Approximation Rates for Schrödinger Bridge Potentials via Fixed-Point ERM

The Schrödinger bridge problem (SBP) provides a principled interpolation between two distributions by selecting, among all path measures matching given endpoint marginals, the one closest in relative entropy to a reference dynamics. In modern applications the marginals are observed only through samples, and standard computational pipelines solve a discretized SBP via {Sinkhorn} iterations and then heuristically extend the resulting dual potentials off-sample, entangling statistical, optimization, and smoothing errors. We study a learning-theoretic alternative based on a fixed-point characterization of a single \emph{transformed} Schrödinger potential $g^\star$, and we focus on quantitative approximation of $g^\star$ by a sample-based estimator $\widehat g$ that is continuous by construction. To address the intrinsic scaling ambiguity of Schrödinger potentials, we introduce a normalized, scale-invariant operator and analyze its local geometry around $g^\star$. Our main theoretical contribution is a stability result linking the error of the fixed-point residual to a distance to the solution $g^\star$ via analysis of spectral-gap property for the {Fréchet} derivative of the operator in a norm $\|\cdot\|$ being the sum of a localized {Hilbert} tangent seminorm and an $L^2$ distance. Combining this stability bound with the excess risk bounds and approximation error yields explicit non-asymptotic rates for $\|\widehat g-g^\star\|$. We illustrate performance of the suggested approach with numerical experiments.