Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Data (2405.19463v1)

Abstract: 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.

• The paper presents two novel algorithms, TOSG-IVaR and OTSG-IVaR, that compute unbiased stochastic gradients for IV regression using streaming data.
• It achieves theoretical convergence rates of O(log T/T) and O(1/T^(1-ι)) by leveraging a two-sample oracle and coupled update strategies.
• Empirical results demonstrate the methods' robust performance across various data scenarios, outperforming traditional approaches like the O2SLS algorithm.

Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Data

Instrumental Variable Regression (IVaR) constitutes a critical tool for resolving endogeneity issues in regression models, a scenario where independent variables are correlated with error terms. This paper contributes to the field by presenting novel streaming algorithms to address IVaR challenges similar to more traditional batch learning approaches.

The authors propose two algorithmic frameworks optimized for streaming data scenarios. These methods are rigorously analyzed and provide efficient means to execute IVaR without requiring matrix inversions, mini-batches, or resorting to complicated approximations via minimax reformulations.

Two-Sample One-Stage Stochastic Gradient-IVaR (TOSG-IVaR)

Concept and Algorithm

The primary paradigm introduced in the paper involves a two-sample oracle capable of producing two independent samples from PP(X|Z) for a given instrumental variable Z and one sample from PP(Y|X). The algorithm leverages these samples to construct an unbiased stochastic gradient estimator for the problem

$$\nabla F(\theta) = \mathbb{E}_Z \left[(\mathbb{E}_{X|Z}[g(\theta;X)] - \mathbb{E}_{Y|Z} [Y])\nabla_\theta \mathbb{E}_{X|Z}[g(\theta;X)]\right],$$

where g parameterizes the model of interest. This approach results in the following gradient update: $$v(\theta) = [g(\theta; X) - Y]\nabla_\theta g(\theta; X'),$$

where X and X' are independent samples from PP(X|Z). This online algorithm avoids the need to explicitly model the relation between X and Z, efficiently overcoming the complications associated with the "forbidden regression."

Convergence and Analysis

Assuming the model is linear, the authors prove that TOSG-IVaR exhibits a convergence rate of $\mathcal{O}(\log T / T)$. This is achieved by utilizing natural moment assumptions in the context of the IVaR problem. Their analysis innovatively deals with issues such as the product of two conditional expectations in gradient computation, surpassing state-of-the-art by avoiding nested sampling techniques.

One-Sample Two-Stage Stochastic Gradient-IVaR (OTSG-IVaR)

Concept and Algorithm

For scenarios lacking the aforementioned two-sample oracle, the authors introduce OTSG-IVaR. This method predicts the mean of X_t given Z_t using a dynamically updated parameter and subsequently computes the stochastic gradient. The updates for theta_t and 'gamma_t` are coupled as below:

$$\begin{aligned} \theta_{t+1} &= \theta_t - \alpha_t \gamma_t^\top Z_t (Z_t^\top \theta_t - Y_t), \ \gamma_{t+1} &= \gamma_t - \beta_t Z_t (Z_t^\top \gamma_t - X_t^\top). \end{aligned}$$

To mitigate potential divergence in updates due to erroneous parameter estimation, the authors provide a choice of stepsizes $\alpha_t$ and $\beta_t$ that ensure convergence in expectation at a rate of $\mathcal{O}(1 / T^{1 - \iota})$.

Convergence and Analysis

Analyzing OTSG-IVaR presents additional challenges due to the coupled nature of updates and the possibly non-uniformly bounded variance of the stochastic gradients. Despite this, the authors successfully establish theoretical guarantees for convergence, addressing complications associated with biases and dependencies in gradient estimates.

Numerical Experiments

Both algorithms are subject to rigorous empirical evaluation. Experiments demonstrate their robust performance under various realistic scenarios, corroborating theoretical results. Specifically:

1. TOSG-IVaR showed consistent performance across linear and quadratic models, even when subjected to different dimensions and noise scales.
2. OTSG-IVaR exhibited superior performance compared to traditional methods such as the O2SLS algorithm, showcasing lower variance and reliable convergence across diverse settings.

Practical and Theoretical Implications

These contributions have profound implications for both theoretical research and practical applications of IVaR:

• Practical Impact: The proposed methods enable real-time processing of streaming data, significantly benefiting fields with continuous data acquisition such as finance, healthcare, and online advertising.
• Theoretical Contributions: The paper advances the theory of stochastic optimization for nested and coupled gradient structures, providing a blueprint for future research in conditional stochastic optimization problems.

Future Directions

Potential avenues for future research include extending these algorithms to non-linear models more rigorously and considering scenarios with more complex dependence structures between the instruments and the endogenous variables. Additionally, integrating these methods with more sophisticated loss functions, like those used in classification tasks, could broaden their applicability and improve their utility in more varied data environments.

In conclusion, this paper provides significant advancements in the development of efficient, streaming-capable IV regression methods, circumventing limitations of existing approaches, and demonstrating promising theoretical and empirical performance.