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Bias and Generalization in Deep Generative Models: An Empirical Study
Shengjia Zhao, Hongyu Ren, Arianna Yuan, Jiaming Song, Noah Goodman, Stefano Ermon
TL;DR
High-dimensional density estimation depends on inductive bias, but the biases of deep generative models are not well understood. The paper introduces a probing framework using carefully designed image datasets, finding generalization patterns that recur across models and architectures and resemble cognitive-psychology results.
Problem
Deep generative models’ inductive biases are poorly understood, although such biases are crucial for density estimation in high-dimensional settings.
Method
The paper probes GANs and VAEs with carefully designed image datasets, projects data onto perceptually relevant features, and compares training and learned distributions.
Results
Generalization patterns are consistent across datasets, models, and hyper-parameter choices, with similarities to cognitive-psychology experiments.
Takeaways & Limitations
The framework reveals systematic single-feature and multi-feature generalization patterns, including novel combinations that depend strongly on the number of existing training combinations.
Abstract
from arXiv · showhide
In high dimensional settings, density estimation algorithms rely crucially on their inductive bias. Despite recent empirical success, the inductive bias of deep generative models is not well understood. In this paper we propose a framework to systematically investigate bias and generalization in deep generative models of images. Inspired by experimental methods from cognitive psychology, we probe each learning algorithm with carefully designed training datasets to characterize when and how existing models generate novel attributes and their combinations. We identify similarities to human psychology and verify that these patterns are consistent across commonly used models and architectures.
1 Introduction
Density estimation in high dimensions depends on inductive bias because unbiased, consistent estimation is impossible and the distribution space grows extremely rapidly. The paper probes deep generative models with designed datasets and finds generalization patterns consistent across settings and resembling cognitive-psychology findings.
- Unbiased and consistent density estimation is impossible, making an algorithm’s inductive bias crucial in practical data regimes.
- The framework probes generative models by varying whether training data has single or multiple modes across one or several features.
- GANs and VAEs trained only on images containing three objects typically generate two to five objects with a log-normal-shaped distribution.
- Across datasets, models, and hyper-parameter choices, the observed generalization patterns remain consistent and include similarities to cognitive-psychology experiments.
2 Density Estimation, Bias, and Generalization
High-dimensional density estimation must generalize from a small dataset to an exponentially larger feature-space support, requiring strong prior assumptions. Studying these inductive biases is difficult because learned and target distributions are hard to compare and lack a common evaluation metric.
- Natural-image attributes create exponentially many feature combinations, so training data can be exponentially smaller than the support of the underlying distribution.
- Generalization produces q(x) from training data D, while inductive bias comprises the assumptions used in producing q(x) from D.
- Systematic bias analysis is difficult because both the input dataset and output distribution are high dimensional.
- There is no commonly accepted metric for evaluating how close a learned distribution is to the target distribution.
3 Exploring Generalization Through Probing Features
The paper analyzes inductive bias by projecting images onto interpretable probing features and comparing the induced training and learned distributions in that feature space. Features are chosen for perceptual relevance and evaluability, while experiments span substantially different model families and architectures.
- The framework defines probing features as functions that map images to values and compares the induced distributions p(z) and q(z) in feature space.
- Feature selection targets attributes important to human perception and studied in cognitive psychology, including numerosity, shape, color, size, and location.
- Numerosity and shape are evaluated independently by three human evaluators, while the other features are evaluated with automated algorithms.
- Experiments use GANs and VAEs with different network architectures and hyper-parameter choices to test whether results depend on modeling decisions.
4 Characterizing Generalization on an Individual Feature
The paper probes how generative models generalize along individual image features, finding structured deviations from training distributions across numerosity, color, size, and location. Multi-modal inputs produce convolution-like outputs when modes are distant but prototype enhancement when modes are close, while feature-wise behavior is largely independent.
- Individual-feature analysis: The framework evaluates generalization by projecting images onto one feature and comparing the learned output distribution q(z) with controlled training distributions.The experiments use toy dots, CLEVR objects, and pie images to isolate numerosity, color proportion, size, and location.
- Numerosity: Training on a single numerosity does not preserve it exactly: generated images center near the training value but show wider, slightly overestimated counts.When trained on six dots, generated images contain four to nine dots; the variance grows with numerosity and the response is asymmetric.
- Color proportion: Color-proportion outputs are approximately Gaussian around the trained value, with sharper distributions at 10% and 90% and symmetric generalization.This pattern is consistent with sensitivity to relative rather than absolute changes described by Weber’s law.
- Size and location: Size and location behave differently: size outputs are skewed toward larger objects, whereas location outputs are fairly symmetric and sharp.Figure 5 compares the learned densities with delta-like training distributions for each isolated feature.
