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Sparse Prediction with the $k$-Support Norm
Andreas Argyriou, Rina Foygel, Nathan Srebro
TL;DR
Sparse prediction needs a convex relaxation that captures both limited support and small ℓ2 norm. The paper derives the k-support norm from that convex hull, shows it is tighter than the elastic net while differing by at most a factor of two, and discusses both its predictive benefits and limitations.
Problem
The paper asks whether the elastic net is the tightest convex relaxation of sparsity combined with an ℓ2 constraint.
Method
The paper defines the k-support norm as the gauge of the convex hull of sparse vectors with bounded ℓ2 norm.
Results
For k > 1, the k-support norm is a tighter convex relaxation than the elastic net, whose gap is at most a factor of 2 and whose tighter alternative improves prediction in experiments.
Takeaways & Limitations
The k-support norm is proposed as a replacement for the elastic net, while the factor-of-two bound also justifies elastic net as a fairly good approximation.
Takeaways & Limitations
The k-support norm may improve prediction without improving sparsity, and the study did not compare with OSCAR, PEN, or trace Lasso because of memory or code-availability constraints.
Abstract
from arXiv · showhide
We derive a novel norm that corresponds to the tightest convex relaxation of sparsity combined with an $\ell_2$ penalty. We show that this new {\em $k$-support norm} provides a tighter relaxation than the elastic net and is thus a good replacement for the Lasso or the elastic net in sparse prediction problems. Through the study of the $k$-support norm, we also bound the looseness of the elastic net, thus shedding new light on it and providing justification for its use.
1 Introduction
The paper argues that combining sparsity with a bounded ℓ2 norm motivates a tighter convex relaxation than the ℓ1-based Lasso or elastic net. It studies the resulting k-support norm and compares its tightness, guarantees, and practical use with existing regularizers.
- Motivation: ℓ1 relaxation can have sample complexity O(k^2 log d), whereas learning directly over k-sparse predictors scales as O(k log d).The paper attributes the quadratic dependence to ℓ1-regularized learning scaling with the squared ℓ1 norm.
- Motivation: When predictor magnitudes and ℓ2 norm are bounded, sparsity can be relaxed using the convex hull of sparse, unit-ℓ2 vectors.This motivates studying sparse predictors with small ℓ2 norm rather than sparsity alone.
- Problem: The elastic net combines ℓ1 and ℓ2 regularization, but the paper asks whether it is the tightest convex relaxation of sparsity plus an ℓ2 constraint.The elastic net interpolates between Lasso and Ridge through the relative values of its penalty weights.
- Method: The k-support norm is the gauge of the convex hull of sparse vectors with bounded ℓ2 norm, yielding a tighter convex relaxation than the elastic net for k > 1.The paper defines the norm through the convex set associated with sparse unit-ℓ2 vectors and advocates it as an elastic-net replacement.
- Results: The gap between the k-support norm and elastic net is at most a factor of 2, corresponding to at most a factor-of-two difference in sample complexity.This bound both limits the possible advantage of the tighter norm and provides justification for elastic-net use as an approximation.
- Implementation: The k-support norm has an efficient O(d log d) computation and a dual equal to the ℓ2 norm of the k largest entries.The paper also describes efficient first-order optimization algorithms despite the exponential group-lasso-overlaps representation.
2 The k-Support Norm
The k-support norm is defined from the convex hull of k-sparse unit vectors, interpolating between ℓ1 and ℓ2 while targeting sparse predictors with small ℓ2 norm. It has tractable computation and supports regularized learning, particularly when feature correlations are unknown.
- Definition: The k-support norm is the gauge of the convex hull of vectors with at most k nonzero entries and ℓ2 norm at most 1.The construction uses nested sets, with k=1 and k=d recovering the ℓ1 and ℓ2 unit balls.
- Dual norm: The dual norm is the ℓ2 norm of the k largest entries and likewise interpolates between ℓ∞ at k=1 and ℓ2 at k=d.This gives a simple characterization of the dual despite the convex-hull definition involving many groups.
- Computation: The k-support norm can be computed in O(d log d) time despite its variational definition involving exponentially many groups.An alternative formula enables efficient norm evaluation.
- Properties: The norm interpolates between ℓ1 and ℓ2, favoring sparse vectors while allowing fitted cardinality larger than k.Its shrinkage combines uniform ℓ2 shrinkage for large components with sparse ℓ1 shrinkage for small components.
- Learning: The paper proposes k-support regularization for sparse, low-ℓ2-norm prediction, with λ and k selected by cross-validation.The parameter k need not equal the sparsity of the fitted coefficients and should be tuned independently.
3 Relation to the Elastic Net
The k-support norm is tighter than the elastic net while remaining close enough to explain the elastic net’s practical usefulness. Their difference can reach a factor of √2, but no more.
