62.62. Softmax Function

Implement the **Softmax Function** for multi-class classification.

Formula:

softmax(z_i) = exp(z_i) / sum(exp(z_j)) for all j

**Numerical Stability:** To avoid overflow with large values, subtract the maximum value before computing exponentials:

z_stable = z - max(z)

softmax(z_i) = exp(z_stable_i) / sum(exp(z_stable_j))

Write a function softmax(z) that takes an array of logits and returns the softmax probabilities.

Example:

z = [2.0, 1.0, 0.1]

exp_z = [exp(2.0), exp(1.0), exp(0.1)] = [7.389, 2.718, 1.105]

sum_exp = 7.389 + 2.718 + 1.105 = 11.212

softmax = [7.389/11.212, 2.718/11.212, 1.105/11.212]

= [0.6590, 0.2424, 0.0986]

# Output sums to 1.0

**Explanation:** Softmax converts raw scores (logits) into a probability distribution. Each output is between 0 and 1, and all outputs sum to exactly 1. It is the standard output activation for multi-class classification.

Constraints: