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Web Simulation

SoftMax

This interactive simulation visualizes the SoftMax function used in classification. You adjust raw logits and a temperature parameter, watch them become probabilities, and pick a true class to highlight in the formula breakdown.

Math behind the Simulation

1. The pipeline

SoftMax turns unconstrained logits z₁, …, z_n into probabilities that sum to 1:

P_i = e^z_i/T / ∑_j e^z_j/T.

Stage 1: Logits — raw scores (any sign). Stage 2: Exponentiation — e^z/T makes values positive and exaggerates differences. Stage 3: Normalization — divide by the sum so probabilities add to 1.

2. Temperature T

Low T sharpens the distribution (almost one-hot); high T softens it toward uniform. T appears in the exponent: z_i/T.

3. True class

Use the True radio under each probability bar to select which class to highlight. The formula breakdown shows P_i for that class.

Formula: P_i = e^z_i/T / ∑_j e^z_j/T

T 1.0

Logits (z)

z₀

2.00

z₁

1.00

z₂

0.10

z₃

-1.00

→ Exp →

e^z/T

→ Normalize →

Probability (P)

True

P₀ = e^2 / (e^2 + …) = 0.00

Usage

Follow these steps to explore the SoftMax pipeline:

Logits (z): Use the four sliders to set raw scores (range −5 to +5). Watch Stage 2 (exponentiation) and Stage 3 (probabilities) update in real time.
Temperature (T): Lower T sharpens the distribution (almost one-hot); higher T softens it. Adjust the T slider and observe the effect on the orange and green bars.
True Class: Select which class to highlight using the True radio under each probability bar. The formula breakdown shows P_i for the selected class.

Tips: Try low T (e.g. 0.2) vs high T (e.g. 3) to compare sharp vs soft distributions. The exponentiation (e^z/T) bars use a dynamic scale: the tallest bar is always 100%, so no clipping; relative heights show the true ratios even when values differ by orders of magnitude.

Parameters