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Optimal Distributed Online Prediction using Mini-Batches

Ofer Dekel, Ran Gilad-Bachrach, Ohad Shamir, Lin Xiao

arXiv:1012.1367v2cs.LGcs.DCmath.OC

TL;DR

The paper tackles distributed stochastic online prediction when serial processors cannot keep pace with high-rate inputs. It introduces distributed mini-batching to convert serial gradient-based methods, proving optimal regret for smooth convex losses despite communication latency and demonstrating the approach on a web-scale task.

  • Problem

    High-rate web-scale prediction can exceed the processing capacity of a single processor, motivating distributed online prediction under latency and system constraints.

  • Method

    The distributed mini-batch algorithm converts serial gradient-based online prediction methods into distributed algorithms by averaging stochastic gradients computed at a common predictor.

  • Results

    For smooth convex losses, the method achieves an asymptotically optimal regret bound, and experiments show DMB outperforms the naive no-communication algorithm with modest latency degradation.

  • Takeaways & Limitations

    Distributed computation can asymptotically perform as well as a hypothetical fast serial computer by reducing stochastic-gradient variance.

  • Takeaways & Limitations

    The analysis assumes bounded predictors can be enforced by restricting the search to a bounded set and assumes non-blocking network operations in the distributed model.

Abstract

from arXiv · show

Online prediction methods are typically presented as serial algorithms running on a single processor. However, in the age of web-scale prediction problems, it is increasingly common to encounter situations where a single processor cannot keep up with the high rate at which inputs arrive. In this work, we present the \emph{distributed mini-batch} algorithm, a method of converting many serial gradient-based online prediction algorithms into distributed algorithms. We prove a regret bound for this method that is asymptotically optimal for smooth convex loss functions and stochastic inputs. Moreover, our analysis explicitly takes into account communication latencies between nodes in the distributed environment. We show how our method can be used to solve the closely-related distributed stochastic optimization problem, achieving an asymptotically linear speed-up over multiple processors. Finally, we demonstrate the merits of our approach on a web-scale online prediction problem.

1. Introduction

The paper addresses high-rate stochastic online prediction by distributing serial gradient-based algorithms across multiple processors. Its distributed mini-batch method targets optimal regret for smooth losses while accounting for communication latency.

  • Motivation: High-rate prediction workloads require distributing computation across cores or cluster nodes to provide real-time service.The paper motivates this setting with web-scale systems such as search engines processing thousands of queries per second.
  • Problem: The no-communication solution gives each node an independent serial algorithm, but its total regret scales poorly with the network size k.With each node processing m/k inputs, the paper identifies this as a basic distributed baseline that must be outperformed.
  • Method: The distributed mini-batch (DMB) algorithm converts any serial gradient-based online prediction algorithm into a parallel or distributed algorithm using the serial update rule as a black box.The approach distributes gradient computation and aggregates mini-batch information across nodes.
  • Results: For smooth convex losses, DMB attains an asymptotically optimal regret bound whose dominant coefficient matches the serial bound and is independent of k, network topology, and latency.The method averages stochastic gradients at the same predictor, while slowly increasing mini-batch size to offset communication latencies.
  • Evaluation and extensions: The paper extends the framework to stochastic optimization, studies its effect on regret, and evaluates both serial mini-batching and DMB on a web-scale prediction problem.The paper also discusses extensions involving nonsmooth convex regularization and reports that large batches and network latency have modest empirical effects.

2. Variance Bounds for Serial Algorithms

The paper abstracts several stochastic gradient-based serial prediction algorithms through a common update rule and derives variance-sensitive regret bounds for smooth convex losses. These bounds motivate distributed mini-batching while preserving the optimal asymptotic rate.

  • Serial update template: The template updates predictor w_j and auxiliary state a_j from the previous state, stochastic gradient g_j, and parameter α_j.Different choices of update rule φ produce different online prediction algorithms.
  • Examples: Projected gradient descent and dual averaging are two concrete update rules that fit the template.Projected gradient descent is a special case of mirror descent, while dual averaging uses an auxiliary state.
  • Variance-sensitive bounds: Smoothness and bounded stochastic-gradient variance yield regret bounds whose leading dependence is O(√m).The variance-sensitive analysis applies to convex losses with L-Lipschitz gradients and σ^2-bounded gradient variance.
  • Generality: The analysis extends beyond Euclidean gradient descent and dual averaging to broader mirror-descent and non-Euclidean dual-averaging families.Composite objectives with simple nonsmooth convex regularization can retain the same regret bounds.
  • Distributed motivation: Without communication, k nodes processing m inputs incur a total regret bound approximately k times worse in its leading term than the ideal serial solution.The DMB approach is introduced to avoid this k-factor penalty.

