A Unified Framework for High-Dimensional Analysis of M-Estimators with Decomposable Regularizers (1010.2731v3)

Abstract: High-dimensional statistical inference deals with models in which the the number of parameters p is comparable to or larger than the sample size n. Since it is usually impossible to obtain consistent procedures unless $p/n\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized M-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\ell_2$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized M-estimators have fast convergence rates and which are optimal in many well-studied cases.

• The paper introduces a framework that leverages decomposability and restricted strong convexity to ensure consistency and optimal error bounds in high-dimensional settings.
• It derives a deterministic inequality that bounds estimation errors, validated for sparse linear models under the restricted eigenvalue condition.
• The framework extends to generalized linear models and weak sparsity settings, demonstrating broad applicability in high-dimensional statistical inference.

A Unified Framework for High-Dimensional Analysis of $M$-Estimators with Decomposable Regularizers

The paper presents a comprehensive framework for evaluating $M$-estimators in high-dimensional settings, primarily when the number of parameters $p$ exceeds the sample size $n$. It addresses the challenge by leveraging decomposable regularizers that facilitate the analysis of consistency and convergence rates of these estimators.

Key Contributions

1. Decomposability and Restricted Strong Convexity: The authors introduce two pivotal concepts: decomposability of regularizers and restricted strong convexity (RSC) of the loss functions. Decomposability is defined regarding a pair of subspaces and ensures that the regularizer penalizes deviations from a low-dimensional subspace structure effectively. RSC, on the other hand, enforces a lower bound on the curvature of the loss function in restricted subspaces, thereby ensuring that the estimator does not drift away significantly from the true parameter.
2. Main Theorem: The principal result in the paper provides a deterministic bound on the estimation error for $M$-estimators:

$$\| \widehat{\theta} - \theta^* \|^2 \leq \frac{9 \lambda^2 \Psi^2(\overline{\mathcal{M}})}{\alpha^2} + \frac{\alpha^{-1}\{\lambda^2 \tau^2(\theta^*) + 4 (\theta^*_{\mathcal{M}^\perp}) \} }{\alpha}$$

Here, $\lambda$ is the regularization parameter, $\Psi$ is a compatibility constant, and $\tau$ is a tolerance function. This deterministic inequality provides a fundamental understanding of the error components due to approximation and estimation.

1. Sparse Linear Models: The framework is applied to sparse linear models (Lasso) where the $\ell_1$ regularization is employed. The authors verify their conditions against the restricted eigenvalue (RE) properties and establish that adequate bounds on estimation errors in $\ell_2$ and $\ell_1$ norms with high probability. Specifically, they demonstrate that for models satisfying the RE condition, the estimation error for the Lasso estimator scales optimally with sample size and sparsity level.
2. Weak Sparsity: The analysis is extended to settings where the underlying model parameters are not exactly sparse but are well-approximated by sparse vectors. Utilizing $\ell_q$-balls, the authors show that the derived rates are minimax-optimal, encompassing a broader regime of high-dimensional models.
3. Group-Structured Norms: For more structured sparsity models, such as those found in group Lasso or multi-task learning, the authors establish generalized restricted eigenvalue conditions. They demonstrate that the results for group-structured penalties, including block $\ell_1/\ell_2$ and $\ell_1/\ell_\infty$ norms, yield similar error bounds, even for weak sparsity in groups.

Practical and Theoretical Implications

• High-Dimensional Statistical Inference:

This unified framework facilitates rigorous analysis for a wide range of high-dimensional models. It shows that certain regularizers can be applied effectively across various statistical tasks, ensuring consistency and optimal convergence under high-dimensional scaling.

• Extensions to Generalized Linear Models:

The framework is not limited to linear regression but extends to GLMs, provided the RSC condition is verified. This broadens the applicability of the findings to logistic regression, Poisson regression, and other settings where the response variable's distribution belongs to the exponential family.

• Future Development:

Future work may explore further extensions to more complex hierarchical and overlapping group regularizers. The provided framework lays the groundwork for investigating other decomposable regularizers, potentially improving on the bounds or discovering new applications in nonparametric regression and other machine learning areas.

This research offers a systematic approach to ensuring that high-dimensional $M$-estimators with decomposable regularizers achieve desirable statistical properties, paving the way for more robust and theoretically grounded applications in machine learning and data analysis.