We develop and analyze algorithms for instrumental variable regression by viewing the problem as a conditional stochastic optimization problem. In the context of least-squares instrumental variable regression, our algorithms neither require matrix inversions nor mini-batches and provides a fully online approach for performing instrumental variable regression with streaming data. When the true model is linear, we derive rates of convergence in expectation, that are of order $\mathcal{O}(\log T/T)$ and $\mathcal{O}(1/T^{1-\iota})$ for any $\iota>0$, respectively under the availability of two-sample and one-sample oracles, respectively, where $T$ is the number of iterations. Importantly, under the availability of the two-sample oracle, our procedure avoids explicitly modeling and estimating the relationship between confounder and the instrumental variables, demonstrating the benefit of the proposed approach over recent works based on reformulating the problem as minimax optimization problems. Numerical experiments are provided to corroborate the theoretical results.

翻译：我们将工具变量回归问题视为条件随机优化问题，进而开发并分析相应算法。在最小二乘工具变量回归框架下，所提算法既无需矩阵求逆运算，也不依赖小批量数据采样，为流数据环境下的工具变量回归提供了完全在线的解决方案。当真实模型为线性时，我们推导出期望收敛速率：在分别具备双样本预言机和单样本预言机的条件下，收敛速率分别为 $\mathcal{O}(\log T/T)$ 和 $\mathcal{O}(1/T^{1-\iota})$（其中 $\iota>0$，$T$ 为迭代次数）。特别值得注意的是，在双样本预言机可用的情况下，我们的方法无需显式建模和估计混淆变量与工具变量之间的关系，这凸显了本方法相对于近期将问题重构为极小极大优化问题的研究方案的优势。数值实验验证了理论结果的有效性。