Stochastic smooth nonconvex minimax problems are prevalent in machine learning, e.g., GAN training, fair classification, and distributionally robust learning. Stochastic gradient descent ascent (GDA)-type methods are popular in practice due to their simplicity and single-loop nature. However, there is a significant gap between the theory and practice regarding high-probability complexity guarantees for these methods on stochastic nonconvex minimax problems. Existing high-probability bounds for GDA-type single-loop methods only apply to convex/concave minimax problems and to particular non-monotone variational inequality problems under some restrictive assumptions. In this work, we address this gap by providing the first high-probability complexity guarantees for nonconvex/PL minimax problems corresponding to a smooth function that satisfies the PL-condition in the dual variable. Specifically, we show that when the stochastic gradients are light-tailed, the smoothed alternating GDA method can compute an $\varepsilon$-stationary point within $\mathcal{O}(\frac{\ell \kappa^2 \delta^2}{\varepsilon^4} + \frac{\kappa}{\varepsilon^2}(\ell+\delta^2\log({1}/{\bar{q}})))$ stochastic gradient calls with probability at least $1-\bar{q}$ for any $\bar{q}\in(0,1)$, where $\mu$ is the PL constant, $\ell$ is the Lipschitz constant of the gradient, $\kappa=\ell/\mu$ is the condition number, and $\delta^2$ denotes a bound on the variance of stochastic gradients. We also present numerical results on a nonconvex/PL problem with synthetic data and on distributionally robust optimization problems with real data, illustrating our theoretical findings.

The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer $D$ for the total number of nestings. Standard Monte Carlo methods typically cost at least $\mathcal{O}(\varepsilon^{-(2+D)})$ and sometimes $\mathcal {O}(\varepsilon^{-2(1+D)})$ to obtain an estimator up to $\varepsilon$-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for $D = 1$. In this paper, we propose a novel Monte Carlo estimator called $\mathsf{READ}$, which stands for ``Recursive Estimator for Arbitrary Depth.'' Our estimator has an optimal computational cost of $\mathcal{O}(\varepsilon^{-2})$ for every fixed $D$ under suitable assumptions, and a nearly optimal computational cost of $\mathcal{O}(\varepsilon^{-2(1 + \delta)})$ for any $0 < \delta < \frac12$ under much more general assumptions. Our estimator is also unbiased, which makes it easy to parallelize. The key ingredients in our construction are an observation of the problem's recursive structure and the recursive use of the randomized multilevel Monte Carlo method.