Sparsity regularization

Consider the linear kernel regularized empirical risk minimization problem with a loss function ${\displaystyle V(y_{i},f(x))}$ and the ${\displaystyle \ell _{0}}$ "norm" as the regularization penalty:

${\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},\langle w,x_{i}\rangle )+\lambda \|w\|_{0},}$

where ${\displaystyle x,w\in \mathbb {R^{d}} }$, and ${\displaystyle \|w\|_{0}}$ denotes the ${\displaystyle \ell _{0}}$ "norm", defined as the number of nonzero entries of the vector ${\displaystyle w}$. ${\displaystyle f(x)=\langle w,x_{i}\rangle }$ is said to be sparse if ${\displaystyle \|w\|_{0}=s<d}$. Which means that the output ${\displaystyle Y}$ can be described by a small subset of input variables.

More generally, assume a dictionary ${\displaystyle \phi _{j}:X\rightarrow \mathbb {R} }$ with ${\displaystyle j=1,...,p}$ is given, such that the target function ${\displaystyle f(x)}$ of a learning problem can be written as:

${\displaystyle f(x)=\sum _{j=1}^{p}\phi _{j}(x)w_{j}}$, ${\displaystyle \forall x\in X}$

The ${\displaystyle \ell _{0}}$ norm ${\displaystyle \|f\|_{0}=\|w\|_{0}}$ as the number of non-zero components of ${\displaystyle w}$ is defined as

${\displaystyle \|w\|_{0}=|\{j|w_{j}\neq 0,j\in \{1,...,p\}\}|}$, where ${\displaystyle |A|}$ is the cardinality of set ${\displaystyle A}$.

${\displaystyle f}$ is said to be sparse if ${\displaystyle \|f\|_{0}=\|w\|_{0}=s<d}$.

However, while using the ${\displaystyle \ell _{0}}$ 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 ${\displaystyle \ell _{1}}$ norm; this has been shown to still favor sparser solutions and is additionally convex.^[4]

Structured sparsity regularization

Structured sparsity regularization extends and generalizes the variable selection problem that characterizes sparsity regularization.^[2][3] Consider the above regularized empirical risk minimization problem with a general kernel and associated feature map ${\displaystyle \phi _{j}:X\rightarrow \mathbb {R} }$ with ${\displaystyle j=1,...,p}$.

${\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},\langle w,\Phi (x_{i})\rangle )+\lambda \|w\|_{0},}$

The regularization term ${\displaystyle \lambda \|w\|_{0}}$ penalizes each ${\displaystyle w_{j}}$ 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.

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 ${\displaystyle w}$ in ${\displaystyle G}$ non-overlapping groups is assumed. Let ${\displaystyle w_{g}}$ be the vector of coefficients in group ${\displaystyle g}$, we can define a regularization term and its group norm as

${\displaystyle \lambda R(w)=\lambda \sum _{g=1}^{G}\|w_{g}\|_{g}}$,

where ${\displaystyle \|w_{g}\|_{g}}$ is the group ${\displaystyle \ell _{2}}$ norm ${\displaystyle \|w_{g}\|_{g}={\sqrt {\sum _{j=1}^{|G_{g}|}(w_{g}^{j})^{2}}}}$ , ${\displaystyle G_{g}}$ is group ${\displaystyle g}$, and ${\displaystyle w_{g}^{j}}$ is the j-th component of group ${\displaystyle G_{g}}$.

The above norm is also referred to as group Lasso.^[2] 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 ${\displaystyle g}$. 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.^[3][8]

There are two types of overlapping group sparsity regularization approaches, which are used to model different types of input variable relationships:

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 ${\displaystyle \ell _{1}}$ 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.^[9]

An example of a way to fix this is to introduce the squared ${\displaystyle \ell _{2}}$ norm of the weight vector as an additional regularization term while keeping the ${\displaystyle \ell _{1}}$ regularization term from the group lasso approach.^[9] If the coefficient of the squared ${\displaystyle \ell _{2}}$ norm term is greater than ${\displaystyle 0}$, then because the squared ${\displaystyle \ell _{2}}$ norm term is strongly convex, the resulting objective function will also be strongly convex.^[9] Provided that the ${\displaystyle \ell _{2}}$ 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 ${\displaystyle \ell _{2}}$ regularization term altogether from the original objective function; the latter scenario corresponds to the group Lasso approach.^[9] Thus this approach allows for simpler optimization while maintaining sparsity.^[9]

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.^{[10][11][12][13][14][15]}

Best subset selection problem

The problem of choosing the best subset of input variables can be naturally formulated under a penalization framework as:^[4]

${\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},w,x_{i})+\lambda \|w\|_{0},}$

Where ${\displaystyle \|w\|_{0}}$ denotes the ${\displaystyle \ell _{0}}$ "norm", defined as the number of nonzero entries of the vector ${\displaystyle w}$.

