Scaled Sparse Linear Regression

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Biometrika (yyyy), 0 , 0, pp . 1–20 Advance Access publication on dd mmm yyyy C  2008 Biometrika T rust Printed in Gr eat Britain Scaled Sparse Linear Regr ession B Y T I N G N I S U N A N D C U N - H U I Z H A N G Department of Statistics and Biostatistics, Hill Center , Busch Campus, Rutg ers University , Piscataway , New Jer se y 08854, U .S.A. tingni@stat.rutgers.edu czhang@stat.rutgers.edu S U M M A RY Scaled sparse linear regression jointly estimates the re gression coef ficients and noise le vel in a linear model. It chooses an equilibrium with a sparse regression method by iterati vely estimating the noise le v el via the mean residual square and scaling the penalty in proportion to the estimated noise lev el. The iterativ e algorithm costs little beyond the computation of a path or grid of the sparse regression estimator for penalty le vels abov e a proper threshold. For the scaled lasso, the algorithm is a gradient descent in a con ve x minimization of a penalized joint loss function for the regression coefficients and noise le vel. Under mild re gularity conditions, we prov e that the scaled lasso simultaneously yields an estimator for the noise level and an estimated coefficient vector satisfying certain oracle inequalities for prediction, the estimation of the noise le vel and the regression coefficients. These inequalities provide sufficient conditions for the consistency and asymptotic normality of the noise le vel estimator , including certain cases where the number of v ariables is of greater order than the sample size. Parallel results are provided for the least squares estimation after model selection by the scaled lasso. Numerical results demonstrate the superior performance of the proposed methods ov er an earlier proposal of joint conv ex minimization. Some ke y words : Con vex minimization; estimation after model selection; iterati ve algorithm; linear regression; oracle inequality; penalized least squares; scale in variance; v ariance estimation. 1 . I N T RO D U C T I O N This paper concerns the simultaneous estimation of the re gression coefficients and noise le vel in a high-dimensional linear model. High-dimensional data analysis is a topic of great current interest due to the growth of applications where the number of unknowns far exceeds the num- ber of data points. Among statistical models arising from such applications, linear regression is one of the best understood. Penalization, con vex minimization and thresholding methods hav e been proposed, tested with real and simulated data, and prov ed to control errors in prediction, estimation and v ariable selection under v arious sets of re gularity conditions. These methods typ- ically require an appropriate penalty or threshold lev el. A larger penalty lev el may lead to a simple model with large bias, while a smaller penalty lev el may lead to a complex noisy model due to overfitting. Scale-in variance considerations and existing theory suggest that the penalty le vel should be proportional to the noise lev el of the regression model. In the absence of kno wl- edge of the latter level, cross-validation is commonly used to determine the former . Howe ver , cross-v alidation is computationally costly and theoretically poorly understood, especially for the purpose of variable selection and the estimation of regression coefficients. The penalty lev el se- lected by cross-validation is called the prediction-oracle in Meinshausen & B ¨ uhlmann (2006), 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 2 T . S U N A N D C . - H . Z H A N G who gav e an example to sho w that the prediction-oracle solution does not lead to consistent model selection for the lasso. Estimation of the noise le vel in high-dimensional regression is interesting in its own right. Examples include quality control in manufacturing and risk management in finance. Our study is moti v ated by St ¨ adler et al. (2010) and the comments on that paper by Anto- niadis (2010) and Sun & Zhang (2010). St ¨ adler et al. (2010) proposed to estimate the regression coef ficients and noise le vel by maximizing their joint log-likelihood with an ` 1 penalty on the regression coef ficients. Their method has a unique solution due to the joint concavity of the log-likelihood under a certain transformation of the unknown parameters. Sun & Zhang (2010) prov ed that this penalized joint maximum likelihood estimator may result in a positive bias for the estimation of the noise le vel and compared it with two alternativ es. The first is a one-step bias correction of the penalized joint maximum likelihood estimator . The second is an iterati ve algorithm that alternates between estimating the noise lev el via the mean residual square and scaling the penalty le vel in a predetermined proportion to the estimated noise lev el in the lasso or minimax concav e penalized selection paths. In a simulation experiment Sun & Zhang (2010) demonstrated the superiority of the iterativ e algorithm, compared with the penalized joint max- imum likelihood estimator and its bias correction. Howe ver , no theoretical results were giv en for the iterativ e algorithm. Antoniadis (2010) commented on the same problem from a different perspecti ve by raising the possibility of adding an ` 1 penalty to Huber’ s concomitant joint loss function. See, for example, section 7.7 of Huber & Ronchetti (2009). Interestingly , the minimizer of this penalized joint con vex loss is identical to the equilibrium of the iterati ve algorithm for the lasso path. Thus, the con ver gence of the iterati ve algorithm is guaranteed by the con vexity . In this paper , we study Sun & Zhang (2010)’ s iterativ e algorithm for the joint estimation of regression coef ficients and the noise level. For the lasso, this is equiv alent to jointly minimizing Huber’ s concomitant loss function with the ` 1 penalty , as Antoniadis (2010) pointed out. For simplicity , we call the equilibrium of this algorithm the scaled version of the penalized regression method, for e xample the scaled lasso or scaled minimax concav e penalized selection, depending on the choice of penalty function. Under mild regularity conditions, we prov e oracle inequalities for prediction and the joint estimation of the noise le vel and re gression coef ficients for the scaled lasso, that imply the consistency and asymptotic normality of the scaled lasso estimator for the noise lev el. In addition, we prov e parallel oracle inequalities for the least squares estimation of the regression coefficients and noise lev el after model selection by the scaled lasso. W e report numerical results on the performance of scaled lasso and other scaled penalized methods, along with that of the corresponding least squares estimator after model selection. These theoretical and numerical results support the use of the proposed method for high-dimensional regression. W e use the following notation throughout the paper . For a vector v = ( v 1 , . . . , v p ) , | v | q = ( P j | v j | q ) 1 /q denotes the ` q norm with the usual extensions | v | ∞ = max j | v j | and | v | 0 = # { j : v j 6 = 0 } . For design matrices X and subsets A of { 1 , . . . , p } , x j denotes column vectors of X and X A denotes the matrix composed of columns with indices in A . Moreover , x + = max( x, 0) . 