Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable (i.e., response, or dependent variable) to be learned can be described by a reduced number of variables in the input space (i.e., the domain, space of features or explanatory variables). Sparsity regularization methods focus on selecting the input variables that best describe the output. Structured sparsity regularization methods generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in .
Common motivation for the use of structured sparsity methods are model interpretability, high-dimensional learning (where dimensionality of may be higher than the number of observations ), and reduction of computational complexity. Moreover, structured sparsity methods allow to incorporate prior assumptions on the structure of the input variables, such as overlapping groups, non-overlapping groups, and acyclic graphs. Examples of uses of structured sparsity methods include face recognition, magnetic resonance image (MRI) processing, socio-linguistic analysis in natural language processing, and analysis of genetic expression in breast cancer.
Definition and related concepts
Sparsity regularization Consider the linear kernel regularized empirical risk minimization problem with a loss function and the "norm" as the regularization penalty:
where , and denotes the "norm", defined as the number of nonzero entries of the vector . is said to be sparse if . Which means that the output can be described by a small subset of input variables.
More generally, assume a dictionary with is given, such that the target function of a learning problem can be written as: , The norm as the number of non-zero components of is defined as , where is the cardinality of set . is said to be sparse if .
However, while using the norm for regularization favors sparser solutions, it is computationally difficult to use and additionally is not convex. A computationally more feasible norm that favors sparser solutions is the norm; this has been shown to still favor sparser solutions and is additionally convex.
Structured sparsity regularization Structured sparsity regularization extends and generalizes the variable selection problem that characterizes sparsity regularization. Consider the above regularized empirical risk minimization problem with a general kernel and associated feature map with .
The regularization term penalizes each component independently, which means that the algorithm will suppress input variables independently from each other.
In several situations we may want to impose more structure in the regularization process, so that, for example, input variables are suppressed according to predefined groups. Structured sparsity regularization methods allow to impose such structure by adding structure to the norms defining the regularization term.
Structures and norms
Non-overlapping groups: group Lasso The non-overlapping group case is the most basic instance of structured sparsity. In it, an a priori partition of the coefficient vector in non-overlapping groups is assumed. Let be the vector of coefficients in group , we can define a regularization term and its group norm as , where is the group norm , is group , and is the j-th component of group .
The above norm is also referred to as group Lasso. This regularizer will force entire coefficient groups towards zero, rather than individual coefficients. As the groups are non-overlapping, the set of non-zero coefficients can be obtained as the union of the groups that were not set to zero, and conversely for the set of zero coefficients.
Overlapping groups Overlapping groups is the structure sparsity case where a variable can belong to more than one group . This case is often of interest as it can represent a more general class of relationships among variables than non-overlapping groups can, such as tree structures or other type of graphs.
There are two types of overlapping group sparsity regularization approaches, which are used to model different types of input variable relationships:
Intersection of complements: group Lasso The intersection of complements approach is used in cases when we want to select only those input variables that have positive coefficients in all groups they belong to. Consider again the group Lasso for a regularized empirical risk minimization problem: , where is the group norm, is group , and is the j-th component of group .
As in the non-overlapping groups case, the group Lasso regularizer will potentially set entire groups of coefficients to zero. Selected variables are those with coefficients . However, as in this case groups may overlap, we take the intersection of the complements of those groups that are not set to zero.
This intersection of complements selection criteria implies the modeling choice that we allow some coefficients within a particular group to be set to zero, while others within the same group may remain positive. In other words, coefficients within a group may differ depending on the several group memberships that each variable within the group may have.
Union of groups: latent group Lasso A different approach is to consider union of groups for variable selection. This approach captures the modeling situation where variables can be selected as long as they belong at least to one group with positive coefficients. This modeling perspective implies that we want to preserve group structure.
The formulation of the union of groups approach is also referred to as latent group Lasso, and requires to modify the group norm considered above and introduce the following regularizer
where , is the vector of coefficients of group g, and is a vector with coefficients for all variables in group , and in all others, i.e., if in group and otherwise.
This regularizer can be interpreted as effectively replicating variables that belong to more than one group, therefore conserving group structure. As intended by the union of groups approach, requiring produces a vector of weights w that effectively sums up the weights of all variables across all groups they belong to.
