# Selection Consistency of Lasso-Based Procedures for Misspecified High-Dimensional Binary Model and Random Regressors

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## Abstract

**:**

## 1. Introduction

- We prove that under misspecification when the sample size grows support ${\widehat{s}}^{*}$ coincides with support of ${\beta}^{*}$ with probability tending to 1. In the general framework allowing for misspecification this means that selection rule ${\widehat{s}}^{*}$ is consistent, i.e., $P({\widehat{s}}^{*}={s}^{*})\to 1$ when $n\to \infty $. In particular, when the model in Equation (1) is correctly specified this means that we recover the support of the true vector $\beta $ with probability tending to 1.
- We also prove approximation result for Lasso estimator when predictors are random and $\rho $ is a convex Lipschitz function (cf. Theorem 1).
- A useful corollary of the last result derived in the paper is determination of sufficient conditions under which active predictors can be separated from spurious ones based on the absolute values of corresponding coordinates of Lasso estimator. This makes construction of nested family containing ${s}^{*}$ with a large probability possible.
- Significant insight has been gained for fitting of parametric model when predictors are elliptically contoured (e.g., multivariate normal). Namely, it is known that in such situation ${\beta}^{*}=\eta \beta $, i.e., these two vectors are collinear [5]. Thus, in the case when $\eta \ne 0$ we have that support ${s}^{*}$ of ${\beta}^{*}$ coincides with support s of $\beta $ and the selection consistency of two-step procedure proved in the paper entails direction and support recovery of $\beta $. This may be considered as a partial justification of a frequent observation that classification methods are robust to misspecification of the model for which they are derived (see, e.g., [5,18]).

## 2. Definitions and Auxiliary Results

- (MC)
- There exist $\vartheta ,\epsilon ,\delta >0$ and non-negative definite matrix $H\in {R}^{{p}_{n}\times {p}_{n}}$ such that for all b with $b-{\beta}^{*}\in {\mathcal{C}}_{\epsilon}\cap {B}_{1}(\delta )$ we have$$R(b)-R({\beta}^{*})\ge \frac{\vartheta}{2}{(b-{\beta}^{*})}^{T}H(b-{\beta}^{*}).$$

- (LL)
- $\exists L>0\phantom{\rule{4pt}{0ex}}\forall {b}_{1},{b}_{2}\in R,y\in \{0,1\}:\phantom{\rule{4pt}{0ex}}|\rho ({b}_{1},y)-\rho ({b}_{2},y)|\le L|{b}_{1}-{b}_{2}|$.

- AIC (Akaike Information Criterion): ${a}_{n}=2$;
- BIC (Bayesian Information Criterion): ${a}_{n}=logn$; and
- EBIC(d) (Extended BIC): ${a}_{n}=logn+2dlog{p}_{n}$, where $d>0$.

- ${C}_{\u03f5}(w)$: $R(b)-R({\beta}^{*})\ge \theta ||b-{\beta}^{*}{||}_{2}^{2}$ for all $b\in {R}^{{p}_{n}}$ such that $suppb\subseteq w$ and $b-{\beta}^{*}\in {B}_{2}(\u03f5).$

**Lemma**

**1.**

**Lemma**

**2.**

- 1.
- $P(S(r)>t)\le \frac{8Lr{s}_{n}\sqrt{log({p}_{n}\vee 2)}}{t\sqrt{n}}$,
- 2.
- $P({S}_{1}(r)\ge t)\le \frac{8Lr{s}_{n}\sqrt{{k}_{n}ln({p}_{n}\vee 2)}}{t\sqrt{n}}$,
- 3.
- $P({S}_{2}(r)\ge t)\le \frac{4Lr{s}_{n}\sqrt{|{s}^{*}|}}{t\sqrt{n}}$.

