls how the accelerated method’s step sizes increase.
- Experimental setup: Experiments compared SGD and AG on astro-physics and CCAT while varying b with fixed training sample size m = nb.The datasets used smoothed hinge loss, with validation data selecting step sizes; curves were averaged over three runs.
- Comparison: For L(w⋆)>0, SGD was initially better in one regime, but degraded faster as b increased and eventually performed worse than AG.AG still benefited when b was large, even under conditions described as least favorable to it.
- Comparison: When suboptimality ϵ was roughly equal to L(w⋆), AG significantly outperformed SGD even with small mini-batches.This observation matched the paper’s theoretical analysis.
- Separable case: With L(w⋆)=0, AG was mostly better at retaining linear parallel speedups for larger mini-batches and could also help with small mini-batches.The modified datasets were created by removing margin violations.
6 Summary
The paper develops L(w⋆)-sensitive theory for stochastic methods, introduces a new accelerated step-size strategy, and evaluates mini-batching more finely. It concludes that acceleration can be necessary for significant speedups, while the tightness of its main bound remains unresolved.
- Contributions: The paper’s theory generalizes prior first-order stochastic optimization results by making the bounds sensitive to L(w⋆).The contributions include Theorems 1 and 2 and a refined analysis of mini-batching.
- Contributions: The accelerated method uses a novel step-size strategy that the experiments found effective in practice.The strategy was developed to obtain the theoretical results and then used empirically.
- Open question: The tightness of Theorem 2’s bound remains an open practical and theoretical question.The endpoint cases b = 1 and b →∞ are tight, but the necessity of the 1/(bn) dependence is unclear.
A Generalizing to Different Norms
The paper extends mini-batch stochastic and accelerated methods from Euclidean spaces to general normed domains. It replaces Euclidean gradient and projection operations with mirror-descent mappings and Bregman projections, under strong convexity of a regularizer.
- General setting: The general setting uses a closed convex domain in a Banach space with a norm and its dual norm.The loss is assumed H-smooth in this normed setting.
- General setting: A non-negative function R that is strongly convex with respect to the norm is the key ingredient for generalization.The paper introduces strong convexity as the condition enabling the mirror-descent construction.
- Algorithmic changes: Generalized SGD and AG replace gradient descent with mappings involving R and R∗, and replace projection with Bregman projection.The Bregman projection minimizes divergence to the mapped point.
- Algorithms: The generalizations are presented as Stochastic Mirror Descent with Mini-Batching and Accelerated Mirror Descent.The corresponding procedures are Algorithms 3 and 4.
- Guarantees: Theorems 3 and 4 provide guarantees for the mirror-descent methods under non-negative strong convexity of R.The Euclidean theorems arise as specializations of these general results.
B Complete Proofs
The appendices prove the stochastic and accelerated mirror-descent guarantees using smoothness, convexity, norm inequalities, conditional expectations, and Jensen’s inequality. They also relate the general proofs back to the Euclidean results.
- Proof structure: The proofs of Theorems 3 and 4 are complete, with Theorems 1 and 2 obtained as Euclidean specializations.This connects the general norm analysis to the earlier algorithms and bounds.
- Proof ingredients: The proof uses H-smoothness and convexity to derive the initial inequalities underlying the stochastic mirror-descent bound.The argument then introduces the regularizer and its associated update representation.
- Stochastic argument: Conditional on previous mini-batches, the current iterate is fixed while within-batch gradient differences are independent mean-zero vectors.This permits the expectation step used in the proof.
- Technical conditions: The analysis enforces a step-size restriction η ≤ b/(32HK^2) so that the parameter α remains at most one.The resulting inequalities are combined with smoothness and Jensen’s inequality.
B.2 Accelerated Mirror Descent
This section establishes conditions for accelerated updates and develops bounds by unrolling recursions, selecting step sizes, and optimizing auxiliary parameters.
- The accelerated update is analyzed under step-size conditions including β1 = 1 and γi+1(βi+1 −1) ≤γiβi.These conditions enable the recursive inequalities used throughout the analysis.
- The update step of accelerated gradient is rewritten in an equivalent form before the main recursion is unrolled.The resulting inequality is repeatedly applied and then averaged by taking expectations.
- The analysis uses mean-zero, independently drawn mini-batch vectors and conditions on the examples preceding the current mini-batch.Under this conditioning, the iterate wi is fixed for the expectation over the current batch.
- For non-negative H-smooth convex functions, the analysis invokes a self-bounding gradient property and uses ∥w⋆∥ ≤ D in a specialization.The latter gives a direct inequality used when applying the preceding theorem.
- The resulting bound contains optimization, smoothness, diameter, and logarithmic terms, with parameter M subsequently optimized.The displayed terms include L0(n −M −2), a term involving R(w⋆), and 64HK2D2 log n.
- The chosen γ must satisfy multiple conditions, including γ ≤ b/(64HK2np), with sufficient sample size also imposed.The text states conditions involving n ≥ 3 and a lower bound on nb.
B.3 Some Technical Lemmas
This section develops technical lemmas for controlling recursive sequences and expected regularizer terms under strong convexity and boundedness assumptions.
- For a 1-strongly convex regularizer R, Fenchel–Young and conjugacy properties are used to control auxiliary dual quantities.The convex conjugate R∗ is identified as 1-strongly smooth with respect to the dual norm.
- With mean-zero independent vectors, expectations of the auxiliary quantities are bounded recursively, starting from S0 = 0.The recursion is substituted back into the main equation to obtain the claimed bound.
- A lemma studies sequences of non-negative numbers bounded by a0 and satisfying a recurrence involving the sequence terms.The recurrence is analyzed by unrolling it and controlling logarithmic factors under an upper-bound condition.
- When A(i) is sufficiently small near the end of the sequence, logarithmic terms are bounded by A(n −i), yielding a cumulative bound.The stated condition is A(i) ≤ 1/2 for the relevant indices.
- The sequence bound concludes under an additional condition involving the cumulative product or sum represented by Pn.The supplied statement indicates that the conclusion holds as long as this condition is satisfied.