How does the chapter propose to modify empirical risk minimization (ERM) to incorporate SME knowledge?

The chapter proposes encoding SME knowledge as a vector v(SME) in the same feature space as the data or its embedded form, then modifying ERM so the model is trained to respect that knowledge. This is operationalized through a directional derivative D(SME)_k = ∇f_k(g(x)) · v(SME), which provides a gradient that optimization algorithms can use during training.

The chapter describes how traditional ERM relies on large, diverse labeled datasets, which are often lacking in high-consequence domains. To incorporate SME knowledge, it assumes that SME knowledge can be codified into a computable vector v(SME) sharing the feature space of X or its embedded representation g(X). The proposed modification to ERM is to train the model to respect this knowledge vector. The chapter defines D(SME)_k as the limit of the change in the k-th class logit f_k(g(x)) when the input embedding is perturbed in the direction v(SME), equivalently ∇f_k(g(x)) · v(SME). This gradient term can be used as the basis for optimization, meaning the modified ERM objective should guide the model so that its decisions are aligned with the SME knowledge embedded in v(SME). The chapter also stresses that evaluating whether SME knowledge has been learned requires metrics beyond simple ERM or accuracy, including assessments of semantic reasoning and generalization across multiple domain-specific target tasks.