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:<math>\min_{w\in\mathbb{R}^d} \frac{1}{n}\sum_{i=1}^n (y_i- \langle w,x_i\rangle)^2+ \lambda \|w\|_1, \quad \text{ where } x_i\in \mathbb{R}^d\text{ and } y_i\in\mathbb{R}.</math>
Proximal gradient methods offer a general framework for solving regularization problems from statistical learning theory with penalties that are tailored to a specific problem application.<ref name=combettes>{{cite journal|last=Combettes|first=Patrick L.|author2=Wajs, Valérie R. |title=Signal Recovering by Proximal Forward-Backward Splitting|journal=Multiscale Model. Simul.|year=2005|volume=4|issue=4|pages=1168–1200|doi=10.1137/050626090|s2cid=15064954
== Relevant background ==
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The general form of Moreau's decomposition states that for any <math>x\in\mathcal{X}</math> and any <math>\gamma>0</math> that
:<math>x = \operatorname{prox}_{\gamma \varphi}(x) + \gamma\operatorname{prox}_{\varphi^*/\gamma}(x/\gamma),</math>
which for <math>\gamma=1</math> implies that <math>x = \operatorname{prox}_{\varphi}(x)+\operatorname{prox}_{\varphi^*}(x)</math>.<ref name=combettes /><ref name=moreau>{{cite journal|last=Moreau|first=J.-J.|title=Fonctions convexes duales et points proximaux dans un espace hilbertien|journal=Comptes Rendus de l'Académie des Sciences, Série A|year=1962|volume=255|pages=2897–2899|mr=144188|zbl=0118.10502}}</ref> The Moreau decomposition can be seen to be a generalization of the usual orthogonal decomposition of a [[vector space]], analogous with the fact that proximity operators are generalizations of projections.<ref name=combettes />
In certain situations it may be easier to compute the proximity operator for the conjugate <math>\varphi^*</math> instead of the function <math>\varphi</math>, and therefore the Moreau decomposition can be applied. This is the case for [[Lasso (statistics)#Group LASSO|group lasso]].
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where <math>x_i\in \mathbb{R}^d\text{ and } y_i\in\mathbb{R}.</math> The <math>\ell_1</math> regularization problem is sometimes referred to as ''lasso'' ([[Lasso (statistics)|least absolute shrinkage and selection operator]]).<ref name=tibshirani /> Such <math>\ell_1</math> regularization problems are interesting because they induce '' sparse'' solutions, that is, solutions <math>w</math> to the minimization problem have relatively few nonzero components. Lasso can be seen to be a convex relaxation of the non-convex problem
:<math>\min_{w\in\mathbb{R}^d} \frac{1}{n}\sum_{i=1}^n (y_i- \langle w,x_i\rangle)^2+ \lambda \|w\|_0, </math>
where <math>\|w\|_0</math> denotes the <math>\ell_0</math> "norm", which is the number of nonzero entries of the vector <math>w</math>. Sparse solutions are of particular interest in learning theory for interpretability of results: a sparse solution can identify a small number of important factors.<ref name=tibshirani>{{cite journal|last=Tibshirani|first=R.|title=Regression shrinkage and selection via the lasso|journal=J. R. Stat. Soc. Ser. B|year=1996|volume=58|series=1|issue=1|pages=267–288|doi=10.1111/j.2517-6161.1996.tb02080.x }}</ref>
=== Solving for L<sub>1</sub> proximity operator ===
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Note here the effective trade-off between the empirical error term <math>F(w) </math> and the regularization penalty <math>R(w)</math>. This fixed point method has decoupled the effect of the two different convex functions which comprise the objective function into a gradient descent step (<math> w^k - \gamma \nabla F\left(w^k\right)</math>) and a soft thresholding step (via <math>S_\gamma</math>).
