Optimal Distributed Online Prediction using Mini-Batches (1012.1367v2)

Abstract: 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.

• The paper introduces the distributed mini-batch (DMB) algorithm, converting serial gradient-based methods into efficient distributed systems.
• It rigorously shows that the method achieves an optimal O(√m) regret bound, matching serial algorithms while accounting for communication latency.
• The work extends its implications to stochastic optimization, offering scalable solutions for high-rate online prediction in web-scale systems.

Overview of Optimal Distributed Online Prediction

The paper by Dekel, Gilad-Bachrach, Shamir, and Xiao presents a significant contribution to the field of distributed computing in stochastic online prediction and optimization. The authors introduce a novel method, termed the "distributed mini-batch (DMB) algorithm," which effectively adapts many serial gradient-based online prediction algorithms to operate within distributed systems. This paper rigorously proves that for smooth convex loss functions, the DMB algorithm achieves an asymptotically optimal regret bound, rivaling what could be achieved in an ideal serial setting.

Key Contributions

1. Distributed Mini-Batch Algorithm: The DMB framework is a method for converting serial gradient-based algorithms into distributed versions. It utilises a strategy where mini-batches of gradients are computed across distributed nodes, effectively reducing the variance of updates without incurring significant communication overheads or latency.
2. Regret Bound Analysis: The authors establish that the regret bound for their distributed algorithm is $O(\sqrt{m})$, matching the optimal bound for serial algorithms. This result is highly significant as it implies that distributed systems can achieve the same efficiency in terms of regret as would be achieved using a hypothetical serial system with infinitely fast computation. The algorithm demonstrates an asymptotically linear speed-up in convergence rates, addressing high-rate input scenarios effectively.
3. Communication Latency Considerations: An important aspect of the analysis is the explicit consideration of communication latencies between distributed nodes. In real-world applications, network latency can heavily influence performance. The analysis shows the algorithm can maintain optimal performance despite such latencies, provided that mini-batch sizes are chosen appropriately.
4. Stochastic Optimization: Beyond online prediction, the paper discusses the application of the DMB method to stochastic optimization problems. It is shown that using mini-batches in this context allows for asymptotically optimal convergence rates for stochastic optimization, with theoretical guarantees of linear speed-up in distributed settings.

Numerical Results and Implications

In empirical evaluations, the DMB algorithm demonstrates superiority over simple no-communication baselines and matches the performance of serial algorithms under distributed scenarios with various parameters. The robustness of the technique in maintaining low communication costs highlights its applicability to web-scale systems, such as search engines that manage high-frequency queries.

Future Directions

The ideas presented in this paper open avenues for further research into:

• Robustness in Heterogeneous Environments: The adaptability to settings with node failures or communication noise could enhance the practical deployment of these algorithms.
• Non-Smooth Loss Functions: Extending the methodology to handle non-smooth convex and non-convex functions could broaden its applicability.
• Asynchronous Implementations: Future work could develop and analyze asynchronous versions of the DMB algorithm to further reduce latency impacts and improve efficiency.

This paper provides an insightful advancement in distributed online prediction, with strong theoretical foundations backed by effective empirical results. The methodologies proposed have the potential to significantly impact the design of scalable, optimal algorithms in modern distributed computing environments, where high rates of data processing are required.