Online Learning with Predictable Sequences (1208.3728v2)

Abstract: We present methods for online linear optimization that take advantage of benign (as opposed to worst-case) sequences. Specifically if the sequence encountered by the learner is described well by a known "predictable process", the algorithms presented enjoy tighter bounds as compared to the typical worst case bounds. Additionally, the methods achieve the usual worst-case regret bounds if the sequence is not benign. Our approach can be seen as a way of adding prior knowledge about the sequence within the paradigm of online learning. The setting is shown to encompass partial and side information. Variance and path-length bounds can be seen as particular examples of online learning with simple predictable sequences. We further extend our methods and results to include competing with a set of possible predictable processes (models), that is "learning" the predictable process itself concurrently with using it to obtain better regret guarantees. We show that such model selection is possible under various assumptions on the available feedback. Our results suggest a promising direction of further research with potential applications to stock market and time series prediction.

• The paper shows how incorporating predictable sequences enhances regret minimization by outperforming traditional worst-case guarantees.
• It introduces a model selection approach that dynamically learns predictive processes in real time for online linear and convex optimization.
• The study extends these methods to partial information settings using novel randomized techniques, establishing robust theoretical bounds.

Analyzing "Online Learning with Predictable Sequences"

This paper, authored by Alexander Rakhlin and Karthik Sridharan, addresses the domain of online linear optimization while factoring in the characteristics of predictable sequences. The primary focus of the research is to formulate algorithms that capitalize on predictable processes, diverging from the traditional approach of managing worst-case scenarios. The paper is positioned within the context of online learning, aiming to embed prior knowledge about the sequence through devoted algorithms that deliver superior regret bounds when sequences are benign.

Key Contributions

The research extends the conventional bounds achieved in no-regret online optimization by introducing predictable processes that provide advantageous performance when sequences exhibit regularity. The paper presents a theoretical framework and bounds for online linear and convex optimization problems, encompassing full and partial information settings. Significant contributions include:

1. Predictable Processes and Regret Minimization: The paper delineates how incorporating predictable sequences into online learning can yield improved regret guarantees. By devising methods to combine these processes with adversarial sequences, the research supplies structures for achieving better performance than standard worst-case guarantees.
2. Model Selection: The paper sets forth an approach to model selection in online learning. It proposes learning the predictable process in real time, thus allowing competitors with a set of predictive models.
3. Extension to Partial Information: The authors extend their findings to settings where only partial or bandit information is available. This scenario is essential for practical applications like financial market predictions.
4. Randomized Methods and Improvements: Included in the paper are novel randomized methods like the Follow the Perturbed Leader (FPL) algorithm and their analysis relative to predictable sequences.

Strong Numerical Results

The paper underscores scenarios where the regret bounds converge quickly, especially when the predictable process matches the sequence behavior well. The strongest results are achieved when the variances or deviations from the predictable trend are minimized, which is often seen in applications of time-series prediction or stock markets. By theoretically demonstrating these outcomes, the paper enhances the understanding of expected losses and provides a practical framework for deploying these algorithms effectively.

Implications and Future Work

The implications of this work resonate across both theoretical and applied fronts. Theoretically, it facilitates a new line of investigation into predictable sequences and their utility in online learning environments. Practically, the concepts could revolutionize analytical practices in industries relying heavily on sequential data. This paper hints at possibilities for further research, such as adaptive online algorithms that could dynamically adjust for different types of predictable sequences or those incorporating non-linear transformations and richer feedback mechanisms.

Conclusion

Rakhlin and Sridharan’s contribution through this research is significant as it bridges the gap between traditional online learning models and the nuanced modeling of predictable sequences. This paper not only consolidates existing knowledge but propels online learning frameworks to new applications by utilizing predictable processes. This research invites further exploration into dynamic model selection techniques and opens avenues for integrative approaches to sequential data processing across varied domains.