Improving Fairness and Mitigating MADness in Generative Models

## Abstract

Abstract We propose autophagy p enalized l ikelihood e stimation (PLE), an unbiased alternative to maximum likelihood estimation (MLE) which is more fair and less susceptible to model autophagy disorder (MADness). Model autophagy refers to models trained on their own output; PLE ensures the statistics of these outputs coincide with the data statistics. This enables PLE to be statistically unbiased in certain scenarios where MLE is biased. When biased, MLE unfairly penalizes minority classes in unbalanced datasets and exacerbates the recently discovered issue of self-consuming generative modeling. Theoretical and empirical results show that 1) PLE is more fair to minority classes and 2) PLE is more stable in a self-consumed setting. Furthermore, we provide a scalable and portable implementation of PLE with a hypernetwork framework, allowing existing deep learning architectures to be easily trained with PLE. Finally, we show PLE can bridge the gap between Bayesian and frequentist paradigms in statistics.

## 1 Introduction

Maximum likelihood is one of the most popular techniques for parameter estimation Johnson (). It is also used to train many popular deep generative models, including variational autoencoders (VAEs) Pu et al. (), normalizing flows (NFs) Rezende and Mohamed (), diffusion models Ho et al. (), and generative adversarial networks (GANs) Goodfellow et al. ()(^1^11These observations apply to models trained on a lower bound of the likelihood function, such as the popular ELBO Kingma and Welling ().). Any deep learning model that uses the negative log likelihood as a loss function is performing maximum likelihood estimation Vapnik (, ).

Despite its ubiquity (and desirable asymptotic properties Cramér ()), maximum likelihood often produces biased estimators with a finite sample size. The most famous example was pointed out by Neyman and Scott (), showing that MLE could produce inconsistent results when the number of parameters was large relative to the amount of data DasGupta (). In effect, there was not enough data relative to the number of parameters to mitigate the bias, leading to what Neyman called “false estimations of the parameters”, or statistics where the stochastic limits were unequal to the values of the parameters to be estimated Stigler (). This overparameterized regime is precisely where most modern deep learning models are trained Zhang et al. (); Belkin et al. (), leading to two problems we discuss: MADness (where models trained on their own output generate poor data Alemohammad et al. ()) and unfairness (resulting from unbalanced datasets consisting of a majority and minority classes). We propose PLE, an alternative to MLE that ensures the statistics of parameters estimated from generated data match those estimated from observed data (See Equation [2](#S3.E2)).
In effect, this constrains MLE in a way that often removes bias, mitigating the above problems to produce estimators that are fairer and less susceptible to self-consumed estimation.

### 1.1 Model Autophagy Disorder (MADness)

Recent work has shown that models trained on their own output, a process called a self-consuming loop, progressively decrease in quality (precision) and diversity (recall) Alemohammad et al. (). Researchers call this phenomenon “going mad” or simply “mad cow,” after bovine spongiform encephalopathy (BSE), the medical term for mad cow disease(^2^22Bovine spongiform encephalopathy (BSE) is a neurological disorder believed to be transmitted by cattle eating the remains of *other* (infected) cattle Prusiner ().). This phenomenon has become a growing concern for the machine learning community due to the availability and ubiquity of synthetic data Nikolenko ().
It often occurs with large language models (LLMs) such as ChatGPT: LLMs trained on their own output suffer in diversity, eventually collapsing to a single point Briesch et al. (). Furthermore, LLMs such as ChatGPT are trained on a massive corpus of text, often collected from the internet Radford et al. ().

Given the popularity and availability of these models, it is almost inevitable that future LLMs will be trained on a corpus containing at least some (if not much) synthetic data, implying that future versions of ChatGPT and similar LLMs may be subject to diversity collapse.
We show that the presence of estimator bias worsens this phenomenon, and we propose a method of removing this bias for deep generative models in Section [4](#S4), effectively slowing down this collapse.

### 1.2 Fairness

The statistical sense of the word bias is distinct from its colloquial usage, so to avoid ambiguity, note that we exclusively use the term “bias” to refer to the statistical bias of an estimator: $b(\hat{\theta})=\mathbb{E}_{\bm{X}|\theta}[\hat{\theta}]-\theta.$ Furthermore, the word bias may invoke a normative undertone we wish to be agnostic towards(^3^33In fact, some argue a more “fair” model is one where bias is purposely introduced to account for unbalanced classes Tyler ().). Recent work has shown that generative models carry and often amplify unbalances present in training data Zhao et al. ().

We look at the quality of generated samples from unbalanced datasets containing a majority and minority class ($C_{\text{Maj}}$ and $C_{\text{Min}}$, respectively), where $|C_{\text{Maj}}|\gg|C_{\text{Min}}|$, observing that generative models are often far worse at representing data from minority classes than the ratio of datapoints $|C_{\text{Maj}}|:|C_{\text{Min}}|$ would suggest. Because PLE removes statistical bias, it improves the representation quality of data from the minority class, leading to more fair outcomes. This is shown in Section [5.2](#S5.SS2)

## 2 Background

### 2.1 Generative Models

Generative models use data to estimate an unknown probability distribution, generating “new” data by sampling from the estimated distribution. A good generative model generates data whose statistics match that of the observed data used to train the model Schwarz et al. (). From a Bayesian perspective, training a generative model amounts to using observed data to update the estimated parameters that are assumed to give rise to the data. The architecture of a generative model can be viewed as a prior on the estimated distribution, constraining the overall shape of the distribution itself Ulyanov et al. ().

Of course, the data are merely samples of (what is usually assumed to be) an underlying stationary stochastic process with fixed parameters. Generative models seek to find these fixed parameters so as to determine the shape and location of the distribution from which our data are assumed to have been drawn. Randomness is introduced in the estimation process in two ways: 1) Each datapoint is random and 2) the sampling process itself contains randomness based on the relationship between the given datapoints. More data implies a given model can better discriminate between competing parameter choices, decreasing estimation uncertainty and thus estimation randomness. In many cases, an infinite amount of data would lead to an estimator that is completely deterministic, as there would be enough data to remove any estimation uncertainty.

### 2.2 Model Estimation

Model estimation involves estimating an unknown probability function $P$ from samples, each sample an outcome from its sample space (see [A](#A1) for more details). We notate this as $\mathbf{X}=[\mathbf{x}_{1},\dots\mathbf{x}_{n}],\;\mathbf{x}_{i}\in\Omega,\;i=1,\dots,n.$ In most cases (and in all practical cases), the number of samples we observe is finite ($|X|\in\mathbb{Z}^{+}$).
Estimating $P$ from a finite number of samples is made difficult by the fact that we not only want to estimate the probability of the observed events (corresponding to the samples), but also estimate the probability of unobserved events that *could have happened*. Since we do not directly observe all events or the probability function itself, we must estimate $\Omega$, $F$ and $P$ from the samples we *do* observe.

Estimating $P$ is especially challenging when $\Omega$ is continuous. In these cases, we are estimating $\Omega$ from a set with zero measure in $\Omega$. When $|\Omega|$ is finite, a sample of $\Omega$ may contain all possible outcomes, and thus the only unknown may be $P$. However, no finite sample can come close to exhausting all outcomes in $\Omega$ when $|\Omega|$ is infinite.

To make the estimation problem more tractable, we often assume that $P$ has a parametric form. Saying $P$ is parameterized by $\theta$ means we can know everything there is to know about $P$ from $\theta$. To estimate $P$, we first estimate (or *a priori* assume) $\Omega$, and then estimate $\theta$.

