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On the Fairness of Disentangled Representations
Francesco Locatello, Gabriele Abbati, Tom Rainforth, Stefan Bauer, Bernhard Schölkopf, Olivier Bachem
TL;DR
The paper asks whether disentangled representations improve fairness when predictions use observations entangling a target with an unobserved sensitive variable. It combines theoretical analysis with large-scale evaluation and finds that disentanglement is consistently associated with fairer downstream predictions, while the relationship depends on the representation models considered.
Problem
The paper addresses limited understanding of how disentangled representations relate to fairness when sensitive variables are unobserved and independent of the target.
Method
The paper analyzes the setting theoretically and evaluates demographic parity across downstream models using more than 10 000 state-of-the-art disentangled representations, relating fairness to six disentanglement scores.
Results
Disentanglement, particularly DCI Disentanglement, appears consistently correlated with increased fairness across representations and data sets, including after accounting for downstream accuracy.
Takeaways & Limitations
Representation choice appears crucial: disentanglement may be a useful property to encourage when seeking fair downstream prediction without observed sensitive variables.
Takeaways & Limitations
The observed correlations depend on the considered methods, hyperparameters, and representations, and may differ for another model distribution.
Abstract
from arXiv · showhide
Recently there has been a significant interest in learning disentangled representations, as they promise increased interpretability, generalization to unseen scenarios and faster learning on downstream tasks. In this paper, we investigate the usefulness of different notions of disentanglement for improving the fairness of downstream prediction tasks based on representations. We consider the setting where the goal is to predict a target variable based on the learned representation of high-dimensional observations (such as images) that depend on both the target variable and an \emph{unobserved} sensitive variable. We show that in this setting both the optimal and empirical predictions can be unfair, even if the target variable and the sensitive variable are independent. Analyzing the representations of more than \num{12600} trained state-of-the-art disentangled models, we observe that several disentanglement scores are consistently correlated with increased fairness, suggesting that disentanglement may be a useful property to encourage fairness when sensitive variables are not observed.
1 Introduction
The paper connects disentangled representation learning with fairness when target and sensitive factors are independent but become entangled in observations. It motivates evaluating whether representation choice affects the accuracy and fairness of downstream classifiers.
- Motivation: Disentangled representations aim to isolate independent factors of variation in separate representation components, supporting interpretability and downstream usefulness.The paper frames representation learning as finding a transformation r(x) useful for downstream classification tasks.
- Motivation: Fairness is difficult because sensitive information can leak into prediction tasks through observations, data collection, causal structure, or algorithmic assumptions.The paper highlights protected attributes such as gender and race as examples of sensitive variables.
- Problem setting: The setting targets factors y while avoiding influence from an unobserved sensitive factor s, although both are entangled in observations x through an unknown mixing mechanism.Representations are learned without supervision, and sensitive variables are unavailable during downstream classifier training.
- Contributions: The paper theoretically shows that predictions can be unfair even with a Bayes optimal classifier when target and sensitive variables are independent.Its empirical study evaluates demographic parity across more than 90 000 downstream models built on more than 10 000 disentangled representations and seven data sets.
- Contributions: Across representations, disentanglement—particularly the DCI Disentanglement score—appears consistently correlated with increased fairness.The paper also investigates how fairness relates to downstream performance and disentanglement scores.
2 Background on learning disentangled representations
The paper situates disentanglement as learning representation components aligned with independent generative factors. It reviews unsupervised VAE-based methods and six scores that measure different aspects of disentanglement.
- Generative setting: Observations x are modeled as generated from k independent sources z1, ..., zk through a generative mechanism.This setup treats the sources as factors underlying the observed data.
- Disentanglement: Disentanglement seeks a representation r(x) with independent components where changes in a generative factor correspond to changes in a representation dimension.The paper notes that this intuition has also been formalized topologically and causally.
- Evaluation scores: The reviewed scores are BetaVAE, FactorVAE, MIG, Modularity, DCI Disentanglement, and SAP.They assess factor prediction, variance-based separation, mutual-information gaps, modularity, feature importance entropy, or predictive-error gaps.
- Unsupervised methods: Unsupervised approaches commonly use VAEs whose losses include regularizers encouraging structure in the aggregate encoder distribution.Examples constrain bottleneck capacity, penalize total correlation, or impose a disentanglement prior on the aggregated posterior.
3 The dangers of general purpose representations for fairness
The paper argues that general-purpose representations can yield unfair downstream predictions even when target and sensitive factors are independent. It analyzes how disentanglement may reduce this risk while noting important scope boundaries.
- Setting: The representation-learning setting observes x, while y is observed only for the downstream task and s remains unobserved.This differs from prior fairness setups and is motivated partly by scarce labels and unknown sensitive factors.
- Why unfairness arises: Independent target and sensitive variables can become conditionally dependent given x or r(x), so prediction from the representation is not guaranteed fair.The paper identifies this conditional dependence as the reason fairness remains non-trivial in the assumed setting.
- Theoretical result: Even a perfect classifier satisfying p(ŷ|x) = p(y|x) need not satisfy demographic parity when x entangles y and s.The theorem states that equal prediction performance across sensitive groups does not follow from optimal prediction.
- Scope of theorem: Worst-case examples may not permit good and fair predictions under any representation, while real-world data may satisfy additional assumptions.The authors therefore advise interpreting the theorem as a worst-case result.
