Asymptotic learning curve and renormalizable condition in statistical learning theory

Asymptotic Learning Curve and Renormalizable Condition in Statistical Learning Theory

Bayes statistics and statistical physics have the common mathematical structure, where the log likelihood function corresponds to the random Hamiltonian. Recently, it was discovered that the asymptotic learning curves in Bayes estimation are subject to a universal law, even if the log likelihood function can not be approximated by any quadratic form. However, it is left unknown what mathematica...

Computational and Statistical Learning Theory

Statistical Decision and Learning Theory

This paper reviews and contrasts the basic elements of statistical decision theory [1–4] and statistical learning theory [5–7]. It is not intended to be a comprehensive treatment of either subject, but rather just enough to draw comparisons between the two. Throughout this paper, let X denote the input to a decision-making process and Y denote the correct response or output (e.g., the value of ...

Computational and Statistical Learning Theory

{0, 1}-valued random variables X1, . . . , Xn are drawn independently each from Bernoulli distribution with parameter p = 0.1. Define Pn := P( 1 n ∑n i=1Xi ≤ 0.2). (a) For n = 1 to 30 calculate and plot the below in the same plot (see [1, section 6.1] for definition of Hoeffding and Bernstein inequalities): i. Exact value of Pn (binomial distribution). ii. Normal approximation for Pn. iii. Hoef...

Computational and Statistical Learning Theory

Specifically for p =∞ ( 1 m ∑m i=1 |f(xi)− vi| )1/p is replaced by maxi∈[m] |f(xi)− vi|. Also define Np(F , α, S) := min{|V | : V is an α-cover of (in `p) ofF on sample S} and Np(F , α, n) := sup x1,...,xm Np(F , α, {x1, . . . , xm}) Definition 2. A function F is said to α-shatter a sample S = {x1, . . . , xm} if there exists a sequence of thresholds, θ1, . . . , θm ∈ R such that ∀ξ ∈ {±1}, ∃f ...