A396366 Decimal expansion of the lambda (scale) constant in the SELU activation function.

1, 0, 5, 0, 7, 0, 0, 9, 8, 7, 3, 5, 5, 4, 8, 0, 4, 9, 3, 4, 1, 9, 3, 3, 4, 9, 8, 5, 2, 9, 4, 6, 0, 6, 3, 5, 1, 9, 5, 8, 9, 8, 1, 2, 0, 4, 3, 7, 6, 4, 0, 4, 5, 4, 6, 9, 7, 7, 8, 7, 8, 3, 8, 7, 8, 0, 1, 9, 9, 4, 6, 2, 0, 3, 9, 7, 7, 0, 0, 2, 6, 3, 7, 2, 0, 8, 3, 0, 0, 3, 2, 9, 1, 6, 1, 4, 9, 9, 3, 6, 6, 5, 4, 1, 8

Offset

1,3

Comments

SELU(x) = scale*(max(0,x)+min(0,alpha*(exp(x)-1))).

SELU is an activation function derived from ELU (Exponential Linear Unit), which uses the identity function for x >= 0 and exp(x)-1 for x <= 0.

It multiplies the negative-region output of ELU by the constant alpha and additionally scales the entire function by the constant lambda so that the outputs of a neural network converge toward zero mean and unit variance.

Links

Table of n, a(n) for n=1..105.

Günter Klambauer, Thomas Unterthiner, Andreas Mayr, and Sepp Hochreiter, Self-Normalizing Neural Networks, arXiv:1706.02515 [cs.LG], 2017.

Formula

Equals sqrt(2/(1 + (sqrt(2/Pi)/(1 - exp(1/2)*erfc(1/sqrt(2))))^2*(1 - 2*exp(1/2)*erfc(1/sqrt(2)) + exp(2)*erfc(sqrt(2))))).

Example

1.0507009873554804934193349852946...

Mathematica

RealDigits[Sqrt[2/(1 + (Sqrt[2/Pi] / (1 - Exp[1/2]*Erfc[1/Sqrt[2]]))^2 * (1 - 2*Exp[1/2] * Erfc[1/Sqrt[2]] + Exp[2]*Erfc[Sqrt[2]]))], 10, 105][[1]] (* Amiram Eldar, May 24 2026 *)

Prog

(PARI) lambda = sqrt(2 / (1 + (sqrt(2/Pi) / (1 - exp(1/2)*erfc(1/sqrt(2.))))^2 * (1 - 2*exp(1/2)*erfc(1/sqrt(2.)) + exp(2)*erfc(sqrt(2.)))))

Crossrefs

Cf. A396365.

Sequence in context: A091685 A349298 A062824 * A292904 A271228 A281528

Adjacent sequences: A396362 A396364 A396365 * A396368 A396369 A396370

Keyword

nonn,cons

Author

Donghwi Park, May 24 2026

Status

approved