- Convolution and prototype enhancement: For multi-modal inputs, convolution predicts outputs well when modes are distant, but close modes merge into a concentrated distribution around their mean.With 30% and 40% red modes, 35% red becomes most likely despite being absent from training; distant modes such as 30% and 90% follow convolution more closely.
- Robustness and feature independence: The observed generalization patterns are consistent across GANs, VAEs, architectures, and hyper-parameter choices, and resemble reported effects in cognitive psychology.The study also finds that each feature can be analyzed largely independently, with only modest variance increases when other dimensions become more random.
5 Characterizing Generalization on Multiple Features
The paper studies how generative models generalize across multiple features and their combinations. Generalization depends on the number of observed combinations, while individual-feature distributions remain comparatively independent of nuisance-feature variation.
- Independence of features: Across red proportion, size, and location, learned feature distributions are fairly independent of nuisance-feature variation, with only a slight variance increase as nuisance features become more random.Greater variance can cause modes that previously remained distinct to merge, consistent with prototype enhancement.
- Generalization across combinations: When training covers few combinations, models largely memorize them; with more combinations, they begin generating novel combinations consistently across settings.This pattern appears across the explored datasets and model configurations.
- Generalization across combinations: Precision remains high-recall while declining as training combinations increase, indicating that GANs and VAEs capture observed combinations but also generate unseen ones.Recall measures coverage of training-support combinations, whereas precision measures how much generated support overlaps the training support.
- CLEVR generalization: For CLEVR, few observed combinations produce little or no generalization, whereas more combinations enable a shape to generalize across all colors while approximately preserving its marginal frequency.The reported settings compare 16 versus 81 combinations.
6 Conclusion and Future Work
The paper introduces carefully designed training sets to study image-generative-model generalization. It identifies feature-specific and combination-level patterns, including marginal preservation and a strong dependence of novel-combination generation on training coverage, while leaving the mechanisms behind these behaviors open.
- The proposed framework uses carefully designed training sets to reveal how image-generative models generalize.
- The study finds feature-specific generalization patterns, including convolution effect, prototype enhancement, and independence, with some similarities to cognitive experiments.
- Novel combinations depend strongly on the number of existing training combinations and can be generated while preserving each feature’s marginal distribution.
- The paper leaves open which modeling ingredient produces the observed behaviors after examining GANs and VAEs.
A Extended Figures for Different Objectives and Architectures
The appendix extends the experiments across objectives, architectures, and feature settings. These additional figures and model variants support qualitative comparisons of the reported generalization behavior.
- WGAN and VAE are the main models, while WGAN-FC and VAE-FC provide substantially different architecture and hyper-parameter settings for robustness checks.
- The extended figures provide corresponding WGAN and VAE plots for several main-paper figures and additional CLEVR color-generalization results.
- The alternative WGAN-GP setting uses a fully connected architecture and Bernoulli latent noise, while retaining the WGAN-GP objective because DCGANs showed severe mode missing.
- The alternative VAE setting uses fully connected encoder and decoder networks, a larger learning rate, and a coefficient of 3 on the DKL(q(z|x)∥p(z)) term.
- For CLEVR color generalization, the appendix compares 4x4 and 9x9 configuration settings, showing training distributions alongside generated-feature frequencies.
A.6 Samples for Three MNIST Dataset
The appendix provides four sample images from the Three MNIST dataset used in the paper’s experiments.
- Four sample images illustrate the Three MNIST dataset used in Section 5.
A.7 Invariance to Architecture Size
The experiments show that memorization is independent of deep-network size, with comparable behavior across small and large architectures.
- A.7 Invariance to Architecture Size: Memorization is independent of the size of the deep network.This result is reported for the experiment shown in Figure 25.
B.1 Datasets
The study uses procedurally generated image datasets with controlled object attributes, including dots, CLEVR objects, and pie images, and compares WGAN architectures by precision-recall curves.
- B.1 Datasets: The dots dataset generates k dots with uniformly random colors and locations, rejecting samples when dots overlap.
- B.1 Datasets: Figure 25 compares WGAN precision-recall curves across networks ranging from 3 million to 24 million parameters.
- B.1 Datasets: The CLEVR dataset generates k objects with random shapes, colors, locations, and sizes, rejecting images with excessive overlap.Images are rejected when any two objects overlap by more than 10%.
- B.1 Datasets: The pie dataset creates images with a specified red-color percentage and three uniformly random non-red colors.
B.2 Evaluation
The evaluation extracts circle attributes from images using measurable properties of the non-background pixels, including area and average location.
- B.2 Evaluation: Circle size is evaluated by computing the non-background area and converting it to the corresponding radius.This works because the images contain a single colored circle on a white background.
- B.2 Evaluation: Circle location is evaluated by computing the average location of the non-background pixels.
- B.2 Evaluation: The evaluation identifies the white background to isolate the circle’s non-background pixels.