Kullback–Leibler divergence

As explained in Bayesian Optimization in Action

Kullback–Leibler divergence is a nonnegative statistical divergence that quantifies how one probability distribution differs from another.

Kullback–Leibler (KL) divergence measures how different one probability distribution is from another. Like Euclidean distance compares points, KL divergence compares distributions. It is nonnegative and equals zero when the distributions match exactly. section 12.2.1

Figure 12.7 Euclidean distance measures the distance between two points on a plane, while the KL divergence measures the distance between two probability distributions.

Use the geometric analogy to contrast ordinary distance between points with divergence between probability distributions. section 12.2.1

In a variational Gaussian process, KL divergence measures the difference between the posterior conditioned on inducing points and the posterior conditioned on the full dataset. Minimizing that difference makes the approximate model resemble the full model. Direct optimization uses the evidence lower bound (ELBO) as a computable proxy for the true loss and marginal log likelihood. section 12.2.1 section 12.5

How Bayesian Optimization in Action covers it

How deep it goes

Defines nonnegativity and matching behavior, then applies the measure to GP posteriors while deferring detailed mathematical computation.

Examples

Uses Euclidean distance between points and posterior Gaussian processes trained on inducing points or complete datasets.

What it assumes

Assumes familiarity with probability distributions and Gaussian processes while defining KL divergence from its distance intuition.

What you can do after

Evaluate/understand how distributional similarity guides approximate Gaussian-process training through ELBO optimization.

How it connects

Connects variational Gaussian processes with inducing points, ELBO, marginal log likelihood, and mini-batch optimization.

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Also explained in

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  Reinforcement Learning from Human FeedbackForward KL divergence is KL divergence from a target distribution to an approximating distribution, with expectation taken under the target distribution.
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  Graph Neural Networks in ActionA measure of the difference between two probability distributions, quantifying the information cost of approximating one with another.
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  Math and Architectures of Deep LearningA loss term based on Kullback-Leibler divergence that penalizes differences between probability distributions, such as an encoder's latent distribution and a chosen prior.
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  Learn Generative AI with PyTorchA divergence measure used as a regularization term to encourage a latent-variable distribution to resemble a normal distribution.
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  Build a Reasoning Model (From Scratch)A measure of how different two probability distributions are, used here for soft distillation.
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  Machine Learning Algorithms in DepthKullback-Leibler divergence is a nonnegative measure of the discrepancy between two probability distributions.
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  Evolutionary Deep Learning: Genetic algorithms and neural networksA divergence-based measure comparing probability distributions through differences in their information content.
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  Causal AIKullback–Leibler divergence quantifies the difference between two probability distributions.
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  Data Without LabelsKullback-Leibler divergence is a directed measure of the difference between two probability distributions based on their relative information content.
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  Optimization Algorithms: AI techniques for design, planning, and control problemsKullback–Leibler divergence is a metric quantifying the dissimilarity between two probability distributions.
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  Deep Learning with R, Third EditionKullback–Leibler divergence is a measure of how one probability distribution differs from a reference probability distribution.

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Related concepts

• Group Relative Policy Optimizationpart of · Group Relative Policy Optimization is a reinforcement-learning algorithm that updates a policy using relative rewards or advantages estimated from groups of sampled outputs or actions.
• t-SNEused by · t-SNE is a nonlinear dimensionality-reduction technique that maps high-dimensional data to a lower-dimensional space while emphasizing local similarities or neighborhood relationships.
• variational inferenceused by · Variational inference is an approximate inference technique that optimizes a tractable family of probability distributions to approximate a target distribution, often a posterior.
• variational Gaussian processused by · A variational Gaussian process is a scalable approximation to a Gaussian process whose posterior is represented through variational inference over a finite set of inducing points.

This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.