[Paper Review] Risk Bounds for Learning Multiple Components with Permutation-Invariant Losses

Fabien Lauer|arXiv (Cornell University)|Apr 16, 2019

Sparse and Compressive Sensing Techniques1 references4 citations

TL;DR

This paper introduces a novel, simple approach to derive tighter generalization error bounds for multiple component learning problems with permutation-invariant losses by leveraging Rademacher complexity decompositions under sparsity-inducing regularization. It achieves mild or even vanishing dependence on the number of components C, especially for nonconvex ℓp-quasi-norm regularization with p∈(0,1), which previous methods could not handle.

ABSTRACT

This paper proposes a simple approach to derive efficient error bounds for learning multiple components with sparsity-inducing regularization. We show that for such regularization schemes, known decompositions of the Rademacher complexity over the components can be used in a more efficient manner to result in tighter bounds without too much effort. We give examples of application to switching regression and center-based clustering/vector quantization. Then, the complete workflow is illustrated on the problem of subspace clustering, for which decomposition results were not previously available. For all these problems, the proposed approach yields risk bounds with mild dependencies on the number of components and completely removes this dependence for nonconvex regularization schemes that could not be handled by previous methods.

Motivation & Objective

• Address the lack of efficient generalization error bounds for multiple component learning problems such as subspace clustering, switching regression, and vector quantization.
• Overcome the limitation of prior methods that yielded linear or square-root dependencies on the number of components C in risk bounds.
• Extend existing theoretical frameworks to handle nonconvex regularization schemes (e.g., ℓp-quasi-norms with p∈(0,1)) that promote sparsity but were intractable under previous analysis.
• Provide a unified, simple method applicable across diverse problems by reusing known decomposition results for Rademacher complexity.
• Enable model selection and theoretical justification for practical algorithms that return non-convex, sparse component models.

Proposed method

• Utilize permutation invariance of the loss function to decompose the Rademacher complexity of the overall loss class into component-wise contributions.
• Apply known decomposition results for product classes of independent component classes to bound the Rademacher complexity of the full loss class.
• Use the duality between ℓp-norms and their duals to derive bounds that scale as O(√C) for p=2 and O(log C) for p=1, with complete removal of C-dependence for p<1.
• Apply the approach to three problems: switching regression, vector quantization, and subspace clustering, using problem-specific decomposition structures.
• Derive uniform risk bounds via empirical Rademacher complexity estimation, avoiding reliance on computationally intractable empirical risk minimizers.
• Leverage trace and Frobenius norm identities for projection matrices to bound the complexity of subspace clustering models.

Experimental results

Research questions

• RQ1 Can we derive tighter generalization error bounds for multiple component learning problems with sparsity-inducing regularization?
• RQ2 How does the dependence on the number of components C affect risk bounds in problems with permutation-invariant losses?
• RQ3 Can the theoretical framework be extended to nonconvex regularization schemes such as ℓp-quasi-norms with p∈(0,1)?
• RQ4 What is the impact of permutation invariance on the Rademacher complexity decomposition and resulting risk bounds?
• RQ5 Can the proposed method be systematically applied to diverse problems like switching regression, clustering, and subspace clustering?

Key findings

• The proposed method yields risk bounds with O(√C) dependence on the number of components C for ℓ2-regularization, improving upon previous linear bounds.
• For ℓ1-regularization, the method achieves O(log C) dependence on C, matching state-of-the-art results but with a simpler derivation.
• For nonconvex ℓp-quasi-norm regularization with p∈(0,1), the method completely removes the dependence on C in the risk bounds, a result unattainable with prior methods.
• The approach is successfully applied to switching regression, tightening bounds from linear to sublinear dependence on C.
• For vector quantization in Hilbert space, the method improves upon existing bounds that had either √C or linear dependence on C.
• In subspace clustering—a problem previously lacking theoretical analysis—the method provides the first complete risk bound derivation using this framework, demonstrating its generality and practicality.