Question

3. How does the complexity of a learnable class depend on the parameters ε and δ? Classify the following as true or false and justify. a) mH increases (in the non-strict sense) as ε decreases with δ fixed; b) rnH increases (in the non-strict sense) as δ increases with ε fixed; c) It seems reasonable that, in a typical situation, lim m H (ε, δ)-+00

Answer 1

(a) False. For ε ≥ 0 we say that function g ε-approximates function f with respect to distribution D if Pr_D[f (x) = g(x)] 1 − ε. We say that an algorithm A efficiently learns concept class C if for every ε > 0, δ > 0, n, c ∈ C

(b) True. For ε ≥ 0 we say that function g ε-approximates function f with respect to distribution D if Pr_D[f (x) = g(x)] 1 − ε. We say that an algorithm A efficiently learns concept class C if for every ε > 0, δ > 0, n, c ∈ C

(c)True. distribution Dn over Xn = {0, 1}^n, A(n, ε, δ), runs in time polynomial in n, 1/δ, 1/ε, |c| and outputs, with probability at least 1 − δ, an efficiently computable hypothesis h from some class of functions H that ε-approximates c.