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Learning to Pivot with Adversarial Networks
Gilles Louppe, Michael Kagan, Kyle Cranmer
TL;DR
Systematic uncertainties and nonrepresentative training data motivate predictive models that remain robust to unknown nuisance parameters. The paper introduces adversarial training to enforce pivotality, with a tunable accuracy–robustness trade-off, and demonstrates effectiveness in toy and particle-physics examples.
Problem
Systematic uncertainties can make training data unrepresentative of real data, creating a need for predictive models robust to unknown nuisance parameters.
Method
The paper jointly trains a predictive model and an adversary to enforce independence from continuous or categorical nuisance parameters, with a hyperparameter controlling accuracy versus robustness.
Results
The approach is theoretically shown to yield an optimal pivotal model when such a solution exists and is empirically effective on toy and particle-physics examples.
Takeaways & Limitations
The method can address nonrepresentative training data and extends to fairness settings requiring prediction independence from observed variables.
Takeaways & Limitations
When accuracy and pivotality cannot be simultaneously optimized, selecting the best trade-off along the Pareto frontier remains an open problem.
Abstract
from arXiv · showhide
Several techniques for domain adaptation have been proposed to account for differences in the distribution of the data used for training and testing. The majority of this work focuses on a binary domain label. Similar problems occur in a scientific context where there may be a continuous family of plausible data generation processes associated to the presence of systematic uncertainties. Robust inference is possible if it is based on a pivot -- a quantity whose distribution does not depend on the unknown values of the nuisance parameters that parametrize this family of data generation processes. In this work, we introduce and derive theoretical results for a training procedure based on adversarial networks for enforcing the pivotal property (or, equivalently, fairness with respect to continuous attributes) on a predictive model. The method includes a hyperparameter to control the trade-off between accuracy and robustness. We demonstrate the effectiveness of this approach with a toy example and examples from particle physics.
1 Introduction
Systematic uncertainties can make training data unrepresentative of real data, challenging robust scientific inference. The paper proposes adversarial networks to enforce pivotal predictions and demonstrates the approach in toy and particle-physics examples.
- Systematic uncertainties affect both the robustness of scientific inference and the metrics used to evaluate analysis strategies.
- Training data may not represent real data when the data-generation process is not uniquely specified.
- The proposed adversarial procedure enforces a predictive model that is pivotal with respect to nuisance parameters.
- The procedure applies regardless of whether nuisance parameters are discrete or continuous or what prior they have.
- The approach is demonstrated with a toy example and examples from particle physics.
2 Problem statement
The problem is to learn predictions for target labels that remain robust when nuisance parameters are unknown. Robustness is formalized by requiring the prediction distribution to be invariant across nuisance-parameter values, making predictions pivotal.
- The model learns a regression or classification function f(X) for target labels Y from observed data X.
- Robust inference requires the distribution of f(X; θf), conditional on Z and possibly Y, to remain invariant across nuisance-parameter values.
- A pivotal predictive function has an output independent of the nuisance parameter Z.
- The basic criterion imposes pivotality after marginalizing the target label Y, while some applications prefer class-conditional independence.
3 Method
The method jointly trains a predictive model and an adversary that models nuisance-parameter distributions from the model output. Their antagonistic objectives push predictions toward independence from nuisance parameters while retaining task performance.
- The predictive model f is trained against adversary r so correlations between predictions and nuisance parameters become detectable through the adversary.
- The adversary estimates the nuisance distribution conditional on the predictive output, using either classification for discrete Z or flexible density modeling for continuous Z.
- The two models are trained simultaneously by optimizing a value function that subtracts the adversarial loss from the predictive loss.
- For binary classification, the predictive loss is the expected negative log-likelihood of Y given X, while the adversarial loss is the expected negative log-likelihood of Z given f(X; θf).
- Training alternates stochastic-gradient updates of the adversary with fixed classifier parameters and of the classifier with fixed adversary parameters.
4 Theoretical results
The theoretical analysis shows that adversarial training can produce a classifier that is both optimal for predicting Y from X and pivotal with respect to nuisance parameters, when such a classifier exists. When optimality and pivotality conflict, the hyperparameter λ controls their trade-off, while convergence guarantees remain limited for finite adversary updates.
- Theoretical guarantee: The minimax solution corresponds to a classifier that is a pivotal quantity when the losses are negative log-likelihoods for Y|X and Z|f(X).The result applies in the setting where the adversary models the nuisance parameter conditional on the classifier output.
- Theoretical guarantee: If a minimax solution reaches H(Y|X) − H(Z), the resulting classifier is both optimal and pivotal.This requires the stated minimax solution to exist at that value.
