Causal treatment effect estimation is a key problem that arises in a variety ofreal-world settings, from personalized medicine to governmental policy making.There has been a flurry of recent work in machine learning on estimating causaleffects when one has access to an instrument. However, to achieve identifiability,they in general require one-size-fits-all assumptions such as an additive error modelfor the outcome. An alternative is partial identification, which provides boundson the causal effect. Little exists in terms of bounding methods that can deal withthe most general case, where the treatment itself can be continuous. Moreover,bounding methods generally do not allow for a continuum of assumptions onthe shape of the causal effect that can smoothly trade off stronger backgroundknowledge for more informative bounds. In this work, we provide a method forcausal effect bounding in continuous distributions, leveraging recent advancesin gradient-based methods for the optimization of computationally intractableobjective functions. We demonstrate on a set of synthetic and real-world datathat our bounds capture the causal effect when additive methods fail, providinga useful range of answers compatible with observation as opposed to relying onunwarranted structural assumptions