2 . A N I T E R AT I V E A L G O R I T H M Suppose we observe a design matrix X = ( x 1 , . . . , x p ) ∈ R n × p and a response vector y ∈ R n . For penalty functions ρ ( · ) , consider penalized loss functions of the form L λ ( β ) = | y − X β | 2 2 2 n + λ 2 p X j =1 ρ ( | β j | /λ ) (1) 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 Scaled Sparse Linear Regr ession 3 where β = ( β 1 , . . . , β p ) 0 is a vector of regression coef ficients. Let the penalty ρ ( t ) be standard- ized to ˙ ρ (0+) = 1 , where ˙ ρ ( t ) = ( d/dt ) ρ ( t ) . A v ector b β = ( b β 1 , . . . , b β p ) 0 is a critical point of the penalized loss (1) if and only if ( x 0 j ( y − X b β ) /n = λ sgn ( b β j ) ˙ ρ ( | b β j | /λ ) , b β j 6 = 0 , x 0 j ( y − X b β ) /n ∈ λ [ − 1 , 1] , b β j = 0 . (2) If the penalized loss (1) is con vex in β , then (2) is the Karush–Kuhn–T ucker condition for its minimization. Gi ven a penalty function ρ ( · ) , one still has to choose a penalty le vel λ to arri ve at a solution of (2). Such a choice may depend on the purpose of estimation, since v ariable selection may require a larger λ than does prediction. Howe ver , scale-inv ariance considerations and theoretical results suggest using a penalty le vel proportional to the noise le vel σ . This motiv ates a scaled penalized least squares estimator as a numerical equilibrium in the follo wing iterati ve algorithm: b σ ← | y − X b β old | 2 / { (1 − a ) n } 1 / 2 , λ ← b σ λ 0 , b β ← b β new , L λ ( b β new ) ≤ L λ ( b β old ) , (3) where λ 0 is a prefixed penalty level, not depending on σ , b σ estimates the noise lev el, and a ≥ 0 provides an option for a degrees-of-freedom adjustment with a > 0 . For p < n and ( a, λ 0 ) = ( p/n, 0) , (3) initialized with the least squares estimator b β (lse) is non-iterati ve and giv es b σ 2 = | y − X b β (lse) | 2 2 / ( n − p ) . For large data sets, one may use a few passes of a gradient descent algorithm to compute b β new from b β old . F or a = 0 , this algorithm was considered in Sun & Zhang (2010). In Sun & Zhang (2010) and the numerical experiments reported in Section 4, b β new is a solution of (2) for the gi ven λ . W e describe this implementation in the follo wing two paragraphs. The first step of our implementation is the computation of a solution path b β ( λ ) of (2) beginning from b β ( λ ) = 0 for λ = | X 0 y /n | ∞ . For quadratic spline penalties ρ ( t ) with m knots, Zhang (2010) de veloped an algorithm to compute a linear spline path of solutions { λ ( t ) ⊕ b β ( t ) : t ≥ 0 } of (2) to cov er the entire range of λ . This extends the least angle re gression solution or lasso path (Osborne et al., 2000a,b; Efron et al., 2004) from m = 1 and includes the minimax conca ve penalty for m = 2 and the smoothly clipped absolute deviation penalty (F an & Li, 2001) for m = 3 . An R package named plus is av ailable for computing the solution paths for these penalties. The second step of our implementation is the iteration (3) along the solution path β ( λ ) com- puted in the first step. That is to use the already computed b β new = b β ( λ ) (4) in (3). For the scaled lasso, we use a = 0 in (3) and ρ ( t ) = t in (1) and (2). For the scaled minimax concav e penalized selection, we use a = 0 and the minimax concave penalty ρ ( t ) = R t 0 (1 − x/γ ) + d x , where γ > 0 regularizes the maximum concavity of the penalty . When γ = ∞ , it becomes the scaled lasso. The algorithm (3) can be easily implemented once a solution path is computed. Consider the ` 1 penalty . As discussed in the introduction, (3) and (4) form an alternating minimization algorithm for the penalized joint loss function L λ 0 ( β , σ ) = | y − X β | 2 2 2 nσ + (1 − a ) σ 2 + λ 0 | β | 1 . (5) 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 4 T . S U N A N D C . - H . Z H A N G Antoniadis (2010) suggested this jointly con vex loss function as a w ay of extending Hu- ber’ s robust regression method to high dimensions. For a = 0 and λ = b σ λ 0 with fixed b σ , b σ L λ 0 ( β , b σ ) = L λ ( β ) + b σ 2 / 2 , so that b β ← b β ( λ ) in (4) minimizes L λ 0 ( β , b σ ) ov er β . For fixed b β , b σ 2 ← | y − X b β | 2 2 / { (1 − a ) n } in (3) minimizes L λ 0 ( b β , σ ) ov er σ . During the revision of this paper , we learned that She & Owen (2011) have considered penalizing Huber’ s concomitant loss function for outlier detection in linear regression. W e summarize some properties of the algorithm (3) with (4) in the follo wing proposition. P RO P O S I T I O N 1 . Let b β = b β ( λ ) be a solution path of (2) with ρ ( t ) = t . The penalized loss function (5) is jointly con vex in ( β , σ ) and the algorithm (3) with (4) con ver ges to ( b β , b σ ) = arg min β ,σ L λ 0 ( β , σ ) . (6) The r esulting estimators b β = b β ( X , y ) and b σ = b σ ( X , y ) ar e scale equivariant in y in the sense that b β ( X , cy ) = c b β ( X , y ) and b σ ( X , cy ) = | c | b σ ( X , y ) . Mor eover , ∂ ∂ σ L λ 0  b β ( σ λ 0 ) , σ  = 1 − a 2 − | y − X b β ( σ λ 0 ) | 2 2 2 nσ 2 . (7) Since (5) is not strictly con vex, the joint estimator may not be unique for some data ( X , y ) . Ho we ver , since (5) is strictly conv ex in σ , b σ is alw ays unique in (6) and the uniqueness of b β follo ws from that of the lasso estimator b β ( λ ) at λ = b σ λ 0 ; b β ( λ ) is unique when the second part of (2) is strict in the sense of not hitting ± λ when b β j = 0 , which holds almost everywhere in ( X , y ) for λ > 0 . See, for example, Zhang (2010). Let b σ ( λ ) = | y − X b β ( λ ) | 2 / { (1 − a ) n } 1 / 2 . For λ 0 = { (2 /n ) log p } 1 / 2 , (7) implies that b σ = b σ ( b λ ) , b λ = min  λ : b σ 2 ( λ ) ≤ nλ 2 / (2 log p )  . (8) While the present paper continues our earlier work (Sun & Zhang, 2010) by providing further theoretical and numerical justifications for (3) and (4), the estimator has appeared in different forms. In addition to (5) and (6) of Antoniadis (2010), (8) appeared in Zhang (2010). While this paper was in revision, a re viewer called our attention to Belloni et al. (2011), who focused on studying b β in an equiv alent form as square-root lasso. W e note that (3) and (4) allow concav e penalties and degrees of freedom adjustments as in Zhang (2010). 3 . T H E O R E T I C A L R E S U LT S 3 · 1 . Analysis of scaled lasso Let β ∗ be a vector of true regression coefficients. An expert with oracular knowledge of β ∗ would estimate the noise le vel by the oracle estimator σ ∗ = | y − X β ∗ | 2 /n 1 / 2 . (9) Under the Gaussian assumption, this is the maximum likelihood estimator for σ when β ∗ is kno wn and n ( σ ∗ /σ ) 2 follo ws the χ 2 n distribution. Due to the scale equiv ariance of b σ in Propo- sition 1, it is natural to use σ ∗ as an estimation target with or without the Gaussian assumption. W e deri ve upper and lower bounds for b σ /σ ∗ − 1 and use them to prov e the consistency and asymptotic normality of b σ . W e deriv e oracle inequalities for the prediction performance and the estimation of β under the ` q loss. Throughout the sequel, pr β ,σ is the probability measure un- der which y − X β ∼ N (0 , σ 2 I n ) . W e assume that | x j | 2 2 = n whene ver pr β ,σ is in voked. The 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 Scaled Sparse Linear Regr ession 5 asymptotic theory here concerns n → ∞ and allows all parameters and v ariables to depend on n , including p ≥ n ≥ | β | 0 → ∞ . W e first provide the consistency for the estimation of σ via an oracle inequality for the pre- diction error of the scaled lasso. In our first theorem, the relativ e error for the estimation of σ is bounded by a quantity τ 0 related to a prediction error bound η ( λ, ξ , w , T ) in (10) below . For λ > 0 , ξ > 1 , w ∈ R p , and T ⊂ { 1 , . . . , p } , define δ w,T = 1 − I ( w = β ∗ , T = ∅ ) and η ( λ, ξ , w , T ) = | X β ∗ − X w | 2 2 /n + (1 + δ w,T )2 λ | w T c | 1 + 4 ξ 2 λ 2 | T | ( ξ + 1) 2 κ 2 ( ξ , T ) (10) where κ ( ξ , T ) , the compatibility factor (van de Geer & B ¨ uhlmann, 2009), is defined as κ ( ξ , T ) = min n | T | 1 / 2 | X u | 2 n 1 / 2 | u T | 1 : u ∈ C ( ξ , T ) , u 6 = 0 o (11) with the cone C ( ξ , T ) = { u : | u T c | 1 ≤ ξ | u T | 1 } . Since the prediction error bound η ( λ, ξ , w , T ) is v alid for all w and T , τ 0 is related to its minimum ov er all w and T at the oracle scale σ ∗ : τ 0 = η 1 / 2 ∗ ( σ ∗ λ 0 , ξ ) /σ ∗ , η ∗ ( λ, ξ ) = inf w,T η ( λ, ξ , w , T ) . (12) T H E O R E M 1 . Let ( b β , b σ ) be the scaled lasso estimator in (6), β ∗ ∈ R p , σ ∗ the oracle noise level in (9), z ∗ = | X 0 ( y − X β ∗ ) /n | ∞ /σ ∗ and ξ > 1 . When z ∗ ≤ (1 − τ 0 ) λ 0 ( ξ − 1) / ( ξ + 1) , max  1 − b σ σ ∗ , 1 − σ ∗ b σ  ≤ τ 0 , | X b β − X β ∗ | 2 n 1 / 2 σ ∗ ≤ 1 σ ∗ η 1 / 2 ∗  σ ∗ λ 0 1 − τ 0 , ξ  ≤ τ 0 1 − τ 0 . (13) In particular , if λ 0 = A { (2 /n ) log p } 1 / 2 with A > ( ξ + 1) / ( ξ − 1) and η ∗ ( σ λ 0 , ξ ) /σ → 0 , then pr β ∗ ,σ  | b σ /σ − 1 | >   → 0 (14) for all  > 0 . Theorem 1 e xtends to the scaled lasso a unification of prediction oracle inequalities for a fixed penalty . W ith λ = σ ∗ λ 0 / (1 − τ 0 ) + , (13) gi ves max { ( σ ∗ τ 0 ) 2 , | X b β − X β ∗ | 2 2 /n } ≤ η ∗ ( λ, ξ ) , or max { ( σ ∗ τ 0 ) 2 , | X b β − X β ∗ | 2 2 /n } ≤ min w n | X w − X β ∗ | 2 2 /n + 4 e C λ p X j =1 min( λ, | w j | ) o (15) for a e C ≥ 1 , if the minimum in (15) is attained at a e w with (1 + 1 /ξ ) 2 κ 2 ( ξ , e T ) ≥ 1 / e C , where e T = { j : | e w j | > λ } . This asserts that for an arbitrary , possibly non-sparse β ∗ , the prediction error of the scaled lasso is no greater than that of the best linear predictor X w with a sparse w for an additional capped- ` 1 cost of the order λ P j min( λ, | w j | ) . A consequence of this prediction error bound for the scaled lasso is the consistency of the corresponding estimator of the noise lev el in (14). Due to the scale equiv ariance in Proposition 1, Theorem 1 and the results in the rest of the section are all scale free. For fixed penalty λ , the upper bound η ( λ, ξ , w, T ) has been previously established for dif- ferent w and T , with possibly dif ferent constant factors. Examples include η ( λ, ξ , β ∗ , ∅ ) = 2 λ | β ∗ | 1 (Greenshtein & Ritov, 2004; Greenshtein, 2006), η ( λ, ξ , β ∗ , S β ∗ ) . λ 2 | β ∗ | 0 with S w = { j : w j 6 = 0 } (van de Geer & B ¨ uhlmann, 2009), and min w η ( λ, ξ , w , S w ) = min w {| X β ∗ − X w | 2 2 /n + O ( λ 2 | w | 0 ) } (K oltchinskii et al., 2011). In (10), the coefficient for | X w − X β ∗ | 2 2 /n is 1 as in K oltchinskii et al. (2011). 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 6 T . S U N A N D C . - H . Z H A N G No w we provide sharper con ver gence rates and the asymptotic normality for the scaled lasso estimation of the noise le vel σ . This sharper rate λµ ( λ, ξ ) /σ 2 , essentially taking the square of the order τ 0 in (13), is based on the follo wing ` 1 error bound for the estimation of β , µ ( λ, ξ ) = ( ξ + 1) min T inf 0 <ν < 1 max h | β ∗ T c | 1 ν , λ | T | / { 2(1 − ν ) } κ 2 { ( ξ + ν ) / (1 − ν ) , T } i . (16) This ` 1 error bound has the interpretation | b β − β ∗ | 1 ≤ µ ( λ, ξ ) ≤ e C p X j =1 min( λ, | β ∗ j | ) , (17) if e C ≥ (1 + ξ ) max { 2 , 1 /κ 2 (2 ξ + 1 , e T ) } with e T = { j : | β ∗ j | > λ } . This allows β ∗ to have many small elements, as in Zhang & Huang (2008), Zhang (2009) and Y e & Zhang (2010). The bound µ ( λ, ξ ) ≤ ( ξ + 1) λ | S β ∗ | / { 2 κ 2 ( ξ , S β ∗ ) } improv es upon its earlier version in v an de Geer & B ¨ uhlmann (2009) by a constant factor 4 ξ / ( ξ + 1) ∈ (2 , 4) . T H E O R E M 2 . Let b β , b σ , β ∗ , σ ∗ , z ∗ and ξ be as in Theorem 1. Set τ ∗ = { λ 0 µ ( σ ∗ λ 0 , ξ ) /σ ∗ } 1 / 2 . (i) The following inequalities hold when z ∗ ≤ (1 − τ 2 ∗ ) λ 0 ( ξ − 1) / ( ξ + 1) , max  1 − b σ /σ ∗ , 1 − σ ∗ / b σ ) ≤ τ 2 ∗ , | b β − β ∗ | 1 ≤ µ ( σ ∗ λ 0 , ξ ) / (1 − τ 2 ∗ ) . (18) (ii) Let λ 0 ≥ { (2 /n ) log ( p/ ) } 1 / 2 ( ξ + 1) / { ( ξ − 1)(1 − τ 2 ∗ ) } . F or all  > 0 and n − 2 > log( p/ ) → ∞ , pr β ∗ ,σ  z ∗ ≤ (1 − τ 2 ∗ ) λ 0 ( ξ − 1) / ( ξ + 1)  ≥ 1 − { 1 + o (1) } / { π log ( p/ ) } 1 / 2 . If λ 0 = A { (2 /n ) log p } 1 / 2 with A > ( ξ + 1) / ( ξ − 1) and λ 0 µ ( σ λ 0 , ξ ) /σ  n − 1 / 2 , then n 1 / 2  b σ /σ − 1  → N (0 , 1 / 2) (19) in distribution under pr β ∗ ,σ . Since σ 2 τ 2 ∗ ≈ µ ( λ, ξ ) ≤ 2( ξ + 1) min T η ( λ, 2 ξ + 1 , β ∗ , T ) with λ = σ λ 0 , the rate τ 2 ∗ in (18) is essentially the square of that in (13), in vie w of (12). It follows that the scaled lasso pro vides a faster con ver gence rate than does the penalized maximum likelihood estimator for the estimation of the noise le vel (St ¨ adler et al., 2010; Sun & Zhang, 2010). In particular , (18) implies that max  1 − b σ /σ ∗ , 1 − σ ∗ / b σ ) ≤ ( ξ + 1) λ 2 0 | S β ∗ | / { 2 κ 2 ( ξ , S β ∗ ) } . | β ∗ | 0 (log p ) /n (20) with S β ∗ = { j : β ∗ j 6 = 0 } , when κ 2 ( ξ , S β ∗ ) can be treated as a constant. The bounds in (20) and its general version (18) lead to the asymptotic normality (19) under proper assumptions. Thus, statistical inference about σ is justified with the scaled lasso in certain large- p -smaller- n cases, for example, when | β ∗ | 0 (log p ) / √ n → 0 under the compatibility condition (v an de Geer & B ¨ uhlmann, 2009). For a fix ed penalty level, oracle inequalities for the ` q error of the lasso hav e been established in Bunea et al. (2007), van de Geer (2008) and van de Geer & B ¨ uhlmann (2009) for q = 1 , Zhang & Huang (2008) and Bick el et al. (2009) for q ∈ [1 , 2] , Meinshausen & Y u (2009) for q = 2 , and Zhang (2009) and Y e & Zhang (2010) for q ≥ 1 . The bounds on b σ/σ ∗ in (18) and (20) allow automatic extensions of these existing ` q oracle inequalities from the lasso with fixed penalty to the scaled lasso. W e illustrate this by extending the oracle inequalities of Y e & Zhang (2010) for the lasso and Candes & T ao (2007) for the Dantzig selector in the following corollary . Y e & 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 Scaled Sparse Linear Regr ession 7 Zhang (2010) used the following sign-restricted cone in vertibility factor to separate conditions on the error y − X β ∗ and design X in the deri v ation of error bounds for the lasso: F q ( ξ , S ) = inf n | S | 1 /q | X 0 X u | ∞ n | u | q : u ∈ C − ( ξ , S ) o , (21) where C − ( ξ , S ) = { u : | u S c | 1 ≤ ξ | u S | 1 6 = 0 , u j x 0 j X u ≤ 0 , for all j 6∈ S } . The quantity (21) can be vie wed as a generalized restricted eigen value comparing the ` q loss and the dual norm of the ` 1 penalty with respect to the inner product for the least squares fit. This giv es a di- rect connection to the Karush–Kuhn–T ucker condition (2). Compared with the restricted eigen- v alue (Bickel et al., 2009) and the compatibility factor (11), a main advantage of (21) is to allo w all q ∈ [1 , ∞ ] . In addition, (21) yields sharper oracle inequalities (Y e & Zhang, 2010). For ( | A | , | B | , | u | 2 ) = ( d a e , d b e , 1) with A ∩ B = ∅ , define δ ± a = max A,u n ±  | X A u/n 1 / 2 | 2 − 1 o , θ a,b = max A,B ,u   X 0 A X B u/n   2 . (22) The quantities in (22) are used in the uniform uncertainty principle (Candes & T ao, 2007) and the sparse Riesz condition (Zhang & Huang, 2008). W e note that 1 − δ − a is the minimum eigen value of X 0 A X A /n among | A | ≤ a , 1 + δ + a is the corresponding maximum eigen value, and θ a,b is the maximum operator norm of size a × b of f-diagonal sub-blocks of the Gram matrix X 0 X/n . C O RO L L A RY 1 . Suppose | β ∗ S c | 1 = 0 . Then, Theor em 2 holds with µ ( λ, ξ ) r eplaced by λ | S | (2 ξ ) / { ( ξ + 1) F 1 ( ξ , S ) } , and for z ∗ ≤ (1 − τ 2 ∗ ) λ 0 ( ξ − 1) / ( ξ + 1) , | b β − β ∗ | q ≤ k 1 /q ( σ ∗ z ∗ + b σ λ 0 ) F q ( ξ , S ) ≤ 2 σ ∗ ξ λ 0 k 1 /q (1 − τ 2 ∗ )( ξ + 1) F q ( ξ , S ) (23) for all 1 ≤ q ≤ ∞ , where k = | S | . In particular , for ξ = √ 2 and z ∗ ≤ (1 − τ 2 ∗ ) λ 0 ( √ 2 − 1) 2 , | b β − β ∗ | 2 ≤ (8 k ) 1 / 2 λ 0 σ ∗ / (1 − τ 2 ∗ ) ( √ 2 + 1) F 2 ( √ 2 , S ) ≤ 4 k 1 / 2 λ 0 σ ∗ / (1 − τ 2 ∗ ) ( √ 2 + 1)(1 − δ − 1 . 5 k − θ 2 k, 1 . 