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The Gaussian activation normalizer is finite and positive

Statement

Assume AC. Let $Z$ have the standard normal law on $R$ . Suppose $σ : R \to R$ is analytic, nonpolynomial, satisfies $∣ σ (0) ∣ \leq B$ for some finite $B$ , and is $L$ -Lipschitz for a finite $L$ . Then $0 < E [σ (Z)^{2}] < \infty$ . Consequently the normalization constant

$c_{σ} : = (E [σ (Z)^{2}])^{- 1}$

is a well-defined positive finite real number.

Statement references

Du et al., Condition 3.1 and equation (2)

Feller, Gaussian distribution and finite second moment

Facts & Assumptions

Given: AC, the activation $σ$ , its constants $B, L$ , and a standard normal random variable $Z$ .

[A1]

AC is assumed by the standard-normal-law and Gaussian-moment suppliers. The Axiom of Choice

[F1]

The standard normal law has second moment $1$ . Standard normal and normal laws Gaussian even moments for Brownian increments

Proof

technique · bound the activation by an affine function of $|Z|$, then use continuity and nonpolynomiality for strict positivity

1.1A1F1givenalgebra

Lipschitz continuity gives $∣ σ (z) ∣ \leq B + L ∣ z ∣$ , hence $σ (z)^{2} \leq 2 B^{2} + 2 L^{2} z^{2}$ . By [F1], $E σ (Z)^{2} \leq 2 B^{2} + 2 L^{2} < \infty$ .

1.2A1F1givenchoose

A nonpolynomial analytic function is not identically zero, so choose $z_{0}$ with $σ (z_{0}) \neq 0$ . Continuity gives an open interval around $z_{0}$ on which $∣ σ ∣ \geq ∣ σ (z_{0}) ∣ / 2$ . The standard normal density is strictly positive on that interval, so its integral of $σ^{2}$ is positive.

2.1step 1.1step 1.2algebra∎

The expectation is both finite and strictly positive by steps 1.1 and 1.2, so its reciprocal exists and is positive and finite.

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Simon S. Du, Jason D. Lee, Haochuan Li, Liwei Wang, and Xiyu Zhai, Gradient Descent Finds Global Minima of Deep Neural Networks, Condition 3.1 and equation (2) (standard reference, not scraped)