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Stable Prediction across Unknown Environments

Kun Kuang, Ruoxuan Xiong, Peng Cui, Susan Athey, Bo Li

arXiv:1806.06270v2cs.LGstat.ML

TL;DR

The paper studies stable prediction when training and unknown test environments differ, a setting where existing approaches may require test-distribution knowledge or diverse training datasets. It proposes DGBR, jointly combining deep auto-encoding and global balancing, and reports theoretically and empirically stable predictions that outperform baselines on synthetic and real-world data.

  • Problem

    Stable prediction is needed when unknown test environments differ from training data, while existing methods may require test-distribution knowledge or diverse training datasets.

  • Method

    DGBR jointly optimizes a deep auto-encoder for dimensionality reduction with global balancing to isolate covariate effects and support prediction.

  • Results

    Theoretical analysis and experiments on synthetic and real-world datasets show stable prediction across unknown environments and outperforming baseline methods.

  • Takeaways & Limitations

    DGBR provides a stable-prediction approach for unknown environments by combining feature-space reduction with global balancing.

  • Takeaways & Limitations

    The method assumes no unobserved confounders and may only approximately balance all covariates.

Abstract

from arXiv · show

In many important machine learning applications, the training distribution used to learn a probabilistic classifier differs from the testing distribution on which the classifier will be used to make predictions. Traditional methods correct the distribution shift by reweighting the training data with the ratio of the density between test and training data. In many applications training takes place without prior knowledge of the testing distribution on which the algorithm will be applied in the future. Recently, methods have been proposed to address the shift by learning causal structure, but those methods rely on the diversity of multiple training data to a good performance, and have complexity limitations in high dimensions. In this paper, we propose a novel Deep Global Balancing Regression (DGBR) algorithm to jointly optimize a deep auto-encoder model for feature selection and a global balancing model for stable prediction across unknown environments. The global balancing model constructs balancing weights that facilitate estimating of partial effects of features (holding fixed all other features), a problem that is challenging in high dimensions, and thus helps to identify stable, causal relationships between features and outcomes. The deep auto-encoder model is designed to reduce the dimensionality of the feature space, thus making global balancing easier. We show, both theoretically and with empirical experiments, that our algorithm can make stable predictions across unknown environments. Our experiments on both synthetic and real world datasets demonstrate that our DGBR algorithm outperforms the state-of-the-art methods for stable prediction across unknown environments.

1. Introduction

The paper addresses stable prediction when unknown test environments differ from training data, especially when retraining is difficult. It proposes DGBR, which combines dimensionality reduction and global balancing to support stable prediction through isolated feature effects.

  • Existing challenges: Distribution-shift methods reweight training data using test-to-training density ratios, requiring prior knowledge of the test distribution.
  • Existing challenges: Unknown-environment methods based on invariant structure rely on diverse training datasets from multiple environments.
  • Problem setting: Stable prediction is framed around outcomes whose conditional expectation given all covariates remains stable across environments.
  • Problem setting: When many causal features exist relative to observations, regularized regression may fail to consistently estimate partial effects.
  • Proposed method: DGBR jointly optimizes a deep auto-encoder for dimension reduction, balancing weights for isolating covariate effects, and a predictive model.
  • Results: Theoretical analysis and experiments on synthetic and real-world datasets establish stable prediction across unknown environments and outperform baseline methods.

2. Related Work

Related work addresses distribution shift through density-ratio reweighting, causal variable balancing, and invariant learning. These approaches differ in whether they require test-distribution knowledge or multiple diverse training environments.

  • Covariate shift: Covariate-shift methods reweight training observations by the ratio of test to training densities.
  • Covariate shift: Density-ratio approaches can correct distribution shift but require prior knowledge of the test distribution.
  • Variable balancing: Covariate-balancing methods were developed to adjust for confounders when estimating causal effects in observational studies.
  • Variable balancing: Prior work includes data-driven decomposition, nonlinear-representation matching, entropy balancing, and approximate residual balancing.
  • Invariant learning: Invariant-learning and domain-generalization methods identify predictors or representations that remain invariant across multiple datasets.

3. Problem and Our Algorithm

The paper formulates stable prediction across unknown environments around stable features whose conditional relationship with the outcome remains invariant, then develops balancing and representation-learning components to estimate that relationship in high dimensions.

  • Problem Formulation: Stable prediction seeks low Average Error and low Stability Error when training on one environment and predicting across unknown environments.Stability Error is defined so that smaller values indicate better stability.
  • Problem Formulation: Assumption 1 requires Pr(Y = y|S = s, V = v) = Pr(Y = y|S = s) = P(y|s) across all environments.Stable features S determine the invariant conditional outcome relationship, while V denotes the remaining noisy features.
  • Global Balancing Regression: Global balancing learns sample weights that successively treat each variable as a treatment and balance it against the remaining variables.The method targets covariate imbalance and can incorporate higher-order moments through interaction features.
  • Global Balancing Regression: After reweighting, stable features can be identified by checking covariate-by-covariate correlations between features and outcomes.The stated rationale is that only stable features remain correlated with Y after global sample reweighting.
  • Deep Global Balancing Regression: Global balancing becomes challenging when features are numerous relative to observations, because estimating feature effects, interactions, and weights may be difficult.The deep auto-encoder is introduced to address these high-dimensional challenges.
  • Deep Global Balancing Regression: DGBR jointly optimizes a deep auto-encoder and Global Balancing Regression to reduce dimensionality, capture nonlinear interactions, and simplify global balancing.The encoder maps inputs to low-dimensional representations and the decoder reconstructs the original input space.

