Sparsity in penalized empirical risk minimization

Soit $(X,Y)$ un couple aléatoire à valeurs dans $S\times T$ et de loi $P$ inconnue. Soient $(X_1,Y_1),...,(X_n,Y_n)$ des répliques i.i.d. de $(X,Y)$, de loi empirique associée $P_n$. Soit $h_1,...,h_N:S\mapsto \left[-1,1\right]$ un dictionnaire composé de $N$ fonctions. Pour tout $\lambda \in {ℝ}^N$, on note $f_{\lambda }:={\sum }_{j=1}^N{\lambda }_j{h}_j$. Soit $\ell :T\times ℝ\mapsto ℝ$ fonction de perte donnée que l’on suppose convexe en la seconde variable. On note $(\ell •f)(x,y):=\ell (y;f(x\left)\right)$. On étudie le problème de minimisation du risque empirique pénalisé suivant

$${\widehat{\lambda }}^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P_n(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right],$$

qui correspond à la version empirique du problème

$${\lambda }^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right]$$

(ici $\epsilon \ge 0$ est un paramètre de régularisation; ${\lambda }^0$ correspond au cas $\epsilon =0$). Ce cadre général englobe un certain nombre de problèmes de régression et de classification. On s’intéresse au cas où $p\ge 1$, mais reste proche de 1 (de sorte que $p-1$ soit de l’ordre $\frac{1}{logN}$, ou inférieur). On montre que la «sparsité» de ${\lambda }^{\epsilon }$ implique la «sparsité» de ${\widehat{\lambda }}^{\epsilon }$. En outre, on étudie les conséquences de la «sparsité» en termes de bornes supérieures sur l’excès de risque $P(\ell •f_{{\widehat{\lambda }}^{\epsilon }})-P(\ell •f_{{\lambda }^0})$ des solutions obtenues pour les différents problèmes de minimisation du risque empirique.

Let $(X,Y)$ be a random couple in $S\times T$ with unknown distribution $P$. Let $(X_1,Y_1),...,(X_n,Y_n)$ be i.i.d. copies of $(X,Y)$, $P_n$ being their empirical distribution. Let $h_1,...,h_N:S\mapsto \left[-1,1\right]$ be a dictionary consisting of $N$ functions. For $\lambda \in {ℝ}^N$, denote $f_{\lambda }:={\sum }_{j=1}^N{\lambda }_j{h}_j$. Let $\ell :T\times ℝ\mapsto ℝ$ be a given loss function, which is convex with respect to the second variable. Denote $(\ell •f)(x,y):=\ell (y;f(x\left)\right)$. We study the following penalized empirical risk minimization problem

$${\widehat{\lambda }}^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P_n(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right],$$

which is an empirical version of the problem

$${\lambda }^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right]$$

(here $\epsilon \ge 0$ is a regularization parameter; ${\lambda }^0$ corresponds to $\epsilon =0$). A number of regression and classification problems fit this general framework. We are interested in the case when $p\ge 1$, but it is close enough to 1 (so that $p-1$ is of the order $\frac{1}{logN}$, or smaller). We show that the “sparsity” of ${\lambda }^{\epsilon }$ implies the “sparsity” of ${\widehat{\lambda }}^{\epsilon }$ and study the impact of “sparsity” on bounding the excess risk $P(\ell •f_{{\widehat{\lambda }}^{\epsilon }})-P(\ell •f_{{\lambda }^0})$ of solutions of empirical risk minimization problems.

[1] A. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301-413. | MR 1679028 | Zbl 0946.62036

[2] P. Bartlett, O. Bousquet and S. Mendelson. Local Rademacher complexities. Ann. Statist. 33 (2005) 1497-1537. | MR 2166554 | Zbl 1083.62034

[3] A. Ben-Tal and A. Nemirovski. Lectures on Modern Convex Optimization. Analysis, Algorithms and Engineering Applications. MPS/SIAM, Series on Optimization, Philadelphia, 2001. | MR 1857264 | Zbl 0986.90032

[4] F. Bunea, A. Tsybakov and M. Wegkamp. Aggregation for Gaussian regression. Ann. Statist. 35 (2007) 1674-1697. | MR 2351101

[5] F. Bunea, A. Tsybakov and M. Wegkamp. Sparsity oracle inequalities for the LASSO. Electron. J. Statist. 1 (2007) 169-194. | MR 2312149 | Zbl 1146.62028

[6] E. Candes and T. Tao. The Dantzig selector statistical estimation when p is much larger than n. Ann. Statist. 35 (2007) 2313-2351. | MR 2382644 | Zbl 1139.62019

[7] E. Candes, M. Rudelson, T. Tao and R. Vershynin. Error correction via linear programming. In Proc. 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS05) 295-308. IEEE, Pittsburgh, PA, 2005.