5 k ) + . (24) The proofs of Theorems 1 and 2 are based on a basic inequality | X b β ( λ ) − X β ∗ | 2 2 /n + | X b β ( λ ) − X w | 2 2 /n ≤ | X w − X β ∗ | 2 2 /n + 2 λ {| w | 1 − | b β ( λ ) | 1 } + 2 σ ∗ z ∗ | w − b β ( λ ) | 1 (25) as a consequence of the Karush–Kuhn–T ucker conditions (2). The version of (25) with w = β ∗ is well-known (v an de Geer & B ¨ uhlmann, 2009) and controls | X b β ( λ ) − X β ∗ | 2 2 for sparse β ∗ . When | X b β ( λ ) − X β ∗ | 2 2 > | X w − X β ∗ | 2 2 , (25) controls the excess for sparse w by the same argument. The general w is taken in Theorem 1, while w = β ∗ is taken in Theorem 2. In both cases, (25) provides the cone condition in (11) and (21). This is used to derive upper and lower bounds for (7), the deriv ativ e of the profile loss function L λ 0 ( b β ( σ λ 0 ) , σ ) with respect to σ , within a neighborhood of σ /σ ∗ = 1 . The bounds for the minimizer b σ then follow from the joint con vexity of the penalized loss (5). 3 · 2 . Estimation after model selection W e hav e proved that without requiring the knowledge of σ , the scaled lasso enjoys prediction and estimation properties comparable to the best kno wn theoretical results for known σ , and the scaled lasso estimate of σ enjoys consistency and asymptotic normality properties under proper conditions. Howe ver , the lasso estimator may have substantial bias (Fan & Peng, 2004; Zhang, 2010), and its bias is significant in our own simulation experiments. Although the smoothly 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 8 T . S U N A N D C . - H . Z H A N G clipped absolute deviation and minimax concave penalized selectors were introduced to remove the bias of the lasso (Fan & Li, 2001; Zhang, 2010), a theoretical study of their scaled version (3) is beyond the scope of this paper . In this subsection, we present theoretical results for another bias removing method: least squares estimation after model selection. Gi ven an estimator b β of the coefficient vector β , the least squares estimator of β and the corresponding estimator of the noise le vel σ in the model selected by b β are β = arg min β n | y − X β | 2 2 : supp ( β ) ⊆ supp ( b β ) o , σ =   y − X β   2  √ n, (26) where supp ( β ) = { j : β j 6 = 0 } . Alternati vely , we may use σ =   y − X β   2  √ ( n − | b β | 0 ) to esti- mate the noise le vel. Howe ver , since the ef fect of this degrees of freedom adjustment is of smaller order than our error bound, we will focus on the simpler (26). In addition to the compatibility factor κ ( ξ , S ) in (11), we use sparse eigen values to study the least squares estimation after the scaled lasso selection. Let λ min ( M ) be the smallest eigen value of a matrix M and λ max ( M ) the largest . For models T ⊂ { 1 , . . . , p } , define κ − ( m, T ) = min B ⊃ T , | B \ T |≤ m λ min ( X 0 B X B /n ) , κ + ( m, T ) = min B ∩ T ∅ , | B |≤ m λ min ( X 0 B X B /n ) , as the sparse lower eigen value of the Gram matrix for models containing T and the sparse up- per eigenv alue for models disjoint with T . Let S = supp ( β ∗ ) and b S = supp ( b β ) . The following theorem pro vides prediction and estimation error bounds for (26) after the scaled lasso selection, along with an upper bound for the false positi ve | b S \ S | , a key element in our study . T H E O R E M 3 . Let ( b β , b σ ) be the scaled lasso estimator in (6) and ( β , σ ) the least squar es estimator (26) in model b S . Let β ∗ , σ ∗ , z ∗ , ξ and τ ∗ be as in Theor em 2 and m be an inte ger satisfying | S | ξ 2 /κ 2 ( ξ , S ) < m/κ + ( m, S ) . If z ∗ ≤ (1 − τ 2 ∗ ) λ 0 ( ξ − 1) / ( ξ + 1) , then | b S \ S | < m, b σ 2 − { σ ∗ m − 1 ,S + √ η ∗ ( b λ, ξ ) } 2 ≤ σ 2 ≤ b σ 2 , (27) with b λ = b σ λ 0 ≤ σ ∗ λ 0 / (1 − τ 2 ∗ ) and σ ∗ m,S = max B ⊇ S, | B \ S | = m | ( y − X β ∗ ) B | 2 / √ n , and κ − ( m − 1 , S ) | β − β ∗ | 2 2 ≤ | X β − X β ∗ | 2 2 /n ≤  σ ∗ m − 1 ,S + 2 √ η ∗ ( b λ, ξ )  2 . (28) Mor eover , in addition to the pr obability bound for z ∗ ≤ (1 − τ 2 ∗ ) λ 0 ( ξ − 1) / ( ξ + 1) in Theor em 2 (ii), for all inte gers 1 ≤ m ≤ p , pr β ∗ ,σ  σ ∗ m,S ( √ n ) /σ ≥ √ ( m + | S | ) + √ { (2 m ) log ( ep/m ) + 2 log (1 / ) }  ≤ /m √ (2 π ) . (29) For Gaussian design matrices, the sparse eigen values κ − ( m, ∅ ) and κ + ( m, ∅ ) can be treated as constants when m (log p ) /n is small and the eigen values of the expected Gram matrix are uniformly bounded aw ay from zero and infinity (Zhang & Huang, 2008). Since κ − ( m, S ) ≥ κ − ( m + | S | , ∅ ) and κ + ( m, S ) ≤ κ + ( m, ∅ ) , the y can be treated as constants in the same sense in Theorem 3. Thus, for suf ficiently small | S | (log p ) /n , we may take an m of the same order as | S | . In this case, the difference between ( β , σ ) and the scaled lasso estimator ( b β , b σ ) is of no greater order than the dif ference between ( b β , b σ ) and the estimation target ( β ∗ , σ ∗ ) . Consequently ,   σ /σ − 1   +   b β − β ∗   2 2 +   X β − X β ∗   2 2 /n = O P (1) | S | (log p ) /n. As we have mentioned earlier, the key element in our analysis of (26) is the bound | b S \ S | < m in (27). Since this is a weaker assertion than v ariable consistency b S = S , the conditions of 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 Scaled Sparse Linear Regr ession 9 T able 1: Performance of fiv e methods in Example 1 at penalty levels λ j = { 2 j − 1 (log p ) /n } 1 / 2 ( j = 1 , 2 , 3) , across 100 replications, in terms of the bias ( × 10 ) and standard error ( × 10 ) of b σ /σ for the selector and σ /σ for the least squares estimator after model selection, the average model size, and the relativ e frequency of sure screening, along with the simulation results in Fan et al. (2012) r 0 = 0 r 0 = 0 · 5 Method b σ /σ σ /σ AMS SSP b σ /σ σ /σ AMS SSP λ 1 1 · 6 ± 0 · 6 − 1 · 1 ± 0 · 7 7 · 6 1 · 0 1 · 5 ± 0 · 6 − 1 · 0 ± 0 · 7 9 · 7 1 · 0 PMLE λ 2 2 · 5 ± 0 · 6 − 0 · 1 ± 0 · 6 3 · 0 1 · 0 2 · 5 ± 0 · 6 − 0 · 3 ± 0 · 6 5 · 2 1 · 0 λ 3 3 · 6 ± 0 · 7 1 · 2 ± 1 · 1 1 · 8 0 · 2 3 · 8 ± 0 · 6 − 0 · 2 ± 0 · 6 3 · 7 1 · 0 λ 1 0 · 5 ± 0 · 6 − 1 · 8 ± 0 · 7 11 · 9 1 · 0 0 · 0 ± 0 · 6 − 1 · 9 ± 0 · 7 15 · 5 1 · 0 BC λ 2 1 · 6 ± 0 · 6 − 0 · 1 ± 0 · 6 3 · 1 1 · 0 0 · 7 ± 0 · 6 − 0 · 3 ± 0 · 6 6 · 1 1 · 0 λ 3 3 · 3 ± 0 · 7 1 · 1 ± 1 · 1 1 · 9 0 · 3 1 · 9 ± 0 · 7 − 0 · 2 ± 0 · 6 4 · 2 1 · 0 Scaled λ 1 0 · 0 ± 0 · 6 − 2 · 1 ± 0 · 8 14 · 6 1 · 0 − 0 · 5 ± 0 · 6 − 2 · 3 ± 0 · 7 18 · 6 1 · 0 lasso λ 2 1 · 3 ± 0 · 7 − 0 · 2 ± 0 · 6 3 · 1 1 · 0 0 · 4 ± 0 · 6 − 0 · 3 ± 0 · 6 6 · 2 1 · 0 λ 3 3 · 1 ± 0 · 7 1 · 0 ± 1 · 1 1 · 9 0 · 3 1 · 2 ± 0 · 7 − 0 · 2 ± 0 · 6 4 · 4 1 · 0 Scaled λ 1 − 1 · 2 ± 0 · 8 − 2 · 4 ± 0 · 8 14 · 1 1 · 0 − 0 · 7 ± 0 · 6 − 2 · 2 ± 0 · 8 13 · 8 1 · 0 MCP λ 2 − 0 · 1 ± 0 · 6 − 0 · 1 ± 0 · 6 3 · 1 1 · 0 0 · 1 ± 0 · 6 − 0 · 2 ± 0 · 6 3 · 2 1 · 0 λ 3 1 · 5 ± 1 · 3 0 · 6 ± 1 · 1 2 · 4 0 · 6 0 · 6 ± 0 · 7 − 0 · 1 ± 0 · 6 3 · 0 1 · 0 Scaled λ 1 − 0 · 6 ± 0 · 6 − 2 · 2 ± 0 · 7 14 · 0 1 · 0 − 0 · 4 ± 0 · 6 − 2 · 2 ± 0 · 8 13 · 9 1 · 0 SCAD λ 2 0 · 8 ± 1 · 0 − 0 · 1 ± 0 · 6 3 · 1 1 · 0 0 · 4 ± 0 · 6 − 0 · 3 ± 0 · 6 3 · 8 1 · 0 λ 3 3 · 1 ± 0 · 7 0 · 9 ± 1 · 1 2 · 0 0 · 3 1 · 2 ± 0 · 7 − 0 · 2 ± 0 · 6 3 · 8 1 · 0 N-LASSO − 5 · 3 ± 2 · 0 36 · 6 1 · 0 − 4 · 6 ± 2 · 0 29 · 6 1 · 0 RCV -SIS 0 · 2 ± 1 · 4 50 · 0 0 · 9 − 0 · 1 ± 1 · 4 50 · 0 1 · 0 RCV -ISIS 0 · 5 ± 1 · 7 30 · 9 0 · 7 0 · 2 ± 1 · 2 29 · 0 0 · 8 RCV -LASSO 0 ± 1 · 3 31 · 1 0 · 9 − 0 · 3 ± 1 · 1 26 · 5 1 · 0 P-SCAD − 1 · 4 ± 1 · 1 30 · 0 1 · 0 − 1 · 2 ± 1 · 7 29 · 9 1 · 0 CV -SCAD 0 · 7 ± 1 · 2 30 · 0 1 · 0 0 · 9 ± 1 · 3 29 · 9 1 · 0 P-LASSO − 0 · 8 ± 2 · 1 36 · 5 1 · 0 − 0 · 9 ± 1 · 5 29 · 6 1 · 0 CV -LASSO 1 · 4 ± 1 · 1 36 · 5 1 · 0 0 · 8 ± 1 · 0 29 · 6 1 · 0 PMLE, ` 1 penalized maximum likelihood estimator; BC, bias-corrected estimator; MCP , minimax concave penalty; SCAD, smoothly clipped absolute deviation penalty; N, naiv e; RCV , refitted cross-validation; SIS, sure independent screening; ISIS, iterative SIS; P , plug-in method with de grees-of-freedom correction; CV , cross- validation; AMS, a verage model size; SSP , relative frequenc y of sure screening Theorem 3 on the design matrix is of a weaker form than the irrepresentability condition for v ariable selection consistency (Meinshausen & B ¨ uhlmann, 2006; Zhao & Y u, 2006). In Zhang & Huang (2008) and Zhang (2010), upper bounds for the false positive were obtained under a sparse Riesz condition on κ − ( m, ∅ ) and κ + ( m, ∅ ) . 