### 2.3 Maximum Likelihood Parameter Estimation

The maximum likelihood parameter estimation procedure chooses parameter values that maximize the *likelihood function*: the conditional probability of the observed data on the parameters, $P(\mathbf{X};\theta)$ Murphy (). The problem with maximum likelihood is that it is too “greedy.” The maximum likelihood parameter estimation method does take into account the randomness associated with the assumed parametric form of the distribution, but not the randomness associated with choosing $\theta$ from $n$ samples. Consequently, maximum likelihood estimates are only guaranteed to be asymptotically unbiased and consistent Johnson () but not unbiased for any $n$. Our method, PLE, seeks to incorporate the randomness associated with sampling $n$ times from a given parametrically defined probability function. This approach effectively de-biases the maximum likelihood estimate proportional to the uncertainty involved in the sampling process itself.

### 2.4 Parameter Estimation and Model Autophagy

Given data $\mathbf{X}$ drawn from a parameterized distribution $P(\mathbf{X};\theta)$, we form an estimator $\hat{\theta}$ of the generative model’s parameters $\theta$.
As shown in figure [1](#S2.F1), the model’s parameters $\theta$ are estimated by some function of the observed data $\hat{\theta}=H(\mathbf{X})$.
MADness (the collapse of generated quality) arises when the estimated parameters are used in the generative model to produce a new dataset $\widehat{\mathbf{X}}$ and this dataset is used to estimate the model’s parameters.

Figure: Figure 1: The self-consuming parameter estimation loop.

## 3 PLE

PLE involves adding a constraint to the maximum likelihood estimator to force it to take into account other possible models that could have generated the data. It can be broken down into several steps:

- 1.
Choose a parametrization $P(\mathbf{X};\theta)$ for our data-generation model where $\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n}]$, with each $\mathbf{x}_{i}\sim P(\mathbf{x};\theta),\;i=1,\dots,n$ are I.I.D. samples from the generative model parameterized by $\theta$.
- 2.
Choose a function $H(\cdot)$ that deterministically produces an estimate of $\theta$ from $\mathbf{X}$: $\hat{\theta}=H(\mathbf{X})$.
- 3.
Generate the candidate set of estimators $C=\left\{\hat{\theta}\text{ s.t. }\mathbb{E}_{\mathbf{X},\mathbf{Y}}[H(\mathbf{Y})-H(\mathbf{X})]=0\right\},$ over all *possible* data $\mathbf{Y}$ generated by $P(\mathbf{Y};H(\mathbf{X}))$.
- 4.
Choose the estimator from $C$ that maximizes the likelihood function: $\hat{\theta}_{\text{PLE}}=\arg\max_{\hat{\theta}\in C}P(\mathbf{X};\hat{\theta})$.

This process can be expressed as a constrained maximum likelihood estimation problem.

$$ $\hat{\theta}_{\text{PLE}}=H^{*}(\mathbf{X}),H^{*}=\arg\max_{H}P(\mathbf{X};\hat{\theta})\text{ s.t. }\mathbb{E}_{\mathbf{X},\mathbf{Y}}[H(\mathbf{Y})-\hat{\theta}]=0.$ (1) $$

Since $\theta=H(\mathbf{X}),$ we can also write this constraint fully in terms of $H$:

$$ $\hat{\theta}_{\text{PLE}}=H^{*}(\mathbf{X}),H^{*}=\arg\max_{H}P(\mathbf{X};H(\mathbf{X}))\text{ s.t. }\mathbb{E}_{\mathbf{X},\mathbf{Y}}[H(\mathbf{Y})-H(\mathbf{X})]=0.$ (2) $$

### 3.1 PLE as Bayesian and Frequentest Estimation

In the field of statistics, there is a divide between Bayesians and frequentists. The Bayesian approach sees the fixed (and unknown) parameters as random variables Wakefield (). In this case, the *data* is fixed while the parameters are random Fornacon-Wood et al. (). The benefit of the Bayesian approach is its ability to find optimal estimators: because parameters are mapped to a probability measure, they can be compared and a maximum can be found (when it exists). The drawback of Bayesian estimation is that it requires an accurate prior; if the prior is inaccurate, the estimator may ignore the data and produce a biased result Dias et al. (). Additionally, detractors point out that the choice of prior (and the form of the posterior itself) is often subjective Gelman ().

The frequentist approach, on the other hand, evaluates a hypothesis by assuming the parameter or hypothesis in question is fixed, and calculating the probability of the data *under* this hypothesis. In this case, the *hypothesis* (or parameter) is fixed while the data is a random variable. As one statistician writes:

> As the name suggests, the frequentist approach is characterized by a frequency view of probability, and the behavior of inferential procedures is evaluated under hypothetical repeated sampling of the data Wakefield ().

Since the true parameters in an estimation problem are often assumed to be non-random, the frequentist is correct to point out that uncertainty in estimation is usually *epistemic*, not *ontic*. This observation, as true as it may be, comes at a cost: there becomes no obvious way of *choosing* optimal parameters. For the Baysian, the a posteriori distribution serves as a “goodness measure” that tells us how well a given hypothesis fits the observed data. The frequentist, on the other hand, has no such measure: since the parameters are fixed, we are unable to make probability statements about the parameters given the observed data Wagenmakers et al. (). This leads to the most fundamental limitation of frequentist inference: it does not condition the observed data Wagenmakers et al. (). Of course, one can *choose* an objective function that a frequentist approach can optimize, such as with kernel density estimation (KDE); however, the Bayesian will point out such assumptions are usually contrived.

The promise of a hybrid view comes from an acknowledgement of the benefits and shortcomings of each approach. As Roderick Little suggests, “inferences under a particular model should be Bayesian, but model assessment can and should involve frequentist ideas Little ()(^4^44See also Gelman (); Rubin ().)” The Bayesian is correct to point out that we are estimating our parameters from observed *data*, so there will be uncertainty in the parameters themselves. The frequentist, on the other hand, is correct to point out that the randomness in estimation results from the data itself and not true parameters we wish to estimate. This is what causes the problem in Section [B.1](#A2.SS1), and with bias in MLE more generally: while the estimated parameters are random under the data (which is true), the randomness of the *data itself* is ignored.

PLE bridges this gap: it treats the true parameters as nonrandom, while acknowledging that the estimated parameters are random *because* the data is random. It incorporates the uncertainty of the estimation process from the uncertainty of the data in a given model class. As a result, the strength of the estimation uncertainty is related to the number of datapoints - with more data, there is less estimation uncertainty as we have more “information” about the true parameters.

### 3.2 Choosing H 𝐻 H

As we show in [B.1](#A2.SS1), as long as $\hat{\theta}_{\text{PLE}}$ (calculated from Equation [1](#S3.E1)) is in the domain of $H$, the choice of $H$ itself does not matter. In other words, PLE is transformation invariant for any transformation that could produce $\hat{\theta}_{\text{PLE}}$. In the absence of any prior information about the form of $H$, the form of the MLE can be used as a reasonable prior for selecting $H$ and thus $\hat{\theta}_{\text{PLE}}.$ In cases when MLE is strictly unbiased (not just asymptotically unbiased, as it is guaranteed to be Johnson ()), PLE will give the same result as MLE. At the very least, the MLE should be in the range of $H$.

## 4 Implementing H 𝐻 H with Hypernetworks

Solving the optimization problem in Equation [2](#S3.E2) in real-world applications is intractable due to two main computational bottlenecks. First, the operator $H$ that produces the weights of the generative model, $P(\mathbf{X};\theta)$, from training data, $\mathbf{X}$, is not an explicit operator. Instead, it is usually an optimization routine, (i.e., training of a generative model given data). As a result, evaluating the PLE constraint in Equation [2](#S3.E2) involves solving an inner optimization problem that trains a secondary generative model on synthetic data. This is clearly intractable as it requires training a new generative model at *every* iteration. Second, it is unclear how the PLE constraint can be strictly imposed.

To address these challenges, we propose parameterizing $H$ as a hypernetwork (Ha et al., ) denoted by $H_{\phi}$, i.e., a neural network that is trained to predict the weights of another neural network. In our case, we use $H_{\phi}$ to learn the parameters of $P(\mathbf{X};\theta)$: $H_{\phi}$ takes as input training data and predicts the weight of a generative model that approximates the distribution of the training data. This allows the PLE constraint to be tractably evaluated by a sample of data through $H_{\phi}$. To address the second challenge, we relax the optimization problem in Equation [2](#S3.E2) so the constraint is turned into a penalty term.