Convergence of this fixed point scheme is well-studied in the literature<ref name=combettes /><ref name=daubechies /> and is guaranteed under appropriate choice of step size <math>\gamma</math> and [[loss function]] (such as the square loss taken here). [[Gradient descent#Extensions|Accelerated methods]] were introduced by Nesterov in 1983 which improve the [[rate of convergence]] under certain regularity assumptions on <math>F</math>.<ref name=nesterov>{{cite journal|last=Nesterov|first=Yurii|title=A method of solving a convex programming problem with convergence rate <math>O(1/k^2)</math>|journal=Soviet Mathematics - Doklady|year=1983|volume=27|issue=2|pages=372–376}}</ref> Such methods have been studied extensively in previous years.<ref>{{cite book|last=Nesterov|first=Yurii|title=Introductory Lectures on Convex Optimization|year=2004|publisher=Kluwer Academic Publisher}}</ref>
For more general learning problems where the proximity operator cannot be computed explicitly for some regularization term <math>R</math>, such fixed point schemes can still be carried out using approximations to both the gradient and the proximity operator.<ref name=bauschke /><ref>{{cite journal|last=Villa|first=S.|author2=Salzo, S. |author3=Baldassarre, L. |author4=Verri, A. |title=Accelerated and inexact forward-backward algorithms|journal=SIAM J. Optim.|year=2013|volume=23|issue=3|pages=1607–1633|doi=10.1137/110844805|citeseerx=10.1.1.416.3633|s2cid=11379846 }}</ref>
== Practical considerations ==
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In the fixed point iteration scheme
:<math>w^{k+1} = \operatorname{prox}_{\gamma R}\left(w^k-\gamma \nabla F\left(w^k\right)\right),</math>
one can allow variable step size <math>\gamma_k</math> instead of a constant <math>\gamma</math>. Numerous adaptive step size schemes have been proposed throughout the literature.<ref name=combettes /><ref name=bauschke /><ref>{{cite journal|last=Loris|first=I. |author2=Bertero, M. |author3=De Mol, C. |author4=Zanella, R. |author5=Zanni, L. |title=Accelerating gradient projection methods for <math>\ell_1</math>-constrained signal recovery by steplength selection rules|journal=Applied & Comp. Harmonic Analysis|volume=27|issue=2|pages=247–254|year=2009|doi=10.1016/j.acha.2009.02.003|arxiv=0902.4424 |s2cid=18093882 }}</ref><ref>{{cite journal|last=Wright|first=S.J.|author2=Nowak, R.D. |author3=Figueiredo, M.A.T. |title=Sparse reconstruction by separable approximation|journal=IEEE Trans. Image Process.|year=2009|volume=57|issue=7|pages=2479–2493|doi=10.1109/TSP.2009.2016892|bibcode=2009ITSP...57.2479W|citeseerx=10.1.1.115.9334|s2cid=7399917 }}</ref> Applications of these schemes<ref name=structSparse /><ref>{{cite journal|last=Loris|first=Ignace|title=On the performance of algorithms for the minimization of <math>\ell_1</math>-penalized functionals|journal=Inverse Problems|year=2009|volume=25|issue=3|doi=10.1088/0266-5611/25/3/035008|page=035008|arxiv=0710.4082|bibcode=2009InvPr..25c5008L|s2cid=14213443}}</ref> suggest that these can offer substantial improvement in number of iterations required for fixed point convergence.
=== Elastic net (mixed norm regularization) ===
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=== Group lasso ===
Group lasso is a generalization of the [[Lasso (statistics)|lasso method]] when features are grouped into disjoint blocks.<ref name=groupLasso>{{cite journal|last=Yuan|first=M.|author2=Lin, Y. |title=Model selection and estimation in regression with grouped variables|journal=J. R. Stat. Soc. B|year=2006|volume=68|issue=1|pages=49–67|doi=10.1111/j.1467-9868.2005.00532.x|s2cid=6162124|
:<math>R(w) =\sum_{g=1}^G \|w_g\|_2,</math>
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* [[Convex analysis]]
* [[Proximal gradient method]]
* [[Regularization (mathematics)#
* [[Statistical learning theory]]
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