Issues with Group Lasso regularization and alternative approaches The objective function using group lasso consists of an error function, which is generally required to be convex but not necessarily strongly convex, and a group regularization term. An issue with this objective function is that it is convex but not necessarily strongly convex, and thus generally does not lead to unique solutions.
An example of a way to fix this is to introduce the squared norm of the weight vector as an additional regularization term while keeping the regularization term from the group lasso approach. If the coefficient of the squared norm term is greater than , then because the squared norm term is strongly convex, the resulting objective function will also be strongly convex. Provided that the coefficient is suitably small but still positive, the weight vector minimizing the resulting objective function is generally very close to a weight vector that minimizes the objective function that would result from removing the group regularization term altogether from the original objective function; the latter scenario corresponds to the group Lasso approach. Thus this approach allows for simpler optimization while maintaining sparsity.
Norms based on the structure over Input variables See: Submodular set function
Besides the norms discussed above, other norms used in structured sparsity methods include hierarchical norms and norms defined on grids. These norms arise from submodular functions and allow the incorporation of prior assumptions on the structure of the input variables. In the context of hierarchical norms, this structure can be represented as a directed acyclic graph over the variables while in the context of grid-based norms, the structure can be represented using a grid.
Hierarchical Norms See: Unsupervised learning
Unsupervised learning methods are often used to learn the parameters of latent variable models. Latent variable models are statistical models where in addition to the observed variables, a set of latent variables also exists which is not observed. Often in such models, "hierarchies" are assumed between the variables of the system; this system of hierarchies can be represented using directed acyclic graphs.
Hierarchies of latent variables have emerged as a natural structure in several applications, notably to model text documents.Bengio, Y. "Learning deep architectures for AI". Foundations and Trends in Machine Learning, 2(1), 2009. Hierarchical models using Bayesian non-parametric methods have been used to learn topic models,Blei, D., Ng, A., and Jordan, M. Latent dirichlet allocation. J. Mach. Learn. Res., 3:993–1022, 2003. which are statistical models for discovering the abstract "topics" that occur in a collection of documents. Hierarchies have also been considered in the context of kernel methods. Hierarchical norms have been applied to bioinformatics,S. Kim and E. Xing. Tree-guided group Lasso for multi-task regression with structured sparsity. In Proc. ICML, 2010. computer vision and topic models.R. Jenatton, J. Mairal, G. Obozinski, and F. Bach. Proximal methods for sparse hierarchical dictionary learning. In Proc. ICML, 2010.
Norms defined on grids If the structure assumed over variables is in the form of a 1D, 2D or 3D grid, then submodular functions based on overlapping groups can be considered as norms, leading to stable sets equal to rectangular or convex shapes. Such methods have applications in computer visionR. Jenatton, G. Obozinski, and F. Bach. Structured sparse principal component analysis. In Proc. AISTATS, 2009.
Algorithms for computation
Best subset selection problem The problem of choosing the best subset of input variables can be naturally formulated under a penalization framework as:L. Rosasco. Lecture 10 of the Lecture Notes for 9.520: Statistical Learning Theory and Applications. Massachusetts Institute of Technology, Fall 2014. Available at https://www.mit.edu/~9.520/fall14/slides/class18/class18_sparsity.pdf
Where denotes the "norm", defined as the number of nonzero entries of the vector .
Although this formulation makes sense from a modeling perspective, it is computationally unfeasible, as it is equivalent to an exhaustive search evaluating all possible subsets of variables.
Two main approaches for solving the optimization problem are: 1) greedy methods, such as step-wise regression in statistics, or matching pursuit in signal processing; and 2) convex relaxation formulation approaches and proximal gradient optimization methods.
Convex relaxation A natural approximation for the best subset selection problem is the norm regularization:
Such a scheme is called basis pursuit or the Lasso, which substitutes the "norm" for the convex, non-differentiable norm.
Proximal gradient methods
Proximal gradient methods, also called forward-backward splitting, are optimization methods useful for minimizing functions with a convex and differentiable component, and a convex potentially non-differentiable component.
As such, proximal gradient methods are useful for solving sparsity and structured sparsity regularization problems of the following form:
Where is a convex and differentiable loss function like the quadratic loss, and is a convex potentially non-differentiable regularizer such as the norm.
Connections to Other Areas of Machine Learning
Connection to Multiple Kernel Learning