Inspired by the form of $H$ obtained analytically for some simple distributions in Appendix [B.1](#A2.SS1), and to impose permutation invariance — with respect to ordering of data points — we propose the following functional form for the hypernetwork $H_{\phi}$ following (Radev et al., ):

$$ $H_{\phi}(\mathbf{X}):=h^{(2)}_{\phi}\left(\frac{1}{n}\sum_{i=1}^{n}h^{(1)}_{\phi}(\mathbf{x}_{i})\right),$ (3) $$

where $h^{(1)}_{\phi}$ and $h^{(2)}_{\phi}$ are two different fully-connected neural networks and $\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n}]$ is a set of data samples. While more expressive architectures can be used, in our experiments we found it sufficient to choose $h^{(2)}_{\phi}$ to be a set of independent fully-connected layers such that for any layer in the target generative model, we predict its weights via applying an independent fully-connected layer to the intermediate representation $\frac{1}{n}\sum_{i=1}^{n}h^{(1)}_{\phi}(\mathbf{x}_{i})$. This functional form has several advantages: (i) it is permutation invariant, which is required since the weights of the generative model $P(\mathbf{X};\theta)$ do not depend on the order of data points $\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n}]$; (ii) it can deal with an arbitrary number of data points, which enables batch training and in turn allows for evaluating $H_{\phi}$ over a large number of data points that otherwise would not fit into memory; and (iii) the inner sum in the functional form of $H_{\phi}$ and the proposed architecture for $h^{(2)}_{\phi}$ (a set of independent linear layers) are highly parallelizable, allowing us to train existing generative models with minimal computational overhead.

In our implementation, we are able to automatically create a hypernetwork architecture given a generative model architecture. As long as the given architecture is a torch.nn.Module, our hypernetwork implementation outputs a named dictionary containing the layer names and weights for the target generative model. In addition, to allow for seamless usage of our PLE formulation as a drop-in replacement for existing objective functions for generative models, we exploit the abstractions provided by PyTorch that allow functional calls to any PyTorch torch.nn.Module that uses the predicted weights from our hypernetwork architecture. This allows us train existing generative models with PLE in just a few additional lines of code. More details regarding our implementation and code to reproduce the results can be found on GitHub(^5^55Link is removed for double-blind review. Please refer to the code uploaded as supplementary material.).

## 5 Experiments

For each experiment, we use one or two NVIDIA Titan X GPUs with 12 GB of RAM. The time of execution for each experiment varies from a few minutes to 14 hours per model trained. For the experiments with multiple models trained, since we must train the models sequentially, the time of execution of the whole experiment is just the number of models times the time necessary to train one model.

### 5.1 MADness Experiments

Our experiments in this section show that models trained with PLE (either analytically or via our hypernetwork approach) are less susceptible to MADness (Model Autophagy Disorder Alemohammad et al. ()) than models trained with MLE. In the following experimental setups, parameters are estimated from fully synthetic data, generated either from a model trained on the ground truth data, or the synthetic data from a previous generation.

An illustrative example involves estimating the parameter $a$ of a one-sided uniform, $U[0,a]$. While the MLE and PLE of $a$ are similar in their closed-form expression (See [B.1](#A2.SS1) for more details), the two quickly diverge and the MLE collapses to 0 0 as a new estimate is produced from data generated from the previous estimate. MLE’s collapse is due to its bias: this bias degrades its result after each generation, as seen in Figure [2](#S5.F2).

We also trained BigGAN Brock et al. () on CIFAR-10 Krizhevsky et al. () and observe a similar result. Specifically, the BigGAN (^6^66We use [https://github.com/ajbrock/BigGAN-PyTorch](this%20repository)) data generation collapses after 3 iterations whereas our PLE method does not. We measure performance here using FID Heusel et al. (), which does not change much for our method over the course of 3 generations yet completely explodes for the BigGAN baseline (see [Figure 3](#S5.F3)). Additionally, both the baseline and the PLE took about the same time, 14 hours on two GPUs.

Figure: Figure 2: Number of generations versus MLE and PLE of one-sided uniform, $n=20.$ The error bars display the standard error for $100$ different initializations. These results use the analytic form of PLE solved in Section [B.1](#A2.SS1).

### 5.2 Fairness Experiments

Suppose we are trying to model the characteristics of a highly unbalanced dataset $\mathbf{X}$ containing two classes of data: $C_{\text{Maj}}$ and $C_{\text{Min}},$ where $\mathbf{X}=\mathbf{X}_{C_{\text{Maj}}}\cup\mathbf{X}_{C_{\text{Min}}}$ and $|C_{\text{Maj}}|\gg|C_{\text{Min}}|.$ We call $C_{\text{Maj}}$ the *majority class* and $C_{\text{Min}}$ the *minority class*. Our generative model $M$ is trained on this unbalanced dataset $\mathbf{X}.$ Note that this dataset is not necessarily *biased*; the class imbalance in $\mathbf{X}$ may accurately represent the true ratio of majority to minority datapoints.
Suppose further that we have a score function $S$ which evaluates the representation quality of samples drawn from $M$. $S(M)$ is a scalar denoting the overall representation quality, which can be broken up into two distinct components: $S(M)_{\text{Maj}}$ and $S(M)_{\text{Min}},$ corresponding to the representation quality of samples from the majority class and minority class, respectively. For generative image models, a common metric for evaluating the distance of generated images to real images is the Frechet Inception Distance (FID) Heusel et al. (); we use the inverse of FID as $S$, since smaller distances correspond to higher quality generated images.

Virtually all variants of empirical risk minimization (including MLE) weight each datapoint equally, and we can expect that for a linear $S(M)$

$$ $S(M)=\frac{|C_{\text{Min}}|}{|C_{\text{Min}}|+|C_{\text{Max}}|}S(M)_{\text{Min}}+\frac{|C_{\text{Max}}|}{|C_{\text{Min}}|+|C_{\text{Max}}|}S(M)_{\text{Max}}.$ (4) $$

This implies that $M$ will more accurately model the density around the majority class than the minority class, and thus $S(M)_{\text{Max}}>S(M)_{\text{Min}}.$ In other words, even if the training dataset accurately represents the relative frequencies of the both classes, this relative imbalance often harms performance when looking only at data from the minority class. This is empirically observed with facial classifiers on unbalanced data Buolamwini and Gebru () and is seen for vanilla generative image models in Table [1](#S5.T1).

In principle, weighting each datapoint equally corresponds to a process considered to be procedurally fair Tyler (), despite the fact that the representation quality for samples in the minority class may be far worse than those in the minority class (with the difference in performance roughly proportional to how unbalanced the dataset is). We introduce a quantity called the *fairness ratio* over a metric $S,$ which is defined as

$$ $R_{\text{Fair}}=S(M)_{\text{Maj}}/S(M)_{\text{Min}}$ (5) $$

Values of $R_{\text{Fair}}$ closer to $1$ correspond to models that do equally well representing majority and minority datapoints, while values much larger than $1$ refer to models that have better performance representing datapoints from the majority class than the minority class. Conventional wisdom suggests that models trained on $\mathbf{X}$ should have $R_{\text{Fair}}\approx|C_{\text{Maj}}|/|C_{\text{Min}}|,$ the ratio of majority to minority datapoints.

Depending on one’s definition of fairness, one can argue that $R_{\text{Fair}}=|C_{\text{Maj}}|/|C_{\text{Min}}|,$ results from a procedurally fair training process, where each datapoint is treated equally(^7^77Unconditional generative models have no “knowledge” of protected attributes.). However $R_{\text{Fair}}\gg|C_{\text{Maj}}|/|C_{\text{Min}}|,$ is clearly unfair, as it penalizes the generation of minority data far more than would be expected by its underrepresentation in the training set. This is precisely what we observe in Table [1](#S5.T1) for generative models trained with a variant of MLE: data from the minority class are modeled far worse than its relative imbalance would suggest.

