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Optimization-based Causal Estimation from Heterogenous Environments

Mingzhang Yin, Yixin Wang, David M. Blei

TL;DR

Co is proposed, an optimization algorithm that provides more accurate estimates of the causal model and more accurate predictions under interventions, and maximizes an objective for which the only solution is the causal solution.

Abstract

This paper presents a new optimization approach to causal estimation. Given data that contains covariates and an outcome, which covariates are causes of the outcome, and what is the strength of the causality? In classical machine learning (ML), the goal of optimization is to maximize predictive accuracy. However, some covariates might exhibit a non-causal association with the outcome. Such spurious associations provide predictive power for classical ML, but they prevent us from causally interpreting the result. This paper proposes CoCo, an optimization algorithm that bridges the gap between pure prediction and causal inference. CoCo leverages the recently-proposed idea of environments, datasets of covariates/response where the causal relationships remain invariant but where the distribution of the covariates changes from environment to environment. Given datasets from multiple environments-and ones that exhibit sufficient heterogeneity-CoCo maximizes an objective for which the only solution is the causal solution. We describe the theoretical foundations of this approach and demonstrate its effectiveness on simulated and real datasets. Compared to classical ML and existing methods, CoCo provides more accurate estimates of the causal model and more accurate predictions under interventions.

Optimization-based Causal Estimation from Heterogenous Environments

TL;DR

Co is proposed, an optimization algorithm that provides more accurate estimates of the causal model and more accurate predictions under interventions, and maximizes an objective for which the only solution is the causal solution.

Abstract

This paper presents a new optimization approach to causal estimation. Given data that contains covariates and an outcome, which covariates are causes of the outcome, and what is the strength of the causality? In classical machine learning (ML), the goal of optimization is to maximize predictive accuracy. However, some covariates might exhibit a non-causal association with the outcome. Such spurious associations provide predictive power for classical ML, but they prevent us from causally interpreting the result. This paper proposes CoCo, an optimization algorithm that bridges the gap between pure prediction and causal inference. CoCo leverages the recently-proposed idea of environments, datasets of covariates/response where the causal relationships remain invariant but where the distribution of the covariates changes from environment to environment. Given datasets from multiple environments-and ones that exhibit sufficient heterogeneity-CoCo maximizes an objective for which the only solution is the causal solution. We describe the theoretical foundations of this approach and demonstrate its effectiveness on simulated and real datasets. Compared to classical ML and existing methods, CoCo provides more accurate estimates of the causal model and more accurate predictions under interventions.

Paper Structure

This paper contains 37 sections, 12 theorems, 64 equations, 9 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Causal optimization from heterogeneous environments
  4. Setup and assumptions
  5. Assumptions for each environment.
  6. Remark.
  7. Assumptions across environments.
  8. Idealized causal optimization
  9. Derivation of a tractable objective
  10. Directional derivatives and feasible directions.
  11. Optimality condition for causal optimization.
  12. Relaxing the first-order condition.
  13. Optimization with multiple environments
  14. Narrowing down the optima set by environments.
  15. Removing the non-informative solution from the optima set.
  16. ...and 22 more sections

Key Result

Lemma 1

[lemma]lem:optimal Under Assumptions assp:sem, for the squared risk function $R(\boldsymbol{\alpha}) = \mathbb{E}[(1/2)(\hat{y}(\boldsymbol{x}; \boldsymbol{\alpha}) - y)^2]$, and linear predictor $\hat{y}(\boldsymbol{x}; \boldsymbol{\alpha})=\boldsymbol{\alpha}^\top\boldsymbol{x}$, the following con has the causal coefficients $\boldsymbol{\alpha} = {\boldsymbol{\beta}}$ as the unique solution.

Figures (9)

  • Figure 1: Geometry of the analytic optima sets for the IRM regularization and CoCo objectives in the 3D space. The causal coefficient is ${\boldsymbol{\beta}} = (3,2,0)$. (a),(b): the optima of the IRM regularization for each environment form a 3D quadric surface; (c): the optima of the IRM regularization with two environments is the intersection of the two surfaces, which forms two elliptic curves; (d): The optima of CoCo objective is a discrete finite set for each environment. The CoCo optima over two environments is the intersection consisting of the zero point and the causal coefficient (the overlap of the black triangle and blue star points); (e): The top view of the IRM regularization and CoCo optima for the two environments. The optima set by CoCo ($\boldsymbol{\alpha} \in \{(0,0,0),(3,2,0)\}$) is a strict subset of that by the IRM regularization (the dashed orange elliptic curves). Better viewed in color.
  • Figure 2: The graphs for the simulation studies in \ref{['sec:linear-sync']}. The case ID of each graph is in the rectangle box. The blue arrow represents a path whose parameter varies across environments, the blue circle of a covariate means its distribution given parents varies across environments, and the blue circle of outcome means the variance of its additive noise varies across environments. Invariance \ref{['eq:invariance']} holds in all cases. The shaded nodes are the variables that are observed.
  • Figure 3: Prediction accuracy for and with linear and nonlinear predictors. The heatmap is the prediction error $(\hat{y} - \mathbb{E}[y|x])^2$, the x-axis, y-axis are the values of input $x_1$ and $x_2$. The orange points are training data from two environments. has better out-of-sample generalization with a wider region of low error (blue region) than ERM for both linear and nonlinear predictors.
  • Figure 4: The change of test predictive error of with different levels of invariance, number of environments, and the hyperparameter. The dashed line is the error rate for reference. The error bar is the standard deviation over 5 trials.
  • Figure 5: Trace plot of training and testing accuracy for , and on GMM and Colored MNIST data. In panel (b), the accuracy is measured on predicting the noised label$y^e$. has the highest prediction accuracy in a new environment.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Lemma 1: Causal Optimality
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • Lemma 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Corollary 9
  • Corollary 10
  • ...and 2 more