- Theoretical guarantee: Optimality minimizes the negative log-likelihood of Y|X, while pivotality maximizes H(Z|f(X)), making f(X) independent of Z when the lower bound is active.The two conditions jointly identify the classifier as both predictive and nuisance-independent.
- Optimization: With sufficiently optimized adversary updates and sufficiently small classifier steps, f should converge to an optimal pivotal classifier if one exists.Formal convergence guarantees for Algorithm 1 with a finite number of adversary steps remain unproven.
- Trade-off and limitations: When no classifier is simultaneously optimal and pivotal, the lower bound is strict, so f must trade off predictive performance against nuisance independence.The paper states that distinct equally good solutions may lie along this Pareto frontier.
- Architectural implication: The adversary needs only the one-dimensional output s of f as input to enforce the pivotal property.Making an intermediate representation pivotal is possible but not necessary for the stated results.
5 Experiments
The experiments test adversarial training on a continuous nuisance parameter in a toy problem and on pileup uncertainty in particle-physics classification. The results show that tuning the accuracy–robustness trade-off can improve hypothesis-test significance, while the ideal jointly optimal and pivotal classifier may be unreachable.
- 5.1 A toy example with a continous nuisance parameter: The toy problem requires classifier scores whose distribution is invariant to the continuous nuisance parameter controlling the second Gaussian’s mean.A classifier trained without adversarial regularization is not pivotal because its score densities differ substantially across nuisance-parameter values.
- 5.1 A toy example with a continous nuisance parameter: Adversarial training pits the classifier against a mixture-density adversary that models the nuisance parameter conditional on the classifier output.The adversary uses a mixture of five Gaussians to model the continuous conditional distribution.
- 5.1 A toy example with a continous nuisance parameter: The ideal classifier that is both optimal and pivotal is unreachable in the toy problem because the optimized objective remains offset from the entropy-based target.The offset is shown between Lf −λLr and the dashed approximation to H(Y |X) −λH(Z).
- 5.2 High energy physics examples: Pileup introduces a nuisance parameter because additional proton-proton interactions add energies unrelated to the underlying discriminating information.The experiment considers no pileup versus an average of 50 overlaid pileup interactions.
- 5.2 High energy physics examples: AMS peaks at 7 without adversarial training and reaches around 7.8 at λ = 10, improving the power of the hypothesis test by trading accuracy for pileup robustness.At λ = 500, AMS decreases because the classifier focuses too strongly on independence from the nuisance parameter at the expense of accuracy.
- 5.2 High energy physics examples: Adversarial training also produced jet classifiers independent of jet mass, with reported studies showing effectiveness on real-world continuous-attribute problems.The cited application links this strategy to enhanced sensitivity in searches for new physics at the LHC.
6 Related work
The paper connects learning pivots to domain adaptation and fairness, while addressing the limitation that much prior work assumes binary nuisance or domain labels. Its adversarial formulation is intended to generalize conditional-dependence modeling to non-binary and continuous nuisance parameters.
- Domain adaptation and fairness: Learning a pivot can be formulated as domain adaptation or fairness: the predictive model is required to be independent of a domain or selected attribute.Examples of selected attributes include gender, color, and age.
- Limitations of prior work: Most related methods assume a binary nuisance variable and minimize a divergence between the score distributions for the two values.This assumption prevents direct generalization to non-binary or continuous nuisance parameters from both practical and theoretical perspectives.
- Limitations of prior work: Estimating p(f(X)|Z = z) empirically for every nuisance value can become impractical as the number of nuisance categories grows.The paper contrasts this with adversarial modeling of conditional dependence, which can generalize without necessarily requiring a growing number of terms.
- Scientific treatment of systematic uncertainty: A nominal-value classifier is not optimal when the true nuisance value differs from the nominal value, motivating alternatives that incorporate uncertainty during training.The paper describes propagation of uncertainty through parametrized estimates of p(f(x)|z) as a common existing approach.
7 Conclusions
The paper proposes adversarial training to make predictive models independent of continuous or categorical nuisance parameters, with a tunable power–robustness trade-off. It reports effectiveness on toy and particle-physics examples and identifies applications to nonrepresentative training data and fairness.
- Adversarially training two neural networks builds predictive models independent of continuous or categorical nuisance parameters.The procedure targets pivotal predictions by jointly training the networks in an adversarial fashion.
- The method’s minimax value corresponds to an optimal and pivotal model when one exists, or permits tuning the trade-off between power and robustness.
- Experiments confirmed the method’s effectiveness on a toy example and a particle-physics example.
- The approach applies when training data may not represent deployment data and also extends to fairness with respect to observed random variables.