While it may not be reasonable to expect $R_{\text{Fair}}=1$ in cases when $|C_{\text{Maj}}|\gg|C_{\text{Min}}|,$ we show that models trained with PLE have values of $R_{\text{Fair}}$ much closer to $1$ than those trained with MLE. Furthermore, our experiments suggest models trained with PLE have $R_{\text{Fair}}<|C_{\text{Maj}}|/|C_{\text{Min}}|,$ implying that PLE helps the generation of minority data beyond what the class imbalance would predict. The results for models trained with PLE versus MLE based training are shown in Table [1](#S5.T1), showing that PLE produces results that are more fair. These results are more pronounced as the classes become more imbalanced (as the ratio $|C_{\text{Maj}}|:|C_{\text{Min}}|$ increases)

**Table 1: FID for Minority and Majority data, trained on MNIST with a Variational Autoencoder (VAE). Lower FID is better, as this corresponds to generated images being more similar to the images in the training set. The VAE consists of an encoder and decoder, each with 5 layers containing a fully connected layer with batchnorm and leaky Relu. This was trained on a single CPU in several hours. The hypernetwork architecture consists of hidden sizes of $32$,$64$, and $96$, which takes several hours to train on a single CPU, taking a few hours longer than the baseline.. The “Majority” class consisted of the digit $3$ and the “Minority” class was the digit $6$ (This choice was arbitrary), with the ratio of majority to minority datapoints used for training shown in the table.**
| Model | Majority Class FID | Minority Class FID | $R_{\text{Fair}}$(^8^88Closer to $1$ is better.) | Overall FID |
| --- | --- | --- | --- | --- |
| $C_{\text{Maj}}:C_{\text{Min}}=10:1$ |  |  |  |  |
| VAE | $\bm{0.3060}$ | $3.9760$ | $12.993$ | $\bm{0.4803}$ |
| Hyper-VAE (Ours) | $0.4482$ | $\bm{2.9806}$ | $\bm{6.650}$ | $0.5856$ |
| $C_{\text{Maj}}:C_{\text{Min}}=20:1$ |  |  |  |  |
| VAE | $\bm{0.2324}$ | $9.9098$ | $42.641$ | $\bm{0.6116}$ |
| Hyper-VAE (Ours) | $0.4122$ | $\bm{6.03186}$ | $\bm{14.633}$ | $0.8603$ |

Figure: Figure 3: PLE is more stable and outperforms the baseline as we train models on their own outputs (MADness). This plot shows the generation versus FID for BigGAN trained on CIFAR-10. The baseline (labeled BigGAN) uses normal BigGAN training and collapses after only three generations. On the other hand, our method is very stable and only sees a slight increase in FID over the course of the three generations.

### 5.3 Stylized example: unbalanced Gaussian mixture model

To showcase the benefits of using PLE when dealing with imbalanced datasets, we estimate the weights of a one-dimensional Gaussian mixture model (GMM) with two components that have considerable overlap. The means of the two Gaussian components are $0.0$ and $2.0$ and the variances are both equal to $1.0$. We vary the responsibility vector such that the contribution of one of the two Gaussian components is decreased. This makes estimating the true GMM parameters challenging, especially in low-data regimes and as the GMM becomes more imbalanced.

We estimate the GMM weights using expectation maximization (EM), the gold standard MLE approach, and compare the results to the estimated GMM via PLE. We rely on sklearn for estimating the GMM via the EM approach, choosing an optimization tolerance smaller than floating point precision, and setting the maximum number of iterations to $10^{5}$ (we did not observe improvement by increasing this number). To implement PLE, we define $H$ to predict the means, variances, and the responsibility vector from the training data in one step. The architecture of $h^{(1)}_{\phi}$ contains three fully-connected layers with hidden dimensions of $8$ and ReLU activation functions. We design $h^{(2)}_{\phi}$ in a similar manner, with the only difference being the last layer, which contains three branches. Each branch includes a linear layer aimed at predicting the mean, variance, and responsibility vector of the GMM. We ensure the predicted variance is positive by using a softplus activation function and ensure the predicted responsibilities are positive and sum to one by using a softmax function. The objective function in the PLE case is to maximize the likelihood of observing the training data under a GMM model whose weights are predicted via the hypernetwork, while also including the PLE constraint as a penalty term with a weight of $0.1$.

To compare the performance between MLE and PLE, after optimization, we calculate the KL divergence between the estimated and the ground truth GMMs using $10^{5}$ samples. We repeat the GMM estimation for 100 different random seeds and average this quantity. Figure [4](#S5.F4) illustrates the difference between the KL divergence between PLE and the true GMM minus the KL divergence between MLE and the true GMM, i.e., $\mathbb{D}_{KL}(q_{\,\text{MLE}}\mid\mid p_{\,\text{GMM}})-\mathbb{D}_{KL}(q_{\,\text{PLE}}\mid\mid p_{\,\text{GMM}})$. The negative values (shown in blue) correspond to settings where PLE outperforms MLE (in terms of KL divergence). We observe that for imbalanced GMMs, and when the training data size is smaller, PLE clearly outperforms MLE. In addition, as expected, as the number of training samples goes to infinity, the performance of PLE and MLE become similar.

Figure: Figure 4: Comparison of KL divergence differences, i.e., $\mathbb{D}_{KL}(q_{\,\text{MLE}}\mid\mid p_{\,\text{GMM}})-\mathbb{D}_{KL}(q_{\,\text{PLE}}\mid\mid p_{\,\text{GMM}})$, for estimating GMM parameters. Positive values indicate scenarios where PLE outperforms MLE (i.e., blue is better), particularly in imbalanced datasets and low-data regimes.
Refer to caption: /html/2405.13977/assets/figs/GMM/kl_comparison_hidden_size-8,8,8_ple_weight-0.1.png

## 6 Conclusion

Autophagy penalized likelihood estimation (PLE) solves many of the problems that result from bias in maximum likelihood estimation (MLE); the two we focus on are MADness (self-consuming generation) and unfairness (resulting from unbalanced datasets). Future work can focus on additional applications of PLE, such as using it as regularization term to mitigate overfitting. Since, in the deep learning case, PLE involves sampling the generative model to evaluate the penalty, future work is needed to allow tractable training of diffusion models, which are expensive to sample from. By combining unbiased statistical estimation methods with deep learning, we believe we can make artificial intelligence more fair and stable.

## Appendix A Sampling, Randomness, and Modal Logic

Probability theory is a way of saying how “likely” something is to happen. A probability space is a $3-$element tuple, $S=(\Omega,F,P),$ consisting of

- •
A sample space $\Omega$ which is a (nonempty) set of all possible outcomes $\omega\in\Omega$
- •
An event space $F$, which is a set of events $f$ which themselves are sets of outcomes. More precisely, $F$ is a $\sigma$-algebra over $\Omega$ Stroock ().
- •
A probability function $P$ which assigns each $f\in F$ to a *probability*, which is a number between 0 0 and $1$ inclusive.

This triple must satisfy a set of probability axioms to be considered a legitimate probability space Papoulis and Pillai (). An event with probability $1$ is said to happen *almost surely* while an event with zero probability is said to happen *almost never*. Note that not all zero probability events are logically impossible (The probability of *any* outcome on a continuous probability distribution is 0 0, even after one *observes* such an outcome). Impossible events are thus those not contained within $F$; these are assigned probability of zero by definition.