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 10 T . S U N A N D C . - H . Z H A N G 4 . N U M E R I C A L R E S U LT S 4 · 1 . Simulation study In this section, we present some simulation results to compare fiv e methods: the scaled pe- nalized methods with the ` 1 penalty , the minimax conca ve penalty and the smoothly clipped absolute de viation penalty , the ` 1 penalized maximum likelihood estimator (St ¨ adler et al., 2010), and its bias correction (Sun & Zhang, 2010). The least squares estimator after model selection by these fi ve methods is also studied. The penalized maximum lik elihood estimator is  b β (pmle) , b σ (pmle)  = arg max β ,σ  − | y − X β | 2 2 2 σ 2 n − log σ − λ 0 | β | 1 σ  , or equiv alently the limit of the iteration b σ ← { y 0 ( y − X b β ) /n } 1 / 2 and b β ← b β ( b σ λ 0 ) . The bias- corrected estimator is one iteration of (3) with (4) from ( b β (pmle) , b σ (pmle) ) with a = 0 , b σ (bc) = | y − b β ( b σ (pmle) λ 0 ) | 2 /n 1 / 2 , b β (bc) = b β ( b σ (bc) λ 0 ) . T wo simulation e xamples are considered. Example 1. W e compare the fiv e estimators at three penalty lev els λ j = √ { 2 j − 1 (log p ) /n } , j = 1 , 2 , 3 . The e xperiment has the setting of Example 2 in Fan et al. (2012), with the small- est signal, b = 1 / √ 3 . W e provide their description of the simulation setting in our notation as follo ws: X has independent and identically distributed Gaussian rows with marginal distri- bution N (0 , 1) , corr( x i , x j ) = r 0 for 1 ≤ i < j ≤ 50 and corr( x i , x j ) = 0 otherwise, ( n, p ) = (200 , 2000) , nonzero coefficients β j = 1 / √ 3 for j ∈ S = { 1 , 2 , 3 } , and y − X β ∼ N (0 , σ 2 I ) with σ = 1 . T wo configurations are considered: independent columns x j with r 0 = 0 and cor- related first 50 columns x j with r 0 = 0 . 5 . W e set γ = 2 / (1 − max | x 0 k x j | /n ) for the conca ve penalties. The top section of T able 1 presents our simulation results, while the bottom section includes the simulation results of Fan et al. (2012) for sev eral joint estimators of ( β , σ ) using cross- v alidation, without repeating their experiment. In addition to the bias and the standard error of the ratios b σ /σ for the fi ve original estimators and σ /σ for the least squares estimation after model selection, we report the av erage model size | b S | and the relativ e frequency of sure screening, b S ⊇ S , as in Fan et al. (2012), where b S = supp ( b β ) is the selected model. W ithout post processing, the scaled minimax concave penalized selector with the univ ersal penalty lev el λ 2 = √ { (2 /n ) log p } clearly outperforms other procedures in this example. How- e ver , the results of the least squares estimation after model selection at penalty level λ 2 are nearly identical to the top performer for all fi ve methods. In view of the results in av erage model size and sure screening proportion, the success of post processing at λ 2 is clearly due to the success of model selection. The fiv e methods select too fe w variables at the larger penalty lev el λ 3 and too man y at the smaller λ 1 , both leading to substantial bias in the estimation of σ for r 0 = 0 . F or r 0 = 0 . 5 , selecting a slightly smaller model does not harm so much since a substantial portion of the effect of the missing v ariables is explained by the selected variables correlated to them. The minimax concave penalized selector is nearly unbiased in this example, so that it does not need post processing. Cross-validation methods select about 30 v ariables when the true model size is 3. This o ver selection is probably the reason for the lar ge bias for most cross-v alidation methods and large standard error for all of them. Example 2. This experiment has the same setting as in the simulation study in Sun & Zhang (2010), where the scaled lasso and the scaled minimax conca ve penalized selection are called the 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 Scaled Sparse Linear Regr ession 11 T able 2: Performance of fiv e methods in Example 2 at penalty levels λ j = { 2 j − 1 (log p ) /n } 1 / 2 ( j = 1 , 2 , 3) , across 100 replications, in terms of the bias ( × 10 ) and standard error ( × 10 ) of b σ /σ and σ /σ , the false positiv e, and the false ne gati ve r 0 = 0 . 1 r 0 = 0 . 9 Method b σ /σ σ /σ FP FN b σ /σ σ /σ FP FN λ 1 5.5 ± 0.3 0.2 ± 0.3 3 11 2.4 ± 0.3 − 0.4 ± 0.3 4 12 PMLE λ 2 7.7 ± 0.4 2.1 ± 0.6 0 19 3.7 ± 0.3 − 0.3 ± 0.3 1 15 λ 3 9.5 ± 0.4 6.3 ± 1.1 0 30 5.7 ± 0.3 − 0.1 ± 0.3 0 20 λ 1 3.2 ± 0.3 − 0.3 ± 0.4 8 9 0.3 ± 0.3 − 0.7 ± 0.3 9 11 BC λ 2 6.1 ± 0.5 1.6 ± 0.5 0 17 1.2 ± 0.3 − 0.3 ± 0.3 2 13 λ 3 9.1 ± 0.5 6.1 ± 1.1 0 29 3.1 ± 0.4 − 0.1 ± 0.3 0 18 Scaled λ 1 1.9 ± 0.4 − 0.8 ± 0.4 14 7 0.0 ± 0.3 − 0.8 ± 0.3 11 11 lasso λ 2 5.0 ± 0.5 1.3 ± 0.5 0 16 0.6 ± 0.3 − 0.3 ± 0.3 2 13 λ 3 8.9 ± 0.6 6.0 ± 1.2 0 29 1.8 ± 0.4 − 0.2 ± 0.3 0 16 Scaled λ 1 − 0.2 ± 0.4 − 1.1 ± 0.4 14 7 0.0 ± 0.3 − 0.7 ± 0.3 10 19 MCP λ 2 1.8 ± 0.5 0.7 ± 0.5 0 13 0.6 ± 0.3 − 0.2 ± 0.3 1 20 λ 3 7.8 ± 1.0 5.6 ± 1.3 0 28 1.7 ± 0.4 0.0 ± 0.3 0 22 Scaled λ 1 0.6 ± 0.4 − 1.0 ± 0.4 14 7 0.0 ± 0.3 − 0.8 ± 0.3 10 15 SCAD λ 2 4.7 ± 0.6 1.3 ± 0.5 0 16 0.6 ± 0.3 − 0.3 ± 0.3 2 15 λ 3 8.9 ± 0.6 6.0 ± 1.2 0 29 1.8 ± 0.4 − 0.2 ± 0.3 0 17 PMLE, ` 1 penalized maximum likelihood estimator; BC, bias-corrected estimator; MCP , min- imax concav e penalty; SCAD, smoothly clipped absolute deviation penalty; FP , false positiv e; FN, false neg ativ e nai ve estimators. W e provide the description of the simulation settings in Sun & Zhang (2010) in our notation as follows: ( n, p ) = (600 , 3000) , the x j are normalized columns from a Gaussian random matrix with independent and identically distrib uted rows and correlation r 0 | k − j | between the j -th and k -th entries within each row , γ = 2 / (1 − max | x 0 k x j | /n ) for the minimax concav e penalty and smoothly clipped absolute deviation penalty , the nonzero β ∗ j are composed of fiv e blocks of β ∗ (1 , 2 , 3 , 4 , 3 , 2 , 1) 0 centered at random multiples j 1 , . . . , j 5 of 25, β ∗ sets | X β ∗ | 2 2 = 3 n , and y − X β ∗ is a vector of independent and identically distributed N (0 , 1) v ariables. Thus, the true noise le vel is σ = 1 . W e set r 0 = 0 . 1 for low correlation between design v ectors and r 0 = 0 . 9 for high correlation. W e summarize the simulation results in T able 2, which provides the bias and standard er- ror of the ratios b σ /σ for the selector and σ /σ for the least squares estimator after model se- lection, the false positive | b S \ S | , and the false negati ve | S \ b S | . Without post processing, the scaled lasso outperforms the penalized maximum likelihood estimator and its bias correction, which are also based on the lasso path. Howe ver , the scaled lasso estimate of σ is still biased, and the lev el of bias is comparable with the order of the error bound ( | S | /n ) log p = 0 . 47 in (20). This and the failure in sure screening by any method reflect the difficulty of this example, where | S | = 35 is not small and the signal is weak, with average β ∗ = 0 . 11 and 0 . 05 respec- ti vely for r 0 = 0 . 1 and r 0 = 0 . 