Isssue in understanding the derivation regarding mean squared error

Question

The following derivation is taken from Chapter 5: Machine Learning Basics from the book titled Deep Learning (by Aaron Courville et al.)

I am facing difficulty in understanding the zero derivative related to minimizing gradient of mean squared error $\nabla_w \text{MSE}_{train} = 0$

$\nabla_w \text{MSE}_{train} = 0$

$\implies \nabla_w(Xw - y)^T(Xw-y) = 0$

$\implies \nabla_w(w^T X^T- y^T)(Xw-y) = 0$

$\implies \nabla_w(w^T X^T- y^T)(Xw-y) = 0$

$\implies \nabla_w(w^T X^TXw - w^T X^Ty -y^TXw+y^Ty) = 0$

$\implies \nabla_w(w^T X^TXw - 2 w^T X^Ty +y^Ty) = 0$

$\implies 2 X^TXw - 2 X^Ty = 0$

$\implies w = 2 (X^TX)^{-1} X^Ty = 0$

I have had difficulty in understanding the flow of the following two lines

$\implies \nabla_w(w^T X^TXw - w^T X^Ty -y^TXw+y^Ty) = 0$

$\implies \nabla_w(w^T X^TXw - 2 w^T X^Ty +y^Ty) = 0$

The first doubt is about in the first two lines, it is possible only if (which I feel is untrue)

$w^T X^Ty = y^TXw$

Note: $X,y$ here refers to input and outputs of train data

Answer 1

The topic you're looking for is called Linear Algebra, and the equation you highlighted at the end is true. This is the reason:

1. If you have a scalar $x \in \mathbb{R}$, then you can take its transpose without changing the value, its transpose is exactly the same number.
2. When you have two vectors $w \in \mathbb{R}^n$ and $y \in \mathbb{R}^n$ and a matrix $X \in \mathbb{R}^{n \times n}$, then $w^TX^Ty \in \mathbb{R}$, i.e. it is a scalar.
3. From Linear Algebra, when you have a product of matrices ($A$, $B$, etc. matrices) you can take the transpose as follows: $(AB)^T = B^TA^T$. You can extend that two three matrices: $(ABC)^T = ((AB)C)^T = C^T(AB)^T = C^TB^TA^T$. You can look at vectors and scalars as special kinds of matrices.
4. When you have $w^TX^Ty$, you can take its transpose without changing the value because it is a scalar, so $w^TX^Ty = (w^TX^Ty)^T = y^TXw$.

Most books and free online resources on Linear Algebra are enough to understand these concepts. Everyone has their favourite, and it really doesn't matter much which one you pick but I like this book: https://link.springer.com/book/10.1007/978-3-319-24346-7.