Train a primal state network and a dual flux network jointly, using the convex primal-dual gap as the main loss and as an a posteriori certificate of state error. Unlike a strong residual, the certificate is based on monotonicity and convex duality, so it can remain informative even when differentiating rapidly oscillatory coefficients would amplify noise by $1/\varepsilon$.
Use neural networks to estimate outcome and treatment nuisances, then edit the resulting debiasing weights so that residualized treatment is conditionally orthogonal to an adversarial class of covariate functions. This should reduce coefficient bias when the two nuisance networks have strongly imbalanced approximation errors, without requiring either network to be correctly specified.
Use the paper's lower bound as a feasibility test for robust interpolation: if a model is asked to fit below the estimated noise floor while maintaining a small Lipschitz constant, automatically increase effective width or relax the fit target. This prevents optimization from wasting compute on an impossible low-sensitivity solution and provides a principled width schedule for noisy regression or classification.