Dual Extrapolation for Sparse Generalized Linear Models
This work addresses computational bottlenecks in high-dimensional statistical inference for researchers and practitioners, but it appears incremental as it builds on existing duality-based techniques.
The paper tackles the challenge of solving sparse Generalized Linear Models (GLMs) in high dimensions by showing that dual iterates exhibit Vector AutoRegressive (VAR) behavior after sign identification, enabling tighter optimality certificates. This enhances screening rules and working set algorithms, though no concrete performance numbers are provided.
Generalized Linear Models (GLM) form a wide class of regression and classification models, where prediction is a function of a linear combination of the input variables. For statistical inference in high dimension, sparsity inducing regularizations have proven to be useful while offering statistical guarantees. However, solving the resulting optimization problems can be challenging: even for popular iterative algorithms such as coordinate descent, one needs to loop over a large number of variables. To mitigate this, techniques known as screening rules and working sets diminish the size of the optimization problem at hand, either by progressively removing variables, or by solving a growing sequence of smaller problems. For both techniques, significant variables are identified thanks to convex duality arguments. In this paper, we show that the dual iterates of a GLM exhibit a Vector AutoRegressive (VAR) behavior after sign identification, when the primal problem is solved with proximal gradient descent or cyclic coordinate descent. Exploiting this regularity, one can construct dual points that offer tighter certificates of optimality, enhancing the performance of screening rules and helping to design competitive working set algorithms.