Kullback–Leibler divergence

As explained in Build a Reasoning Model (From Scratch)

A measure of how different two probability distributions are, used here for soft distillation.

Kullback–Leibler divergence measures how different two probability distributions are. Soft distillation minimizes it to make a student model match the teacher’s full token distribution across the vocabulary. section 8.1

In the distillation workflow, KL divergence marks the difference between two supervision strategies:

• Hard distillation matches teacher-generated tokens and needs only generated text. section 8.1
• Soft distillation matches the teacher’s complete output distribution, including alternative tokens the teacher considers plausible, and requires logits or log probabilities. section 8.1

Teacher and student distributions

A large teacher produces a probability distribution for each next token. The smaller student is trained to reduce the KL divergence between its distribution and the teacher’s distribution. section 8.1

Figure 8.2 Hard distillation trains the student on teacher-generated tokens; soft distillation trains the student on the teacher’s full output distribution.

Compare hard distillation’s token targets with soft distillation’s full probability distributions. section 8.1

How Build a Reasoning Model (From Scratch) covers it

How deep it goes

Defines the metric and contrasts its required probability information with hard distillation, without deriving its formula.

Examples

Uses teacher-generated tokens, full vocabulary distributions, and alternative token probabilities as contrasting supervision examples.

What it assumes

Assumes familiarity with probability distributions and distinguishes teacher-generated text from teacher logits or log probabilities.

What you can do after

Apply it via soft distillation by training a student to match teacher token distributions when logits or log probabilities are available.

How it connects

Connects probability distributions with soft distillation, hard distillation, teacher models, student models, logits, and log probabilities.

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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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  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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  Bayesian Optimization in ActionKullback–Leibler divergence is a nonnegative statistical divergence that quantifies how one probability distribution differs from another.
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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.

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