3. Serial Online Prediction using Mini-Batches

Serial mini-batching keeps predictions fixed across each batch, averages the batch gradients, and updates only between batches. Averaging reduces stochastic-gradient variance by a factor of b without significantly worsening the asymptotic regret rate.

  • Algorithm: A serial mini-batch algorithm holds w_j constant for b inputs, accumulates their gradients, and updates once at the batch boundary.The averaged gradient is supplied to the same update rule φ for the next batch.
  • Regret analysis: The mini-batch regret bound is obtained by applying the serial bound to m/b averaged gradients and multiplying the result by b.The update parameters must be adjusted to reflect the reduced variance.
  • Variance reduction: Each batch average has stochastic-gradient variance bounded by σ^2/b.The reduction follows from independent inputs and properties of the Euclidean norm.
  • Conclusion: The resulting bound is asymptotically equivalent to the basic serial O(√m) regret bound.Serial mini-batching does not improve the asymptotic rate because computation remains serial.

4. Distributed Mini-Batch for Stochastic Online Prediction

The distributed mini-batch algorithm synchronizes averaged stochastic gradients across nodes while overlapping communication with additional prediction work. Under smooth convex losses, it achieves an asymptotically optimal regret bound despite network latency.

  • Communication: A non-blocking distributed vector sum aggregates gradients across nodes, after which every node applies the same deterministic update rule.The network model permits continued input processing during communication.
  • DMB workflow: DMB divides each batch into b gradient-producing inputs and μ latency-hiding inputs processed with the unchanged predictor w_j.Only the first b inputs contribute gradients; the μ additional inputs are processed while the vector sum runs.
  • Guarantee: Theorem 4 gives an expected regret bound for DMB under L-smooth convex losses and σ^2-bounded stochastic-gradient variance.The bound applies when the distributed algorithm uses a serial update rule with regret bound ψ(σ^2,m).
  • Parameter choice: For b=m^ρ with ρ∈(0,1/2), DMB retains the asymptotic rate, while fixed b changes the dominant regret term.Choosing b as a function of m can require advance knowledge of the sequence length; a doubling trick relaxes this requirement.
  • Comparison: The DMB bound is asymptotically equivalent to the ideal serial bound, with identical constants in the dominant √m term.The dominant term does not contain μ, which usually scales logarithmically with k.
  • Short streams: When m is small relative to latency, the asymptotically optimal batch size can discard many inputs during communication.For m=15,000 and μ=100, the stated parameters yield b∼25, so 100 inputs are discarded for every 25 used in an update.

5. Stochastic Optimization

The paper transfers DMB from online prediction to stochastic optimization by applying averaged gradients in batches and measuring expected optimality gap. Under smooth convex assumptions, the method preserves convergence rates and can achieve asymptotically linear speed-up.

  • Problem: Stochastic optimization seeks a final output whose expected loss is close to that of the true minimizer, unlike online prediction, which measures cumulative regret.The optimization objective is based on the expected loss F(w)=E_z[f(w,z)].
  • Method: The stochastic-optimization DMB template processes b/k samples per node, averages their gradients, and applies the update rule once per batch.Unlike online prediction, it need not process additional inputs while the vector sum completes.
  • Guarantee: Theorem 5 bounds DMB’s expected optimality gap using the serial bound evaluated at variance σ^2/b and sample count m/b.The result assumes L-smooth convex losses and σ^2-bounded stochastic-gradient variance.
  • Rate: The dominant convergence-rate factor is unaffected by batch size, allowing large batches without significantly changing the convergence rate.This makes serial stochastic-optimization algorithms amenable to parallel execution.
  • Speed-up: DMB achieves asymptotically linear speed-up as the batch size grows sufficiently with m.The speed-up approaches k processors in the limit.
  • Speed-up measure: For b(m)=Θ(m^ρ) with ρ∈(0,1/2), the expected-gap-based speed-up remains asymptotically linear.The speed-up ratio is a conceptual guide because its numerator and denominator are upper bounds rather than precise actual running times.

6. Experiments

The experiments evaluate serial mini-batching and DMB on a web-scale binary-classification problem, including cluster size, communication latency, and batch-size effects. Large batches improve early learning, while smaller batches can perform better later; latency causes modest degradation.

