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Invariant Representations without Adversarial Training
Daniel Moyer, Shuyang Gao, Rob Brekelmans, Greg Ver Steeg, Aram Galstyan
TL;DR
The paper addresses how to remove nuisance or protected-covariate information from representations without sacrificing task usefulness, replacing iterative adversarial optimization with a directly optimized information-theoretic objective. Its mutual-information upper-bound approach supports c-agnostic learning and controllable generation, and the authors report performance matching or exceeding state-of-the-art adversarial methods while avoiding adversarial-training pitfalls.
Problem
Learning representations invariant to arbitrary nuisance factors while retaining usefulness for other tasks is challenging, including in fair prediction and covariate control.
Method
The paper penalizes mutual information I(z, c), derives a variational upper bound, and combines it with VAE and VIB frameworks for tractable direct optimization.
Results
The approach matches or exceeds state-of-the-art adversarial methods while avoiding the pitfalls inherent in adversarial learning for invariant representations.
Takeaways & Limitations
Covariate labels are needed during training but not in production, and the conditional decoder can manipulate specified factors when generating realistic data.
Takeaways & Limitations
The analysis assumes p(z|c) = p(z), an intentional model misspecification that encodes a prior belief that latent factors should be independent of c.
Abstract
from arXiv · showhide
Representations of data that are invariant to changes in specified factors are useful for a wide range of problems: removing potential biases in prediction problems, controlling the effects of covariates, and disentangling meaningful factors of variation. Unfortunately, learning representations that exhibit invariance to arbitrary nuisance factors yet remain useful for other tasks is challenging. Existing approaches cast the trade-off between task performance and invariance in an adversarial way, using an iterative minimax optimization. We show that adversarial training is unnecessary and sometimes counter-productive; we instead cast invariant representation learning as a single information-theoretic objective that can be directly optimized. We demonstrate that this approach matches or exceeds performance of state-of-the-art adversarial approaches for learning fair representations and for generative modeling with controllable transformations.
1 Introduction
Invariant representation learning seeks encodings that remove information about an unwanted covariate while preserving usefulness for the original task. The paper replaces adversarial formulations with a mutual-information objective and derives a tractable approach applicable to fair prediction, nuisance removal, and controllable generation.
- Invariant representations encode x into z that is uninformative of covariate c while remaining useful for the original task.
- The approach applies a shared invariance objective across transform-invariant vision features, covariate control, and fair representations.
- The paper relaxes the independence condition z ⊥c into a penalty on mutual information I(z, c).
- I(z, c) admits a variational upper bound, unlike the usual lower bounds used for related mutual-information objectives.
- Combining this upper bound with VAE and VIB frameworks yields computationally tractable learning methods for c-agnostic encodings and predictors.
- The proposed methods require covariate c during training but not at test time, and their conditional decoder can generate transformations resembling Fader Networks.
2 Model
The model replaces the difficult independence constraint between representations z and covariates c with a mutual-information penalty, yielding directly optimized invariant-learning objectives. Variational bounds connect this penalty to conditional reconstruction, VAE-style losses, and supervised Information Bottleneck formulations.
- Invariant objective: The model seeks an encoder q(z|x) whose representation is useful for the original task while independent of covariate c.The observed covariate may depend nonlinearly on x, making the independence-constrained optimization difficult.
- Invariant objective: Minimizing mutual information I(z, c) provides a tractable relaxation of the strong independence constraint.The objective combines the original task loss L with the invariance penalty using trade-off parameter λ.
- Variational bound: A variational inequality bounds I(z, c) using a decoder p(x|z, c), making the penalty resemble a VAE objective with conditional reconstruction.The bound is expressed through divergence and reconstruction terms, while conditional reconstruction prevents discarded covariate information from being penalized.
- Invariant codes through VAE: The resulting unsupervised VAE loss combines prior regularization, a KL penalty toward the marginal q(z), and conditional reconstruction likelihood.Its stated form is E(x,c)[ −KL[ q(z|x) ∥p(z)] −λKL[ q(z|x) ∥q(z) ] + (1 + λ)Ez∼q[log p(x|z, c)] ].
