A Newton Frank-Wolfe Method for Constrained Self-Concordant Minimization
This provides an incremental improvement for optimization researchers and practitioners working on constrained convex problems with self-concordant functions.
The authors tackled constrained self-concordant minimization problems by developing a Newton Frank-Wolfe method that uses linear minimization oracles, achieving nearly the same number of oracle calls as Frank-Wolfe for L-smooth cases with O(ε^{-ν}) complexity where ν:= 1 + o(1), and demonstrated superior performance in portfolio design, D-optimal experimental design, and logistic regression with elastic net.
We demonstrate how to scalably solve a class of constrained self-concordant minimization problems using linear minimization oracles (LMO) over the constraint set. We prove that the number of LMO calls of our method is nearly the same as that of the Frank-Wolfe method in the L-smooth case. Specifically, our Newton Frank-Wolfe method uses $\mathcal{O}(ε^{-ν})$ LMO's, where $ε$ is the desired accuracy and $ν:= 1 + o(1)$. In addition, we demonstrate how our algorithm can exploit the improved variants of the LMO-based schemes, including away-steps, to attain linear convergence rates. We also provide numerical evidence with portfolio design with the competitive ratio, D-optimal experimental design, and logistic regression with the elastic net where Newton Frank-Wolfe outperforms the state-of-the-art.