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ALICE: Towards Understanding Adversarial Learning for Joint Distribution Matching
Chunyuan Li, Hao Liu, Changyou Chen, Yunchen Pu, Liqun Chen, Ricardo Henao, Lawrence Carin
TL;DR
The paper addresses non-identifiability in bidirectional adversarial joint-distribution matching, which can yield undesirable mappings despite matched joints. It uses conditional entropy to control conditionals through adversarial and non-adversarial approaches, obtaining more stable and faithful solutions while extending the framework to semi-supervised learning.
Problem
Bidirectional adversarial methods can match joint distributions without identifying desirable conditional dependencies, causing unfaithful reconstructions and incorrect pairings.
Method
ALICE regularizes joint distribution matching with conditional entropies and provides adversarial and non-adversarial solutions for unsupervised, supervised, and semi-supervised learning.
Results
ALICE is more consistent and quantitatively reliable than ALI in synthetic experiments, more stable to train, and produces more faithful reconstructions on real data.
Takeaways & Limitations
The joint-matching perspective reduces ambiguity in conditionals and supports weakly supervised pairing, including prediction of unseen pairs.
Takeaways & Limitations
The conditional entropy is intractable during training and must be approximated; the presented theoretical derivation for one direction has a symmetric counterpart discussed separately.
Abstract
from arXiv · showhide
We investigate the non-identifiability issues associated with bidirectional adversarial training for joint distribution matching. Within a framework of conditional entropy, we propose both adversarial and non-adversarial approaches to learn desirable matched joint distributions for unsupervised and supervised tasks. We unify a broad family of adversarial models as joint distribution matching problems. Our approach stabilizes learning of unsupervised bidirectional adversarial learning methods. Further, we introduce an extension for semi-supervised learning tasks. Theoretical results are validated in synthetic data and real-world applications.
1 Introduction
The paper identifies non-identifiability in bidirectional adversarial learning and introduces ALICE to regularize joint matching with conditional entropy. It presents unified adversarial formulations, empirical stability and reconstruction gains, and a semi-supervised extension.
- Motivation: ALI can match joint distributions while producing unfaithful reconstructions because the within-joint dependency structure is not constrained.The same issue can make domain-transformation pairings incorrect.
- Contributions: ALICE regularizes ALI using Conditional Entropy to address its non-identifiability problem.
- Contributions: The paper unifies ALI, CycleGAN, and Conditional GAN as models for joint distribution matching.
- Empirical findings: Experiments on synthetic and real data find ALICE more stable to train than ALI and more faithful in image reconstruction.The reported stability includes less sensitivity to architectural hyperparameters.
- Contributions: ALICE extends the framework to semi-supervised learning by leveraging paired data when available.
2 Background
The framework represents two domains through marginal, conditional, and joint distributions, then uses adversarial generators and discriminators to match these distributions. GAN matches a marginal distribution, whereas ALI matches joint distributions but leaves conditional identifiability unresolved.
- Distribution matching: Two domains are characterized by marginals q(x) and p(z), conditional distributions q(z|x) and p(x|z), and joint distributions q(x,z) and p(x,z).The joint distributions factor as q(x)q(z|x) and p(z)p(x|z).
- Adversarial generation: Adversarial generators produce conditional samples as deterministic transformations of a source variable and independent Gaussian noise.The generators gθ and gφ are neural networks receiving concatenated inputs [z ϵ] and [x ζ].
- Marginal matching: GAN trains a discriminator to distinguish generated samples from pθ(x) and data samples from q(x), with the desired marginal match characterized by pθ*(x) = q(x).The data distribution q(x) is sampled directly, while p(z) is specified as a simple parametric distribution.
- Joint matching: ALI matches the joint distributions pθ(x,z) and qφ(x,z) using an adversarial discriminator over pairs (x,z).Its generator and discriminator optimum is a saddle point if and only if pθ*(x,z) = qφ*(x,z).