## 3. Properties of Lasso for a General Loss Function and Random Predictors

**Lemma**

**3.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 4. GIC Consistency for a a General Loss Function and Random Predictors

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Remark**

**1.**

- (1)
- ${a}_{n}=logn$ and ${p}_{n}<\frac{{n}^{\frac{A}{{k}_{n}(1+u)}}}{2}$ for some $u>0$.
- (2)
- ${a}_{n}=logn+2\gamma log{p}_{n}$, ${k}_{n}\le C$ and $2A\gamma -(1+u)C\ge 0$, where $C,u>0$.
- (3)
- ${a}_{n}=logn+2\gamma log{p}_{n}$, ${k}_{n}\le C$, $2A\gamma -(1+u)C<0$, ${p}_{n}<B{n}^{\delta}$, where $\delta =\frac{A}{(1+u)C-2A\gamma}$ and $B={2}^{-(1+u)C}$.

**Theorem**

**3.**

**Proof.**

**Corollary**

**3.**

**Proof.**

## 5. Selection Consistency of SS Procedure

- Choose some $\lambda >0$.
- Find ${\widehat{\beta}}_{L}=\underset{b\in {R}^{{p}_{n}}}{arg\; min}{R}_{n}(b)+{\lambda ||b||}_{1}$.
- Find ${\widehat{s}}_{L}=supp{\widehat{\beta}}_{L}=\{{j}_{1},\dots ,{j}_{k}\}$ such that $|{\widehat{\beta}}_{L,{j}_{1}}|\ge \dots \ge |{\widehat{\beta}}_{L,{j}_{k}}|>0$ and ${j}_{1},\dots ,{j}_{k}\in \{1,\dots ,{p}_{n}\}$.
- Define ${\mathcal{M}}_{SS}=\{\varnothing ,\{{j}_{1}\},\{{j}_{1},{j}_{2}\},\dots ,\{{j}_{1},{j}_{2},\dots ,{j}_{k}\}\}$.
- Find ${\widehat{s}}^{*}=\underset{w\in {\mathcal{M}}_{SS}}{arg\; min}GIC(w)$.

**Corollary**

**4.**

- $|{s}^{*}|\le {k}_{n}$,
- $P(\forall w\in {\mathcal{M}}_{SS}:|w|\le {k}_{n})\to 1$,
- $\underset{n}{lim\; inf}{\kappa}_{H}(\epsilon )>0$ for some $\epsilon >0$, where H is non-negative definite matrix and ${\kappa}_{H}(\epsilon )$ is defined in Equation (12),
- $log({p}_{n})=o(n{\lambda}^{2})$,
- ${k}_{n}\lambda =o(min\{{\beta}_{min}^{*},1\})$,
- ${k}_{n}log{p}_{n}=o(n)$,
- ${k}_{n}log{p}_{n}=o({a}_{n})$,
- ${a}_{n}{k}_{n}=o(nmin{\{{\beta}_{min}^{*},1\}}^{2})$,