This idea of probability is closely related to the idea possible worlds, other ways the world *could have been*. We use the semantics of modal logic to write the different “modes” of truth, including *necessary* propositions ($\Box x$), *possible* propositions ($\Diamond y$) and *impossible* propositions ($\neg\Diamond z$) Garson (). Consider a fair dice roll on a six-sided die, where each outcome is the corresponding number on top of the die (from 1-6). Let $f_{n<10}$ be the event that one rolls a number less than $10$. Since all possible outcomes have a value less than $10$, we say it is *necessary* that one rolls a number less than $10$, or $\Box f_{n<10}.$ Let the $f_{4}$ be the event that one rolls a $4$. It is *possible* that one rolls a $4$, so we say $\Diamond f_{4},$ however it is not necessary, because one could have rolled a $5$ instead. It is *impossible* to roll a $7$, so we say $\neg\Diamond\text{roll a seven}.$ Rolling a seven is not an event because it does not exist in $\Omega$; $7$ is not a legitimate outcome as constructed(^9^99 It is common to notate such events using the empty set.).

An outcome $x\sim P(\theta)$ from a probability distribution $P$ cannot feature all possibilities from the distribution except in the case of a trivial distribution (which has no randomness). For nontrivial distributions, or ones with infinite support, there will always be at least some other outcome with nonzero probability $\tilde{x}$ that *could have happened* if our observed outcome was different. Semantically, we say there is a *possible world* in which $\tilde{x}$ happens as long as $\tilde{x}\in F$ Menzel ().

## Appendix B One-Sided Uniform

### B.1 MLE

Consider a $n$ samples drawn from the following uniform distribution

$$ $\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n}],\;\mathbf{x}_{i}\sim U[0,a]\;i=1,\dots,n.$ $$

We wish to estimate the parameter $\hat{a}$ from $\mathbf{X}$ so $\hat{a}=a.$ First, we write out the likelihood function: $P_{\mathbf{X}|\hat{a}}(\mathbf{X}|\hat{a})$ as

$$ $P_{\mathbf{X}|\hat{a}}(\mathbf{X}|\hat{a})=\begin{cases}0&\text{if }\hat{a}<\max(\mathbf{X})\\ \frac{1}{\alpha\hat{a}^{n}}&\text{else}\end{cases}$ $$

Where $\alpha$ is a scaling factor that ensures the conditional PDF integrates to $1$. Since this function is monotonic with respect to $\hat{a},$ the MLE is easily found as $\hat{a}_{\text{MLE}}=\arg\max_{\hat{a}}P_{\mathbf{X}|\hat{a}}(\mathbf{X}|\hat{a})=\max(\mathbf{X}).$ This also corresponds to the $n$-th order statistic.

### B.2 MLE Bias

Now that we have $\hat{a}_{\text{MLE}},$ we can calculate the bias as follows:

$$ $b(\hat{a}_{\text{MLE}})=\mathbb{E}_{\mathbf{X}|a}[\hat{a}_{\text{MLE}}]-a=\mathbb{E}_{\mathbf{X}|a}[\max{(\mathbf{X})}]-a$ (6) $$

The expected value of the maximum of $\mathbf{X}$ (the $n$-th order statistic of $\mathbf{X}$) can be calculated by taking the derivative of the CDF of the maximum value with respect to the parameter in question:

$$ $F(\max(\mathbf{X}))=P(\max(\mathbf{X})\leq\hat{a})=\begin{cases}0&a<0\\ \left(\frac{\hat{a}}{a}\right)^{n}&\hat{a}\in[0,a]\\ 1&\hat{a}>a\end{cases}$ $$

$$ $f(\max(X))=P(\max(\mathbf{X})=\hat{a})\begin{cases}0&a<0\\ \frac{n\hat{a}^{n-1}}{a^{n}}&\hat{a}\in[0,a]\\ 0&\hat{a}>a\end{cases}$ $$

Now we can calculate $\mathbb{E}[\max(\mathbf{X})]$ as follows:

$$ $\mathbb{E}_{\mathbf{X}|a}[\max(\mathbf{X})]=\frac{n}{a^{n}}\int_{0}^{a}\hat{a}^{n}d\hat{a}=\frac{n}{n+1}a.$ (7) $$

Therefore, the bias of the MLE is

$$ $b(\hat{a}_{\text{MLE}})=\frac{n}{n+1}a-a=-\frac{1}{n+1}a.$ $$

Note that the bias here is negative, implying that the MLE $\hat{a}_{\text{MLE}}$ will consistently *underestimate* $a$.

### B.3 One-Sided Uniform Example - Linear Function Class

Suppose we have data $\mathbf{X}=[\mathbf{x}_{1},\dots\mathbf{x}_{n}],\mathbf{x}_{i}\sim U[0,a],i=1,\dots,n$ and we want to estimate $\hat{a}$ from $\mathbf{X}$. We assume that $H$ is linear, so

$$ $\hat{a}=H(\mathbf{X})=\sum_{i=1}^{n}\alpha_{i}\mathbf{x}_{i}$ $$

Now we calculate $H(\mathbf{Y}),$ via

$$ $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})-H(\mathbf{X})\right]=0\iff{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]={\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]$ First, we look at ${\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]$ : $\displaystyle{\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]=\sum_{i=1}^{n}\alpha_{i}{\mathbb{E}}\left[\mathbf{x}_{i}\right]=\mu\sum_{i=1}^{n}\alpha_{i}$ Now we look at ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[sH(\mathbf{Y})\right]$ : $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[sH(\mathbf{Y})\right]=\sum_{i=1}^{n}\alpha_{i}{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[\mathbf{y}_{i}\right]=\sum_{i=1}^{n}\alpha_{i}\cdot\left(\frac{\mu}{2}\sum_{j=1}^{n}\alpha_{j}\right)=\frac{\mu}{2}\left(\sum_{j=1}^{n}\alpha_{j}\right)^{2}$ Combining these results gives: $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]={\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]\implies\mu\left(\sum_{i=1}^{n}\alpha_{i}\right)=\frac{\mu}{2}\left(\sum_{j=1}^{n}\alpha_{j}\right)^{2}.$ $$

This equality only holds if $\sum_{j=1}^{n}\alpha_{j}=\frac{1}{2}\left(\sum_{j=1}^{n}\alpha_{j}\right)^{2}$. So $\sum_{j=1}^{n}\alpha_{j}$ must be 2 or 0. It can only sum to 0 if the mean is zero, otherwise ${\mathbb{E}}\left[H(\mathbf{X})\right]=0$ always (this is the degenerate case). Thus, it must sum to 2. Also, $\alpha_{i}$ must be equal to $\frac{2}{n}$ because otherwise, permutations of the data would yield different results, and we are assuming they are independent. As a result,

$$ $\hat{a}_{\text{PLE}}=H(\mathbf{X})=\frac{2}{n}\sum_{i=1}^{n}\mathbf{x}_{i}=2m_{\mathbf{X}}.$ $$

Where $m_{\mathbf{X}}$ is the sample mean, $m_{\mathbf{X}}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{x}_{i}$. We calculate the bias by computing

$$ ${\mathbb{E}}_{\mathbf{X}|a}\left[\hat{a}_{\text{PLE}}\right]={\mathbb{E}}_{\mathbf{X}|a}\left[2{\mathbb{E}}\left[\mathbf{X}\right]\right]=2\cdot\frac{a}{2}=a\implies b(\hat{a})=a-a=0.$ $$

Thus the PLE estimate of the one-sided uniform is unbiased.