- Identifiability: Although ALI joint matching guarantees matching marginals and conditionals of the pair, its objective imposes no restrictions on the two cross-conditionals pθ(z|x) and qφ(z|x).This unconstrained relationship is identified as central to ALI's identifiability issues.
3 Adversarial Learning with Information Measures
ALI can match joint distributions without identifying a desirable relationship between domains. ALICE regularizes this ambiguity with conditional entropy, approximated through cycle-consistency or adversarial objectives for unsupervised, supervised, and semi-supervised settings.
- Non-identifiability: ALI permits multiple matched joint distributions because its objective constrains marginals and joint matching but not the relationship between x and z.Different saddle points can satisfy the objective while producing undesirable mappings and poor reconstructions.
- Conditional Entropy: Conditional entropy measures uncertainty between x and z and vanishes when one variable is a deterministic mapping of the other.ALICE adds Hπ(x|z) or Hπ(z|x) as a regularization term to select mappings with better reconstruction ability.
- Unsupervised Learning: Because conditional entropy is intractable during training, ALICE uses cycle-consistency as an upper bound that can be approximated through Monte Carlo simulation.The bound approaches the conditional-entropy objective as ALI reaches its optimum and can be added without changing the original training procedure.
- Unsupervised Learning: When cycle-consistency is satisfied, deterministic mappings match the relevant conditionals, while matched conditionals imply zero conditional entropy and deterministic correspondence.The fully adversarial variant uses paired reconstruction samples; its optimum enforces the reconstruction distribution to concentrate on the original sample.
- Supervised and Semi-supervised Learning: Paired samples specify a desirable joint distribution, allowing supervised ALICE to approximate conditional entropy and recover the paired joint and conditional distributions.A small paired subset can guide configuration selection when complete pairing is expensive or impossible, motivating semi-supervised classification.
4 Related Work: A Unified Perspective for Joint Distribution Matching
The paper interprets cycle-consistency and conditional GANs within a unified joint-distribution-matching perspective. It connects ALI, CycleGAN, and conditional GAN formulations while using paired data to address ambiguity in unsupervised variants.
- ALI and CycleGAN: Cycle-consistency is interpreted as controlling conditional entropies and matching conditional distributions, complementing marginal matching through GAN objectives.The paper further proposes adversarially learned cycle-consistency when high-quality samples are important.
- Conditional GANs: The paper formulates conditional GANs as joint distribution matching problems and uses paired-data distributions to resolve ambiguity in unsupervised ALI variants.This is presented as the first such study to the authors’ knowledge.
5 Experimental Results
Experiments show that ALICE improves the stability and quality of bidirectional learning across toy and real image-translation tasks, supporting reconstruction, generation, and weakly supervised pairing.
- 5.1 Effectiveness and Stability of Cycle-Consistency: ALICE yields more consistent and quantitatively reliable toy-data solutions than ALI across 576 experiments.ALICE achieves ICP > 4.5 in 77% of experiments, while ALI’s ICP varies substantially across runs.
- 5.1 Effectiveness and Stability of Cycle-Consistency: ALICE provides acceptable reconstruction while retaining stronger generation ability than the DAE baseline.The DAE has the lowest MSE but weakest generation ability; ALICE’s MSE is comparatively low and substantially better than ALI.
- 5.1 Effectiveness and Stability of Cycle-Consistency: On toy data, ALICE produces a more consistent latent representation and smoother interpolation than ALI.Its latent samples preserve mixture-component clustering and approximate a Gaussian distribution, while interpolations remain consistent with the ground-truth distribution.
- 5.1 Effectiveness and Stability of Cycle-Consistency: The regularizer improves ALI’s image generation and reconstruction across a broad range of weighting hyperparameters on toy data, MNIST, and CIFAR-10.This result indicates that the improvement is not confined to one regularization setting or dataset.
- 5.2 Reconstruction and Cross-Domain Transformation on Real Datasets: The real-data experiments target cross-domain prediction while maintaining reconstruction within each domain.Car-to-Car uses weak supervision, and Edge-to-Shoe provides quantitative SSIM comparisons with cycle consistency applied in both domains.