4. Theoretical Analysis

The analysis establishes conditions under which global balancing supports stable prediction and examines how finite samples and dimensionality affect imbalance and risk. It also explains how DGBR uses dimensionality reduction and nonlinear modeling to extend these properties.

  • 4.1 Analysis on Stable Prediction: Under overlap, exact balancing weights exist with sufficient data and make covariates mutually independent after reweighting.The result requires all covariate realizations to appear in the sample.
  • 4.1 Analysis on Stable Prediction: After reweighting, only stable features remain correlated with outcomes, while p(y|s) is unchanged across environments.This supports stable prediction when the conditional outcome given all covariates is stable across environments.
  • 4.2.4 Properties of DGBR: DGBR reduces dimensionality to ease overlap and balancing, while nonlinear mappings improve estimation of p(y|s) with many stable features.The dimensionality reduction is motivated by the Johnson–Lindenstrauss lemma.
  • 4.2.2 Upper Bound of Global Balancing Regularizer: As p grows with fixed n, overlap becomes harder to satisfy empirically because more covariate values are absent from the sample.The number of missing covariate values m can increase, raising imbalance α.
  • 4.2.2 Upper Bound of Global Balancing Regularizer: The expected maximum imbalance E[α] decreases with sample size n but increases with feature dimension p.Figure 3 illustrates this dependence.
  • 4.2.3 Upper Bound of the Risk in Approximate Balancing: When exact balancing is unattainable, the risk bound worsens with covariate imbalance and with more complex auto-encoder or regression model classes.The analysis treats the learned weights as an approximate solution when the balancing objective cannot reach zero.

5. Optimization and Discussion

DGBR is optimized by alternating updates to balancing weights, regression coefficients, and auto-encoder parameters. The procedure has per-iteration complexity O(npd) and uses diverse validation distributions to assess stability.

  • 5.1 Optimization: The objective is minimized iteratively over balancing weights W, regression coefficients β, and auto-encoder parameters θ.Each parameter block is updated while the other two are held fixed, until convergence or the iteration limit.
  • 5.1 Optimization: Regression coefficients use an ℓ1-regularized least-squares update, while nonnegative balancing weights are parameterized as W = ω ⊙ω.The weight update uses derivatives of the mixed objective.
  • 5.1 Optimization: With optimized β and θ, DGBR makes stable predictions on agnostic test datasets.This is the stated operational endpoint of the optimization procedure.
  • 5.2 Complexity: O(npd) is the complexity of each Algorithm 1 iteration, where n is sample size, p is observed-variable dimension, and d is maximum hidden-layer dimension.The dominant costs are evaluating the loss and updating W, β, and θ.
  • 5.3 Parameter Tuning: Validation datasets are generated by resampling training data to create distributions diverse from both one another and the training distribution.Average Error and Stability Error are computed using RMSE for parameter tuning.

6. Experiments

Synthetic experiments vary causal relationships, sample size, dimensionality, and bias across environments. DGBR consistently provides more stable and precise predictions than the evaluated baselines, with larger gains when data are relatively scarce or biased.

  • Experimental design: The experiments vary relationships between stable and noisy features, sample size n, variable dimension p, and training bias rate r across synthetic environments.The settings include S ⊥V, S →V, and V →S, with n={1000,2000,4000}, p={20,40,80}, and r={0.65,0.75,0.85}.
  • Baseline comparisons: LR and DLR cannot address stable prediction because they retain spurious correlations between noisy features and the response.These methods therefore estimate large noisy-feature effects, producing instability across environments.
  • Baseline comparisons: GBR is more stable than LR, while DGBR is more stable than DLR across the evaluated settings.Global balancing helps estimate stable-feature effects accurately and reduce noisy-feature effects.
  • Model comparison: DGBR produces more precise and stable predictions than GBR because its deep embedding makes global balancing weights less noisy.The embedding also simplifies estimation of stable-feature effects.
  • Robustness: DGBR’s RMSE remains consistently small and stable across environments as n, p, and r vary.Its prediction improvements become more significant when n is small relative to p and r.
  • Parameter analysis: Increasing balancing strength initially lowers Average Error and Stability Error, while feature-representation strength requires an appropriate choice and weight-variance constraints can become harmful when excessive.The reported hyperparameters are λ1, λ2, and λ3, respectively.

6.3 Experiments on Real World Data

Real-world experiments use WeChat advertising feedback and test robustness under biased sampling and age-separated environments. DGBR is competitive in the training-age environment and outperforms baselines when outcome prevalence is fixed across environments.