9 . From this perspectiv e, the scaled minimax concave penalized 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 12 T . S U N A N D C . - H . Z H A N G T able 3: Estimated coef ficients ( × 10 3 ) of selected probe sets by four methods in the real data example: the lasso with cross-v alidation, the lasso with adjusted cross-v alidation, and the scaled lasso and minimax concav e penalized selection at λ 0 = λ 2 = { 2(log p ) /n } 1 / 2 Probe ID C-V lasso C-V lasso/LSE Scaled lasso Scaled MCP #cov 200 3000 200 3000 200 3000 200 3000 1369353 at − 9 · 12 − 7 · 13* − 7 · 09 − 2 · 79* − 7 · 3 − 4 · 03* 1370052 at M 3 · 65 1370429 at − 3 · 22 − 8 · 94* − 11 · 06 − 8 · 78* − 9 · 36 − 16 · 37* 1371242 at − 6 · 66 1374106 at 8 · 88* 10 · 58* 7 · 33* 6 · 14* 7 · 45* 7 · 01* 8 · 47* 10 · 02* 1374131 at 4 · 07 0 · 80 1375585 at M 0 · 58 1384204 at 0 · 70 0 · 70 1387060 at M 3 · 50* 1388538 at M 1 · 42 1389584 at 17 · 16* 25 · 39* 20 · 07* 19 · 61* 19 · 97* 21 · 18* 45 · 75* 50 · 49* 1393979 at − 1 · 81 − 0 · 22 − 0 · 4 1379079 at M − 1 · 43* 1379495 at 4 · 84 1 · 73 1 · 71 1 · 00 1379971 at 13 · 56* 13 · 1 11 · 19* 8 · 81 11 · 25* 9 · 52 1380033 at 8 · 69 2 · 76 2 · 97 6 · 75* 1380070 at M 0 · 19 1381787 at − 2 · 05 − 2 · 01 − 2 · 11 1382452 at M 12 · 93 1 · 63 12 · 91* 1382835 at 12 · 64 5 · 79 3 · 73 4 · 15 1383110 at 9 · 03* 19 · 99 15 · 10* 16 · 43 14 · 97* 16 · 69 15 · 80* 23 · 01* 1383522 at 3 · 03* * * 1383673 at 5 · 54 6 · 12* 6 · 07 6 · 15* 6 · 08 6 · 47* 1383749 at − 13 · 86 − 10 · 85* − 10 · 84 − 6 · 7* − 11 · 02 − 8 · 07* − 2 · 74 * − 1 · 11* 1383996 at 25 · 01* 17 · 82* 18 · 61* 14 · 30* 18 · 88* 15 · 52* 25 · 07* 19 · 19* 1385687 at M − 0 · 99 1386683 at 4 · 60* 2 · 90* 1390788 a at 0 · 92 1392692 at M 1 · 74 1393382 at 2 · 43 1393684 at 1 · 59 1395076 at M 0 · 23 1397489 at M 3 · 33 Model size 19 20 15 10 15 14 7 6 b λ = b σ λ 0 0.0103 0.0163 0.025 0.035 0.0243 0.0315 0.0244 0.0304 C-V lasso/LSE, the lasso estimator with the adjusted cross-validation; MCP , minimax concave penalty; #cov , the number of cov ariates considered; M , probes not in the smaller set of 200 probes; * , covariates selected by stability selection selection, designed to reduce the bias of the lasso, performs quite well at the uni versal penalty le vel λ 2 = √ { (2 /n ) log p } , especially with post processing. The least squares estimation af- ter model selection reduces the bias substantially in all cases, e ven without successful model selection. This example seems to suggest the possibility of improving the performance of the scaled estimators at a penalty lev el λ smaller than the universal penalty le vel λ 2 , a simple upper bound for | X 0 ( y − X β ) / ( σ ∗ n ) | ∞ under pr β ,σ . Ho we ver , consistent v ariable selection requires 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 Scaled Sparse Linear Regr ession 13 0.00 0.02 0.04 0.06 0.08 0.10 0.000 0.005 0.010 0.015 0.020 200 covariates being considered lambda mean squared error 0.00 0.02 0.04 0.06 0.08 0.10 0.000 0.005 0.010 0.015 0.020 3000 covariates being considered lambda mean squared error Fig. 1: Mean squared prediction error against penalty le vel: solid curv e, testing error of the lasso for fixed λ ; dotted curve, training error of the lasso for fixed λ ; dot-dashed line, testing error of the scaled lasso with fixed λ 0 = √ { (2 /n ) log p } , or equiv alently the lasso at penalty lev el b σ λ 0 . λ ≥ | X 0 ( y − X β ∗ ) / ( σ ∗ n ) | ∞ as Example 1 demonstrates. Since the scaled lasso estimator b σ is an increasing function of the penalty lev el by Proposition 1, it is always possible to reduce the bias of b σ to zero by taking a specific λ for each specific example. Ho wev er, the two examples in our simulation experiment demonstrate the dif ficulty of picking such a penalty le vel consistently . 4 · 2 . Real data example W e study a data set containing 18976 probes for 120 rats, which is reported in Scheetz et al. (2006). Our goal is to find probes that are related to that of gene TRIM32, which has been found to cause Bardet–Biedl syndrome, a genetically heterogeneous disease of multiple org an systems including the retina. W e consider linear re gression with the probe from TRIM32, 1389163 at, as the response variable. As in Huang et al. (2008), we focus on 3000 probes with the largest v ari- ances among the 18975 covariates and consider two approaches. The first approach is to regress on these p = 3000 probes. The second approach is to regress on the 200 probes among the 3000 with the largest marginal correlation coefficients with TRIM32. For the cross-validation lasso, we randomly partition the data 1000 times, each with a training set of size 80 and a v alidation set of size 40. For each partition, the penalty level λ is selected by minimizing the prediction mean squared error in the validation set. Then we compute the lasso estimator with all 120 observa- tions at the penalty le vel equal to the median of the selected penalty le vels with the 1000 random partitions. Since cross-validation tends to choose a larger model, we also consider an adjusted version using the cross-v alidated error of the least squares estimator after the lasso selection. For the minimax conca ve penalty , we set γ = 2 / (1 − σ 0 . 95 ) = 6 . 37 , where σ 0 . 95 is the 95% quantile of | x 0 k x j | /n . T able 3 shows the probe sets identified by four methods: the cross-validation lasso, its adjusted version, the scaled lasso at at uni versal penalty le vel λ 2 = { 2(log p ) /n } 1 / 2 , and the minimax concav e penalized selection at the same penalty level. W e apply stability selection (Meinshausen & Buhlmann, 2010) to check the reliability of selection. Let W 1 , . . . , W p be independent vari- 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 14 T . S U N A N D C . - H . Z H A N G T able 4: Prediction performance of eight methods in the real data e xam- ple at penalty levels λ 0 = λ j = { 2 j − 1 (log p ) /n } 1 / 2 ( j = 1 , 2 , 3) , in terms of the prediction mean squared error ( × 10 2 ), estimated model size, and correlation coef ficient ( × 10 2 ) between fitted and observed responses #cov = 200 #cov = 3000 Method P-MSE | b β | 0 corr P-MSE | b β | 0 corr PMLE λ 1 0 · 94 12 67 · 1 0 · 97 12 63 · 5 λ 2 0 · 97 9 63 · 5 1 · 04 7 59 · 8 λ 3 1 · 09 6 57 · 6 1 · 23 3 52 · 2 BC λ 1 0 · 93 13 68 · 2 0 · 96 15 64 · 6 λ 2 0 · 95 10 64 · 7 1 · 01 9 60 · 9 λ 3 1 · 04 7 59 · 4 1 · 17 4 53 · 1 Scaled lasso λ 1 0 · 93 13 68 · 4 0 · 96 17 64 · 3 λ 2 0 · 94 10 65 · 2 0 · 98 10 61 · 7 λ 3 1 · 02 7 60 · 8 1 · 13 5 53 · 9 Scaled MCP λ 1 1 · 03 6 66 · 4 1 · 08 8 62 · 3 λ 2 1 · 03 5 63 · 4 1 · 06 5 60 · 0 λ 3 1 · 12 3 59 · 1 1 · 18 2 54 · 9 Scaled SCAD λ 1 1 · 00 11 68 · 9 1 · 01 14 65 · 1 λ 2 0 · 95 10 68 · 8 0 · 98 10 65 · 9 λ 3 1 · 01 8 65 · 0 1 · 09 5 59 · 7 C-V lasso 0 · 94 15 69 · 0 0 · 99 25 63 · 8 C-V lasso/LSE1 0 · 97 11 64 · 8 0 · 98 12 62 · 5 C-V lasso/LSE2 0 · 97 11 66 · 8 1 · 09 12 62 · 6 PMLE, ` 1 penalized maximum likelihood estimator; BC, bias-corrected PMLE; MCP , minimax concave penalty; SCAD, smoothly clipped absolute devia- tion penalty; C-V lasso/LSE1, the lasso with adjusted cross-validation; C-V lasso/LSE2, the least squares estimator after the lasso selection with adjusted cross-validation, #cov , the number of cov ariates considered; corr , the correla- tion coefficient between fitted and observed responses; P-MSE, prediction mean squared error ables with P ( W = 0 . 