CS 285: Lecture 18, Variational Inference, Part 1

Summary

In this lecture, the focus shifts from traditional reinforcement learning algorithms to variational inference and generative models. The discussion covers probabilistic latent variable models, their connection to reinforcement learning, and how variational inference can be used to approximate training for such models. Key concepts include amortized variational inference, function approximators like deep neural networks, and applications such as variational autoencoders. The lecture highlights how latent variable models are employed in model-based RL, human behavior modeling, and exploration, and concludes by outlining the challenges of training these models efficiently using expected log likelihood.

Takeaways

• 😀 Latent variable models are probabilistic models where some variables are hidden and need to be integrated out to compute the model's probability.
• 😀 Mixture models are a classic example of latent variable models, where the latent variable represents unobserved cluster identities in the data.
• 😀 Probabilistic models can be either unconditional (modeling p(x)) or conditional (modeling p(y|x)) distributions.
• 😀 In reinforcement learning, latent variable models are useful for tasks like model-based reinforcement learning, inverse reinforcement learning, and exploration.
• 😀 Variational inference is a method to approximate difficult-to-compute integrals and likelihoods in probabilistic models, making it easier to train latent variable models.
• 😀 The expected log likelihood is a more tractable objective compared to direct likelihood maximization, as it allows for sampling from the posterior distribution of latent variables.
• 😀 Amortized variational inference combines variational inference with function approximators, such as neural networks, to efficiently approximate the posterior distribution over latent variables.
• 😀 The goal of variational inference is to estimate the distribution of latent variables (p(z|x)) and use this to compute the expected log likelihood.
• 😀 Generative models are typically represented as latent variable models because they can represent complex distributions as products of simpler ones.
• 😀 Latent variable models are central in various fields, such as imitation learning, human behavior modeling, and exploration in reinforcement learning, due to their ability to handle hidden structures in data.

Q & A

What is the main topic of today's lecture?

The main topic of the lecture is variational inference and generative models, which are closely related to reinforcement learning concepts.

Why are variational inference and generative models important in reinforcement learning?

Variational inference and generative models are important in reinforcement learning because they help in model-based reinforcement learning, inverse reinforcement learning, and exploration. These concepts are used to approximate complex models and to better understand learning processes.

What is a probabilistic model?

A probabilistic model is a model that represents a probability distribution over some random variable(s), such as p(x) for a random variable x, or p(y|x) for a conditional model where y is dependent on x.

What is the key difference between latent variable models and other probabilistic models?

Latent variable models include variables that are not directly observed (latent variables), and these need to be integrated out to evaluate the probability of the observed data. This is in contrast to regular probabilistic models, where all variables are typically observed.

What is a classic example of a latent variable model?

A classic example of a latent variable model is a mixture model, where there is a latent categorical variable (such as the cluster identity) that helps model the distribution over the observed data (x).

How do mixture models work as latent variable models?

In mixture models, the observed data (x) is modeled as a sum over latent variables (z), where each z corresponds to a different component of the mixture. The likelihood of x is computed by summing over all possible values of z, weighted by the probability of each z.

What is the role of the latent variable z in the mixture density network?

In a mixture density network, the latent variable z represents the cluster identity or the underlying factors that generate the observed data y. The network outputs parameters such as means, covariances, and mixture probabilities, which together define the distribution over y given x.

Why is variational inference needed in latent variable models?

Variational inference is needed because directly computing the probability or likelihood of the data in latent variable models is intractable due to the integration over the latent variables. Variational inference provides a way to approximate these complex calculations efficiently.

What is the expected log likelihood objective in variational inference?

The expected log likelihood objective is used to estimate the log likelihood of the data by averaging over possible values of the latent variable z, weighted by its probability given the data. This helps avoid the direct calculation of complex integrals.

How can we calculate the posterior p(z|x) in latent variable models?

The posterior p(z|x) can be calculated using variational inference techniques, which approximate the true posterior distribution by using simpler distributions, such as a Gaussian or other parameterized distributions. Once the posterior is known, it can be sampled from to estimate the expected log likelihood and its gradient.