[8] E. Candes, J. Romberg and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (2006) 1207-1223. | MR 2230846 | Zbl 1098.94009

[9] O. Catoni. Statistical Learning Theory and Stochastic Optimization. Springer, New York, 2004. | MR 2163920 | Zbl 1076.93002

[10] D. L. Donoho. For most large underdetermined systems of equations the minimal ℓ1-norm near-solution approximates the sparsest near-solution. Preprint, 2004. | Zbl 1105.90068

[11] D. L. Donoho. For most large underdetermined systems of linear equations the minimal ℓ1-norm solution is also the sparsest solution. Comm. Pure Appl. Math. 59 (2006) 797-829. | MR 2217606 | Zbl 1113.15004

[12] D. L. Donoho. Compressed sensing. IEEE Trans. Inform. Theory 52 (2006) 1289-1306. | MR 2241189

[13] D. L. Donoho, M. Elad and V. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory 52 (2006) 6-18. | MR 2237332

[14] Van De S. Geer. High-dimensional generalized linear models and the Lasso. Ann. Statist. 36 (2008) 614-645. | MR 2396809 | Zbl 1138.62323

[15] V. Koltchinskii. Model selection and aggregation in sparse classification problems. Oberwolfach Reports Meeting on Statistical and Probabilistic Methods of Model Selection, October, 2005.

[16] V. Koltchinskii. Local Rademacher complexities and oracle inequalities in risk mnimization. Ann. Statist. 34 (2006) 2593-2656. | MR 2329442 | Zbl 1118.62065

[17] V. Koltchinskii and D. Panchenko. Complexities of convex combinations and bounding the generalization error in classification. Ann. Statist. 33 (2005) 1455-1496. | MR 2166553 | Zbl 1080.62045

[18] M. Ledoux and M. Talagrand. Probability in Banach Spaces. Springer, New York, 1991. | MR 1102015 | Zbl 0748.60004

[19] P. Massart. Some applications of concentration inequalities to statistics. Ann. Fac. Sci. Tolouse (IX) 9 (2000) 245-303. | Numdam | MR 1813803 | Zbl 0986.62002

[20] P. Massart. Concentration Inequalities and Model Selection. Springer, Berlin, 2007. | MR 2319879 | Zbl 1170.60006

[21] S. Mendelson, A. Pajor and N. Tomczak-Jaegermann. Reconstruction and subgaussian operators in Asymptotic Geometric Analysis. Geom. Funct. Anal. 17 (2007) 1248-1282. | MR 2373017 | Zbl 1163.46008

[22] N. Meinshausen and P. Bühlmann. High-dimensional graphs and variable selection with the LASSO. Ann. Statist. 34 (2006) 1436-1462. | MR 2278363 | Zbl 1113.62082

[23] A. Nemirovski. Topics in non-parametric statistics. In Ecole d'Et'e de Probabilités de Saint-Flour XXVIII, 1998 85-277. P. Bernard (Ed). Springer, New York, 2000. | MR 1775640 | Zbl 0998.62033

[24] M. Rudelson and R. Vershynin. Geometric approach to error correcting codes and reconstruction of signals. Int. Math. Res. Not. 64 (2005) 4019-4041. | MR 2206919 | Zbl 1103.94014

[25] R. Tibshirani. Regression shrinkage and selection via Lasso. J. Royal Statist. Soc. Ser. B 58 (1996) 267-288. | MR 1379242 | Zbl 0850.62538

[26] A. Tsybakov. Optimal rates of aggregation. In Proc. 16th Annual Conference on Learning Theory (COLT) and 7th Annual Workshop on Kernel Machines, 303-313. Lecture Notes in Artificial Intelligence 2777. Springer, New York, 2003.

[27] Van Der A. Vaart and J. Wellner. Weak Convergence and Empirical Processes. Springer, New York, 1996. | MR 1385671 | Zbl 0862.60002

[28] Y. Yang. Mixing strategies for density estimation. Ann. Statist. 28 (2000) 75-87. | MR 1762904 | Zbl 1106.62322

[29] Y. Yang. Aggregating regression procedures for a better performance. Bernoulli 10 (2004) 25-47. | MR 2044592 | Zbl 1040.62030

[30] P. Zhao and B. Yu. On model selection consistency of LASSO. J. Mach. Learn. Res. 7 (2006) 2541-2563. | MR 2274449