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Extra resources for Pac-Bayesian supervised classification: The thermodynamics of statistical learning
Moreover we have also used P = P(P ) = 1 N N Pi , i=1 where it should be remembered that the joint distribution of the process (Xi , Yi )N i=1 N is P = P . We have considered ψ(θ, θ) as a function deﬁned on X × Y as i i=1 ψ(θ, θ)(x, y) = 1 y = fθ (x) − 1 y = fθ (x) , (x, y) ∈ X × Y so that it should be understood that P (ψ) = 1 N N P ψi (θ, θ) i=1 = 1 N N P 1 Yi = fθ (Xi ) − 1 Yi = fθ (Xi ) = R (θ, θ). i=1 In the same way P log(1 − λψ) = 1 N N log 1 − λψi (θ, θ) . i=1 Moreover integration with respect to ρ bears on the index θ, so that ρ log 1 − λP (ψ) log 1 − = θ∈Θ ρ P log(1 − λψ) = θ∈Θ 1 N λ N N P ψi (θ, θ) ρ(dθ), log 1 − λψi (θ, θ) ρ(dθ).
Another reassuring remark is that the empirical margin functions ϕ and ϕ behave well in the case when inf Θ r = 0. Indeed in this case m (θ, θ) = r (θ, θ) = r(θ), θ ∈ Θ, and thus ϕ(1) = ϕ(1) = 0, and ϕ(x) ≤ −(x − 1) inf Θ1 r, x ≥ 1. This shows that in this case we recover the same accuracy as with non-relative local empirical bounds. 11 does not collapse in presence of massive over-ﬁtting in the larger model, causing r(θ) = 0, which is another hint that this may be an accurate bound in many situations.
1/2 π[(1−α)N r ] π[(1+γ)−N r ] and ρ = π(1−α)N r , we get as desired a bound that is adaptively local in all the Θm (at least when M is countable and μ is atomic): 2 log μ exp B(ν, ρ) ≤ − N (α−γ) N 2 log (1 + γ)(1 − α) r − log(1−α) log(1+γ) − log ≤ inf m∈M − log (1−α)(1+γ) α−γ de 2 − 2 log( ) N (α − γ) r (m) − log(1−α) log(1+γ) + log de (m) N (α−γ) −2 log μ(m) N (α−γ) . The penalization by the empirical dimension de (m) in each sub-model is as desired linear in de (m). Non random partially local bounds could be obtained in a way that is easy to imagine.