Decision boundaries — Hinglish notes

2.2.11 · AI-ML › Linear & Logistic Regression

WHAT is a decision boundary?

WHY 0.5? Kyunki do classes ki probabilities ka sum 1 hota hai. Woh tab barabar hoti hain jab dono $0.5$ hon. Ek taraf $P (y = 1) > 0.5$ hota hai (predict 1), doosri taraf $P (y = 1) < 0.5$ (predict 0). Yeh knife-edge hi boundary hai.

HOW to derive the boundary for Logistic Regression

Logistic regression model karta hai $P (y = 1 ∣ x) = σ (z), z = w^{⊤} x + b, σ (z) = \frac{1}{1 + e ^{- z}} .$

Step 1 — Boundary condition set karo. Hum chahte hain $P (y = 1 ∣ x) = 0.5$ . Yeh step kyun? Definition ke hisaab se boundary wahan hoti hai jahan dono classes tie karti hain, aur tie $0.5$ par hoti hai.

Step 2 — Sigmoid ko invert karo. $σ (z) = \frac{1}{2} ⟹ \frac{1}{1 + e ^{- z}} = \frac{1}{2} ⟹ 1 + e^{- z} = 2 ⟹ e^{- z} = 1 ⟹ z = 0.$ Yeh step kyun? Sigmoid monotonic hai, isliye uski value 0.5 exactly ek hi input se correspond karti hai, $z = 0$ . Yeh ek probability condition ko ek clean linear condition mein convert karta hai.

Step 3 — Boundary equation likho. $w^{⊤} x + b = 0$ Yeh step kyun? $z = 0$ hi boundary hai. Yeh ek hyperplane ki equation hai (2D mein line, 3D mein plane).

WHY linear? Kyunki $σ$ sirf linear score $z$ ko probability mein reshape karta hai. Boundary ki shape poori tarah se decide hoti hai jahan $z = 0$ ho, aur $z$ , $x$ ka ek straight-line function hai. Sigmoid probabilities ko bend karta hai, boundary ko nahi.

Making the boundary curved

Agar true separation curved hai, toh same linear machinery mein nonlinear features daalo. Example: $x_{1}^{2}, x_{2}^{2}, x_{1} x_{2}$ add karo. Tab $z = w_{0} + w_{1} x_{1} + w_{2} x_{2} + w_{3} x_{1}^{2} + w_{4} x_{2}^{2} + w_{5} x_{1} x_{2} = 0$ weights mein linear hai lekin original $(x_{1}, x_{2})$ space mein ek conic (circle/ellipse/curve) hai.

Worked examples

Common mistakes

Recall Feynman: 12-saal ke bacche ko samjhao

Socho playground par chalk se ek line kheench rahe ho. Line ke left mein bacche "Team Red" hain, right mein "Team Blue". Chalk line hi decision boundary hai — yeh woh fence hai jo decide karti hai tum kis team mein ho. Computer ka "main kitna sure hoon?" meter exactly line par 50/50 read karta hai, aur jitna door jaate ho utna confident hota jaata hai. Agar ek seedhi chalk line teams ko separate nahi kar sakti (maano Reds, Blues ke around ek ring mein hain), toh hum computer ko circle kheenchne dete hain center se "distance" jaisi fancier clues dekh ke.

Active recall

Probability terms mein do-class decision boundary kaunsi condition define karti hai?

P (y = 1 ∣ x) = 0.5

(dono classes tie karti hain).

Logistic regression ke liye kaunsi equation decision boundary hai?

w^{⊤} x + b = 0

(ek hyperplane).

P = 0.5

kyun

z = 0

reduce hota hai?

Kyunki

σ (z) = 0.5

sirf tab hota hai jab

z = 0

(sigmoid monotonic hai).

Kya logistic regression boundary curved hoti hai?

Nahi — yeh ek linear hyperplane hai; sirf probability surface curved hoti hai.

Linear model se curved boundary kaise milti hai?

Nonlinear features add karo (jaise

x^{2}

x_{1} x_{2}

); boundary feature space mein linear hai, input space mein curved.

Vector

w

geometrically kya represent karta hai?

Boundary ka normal, class-1 region ki taraf point karta hua.

w

aur

b

c > 0

se scale karne par boundary move hoti hai?

Nahi — set

w^{⊤} x + b = 0

unchanged rehta hai; sirf confidence steepness badlti hai.

Classification threshold change karna boundary ko kya karta hai?

Hyperplane ko apne aap ke parallel shift karta hai (offset

b

change karta hai), orientation same rehti hai.

(1, 1)

z = x_{1} + x_{2} - 3

ke liye classify karo.

z = - 1 < 0

→ class 0.

z = x_{1}^{2} + x_{2}^{2} - 4

ki boundary kya hai?

Origin par centered radius 2 ka circle.

Connections

Logistic Regression — jahan se $z = 0$ boundary aati hai.
Sigmoid Function — monotonicity ki wajah se 0.5 ⇔ $z = 0$ .
Linear Regression — same linear score $z$ , alag output/loss.
Feature Engineering — linear boundaries ko curved mein baadalna.
Support Vector Machines — same hyperplane idea ka maximum-margin version.
Softmax Classification — kai boundaries wala multi-class generalisation.