**Proof.**

#### 5.1. Case of Misspecified Semi-Parametric Model

**Corollary**

**5.**

**Remark**

**2.**

**Remark**

**3.**

## 6. Numerical Experiments

#### 6.1. Selection Procedures

- Choose some ${\lambda}_{1}>\dots >{\lambda}_{m}>0$.
- Find ${\widehat{\beta}}_{L}^{(i)}=\underset{b\in {R}^{{p}_{n}+1}}{arg\; min}{R}_{n}(b)+{\lambda}_{i}||\tilde{b}{||}_{1}$ for $i=1,\dots ,m$.
- Find ${\widehat{s}}_{L}^{(i)}=supp{\widehat{\tilde{\beta}}}_{L}^{(i)}=\{{j}_{1}^{(i)},\dots ,{j}_{{k}_{i}}^{(i)}\}$ where ${j}_{1}^{(i)},\dots ,{j}_{{k}_{i}}^{(i)}$ are such that $|{\widehat{\beta}}_{L,{j}_{1}^{(i)}}^{(i)}|\ge \dots \ge |{\widehat{\beta}}_{L,{j}_{{k}_{i}}^{(i)}}^{(i)}|>0$ for $i=1,\dots ,m$.
- Define ${\mathcal{M}}_{i}=\{\{{j}_{1}^{(i)}\},\{{j}_{1}^{(i)},{j}_{2}^{(i)}\},\dots ,\{{j}_{1}^{(i)},{j}_{2}^{(i)},\dots ,{j}_{{k}_{i}}^{(i)}\}\}$ for $i=1,\dots ,m$.
- Define $\mathcal{M}=\{\varnothing \}\cup {\displaystyle \bigcup _{i=1}^{m}}{\mathcal{M}}_{i}$.
- Find ${\widehat{s}}^{*}=\underset{w\in \mathcal{M}}{arg\; min}GIC(w)$, where$$GIC(w)=\underset{b\in {R}^{{p}_{n}+1}:supp\tilde{b}\subseteq w}{min}n{R}_{n}(b)+{a}_{n}(|w|+1).$$

- SSnet with logistic or quadratic loss:
`ncvreg`; - SSCV or LFT with logistic or quadratic loss:
`glmnet`; and - SSnet, SSCV or LFT with Huber loss (cf. [12]):
`hqreg`.

- logistic loss:
`glm.fit`(package`stats`); - quadratic loss:
`.lm.fit`(package`stats`); and - Huber loss:
`rlm`(package`rlm`).

- $ANGLE=\frac{1}{L}{\displaystyle \sum _{k=1}^{L}}arccos|cos\angle ({\tilde{\beta}}_{0},\widehat{\tilde{\beta}}({\widehat{s}}_{k}^{*}))|$, where$$cos\angle (\tilde{\beta},\widehat{\tilde{\beta}}({\widehat{s}}_{k}^{*}))=\frac{{\displaystyle \sum _{j=1}^{{p}_{n}}}{\beta}_{j}{\widehat{\beta}}_{j}({\widehat{s}}_{k}^{*})}{||\tilde{\beta}{||}_{2}||\widehat{\tilde{\beta}}({\widehat{s}}_{k}^{*}){||}_{2}}$$
- ${P}_{inc}=\frac{1}{L}{\displaystyle \sum _{k=1}^{L}}I({s}^{*}\in {\mathcal{M}}^{(k)})$,
- ${P}_{equal}=\frac{1}{L}{\displaystyle \sum _{k=1}^{L}}I({\widehat{s}}_{k}^{*}={s}^{*})$.
- ${P}_{supset}=\frac{1}{L}{\displaystyle \sum _{k=1}^{L}}I({\widehat{s}}_{k}^{*}\supseteq {s}^{*})$.

#### 6.2. Regression Models Considered

- ${a}_{n}=logn$ (BIC); and
- ${a}_{n}=logn+2log{p}_{n}$ (EBIC1).

#### 6.3. Results for Models M1 and M2

## 7. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof.**

**Lemma**

**A1.**

**Proof.**

**Proof.**

**Proof.**

**Proof.**

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**MDPI and ACS Style**

Kubkowski, M.; Mielniczuk, J.
Selection Consistency of Lasso-Based Procedures for Misspecified High-Dimensional Binary Model and Random Regressors. *Entropy* **2020**, *22*, 153.
https://doi.org/10.3390/e22020153

**AMA Style**

Kubkowski M, Mielniczuk J.
Selection Consistency of Lasso-Based Procedures for Misspecified High-Dimensional Binary Model and Random Regressors. *Entropy*. 2020; 22(2):153.
https://doi.org/10.3390/e22020153

**Chicago/Turabian Style**

Kubkowski, Mariusz, and Jan Mielniczuk.
2020. "Selection Consistency of Lasso-Based Procedures for Misspecified High-Dimensional Binary Model and Random Regressors" *Entropy* 22, no. 2: 153.
https://doi.org/10.3390/e22020153