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Primary Submission Category: Machine Learning and Causal Inference

Penalized Minimax Instrumental Variable Estimation with General Function Approximation

Authors: Masatoshi Uehara, Andrew Bennett, Nathan Kallus, Xiaojie Mao, Whitney Newey, Vasilis Syrgkanis, Masatoshi Uehara,

Presenting Author: Masatoshi Uehara*

We investigate instrumental variable estimation, which involves finding a solution for the equation E[Y-h(X)|Z]=0 with respect to h. It is widely utilized across a variety of fields and has seen significant progress with the use of flexible machine learning techniques. However, current approaches often have one or more of the following limitations: (1) a requirement for a unique solution, (2) an inability to accurately identify the solution with L2 rate guarantees, (3) a need for the conditional expectation operator to have smoothness in addition to the widely-accepted source condition, also known as the closedness assumption. We present the first method that avoids all of these limitations. Specifically, we propose a new minimax algorithm that has an L2 convergence guarantee under the source condition and realizability assumptions, and does not require uniqueness. This is achieved by viewing the least norm solution of E[Y-h(X)|Z]=0 as a saddle point of certain constrained minimax optimization problems, a novel perspective not taken in prior works such as Dikkala et al. 2020 and Liao et al. 2020. Finally, we demonstrate the applicability of our proposed estimator and theory to policy learning in the infinite horizon setting, a goal of great interest in fields such as mobile health and robotics.