We derive and implement nonlinear extensions of the classical instrumental variable regression (IVR) technique. Our key insight is that even in the nonlinear setting, finding a causally consistent estimate of a structural equation is equivalent to satisfying constraints on conditional outcome moments. This insight allows us to leverage standard constrained optimization techniques to reframe the work of Dikkala et al. as optimizing a regularized Lagrangian and reveal underlying smoothness assumptions. We then propose a new algorithm, CausAL, that instead optimizes an augmented Lagrangian, requiring a different definition of smoothness and no adversarial training. We then extend our method to handle matching outcome distributions instead of just expected values, propose an efficient no-regret procedure, and implement a practical realization via a modification of an Integral Probability Metric (IPM) GAN which we call ACADIMI.