Online Structured Prediction with Fenchel--Young Losses and Improved Surrogate Regret for Online Multiclass Classification with Logistic Loss (2402.08180v3)

Abstract: This paper studies online structured prediction with full-information feedback. For online multiclass classification, Van der Hoeven (2020) established \emph{finite} surrogate regret bounds, which are independent of the time horizon, by introducing an elegant \emph{exploit-the-surrogate-gap} framework. However, this framework has been limited to multiclass classification primarily because it relies on a classification-specific procedure for converting estimated scores to outputs. We extend the exploit-the-surrogate-gap framework to online structured prediction with \emph{Fenchel--Young losses}, a large family of surrogate losses that includes the logistic loss for multiclass classification as a special case, obtaining finite surrogate regret bounds in various structured prediction problems. To this end, we propose and analyze \emph{randomized decoding}, which converts estimated scores to general structured outputs. Moreover, by applying our decoding to online multiclass classification with the logistic loss, we obtain a surrogate regret bound of $O(| \mathbf{U} |\mathrm{F}^2)$, where $\mathbf{U}$ is the best offline linear estimator and $| \cdot |\mathrm{F}$ denotes the Frobenius norm. This bound is tight up to logarithmic factors and improves the previous bound of $O(d| \mathbf{U} |_\mathrm{F}^2)$ due to Van der Hoeven (2020) by a factor of $d$, the number of classes.

• The paper introduces a novel randomized decoding procedure that efficiently maps scores to structured outputs, achieving tight surrogate regret bounds.
• It extends the exploit-the-surrogate-gap framework to online structured prediction by integrating Fenchel–Young losses, including logistic loss for multiclass classification.
• The method's efficiency and theoretical guarantees pave the way for robust online learning applications in diverse fields like NLP and bioinformatics.

Enhancing Online Structured Prediction with Fenchel–Young Losses

Introduction

Structured prediction tasks have become indispensable in fields ranging from natural language processing to bioinformatics, due to their ability to predict complex outputs like trees or sequences. Though theoretical frameworks like surrogate losses have facilitated advancements in structured prediction, the leap to online settings – especially with general structured targets – introduces new challenges. This paper acknowledges the limitations of existing exploit-the-surrogate-gap strategies in online structured prediction and extends the framework to incorporate Fenchel–Young losses, encompassing a broad class of surrogate losses, including the logistic loss for multiclass classification.

Methodology and Theoretical Contributions

The authors propose a methodology that extends the exploit-the-surrogate-gap framework to online structured prediction tasks by incorporating Fenchel–Young losses. Key to their approach is a novel randomized decoding procedure which efficiently maps estimated scores to structured outputs, addressing the challenge of converting scores to structured predictions in a non-trivial manner. The paper details this procedure along with an efficient implementation, leveraging a fast Frank–Wolfe-type algorithm for decoding. The authors undertake a rigorous analysis, revealing conditions under which finite surrogate regret bounds can be achieved. Highlighting the theoretical contributions, the paper demonstrates that the methodology achieves tight surrogate regret bounds, signifying an improvement over existing bounds in the context of online multiclass classification with logistic loss.

Practical Implications and Future Prospects

From a practical standpoint, the extended framework and the introduction of the randomized decoding procedure offer new avenues for applying online structured prediction across various fields. The efficiency of the method, backed by strong numerical results, makes it particularly applicable to real-world scenarios where structured outputs are common. Looking ahead, the research opens up potential for further exploration in several directions, including applying the framework to other forms of online learning settings, adapting the approach to different surrogate losses, and extending the findings to more complex structured prediction tasks.

Conclusion

By successfully extending the exploit-the-surrogate-gap framework to encompass online structured prediction with Fenchel–Young losses, this paper contributes significantly to the field of online learning. The introduction of a novel randomized decoding procedure, combined with comprehensive theoretical analysis that achieves improved surrogate regret bounds, marks a notable advancement in the theory and application of online structured prediction. The implications of this research not only enhance our understanding of structured prediction but also promise to broaden the application of online learning methodologies in various domains.