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Regularization Techniques for Learning with Matrices

Sham M. Kakade, Shai Shalev-Shwartz, Ambuj Tewari

arXiv:0910.0610v2cs.LGstat.ML

TL;DR

Learning problems with matrix-organized parameters need regularizers that encode sophisticated prior knowledge, but choosing among them requires problem-sensitive guidance. The paper uses strong-convexity/strong-smoothness duality to construct and analyze matrix-based regularization methods, deriving regret and generalization bounds across several learning settings. It concludes that statistical structure can guide regularizer choice, with group or spectral norms suited to different assumptions.

  • Problem

    Matrix-organized learning problems require systematic tools for designing regularization schemes that impose sophisticated prior knowledge.

  • Method

    The paper uses strong-convexity/strong-smoothness duality to characterize matrix regularizers and derive learning algorithms with associated regret and generalization analyses.

  • Results

    The framework yields regret and generalization bounds for multi-task, multi-class, and kernel learning, while simplifying previous analyses and sometimes sharpening their bounds.

  • Takeaways & Limitations

    Underlying statistical properties can guide the choice of regularization function, including group or spectral norms for different matrix structures.

Abstract

from arXiv · show

There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge). This work describes and analyzes a systematic method for constructing such matrix-based, regularization methods. In particular, we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate. Our methodology is based on the known duality fact: that a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm. This result has already been found to be a key component in deriving and analyzing several learning algorithms. We demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi-task learning, multi-class learning, and kernel learning.

1 Introduction

The paper develops a unified strong-convexity/strong-smoothness framework for designing and analyzing matrix-based regularization, using problem statistics to select appropriate regularizers. It applies this framework to multi-task, multi-class, and kernel learning while simplifying or sharpening prior analyses.

  • Motivation: Matrix-based regularization organizes parameters as matrices to impose sophisticated prior knowledge in challenging learning problems.Examples include group Lasso, kernel learning, multi-class prediction, and multi-task learning.
  • Method: Strong-convexity/strong-smoothness duality provides a systematic methodology for deriving regret and generalization analyses.The framework emphasizes a key inequality that supports algorithm design and complexity control.
  • Method: The framework characterizes matrix regularization functions and links their performance bounds to the statistical properties of the learning problem.This supports deciding which regularization function is appropriate for a given problem.
  • Applications: Template algorithms and bounds cover online and batch multi-task learning, online multi-class categorization, and kernel learning.The analyses use group or spectral norms where applicable and include generalization or mistake bounds.
  • Results: The unified analysis simplifies prior regret analyses and sometimes produces sharper bounds, including for multiple kernel learning.The paper also describes a group Perceptron that significantly outperforms previously proposed algorithms under shared class structure.
  • Scope: The work distinguishes agnostic error-rate analysis from multi-task union support recovery, which focuses on identifying relevant features.Its multi-task analysis studies sample size needed to obtain a given error rate rather than recovering the feature set.

2 Preliminaries and Techniques

The paper develops learning guarantees from strong-convexity/strong-smoothness duality and applies them to matrix-based regularization. It characterizes strongly convex penalties for vectors and matrices, yielding online regret and batch generalization bounds under several norms.

  • Notation and convex analysis: The presentation uses convex analysis, dual norms, Fenchel conjugates, singular values, and eigenvalues to formulate vector and matrix regularization.Matrices use the trace inner product; singular values and eigenvalues are arranged in non-increasing order.
  • Strong convexity–strong smoothness duality: Strong convexity and strong smoothness are dual properties for closed convex functions, with the conjugate using the reciprocal smoothness parameter.The paper also notes that a strongly convex function may have a restricted domain, whereas its conjugate is finite and everywhere differentiable.
  • Learning implications: A central corollary converts strong convexity into the inequality used to derive both online regret bounds and batch generalization bounds.The framework applies the same analytical mechanism across the two learning models.
  • Online learning: Online Mirror Descent achieves regret guarantees when the penalty is strongly convex and losses are convex and Lipschitz with respect to the dual norm.The guarantee is stated for any positive step-size parameter η under the theorem’s assumptions.
  • Batch learning: Rademacher analysis and Lipschitz-loss arguments yield generalization guarantees for linear predictors constrained by a strongly convex function.The result covers both a Rademacher complexity theorem and an empirical-risk-minimization corollary.

3 Matrix Regularization

The paper compares matrix norm classes and links their suitability to sparsity, group structure, and spectral structure in the data and predictor. These comparisons provide a basis for selecting regularization functions and deriving bounds.