2) = P ( W = 1) = 1 / 2 and b β W = arg min b | y − X b | 2 2 2 n + b λ p X j =1 | β j | /W j , where b λ is the penalty lev el chosen by individual methods. Stability selection selects variables with nonzero estimated b β W j ov er 50 times in 100 replications. W e observe that the scaled min- imax concav e penalized selector produces most sparse and most stable selection, follo wed by the adjusted cross-validation, the scaled lasso and then the plain cross-validation. The selection results are consistent among the four methods in the sense that the selected models are almost nested. Since the model size is between 6 and 8 by stability selection in all 8 cases and by the scaled minimax conca ve penalized selection for both p = 200 and p = 3000 , these two methods 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 Scaled Sparse Linear Regr ession 15 provide most consistent results. The scaled lasso and the adjusted cross-v alidation yield identical lasso and stability selections for p = 200 and identical stability selection for p = 3000 . W e also compare the prediction performance of the scaled lasso with that of the lasso with the best fixed penalty lev el. W e compute the scaled estimators in 1000 replications. In each replication, the dataset is split at random into a training set with 80 observations and a test set with 40 observations. The prediction mean squared error is computed within the test set, while the scaled estimators and the lasso estimator with fixed penalty lev el λ are computed based on the training set. Figure 1 demonstrates that in prediction, the scaled lasso with λ 0 chosen as λ 2 = { 2(log p ) /n } 1 / 2 performs almost as well as the lasso with the optimal fixed λ . In addition, we compare the prediction performance of all the estimators mentioned in this section. In each replication, we compute the penalized maximum likelihood estimator , its bias- correction, and scaled penalization methods based on the training set of 80 observ ations. For cross-v alidation, the training set of 80 observ ations is further partitioned at random 100 times into two groups of sizes 60 and 20, and a penalty le vel is selected by minimizing the estimated loss in the smaller group for the lasso estimator based on the lar ger group. This selected penalty le vel is then used for the lasso with the entire training set. Thus, the cross-validation lasso is also based on the training set with 80 observations. For the penalty le vel selected by the adjusted cross- v alidation, two estimators are considered: the lasso estimator and the least squares estimator after the lasso selection. In T able 4, we present the medians of the prediction mean squared error and the selected model size in the 200 replications. The scaled lasso has comparable prediction performance as cross-validation. Again, T able 4 suggests that original cross-validation tends to choose lar ger models, while adjusted cross-v alidation leads to results comparable with the scaled lasso. 5 . D I S C U S S I O N In the theoretical analysis, we have considered λ 0 = A { (2 /n ) log p } 1 / 2 with A > 1 . This choice is somewhat conservati ve from a number of points of vie w . Simulation results suggest that the requirement A > 1 is a mathematical technicality . If | X 0 ε/n | ∞ ≤ λ ∗ with large prob- ability for a standard normal vector ε , the theoretical results in this paper are all valid under pr β ,σ when λ 0 is replaced by the smaller min( λ 0 , Aλ ∗ ) . The value of λ ∗ can be estimated by simulation with the giv en X and separately generated ε . A somewhat sharper theoretical choice of λ 0 is A { (2 /n ) log ( p/s ) } 1 / 2 with the unknown s = | β ∗ | 0 (Zhang, 2010), or its simulated ver- sion with λ ∗ = max | T | = s | X 0 T ε | 2 / | T | 1 / 2 . The dif ference between the two λ 0 is limited unless log p = { 1 + o (1) } log n . A revie wer called our attention to an unpublished 2011 report by Ba- raud, Giraud and Huet, a v ailable at http://arxi v .org/abs/1007.2096, whose method can be used to select a penalty le vel to nearly minimize the order of a penalized prediction error . This may also justify the use of smaller estimated penalty le vels. In the proof of our theoretical results for the scaled lasso, we use oracle inequalities for fixed penalty which unify and some what sharpen existing results. W e now present this result. Define η ∗ ( λ, ξ ) = min T 2 − 1 h η ( λ, ξ , β ∗ , T ) +  η 2 ( λ, ξ , β ∗ , T ) − 16 λ 2 | β ∗ T c | 2 1  1 / 2 i (30) as a sharper version of η ( λ, ξ , β ∗ , T ) in (10). T H E O R E M 4 . Let b β ( λ ) be the minimizer of (1) with ρ ( t ) = t . Let β ∗ ∈ R p be a targ et vector and ξ > 1 . Then, in the event | X 0 ( y − X β ∗ ) | ∞ /n ≤ λ ( ξ − 1) / ( ξ + 1) , we have | X b β ( λ ) − X β ∗ | 2 2 /n ≤ min  η ∗ ( λ, ξ ) , η ∗ ( λ, ξ )  (31) 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 16 T . S U N A N D C . - H . Z H A N G with η ∗ ( λ, ξ ) in (12). Mor eover , in the same event and with µ ( λ, ξ ) in (16), | b β ( λ ) − β ∗ | 1 ≤ µ ( λ, ξ ) . (32) The interpretations of (31) and (32) are giv en in (15) and (17), along with their relationship to se veral existing results. W e note here that the condition κ ( ξ , S )  1 for (15) and (17), weaker than the parallel condition on the restricted eigenv alue (Bickel et a l., 2009), can be slightly weak- ened by using F 1 ( ξ , S ) in (21) (Y e & Zhang, 2010). A C K N O W L E D G E M E N T This research was supported by the National Science F oundation and the National Security Agency . W e thank Jian Huang for sharing the gene expression data, and re vie wers for valuable suggestions. A P P E N D I X Here we prov e Proposition 1, Theorem 4, Theorem 1, Theorem 2 and then Theorem 3. Pr oof of Proposition 1. (i) Since b β = b β ( σ λ 0 ) is a solution of (2) at λ = σ λ 0 , n ( ∂ /∂ w ) L λ 0 ( w , σ )    w = b β ( σ λ 0 ) o j = 0 , for all b β j ( σ λ 0 ) 6 = 0 . Since { j : b β j ( λ ) } is unchanged in a neighborhood of σ λ 0 , [( ∂ /∂ σ ) { b β ( σ λ 0 ) /σ } ] j = 0 for b β j ( σ λ 0 ) = 0 . Thus, ∂ ∂ σ L λ 0 { b β ( σ λ 0 ) , σ } = ∂ ∂ t L λ 0 { b β ( σ λ 0 ) , t }    t = σ = 1 − a 2 − | y − X b β ( σ λ 0 ) | 2 2 2 nσ 2 . (ii) The conv ergence of (3) and (4) follows from the joint con vexity of L λ 0 ( β , σ ) . The scale in variance follows from L 0 ( cβ , cσ ; X , cy ) = cL 0 ( β , σ ; X , y ) , where L 0 ( β , σ ; X , y ) expresses the dependence of (5) on the data ( X, y ) .  Pr oof of Theorem 4. (i) Let b β = b β ( λ ) . Since σ ∗ z ∗ = | X 0 ( y − X β ∗ ) | ∞ /n and ˙ ρ ( | b β j | /λ ) = 1 for b β j 6 = 0 , the inner product of w − b β and the Karush–Kuhn–T ucker condition (2) yield ( X b β − X w ) 0 ( X b β − X β ∗ ) /n ≤ λ ( | w | 1 − | b β | 1 ) + σ ∗ z ∗ | w − b β | 1 . Since 2( X b β − X w ) 0 ( X b β − X β ∗ ) = | X b β − X w | 2 2 + | X h | 2 2 − | X β ∗ − X w | 2 2 , this gives the basic in- equality (25). Let h = b β − β ∗ . Since σ ∗ z ∗ ≤ λ ( ξ − 1) / ( ξ + 1) , λ {| w | 1 − | b β | 1 } + σ ∗ z ∗ | w − b β | 1 is no greater than b | ( w − b β ) T | 1 + 2 λ | w T c | 1 − ( b/ξ ) | ( w − b β ) T c | 1 with b = 2 ξ λ/ ( ξ + 1) . Thus, (25) implies | X b β − X w | 2 2 /n + | X h | 2 2 /n + (2 b/ξ ) | ( w − b β ) T c | 1 ≤ 2 c + 2 b | ( w − b β ) T | 1 (A1) with c = | X β ∗ − X w | 2 2 / (2 n ) + 2 λ | w T c | 1 . For T = ∅ and w = β ∗ , (A1) directly yields | X h | 2 2 /n ≤ c = 2 λ | β ∗ | 1 . For general { w, T } , we want to prove | X h | 2 2 /n ≤ η ( λ, ξ , w , T ) = 2 c + b 2 /a, a = κ 2 ( ξ , T ) / | T | . It suffices to consider | X h | 2 2 /n ≥ 2 c . In this case, b β − w ∈ C ( ξ , T ) by (A1), so that by (11) a | ( w − b β ) T | 2 1 ≤ | X b β − X w | 2 2 /n. (A2) 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 Scaled Sparse Linear Regr ession 17 Let x = | ( w − b β ) T | 1 and y = | X h | 2 2 /n . It follows from (A1) and (A2) that ax 2 + y ≤ 2 c + 2 bx . For such ( x, y ) , y − 2 c ≤ max x { 2 bx − ax 2 } = b 2 /a . This giv es y ≤ 2 c + b 2 /a = η ( λ, ξ , w , T ) . For w = β ∗ , it suf fices to consider the case y > c = 2 λ | β ∗ T c | 1 , where the cone condition holds for b β − β ∗ . Now , ( x, y ) satisfies ax 2 ≤ y ≤ c + bx . The maximum of y , attained at ax 2 = c + bx , is c + b { b + ( b 2 + 4 ac ) 1 / 2 } / (2 a ) =  η ( λ, ξ , β ∗ , T ) + { η 2 ( λ, ξ , β ∗ , T ) − 4 c 2 } 1 / 2  / 2 . (A3) (ii) Let 0 < ν < 1 and T ⊂ { 1 , . . . , p } . It follows from (A1) with w = β ∗ that (1 + ξ ) | X h | 2 2 /n + 2 λ | h T c | 1 ≤ 2 λ ( ξ + 1) | β ∗ T c | 1 + 2 ξ λ | h T | 1 . It suffices to consider ν | h | 1 ≥ ( ξ + 1) | β ∗ T c | 1 . In this case (1 + ξ ) | X h | 2 2 /n + 2 λ (1 − ν ) | h T c | 1 ≤ 2 λ ( ξ + ν ) | h T | 1 . Thus, (1 − ν ) | h T c | 1 ≤ ( ξ + ν ) | h T | 1 , or equiv alently h ∈ C { ( ξ + ν ) / (1 − ν ) , T } . It follows from (11) that | X h | 2 2 /n ≥ | h T | 2 1 κ 2 { ( ξ + ν ) / (1 − ν ) , T } / | T | , so that (1 + ξ ) | h T | 2 1 κ 2 { ( ξ + ν ) / (1 − ν ) , T } / | T | + 2(1 − ν ) λ | h T c | 1 ≤ 2( ξ + ν ) λ | h T | 1 . (A4) Let x = | h T | 1 and y = | h T c | 1 . Write (A4) as ax 2 + by ≤ cx . Subject to this inequality , the maximum of x + y is max x ≥ 0 { x + ( cx − ax 2 ) /b } . This maximum, attained at 2 ax = b + c , is x ( b + c ) / (2 b ) = ( b + c ) 2 / (4 ab ) . Thus, | h | 1 ≤ { 2( ξ + 1) λ } 2 | T | 4(1 + ξ ) κ 2 { ( ξ + ν ) / (1 − ν ) , T }{ 2(1 − ν ) λ } = ( ξ + 1) λ | T | / (1 − ν ) 2 κ 2 { ( ξ + ν ) / (1 − ν ) , T } . This giv es | h | 1 ≤ µ ( λ, ξ ) for ν | h | 1 ≥ ( ξ + 1) | β ∗ T c | 1 .  