Understanding Loss Functions in Machine Learning

Kullback–Leibler (KL) Divergence: Information-Theoretic Loss

The Kullback–Leibler (KL) divergence is a fundamental concept in information theory and machine learning, measuring how one probability distribution diverges from a second, expected distribution. Mathematically, it is defined as:

D_{KL}(P \| Q) = \sum_{i} P(i) \log \frac{P(i)}{Q(i)}

Here, $P$ and $Q$ are discrete probability distributions over the same set of events or outcomes, and the sum runs over all possible events $i$ . In this formula, $P(i)$ represents the true probability of event $i$ , while $Q(i)$ represents the probability assigned to event $i$ by the model or approximation.

Note

KL divergence quantifies the inefficiency of assuming $Q$ when the true distribution is $P$ . It can be interpreted as the extra number of bits needed to encode samples from $P$ using a code optimized for $Q$ instead of the optimal code for $P$ .

KL divergence has several important properties. First, it is asymmetric:

D_{KL}(P \| Q) \neq D_{KL}(Q \| P)

This means the divergence from $P$ to $Q$ is not the same as from $Q$ to $P$ , reflecting that the "cost" of assuming $Q$ when $P$ is true is not the same as the reverse.

Second, KL divergence is non-negative:

D_{KL}(P \| Q) \geq 0

for all valid probability distributions $P$ and $Q$ , with equality if and only if $P = Q$ everywhere.

In machine learning, KL divergence is widely used as a loss function, especially in scenarios involving probability distributions. It plays a central role in variational inference, where it measures how close an approximate distribution is to the true posterior. Additionally, KL divergence often appears as a regularization term in models that seek to prevent overfitting by encouraging distributions predicted by the model to remain close to a prior or reference distribution.

Which of the following statements about KL divergence are true?

Select the correct answer

KL divergence is always non-negative and equals zero only when $P = Q$ .

KL divergence measures the inefficiency of assuming $Q$ when the true distribution is $P$ .

KL divergence is always symmetric: $D_{KL}(P \| Q) = D_{KL}(Q \| P)$ .

KL divergence can take negative values if $Q(i) > P(i)$ for some $i$ .

KL divergence is commonly used in variational inference as a loss function.

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Abschnitt 4. Kapitel 1

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