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.^[4]

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 ${\displaystyle \ell _{1}}$ norm regularization:^[4]

${\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},w,x_{i})+\lambda \|w\|_{1}}$

Such a scheme is called basis pursuit or the Lasso, which substitutes the ${\displaystyle \ell _{0}}$ "norm" for the convex, non-differentiable ${\displaystyle \ell _{1}}$ 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^[9] of the following form:

${\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},w,x_{i})+R(w)}$

Where ${\displaystyle V(y_{i},w,x_{i})}$ is a convex and differentiable loss function like the quadratic loss, and ${\displaystyle R(w)}$ is a convex potentially non-differentiable regularizer such as the ${\displaystyle \ell _{1}}$ norm.

Connection to Multiple Kernel Learning

Structured Sparsity regularization can be applied in the context of multiple kernel learning.^[16] Multiple kernel learning refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal linear or non-linear combination of kernels as part of the algorithm.

In the algorithms mentioned above, a whole space was taken into consideration at once and was partitioned into groups, i.e. subspaces. A complementary point of view is to consider the case in which distinct spaces are combined to obtain a new one. It is useful to discuss this idea considering finite dictionaries. Finite dictionaries with linearly independent elements - these elements are also known as atoms - refer to finite sets of linearly independent basis functions, the linear combinations of which define hypothesis spaces. Finite dictionaries can be used to define specific kernels, as will be shown.^[16] Assume for this example that rather than only one dictionary, several finite dictionaries are considered.

For simplicity, the case in which there are only two dictionaries ${\displaystyle A=\{a_{j}:X\rightarrow \mathbb {R} ,j=1,...,p\}}$ and ${\displaystyle B=\{b_{t}:X\rightarrow \mathbb {R} ,t=1,...,q\}}$ where ${\displaystyle q}$ and ${\displaystyle p}$ are integers, will be considered. The atoms in ${\displaystyle A}$ as well as the atoms in ${\displaystyle B}$ are assumed to be linearly independent. Let ${\displaystyle D=\{d_{k}:X\rightarrow \mathbb {R} ,k=1,...,p+q\}=A\cup B}$ be the union of the two dictionaries. Consider the linear space of functions ${\displaystyle H}$ given by linear combinations of the form

${\displaystyle f(x)=\sum _{i=1}^{p+q}{w^{j}d_{j}(x)}=\sum _{j=1}^{p}{w_{A}^{j}a_{j}(x)}+\sum _{t=1}^{q}{w_{B}^{t}b_{t}(x)},x\in X}$

for some coefficient vectors ${\displaystyle w_{A}\in \mathbb {R} ^{p},w_{B}\in \mathbb {R} ^{q}}$, where ${\displaystyle w=(w_{A},w_{B})}$. Assume the atoms in ${\displaystyle D}$ to still be linearly independent, or equivalently, that the map ${\displaystyle w=(w_{A},w_{B})\mapsto f}$ is one to one. The functions in the space ${\displaystyle H}$ can be seen as the sums of two components, one in the space ${\displaystyle H_{A}}$, the linear combinations of atoms in ${\displaystyle A}$ and one in ${\displaystyle H_{B}}$, the linear combinations of the atoms in ${\displaystyle B}$.

One choice of norm on this space is ${\displaystyle ||f||=||w_{A}||+||w_{B}||}$. Note that we can now view ${\displaystyle H}$ as a function space in which ${\displaystyle H_{A}}$, ${\displaystyle H_{B}}$ are subspaces. In view of the linear independence assumption, ${\displaystyle H}$ can be identified with ${\displaystyle \mathbb {R} ^{p+q}}$ and ${\displaystyle H_{A},H_{B}}$ with ${\displaystyle \mathbb {R} ^{p},\mathbb {R} ^{q}}$ respectively. The norm mentioned above can be seen as the group norm in ${\displaystyle H}$associated to the subspaces ${\displaystyle H_{A}}$, ${\displaystyle H_{B}}$, providing a connection to structured sparsity regularization.

Here, ${\displaystyle H_{A}}$, ${\displaystyle H_{B}}$ and ${\displaystyle H}$ can be seen to be the reproducing kernel Hilbert spaces with corresponding feature maps ${\displaystyle \Phi _{A}:X\rightarrow \mathbb {R} ^{p}}$, given by ${\displaystyle \Phi _{A}(x)=(a_{1}(x),...,a_{p}(x))}$, ${\displaystyle \Phi _{B}:X\rightarrow \mathbb {R} ^{q}}$, given by ${\displaystyle \Phi _{B}(x)=(b_{1}(x),...,b_{q}(x))}$, and ${\displaystyle \Phi :X\rightarrow \mathbb {R} ^{p+q}}$, given by the concatenation of ${\displaystyle \Phi _{A},\Phi _{B}}$, respectively.

In the structured sparsity regularization approach to this scenario, the relevant groups of variables which the group norms consider correspond to the subspaces ${\displaystyle H_{A}}$ and ${\displaystyle H_{B}}$. This approach promotes setting the groups of coefficients corresponding to these subspaces to zero as opposed to only individual coefficients, promoting sparse multiple kernel learning.

The above reasoning directly generalizes to any finite number of dictionaries, or feature maps. It can be extended to feature maps inducing infinite dimensional hypothesis

spaces.^[16]