Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima (1305.2436v2)

Abstract: We provide novel theoretical results regarding local optima of regularized $M$-estimators, allowing for nonconvexity in both loss and penalty functions. Under restricted strong convexity on the loss and suitable regularity conditions on the penalty, we prove that \emph{any stationary point} of the composite objective function will lie within statistical precision of the underlying parameter vector. Our theory covers many nonconvex objective functions of interest, including the corrected Lasso for errors-in-variables linear models; regression for generalized linear models with nonconvex penalties such as SCAD, MCP, and capped-$\ell_1$; and high-dimensional graphical model estimation. We quantify statistical accuracy by providing bounds on the $\ell_1$-, $\ell_2$-, and prediction error between stationary points and the population-level optimum. We also propose a simple modification of composite gradient descent that may be used to obtain a near-global optimum within statistical precision $\epsilon$ in $\log(1/\epsilon)$ steps, which is the fastest possible rate of any first-order method. We provide simulation studies illustrating the sharpness of our theoretical results.

• The paper extends M-estimators to nonconvex settings, proving that all local optima lie within a tight statistical neighborhood of the true parameter.
• It establishes statistical error bounds under restricted strong convexity, ensuring local optima have accuracy comparable to global solutions.
• The work demonstrates that standard first-order methods, like composite gradient descent, achieve linear convergence in high-dimensional simulations.

Overview of Regularized M-estimators with Nonconvexity

This paper presents a comprehensive theoretical examination of regularized M-estimators, emphasizing nonconvexity in loss and penalty functions. It offers new insights into the behavior of local optima, supported by rigorous proofs and practical simulations.

Key Contributions

1. Nonconvex Objectives: The paper extends the understanding of M-estimators to nonconvex settings, such as the corrected Lasso, generalized linear models with SCAD, MCP, capped-ℓ1 penalties, and high-dimensional graphical models. The authors provide theoretical guarantees that all local optima reside within a small statistical neighborhood of the true parameter vector.
2. Theoretical Guarantees: Under restricted strong convexity (RSC) conditions and suitable penalty regularity, the authors demonstrate that local optima offer statistical precision comparable to global optima. Bounds are established for ℓ1, ℓ2, and prediction errors between stationary points and the true parameter.
3. Algorithmic Implications: The paper proposes employing standard first-order methods like composite gradient descent to efficiently reach near-global optima. This approach is validated through simulations showing linear convergence to statistical accuracy.
4. Simulation Studies: Empirical results support the theory, illustrating that local optima from nonconvex regularizers perform reliably within statistical error margins. The paper also discusses intuitive parameter settings, aligning well with empirical findings (e.g., SCAD parameter a = 3.7 for linear regression).

Practical and Theoretical Implications

The insights and methods provided by the paper have multifaceted implications:

• Statistical Application: The results apply to numerous statistical models with nonconvex penalties, broadening the toolset for high-dimensional data analysis and enhancing robustness in parameter estimation despite nonconvexity.
• Algorithm Development: By showing that general-purpose optimization methods suffice for nonconvex M-estimators, the paper reduces reliance on bespoke algorithms for specific nonconvex penalties. This has the potential to streamline computational strategies in high-dimensional statistics.
• Nonconvex Regularization: The paper reinforces the potential and practicality of nonconvex regularization techniques by presenting conditions under which they maintain desirable statistical properties, even when the underlying optimization landscape is complex.

Future Directions

This work opens several avenues for further research:

• Non-Decomposable and Nonsmooth Regularizers: Extending the theory to other nonconvex regularizers, such as those not decomposing over coordinates or nonsmooth loss functions like the hinge loss, remains an open challenge.
• Advanced Algorithmics: While the paper proposes composite gradient descent, exploring other algorithms' potentials, particularly for more complex problems, would be beneficial.
• RSC/RSM Conditions: Developing general frameworks to establish RSC and restricted smoothness (RSM) beyond the specific scenarios analyzed could enable broader application.

Overall, this paper contributes significantly to our understanding of nonconvex M-estimators, providing both theoretical foundations and practical algorithms that ensure efficient and accurate high-dimensional statistical estimations.