Contours of statistical penalty functions as GIF images

March 17, 2017
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Many statistical modeling problems reduce to a minimization problem of the general form:

or

where is some type of loss function, denotes the data, and is a penalty, also referred to by other names, such as “regularization term” (problems (1) and (2-3) are often equivalent by the way). Of course both, and , may depend on further parameters.

There are multiple reasons why it can be helpful to check out the contours of such penalty functions :

  1. When is two-dimensional, the solution of problem (2-3) can be found by simply taking a look at the contours of and .
  2. That builds intuition for what happens in more than two dimensions, and in other more general cases.
  3. From a Bayesian point of view, problem (1) can often be interpreted as an MAP estimator, in which case the contours of are also contours of the prior distribution of .

Therefore, it is meaningful to visualize the set of points that maps onto the unit ball in , i.e., the set

B_{g} := \{ \mathbf{x}\in\mathbb{R}^2 : g(\mathbf{x}) \leq 1 \}.

Below you see GIF images of such sets for various penalty functions in 2D, capturing the effect of varying certain parameters in . The covered penalty functions include the family of -norms, the elastic net penalty, the fused penalty, the sorted norm, and several others.

:white_check_mark: R code to reproduce the GIFs is provided.

p-norms in 2D

First we consider the -norm,

g_{p}(\boldsymbol{\beta}) = \lVert\boldsymbol{\beta}\rVert_{p}^{p} = \lvert\beta_{1}\rvert^p + \lvert\beta_{2}\rvert^p,

with a varying parameter (which actually isn’t a proper norm for ). Many statistical methods, such as LASSO (Tibshirani 1996) and Ridge Regression (Hoerl and Kennard 1970), employ -norm penalties. To find all on the boundary of the 2D unit -norm ball, given (the first entry of ), is easily obtained as

\beta_2 = \pm (1-|\beta_1|^p)^{1/p}, \quad \forall\beta_1\in[-1, 1].

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Elastic net penalty in 2D

The elastic net penalty can be written in the form

g_{\alpha}(\boldsymbol{\beta}) = \alpha \lVert \boldsymbol{\beta} \rVert_{1} + (1 - \alpha) \lVert \boldsymbol{\beta} \rVert_{2}^{2},

for . It is quite popular with a variety of regression-based methods (such as the Elastic Net, of course). We obtain the corresponding 2D unit “ball”, by calculating from a given as

\beta_{2} = \pm \frac{-\alpha + \sqrt{\alpha^2 - 4 (1 - \alpha) ((1 - \alpha) \beta_{1}^2 + \alpha \beta_{1} - 1)}}{2 - 2 \alpha}.

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Fused penalty in 2D

The fused penalty can be written in the form

g_{\alpha}(\boldsymbol{\beta}) = \alpha \lVert \boldsymbol{\beta} \rVert_{1} + (1 - \alpha) \sum_{i = 2}^m \lvert \beta_{i} - \beta_{i-1} \rvert.

It encourages neighboring coefficients to have similar values, and is utilized by the fused LASSO (Tibshirani et. al. 2005) and similar methods.

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(Here I have simply evaluated the fused penalty function on a grid of points in , because figuring out equations in parametric form for the above polygons was too painful for my taste… :stuck_out_tongue:)

Sorted L1 penalty in 2D

The Sorted penalty is used in a number of regression-based methods, such as SLOPE (Bogdan et. al. 2015) and OSCAR (Bondell and Reich 2008). It has the form

g_{\boldsymbol{\lambda}}(\boldsymbol{\beta}) = \sum_{i = 1}^m \lambda_{i} \lvert \beta \rvert_{(i)},

where are the absolute values of the entries of arranged in a decreasing order. In 2D this reduces to

g_{\boldsymbol{\lambda}}(\boldsymbol{\beta}) = \lambda_{1} \max\{|\beta_{1}|, |\beta_{2}|\} + \lambda_{2} \min\{|\beta_{1}|, |\beta_{2}|\}.

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Difference of p-norms

It holds that

\lVert \boldsymbol{\beta} \rVert_{1} \geq \lVert \boldsymbol{\beta} \rVert_{2},

or more generally, for all -norms it holds that

(\forall p \leq q) : \lVert \boldsymbol{\beta} \rVert_{p} \geq \lVert \boldsymbol{\beta} \rVert_{q}.

Thus, it is meaningful to define a penalty function of the form

g_{\alpha}(\boldsymbol{\beta}) = \lVert \boldsymbol{\beta} \rVert_{1} - \alpha \lVert \boldsymbol{\beta} \rVert_{2},

for , which results in the following.

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We visualize the same for varying fixing , i.e., we define

g_{\alpha}(\boldsymbol{\beta}) = \lVert \boldsymbol{\beta} \rVert_{1} - 0.6 \lVert \boldsymbol{\beta} \rVert_{p},

and we obtain the following GIF.

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Hyperbolic tangent penalty in 2D

The hyperbolic tangent penalty, which is for example used in the method of variable selection via subtle uprooting (Su, 2015), has the form

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