### B.4 One-Sided Uniform Example - Function of the n-th order statistic

As before, we have a dataset $\mathbf{X}=[\mathbf{x}_{1},\dots\mathbf{x}_{n}],\mathbf{x}_{i}\sim U[0,a],i=1,\dots,n,$ However this time, knowing the MLE estimate of $a$ is $\hat{a}_{\text{MLE}}=\max(\mathbf{X}),$ we assume the form of PLE is a linear function of the maximum of the data; i.e. $\hat{a}_{\text{PLE}}=H(\mathbf{X})=c\cdot\max{\mathbf{X}}$.
We want to estimate $\hat{a}$ from $U[0,a]$. We calculate $H(\mathbf{Y}),$ via $H(\mathbf{X}),$ since $\mathbf{Y}\sim U[0,H(\mathbf{X})]$

We calculate $H(Y)$ as

$$ $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})-H(\mathbf{X})\right]=0\implies{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]={\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]$ Looking at ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]$ , we have $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]=c\cdot{\mathbb{E}}_{\mathbf{Y}}\left[\max{\mathbf{Y}}\right]=c\cdot\frac{n}{n+1}{\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]$ Combining this with ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]$ , we have $\displaystyle\frac{cn}{n+1}{\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]={\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]\implies c=\frac{n+1}{n}$ $$

As a result, $\hat{a}_{\text{PLE}}=H(\mathbf{X})=\frac{n+1}{n}\max{\mathbf{X}}$. We show the PLE estimate is unbiased by taking the expected value of this estimator,

$$ ${\mathbb{E}}_{\mathbf{X}|a}\left[\hat{a}_{\text{PLE}}\right]={\mathbb{E}}_{\mathbf{X}|a}\left[\frac{n+1}{n}\max{\mathbf{X}}\right]=\frac{n+1}{n}\cdot\frac{n}{n+1}a\implies{\mathbb{E}}_{\mathbf{X}|a}\left[\hat{a}_{\text{PLE}}\right]=a,$ $$

Since $b(\hat{a})=a-a=0$), the PLE estimate of $a$ is unbiased. Furthermore, notice how we get the same result (in expectation) as the parametrization considered in section [B.3](#A2.SS3). In other words, as long as different $H$’s contain the optimal value, PLE is transformation invariant and will select this optimal value regardless of the parametrization.

## Appendix C Univariate Gaussian Parameters

### C.1 Maximum Likelihood Estimation

The probability density function for a univariate Gaussian random variable can be written as:

$$ $f(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)}{2\sigma}}.$ (8) $$

The likelihood function of a dataset $\mathbf{X}$ of $n$ such datapoints $x_{i},i=1,\dots,n$ can be written as:

$$ $f(x_{1},\dots,x_{n}|\mu,\sigma)=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x_{i}-\mu)^{2}}{2\sigma^{2}}}$ (9) $$

We find our maximum likelihood estimate of $\mu$ and $\sigma$ as

$$ $\mu_{\text{MLE}}=\arg\max_{\mu}f(x_{1},\dots,x_{n}|\mu)$ $$

and

$$ $\sigma_{\text{MLE}}=\arg\max_{\sigma}f(x_{1},\dots,x_{n}|\sigma).$ $$

Because we will be maximizing the likelihood, we can apply a logarithm (which is monotonic) to the likelihood function to get the log likelihood (ll) function:

$$ $ll(x_{1},\dots,x_{n}|\mu,\sigma)=\ln f(x_{1},\dots,x_{n})=-\frac{n}{2}\ln{2\pi}-\frac{n}{2}\ln{\sigma^{2}}-\frac{1}{2\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\mu)^{2}$ $$

First, the maximum likelihood of the mean can be found by taking the gradient of the above with respect to $\mu$ and solving when the gradient is 0:

$$ $\frac{\partial ll}{\partial\mu}=0\implies\frac{1}{\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\mu)=0\implies\mu_{\text{MLE}}=\frac{1}{n}\sum_{i=1}^{n}x_{i}.$ $$

Similarly, we can calculate the maximum likelihood of the variance as

$$ $\frac{\partial ll}{\partial\sigma^{2}}=0\implies-\frac{n}{2\sigma^{2}}+\frac{1}{2(\sigma^{2})^{2}}\sum_{i=1}^{n}(x_{i}-\mu)^{2}=0\implies\sigma^{2}_{\text{MLE}}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\mu)^{2}.$ $$

Substituting in $\mu_{\text{MLE}}$ for $\mu$ in our MLE for $\sigma_{\text{MLE}}^{2},$ we have

$$ $\displaystyle\sigma^{2}_{\text{MLE}}$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-\frac{1}{n}\sum_{j=1}^{n}x_{j}\right)^{2}$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}-\frac{2}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}+\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}x_{i}x_{j}$ $\displaystyle=\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}-\left(\frac{1}{n}\sum_{j=1}^{n}x_{j}\right)^{2}.$ $$

### C.2 Estimating the mean of a Gaussian with unknown parameters

Suppose $\mathbf{X}=[\mathbf{x}_{1},\dots,\mathbf{x}_{n}],\mathbf{x}_{i}\sim N(\mu,\sigma^{2}),i=1,\dots,$ and we wish to estimate $\mu$. Assume that $H$ is linear, so

$$ $\hat{\mu}=H(\mathbf{X})=\sum_{i=1}^{n}\alpha_{i}\mathbf{x}_{i}$ $$

Now we solve for $H(\mathbf{Y})$

$$ $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})-H(\mathbf{X})\right]=0\iff{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]={\mathbb{E}}_{\mathbf{X}}\left[H(\mathbf{X})\right]$ Looking at ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{X})\right]$ , we have: $\displaystyle{\mathbb{E}}_{\mathbf{x}}\left[H(\mathbf{X})\right]=\sum_{i=1}^{n}\alpha_{i}{\mathbb{E}}\left[\mathbf{x}_{i}\right]=n\mu_{\mathbf{X}}\sum_{i=1}^{n}\alpha_{i}$ Now we look at ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H(\mathbf{Y})\right]$ : $\displaystyle{\mathbb{E}}_{\mathbf{x},\mathbf{y}}\left[H(\mathbf{y}_{1},\dots,\mathbf{y}_{n})\right]=\sum_{i=1}^{n}\alpha_{i}{\mathbb{E}}\left[\mathbf{y}_{i}\right]=\sum_{i=1}^{n}\alpha_{i}\left(\mu_{\mathbf{X}}\sum_{j=1}^{n}\alpha_{j}\right)$ Setting these two equal to each other, we have $\displaystyle\sum_{i=1}^{n}\alpha_{i}\left(\mu_{\mathbf{X}}\sum_{j=1}^{n}\alpha_{j}\right)=n\mu_{\mathbf{X}}\left(\sum_{j=1}^{n}\alpha_{j}\right)^{2}$ $$

The only way that this works is if $\sum_{j=1}^{n}\alpha_{j}=\left(\sum_{j=1}^{n}\alpha_{j}\right)^{2}$. So $\sum_{j=1}^{n}\alpha_{j}$ must be 1 or 0. It can only sum to 0 if the mean is zero, otherwise ${\mathbb{E}}\left[H(\mathbf{x}_{1},\dots,\mathbf{x}_{n})\right]=0$ always (this is the degenerate case). Thus, it must sum to 1. Also, $\alpha_{i}$ must be equal to $\frac{1}{n}$ because otherwise, permutations of the data would yield different results, and we are assuming they are independent.
Thus,

$$ $\mu_{\text{PLE}}=\mu_{\mathbf{X}}.$ $$

Obviously, the sample mean is an unbiased estimator of the expected value, so this result is unbiased.

Since the above derivation does not use the Gaussian assumption, it can be repeated for any distribution to estimate its mean. Therefore, distributions whose mean characterize them are estimated unbiasedly with PLE. Such distributions include exponential, Bernoulli, Borel, Irwin–Hall, etc.