  • Experimental Setup: The evaluation uses one billion Bing queries to construct a web-scale binary-classification problem predicting whether queries are highly monetizable.Inputs include query text, timestamps, and temporary user identifiers; predictions use recent query history.
  • Experimental Setup: The model represents recent user query history as a binary vector and predicts monetizability with a logistic loss under a norm constraint.The vector records queries issued during the previous two hours, and the predictor is constrained by ∥w_t∥≤C.
  • Experimental Setup: The experiments simulate synchronous DMB with Euclidean dual averaging while varying cluster size and network latency.The simulation models a binary-tree communication structure over a 1Gbs Ethernet network and reports average online loss because the offline risk minimizer is unknown.
  • Optimal Mini-Batch Size: For k = 32, b = 512 is beneficial after 10^7 inputs, whereas b = 128 performs better after 10^9 inputs.The experiment reports average loss after 10^7, 10^8, and 10^9 inputs across batch sizes from 8 through 4096.
  • Evaluating DMB: DMB outperforms the naive no-communication algorithm, and the comparison covers clusters of k = 1024 and k = 32 machines.The no-communication baselines use independent serial algorithms, including a serial mini-batch baseline with batch size 128.
  • Discussion: Large batches improve performance even serially, while the optimal batch size generally decreases over time.The conclusion summarizes the observed temporal trade-off in batch-size selection.

7. Related Work

Related work addresses distributed online learning and optimization through pipelines, consensus, or final model averaging. The paper distinguishes DMB by combining a general gradient-based framework with asymptotically optimal regret analysis and explicit latency treatment.

  • Distributed Online Learning: Distributed online learning research targets the scalability limits of inherently sequential online algorithms.The cited motivation is shared with prior work on distributed online learning.
  • Distributed Online Learning: Langford et al. use a pipeline model that overlaps gradient computation with serial updates, but their scalability depends on the gradient-to-update time ratio.Their proofs also apply only to unconstrained convex optimization without projection.
  • Distributed Optimization: Duchi et al. distribute dual averaging through asynchronous consensus over gradients computed at different points.Their assumptions use Lipschitz-continuous losses, while the resulting online bound is no better than independent machines without communication.
  • Distributed Optimization: Zinkevich et al. average final weight vectors from independently trained machines, which is equivalent to no communication for online regret.Their approach is instead evaluated using the optimality gap of the final averaged predictor.
  • DMB Contributions: DMB provides a general framework for distributing many gradient-based update rules, including mirror descent and dual averaging.The paper presents this framework as a distinction from prior distributed methods.
  • DMB Contributions: DMB explicitly incorporates network latency into regret analysis and guarantees that large latency can be overcome by appropriate parameter settings.The framework treats distributed computation as an opportunity to exploit variance-based regret bounds rather than merely as a constraint.

8. Conclusions and Further Research

The paper develops distributed online algorithms with asymptotically optimal guarantees and evaluates the distributed mini-batch method on a large-scale real-world problem. It identifies smoothness, synchronization, and heterogeneous or asynchronous systems as important scope boundaries and future directions.

  • The distributed mini-batch algorithm achieves asymptotically optimal regret and optimality-gap guarantees for distributed stochastic online prediction and optimization.The method reduces stochastic-gradient variance, allowing distributed computation to asymptotically match a hypothetical fast serial computer.
  • Experiments on a large-scale real-world problem found that DMB outperforms other simple solutions.The experiments also suggest optimizing batch size and adjusting learning rate using empirical measures.
  • The formal results rely on smooth loss functions and variance-dependent regret bounds for stochastic online update rules.The paper leaves open whether smoothness can be weakened and states that non-smooth extensions remain an open problem.
  • The theoretical result is independent of network size k and network latency µ.This independence is stated for the resulting regret bound under the paper’s assumptions.
  • Heterogeneous clusters, asynchronous environments, and dynamic networks are identified as important directions for extending distributed learning.A master-workers reformulation lets workers process inputs at their own pace and periodically send accumulated gradients to a master.

Appendix A. Smooth Stochastic Online Prediction in the Serial Setting

The appendix formalizes smooth stochastic online prediction in a general normed setting, defining the optimization objective, smoothness and variance assumptions, regret, and Bregman-divergence tools used by the algorithms.

  • The appendix extends the main analysis from the Euclidean case to general finite-dimensional normed spaces and full algorithmic generality.It covers stochastic dual averaging and stochastic mirror descent for smooth loss functions.
  • The prediction domain W is closed and convex, while f(w,z) is assumed convex and differentiable in w.Gradients are taken with respect to the prediction variable and belong to the dual space.
  • L-smoothness requires gradients at any two prediction points to differ by at most L times their distance.The condition is stated for every realization z and uses the primal and dual norms.
  • The population objective is F(w) = E_z[f(w, z)], with gradient ∇F(w) = E_z[∇f(w, z)].The analysis assumes an existing minimizer w⋆ of F.
  • Expected regret compares cumulative stochastic losses with the loss of the fixed population minimizer w⋆.The appendix focuses on bounding E[R(m)] under stochastic inputs.
  • Bregman divergence supplies the general geometry used by mirror-based updates and is nonnegative but generally asymmetric.Its three-point identity supports the appendix’s algorithmic analysis.

