Generative Models - Introduction


CSE 891: Deep Learning

Vishnu Boddeti

Generative Models

  • Unsupervised Learning: only use the inputs $\mathbf{x}^{(t)}$ for learning.
    • automatically extract meaningful features for your data
    • leverage the availability of unlabeled data
    • add a data-dependent regularizer to trainings
  • Many neural network based unsupervised learning approaches exist
    • Autoregressive Models
    • Autoencoders (Variational Autoencoders)
    • Generative Adversarial Networks
    • Normalizing Flows
    • Diffusion Models

Why Generative Models?

  • Move beyond associating inputs to outputs
  • Recognize objects in the world and their factors of variation
  • Understand and imagine how the world evolves
  • Detect surprising events in the world
  • Imagine and generate rich plans for the future
  • Establish concepts as useful for reasoning and decision making

Desiderata

  • We want to estimate distributions of complex, high-dimensional data
    • A $128\times 128\times 3$ image lies in a ~50,000-dimensional space
  • We also want computational and statistical efficiency
    • Efficient training and model representation
    • Expressiveness and generalization
    • Sampling quality and speed
    • Compression rate and speed

Simplest Generative Model

Learning Through Histograms

  • Our goal: estimate $p_{data}$ from samples $x^{(1)}, \dots, x^{(n)} \sim p_{data}(x)$
  • Suppose the samples take on values in a finite set $\{1, \dots, k\}$
  • The model: a histogram
    • (Redundantly) described by k nonnegative numbers: $p_1, \dots, p_k$
    • To train this model: count frequencies $$\begin{equation} p_i = \frac{\mbox{# times i appears in the dataset}}{\mbox{# points in the dataset}} \end{equation}$$

Inference and Sampling

  • Inference (querying $p_i$ for arbitrary $i$): simply a lookup into the array $p_1, \dots, p_k$
  • Sampling (lookup into the inverse cumulative distribution function)
    • From the model probabilities $p_1, \dots, p_k$, compute the cumulative distribution
    • Draw a uniform random number $u \sim [0, 1]$
    • Return the smallest $i$ such that $u \leq F_i$
  • Are we done?

Failure in High Dimensions

  • No, because of the curse of dimensionality. Counting fails when there are too many bins.
    • (Binary) MNIST: 28x28 images, each pixel in {0, 1}
    • There are $2^{784} \approx 10^{236}$ probabilities to estimate
    • Any reasonable training set covers only a tiny fraction of this
    • Each image influences only one parameter. No generalization whatsoever!

Poor Generalization

  • learned histogram = training data distribution
    • poor generalization

Parameterized Distributions

Status

  • Issues with Histograms
    • High dimensions: won't work
    • Even 1-d: if many values in the domain, prone to overfitting
  • Solution: function approximation. Instead of storing each probability, store a parameterized function $p_{\mathbf{\theta}}(\mathbf{x})$

Applications

Applications: Data Generation

Applications: Data Generation

Applications: Data Imputation, Denoising, In-painting

Applications: Communication and Compression

Learning Generative Models

Likelihood-based generative models

  • Our goal: estimate $p_{data}$ from samples $x^{(1)}, \dots, x^{(n)} \sim p_{data}(x)$
  • Now we introduce function approximation: learn $\mathbf{\theta}$ so that $p_{\mathbf{\theta}}(x) \approx p_{data}(x)$
    • How do we design function approximators to effectively represent complex joint distributions over $\mathbf{x}$, yet remain easy to train?
    • There will be many choices for model design, each with different trade-offs and different compatibility criteria.
  • Designing the model and the training procedure go hand-in-hand.

Fitting Distributions

  • Given data $x^{(1)}, \dots, x^{(n)}$ sampled from a "true" distribution $p_{data}$
  • Set up a model class: a set of parameterized distributions $p_{\mathbf{\theta}}$
  • Pose a search problem over parameters $$\begin{equation} \underset{\mathbf{\theta}}{\operatorname{arg min}} loss(\mathbf{\theta},\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(n)}) \end{equation}$$
  • Want the loss function + search procedure to:
    • Work with large datasets ($n$ is large, say millions of training examples)
    • Yield $\mathbf{\theta}$ such that $p_{\mathbf{\theta}}$ matches $p_{data}$ - i.e. the training algorithm works. Think of the loss as a distance between distributions.
    • Note that the training procedure can only see the empirical data distribution, not the true data distribution: we want the model to generalize.

Maximum Likelihood

  • Maximum likelihood: given a dataset $x^{(1)}, \dots, x^{(n)}$, find $\mathbf{\theta}$ by solving the optimization problem
  • $$\begin{equation} \underset{\mathbf{\theta}}{\operatorname{arg min}} loss(\mathbf{\theta},\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(n)}) = \frac{1}{n}\sum_{i=1}^n -\log p_{\mathbf{\theta}}(\mathbf{x}^{(i)}) \end{equation}$$
  • Statistics tells us that if the model family is expressive enough and if enough data is given, then solving the maximum likelihood problem will yield parameters that generate the data
  • Equivalent to minimizing KL divergence between the empirical data distribution and the model $$\begin{eqnarray} \hat{p}_{data}(\mathbf{x}) &=& \frac{1}{n}\sum_{i=1}^n\mathbf{1}[\mathbf{x}==\mathbf{x}^{(i)}] \\ KL(\hat{p}_{data}||p_{\mathbf{\theta}}) &=& \mathbb{E}_{\mathbf{x}\sim\hat{p}_{data}}[-\log p_{\mathbf{\theta}}(\mathbf{x})] - H(\hat{p}_{data}) \end{eqnarray}$$