  • Compared regularizers: The learning setup uses matrix predictors W and compares four norm-constrained classes: W1,1, W2,2, W2,1, and WS(1).The corresponding data norms determine the bounds summarized in Table 1.
  • Elementwise structure: W1,1 and W2,2 reproduce the usual ℓ1-versus-ℓ2 trade-off because both treat the matrix as a vector of dimension kd.The preferred class depends on the relative sizes of the predictor and data norms.
  • Elementwise structure: If X is sparse, X2,2 is close to X∞,∞; if X is dense with bounded nonzero entries, the two norms can differ by a factor of √kd.Analogous sparsity considerations apply to W, helping determine which bound is smaller.
  • Group structure: W2,1 is appropriate for group sparsity, where only some columns of W are nonzero and active columns are dense.This structure is associated with group Lasso and multi-task learning.
  • Spectral structure: WS(1) is appropriate when W has a sparse spectrum, meaning W is low rank, while X has a dense spectrum.The distinction from W2,2 concerns sparsity in singular values rather than entries.
  • Selection principle: The methodology uses these norm-dependent bounds to decide which regularization function better matches the underlying statistical properties of a problem.The paper applies the framework to derive generalization and regret bounds for problems of recent interest.

4 Multi-task learning

The multi-task formulation learns a matrix whose rows predict multiple, potentially heterogeneous tasks, while matrix regularizers encode shared sparsity or low-dimensional structure. The resulting online and batch bounds show when group or spectral norms can be preferable.

  • Problem formulation: Multi-task learning represents k prediction problems with a matrix W whose row wj predicts task j.The formulation permits different tasks and loss functions, including mixtures of regression and classification.
  • Structural assumptions: Related tasks motivate regularizers that couple them, with group norms encoding common feature sparsity and unitarily invariant norms encoding low-rank structure.These alternatives exploit different forms of similarity across tasks.
  • Online learning: The analysis supplies online multi-task algorithms with regret bounds for W1,1, W2,2, W2,1, and WS(1).The same norm classes and corresponding data radii are used in the bound comparison.
  • Bound implications: The group norm can outperform nongrouped norms when the number of active columns and feature sparsity satisfy the stated inequalities.Specifically, its bound is better than W2,2 if sg ln(d) < d and better than W1,1 if sgsx/d < sw.
  • Bound implications: WS(1) can be best when the rows of W lie in a low-dimensional subspace and the energy of X is spread across its spectrum.This corresponds to sparse singular values for W and a small maximal singular value for X.
  • Batch learning: The framework yields expected excess multitask-risk bounds for empirical risk minimizers matching the matrix-class bounds in Table 1.The batch result follows from a matrix Rademacher-complexity analysis under strong convexity and row-sign invariance.

5 Multi-class learning

For multi-class prediction, the paper applies matrix-norm analysis to online classifiers and derives regret and mistake bounds for several regularizers. The group norm benefits shared sparsity, while the spectral norm benefits shared low-dimensional structure.

  • Problem formulation: The multi-class predictor uses Wxt and chooses the index of its largest coordinate, with a matrix-adapted hinge loss.The setting has k labels and d-dimensional instances.
  • Guarantees: The analysis provides online multi-class regret bounds for W1,1, W2,2, W2,1, and WS(1), together with a mistake bound for the group algorithm.The hinge loss upper-bounds zero-one loss, allowing regret control to imply mistake control.
  • Bound implications: When instances are sparse, W2,2 is preferable to W1,1 and W2,1; WS(1) can be better when W lies in a low-dimensional space with unit-norm rows.The analysis provides a rigorous explanation for earlier empirical advantages of WS(1) under shared class structure.
  • Bound implications: When X2 is much larger than X∞ and W's columns share a sparsity pattern, W2,1 can be substantially better than the alternatives.The comparison depends on the relationship between instance geometry and shared sparsity in W.
  • Bound implications: With a combination of s ≪ d experts predicting the label, W has only s nonzero columns, favoring W2,1 under the stated bound comparisons.The example contrasts grouped sparsity with dense input vectors and shows why regularizer choice is structure-dependent.
  • Algorithm: The group multi-class Perceptron is a new mirror-descent algorithm using a group-norm regularizer and conservative updates on prediction mistakes.Its update is expressed through the Fenchel dual of the regularizer.

6 Kernel learning

The kernel-learning analysis applies the matrix-regularization framework to convex combinations of base kernels, yielding generalization bounds with mild dependence on the number of kernels.

  • Kernel-learning setting: Kernel learning considers function classes formed from a kernel family, including convex combinations of finitely many base kernels.The setting constrains the resulting function class using parameters such as a margin and the Gram matrix on the training data.
  • Kernel-learning bound: Theorem 23 provides a generalization result for the constrained kernel-learning class defined in Eq. (3).The theorem assumes each base kernel satisfies Kj(x, x) ≤B for 1 ≤j ≤k.
  • Proof strategy: The kernel-learning proof follows from equivalence with group Lasso and the strong-convexity/strong-smoothness framework.The authors contrast this derivation with a prior proof using a Rademacher chaos process of order two.
  • Kernel-learning bound: The dependence on the number of base kernels, k, is logarithmic, allowing convex combinations of a rather large number of base kernels.This dependence is described as rather mild compared with earlier bounds lacking logarithmic k dependence.
  • Comparison with prior bounds: If k ≥n, the earlier Srebro and Ben-David bound is vacuous while the paper’s bound remains meaningful; if k ≤n, the paper’s bound is better.The comparison is made through resulting generalization error bounds because the prior result does not use Rademacher complexities.

