Abstract: Maximum Inner Product Search (MIPS) is a ubiquitous task in machine learning applications. Given a query vector and $n$ other vectors in $d$ dimensions, the MIPS problem is to find the atom that has the highest inner product with the query vector. Existing MIPS algorithms scale at least as $O(\sqrt{d})$ with respect to $d$, which becomes computationally prohibitive in high-dimensional settings. In this work, we present BanditMIPS, a novel randomized algorithm that provably improves the state-of-the-art complexity from $O(\sqrt{d})$ to $O(1)$ with respect to $d$. We validate the scaling of BanditMIPS and demonstrate that BanditMIPS outperforms prior state-of-the-art MIPS algorithms in sample complexity, wall-clock time, and precision/speedup tradeoff across a variety of experimental settings. Furthermore, we propose a variant of our algorithm, named BanditMIPS-$\alpha$, which improves upon BanditMIPS by employing non-uniform sampling across coordinates. We also demonstrate the usefulness of BanditMIPS in problems for which MIPS is a subroutine, including Matching Pursuit and Fourier analysis. Finally, we demonstrate that BanditMIPS can be used in conjunction with preprocessing techniques to improve its complexity with respect to $n$. All of our experimental results are reproducible via a 1-line script at github.com/ThrunGroup/BanditMIPS.