Research proof index

For positive parameters $A,B,C,R$, the classical error-bound function $A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ is nonincreasing once $x\ge \exp\!\left(R(2B/C)^2\right)$.

Under a realizable distribution P.map (x ↦ (x, c(x))), the general 0-1 error coincides with the binary hypothesisError P h c, where h and c are viewed as subsets of α via the characteristic function decide (· ∈ ·).

The deterministic PAC learner predicate is antitone in the concept class: a learner for a larger class C' is also a learner for any subclass C ⊆ C', since the agnostic benchmark optimalError _ C ≥ optimalError _ C' makes the error requirement easier.

A PAC learner with accuracy ε₁ is also a PAC learner with any weaker accuracy ε₂ ≥ ε₁: the bad event {error > opt + ε} only shrinks.

If every tile in $P$ lies at the fixed generation $k$, then the first global density of $P$ is at most the generation-specific density $\mathrm{dens}'(k,P)$.

Reconstructing a concrete coset FFT domain from its class instance gives back the original domain.