[Paper Review] Structured Sparsity and Generalization

Andreas Maurer, Massimiliano Pontil|arXiv (Cornell University)|Aug 17, 2011

Statistical Methods and Inference26 references59 citations

TL;DR

This paper introduces a general data-dependent generalization bound for regularized learning algorithms that enforce structured sparsity via an infimum convolution norm defined over a set of bounded linear operators on a Hilbert space. The key contribution is a dimension-free Rademacher complexity bound that applies to infinite-dimensional settings—such as the Lasso in a separable Hilbert space or multiple kernel learning with countably many kernels—thereby enabling tighter, more flexible generalization guarantees without the typical logarithmic dependence on dimensionality.

ABSTRACT

We present a data dependent generalization bound for a large class of regularized algorithms which implement structured sparsity constraints. The bound can be applied to standard squared-norm regularization, the Lasso, the group Lasso, some versions of the group Lasso with overlapping groups, multiple kernel learning and other regularization schemes. In all these cases competitive results are obtained. A novel feature of our bound is that it can be applied in an infinite dimensional setting such as the Lasso in a separable Hilbert space or multiple kernel learning with a countable number of kernels.

Motivation & Objective

• To develop a general, data-dependent generalization bound applicable to a wide class of regularized learning algorithms that enforce structured sparsity.
• To extend existing Rademacher complexity bounds to infinite-dimensional Hilbert spaces, particularly for the Lasso and multiple kernel learning with countable kernel sets.
• To eliminate the dimension-dependent log(d) factor present in classical bounds, achieving dimension-free generalization guarantees under finite second-moment conditions.
• To unify and generalize existing bounds for algorithms like ridge regression, Lasso, group Lasso, and multiple kernel learning under a single theoretical framework.

Proposed method

• Define a structured sparsity regularizer as an infimum convolution over a set M of symmetric bounded linear operators on a Hilbert space H.
• Introduce the dual norm ‖z‖_M* = sup_{M∈M} ‖Mz‖, which simplifies the analysis of the Rademacher complexity.
• Derive a bound on the empirical Rademacher complexity R_M(x) using duality and moment inequalities, leading to R_M(x) ≤ (2^{3/2}/n) × √[sup_M ∑_i ‖Mx_i‖²] × (2 + √(ln(∑_M ‖Mx_i‖² / sup_N ∑_j ‖Nx_j‖²)))
• Establish a tighter, distribution-dependent bound when M is finite: R_M(X) ≤ (2^{3/2}C / √n) × (2 + √(ln|M|)) under the condition ‖X‖_M* ≤ C.
• Apply the bound to specific algorithms including the Lasso, group Lasso, multiple kernel learning, and mixed-norm regularization by choosing appropriate operator sets M.
• Use moment bounds on Gaussian and Rademacher chaos to control the expected suprema of empirical processes, leveraging Hilbert-Schmidt norms and ℓ_p/ℓ_{p/2} triangle inequalities.

Experimental results

Research questions

• RQ1 Can a general, data-dependent generalization bound be derived for structured sparsity regularization that avoids the log(d) dependence on dimension?
• RQ2 Is it possible to extend Rademacher complexity bounds to infinite-dimensional settings such as the Lasso in a separable Hilbert space or multiple kernel learning with countably many kernels?
• RQ3 How does the proposed bound compare to existing bounds in terms of tightness and applicability across standard regularization schemes like Lasso and group Lasso?
• RQ4 Can the bound be made dimension-free while still capturing the intrinsic complexity of structured sparsity patterns?

Key findings

• The proposed bound is dimension-free and applies to infinite-dimensional settings, such as the Lasso in a separable Hilbert space or multiple kernel learning with a countable number of kernels, provided the second moment condition ∑_M ‖M‖_HS^p < ∞ holds.
• For finite M, the bound R_M(X) ≤ (2^{3/2}C / √n)(2 + √(ln|M|)) is distribution-dependent and avoids the log(d) factor common in prior bounds.
• The bound recovers and improves upon existing results for standard regularization schemes, including ridge regression, Lasso, group Lasso, and multiple kernel learning, with only minor constant differences.
• The bound is tight in the sense that the log(d) factor is unavoidable in general, and the proposed bound matches this lower bound when d is replaced by the effective dimension R² = ∑_M ‖M‖_HS^2.
• The method enables generalization guarantees for multiple kernel learning with countably infinite kernels, provided the sum of the Hilbert-Schmidt norms raised to power p is finite.
• The analysis demonstrates that the Rademacher complexity can be controlled via the dual norm and operator norms, leading to a unified framework for structured sparsity.