  • Dataset: The online advertising dataset contains 14,891 Likes and 93,108 Dislikes from WeChat users, with demographic, social, device, and settings features.The campaign concerns LONGCHAMP handbags for young women.
  • Experimental settings: The real-world evaluation creates environments either by biased sampling on outcome-associated noisy features or by separating users according to age.The age groups are [20,30), [30,40), [40,50), and [50,100).
  • Biased-sampling results: Under biased sampling, models are trained at bias rate r=0.6 and tested across different bias rates while RMSE, Average Error, and Stability Error are reported.The comparison includes DGBR and the baseline methods.
  • Age-separated results: DGBR achieves comparable performance to baselines on test users aged 20–30, the age range used for training.The age-separated evaluation tests generalization to the other three age environments.
  • Scope of stability: DGBR is less stable than baselines when outcome prevalence varies across age environments, but outperforms them after P(Y) is fixed globally.The fixed prevalence is P(Y=1)=14,891/(14,891+93,108).

7. Conclusion

The conclusion presents DGBR as a method for stable prediction when future environments are unknown. It combines global balancing with deep representation learning and reports theoretical and empirical support for outperforming baselines.

  • Conclusion: DGBR targets stable prediction when unknown environments may differ substantially from the training distribution.The paper contrasts this setting with methods requiring test-distribution knowledge or diverse training datasets.
  • Conclusion: DGBR jointly optimizes a deep auto-encoder and a global balancing model.The balancing model identifies causal relationships, while the auto-encoder captures nonlinear structure and makes balancing easier and less noisy.
  • Conclusion: Theoretical analysis and experiments on synthetic and real-world datasets show stable prediction and outperforming baseline performance.Both forms of evidence are explicitly reported in the conclusion.

Appendix A. Proof of Lemma 1

The appendix proof establishes the lemma by applying the overlap condition to each feature in turn. It treats each Xi,k as a possible treatment variable and repeats the argument across variables.

  • Proof: The proof assumes treatment variable T=Xi,j and covariates Xi,−j, then uses propensity scores bounded away from zero and one.The displayed proof also invokes a positive joint-probability condition for feature configurations.
  • Proof: Each other feature Xi,k can likewise be treated as the treatment variable in the same argument.The proof explicitly repeats the reasoning for all variables one by one.

Appendix B. Proof of Proposition 3

The proof uses a balancing weight satisfying equation (4) to derive a factorized weighted joint distribution. This factorization implies independence among the covariates.

  • The proof first establishes that the constructed W ∗ satisfies equation (4) when 0 < ˆP(Xi = x) < 1.
  • A weight satisfying equation (4) makes the weighted joint distribution factorize into the product of weighted marginal distributions.
  • The factorization directly implies that the covariates in X are independent after balancing by W ∗.

Appendix C. Proof of Lemma 5

The proof establishes the bound through cases indexed by m, the number of selected values, using symmetry assumptions and Theorem 2 for the base case.

  • When m = 0, α = 0 follows directly from Theorem 2.
  • For 0 < m ≤ 2p−2, the proof assumes without loss of generality that m2 ≥ m4.
  • For 2p−2 < m < 2p−1, the proof again assumes m2 ≥ m4 and distinguishes the case m2 ≤ 2p−2.
  • The resulting expression is written as 1 − 2p−1−m divided by 3 × 2p−2−m.
  • The case construction satisfies m2 + m4 = 1, m1 ≤ m2, and m3 ≤ m4.

Appendix D. Proof of Theorem 6

The proof expresses the probability that m distinct values in X do not appear as a ratio of constrained to unconstrained solution counts. It then identifies the resulting probability using g(p, m).

  • The probability that m different values in X do not appear is represented as a ratio of solution counts.
  • The denominator counts all solutions to Eq. (31) without constraints.
  • The numerator selects m distinct indices with yi = 0 and counts solutions to y1 + y2 + · · · + y2p−m = n.
  • The resulting probability is expressed using g(p, m), defined in equation (12).

Appendix E. Proof of Theorem 7

The proof bounds prediction loss under the weighted distribution using the model class’s complexity and the relationship between weighted and original risks. The auto-encoder’s decoder does not contribute to prediction complexity.

  • The weighted distribution preserves P(Yi = y|Xi = x) while replacing the covariate marginal with ˜px.
  • The predictor ˜f∗ minimizes weighted loss L ˜P (f), while ˆf is evaluated against the target loss LP ( ˆf).
  • Because X and Y are binary and model weights are bounded, maxx E[l( ˆf(x), y)|x] is bounded.
  • The generalization bound uses the empirical Rademacher complexity of the cross-entropy loss class with probability at least 1 − δ.
  • The auto-encoder constructs φ(Xi), predicts Yi by logistic regression on φ(Xi), and reconstructs Xi through a decoder.
  • The decoder does not predict Yi and therefore does not affect the complexity Rn(A).
  • The complexity bound is developed under layer-wise weight constraints and combines two norm cases before substitution into the preceding inequalities.
  • The proof separately bounds the difference between L ˜P ( ˜f∗) and LP (f∗) before combining the inequalities.
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