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STRONG ORACLE OPTIMALITY OF FOLDED CONCAVE PENALIZED ESTIMATION.

ABSTRACT: Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, i.e., sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.

SUBMITTER: Fan J

PROVIDER: S-EPMC4295817 | biostudies-literature | 2014 Jun

REPOSITORIES: biostudies-literature

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STRONG ORACLE OPTIMALITY OF FOLDED CONCAVE PENALIZED ESTIMATION.

Fan Jianqing J Xue Lingzhou L Zou Hui H

Annals of statistics 20140601 3

Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important ...[more]

PMID: 25598560

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Project description:This paper concerns the folded concave penalized sparse linear regression (FCPSLR), a class of popular sparse recovery methods. Although FCPSLR yields desirable recovery performance when solved globally, computing a global solution is NP-complete. Despite some existing statistical performance analyses on local minimizers or on specific FCPSLR-based learning algorithms, it still remains open questions whether local solutions that are known to admit fully polynomial-time approximation schemes (FPTAS) may already be sufficient to ensure the statistical performance, and whether that statistical performance can be non-contingent on the specific designs of computing procedures. To address the questions, this paper presents the following threefold results: (i) Any local solution (stationary point) is a sparse estimator, under some conditions on the parameters of the folded concave penalties. (ii) Perhaps more importantly, any local solution satisfying a significant subspace second-order necessary condition (S3ONC), which is weaker than the second-order KKT condition, yields a bounded error in approximating the true parameter with high probability. In addition, if the minimal signal strength is sufficient, the S3ONC solution likely recovers the oracle solution. This result also explicates that the goal of improving the statistical performance is consistent with the optimization criteria of minimizing the suboptimality gap in solving the non-convex programming formulation of FCPSLR. (iii) We apply (ii) to the special case of FCPSLR with minimax concave penalty (MCP) and show that under the restricted eigenvalue condition, any S3ONC solution with a better objective value than the Lasso solution entails the strong oracle property. In addition, such a solution generates a model error (ME) comparable to the optimal but exponential-time sparse estimator given a sufficient sample size, while the worst-case ME is comparable to the Lasso in general. Furthermore, to guarantee the S3ONC admits FPTAS.

Project description:BackgroundThe rational design of modified proteins with controlled stability is of extreme importance in a whole range of applications, notably in the biotechnological and environmental areas, where proteins are used for their catalytic or other functional activities. Future breakthroughs in medical research may also be expected from an improved understanding of the effect of naturally occurring disease-causing mutations on the molecular level.ResultsPoPMuSiC-2.1 is a web server that predicts the thermodynamic stability changes caused by single site mutations in proteins, using a linear combination of statistical potentials whose coefficients depend on the solvent accessibility of the mutated residue. PoPMuSiC presents good prediction performances (correlation coefficient of 0.8 between predicted and measured stability changes, in cross validation, after exclusion of 10% outliers). It is moreover very fast, allowing the prediction of the stability changes resulting from all possible mutations in a medium size protein in less than a minute. This unique functionality is user-friendly implemented in PoPMuSiC and is particularly easy to exploit. Another new functionality of our server concerns the estimation of the optimality of each amino acid in the sequence, with respect to the stability of the structure. It may be used to detect structural weaknesses, i.e. clusters of non-optimal residues, which represent particularly interesting sites for introducing targeted mutations. This sequence optimality data is also expected to have significant implications in the prediction and the analysis of particular structural or functional protein regions. To illustrate the interest of this new functionality, we apply it to a dataset of known catalytic sites, and show that a much larger than average concentration of structural weaknesses is detected, quantifying how these sites have been optimized for function rather than stability.ConclusionThe freely available PoPMuSiC-2.1 web server is highly useful for identifying very rapidly a list of possibly relevant mutations with the desired stability properties, on which subsequent experimental studies can be focused. It can also be used to detect sequence regions corresponding to structural weaknesses, which could be functionally important or structurally delicate regions, with obvious applications in rational protein design.

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STRONG ORACLE OPTIMALITY OF FOLDED CONCAVE PENALIZED ESTIMATION.

Publications

STRONG ORACLE OPTIMALITY OF FOLDED CONCAVE PENALIZED ESTIMATION.

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