Analyzing and Improving Greedy 2-Coordinate Updates for Equality-Constrained Optimization via Steepest Descent in the 1-Norm
This work addresses optimization efficiency for problems like support vector machines, offering incremental improvements in convergence speed and computational cost.
The paper tackles the problem of minimizing a smooth function under summation and bound constraints, showing that a greedy 2-coordinate update method achieves a convergence rate faster than random selection and independent of dimension, and improves computational efficiency from O(n^2) to O(n log n) for constrained cases.
We consider minimizing a smooth function subject to a summation constraint over its variables. By exploiting a connection between the greedy 2-coordinate update for this problem and equality-constrained steepest descent in the 1-norm, we give a convergence rate for greedy selection under a proximal Polyak-Lojasiewicz assumption that is faster than random selection and independent of the problem dimension $n$. We then consider minimizing with both a summation constraint and bound constraints, as arises in the support vector machine dual problem. Existing greedy rules for this setting either guarantee trivial progress only or require $O(n^2)$ time to compute. We show that bound- and summation-constrained steepest descent in the L1-norm guarantees more progress per iteration than previous rules and can be computed in only $O(n \log n)$ time.