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Bound error in approximating $E_x [H(f(x))]$ with random $(1/n) \sum_{i=1}^n \Phi(f(x_i)/h)$ where $H$ is Heaviside function and $\Phi$ is normal CDF
Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently smooth" function. For simplicity, we may consider $f$ to be an affine function, i.e $f(x) \equiv b-x^\top w$, for some $(w,b) \in \mathbb ...
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Does log-concave approximable distribution satisfy transportation-cost inequality?
Definition: Recall that a distribution $\mu$ on $\mathbb R^d$ is said to be log-convave with constant $c > 0$, if density $d\nu \propto e^{-V}dvol$ satisfying the curvature condition
$$
\...
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