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About this paper
Sahand N. Negahban, Pradeep Ravikumar, Martin J. Wainwright, Bin Yu
A unified theoretical framework is introduced for analyzing high-dimensional M-estimators with decomposable regularizers, demonstrating that restricted strong convexity of the loss function and decomposability of the regularizer are sufficient to establish finite-sample convergence rates for a broad class of models. This framework successfully recovers and extends optimal error bounds for methods like Lasso and Group Lasso.
Key insights
- Decomposability of a norm-based regularizer ensures that the error vector (difference between true and estimated parameters) lies within a specific structured set or cone.
- Restricted Strong Convexity (RSC) of the loss function, defined over the specific error cone induced by decomposability, is a sufficient condition for consistent estimation in high dimensions.
- A 'subspace compatibility constant' quantifies the alignment between the regularizer and error norm over relevant subspaces, critically influencing the RSC condition and overall convergence rates.
Summary by alphaXiv.
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