Stochastic Gradient Descent

  • Maximum likelihood is an optimization problem. How do we solve it?
  • Stochastic gradient descent (SGD).
    • SGD minimizes expectations: for a differentiable function of $\theta$, it solves $$\begin{equation} \underset{\mathbf{\theta}}{\operatorname{arg min}} \mathbb{E}[f(\mathbf{\theta})] \end{equation}$$
    • With maximum likelihood, the optimization problem is $$\begin{equation} \underset{\mathbf{\theta}}{\operatorname{arg min}} \mathbb{E}_{\mathbf{x}\sim\hat{p}_{data}}[-\log p_{\mathbf{\theta}}(\mathbf{x})] \end{equation}$$
    • Why maximum likelihood + SGD? It works with large datasets and is compatible with neural networks.

Designing The Model

  • Key requirement for maximum likelihood + SGD: efficiently compute $\log p(x)$ and its gradient
  • We will choose models $p_{\mathbf{\theta}}$ to be deep neural networks, which work in the regime of high expressiveness and efficient computation (assuming specialized hardware)
  • How exactly do we design these networks?
    • Any setting of $\mathbf{\theta}$ must define a valid probability distribution over $\mathbf{x}$: $$\begin{equation} \mbox{for all } \mathbf{\theta}, \mbox{ } \sum_{\mathbf{x}}p_{\mathbf{\theta}}(\mathbf{x}) = 1 \mbox{ and } p_{\mathbf{\theta}}(\mathbf{x})\geq 0 \mbox{ for all } \mathbf{x} \end{equation}$$
    • $\log p_{\mathbf{\theta}}(\mathbf{x})$ should be easy to evaluate and differentiate with respect to $\mathbf{\theta}$
    • This can be tricky to set up!

Simple Generative Models

Example: Autoencoder

  • Autoencoders consists of:
    • encoder
    • decoder
    • Encoder and decoder are trained to reproduce its input at the output layer.

Loss Function

  • Binary Inputs
    • cross-entropy \begin{equation} L = -\sum_{k} x_k\log(\hat{x}_k) + (1-x_k)\log(1-\hat{x}_k) \nonumber \end{equation}
  • Real-Valued Inputs
    • sum-of-squared differences
    • we use a linear activation function at the output \begin{equation} L = \frac{1}{2}\sum_{k}(x_k-\hat{x}_k)^2 \nonumber \end{equation}
  • Gradient: $\nabla_{\mathbf{\hat{a}}(\mathbf{x}^{t})}L = \mathbf{\hat{x}}^{t} - \mathbf{x}^{t}$
  • when weights are tied, $\nabla_{\mathbf{W}}L$ is the sum of two gradients.

Under-complete Hidden Layers

  • Hidden layer is under-complete if smaller than the input layer
    • hidden layer "compresses" the input
    • will compress well only for the training distribution
  • Hidden units will be
    • good features for the training distribution
    • but bad for other types of input
  • Training Distribution:
  • Other Distribution:

Over-complete Hidden Layer

  • Hidden layer is over-complete if greater than the input layer
    • no compression in hidden layer
    • each hidden unit could copy a different input component
  • No guarantee that the hidden units will extract meaningful structure.

Denoising Autoencoder

  • Representation should be robust to introduction of noise.
    • randomly assign subset of inputs to 0, with probability $\nu$
    • Gaussian additive noise
  • reconstruction $\hat{\mathbf{x}}$ computed from the corrupted input $\tilde{\mathbf{x}}$.
  • Loss function compares $\hat{\mathbf{x}}$ reconstruction with the noiseless input $\mathbf{x}$.

Linear Autoencoder

  • Given a linear decoder, what is the best encoder for MSE loss?
  • Theorem:
    • let $\mathbf{A}$ be any matrix, with SVD $\mathbf{A}=\mathbf{U\Sigma V}^T$
    • Decomposition of rank $K$: $\mathbf{U}_{.,\leq K}\Sigma_{\leq K, \leq K}\mathbf{V}^T_{.,\leq K}$
    • then, the matrix $\mathbf{B}$ of rank $K$ that is closest to $\mathbf{A}$: \[\mathbf{B}^{*} = \underset{\mathbf{B} \mbox{ s.t. rank}(\mathbf{B})=K}{\operatorname{arg min}} \|A-B\|_{F}\] is $\mathbf{B}^{*}=\mathbf{U}_{.,\leq K}\Sigma_{\leq K, \leq K}\mathbf{V}^T_{.,\leq K}$
  • If inputs are normalized as follows: $\mathbf{x}^t = \frac{1}{\sqrt{T}}\left(\mathbf{x}^t-\frac{1}{T}\sum_{t'=1}^T\mathbf{x}^{t'}\right)$
    • encoder corresponds to Principal Component Analysis (PCA)
    • singular values and (left) vectors = the eigenvalues/eigenvectors of covariance matrix

What you will learn

  • $p_{\mathbf{\theta}}(\mathbf{x})$ Use deep networks as function approximators.
    • fully connected
    • convolutional networks
    • recurrent networks
  • Machinery
    • Design of probabilistic models
    • Memoryless and Amortized Inference
    • Stochastic Optimization
  • Underlying goal: Density Estimation

A Probabilistic Viewpoint

  • Goal: modeling $p_{data}$
Fully Observed Models
Transformation Models (likelihood free)