Extensions to the Proximal Distance Method of Constrained Optimization
This work addresses optimization under general constraints for researchers and practitioners in machine learning and statistics, offering incremental improvements to existing methods.
The paper tackles the problem of minimizing a loss function subject to fusion constraints, such as smoothness or sparsity, by extending the proximal distance method with a steepest descent variant to avoid costly linear solves. The result shows superior speed and acceptable accuracy in high-dimensional problems compared to ADMM, as demonstrated in numerical tests on tasks like image denoising and convex clustering.
The current paper studies the problem of minimizing a loss $f(\boldsymbol{x})$ subject to constraints of the form $\boldsymbol{D}\boldsymbol{x} \in S$, where $S$ is a closed set, convex or not, and $\boldsymbol{D}$ is a matrix that fuses parameters. Fusion constraints can capture smoothness, sparsity, or more general constraint patterns. To tackle this generic class of problems, we combine the Beltrami-Courant penalty method with the proximal distance principle. The latter is driven by minimization of penalized objectives $f(\boldsymbol{x})+\fracρ{2}\text{dist}(\boldsymbol{D}\boldsymbol{x},S)^2$ involving large tuning constants $ρ$ and the squared Euclidean distance of $\boldsymbol{D}\boldsymbol{x}$ from $S$. The next iterate $\boldsymbol{x}_{n+1}$ of the corresponding proximal distance algorithm is constructed from the current iterate $\boldsymbol{x}_n$ by minimizing the majorizing surrogate function $f(\boldsymbol{x})+\fracρ{2}\|\boldsymbol{D}\boldsymbol{x}-\mathcal{P}_{S}(\boldsymbol{D}\boldsymbol{x}_n)\|^2$. For fixed $ρ$ and a subanalytic loss $f(\boldsymbol{x})$ and a subanalytic constraint set $S$, we prove convergence to a stationary point. Under stronger assumptions, we provide convergence rates and demonstrate linear local convergence. We also construct a steepest descent (SD) variant to avoid costly linear system solves. To benchmark our algorithms, we compare against the alternating direction method of multipliers (ADMM). Our extensive numerical tests include problems on metric projection, convex regression, convex clustering, total variation image denoising, and projection of a matrix to a good condition number. These experiments demonstrate the superior speed and acceptable accuracy of our steepest variant on high-dimensional problems.