### C.3 Estimating the standard deviation of a Gaussian with unknown parameters

Now that we have the mean estimated, let us estimate the standard deviation. Since we used $H$ above to estimate the mean (which we now notate $H_{\mu}$), we will use $H_{\sigma}$ to estimate the variance to distinguish between the two. Suppose that $H_{\sigma}$ has the following quadratic form:

$$ $\hat{\sigma}^{2}=H_{\sigma}(\mathbf{X})=\sum_{i=1}^{n}\beta_{i}\left(\mathbf{x}_{i}-H_{\mu}(\mathbf{X})\right)^{2}=\sum_{i=1}^{n}\beta_{i}\left(\mathbf{x}_{i}-\frac{1}{n}\sum_{j=1}^{n}\mathbf{x}_{j}\right)^{2}.$ $$

Now, we would like to evaluate ${\mathbb{E}}_{\mathbf{X}}\left[H_{\sigma}(\mathbf{X})\right]$ and ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H_{\sigma}(\mathbf{Y})\right]$ so that we can set them equal to each other and solve. Let us start with the former:

$$ $\displaystyle{\mathbb{E}}_{\mathbf{X}}\left[H_{\sigma}(\mathbf{X})\right]$ $\displaystyle={\mathbb{E}}_{\mathbf{X}}\left[\sum_{i=1}^{n}\beta_{i}\left(\mathbf{x}_{i}-\frac{1}{n}\sum_{j=1}^{n}\mathbf{x}_{j}\right)^{2}\right]$ $\displaystyle={\mathbb{E}}_{\mathbf{X}}\left[\sum_{i=1}^{n}\beta_{i}\left(\mathbf{x}_{i}^{2}-\frac{2}{n}x_{i}\sum_{j=1}^{n}\mathbf{x}_{j}+\frac{1}{n^{2}}\sum_{j=1}^{n}\sum_{k=1}^{n}\mathbf{x}_{j}\mathbf{x}_{k}\right)\right]$ $\displaystyle=\sum_{i=1}^{n}\beta_{i}\left({\mathbb{E}}_{\mathbf{X}}\left[\mathbf{x}_{i}^{2}\right]-\frac{2}{n}{\mathbb{E}}_{\mathbf{X}}\left[\mathbf{x}_{i}\sum_{j=1}^{n}\mathbf{x}_{j}\right]+\frac{1}{n^{2}}\sum_{j=1}^{n}\sum_{k=1}^{n}{\mathbb{E}}_{\mathbf{X}}\left[\mathbf{x}_{j}\mathbf{x}_{k}\right]\right)$ $\displaystyle=\sum_{i=1}^{n}\beta_{i}\left(\Big{(}\sigma^{2}+\mu^{2}\Big{)}-\frac{2}{n}\Big{(}[\sigma^{2}+\mu^{2}]+[n-1]\mu^{2}\Big{)}+\frac{1}{n^{2}}\Big{(}n[\sigma^{2}+\mu^{2}]+[n^{2}-n]\mu^{2}\Big{)}\right)$ $\displaystyle=\sum_{i=1}^{n}\beta_{i}\left(\Big{(}1-\frac{2}{n}+\frac{1}{n}\Big{)}\Big{(}\sigma^{2}+\mu^{2}\Big{)}-\Big{(}\frac{2n-2}{n}-\frac{n-1}{n}\Big{)}\mu^{2}\right)$ $\displaystyle=\sum_{i=1}^{n}\beta_{i}\left(\frac{n-1}{n}\Big{(}\sigma^{2}+\mu^{2}\Big{)}-\frac{n-1}{n}\mu^{2}\right)$ $\displaystyle=\frac{n-1}{n}\sum_{i=1}^{n}\beta_{i}\sigma^{2}.$ $$

On the other hand, we have

$$ $\displaystyle{\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H_{\sigma}(\mathbf{X})\right]$ $\displaystyle=\frac{n-1}{n}\sum_{i=1}^{n}\beta_{i}{\mathbb{E}}\left[H_{\sigma}(\mathbf{X})\right]$ $\displaystyle(\ast)$ $\displaystyle=\frac{n-1}{n}\sum_{i=1}^{n}\beta_{i}\left(\frac{n-1}{n}\sum_{j=1}^{n}\beta_{j}\sigma^{2}\right),$ $$

where $(\ast)$ is obtained by repeating the ${\mathbb{E}}_{\mathbf{X}}\left[H_{\sigma}(\mathbf{X})\right]$ derivation for ${\mathbb{E}}_{\mathbf{X},\mathbf{Y}}\left[H_{\sigma}(\mathbf{Y})\right]$. Setting these two equal to one another, we see that a necessary condition for $H[\mathbf{X}]_{\sigma}$ is that

$$ $\frac{n-1}{n}\sum_{i=1}^{n}\beta_{i}=1.$ $$

Using similar arguments that we used with $H_{\mu}$, we see that $\beta_{i}=\beta_{j}$ for all $i,j\in\{1,\dots,n\}$. Therefore, we have that $\beta_{i}=\frac{1}{n-1}$ for all $i$, leading us to the unbiased MLE solution for the variance:

$$ $H(\mathbf{X})=\frac{1}{n-1}\sum_{i=1}^{n}\left(\mathbf{x}_{i}-\frac{1}{n}\sum_{j=1}^{n}\mathbf{x}_{j}\right)^{2}.$ $$

If, instead, the mean $\mu$ is known and we do not have to solve for $H_{\mu}$, then $\beta_{j}=\frac{1}{n}$ and the proof is similar to the one in [C.2](#A3.SS2).

Figure: Figure 5: Generation index versus MLE and PLE of standard deviation for standard normal, $U[0,1]$. The error bars display the standard error for $100$ different initializations. These results use the analytic form of the Gaussian standard deviation derived in Section [C.3](#A3.SS3).

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 5
presentation: 1
questions: 1. How sensitive is the method to the choice of $\lambda=0.1$? Was this value chosen empirically or is there theoretical justification?
2. In Figure 2, why does FID still increase with PLE, albeit more slowly? Is this related to the hyperparameter choice (e.g. number of data points in equation 6)?
3. In Figure 4, some distributions show stable performance while others show increasing MADness---why?
4. How does the hypernetwork architecture choice and hyperparameters like $\lambda$ affect performance? I think we need a systematic study on this?

In general I really like the paper and I'd like to increase my rating if the presentation is improved and my question on hyperparameter is answered.
soundness: 3
strengths: - Novel theoretical contribution connecting statistical bias in MLE to fairness and MADness issues in generative models
- Comprehensive empirical validation across multiple types of distributions and models (VAE, BigGAN)
- Strong technical foundation with clear connections to existing statistical theory
- Results show meaningful improvements in both fairness metrics and stability against MADness
summary: The paper proposes penalized autophogy estimation (PLE), which is a method that impproves fairness and reduce MADness of generative models.
The key idea is to make the model recursively stable via a regularization term on the MLE loss.
A naive formulation of this loss is intractable and the author(s) solve this issue by a hypernetwork that generates the parameters of the generative model
Experiments show that the proposed method can improve fairness and reduce MADness of generative models on various datasets.
weaknesses: - The motivation and problem setup in the introduction is not well structured, making it difficult to grasp the core contribution initially
- Limited ablation studies on the choice of hyperparameters (e.g., PLE penalty $\lambda$=0.1)
- Some experimental results show inconsistent trends across different distributions without sufficient explanation
- The presentation could be more accessible to readers less familiar with statistical estimation theory

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 3
questions: Please see the weaknesses above. In addition, the paper needs proofreading to fix minor typos. Here is a non-exhaustive list of them:
- Line 060: Autophogy Estimation => Autophagy
- Line 078: [...] some events events[...] => some events
- Line 088: [...] and is is described => and is described
- Line 322: [...] S ompares the representation => S compares the representation
- Line 414: For hyeprnetwork training [...] => For hypernetwork
soundness: 2
strengths: 1. The paper is well-written and addresses important problems in generative models. 
2. The proposed method is intuitive and easy to understand. 
3. The experiments are well conducted and demonstrate the effectiveness of the method.
summary: The paper sets out to address the important problem of representation bias in generative models and MADness (decreased models’ performance when trained using their outputs). The authors propose a method that enforces the parameters of the generative model to remain consistent when trained on the original or synthetic data. To alleviate the challenges of intractable search over generative models’ parameters trained on different synthetic data, the proposed method uses hypernetworks to sample generative model weights. Experiments show the ability of the proposed method to mitigate unfairness and MADness.
weaknesses: The paper lacks discussion and comparisons with existing bias mitigation methods in generative models. The authors could consider the following methods and explain how their method differs conceptually from these existing approaches and provide empirical comparisons that can support the benefit of the proposed method in the considered setup:

