Threshold — VC = 1

Parent: examples Prereq: learning-theory — H, h, shattering; ℝ.

Class $H = \{\, h_t : t \in \mathbb{R} \,\}$, one parameter:

Let's count. (Recall $h_t(p) = {+}$ iff $t \le p$.)

Shatter 1 point p? label +: take any t ≤ p; label −: take any t > p. Both realizable → shatters 1 → VC ≥ 1.

Shatter 2 points p₁ < p₂? write the labeling as (h(p₁), h(p₂)) and solve for t in each of the 2²=4 cases:

(+,−) has no solution → no 2 points can be shattered → VC < 2. So VC = 1.

Breaks on any non-monotone pattern, e.g. − + − ("positive only in the middle"): that rule is not in H at all → underfit. Climb one step → interval.