In the context of Empirical Risk Minimization, how does the choice of a loss function affect the consistency of estimators within a given hypothesis space?

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Seekh · Accepted Answer

Consistency means that as more data are collected the estimator will converge to the best function in the hypothesis space. The loss function defines what “best” means by weighting different prediction errors, so a loss that matches the true data‑generation mechanism will guide the estimator toward the correct target. If the loss is mis‑specified or too insensitive, the empirical minimizer may converge to a wrong function even though the hypothesis space contains the true one. For example, using squared loss on a binary classification problem forces the model to predict probabilities, whereas the 0‑1 loss directly targets correct class labels; the former can be consistent with a probability‑modeling hypothesis space, while the latter may not be. Thus, choosing a loss that is aligned with the task and the hypothesis space is essential for achieving consistency.

In the context of Empirical Risk Minimization, how does the choice of a loss function affect the consistency of estimators within a given hypothesis space?

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