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Causality-Inspired Robustness for Nonlinear Models via Representation Learning

Marin Šola, Peter Bühlmann, Xinwei Shen

TL;DR

This work tackles distributional shifts in nonlinear prediction by introducing CIRRL, a two-step framework that fuses nonlinear latent representation learning with causality-inspired robustness. The first step uses Distributional Principal Autoencoder (DPA) augmented with an environment-aware prior to learn $\widehat{\phi}$ up to an affine equivalence, effectively capturing a stable latent structure. The second step applies Distributional Robustness via Invariant Gradients (DRIG) on centered latent features to obtain a robust predictor $\widehat{f}(x)=\widehat{b}^T\widehat{\phi}_c(x)$ with a tunable radius $\gamma$, yielding a finite-radius robustness guarantee in nonlinear settings. Theoretical results characterize the worst-case risk $\mathcal{L}_\gamma(f)$ and establish affine identifiability under SCMs, while experiments on synthetic and single-cell perturbation data show CIRRL outperforms ERM and IRM and benefits from finite-radius robustness.

Abstract

Distributional robustness is a central goal of prediction algorithms due to the prevalent distribution shifts in real-world data. The prediction model aims to minimize the worst-case risk among a class of distributions, a.k.a., an uncertainty set. Causality provides a modeling framework with a rigorous robustness guarantee in the above sense, where the uncertainty set is data-driven rather than pre-specified as in traditional distributional robustness optimization. However, current causality-inspired robustness methods possess finite-radius robustness guarantees only in the linear settings, where the causal relationships among the covariates and the response are linear. In this work, we propose a nonlinear method under a causal framework by incorporating recent developments in identifiable representation learning and establish a distributional robustness guarantee. To our best knowledge, this is the first causality-inspired robustness method with such a finite-radius robustness guarantee in nonlinear settings. Empirical validation of the theoretical findings is conducted on both synthetic data and real-world single-cell data, also illustrating that finite-radius robustness is crucial.

Causality-Inspired Robustness for Nonlinear Models via Representation Learning

TL;DR

This work tackles distributional shifts in nonlinear prediction by introducing CIRRL, a two-step framework that fuses nonlinear latent representation learning with causality-inspired robustness. The first step uses Distributional Principal Autoencoder (DPA) augmented with an environment-aware prior to learn up to an affine equivalence, effectively capturing a stable latent structure. The second step applies Distributional Robustness via Invariant Gradients (DRIG) on centered latent features to obtain a robust predictor with a tunable radius , yielding a finite-radius robustness guarantee in nonlinear settings. Theoretical results characterize the worst-case risk and establish affine identifiability under SCMs, while experiments on synthetic and single-cell perturbation data show CIRRL outperforms ERM and IRM and benefits from finite-radius robustness.

Abstract

Distributional robustness is a central goal of prediction algorithms due to the prevalent distribution shifts in real-world data. The prediction model aims to minimize the worst-case risk among a class of distributions, a.k.a., an uncertainty set. Causality provides a modeling framework with a rigorous robustness guarantee in the above sense, where the uncertainty set is data-driven rather than pre-specified as in traditional distributional robustness optimization. However, current causality-inspired robustness methods possess finite-radius robustness guarantees only in the linear settings, where the causal relationships among the covariates and the response are linear. In this work, we propose a nonlinear method under a causal framework by incorporating recent developments in identifiable representation learning and establish a distributional robustness guarantee. To our best knowledge, this is the first causality-inspired robustness method with such a finite-radius robustness guarantee in nonlinear settings. Empirical validation of the theoretical findings is conducted on both synthetic data and real-world single-cell data, also illustrating that finite-radius robustness is crucial.
Paper Structure (21 sections, 7 theorems, 40 equations, 5 figures, 1 table)

This paper contains 21 sections, 7 theorems, 40 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Distributionally robust opimization
  4. Latent representation learning
  5. Method
  6. Model setup
  7. Two-step approach
  8. Representation learning step
  9. Causality-inspired robustness step
  10. Final algorithm
  11. Theoretical guarantees
  12. Experiments
  13. Synthetically generated dataset
  14. Single-cell datatset
  15. Conclusion
  16. ...and 6 more sections

Key Result

Proposition 1

Assume that the SCMs (Equations scm2, scm3) hold as described above. Let $\phi: \mathbb{R}^d \rightarrow \mathbb{R}^k$ be an affine transform of $\phi^*$, i.e. $\phi(x)=N\phi^*(x)+m$ for an invertible $k\times k$ matrix $N$, and a vector $m$. Then, the loss function $\mathcal{L}_\gamma(b^\top\phi_c) where $\omega^e>0$ denote environment weights such that $\sum_{e\in \mathcal{E}} \omega^e = 1$.

Figures (5)

  • Figure 1: Single-cell dataset. Figure presents boxplots of the MSE across environments. In each group of four boxes—each corresponding to a specific value of $\gamma$ (as indicated on the $x$-axis). Note that both IRM and ERM do not depend on $\gamma$, which is why their boxplots remain unchanged across different groups. The dotted lines overlaid on each group indicate the worst performing environment, marking the maximum MSE error (or worst quantile) observed.
  • Figure 2: Simulated (left), and single-cell (right) dataset - values of optimized loss function $L$ (equation \ref{['finallossfct']}). In case there is a clear elbow point, as in the left figure, one should choose it as the latent dimension for the model. However, if the elbow point is less pronounced as it is the case in the second figure, we recommend choosing a value slightly to its right, in this example three or four.
  • Figure 3: Synthetic dataset for well-specified (left) & misspecified setting (right)- final OOD MSEs for IRM, CIRRL, and ERM for various perturbation strengths $\eta$ (Appendix \ref{['generatingscheme']}). The inferiority of IRM to ERM could be attributed to the finite nature of the perturbations.
  • Figure 4: Synthetically generated dataset of latent dimension two (left) & single-cell dataset (right) - the panels illustrate the evolution of OOD MSE error of the proposed model in terms of chosen robustness radius $\gamma$. Notably, in the first case ,the finite nature of the perturbation is clear, as the performance degrades for overly conservative values of $\gamma$. This occurrence is less clear in the second case, but still visible in the median. For the right panel, from top to bottom in shades of red: maximum, 90th, 75th quantile, median, 25th, 10th quantile, and minimum of the MSE across test environments. The black dashed line represents the mean.
  • Figure 5: Simulated (left), single-cell (right), learning curves over 1000 epochs for different values of $\alpha \in \{ 0, \frac{1}{1000}, \frac{1}{100}, \frac{1}{10}, 1, 10\}$ colored blue, orange, green, red, purple, brown, respectively. Considering only the lower pairs of plots, it is evident that $\alpha=\frac{1}{10}$ (red) achieves the best trade-off among the selected values for the optimized loss $L$. The upper row depicts performance in terms of training and test MSE, respectively.

Theorems & Definitions (14)

  • Proposition 1
  • Lemma 2
  • Theorem 3
  • Remark 1
  • Definition 1
  • Lemma 4
  • Lemma 5
  • proof : Proof of Propositon \ref{['driglemma']}
  • Definition 2
  • Lemma 6
  • ...and 4 more