This document presents a formula for calculating the loss function J(θ) in machine learning models. The formula averages the negative log likelihood of the predicted probabilities being correct over all samples S, and includes a regularization term λ that penalizes predicted embeddings being dissimilar from actual embeddings. It also defines the cosine similarity term used in the regularization.

27. J(θ) = −
1
|S| ∑
s∈S
[(puc,a ⋅ log( ̂puc,a) + (1 − puc,a) ⋅ log(1 − ̂puc,a)
−λ ⋅ (puc,a ⋅ (1 − cos(euc, ea)) + (1 − puc,a) ⋅ cos(euc, ea))],
where cos(euc, ea) :=
cos(euc, ea) + 1
2
.

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