- Solution sets: Elastic-net solutions can be compared through a parameter k=(∥ŵ∥1/∥ŵ∥2)^2, allowing the elastic-net solution set to align with the interpolating norm formulation.The parameter lies in [1,d].
- Geometric relation: The elastic-net and k-support unit balls are nested, but the norms are not equal; the k-support ball is more rounded.The k-support norm is strictly tighter for k>1.
- Tightness bound: The elastic net and k-support norm differ by at most a factor of √2.The paper constructs an example attaining this maximum difference.
- Proof: The strict comparison holds because the dual-norm inequalities are strict across the relevant cases.The proof distinguishes whether the largest entry exceeds the (k+1)-th entry or all entries through k+1 are equal.
4 Optimization
The paper develops efficient proximal optimization for squared-loss learning with k-support regularization. The proximity operator can be computed efficiently and combined with accelerated first-order methods.
- Proximal formulation: The learning problem is solved with a first-order proximal algorithm for a smooth convex loss plus squared k-support regularization.The proximity operator is applied alongside loss-gradient computation.
- Proximity operator: The proximity map of 1/2 L(∥·∥sp_k)^2 can be computed in O(d(k + log d)) steps.Algorithm 1 identifies the relevant indices and rescales, thresholds, and reorders the result.
- Derivation: The proximal derivation exploits sign and permutation invariance to work with ordered nonnegative vectors without loss of generality.The resulting proximal vector preserves the input ordering and signs.
- Acceleration: Accelerated first-order methods such as FISTA combine the gradient step with the k-support proximity step.With appropriate step sizes, the method has an optimal O(1/T^2) objective convergence rate after T iterations.
5 Empirical Comparisons
The experiments compare the k-support norm with Lasso and elastic net across synthetic, heart-disease, and 20 Newsgroups data. Theory predicts at most a factor-of-two difference in sample complexity, while experiments test whether the tighter relaxation yields practical gains.
- Theoretical comparison: At most a factor of two separates the k-support norm and elastic net in sample complexities and generalization guarantees.The analysis therefore predicts limited performance differences while motivating empirical tests of the tighter relaxation.
- Synthetic data: Fifty synthetic data sets used 50 training, 50 validation, and 350 test points to evaluate prediction against an oracle vector.Parameters were selected by validation mean squared error across the compared methods.
- Synthetic data: The coefficient patterns from k-support regularization showed less variability than elastic net solutions across the synthetic data sets.Elastic net could attain higher values at relevant features, whereas k-support produced a more consistent feature pattern.
- Evaluation: The empirical comparisons report mean squared errors and classification accuracy for synthetic, South African heart, and 20 Newsgroups data.The table summarizes medians over repeated synthetic and heart-data runs.
- South African Heart Data: The South African heart-data experiment found identical performance for k-support, Lasso, and elastic net.The task used 50 random train-validation-test splits and reported test MSE and accuracy.
20 Newsgroups
The 20 Newsgroups experiment evaluates binary classification after reducing the vocabulary and splitting the data into training, validation, and test sets. K-support regularization achieved higher prediction accuracy than Lasso and elastic net.
- Dataset and protocol: The binary 20 Newsgroups task used 14,000 training, 1,000 validation, and 4,996 test examples after removing words appearing in fewer than three documents.The positive class comprised sci.*, comp.*, and misc.forsale groups; the negative class comprised the other groups.
- Results: K-support regularization improved prediction accuracy over both Lasso and elastic net on the 20 Newsgroups data.Mean squared error and accuracy were reported on the test set.
6 Summary
The paper introduces the k-support norm as the tightest convex relaxation of sparsity with an ℓ2 constraint and compares it with elastic net. It reports a tighter relaxation, bounded approximation gap, and empirical prediction benefits, while noting a sparsity trade-off and unresolved comparisons.
- Contribution: The k-support norm is introduced as the tightest convex relaxation of sparsity combined with ℓ2 regularization.The paper presents it as a norm whose unit ball captures the combined constraint.
- Contribution: The k-support norm is tighter than the elastic net, while the elastic net remains a close approximation to the tightest relaxation.The paper uses this relationship both to motivate k-support and to justify continued use of elastic net.
- Scope: Comparisons with OSCAR, PEN, and trace Lasso were not completed because of memory limitations or unavailable code.These omissions constrain the empirical comparison set.
- Trade-offs: The k-support norm has better prediction properties but not necessarily better sparsity-inducing properties than elastic net.Its more rounded unit ball can produce solutions that are even less sparse but more predictive, especially with correlated features.
- Future direction: The paper suggests that k-support regularization may refine compressed-sensing results by handling more correlation structure among features.This is presented as a direction for further consideration rather than an established result.