Pr oof of Theorem 1. Assume τ 0 < 1 without loss of generality . Consider t ≥ σ ∗ (1 − τ 0 ) and the penalty le vel λ = tλ 0 for the lasso. Since z ∗ σ ∗ ≤ σ ∗ (1 − τ 0 ) λ 0 ( ξ − 1) / ( ξ + 1) ≤ λ ( ξ − 1) / ( ξ + 1) and σ ∗ = | y − X β ∗ | 2  n 1 / 2 , the Cauchy–Schwarz inequality and (31) imply   | y − X b β ( tλ 0 ) | 2  n 1 / 2 − σ ∗   ≤ | X b β ( tλ 0 ) − X β ∗ | 2  n 1 / 2 ≤ η 1 / 2 ∗ ( tλ 0 , ξ ) . Since η 1 / 2 ∗ ( tλ 0 , ξ ) ≤ σ ∗ τ 0 for t < σ ∗ , the deriv ativ e (7) of the loss with a = 0 satisfies 2 t 2 ∂ ∂ t L λ 0 { b β ( tλ 0 ) , t } = t 2 − | y − X b β ( tλ 0 ) | 2 2 /n ≤ t 2 − ( σ ∗ ) 2 (1 − τ 0 ) 2 = 0 at t = σ ∗ (1 − τ 0 ) . This implies b σ ≥ σ ∗ (1 − τ 0 ) by the strict con vexity of the profile loss (5) in σ . For t > σ ∗ , η 1 / 2 ∗ ( tλ 0 , ξ ) ≤ tτ 0 by (10) and (12), so that at t = σ ∗ / (1 − τ 0 ) , t 2 − | y − X b β ( tλ 0 ) | 2 2 /n ≥ t 2 −  σ ∗ + tτ 0  2 ≥ 0 . This implies σ ∗ ≥ b σ (1 − τ 0 ) by the strict conv exity of (5) in σ . Thus, the first part of (13) holds. Moreover , | X b β − X β ∗ | 2  n 1 / 2 ≤ η 1 / 2 ∗ ( b σ λ 0 , ξ ) ≤ η 1 / 2 ∗ { σ ∗ λ 0 / (1 − τ 0 ) , ξ } ≤ σ ∗ τ 0 / (1 − τ 0 ) . Finally , since pr β ,σ [ | X 0 ( y − X β ) /n | ∞ ≤ σ { (2 /n ) log p } 1 / 2 ] → 1 , (14) follows from (13).  The proof of Theorem 2 requires the following lemma. L E M M A 1 . Let T m have the t-distribution with m de grees of fr eedom. Then, there e xists  m → 0 suc h that for all t > 0 pr  T 2 m > m { e 2 t 2 / ( m − 1) − 1 }  ≤ (1 +  m ) e − t 2 / ( π 1 / 2 t ) . (A5) 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 18 T . S U N A N D C . - H . Z H A N G Pr oof of Lemma 1. Let x = [ m { e 2 t 2 / ( m − 1) − 1 } ] 1 / 2 . Since T m has the t -distribution, pr  T 2 m > x 2  = 2Γ { ( m + 1) / 2 } Γ( m/ 2)( mπ ) 1 / 2 Z ∞ x  1 + u 2 m  − ( m +1) / 2 d u ≤ 2Γ { ( m + 1) / 2 } x Γ( m/ 2)( mπ ) 1 / 2 Z ∞ x  1 + u 2 m  − ( m +1) / 2 u d u = 2Γ { ( m + 1) / 2 } m x Γ( m/ 2)( mπ ) 1 / 2 ( m − 1)  1 + x 2 m  − ( m − 1) / 2 . Since x ≥ t { 2 m/ ( m − 1) } 1 / 2 , pr  T 2 m > x 2  ≤ √ 2Γ { ( m + 1) / 2 } Γ( m/ 2)( m − 1) 1 / 2 e − t 2 tπ 1 / 2 = (1 +  m ) e − t 2 tπ 1 / 2 , where  m = { 2 / ( m − 1) } 1 / 2 Γ { ( m + 1) / 2 } / Γ( m/ 2) − 1 → 0 as m → ∞ .  Pr oof of Theorem 2. W e need to express τ 2 ∗ as a function of σ at σ = σ ∗ in the proof. Define φ ( σ ) = λ 0 µ ( σ λ 0 , ξ ) /σ, φ + = φ ( σ ∗ ) ξ ( ξ + 1) { 1 − φ ( σ ∗ ) } + , φ − = φ ( σ ∗ )( ξ − 1) ξ + 1 . W e hav e τ 2 ∗ = φ ( σ ∗ ) < 1 , φ − ≤ φ ( σ ∗ ) and φ + ≤ φ ( σ ∗ ) / (1 − φ ( σ ∗ ) . (i) Consider z ∗ ≤ (1 − φ − ) λ 0 ( ξ − 1) / ( ξ + 1) . Let λ = tλ 0 and h = b β ( λ ) − β ∗ . Since | X 0 ( y − X β ∗ ) /n | ∞ = z ∗ σ ∗ , the Karush–Kuhn–T ucker condition (2) giv es − ( z ∗ σ ∗ + λ ) | h | 1 ≤ ( X h ) 0 { y − X β ∗ + y − X b β ( λ ) } /n = ( σ ∗ ) 2 − | y − X b β ( λ ) | 2 2 /n = ( X h ) 0 { 2( y − X β ∗ ) − X h } /n ≤ 2 z ∗ σ ∗ | h | 1 (A6) as lower and upper bounds for ( σ ∗ ) 2 − | y − X b β ( λ ) | 2 2 /n . This is a key point in the proof. For t ≥ σ ∗ (1 − φ − ) , z ∗ σ ∗ ≤ tλ 0 ( ξ − 1) / ( ξ + 1) = λ ( ξ − 1) / ( ξ + 1) , so that (32) in Theorem 4 im- plies | h | 1 ≤ µ ( tλ 0 , ξ ) . It follows (A6) that for t = σ ∗ (1 − φ − ) , t 2 − | y − X b β ( tλ 0 ) | 2 2 /n ≤ t 2 − ( σ ∗ ) 2 + 2 z ∗ σ ∗ µ ( tλ 0 , ξ ) ≤ 2 t ( t − σ ∗ ) + 2 tλ 0 ( ξ − 1)( ξ + 1) − 1 µ ( σ ∗ λ 0 , ξ ) = 0 , due to φ − = ( ξ − 1)( ξ + 1) − 1 φ ( σ ∗ ) = ( ξ − 1)( ξ + 1) − 1 λ 0 µ ( σ ∗ λ 0 , ξ ) /σ ∗ . As in the proof of Theorem 1, we find b σ /σ ∗ ≥ 1 − φ − by (7) and the strict con vexity of (5) in σ . Now we prove that b σ /σ ∗ ≤ 1 + φ + . For t > σ ∗ , µ ( tλ 0 , ξ ) ≤ ( t/σ ∗ ) µ ( σ ∗ λ 0 , ξ ) by (16). Thus, since ( ξ − 1) / ( ξ + 1) + 1 = 2 φ + { 1 − φ ( σ ∗ ) } /φ ( σ ∗ ) and φ + ≤ (1 + φ + ) φ ( σ ∗ ) , for t/σ ∗ = 1 + φ + , (A6) and (32) imply that t 2 − | y − X b β ( tλ 0 ) | 2 2 /n ≥ t 2 − ( σ ∗ ) 2 − ( z ∗ σ ∗ + tλ 0 ) µ ( tλ 0 , ξ ) ≥ ( t + σ ∗ ) σ ∗ φ + − { ( ξ − 1) / ( ξ + 1) + 1 + φ + } tλ 0 µ ( σ ∗ λ 0 , ξ ) = ( σ ∗ ) 2  (2 + φ + ) φ + − [2 φ + { 1 − φ ( σ ∗ ) } /φ ( σ ∗ ) + φ + ](1 + φ + ) φ ( σ ∗ )  = ( σ ∗ ) 2 φ + { φ ( σ ∗ )(1 + φ + ) − φ + } > 0 . It follows that b σ /σ ∗ ≤ 1 + φ + by con vexity . Since 1 − φ − ≤ b σ /σ ∗ ≤ 1 + φ + , | b β ( b σ λ 0 ) − β ∗ | 1 ≤ µ ( b σ λ 0 , ξ ) ≤ µ ( σ ∗ λ 0 , ξ )(1 + φ + ) . This com- pletes the proof of (18). (ii) Let z j = x 0 j ( y − X β ∗ ) / ( nσ ∗ ) with z ∗ = max j ≤ p | z j | . Under pr β ∗ ,σ , ε ∗ = y − X β ∗ is a v ector of independent and identically distributed normal variables with zero mean. Since σ ∗ = | y − X β ∗ | /n 1 / 2 , z j / { (1 − z 2 j ) / ( n − 1) } 1 / 2 follows a t -distribution with n − 1 degrees of freedom. Lemma 1 with m = 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 Scaled Sparse Linear Regr ession 19 n − 1 and t 2 = log( p/ ) > 2 implies pr β ∗ ,σ h ( n − 1) z 2 j 1 − z 2 j > ( n − 1) { e 2 t 2 / ( n − 2) − 1 } i ≤ 1 +  n − 1 π 1 / 2 t e − t 2 = (1 +  n − 1 ) /p { π log( p/ ) } 1 / 2 . (A7) Since e a − 1 ≤ P ∞ k =1 a k / 2 k − 1 = a/ (1 − a/ 2) for any 0 < a < 2 , ( n − 1) { e 2 t 2 / ( n − 2) − 1 } ≤ 2( n − 1) t 2 / ( n − 2) 1 − t 2 / ( n − 2) ≤ 2( n − 1) t 2 /n 1 − 2 t 2 /n . (A8) The combination of (A7) and (A8) yields pr β ∗ ,σ  | z j | > { 2 log( p/ ) /n } 1 / 2  = pr β ∗ ,σ n ( n − 1) z 2 j 1 − z 2 j > 2( n − 1) t 2 /n 1 − 2 t 2 /n o ≤ pr β ∗ ,σ n ( n − 1) z 2 j 1 − z 2 j > ( n − 1)( e 2 t 2 n − 2 − 1) o ≤ (1 +  n − 1 )( /p ) / { π log( p/ ) } 1 / 2 . Since λ 0 ≥ { (2 /n ) log( p/ ) } 1 / 2 ( ξ + 1) / { ( ξ − 1)(1 − φ − ) } , this bounds the tail probability of z ∗ = max j ≤ p | z j | by the union bound. Since n ( σ ∗ /σ ) 2 follows the χ 2 n distribution, n 1 / 2 ( σ ∗ /σ − 1) con ver ges to N (0 , 1 / 2) in distrib ution, which then implies (19) by (18) under φ ( σ ) = o ( n − 1 / 2 ) .  Pr oof of Theorem 3. Let h = b β − β ∗ . It follows from the proof of Theorem 2 (i) that z ∗ ≤ (1 − φ − ) λ 0 ( ξ − 1) / ( ξ + 1) ≤ ( b σ /σ ∗ ) λ 0 ( ξ − 1) / ( ξ + 1) , so that b λ + z ∗ σ ∗ ≤ ξ ( b λ − z ∗ σ ∗ ) . By (2), | x 0 j X h/n | = | x 0 j ( y − X b β − ε ∗ ) /n | ≥ b λ − z ∗ σ ∗ for b β j 6 = 0 . Let B ⊆ b S \ S with | B | ≤ m . Since κ + ( m, S ) is the upper sparse eigen value, ( b λ − z ∗ σ ∗ ) 2 | B | ≤ κ + ( m, S ) | X h | 2 2 /n . By the basic in- equality (25) with w = β ∗ , | X h | 2 2 /n ≤ ( b λ + z ∗ σ ∗ ) | h S | 1 and h is in the cone C ( ξ , S ) . Thus, since | h S | 2 1 κ 2 ( ξ , S ) ≤ | X h | 2 2 | S | /n by (11), | X h | 2 2 /n ≤ ( b λ + z ∗ σ ∗ ) 2 | S | /κ 2 ( ξ , S ) . It follows that | B | ≤ κ + ( m, S ) ξ 2 | S | /κ 2 ( ξ , S ) < m . Since all B ⊆ b S \ S of size | B | ≤ m have size | B | < m , b S \ S does not hav e a subset of size m . This gives the first inequality in (27). Let P B be the orthogonal projection to the linear span of ( x j , j ∈ B ) . By the definition of σ ∗ m,S and the prediction error bound | X h | 2 2 /n ≤ η ∗ ( b λ, ξ ) in Theorem 4, b σ 2 − σ 2 = | P b S ( y − X b β ) | 2 2 /n ≤  | P b S ε ∗ | 2 + | P b S X h | 2  2 /n ≤ { σ ∗ m − 1 ,S + √ η ∗ ( b λ, ξ ) } 2 . This giv es the second inequality in (27). 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Journal of Machine Learning Research 7 , 2541–2567. [ Received mmm y y y y . Revised mmm y y y y ]