Luo Luo

Lesi Chen, Jing Xu, Luo Luo

We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_ξ [F(x; ξ)]$, where the component $F(x;ξ)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} ε^{-4} + ΔL^3 d^{3/2} δ^{-1} ε^{-4})$ stochastic zeroth-order oracle complexity to find a $(δ,ε)$-Goldstein stationary point of objective function, where $Δ= f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ and $x_0$ is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to $\mathcal{O}(L^3 d^{3/2} ε^{-3}+ ΔL^2 d^{3/2} δ^{-1} ε^{-3})$.

Lesi Chen, Haishan Ye, Luo Luo

This paper studies the stochastic nonconvex-strongly-concave minimax optimization over a multi-agent network. We propose an efficient algorithm, called Decentralized Recursive gradient descEnt Ascent Method (DREAM), which achieves the best-known theoretical guarantee for finding the $ε$-stationary points. Concretely, it requires $\mathcal{O}(\min (κ^3ε^{-3},κ^2 \sqrt{N} ε^{-2} ))$ stochastic first-order oracle (SFO) calls and $\tilde{\mathcal{O}}(κ^2 ε^{-2})$ communication rounds, where $κ$ is the condition number and $N$ is the total number of individual functions. Our numerical experiments also validate the superiority of DREAM over previous methods.

Lesi Chen, Luo Luo

We study the problem of finding a near-stationary point for smooth minimax optimization. The recently proposed extra anchored gradient (EAG) methods achieve the optimal convergence rate for the convex-concave minimax problem in the deterministic setting. However, the direct extension of EAG to stochastic optimization is not efficient. In this paper, we design a novel stochastic algorithm called Recursive Anchored IteratioN (RAIN). We show that the RAIN achieves near-optimal stochastic first-order oracle (SFO) complexity for stochastic minimax optimization in both convex-concave and strongly-convex-strongly-concave cases. In addition, we extend the idea of RAIN to solve structured nonconvex-nonconcave minimax problem and it also achieves near-optimal SFO complexity.

Zhuanghua Liu, Luo Luo

This paper considers the smooth bilevel optimization in which the lower-level problem is strongly convex and the upper-level problem is possibly nonconvex. We focus on the stochastic setting that the algorithm can access the unbiased stochastic gradient evaluation with heavy-tailed noise, which is prevalent in many machine learning applications such as training large language models and reinforcement learning. We propose a nested-loop normalized stochastic bilevel approximation (N$^2$SBA) for finding an $ε$-stationary point with the stochastic first-order oracle (SFO) complexity of $\tilde{\mathcal{O}}\big(κ^{\frac{7p-3}{p-1}} σ^{\frac{p}{p-1}} ε^{-\frac{4 p - 2}{p-1}}\big)$, where $κ$ is the condition number, $p\in(1,2]$ is the order of central moment for the noise, and $σ$ is the noise level. Furthermore, we specialize our idea to solve the nonconvex-strongly-concave minimax optimization problem, achieving an $ε$-stationary point with the SFO complexity of $\tilde{\mathcal O}\big(κ^{\frac{2p-1}{p-1}} σ^{\frac{p}{p-1}} ε^{-\frac{3p-2}{p-1}}\big)$. All above upper bounds match the best-known results under the special case of the bounded variance setting, i.e., $p=2$.