Reply the following questions using the empty boxes.
(a) (5 points) Consider the target value y and the linear model f(x) with x as
input. (A) Define the mean absolute error for m training samples as the loss
function (L1 loss). (B) Draw the derivative of the loss function.

ChatGPT said:
ChatGPT
Here are the answers to your questions:

(A) Definition of Mean Absolute Error (L1 loss):
The Mean Absolute Error (MAE) for 
𝑚
m training samples is defined as:

𝐿
=
1
𝑚
∑
𝑖
=
1
𝑚
∣
𝑦
𝑖
−
𝑓
(
𝑥
𝑖
)
∣
L= 
m
1
​
  
i=1
∑
m
​
 ∣y 
i
​
 −f(x 
i
​
 )∣
where 
𝑦
𝑖
y 
i
​
  is the target value, 
𝑓
(
𝑥
𝑖
)
f(x 
i
​
 ) is the model's prediction, and 
𝑚
m is the total number of samples.

(B) Derivative of the Loss Function:
The derivative of 
∣
𝑦
−
𝑓
(
𝑥
)
∣
∣y−f(x)∣ with respect to 
𝑓
(
𝑥
)
f(x) is:

∂
𝐿
∂
𝑓
(
𝑥
)
=
{
−
1
if 
𝑦
>
𝑓
(
𝑥
)
,
1
if 
𝑦
<
𝑓
(
𝑥
)
,
0
if 
𝑦
=
𝑓
(
𝑥
)
.
∂f(x)
∂L
​
 = 
⎩
⎨
⎧
​
  
−1
1
0
​
  
if y>f(x),
if y<f(x),
if y=f(x).
​
 
Would you like a graph to visualize the derivative of the loss function?






You said:
Consider the target value y and the linear model f(x) with x as
input. Given m training samples, define the mean squared error loss function.
ChatGPT said:
ChatGPT
The Mean Squared Error (MSE) loss function for 
𝑚
m training samples is defined as:

𝐿
=
1
𝑚
∑
𝑖
=
1
𝑚
(
𝑦
𝑖
−
𝑓
(
𝑥
𝑖
)
)
2
L= 
m
1
​
  
i=1
∑
m
​
 (y 
i
​
 −f(x 
i
​
 )) 
2
 
where:

𝑦
𝑖
y 
i
​
  is the target value for the 
𝑖
i-th sample,
𝑓
(
𝑥
𝑖
)
f(x 
i
​
 ) is the predicted value of the linear model for the 
𝑖
i-th input 
𝑥
𝑖
x 
i
​
 ,
𝑚
m is the total number of samples.
This loss function measures the average of the squared differences between the predicted and actual target values.






You said:
I just finished reading Linear regression 
ChatGPT said:
ChatGPT
Memory updated
That's great! Linear regression is a fundamental concept in machine learning and statistics. If you'd like, we can explore related topics like polynomial regression, regularization techniques (Ridge or Lasso), or move on to more complex algorithms. Let me know how you'd like to proceed!






You said:
So MSE is = 1/M [ Actual Y - W^T Xi)^2
ChatGPT said:
ChatGPT
Yes, you're almost correct! The formula for Mean Squared Error (MSE) loss function is as follows:

𝐿
=
1
𝑀
∑
𝑖
=
1
𝑀
(
𝑌
𝑖
−
𝑊
𝑇
𝑋
𝑖
)
2
L= 
M
1
​
  
i=1
∑
M
​
 (Y 
i
​
 −W 
T
 X 
i
​
 ) 
2
 
where:

𝑀
M: Total number of training samples.
𝑌
𝑖
Y 
i
​
 : Actual target value for the 
𝑖
i-th sample.
𝑊
𝑇
𝑋
𝑖
W 
T
 X 
i
​
 : Prediction from the linear model, where 
𝑊
𝑇
W 
T
  is the transpose of the weight vector, and 
𝑋
𝑖
X 
i
​
  is the input feature vector for the 
𝑖
i-th sample.
This measures the average squared difference between the actual and predicted values. Would you like help with deriving or implementing this?






You said:
then what is this my professor did this =
1
𝑚
σ𝑖=1
𝑚 𝐰𝑇𝐱𝑖
𝑡𝑟𝑎𝑖𝑛 − y𝑖
𝑡𝑟𝑎𝑖𝑛 2
ChatGPT said:
ChatGPT
Your professor's formula seems to represent the Mean Squared Error (MSE) as well, but with slight variations in notation for clarity. Let me rewrite and explain it:

𝐿
=
1
𝑚
∑
𝑖
=
1
𝑚
(
𝑤
𝑇
𝑥
𝑖
train
−
𝑦
𝑖
train
)
2
L= 
m
1
​
  
i=1
∑
m
​
 (w 
T
 x 
i
train
​
 −y 
i
train
​
 ) 
2
 
Explanation:
𝑚
m: Total number of training samples.
𝑤
𝑇
w 
T
 : Transpose of the weight vector (parameters of the linear model).
𝑥
𝑖
train
x 
i
train
​
 : Feature vector of the 
𝑖
i-th training sample.
𝑦
𝑖
train
y 
i
train
​
 : Actual target value (ground truth) of the 
𝑖
i-th training sample.
𝑤
𝑇
𝑥
𝑖
train
w 
T
 x 
i
train
​
 : Predicted value for the 
𝑖
i-th sample using the linear model.
This formula is the same as the traditional MSE formula, but it's explicitly written to emphasize that it's calculated over the training dataset and uses a linear model 
𝑤
𝑇
𝑥
𝑖
w 
T
 x 
i
​
 . The difference 
𝑤
𝑇
𝑥
𝑖
train
−
𝑦
𝑖
train
w 
T
 x 
i
train
​
 −y 
i
train
​
  is squared and averaged over 
𝑚
m samples.

