Here’s a clean, intuitive explanation of argmax, softmax, and cross-entropy --- 1. Argmax Argmax = “which class has the highest score?” If your model outputs something like: ` scores = [2.1, 0.4, 5.3] ` Then: ` argmax(scores) = 2 because 5.3 is the largest ` * It does NOT give the value (5.3) * It gives the index of the maximum value Used in: * Classification prediction * Q-learning (choose best action = argmax(Q-values)) * Choosing best probability class after softmax --- 2. Softmax Softmax turns raw model scores (logits) into probabilities: pᵢ = e^(zᵢ) / Σⱼ e^(zⱼ) Properties: * All probabilities are between 0 and 1 * Sum to 1 * Larger scores → larger probability (exponentially) Example: Scores: ` z = [2, 1, 0] ` Softmax: ` e^2 = 7.389 e^1 = 2.718 e^0 = 1 sum = 11.107 softmax = [7.389/11.107, 2.718/11.107, 1/11.107] ≈ [0.665, 0.245, 0.090] ` So the model thinks class 0 is most likely. --- ✅ 3. Cross-Entropy Loss Cross-entropy measures how good the predicted probability distribution is. For classification, if the true class is class ( y ): $$\text{CE} = -\log(p_y)$$ * If the model gives the true class high probability, loss is small. * If the model gives the true class low probability, loss is huge. Examples: If true class = 0 Model predicts: ` p = [0.8, 0.1, 0.1] Loss = -log(0.8) = 0.223 (very good) ` Bad prediction: ` p = [0.2, 0.3, 0.5] Loss = -log(0.2) = 1.609 (bad) ` Cross-entropy punishes wrong, confident predictions the most. --- 🎯 How these three work together In neural networks: 1. Model outputs scores (logits), e.g. ` [2.1, 0.4, 5.3] ` 2. Softmax converts to probabilities ` [0.03, 0.01, 0.96] ` 3. Cross-entropy checks how good the probability for the correct class is e.g. if true class = 2 → loss = −log(0.96) 4. At prediction time, use argmax to pick the most likely class. --- 🔥 Concrete Numerical Example (Everything together) Suppose: * Model outputs logits ` z = [1.0, 3.0, 2.0] ` * True label = class 1 Step 1 — Softmax ` e^1 = 2.718 e^3 = 20.085 e^2 = 7.389 sum = 30.192 probabilities = [2.718/30.192, 20.085/30.192, 7.389/30.192] ≈ [0.09, 0.66, 0.24] ` Step 2 — Cross-entropy True class = 1 → p_y = 0.66 Loss: ` CE = -log(0.66) ≈ 0.415 ` Step 3 — Prediction (argmax) argmax(z) = 1 because "3.0" is the largest logit. → model predicts class 1. --- Summary Table | Concept | What it does | Formula | Example | | ----------------- | ----------------------------------------------- | -------------------------------- | ---------------------------- | | Argmax | Picks largest score | argmax(z) | [1,5,3] → 1 | | Softmax | Converts logits → probabilities | ( \frac{e^{z_i}}{\sum e^{z_j}} ) | [2,1,0] → [0.66,0.24,0.09] | | Cross-Entropy | Measures how wrong the predicted probability is | −log(p_y) | true=1, p=0.66 → loss=0.415 |