- Xu, Depeng, et al. "Fairgan: Fairness-aware generative adversarial networks." 2018 IEEE international conference on big data (big data). IEEE, 2018.
- Choi, Kristy, et al. "Fair generative modeling via weak supervision." ICML 2020.
- Sabbagh, Kamil, et al. "RepFair-GAN: Mitigating Representation Bias in GANs Using Gradient Clipping." Tiny Papers @ ICLR (2023).
- Rajabi, Amirarsalan, and Ozlem Ozmen Garibay. "Tabfairgan: Fair tabular data generation with generative adversarial networks." Machine Learning and Knowledge Extraction 4.2 (2022): 488-501

The experiments are not consistent enough; specifically, the datasets used in the paper are considered for different types of experiments. For example, fairness is evaluated only on MNIST using VAE, and MADness mitigation is evaluated on CIFAR 10 using GANs. The authors should provide justifications for the experimental design or consider evaluating different datasets under the same set of experiments, i.e., show that the method also mitigates MADness/fairness on MNIST with GANs, and similarly for CIFAR 10. This would demonstrate the generalizability of the proposed method under the metric being evaluated (e.g., fairness or MADness).   

Another important concern is the authors did not provide sufficient justification or insights into how the method can mitigate bias in generated data, particularly when a generative model trained on the original data is already biased, which can be exacerbated in the follow-up generation. More specifically, it is unclear how constraining the MLE to find the model parameters that remain consistent with synthetic and original data (Eq. 4) mitigates unfairness. The authors could provide a more detailed explanation or intuition for how their method addresses bias, particularly in cases where the original training data is biased.

The authors did not provide ablation on the impact of the parameter $\lambda$ in Equation 5. As the regularization term proposed is the core contribution of the paper, providing these experiments would provide more insights into how the regularization term influences several aspects of the work: the quality of the generated data, the number of generations after which “MADness” occurs, and the fairness of the synthetic data. These experiments could be provided for synthetic distributions used in Figure 4.  In addition, the authors can discuss the range of $\lambda$ values they think would be most interesting to explore and why.  

The Appendix contains unnecessary materials (e.g., B, C, D, E). While this material is interesting and well-written, it can confuse the reader since it is not directly linked to the paper's main contribution. Instead, the authors should consider referring the readers to books/papers that contain this background information.  

For the experiments on the MNIST dataset, Line 334 reads: _The majority class was the digit 3, and the minority class was the digit 6 (this choice was arbitrary)_. Instead of arbitrarily choosing the minority class, the authors should consider the class with the highest false negative rate when classified as the minority. This means the minority class confuses the most with other classes and can be harder to learn when it is underrepresented, for example, see how reference [1] chooses the class to artificially under-represent. 

The paper highly depends on hypernetworks, which increase the complexity of the method and its practical usage. 

[1]Bagdasaryan, Eugene, Omid Poursaeed, and Vitaly Shmatikov. "Differential privacy has disparate impact on model accuracy." NeurIPS 2019

---

confidence: 2
contribution: 2
initial_rating: 5
presentation: 2
questions: 1. In Section 1.1, why does it say that GAN is trained with MLE?

2. In Line 087-088, should $ R_I = C_{Maj} / C_{Min} $ actually be $ R_I = |C_{Maj}| / |C_{Min}| $?

3. In Equation 2, why is $S(M)$ linear? Could the authors provide a detailed explanation for this?

4. In Line 316, how should we interpret $ R_{Fair} < |C_{Maj}| / |C_{Min}| $? According to your conclusion about the linearity of $ S(M) $, shouldn’t the optimal $ R_{Fair} $ that fits the data be equal to $ |C_{Maj}| / |C_{Min}| $?

5. How are the hypernetworks $H_\phi$ trained?
soundness: 2
strengths: + The overall paper is well-structured and clearly written, with concise explanations that make concepts like hypernetworks and MADness accessible to a broader readers.

+ This paper tackles an important and timely issue by addressing fairness and bias in generative models, particularly focusing on challenges such as minority class penalization and model autophagy disorder (MADness).

+ The discussion related to large language models (LLMs) adds practical relevance to this research, as the content generated by LLMs is widely distributed across the internet.
summary: This paper addresses fairness and bias in generative models, which often penalize minority classes and suffer from model autophagy disorder (MADness). The authors propose training generative models with hypernetworks to make them more fair, stable, and less biased. They introduce a regularization term to reduce discrepancies between a model's performance on real data versus synthetic data, helping mitigate bias and improve representation fairness. The experimental results show that their framework is scalable, supporting integration with existing deep generative models such VAEs and GANs.
weaknesses: - The experiments are relatively weak, especially given the small dataset and the older models used (VAE and BigGAN). It would strengthen the paper significantly if the method were tested on diffusion models.
- There is extensive discussion of ChatGPT and other LLMs in Section 1.3. It would enhance the paper if the proposed method could be applied directly to LLMs.

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 3
questions: See the weaknesses section.

Also:
- The definition of fairness seems overly simple-- it would be helpful to understand how statistics deals with this issue and what the existing metrics are.  For instance, how do you adapt the current definition when there are multiple classes?
- How does the method compare to more standard statistical techniques (i.e. upweighting the loss from the minority classes)?
soundness: 2
strengths: The proposed method is certainly interesting, and if it truly works as claimed it would be an impressive and efficient way to train new networks.  The paper is clear and well-motivated.  It poses an original solution to mitigate MADness in generative models.  The results regarding fairness seem experimentally compelling.
summary: The paper contains two new innovations:
- A metric of fairness when there is a minority and majority class in the training data.
- A hyperparameter network and corresponding loss function intended to prevent autophagous collapse (MADness)

The authors demonstrate the efficacy of their method to mitigate MADness over multiple generate-train loops and improve fairness.  They do this in both a deep generative model and a statistical context.
weaknesses: - Concerns about the validity of the method: Using a 3-layer FC neural network with the objective stated in the paper will likely result in pathologies.  Suppose that $\theta*$ is the MLE.  It is likely that the FC network H is learning to set its weights close to 0 and its biases close to $\theta*$.  In this way, changing the data is unlikely to alter the parameters very much, making any additional training/retraining using $H$ redundant.  One experiment that you could do to test this (which would alleviate my concerns and requires no additional training) is as follows.  Let G be the BigGAN from figure 2, and let H be the corresponding hyper-network.  If H is really doing it's job, then when you fit $\theta = \sum_{i=1}^n H(x_i)/n for x_i$ only in the category of "airplanes", the resulting generative network $G(x|\theta)$ should produce only images of airplanes.  I suspect that instead one of two things will happen: either the network will barely change, or it will produce images that are not discernible as anything.

- Quality concerns: although the authors report an FID score, there are no generated images from CIFAR in the paper or the appendix.  There also is no clear explanation for why there is a sudden spike in MADness at iteration 4.  

- The evaluations are rather limited: The deep generative models are restricted to CIFAR-10.

- There is no theoretical explanation for why this method improves what the authors call fairness.  It is non-obvious to me why the method should improve fairness.

Meta-review: The paper aims to address the problem of representation bias in generative models, which is the models’ decreased performance when trained using their own synthetic outputs. The main idea is to use a regularization to make sure that model parameters are stable during training. The authors also proposed to use a hypernetwork to tackle the intractable regularization. The reviewers are concerned about the lack of both conceptual and empirical comparisons against existing methods in the literature of generative modeling that achieve the same goal. Furthermore, the evaluation could be significantly improved both qualitatively (by showing the models' generated images) as well as quantitatively (by comparing against other baseline methods). The authors responded to the reviewers' questions but they do not address the above concerns. 

The authors are encouraged to take the reviewers' comments to strengthen the work further.

Decision: Reject