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Stable Prediction with Model Misspecification and Agnostic Distribution Shift
Kun Kuang, Ruoxuan Xiong, Peng Cui, Susan Athey, Bo Li
TL;DR
The paper studies stable prediction when model misspecification and agnostic distribution shift make parameter estimation and prediction unstable across unknown test environments. It proposes Decorrelated Weighting Regression, which learns sample weights that decorrelate variables before weighted regression. The authors report improved parameter estimation accuracy and prediction stability across unknown test environments.
Problem
The paper asks how to achieve stable prediction across unknown environments when model misspecification and agnostic distribution shift jointly affect estimation and predictive stability.
Method
DWR jointly optimizes a variable decorrelation regularizer and weighted regression, using sample weights that reduce correlations among covariates.
Results
Theoretical analysis and empirical experiments show that DWR improves parameter estimation accuracy and prediction stability across unknown test environments.
Takeaways & Limitations
Reducing correlations among predictors helps prevent unstable features from proxying for omitted nonlinear functions of stable features.
Takeaways & Limitations
DWR performs worse than the baseline when rtest > 1.3, where unstable-feature correlations remain similar between training and test data and can aid prediction.
Abstract
from arXiv · showhide
For many machine learning algorithms, two main assumptions are required to guarantee performance. One is that the test data are drawn from the same distribution as the training data, and the other is that the model is correctly specified. In real applications, however, we often have little prior knowledge on the test data and on the underlying true model. Under model misspecification, agnostic distribution shift between training and test data leads to inaccuracy of parameter estimation and instability of prediction across unknown test data. To address these problems, we propose a novel Decorrelated Weighting Regression (DWR) algorithm which jointly optimizes a variable decorrelation regularizer and a weighted regression model. The variable decorrelation regularizer estimates a weight for each sample such that variables are decorrelated on the weighted training data. Then, these weights are used in the weighted regression to improve the accuracy of estimation on the effect of each variable, thus help to improve the stability of prediction across unknown test data. Extensive experiments clearly demonstrate that our DWR algorithm can significantly improve the accuracy of parameter estimation and stability of prediction with model misspecification and agnostic distribution shift.
Introduction
The paper addresses stable prediction when model misspecification coincides with distribution shifts toward unknown test environments. It proposes DWR, which jointly decorrelates variables and fits a weighted regression model.
- Domain adaptation and transfer-learning methods adjust training data toward test data but require test data as prior knowledge.
- Domain generalization and related methods use invariant structures across training datasets but may fail under unobserved distribution shifts.
- The setting distinguishes stable features with invariant causal effects from unstable features whose correlations can vary across environments.
- Model misspecification can arise from omitted nonlinear or interaction terms, causing parameter estimates to vary across environments with different covariate distributions.
- DWR jointly optimizes variable decorrelation and weighted regression to approximately isolate each variable’s effect when stable features are not known in advance.
- The paper reports extensive synthetic and real-world experiments demonstrating advantages for stable prediction under model misspecification and agnostic distribution shift.
Problem and Our Algorithm
The paper formulates stable prediction across unknown environments when model misspecification and distribution shifts can bias parameter estimates and destabilize predictions. It proposes decorrelating predictors through sample reweighting and combining the learned weights with weighted regression.
- Problem: Stable prediction seeks low Average Error and Stability Error when predicting across unknown environments from one training environment.Average Error and Stability Error are the mean and variance of predictive error across environments.
- Problem: Under model misspecification, omitted nonlinear or interaction terms can make unstable-feature correlations bias coefficient estimates and vary predictions across environments.The problem arises when unstable features correlate with omitted functions of stable features or with stable features themselves.
- Variable Decorrelation: When covariates are mutually independent, the paper states that correlations with omitted terms vanish, motivating reweighting to improve parameter estimation.The theoretical motivation links independence to unbiased estimation of the unstable-feature coefficient even when g(S) is omitted.
- Variable Decorrelation: The variable decorrelation regularizer learns sample weights that reduce correlations among predictors, with higher-order moments optionally incorporated to improve stability.The paper focuses on first moments, which are sufficient for variable decorrelation in its formulation.
- Analysis: The regularizer has a unique solution under stated growth and boundedness conditions, and its learned weights remove training-environment correlations asymptotically.The stated conditions include fixed p, λ3n ≫ p2 + λ4, p2 ≫ max(λ3, λ4), and bounded covariates.
- Our Algorithm: DWR jointly optimizes the variable decorrelation regularizer and weighted regression, and its weighted least-squares estimate is expected to have less bias than OLS under the decorrelation property.The learned weights are applied in weighted least squares to estimate regression coefficients.
Optimization and Analysis
DWR is optimized iteratively by alternating updates of sample weights and regression coefficients until convergence. Each iteration has computational complexity O(np2).
- Optimization: DWR initializes all sample weights to 1 and regression coefficients to zero, then alternates weight and coefficient updates until the objective converges.Each weight update fixes β, and each coefficient update fixes W.
