Direct Acceleration of Stochastic Root-Finding Without Variance Reduction and Regularization

We study Polyak-type step-size selection for extragradient methods for solving deterministic and stochastic monotone root-finding problems. We show that the known projection-type correction for deterministic extragradient arises from minimizing an upper bound on the distance to a solution, paralleling the classical Polyak step-size construction. Using this viewpoint, we provide a unified deterministic analysis of the Polyak-type Extragradient Method (PolyakEG), based on a local critical condition controlling the variation of operator $F$ along the extrapolation direction. This analysis does not require global Lipschitz continuity, and covers sublinear convergence under broader conditions such as Hölder continuity or $(L_0, L_1)$-Lipschitzness and linear convergence under additional strong monotonicity, all through a single framework. We then study the stochastic extensions of this approach. We first prove convergence of a direct stochastic variant, PolyakSEG, when all stochastic component operators share a common solution. We also show that, without this condition, PolyakSEG with nonvanishing step-sizes may fail to converge to a zero of the mean operator. To address this limitation, we propose DecPolyakSEG, which combines decreasing step-sizes with Polyak-type updates, and establish a sublinear residual convergence result without requiring a common solution across the component operators. These results parallel recent developments in stochastic Polyak step-sizes from the convex minimization literature and establish an analogous research avenue in the broader root-finding regime.

Many central machine learning tasks, from entropy tuning in reinforcement learning to equilibrating generative adversarial networks, are fundamentally stochastic root-finding problems rather than loss minimization. Yet, they are frequently forced into a minimization framework via squared residuals, introducing a critical flaw we identify as the Variance Trap. Standard bilevel minimization algorithms require estimating hypergradients involving implicit Jacobians; in stochastic settings, these terms act as noise amplifiers, destabilizing convergence. We formalize Root-Finding Bilevel Optimization (RF-BO) as a distinct problem class that bypasses this pathology. We propose a Jacobian-free solution using Two-Time-Scale Stochastic Approximation (TTSA) that updates directly along the root error, structurally avoiding variance amplification. We provide the first non-asymptotic convergence guarantees for TTSA in this setting under Markovian noise. Extensive experiments demonstrate the decisive advantage of this paradigm: compared to squared-residual and implicit-gradient baselines, our framework achieves a 2.6% top-1 accuracy gain in SimCLR, 17$\times$ faster convergence in non-linear ODE control where baselines fail, significantly improved entropy stability in reinforcement learning, and an 11.1% quality improvement in generative modeling.

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $ε> 0, δ\in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq ε$ with probability at least $1-δ$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $τ(\mathbf{x}; ξ)$ itself, we clip stochastic differences at the Lipschitz scale $γ\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - δ$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $ε$ and Lipschitz constant $γ\in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{ε^{-5}, (1-γ)^{-3}ε^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $ε^{-3}$ nonexpansive rate (i.e., for $γ= 1$), and under samplewise nonexpansiveness to $ε^{-2}$.