A.1 Convex analysis

This section reviews convex functions, subdifferentials, dual norms, Fenchel conjugates, and the Fenchel-Young inequality as tools for the paper’s analysis.

  • Convex-analysis definitions: The subdifferential ∂f(x) contains vectors defining global affine lower bounds to the convex function f at x.It is defined by requiring f(x + z) ≥f(x) + ⟨y, z⟩ for every z.
  • Norm duality: The dual norm is defined by ∥y∥⋆:= sup{⟨x, y⟩: ∥x∥≤1}.The section uses dual norms in describing conjugates and smoothness.
  • Fenchel conjugacy: The Fenchel-Young inequality states f(x) + f⋆(y) ≥⟨x, y⟩.Equality characterizes membership in the subdifferential through ∂f(x) = {y : f(x) + f⋆(y) = ⟨x, y⟩}.

A.2 Convex analysis of matrix functions

Matrix functions are constructed from symmetric functions of singular values or eigenvalues, allowing vector norm properties, duality, and subdifferentials to transfer to matrix spaces.

  • Matrix spectral representations: Matrices are represented through ordered singular values, while symmetric matrices are handled through eigenvalues under the trace inner product.The matrix spaces considered are Rm×n and Sn.
  • Symmetry conditions: Absolutely symmetric functions are invariant to arbitrary permutations and sign changes, whereas symmetric functions are invariant to permutations.These invariance properties support defining functions through singular values or eigenvalues.
  • Matrix-function construction: The constructions f ◦σ and g ◦λ define functions on rectangular and symmetric matrices from functions on singular values and eigenvalues.The resulting domains are Rm×n for f ◦σ and Sn for g ◦λ.
  • Convex analysis of matrix functions: Theorems 26 and 28 transfer convex-analytic structure and subdifferential calculations from vector functions to singular-value and eigenvalue matrix functions.The proofs use von Neumann’s inequality for singular values and Fan’s inequality for eigenvalues.
  • Matrix norms: If f is a norm on Rl, then f ◦σ is a matrix norm whose dual is ∥σ(·)∥⋆; the analogous eigenvalue construction holds on Sn.This provides matrix norms from absolutely symmetric or symmetric vector norms.

B.1 Proof of Theorem. 3

The proof establishes the reverse direction of strong-convexity/strong-smoothness duality by relating gradients of the conjugate to subgradients of the original function.

  • Strong smoothness to strong convexity: The proof assumes the conjugate f⋆ is strongly smooth and uses this assumption to bound its deviation from a first-order approximation.The argument initially simplifies to β = 1.
  • Conjugate identities: Fenchel conjugacy and Fenchel-Young equality convert the smoothness bound for f⋆ into an inequality involving f and its subgradients.The proof uses f⋆⋆=f and the relation ⟨x, u⟩= f⋆(x) + f(u).
  • Subgradient-gradient correspondence: For points in the relative interior of f’s domain, x ∈∂f(u′) implies u′ = ∇f⋆(x).The proof compares Fenchel-Young equalities for u′ and ∇f⋆(x).
  • Strong convexity conclusion: The argument applies the derived inequality at two points around a convex combination and then weights the results by α and 1 −α.This combines the two inequalities for u1 and u2 around u = αu1 + (1 −α)u2.
  • Strong convexity conclusion: Summing the weighted inequalities proves that f is strongly convex.This completes the reverse implication under the stated domain conditions.

B.2 Proof of Theorem. 12

The proof uses an equivalent characterization of σ-smoothness and a convexity inequality, then substitutes the inequality into Eq. (6) to establish Eq. (5).

  • σ-smoothness is characterized equivalently through an inequality involving arbitrary x, y, and α ∈ [0, 1].
  • The argument applies the setup to arbitrary matrices X and Y with columns Xi and Yi.
  • The proof evaluates Ψ,Φ at interpolated columns αXi + (1 − α)Yi and invokes an inequality for nonnegative x and y.
  • The resulting bound combines Φ2 evaluated at Ψ(Xi) and Ψ(Yi) with weights α and 1 − α.
  • Substitution into Eq. (6) proves Eq. (5).

B.3 Proof of Theorem. 23

The proof works in a product Hilbert space equipped with an inner product and a ∥·∥2,1 norm, uses a kernel feature map, and concludes from results in Section 2.

  • The construction considers H = H1 × . . . × Hk with an inner product formed by summing the component-space inner products.
  • The vector norm ∥·∥2,1 is defined on H by summing component norms and constraining the result by 1/γ.
  • The feature map sends x to the tuple of kernel sections (K1(x, ·), . . . , Kk(x, ·)) in H.
  • The final claim follows directly from the results derived in Section 2.
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