- Computational approximation: The method approximates the difficult marginal q(z) with pairwise KL divergences between encoder distributions, requiring O(b^2) operations before matrix-algebra acceleration.The pairwise Gaussian divergences have closed form under the re-parameterization trick.
- Supervised invariant codes: The same invariance principle extends to supervised prediction by adding I(z, c) minimization to the Information Bottleneck objective while retaining information about labels y.The resulting formulation adds a decoder branch that maximizes p(x|z, c).
3 Computation and Empirical Evaluation
The paper evaluates its invariance penalty on fair-classification datasets against prior methods and demonstrates controllable image generation on MNIST.
- Implementation: The implementation uses encoder-decoder networks, a 30-dimensional latent space, Adam optimization, and a batch size of 128.The method uses 64 hidden units for both datasets, while VFAE uses its described architecture.
- Generative Modeling: MNIST experiments test Fader Network-like manipulation by removing digit-label information from z while retaining non-class stylistic information.The conditional decoder generates artificial digits from a style representation and a specified class.
- Fair Classification: The proposed method is evaluated on fair classification using prediction accuracy and adversarial error, alongside VFAE and Xie et al. baselines.Both datasets come from the UCI repository and use preprocessing following Zemel et al. 2013.
- Fair Classification: The German dataset contains 1000 financial records for predicting good credit scores, with binarized Age as the protected class.
- Fair Classification: The Adult dataset contains 45,222 census records for predicting savings above $50,000, with Gender as the protected factor.
- Evaluation: Figure 1 measures adversarial loss and predictive accuracy on hold-out test sets, while Figure 2 visualizes latent encodings colored by covariate class.
4 Results
Results show comparable predictive accuracy across methods, stronger invariance performance for the proposed method on Adult, heterogeneous latent embeddings, and style transfer with some failures.
- Fair Classification: On German, methods have comparable predictive accuracy and the VFAE and proposed method have competitive adversarial loss.The smaller dataset generally does not differentiate the methods.
- Fair Classification: On Adult, all three methods again have comparable predictive accuracy, while the proposed method provides the best adversarial error across four difficulty levels.Its accuracy is comparable with VFAE, whereas each baseline has very high loss against stronger adversaries.
- Latent Representations: The proposed Adult latent embedding has only a heterogeneous region, whereas both baseline embeddings contain blue-only regions.All methods also contain large red-only regions, which is attributed to the majority class.
- Generative Modeling: MNIST generation transfers stylistic properties such as azimuth and thickness across digit classes, but some styles fail to transfer between curved and straight digits.
5 Discussion
The discussion argues that adversarial training is unnecessary for invariant representations, while acknowledging that its practical performance depends on optimization and adversary strength.
- Adversarial training can match the derived method in the optimal case, but adds a second optimization problem and is unnecessary for invariant representation learning.The authors do not claim adversarial training is strictly worse in practice.
- Adversarial training may still be easier or less restrictive in some cases, and shared GAN literature may offer heuristics that improve its performance.
- Earlier severely contorted Xie et al. baseline latent codes were attributed to a convergence issue, with mild performance improvements observed after investigation.
- Weak or poorly trained adversaries may provide bad gradient information, causing poor encoder performance against a post-hoc adversary.
- The experiments found comparable predictive accuracy for Xie et al.’s method but local-minimum behavior and poor generalization from jointly trained to post-hoc adversaries.The jointly trained adversary predicted c poorly, whereas a post-hoc adversary predicted c easily.
- The authors could not replicate Xie et al.’s reported adversarial errors, although they reproduced Louizos et al.’s results using logistic regression.They note that their adversary may have been inadvertently stronger, while relative rankings should remain roughly similar.
6 Conclusion
The conclusion presents a variational mutual-information upper bound for learning covariate-invariant representations without adversarial training. The resulting approach supports production use without covariate labels and enables controllable generation while matching or exceeding state-of-the-art adversarial methods.
- The paper derives a variational upper bound for mutual information between latent representations and covariate factors.
- Using labeled covariates during training, the method supports supervised and unsupervised invariant learning without adversarial training.
- After training, the method can be used in production without requiring covariate labels.
- The approach enables manipulation of specified factors when generating realistic data and produces results that match or exceed state-of-the-art methods.