- 5.2 Reconstruction and Cross-Domain Transformation on Real Datasets: On Car-to-Car, 1% true-pair supervision gives significantly higher accuracy than DiscoGAN and BiGAN, while sparse angle supervision approaches full-supervision accuracy.Using 10% supervision in only 2 or 6 of 11 angles yields comparable test accuracy to full angle supervision, demonstrating zero-shot prediction of unseen pairs.
- 5.2 Reconstruction and Cross-Domain Transformation on Real Datasets: On Edge-to-Shoe, cycle consistency improves unsupervised reconstruction, while 10% supervision significantly increases prediction SSIM.ALICE performs similarly with 50% and full supervision, and adversarial supervision produces more detailed generated edges than ℓ2 supervision.
6 Conclusion
The conclusion frames bidirectional adversarial learning as a joint-distribution-matching problem with non-identifiability, and presents ALICE as a way to reduce ambiguity and control conditionals.
- 6 Conclusion: The paper studies non-identifiability in bidirectional adversarial networks through a unified joint-distribution-matching perspective.This perspective covers multiple GAN models and motivates the proposed treatment of ambiguity.
- 6 Conclusion: ALICE uses adversarial and non-adversarial approaches to reduce ambiguity and control conditionals in unsupervised and semi-supervised learning.The conclusion identifies conditional control as the scope of the proposed framework.
Supplementary Material of ALICE: Towards Understanding Adversarial Learning for
The supplementary material develops information-theoretic relationships used to analyze joint-distribution matching, reconstruction, and adversarial proofs.
- A Information Measures: The supplement introduces mutual information, variation of information, and conditional entropy for analyzing correlations between x and z.These measures are presented through additive and subtractive relationships under a probability measure π.
- A Information Measures: Reconstruction loss is related to variation of information and mutual information through encoder and decoder probability measures.The z-reconstruction loss is written in log-likelihood form as LR = −E_z∼p(z),x∼p(x|z)[log q(z|x)].
- A Information Measures: Lemma 4 expresses a relationship involving conditional entropy for random variables with distinct joint probability measures p(x,z) and q(x,z).The following corollaries derive mutual-information and variation-of-information forms under a single probability measure.
- A Information Measures: With an isotropic Gaussian prior p(z), the entropy H(z) is constant.This simplifies information-theoretic expressions involving the latent variable.
- Joint Distribution Matching: The supplement proves cycle-consistency and conditional-GAN results by analyzing optimal discriminators and the corresponding optimal generators.The proof follows the strategy of the original GAN paper.
B.1 Proof of Proposition 1: Adversarially Learned Cycle-Consistency for Unpair Data
The adversarial reconstruction game matches generated reconstructions to identity reconstructions, yielding cycle-consistency for both data and latent variables at the optimum.
- Adversarial reconstruction: The reconstruction game contrasts identity pairs (x, x) with generated pairs (x, x̂) to train an adversarial discriminator.The generated reconstruction samples x̂ are obtained by encoding x and decoding the resulting latent variable.
- Optimal discriminator: At the discriminator optimum, the generated conditional reconstruction distribution matches the identity target distribution.The target conditional distribution is δ(x̂ − x).
- Adversarial equilibrium: When the generator and encoder confuse the discriminator, the optimal discriminator output is 1/2 for generated and identity pairs.This equality forces the two compared joint distributions to coincide.
- Cycle-consistency: The resulting optimum satisfies perfect reconstruction in data space: Eqφ*(z|x)pθ*(x̂|z) = δ(x̂ − x).The encoder-decoder composition returns the original observation.
- Cycle-consistency: The analogous reverse composition satisfies latent-space cycle consistency: Epθ*(x|z)qφ*(ẑ|x) = δ(z − ẑ).Thus the learned mappings reconstruct latent variables as well as observations.
B.2 Proof of Proposition 2: Adversarially Learned Conditional Generation for Paired Data
For paired data, an adversarial conditional-generation game matches generated conditionals to the empirical paired conditionals, while the supplied qualitative results compare reconstruction and sampling behavior.
