Threshold — VC = 1

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

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

h_t(x) = \begin{cases} + & x \ge t \\ - & x < t \end{cases}

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:

labeling	condition on `t`	realizable?
`(−,−)`	`t > p₂`	✓
`(−,+)`	`p₁ < t ≤ p₂`	✓
`(+,+)`	`t ≤ p₁`	✓
`(+,−)`	`t ≤ p₁` and `t > p₂`, i.e. `p₂ < t ≤ p₁` — empty since `p₁ < p₂`	✗

(+,−) 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.