A.1 Stochastic Dual Averaging

The stochastic dual averaging appendix section specifies the update framework and establishes expected-regret bounds for smooth stochastic online prediction, including a simplification when the optimum has zero gradient.

  • The dual-averaging proof is adapted from accelerated-algorithm analyses while being applied to online-prediction regret.The cited proof lineage includes Tseng and Xiao.
  • Stochastic dual averaging predicts with a strongly convex regularizer and accumulates stochastic gradients using positive nondecreasing parameters β_i.The regularizer is normalized so its minimum over W is zero.
  • The section establishes an expected-regret theorem for dual averaging in the smooth stochastic setting.The proof uses a fundamental convex-analytic lemma, smoothness, conditional expectations, and telescoping inequalities.
  • When ∇F(w⋆) = 0, the general expected-regret bound admits a simplified form.This condition certainly holds when W is the whole space.
  • The proof treats each prediction w_i as a deterministic function of earlier observations, enabling unbiased conditional-expectation arguments.The stochastic-gradient noise is represented by q_i = ∇f(w_i,z_i) − ∇F(w_i).

A.2 Stochastic Mirror Descent

The stochastic mirror descent section gives expected-regret bounds under smoothness and bounded-geometry assumptions, then presents a known-horizon variant that relaxes boundedness requirements.

  • The mirror-descent analysis relies on variance-based convergence results and adapts their proofs to online prediction regret.The section connects this analysis to prior accelerated stochastic mirror-descent work.
  • Stochastic mirror descent uses a differentiable 1-strongly convex regularizer and its generated Bregman divergence for prediction updates.The initialization matches the dual-averaging setup.
  • The main mirror-descent theorem assumes W is closed and bounded and that d(u,v) is bounded on W.The parameters β_i are positive and nondecreasing.
  • The section establishes an expected-regret bound for stochastic mirror descent under these assumptions.The proof combines smoothness inequalities with Bregman-divergence telescoping terms.
  • For unconstrained optima satisfying ∇F(w⋆) = 0, the initial objective gap can be bounded using smoothness and the regularizer geometry.This yields a simplified expected-regret bound.
  • The mirror-descent assumptions are stronger than those for dual averaging, although they hold for the standard Euclidean distance.The Euclidean divergence is d(u,v) = (1/2)||u − v||^2.
  • The standard mirror-descent treatment excludes KL divergence on the simplex because that divergence is unbounded there.Other variants can weaken this restriction, including a constant parameter based on prior knowledge of the total horizon m.
  • Knowing the total number of steps m permits a constant β_i and removes the need for bounded W or bounded Bregman divergence.The resulting bound uses the initialization at the minimizer of h and a parameter chosen from σ and m.

Appendix B. High-Probability Bounds

Appendix B extends the paper’s expected-regret results to high-probability guarantees using martingale concentration. The analysis preserves variance-sensitive terms and accounts for reduced variance from mini-batching, while adding communication-related and confidence-dependent terms.

  • High-probability guarantees: High-probability regret bounds hold with probability at least 1 −δ, rather than only bounding expected regret.The appendix develops this extension for stochastic dual averaging and indicates the same technique applies to stochastic mirror descent.
  • Assumptions: The high-probability analysis assumes bounded losses, bounded stochastic gradients, bounded loss variance, and a prediction domain with bounded diameter.These assumptions introduce constants B, G, ˆσ, and D that control the relevant deviations and regret terms.
  • Proof technique: The proofs use a Bernstein-type inequality for bounded martingale-difference sequences to control stochastic deviations.The argument separately constructs martingale differences for gradient and loss terms, then combines the resulting bounds with a union bound.
  • Mini-batch effects: Mini-batching reduces the variances of averaged losses and gradients to at most ˆσ2/b and σ2/b, respectively.The resulting regret bound remains variance-sensitive, while also depending on the function variance ˆσ2 and confidence parameter δ.
  • Distributed batching: For batches of size b with communication latency µ, the update rule receives ¯m inputs, and the proof handles both divisible and non-divisible input counts.The batch interval contains b + µ examples overall, and the non-divisible case is treated using the earlier mini-batch argument.
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