- Conditional generation: The supervised game contrasts paired observations (x, z) with generated observations x̃ conditioned on z.The discriminator receives real pairs from π(x, z) and generated samples from pθ(x̃|z).
- Generator optimum: At the saddle point, the generator conditional equals the empirical conditional: pθ*(x|z) = π(x|z).The corresponding joint distribution also satisfies pθ*(x, z) = π(x, z).
- Data assumption: The paired-data proof uses the empirical pairing relation π(x̃|z) = δ(x̃ − x) for each observed pair.This relation supplies the conditional target used in the adversarial objective.
- Joint matching: Using the paired-data marginal q(x) = π(x), the learned generator and conditional encoder match the same joint distribution π(x, z).The stated result is pθ*(x, z) = π(x, z) = qφ*(x, z).
- Qualitative results: On toy data, ALICE shows good sampling across the five Gaussian components, whereas ALI samples are concentrated and DAE has weak generation ability.The comparison concerns reconstruction of z and sampling in x.
C.3 Summary of the four variants of ALICE
ALICE uses explicit or implicit reconstruction and mapping constraints across unsupervised and semi-supervised settings, with toy experiments exposing trade-offs among reconstruction, sampling, and correspondence.
- Four variants: ALICE has four variants: explicit or implicit cycle-consistency for unsupervised learning, and explicit or implicit mappings for semi-supervised learning.Explicit methods use ℓk-norm objectives; implicit methods learn the relation adversarially.
- Explicit methods: Explicit reconstruction methods are easy to implement and optimize but may produce visually low-quality reconstructions in high dimensions.The discussion specifically considers ℓk losses with k = 1, 2.
- Implicit methods: Implicit adversarial methods theoretically guarantee perfect reconstruction but are hard to achieve in practice, especially in high-dimensional spaces.They effectively measure reconstruction closeness using an ℓ0 metric.
- Correspondence ambiguity: In the supervised toy setting, symmetric domains create correspondence ambiguity for purely unsupervised CycleGAN variants, producing random component switching across runs.The z-domain has two GMM components and the x-domain has five.
- Mapping trade-off: Deterministic mappings yield higher reconstruction ability but lower sampling ability, while more deterministic mappings produce more concentrated reconstructions.These comparisons use reconstruction and generation experiments on toy data.
D More Results on the Effectiveness of CE Regularizers
The paper evaluates cycle-consistency regularization across toy data, MNIST, and CIFAR-10 over a broad weighting range, comparing ALICE against unregularized ALI and an ideal reference.
- Effectiveness: Across the three datasets, the regularizer improves ALI’s image generation and reconstruction over a large range of weighting hyperparameter values.The reported evaluation uses inception scores on the datasets and MSE for reconstruction.
- Comparison: Figure 10 compares the perfect reference, unregularized ALI, ALICE at different regularization levels, and ALICE’s best performance.The perfect performance is shown as a solid line, while ALI is shown as a dashed line.
E.1 Car to Car Experiment
The car-to-car experiment evaluates ALICE on cross-domain translation, reconstruction, and controllable correspondence under weak supervision. Very weak supervision improves convergence and enables coherent or opposite cross-domain relationships.
- Experimental setup: The dataset contains rendered 3D car images across 11 azimuth views, split into training and test sets and two domains.The training set has 1,859 images and the test set has 154 images.
- Evaluation: ALICE evaluates translation using cross-domain prediction and one-domain reconstruction, with MSE and prediction accuracy reported under varying supervision levels.The evaluation uses a regressor and classifier trained to predict azimuth angles.
- Weakly supervised correspondence: Only 1% supervision per angle can enforce either coherent or opposite correspondence configurations in car-to-car translation.Scatter plots compare predicted original and transformed angles; supervised points concentrate more closely on the diagonals.
- Weakly supervised correspondence: Very weak supervision substantially improves convergence results and speed in the car-to-car experiment.Learning curves report RMSE in angle prediction, with all compared plots evaluated at the 10th epoch.