Your professor's notation is often used in machine learning classes to link the theoretical MSE with the linear model parameters 
𝑤
w.






You said:
Doe the value changes or not
ChatGPT said:
ChatGPT
No, the value of the Mean Squared Error (MSE) does not change.

The formula your professor wrote is mathematically identical to the standard MSE formula:

𝐿
=
1
𝑚
∑
𝑖
=
1
𝑚
(
𝑌
𝑖
−
𝑤
𝑇
𝑥
𝑖
)
2
L= 
m
1
​
  
i=1
∑
m
​
 (Y 
i
​
 −w 
T
 x 
i
​
 ) 
2
 
The only difference is notation. Your professor explicitly highlighted the use of:

Training samples (
𝑥
𝑖
train
,
𝑦
𝑖
train
x 
i
train
​
 ,y 
i
train
​
 ),
The linear model 
𝑤
𝑇
𝑥
𝑖
w 
T
 x 
i
​
 .
The underlying calculation and result are the same, as long as you're working with the same dataset, model, and weights.






You said:
Beacuse of the square the values remains the same ryt??
ChatGPT said:
ChatGPT
Not quite! The square in the MSE formula ensures that all differences (residuals) are positive, but it does not guarantee that the value remains the same. Here's why:

Squaring changes the magnitude of differences:

When you square a number, small differences become even smaller, and large differences become significantly larger.
For example:
(
2
)
2
=
4
(2) 
2
 =4
(
−
2
)
2
=
4
(−2) 
2
 =4
(
5
)
2
=
25
(5) 
2
 =25
The squaring operation amplifies larger residuals, making the loss more sensitive to large errors.
Values don't "stay the same":

Squaring changes the scale of the values. For example:
Without squaring: Residual 
𝑟
=
−
3
r=−3.
After squaring: 
𝑟
2
=
9
r 
2
 =9.
If you add up squared residuals, the value will generally be larger than the sum of absolute residuals.
Why does MSE use squaring?
The squaring operation penalizes larger errors more heavily, which is often desirable when training models. If you don't square (e.g., use Mean Absolute Error instead), all residuals contribute linearly.

So, while squaring ensures all values are non-negative, it does change the value and increases the influence of larger errors.






You said:
thats what seee (Actual - predicted)^2 = (Predicted - Actual)^2
ChatGPT said:
ChatGPT
Ah, now I see what you're pointing out! Yes, you're absolutely correct:

(
Actual
−
Predicted
)
2
=
(
Predicted
−
Actual
)
2
(Actual−Predicted) 
2
 =(Predicted−Actual) 
2
 
This equality holds because of the properties of squaring:

Squaring a number removes the sign difference (
𝑥
2
=
(
−
𝑥
)
2
x 
2
 =(−x) 
2
 ).
In other words:

The order of subtraction does not matter in the Mean Squared Error formula, as long as you're consistent in how you compute the difference.
Whether you calculate 
(
𝑦
−
𝑦
^
)
2
(y− 
y
^
​
 ) 
2
  or 
(
𝑦
^
−
𝑦
)
2
( 
y
^
​
 −y) 
2
 , the squared result will be the same.
This symmetry is why MSE works seamlessly regardless of how you frame the subtraction. Thanks for pointing that out! 😊






You said:
So the prof's formula is ryt and ur too
ChatGPT said:
ChatGPT
Yes, exactly! Both your professor's formula and mine are mathematically correct. The difference lies only in the notation and emphasis:

My formula:
𝐿
=
1
𝑚
∑
𝑖
=
1
𝑚
(
𝑌
𝑖
−
𝑊
𝑇
𝑋
𝑖
)
2
L= 
m
1
​
  
i=1
∑
m
​
 (Y 
i
​
 −W 
T
 X 
i
​
 ) 
2
 
is the standard way of expressing the Mean Squared Error.

Your professor's formula:
𝐿
=
1
𝑚
∑
𝑖
=
1
𝑚
(
𝑊
𝑇
𝑋
𝑖
train
−
𝑌
𝑖
train
)
2
L= 
m
1
​
  
i=1
∑
m
​
 (W 
T
 X 
i
train
​
 −Y 
i
train
​
 ) 
2
 
explicitly includes training data notation (e.g., 
train
train) to make it clear that the computation is performed on the training set, and it uses 
𝑊
𝑇
𝑋
𝑖
W 
T
 X 
i
​
  to emphasize the linear model prediction.

Both ultimately compute the same thing because of the property you pointed out:

(
Actual
−
Predicted
)
2
=
(
Predicted
−
Actual
)
2
(Actual−Predicted) 
2
 =(Predicted−Actual) 
2
 
So, yes, both formulas are correct and equivalent!






You said:
great thanks
ChatGPT said:
ChatGPT
You're welcome! 😊 Glad we clarified that. If you have more questions about regression, loss functions, or anything else, feel free to ask. Happy learning and good luck with your studies! 🚀📚




Remeber all for this responeses i did with previous gpt 4