Convergence Rate Analysis of the Multiplicative Gradient Method for PET-Type Problems

TL;DR

This paper analyzes the convergence rate of the multiplicative gradient method for PET-type problems, showing it has an $O(rac{ ext{ln}(n)}{t})$ rate and converges to optimal solutions at $O(1/ extsqrt{t})$, with implications for computational complexity.

Contribution

The paper provides the first convergence rate analysis of the multiplicative gradient method for PET-type problems, demonstrating its efficiency compared to other methods in high-dimensional regimes.

Findings

MG method has $O( ext{ln}(n)/t)$ convergence rate.

Distances to optimal solutions decrease at $O(1/ extsqrt{t})$.

MG method is computationally more efficient in certain regimes.

Abstract

We analyze the convergence rate of the multiplicative gradient (MG) method for PET-type problems with $m$ component functions and an $n$-dimensional optimization variable. We show that the MG method has an $O(\ln(n)/t)$ convergence rate, in both the ergodic and the non-ergodic senses. Furthermore, we show that the distances from the iterates to the set of optimal solutions converge (to zero) at rate $O(1/\sqrt{t})$. Our results show that, in the regime $n=O(\exp(m))$, to find an $\varepsilon$-optimal solution of the PET-type problems, the MG method has a lower computational complexity compared with the relatively-smooth gradient method and the Frank-Wolfe method for convex composite optimization involving a logarithmically-homogeneous barrier.