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Hierarchical Sparse Modeling: A Choice of Two Group Lasso Formulations

Demanding sparsity in estimated models has become a routine 
practice in statistics. In many situations, we wish to require that the sparsity 
patterns attained honor certain problem-specific constraints. Hierarchical 
sparse modeling (HSM) refers to situations in which these constraints 
specify that one set of parameters be set to zero whenever another is set to 
zero. In recent years, numerous papers have developed convex regularizers 
for this form of sparsity structure, which arises in many areas of statistics 
including interaction modeling, time series analysis, and covariance estimation. 
In this paper, we observe that these methods fall into two frameworks, 
the group lasso (GL) and latent overlapping group lasso (LOG), which have 
not been systematically compared in the context of HSM. The purpose of this 
paper is to provide a side-by-side comparison of these two frameworks for 
HSM in terms of their statistical properties and computational efficiency. We 
call special attention to GL’s more aggressive shrinkage of parameters deep 
in the hierarchy, a property not shared by LOG. In terms of computation, 
we introduce a finite-step algorithm that exactly solves the proximal operator 
of LOG for a certain simple HSM structure; we later exploit this to develop 
a novel path-based block coordinate descent scheme for general HSM 
structures. Both algorithms greatly improve the computational performance 
of LOG. Finally, we compare the two methods in the context of covariance 
estimation, where we introduce a new sparsely-banded estimator using LOG, 
which we show achieves the statistical advantages of an existing GL-based 
method but is simpler to express and more efficient to compute.