@inproceedings{
ding2024optimizing,
title={Optimizing over Multiple Distributions under Generalized Quasar-Convexity Condition},
author={Shihong Ding and Long Yang and Luo Luo and Cong Fang},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=lOV9kSX3Uo}
}
We study a typical optimization model where the optimization variable is composed of multiple probability distributions. Though the model appears frequently in practice, such as for policy problems, it lacks specific analysis in the general setting. For this optimization problem,  we propose a new structural condition/landscape description named  generalized quasar-convexity (GQC) beyond the realms of convexity. In contrast to original quasar-convexity \citep{hinder2020near}, GQC allows an individual quasar-convex parameter $\gamma_i$ for each variable block $i$ and the smaller of $\gamma_i$ implies less block-convexity. To minimize the objective function, we consider a generalized oracle termed as the internal function that includes the standard gradient oracle as a special case. We provide optimistic mirror descent (OMD) for multiple distributions and prove that the algorithm can achieve an adaptive $\tilde{\mathcal{O}}((\sum_{i=1}^d1/\gamma_i)\epsilon^{-1})$ iteration complexity to find an $\varepsilon$-suboptimal global solution without pre-known the exact values of $\gamma_i$ when the objective admits ``polynomial-like'' structural. Notably, it achieves iteration complexity that does not explicitly depend on the number of distributions and strictly faster $(\sum_{i=1}^d 1/\gamma_i \text{ v.s. } d\max_{i\in[1:d]} 1/\gamma_i)$ than mirror decent methods. We also extend GQC to the minimax optimization problem proposing the generalized quasar-convexity-concavity (GQCC) condition and a decentralized variant of OMD with regularization. Finally, we show the applications of our algorithmic framework on discounted Markov Decision Processes problem and Markov games, which bring new insights on the landscape analysis of reinforcement learning.
Optimizing over Multiple Distributions under Generalized Quasar-Convexity Condition
Shihong Ding Long Yang Luo Luo Cong Fang