- Complexity Analysis: O(np2) is the total complexity of each DWR iteration.Loss evaluation and weight updates each cost O(np2), while coefficient updates cost O(np).
Experiments
The paper evaluates DWR on synthetic and real-world datasets to examine its performance.
- Experiments: Experiments assess the algorithm on both synthetic and real-world datasets.
Baselines
The paper uses OLS, Lasso, Ridge Regression, and IILasso as baseline methods, with regularization settings intended to avoid degeneration.
- The four baselines are Ordinary Least Square, Lasso, Ridge Regression, and Independently Interpretable Lasso.
- The baseline formulations include a correlation-based matrix R with elements R_jk = |r_jk|/(1 − |r_jk|).
- The baseline hyperparameters λ1 and λ2 are set nonzero to avoid degeneration.
Evaluation Metrics
Performance is evaluated using prediction error, coefficient estimation error, average error, and stability error metrics.
- The evaluation uses RMSE, β Error, Average Error, and Stability Error.
- RMSE compares predicted outcomes ˆY_k with true outcomes Y_k across samples.
- β Error compares estimated regression coefficients ˆβ with true coefficients β.
Experiments on Synthetic Data
Synthetic experiments evaluate DWR under multiple feature relationships, outcome misspecifications, environment shifts, and training settings. DWR reduces predictor correlations through weighting and generally improves parameter estimation and prediction stability, with a setting-dependent limitation.
- Experimental settings: The experiments cover independent, causal, and reverse-causal relationships between stable features S and unstable features V.These settings are evaluated under synthetic data generation schemes involving S ⊥V, S →V, and V →S.
- Experimental settings: The synthetic outcomes include polynomial and exponential functions with omitted nonlinear or interaction terms of stable features.The outcome-generating functions combine linear effects with nonlinear terms involving stable variables.
- Experimental settings: Different environments are generated by varying the conditional distribution P(Vb|S) through bias rates r, changing the unstable correlation between Vb and Y.Positive and negative values of r represent positive and negative unstable correlations, with larger |r| indicating stronger correlation.
- Evaluation: The evaluation measures parameter-estimation error and predictive stability across unknown test environments using repeated training and test datasets.Training uses 50 independent datasets, while test performance is summarized with Average Error and Stability Error across environments.
- Results: DWR weighting removes correlations among predictors and reduces unstable correlations with omitted functions, improving estimation of stable and unstable effects.The weighted data strengthen the conditional relationship between stable features and the outcome while reducing reliance on unstable proxies.
- Results: DWR provides more stable prediction than baselines across settings, while OLS, Lasso, Ridge, and IILasso can assign excessive effects to unstable features.The reported advantage is attributed to reducing the use of unstable features as proxies for omitted nonlinear functions.
- Results: DWR is worse than baselines when rtest > 1.3 but much better when rtest < −1.3 under the reported training setting.When training and test correlations are similar, unstable features can be useful proxies; opposite correlations create greater instability.
- Results: Across varying sample size, dimension, training bias rate, and missing-term form, DWR consistently outperforms baselines on estimation and stable prediction.The paper concludes that DWR addresses stable prediction with model misspecification and agnostic distribution shift.
Experiments on Real World Data
The real-world experiment uses PM10 and meteorological data from 10 U.S. states, training models on one state and evaluating prediction stability across others. DWR is less accurate than baselines for small distribution shifts but improves as the shift increases and achieves the most stable predictions.
- Data and setting: The study uses PM10 pollution as the outcome and temperature, humidity, pressure, wind speed, and wind direction as observed meteorological features.The data come from the U.S. EPA’s Air Quality System database.
- Data and setting: Models are trained on State 1 and evaluated across 10 U.S. states representing different environments.The setup reflects training on one dataset before applying the model to related settings.
- Distribution shift: Distribution distance between environments is measured using differences in the means of observed variables X across state pairs.The metric uses the first moment and sums distances across variable dimensions.
- Prediction results: DWR performs worse than baselines when training and test distributions are close because reweighting introduces variance away from their similar distribution.This disadvantage occurs at small distribution distances.
- Limitations: The distribution-distance metric considers only the first moment, while other measures such as KL-divergence are left for future work.The passage notes that all moments can uniquely determine a variable’s distribution.
- Prediction results: DWR improves relative to the baseline and ultimately becomes better as distribution distance increases, while producing the most stable predictions under agnostic shift.The evaluation reports RMSE, Average Error, and Stability Error across the states.
Conclusion
The paper addresses stable prediction when model misspecification combines with unknown distribution shifts between training and test data. Theoretical analysis and experiments show that the proposed algorithm improves parameter-estimation accuracy and prediction stability, outperforming baselines when covariate correlations vary substantially across environments.
- The paper targets instability caused jointly by model misspecification and agnostic distribution shift across unknown test data.
- The algorithm is supported by theoretical analysis and empirical experiments on synthetic and real-world datasets.
- The experiments show improved parameter-estimation accuracy and prediction stability across unknown test environments.
- The algorithm outperforms baselines when covariate correlations vary substantially across environments.