In Gradient Descent, Is Gradient a vector?

 In Gradient Descent - Is the Gradient (Slope) a vector and related question

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Question 1: Is the Gradient a Vector?

Yes, absolutely. In multi-dimensional space, the gradient is a vector with both magnitude and direction.

The Gradient Vector

For a function f(w₁, w₂, ..., wₙ), the gradient is:

        ┌ ∂f/∂w₁ ┐
        │ ∂f/∂w₂ │
∇f  =   │   ⋮    │
        └ ∂f/∂wₙ ┘

Each component tells you: "How much does the loss change if I nudge this particular weight?"

Direction and Magnitude

Property	Meaning
Direction	Points toward steepest ascent (we move opposite for descent)
Magnitude	How steep the slope is (larger = steeper terrain)

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Question 2: One Shot or One-at-a-Time?

One shot — all dimensions simultaneously.

This is crucial: standard gradient descent updates ALL parameters together in a single step, not sequentially.

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Concrete Example: 2D Landscape

Consider a simple loss function with two weights:

L(w₁, w₂) = w₁² + 4w₂²

This creates an elliptical bowl (minimum at origin).

Step-by-Step Walkthrough

Starting point: (w₁, w₂) = (4, 2)

Step 1: Compute partial derivatives

∂L/∂w₁ = 2w₁ = 2(4) = 8

∂L/∂w₂ = 8w₂ = 8(2) = 16

Step 2: Form the gradient vector

       ┌  8 ┐
∇L  =  └ 16 ┘

This vector points "uphill" — toward increasing loss.

Step 3: Compute magnitude and direction

Magnitude = √(8² + 16²) = √320 ≈ 17.9

                            1    ┌  8 ┐       ┌ 0.45 ┐
Direction (unit vector) = ──── × └ 16 ┘   ≈   └ 0.89 ┘
                          17.9

Step 4: Update ALL weights simultaneously

With learning rate η = 0.1:

┌ w₁ⁿᵉʷ ┐     ┌ 4 ┐         ┌  8 ┐     ┌ 4 - 0.8 ┐     ┌ 3.2 ┐
└ w₂ⁿᵉʷ ┘  =  └ 2 ┘ - 0.1 × └ 16 ┘  =  └ 2 - 1.6 ┘  =  └ 0.4 ┘

Both weights move together in one atomic step.

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Visual Intuition

        w₂
        ↑
        |     ● (4, 2) ← Current position
        |      ↖
        |       ↖  Gradient points this way (uphill)
        |        ↖
        |         ○ We move OPPOSITE (downhill)
        |            to (3.2, 0.4)
        |
        +------------------→ w₁
                    ★ Minimum at (0, 0)

The gradient vector [8, 16] points toward the steepest uphill direction. Gradient descent moves in the opposite direction (steepest downhill).

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Why Simultaneous, Not Sequential?

What would happen if we updated one-at-a-time?

Sequential approach (Coordinate Descent):

1. Fix w₂, update w₁
2. Fix w₁, update w₂
3. Repeat

This is a different algorithm called Coordinate Descent. It works, but:

Simultaneous (Gradient Descent)	Sequential (Coordinate Descent)
Moves directly toward minimum	Moves in axis-aligned zigzags
Uses full gradient information	Uses one partial derivative at a time
Standard for neural networks	Used in some optimization problems

Geometric Comparison

Simultaneous (Gradient Descent):     Sequential (Coordinate Descent):
        
    ●                                     ●
     ↘                                    ↓
      ↘                                   ●→●
       ↘                                    ↓
        ★                                   ●→★
    
  (Direct diagonal path)              (Zigzag staircase path)

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Python Demonstration

import numpy as np

def loss(w):
    """Elliptical bowl: L = w1² + 4w2²"""
    return w[0]**2 + 4*w[1]**2

def gradient(w):
    """Gradient vector: [2w1, 8w2]"""
    return np.array([2*w[0], 8*w[1]])

# Starting point
w = np.array([4.0, 2.0])
learning_rate = 0.1

print("Gradient Descent (Simultaneous Update)")
print("=" * 50)
print(f"{'Step':<6} {'w1':<10} {'w2':<10} {'Loss':<12} {'Gradient'}")
print("-" * 50)

for step in range(6):
    grad = gradient(w)
    magnitude = np.linalg.norm(grad)
    print(f"{step:<6} {w[0]:<10.4f} {w[1]:<10.4f} {loss(w):<12.4f} [{grad[0]:.2f}, {grad[1]:.2f}] |{magnitude:.2f}|")
    
    # SIMULTANEOUS update - both weights at once
    w = w - learning_rate * grad

print("-" * 50)
print("Note: Both w1 and w2 update TOGETHER each step")

Output:

Gradient Descent (Simultaneous Update)
==================================================
Step   w1         w2         Loss         Gradient
--------------------------------------------------
0      4.0000     2.0000     32.0000      [8.00, 16.00] |17.89|
1      3.2000     0.4000     10.8800      [6.40, 3.20] |7.16|
2      2.5600     0.0800     6.5792       [5.12, 0.64] |5.16|
3      2.0480     0.0160     4.1953       [4.10, 0.13] |4.10|
4      1.6384     0.0032     2.6844       [3.28, 0.03] |3.28|
5      1.3107     0.0006     1.7180       [2.62, 0.01] |2.62|
--------------------------------------------------
Note: Both w1 and w2 update TOGETHER each step

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Summary

Question	Answer
Is gradient a vector?	Yes — has magnitude (steepness) and direction (steepest ascent)
Update method?	Simultaneous — all parameters move together in one step
Why simultaneous?	Follows true steepest descent path; sequential would zigzag

The gradient vector captures the complete "slope information" for the entire landscape at that point, and moving along its negative direction is the most efficient single step toward the minimum.

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Quick Reference: Math Symbols Used

Symbol	Meaning
∇	Nabla (gradient operator)
∂	Partial derivative
η	Learning rate (eta)
√	Square root
²	Squared
≈	Approximately equal
w₁, w₂	Subscript notation for weights

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References:

https://www.youtube.com/watch?v=TYLyAfFn_ME

https://www.youtube.com/watch?v=GkB4vW16QHI