Kullback–Leibler divergence

As explained in Optimization Algorithms: AI techniques for design, planning, and control problems

Kullback–Leibler divergence is a metric quantifying the dissimilarity between two probability distributions.

Kullback–Leibler divergence, or relative entropy, measures the dissimilarity between probability distributions P and Q. It is defined as KL(P || Q) = ∫ P(x) · log(P(x) / Q(x)) dx across the variable’s support. section 12.1.5

In PPO-penalty, KL divergence penalizes updates that move a policy too far from its previous version, supporting a manageable trust region and reducing large policy shifts. section 12.1.5

Comparing Gaussian policies

A comparison between two Gaussian distributions with different means and variances illustrates the divergence. KL divergence equals zero exactly when the two distributions are identical. section 12.1.5

KL divergence between two Gaussian distributions

Inspect the figure comparing KL divergence between two Gaussian distributions. section 12.1.5

How Optimization Algorithms: AI techniques for design, planning, and control problems covers it

How deep it goes

Defines the integral and zero condition, but stops short of deriving the measure or detailing numerical computation.

Examples

Uses Gaussian distributions with different means and variances as a visual comparison.

What it assumes

Assumes familiarity with probability distributions, probability density functions, and policy updates.

What you can do after

Evaluate how far an updated policy departs from an earlier policy within PPO-style optimization.

How it connects

Connects probability distributions, PPO, policy updates, trust regions, and TRPO.

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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.

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