 Solution

ID: past-exam-of-the-computer-science-course-of-the-university-of-oxford/2015/computational-learning-theory/1/a/iv/solution

d

satisfies

Π_{H} (m) \leq j = 0 \sum m i n (d, m) (j m) .

In particular, for

m \geq d \geq 1

Π_{H} (m) \leq (\frac{e m}{d})^{d} .

Here the binomial coefficients count small subsets. One way to obtain the last inequality is to set

t = d / m \leq 1

: since

t^{j} \geq t^{d}

for

j \leq d

t^{d} \sum_{j = 0}^{d} (j m) \leq \sum_{j = 0}^{d} (j m) t^{j} \leq (1 + t)^{m} \leq e^{m t} = e^{d} .

Thus finite VC dimension replaces the unrestricted exponential number of labellings by a polynomial bound in the sample size.

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