Abstract: We develop an asymptotic optimality theory for a wide range of statistical decision problems in which the decision maker uses data to select an action from a continuum of alternatives. Using Le Cam's limits of experiments approach, we jointly approximate the loss function and data distribution, which leads to a limiting decision problem that is asymptotic equivalent to the original decision problem. We then derive asymptotically optimal decision rules under average and minmax risk criteria. Our framework accommodates non-smooth loss functions and semiparametric models that are common in empirical applications. We apply this framework to study a treatment choice problem where the treatment effect is only partially identified. A simulation study shows that the proposed asymptotically optimal rules significantly reduce risk compared to standard plug-in methods when the loss function is non-smooth. We apply our findings to an empirical treatment choice problem based on McKenzie (2017).