Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) are two popular generative models in deep learning that are capable of generating new data instances, such as images, that plausibly could have been drawn from the original data distribution. There are some key differences in how they work and what types of problems they are best suited for.

GANs are based on a game-theoretic framework where there are two competing neural networks – a generator and a discriminator. The generator produces synthetic data instances that are meant to fool the discriminator into thinking they are real (coming from the original training data distribution). The discriminator is trained to detect synthetic data from the generator versus real data. Through this adversarial game, the generator is incentivized to produce synthetic data that is indistinguishable from real data. The goal is for the generator to eventually learn the true data distribution well enough to fool even a discriminator that has also been optimized.

VAEs, on the other hand, are based on a probabilistic framework that leverages variational inference. VAEs consist of an encoder network that learns an underlying latent representation of the data, and a decoder network that learns to reconstruct the original data from this latent representation. To ensure the latent space accurately captures the underlying structure of the data, a regularization term is added based on latent space density estimation. This forces the latent representation to follow a prior conditional Gaussian distribution (typically standard normal). During training, VAEs optimize both the reconstruction loss as well as the KL divergence loss between the posterior and the prior on the latent space.

Some key differences between GANs and VAEs include:

Model architecture: GANs consist of separate generator and discriminator networks that compete against each other in a two-player mini-max game. VAEs consist of an encoder-decoder model trained using variational inference to maximize a variational lower bound.

Training objectives: GAN generators are trained to minimize log(1 – D(G(z))) to fool the discriminator, while discriminators minimize log(D(x)) + log(1 – D(G(z))) to detect real vs. fake. VAEs are trained to maximize the evidence lower bound (ELBO) which consists of reconstruction loss – KL divergence loss.

Latent space: GANs do not explicitly learn a latent space and conditioning must be done by manipulating latent vectors directly. VAEs learn an explicitly conditioned latent space through the encoder that can be sampled from or interpolated in.

Mode dropping: Due to only playing an adversarial game, GANs more easily suffer from mode dropping where certain modes in the data are not captured by the generator. VAEs directly regularize the latent space to mitigate this.

Stability: GAN training is notoriously unstable and difficult, often not converging or convergence to degenerate solutions. VAE training is much more stable via standard backpropagation and regularization.

Evaluation: It is difficult to formally evaluate GANs since their goal is to match the data distribution rather than just minimize a cost function. VAEs can be directly evaluated via reconstruction error and their latent space density.

Applications: GANs tend to produce higher resolution, sharper images but struggle with complex, multimodal data. VAEs work better on more structured data like text where their probabilistic framework is advantageous.

To summarize some key differences:

GANs rely on an adversarial game between generator and discriminator while VAEs employ variational autoencoding.
GANs do not explicitly learn a latent space while VAEs do.
VAE training directly optimizes a regularized objective function while GAN training is notoriously unstable.
GANs can generate higher resolution images but struggle more with multimodal data; VAEs work better on structured data.

Overall, GANs and VAEs both allow modeling generative processes and generating new synthetic data instances, but have different underlying frameworks, objectives, strengths, and weaknesses. The choice between them depends heavily on the characteristics of the data and objectives of the task at hand. GANs often work best for high-resolution image synthesis while VAEs excel at structured data modeling due to their stronger inductive biases. A combination of the two approaches may also be beneficial in some cases.