Wednesday, 15:45 - 16:10 h, Room: H 2038

 

Sahand Negahban
Fast global convergence of composite gradient methods for high-dimensional statistical recovery

Coauthors: Alekh Agarwal, Martin J. Wainwright

 

Abstract:
Many statistical M-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension d to grow with (and possibly exceed) the sample size n. This high-dimensional structure precludes the usual global assumptions-namely, strong convexity and smoothness conditions-that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that composite gradient descent has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical distance between the true unknown parameter θ* and an optimal solution \hat{θ}. This result is substantially sharper than previous convergence guarantees. These results extend existing ones based on constrained M-estimators.

 

Talk 2 of the invited session Wed.3.H 2038
"Conic and convex programming in statistics and signal processing IV" [...]
Cluster 4
"Conic programming" [...]

 

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