Refined Risk Bounds for Unbounded Losses via Transductive Priors (2410.21621v2)

Abstract: We revisit the sequential variants of linear regression with the squared loss, classification problems with hinge loss, and logistic regression, all characterized by unbounded losses in the setup where no assumptions are made on the magnitude of design vectors and the norm of the optimal vector of parameters. The key distinction from existing results lies in our assumption that the set of design vectors is known in advance (though their order is not), a setup sometimes referred to as transductive online learning. While this assumption seems similar to fixed design regression or denoising, we demonstrate that the sequential nature of our algorithms allows us to convert our bounds into statistical ones with random design without making any additional assumptions about the distribution of the design vectors--an impossibility for standard denoising results. Our key tools are based on the exponential weights algorithm with carefully chosen transductive (design-dependent) priors, which exploit the full horizon of the design vectors. Our classification regret bounds have a feature that is only attributed to bounded losses in the literature: they depend solely on the dimension of the parameter space and on the number of rounds, independent of the design vectors or the norm of the optimal solution. For linear regression with squared loss, we further extend our analysis to the sparse case, providing sparsity regret bounds that additionally depend on the magnitude of the response variables. We argue that these improved bounds are specific to the transductive setting and unattainable in the worst-case sequential setup. Our algorithms, in several cases, have polynomial time approximations and reduce to sampling with respect to log-concave measures instead of aggregating over hard-to-construct $\varepsilon$-covers of classes.

• The paper introduces transductive priors to achieve improved risk bounds for unbounded losses in prediction tasks.
• It refines regret bounds in logistic and linear regression by eliminating dependence on bounded norm assumptions while ensuring computational efficiency.
• It distinguishes transductive from inductive setups, demonstrating significant improvements in sequential decision-making under uncertainty.

Refined Risk Bounds for Unbounded Losses via Transductive Priors: An Overview

This paper addresses the challenge of providing refined risk bounds for prediction tasks involving unbounded losses, specifically focusing on tasks such as linear regression, logistic regression, and classification with hinge loss. The paper's primary contribution is identifying that by considering a transductive framework—where the order of design vectors is unknown, but their set is known—improved risk bounds can be achieved. This refinement is applicable even in cases of unbounded losses, which traditionally pose significant hurdles in deriving efficient risk bounds.

The authors introduce the concept of transductive online learning, which stands apart from traditional online learning frameworks where sequences are revealed in a particular order, or batch learning setups, where the entire dataset is available ahead of time. The novel approach presented leverages the prior knowledge about the set of design vectors to form transductive priors, which are pivotal in achieving the stated improvements in risk bounds.

Key Contributions

1. Transductive Priors and Exponential Weights: The paper introduces transductive priors, which are design-dependent priors used within the exponential weights algorithm framework. By incorporating the complete set of design vectors into the prior, the algorithm can leverage additional information, enhancing prediction accuracy and risk bounds beyond traditional methods.
2. Refinement of Regret Bounds: For logistic regression, the approach yields regret bounds that are independent of the norm of the optimal solution. This contrasts with existing methods, which typically require bounded norms. Additionally, for linear regression, including sparse cases, the authors not only improve the regret bounds but also maintain computational efficiency through log-concave sampling.
3. Separation of Transductive and Inductive Setups: The work demonstrates clear separations in learnability when applying transductive setups versus traditional online learning frameworks. The results indicate scenarios where transductive learning methods provide sublinear regret bounds, highlighting their enhanced capability in handling unbounded losses.
4. Computational Efficiency and Practicality: While maintaining theoretical rigor, the algorithms developed exhibit polynomial time computation in many instances, another key advantage over conventional densification or $L^p$ covering strategies, which are computationally prohibitive.
5. Statistical Implications and Batch Learning: By introducing a variant of the online-to-batch conversion technique, the authors connect their findings to classical statistical risk bounds. This bridges the theoretical improvements achieved in online learning to practical application within batch settings.

Implications and Future Directions

The implications of this research are substantial for fields relying on sequential decision-making under uncertainty, such as finance, healthcare, and autonomous systems. By mitigating the dependency on loss constraint assumptions, the models can be more robust and widely applicable.

Theoretically, this work also stimulates further inquiries into the potential of less-constrained models in other prediction scenarios, possibly extending beyond linear problems to broader, non-parametric settings. As such, future work could explore the universality of transductive priors across various other loss types and their impact on regret minimization strategies.

In summary, this paper pioneers a path toward harmonizing the robustness of machine learning models under unbounded losses by leveraging transductive priors. This not only pushes the frontier of theoretical machine learning but also opens doors to pragmatic solution strategies involving extensive decision datasets.