- Practical complications: Complex unknown mixing mechanisms and downstream model biases may prevent empirical predictors from separating target and sensitive factors.The downstream model may not be incentivized or able to invert the mixing mechanism successfully.
- Why disentanglement might help: Disentanglement may help by placing information about different generative factors in different latent dimensions, limiting dependence between code dimensions.This can encourage predictions to rely on target-related dimensions rather than the dimension associated with the sensitive factor.
4 Do disentangled representations matter?
Across seven datasets and more than 12,600 pretrained representations, unfairness varies substantially and is generally associated with disentanglement, including after adjusting for downstream performance, though important metric- and dataset-specific exceptions remain.
- Experimental setup: The benchmark uses seven datasets and 12,600 pretrained models spanning six prominent unsupervised disentanglement methods.The models cover varied hyperparameters and random seeds, and downstream predictions are evaluated across the learned representations.
- The unfairness of general purpose representations: Unfairness can reach 15%–25% on five of seven datasets even when target and sensitive variables are independent.Unfairness is measured through average total variation of predictions after intervening on the sensitive variable.
- The unfairness of general purpose representations: Unfairness varies considerably across learned representations, indicating that some representations support predictions with low unfairness.The observed spread motivates examining whether disentanglement scores identify less unfair representations.
- The unfairness of general purpose representations: All disentanglement scores except Modularity are consistently correlated with lower unfairness across datasets, with DCI Disentanglement showing the strongest correlation.The reported ordering is DCI Disentanglement, MIG, BetaVAE, FactorVAE, SAP, then Modularity; Figure 3 confirms the DCI relationship.
- Adjusting for downstream performance: The fairness–disentanglement relationship depends on the evaluated model distribution, so it may differ for other methods, hyperparameters, and representations.The authors therefore interpret the observed correlations with care rather than as a universal guarantee.
- Adjusting for downstream performance: After adjustment for downstream performance, higher disentanglement still appears correlated with increased fairness, but correlations weaken and several metric–dataset exceptions emerge.Exceptions include adjusted Modularity broadly, selected scores on Shapes3D, and adjusted MIG, DCI Disentanglement, Modularity, and SAP on SmallNORB.
5 Related Work
The paper situates disentanglement within longstanding representation-learning work and contrasts its unlabeled setting with fairness methods using labeled sensitive factors. It also frames disentanglement as a potential proxy for fairness under explicit causal assumptions.
- Disentanglement builds on nonlinear ICA and classical efforts to separate factors such as pose, content, and motion.
- The paper uses disentanglement in the sense of representations whose components correspond to independent generative factors.
- Figure 8 proposes disentanglement as a proxy for downstream performance and fairness when it is a causal parent and hidden confounders are absent.
- Fair representation-learning methods discussed in prior work use labeled sensitive factors, unlike this paper’s unlabeled representation-learning setting.
6 Conclusion
The paper finds that general-purpose representations can produce unfair predictions even when target and sensitive variables are independent, while greater disentanglement is consistently associated with greater fairness. This relationship persists across diverse representations and datasets and remains after accounting for downstream accuracy.
- General-purpose representations can yield substantial unfairness even when target and sensitive variables are independent and only mixed observations are available.
- Increased disentanglement is consistently correlated with increased fairness across downstream prediction tasks, representations, and datasets.
- The fairness–disentanglement correlation appears to hold after accounting for downstream accuracy.
- The findings motivate further study of fairness as a desired property of general-purpose representations beyond VAEs.
A Proof of Theorem 1
The proof uses a Bernoulli counterexample with mixing mechanism x = min(y, s) to show that a perfect classifier need not satisfy demographic parity. Applying demographic parity, the causal Markov condition, and perfect-classifier assumptions leads to a contradiction.
- The proof constructs a Bernoulli counterexample with p(s = 1) = q, p(y = 1) = b, and x = min(y, s).
- Demographic parity is translated into equality of prediction probabilities across sensitive groups and then into joint-probability equalities over x.
- Using the causal Markov condition, the joint terms factor as p(ŷ = 1|x)p(x|s), linking predictions to the representation distribution within each sensitive group.
- Assuming a perfect classifier sets p(ŷ|x) = p(y|x), producing an equality involving target-conditionals and sensitive-group representation probabilities.
- The mixing mechanism supplies the conditional probabilities needed to reduce the demographic-parity equation to a contradiction.
- The contradiction forces b = 1, violating the assumption b < 1 and proving that perfect prediction does not imply demographic parity.
B Additional Results
The additional experiments compare unfairness with multiple disentanglement metrics and examine correlations among adjusted metrics across datasets. Unfairness generally correlates with disentanglement, while Modularity and partly SAP are exceptions, and adjusted metrics support fairness analysis independently of downstream accuracy.
- Unfairness and disentanglement are generally correlated across the evaluated metrics, except for Modularity and partially the SAP score.
- Figure 9 extends the unfairness-versus-disentanglement analysis to all disentanglement scores.
- Adjusted metrics show rank correlations similar to those previously observed for non-adjusted metrics.
- The correlation between disentanglement metrics is not exclusively driven by downstream performance.
- Adjusted metrics are suitable for discussing representation fairness independently of downstream classification accuracy.