Accelerated Variance-Reduced Forward-Reflected Methods for Root-Finding Problems

We propose a novel class of Nesterov's stochastic accelerated forward-reflected-based methods with variance reduction to solve root-finding problems under $\frac{1}{L}$-co-coerciveness. Our algorithm is single-loop and leverages a new family of unbiased variance-reduced estimators specifically designed for root-finding problems. It achieves both $\mathcal{O}(L^2/k^2)$ and $o(1/k^2)$-last-iterate convergence rates in terms of expected operator squared norm, where $k$ denotes the iteration counter. We instantiate our framework for two prominent estimators: SVRG and SAGA. By an appropriate choice of parameters, both variants attain an oracle complexity of $\mathcal{O}( n + Ln^{2/3}\epsilon^{-1})$ to reach an $\epsilon$-solution, where $n$ represents the number of summands in the finite-sum operator. Furthermore, under $\mu$-strong quasi-monotonicity, our method achieves a linear convergence rate and an oracle complexity of $\mathcal{O}(n+ \kappa n^{2/3}\log(\epsilon^{-1}))$, where $\kappa := \frac{L}{\mu}$. We extend our approach to solve a class of finite-sum monotone inclusions, demonstrating that our schemes retain the same theoretical guarantees as in the equation setting. Finally, numerical experiments validate our algorithms and demonstrate their promising performance compared to state-of-the-art methods.