Precise Error Analysis of Regularized M-estimators in High-dimensions (1601.06233v1)

Abstract: A popular approach for estimating an unknown signal from noisy, linear measurements is via solving a so called \emph{regularized M-estimator}, which minimizes a weighted combination of a convex loss function and of a convex (typically, non-smooth) regularizer. We accurately predict the squared error performance of such estimators in the high-dimensional proportional regime. The random measurement matrix is assumed to have entries iid Gaussian, only minimal and rather mild regularity conditions are imposed on the loss function, the regularizer, and on the noise and signal distributions. We show that the error converges in probability to a nontrivial limit that is given as the solution to a minimax convex-concave optimization problem on four scalar optimization variables. We identify a new summary parameter, termed the Expected Moreau envelope to play a central role in the error characterization. The \emph{precise} nature of the results permits an accurate performance comparison between different instances of regularized M-estimators and allows to optimally tune the involved parameters (e.g. regularizer parameter, number of measurements). The key ingredient of our proof is the \emph{Convex Gaussian Min-max Theorem} (CGMT) which is a tight and strengthened version of a classical Gaussian comparison inequality that was proved by Gordon in 1988.

• The paper derives the asymptotic squared error of regularized M-estimators using a minimax convex-concave optimization framework.
• The paper introduces the Expected Moreau envelope to quantify the combined effects of loss functions and regularizers on estimator performance.
• The analysis leverages the Convex Gaussian Min-max Theorem to simplify high-dimensional problems to scalar optimization, enabling optimal parameter tuning.

Precise Error Analysis of Regularized M-estimators in High-dimensions

The paper presents an analytical examination of regularized M-estimators within the context of high-dimensional signal estimation, where both the number of observations, $m$, and the dimension of the signal, $n$, approach infinity with a constant ratio $m/n \rightarrow \delta$. The authors focus on the squared error performance of these estimators when the design matrix $A$ has entries that are independent and identically distributed (i.i.d.) Gaussian.

Core Contributions

1. Asymptotic Error Characterization: The paper rigorously derives the asymptotic performance of regularized M-estimators using a minimax convex-concave optimization framework. The error expression converges to a deterministic value, $\alpha_*^2$, as $m, n \rightarrow \infty$.
2. Influence of Expected Moreau Envelopes: A new parameter, termed the Expected Moreau envelope, encapsulates the impact of both the loss function $L$ and the regularizer $f$. This parameter plays a vital role in characterizing the error behavior and facilitating performance comparisons between different estimator configurations.
3. Application of CGMT: The analysis leverages the Convex Gaussian Min-max Theorem (CGMT), enabling a reduction of the high-dimensional problem to one involving scalar optimization. This methodological innovation simplifies error analysis significantly without losing precision.
4. Generality and Flexibility: The framework accommodates a variety of convex loss functions, regularizers, and does not require loss function or regularizer separability. Moreover, it applies to noise models with unbounded moments, provided certain regularity conditions hold.
5. Characterization of Performance Metrics: The paper provides a comprehensive performance metric involving not only the number of measurements $\delta$ but also the regularization parameter $\lambda$. This enables an understanding of the conditions under which M-estimators achieve minimal error.

Numerical Results and Theoretical Implications

• Boundedness and Stability: The analysis identifies conditions under which the squared error is bounded, a critical consideration for estimator reliability in finite-sample regimes.
• Optimal Tuning and Performance Comparison: The characterization allows for optimal tuning of the regularizer $\lambda$ and supports rigorous performance comparisons across different estimator configurations.
• Potential for Consistent Estimation: The thorough examination provides insight into conditions necessary for consistent estimation, where the estimation error approaches zero.

Future Directions and Impact

The precise characterization of M-estimators' error performance underpins a robust framework for the comparison and optimization of statistical estimators in high-dimensional environments. Moreover, extending beyond Gaussian designs, examining optimal loss and regularizer formulations, and applying the framework to other performance metrics are fruitful areas for further research. By setting a new standard for precise performance analysis, this paper has the potential to significantly influence the design and application of statistical methods in machine learning and signal processing arenas.