Solving Stochastic Fixed-Point Equations with High Probability

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 $\epsilon > 0, \delta \in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq \epsilon$ with probability at least $1-\delta$. 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 $\tau(\mathbf{x}; \xi)$ itself, we clip stochastic differences at the Lipschitz scale $\gamma\|\mathbf{x} - \mathbf{y}\|$. 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 $1 - \delta$, 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 $\epsilon$ and Lipschitz constant $\gamma \in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{\epsilon^{-5}, (1-\gamma)^{-3}\epsilon^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $\epsilon^{-3}$ nonexpansive rate (i.e., for $\gamma = 1$), and under samplewise nonexpansiveness to $\epsilon^{-2}$.