[Paper Review] Leveraging Low-Rank Relations Between Surrogate Tasks in Structured Prediction

Giulia Luise, Dimitrios Stamos+2 more|arXiv (Cornell University)|Mar 2, 2019

Machine Learning and Data Classification4 citations

TL;DR

This paper proposes a novel trace norm regularization method for structured prediction that leverages low-rank relations among surrogate outputs without requiring explicit knowledge of the coding/decoding functions. By using a variational formulation of trace norm regularization, the approach enables efficient learning in high- or infinite-dimensional surrogate spaces and establishes excess risk bounds that demonstrate improved generalization over standard $μat$ regularization in specific regimes, with empirical validation on ranking tasks showing significant performance gains.

ABSTRACT

We study the interplay between surrogate methods for structured prediction and techniques from multitask learning designed to leverage relationships between surrogate outputs. We propose an efficient algorithm based on trace norm regularization which, differently from previous methods, does not require explicit knowledge of the coding/decoding functions of the surrogate framework. As a result, our algorithm can be applied to the broad class of problems in which the surrogate space is large or even infinite dimensional. We study excess risk bounds for trace norm regularized structured prediction, implying the consistency and learning rates for our estimator. We also identify relevant regimes in which our approach can enjoy better generalization performance than previous methods. Numerical experiments on ranking problems indicate that enforcing low-rank relations among surrogate outputs may indeed provide a significant advantage in practice.

Motivation & Objective

• To address the challenge of structured prediction where output spaces are non-vectorial (e.g., permutations, graphs), by extending surrogate methods to exploit inter-output relationships.
• To develop an efficient learning algorithm that does not require explicit knowledge of the surrogate coding/decoding functions, enabling application to large or infinite-dimensional surrogate spaces.
• To establish theoretical excess risk bounds for the proposed estimator, demonstrating consistency and improved learning rates under trace norm regularization.
• To identify conditions under which trace norm regularization outperforms standard $μat$ regularization in structured prediction settings.
• To empirically validate the method on learning-to-rank problems, showing practical advantages of enforcing low-rank structure in surrogate outputs.

Proposed method

• The method employs trace norm regularization on the surrogate output operator to encourage low-rank structure among predictions, without relying on explicit knowledge of the coding function.
• It uses a variational formulation of trace norm regularization to derive an efficient optimization procedure, even in infinite-dimensional reproducing kernel Hilbert spaces.
• The approach formulates the learning problem as a Tikhonov-regularized vector-valued regression in a Hilbert space of operators, with the trace norm acting as a low-rank inducing penalty.
• Theoretical analysis establishes equivalence between Ivanov and Tikhonov formulations of trace norm regularization, enabling stable optimization and generalization guarantees.
• The representer theorem is applied to show that the minimizer lies in a finite-dimensional subspace, enabling computational tractability.
• The method is applied to structured prediction via surrogate loss minimization, with decoding applied post-optimization to recover structured outputs.

Experimental results

Research questions

• RQ1 Can low-rank regularization among surrogate outputs improve generalization in structured prediction without explicit knowledge of the coding function?
• RQ2 In what settings does trace norm regularization lead to better learning rates and excess risk bounds compared to standard $μat$ regularization?
• RQ3 Is it possible to derive an efficient optimization algorithm for trace norm-regularized structured prediction in infinite-dimensional surrogate spaces?
• RQ4 Does enforcing low-rank structure in surrogate outputs lead to measurable performance gains in real-world structured prediction tasks?
• RQ5 How do the theoretical excess risk bounds of the proposed estimator compare to existing results in vector-valued and multitask learning?

Key findings

• The proposed method achieves better generalization performance than standard $μat$ regularization in specific regimes, particularly when the true output structure is low-rank.
• Excess risk bounds are derived for the trace norm regularized estimator, establishing consistency and learning rates that extend prior results in vector-valued regression.
• Theoretical analysis shows that trace norm regularization can provide significant advantages over $μat$ regularization even for least-squares loss, a novel result in this context.
• Numerical experiments on learning-to-rank problems demonstrate that enforcing low-rank relations among surrogate outputs leads to substantial performance improvements over all competitors.
• The algorithm is applicable to surrogate spaces of arbitrary dimension, including infinite-dimensional ones, due to the implicit handling of coding functions via variational formulations.
• The equivalence between Ivanov and Tikhonov formulations of trace norm regularization is proven, enabling stable and efficient optimization.