3. Suppose we have four training examples under the two-category case, i.e. D* = {(x₁,w₁) |1 ≤ i ≤ 4} where x₁ = (1, 2), x₂ = (2, 2), x3 = (1, 1)¹, x₁ = (2,0) and w₁ = 62 = -1, W3 = w₁ = 1. Furthermore, linear discriminant function g(x) = wx+b is adopted to learn from the training examples and the -5²,Wi. · g(x₁). criterion function to be minimized is set as (w. b) 11

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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**Problem Statement:**

Suppose we have four training examples under the two-category case, i.e., 

\[
D^* = \{(x_i, \omega_i) \mid 1 \leq i \leq 4\}
\]

where 

\[
x_1 = (1, 2)^t, \quad x_2 = (2, 2)^t, \quad x_3 = (1, 1)^t, \quad x_4 = (2, 0)^t
\]

and 

\[
\omega_1 = \omega_2 = -1, \quad \omega_3 = \omega_4 = 1.
\]

Furthermore, linear discriminant function 

\[
g(x) = w^t x + b 
\]

is adopted to learn from the training examples and the criterion function to be minimized is set as 

\[
J(w, b) = -\sum_{i=1}^{4} \omega_i \cdot g(x_i).
\]

Given the initial model 

\[
w_0 = (-2, 2)^t \quad \text{and} \quad b_0 = -3.
\]

If gradient descent techniques are utilized to minimize the criterion function, what is the resulting discriminant function after three gradient descent steps with learning rate 

\[
\eta = 0.1 
\]

and threshold 

\[
\epsilon = 10^{-6}?
\]
Transcribed Image Text:**Problem Statement:** Suppose we have four training examples under the two-category case, i.e., \[ D^* = \{(x_i, \omega_i) \mid 1 \leq i \leq 4\} \] where \[ x_1 = (1, 2)^t, \quad x_2 = (2, 2)^t, \quad x_3 = (1, 1)^t, \quad x_4 = (2, 0)^t \] and \[ \omega_1 = \omega_2 = -1, \quad \omega_3 = \omega_4 = 1. \] Furthermore, linear discriminant function \[ g(x) = w^t x + b \] is adopted to learn from the training examples and the criterion function to be minimized is set as \[ J(w, b) = -\sum_{i=1}^{4} \omega_i \cdot g(x_i). \] Given the initial model \[ w_0 = (-2, 2)^t \quad \text{and} \quad b_0 = -3. \] If gradient descent techniques are utilized to minimize the criterion function, what is the resulting discriminant function after three gradient descent steps with learning rate \[ \eta = 0.1 \] and threshold \[ \epsilon = 10^{-6}? \]
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