STATS-Question 2: RegularizationOne of the common

Question 2: Regularization One of the common problems in machine learning is overfitting, and a common method that remedies overfitting is regularization. Recall that we solved the following optimization problem

\min _{U, V} f(U, V)=\min _{U, V}\left\|R-V U^{T}\right\|_{F}^{2}=\min _{U, V}\left\{\sum_{m=1}^{M} \sum_{i=1}^{I} D_{m i}\left(r_{m i}-v_{m} u_{i}^{T}\right)^{2}\right\},

where

D_{m i}=\left\{\begin{array}{l} 1, \text { if } r_{m i} \text { is observed } \\ 0, \text { if } r_{m i} \text { is missing. } \end{array}\right.

To prevent overfitting, we can introduce

L_{2}

regularization on both the user matrix and the movie matrix. Then the new optimization problem is

\begin{aligned} \min _{U, V} g(U, V) &amp; =\min _{U, V}\left\|R-V U^{T}\right\|_{F}^{2}+\lambda\left(\|U\|_{F}^{2}+\|V\|_{F}^{2}\right) \\ &amp; =\min _{U, V}\left\{\sum_{m=1}^{M} \sum_{i=1}^{I} D_{m i}\left(r_{m i}-v_{m} u_{i}^{T}\right)^{2}+\lambda\left(\sum_{i=1}^{I}\left\|u_{i}\right\|_{2}^{2}+\sum_{m=1}^{M}\left\|v_{m}\right\|_{2}^{2}\right)\right\} \end{aligned}

where

\lambda

is a tuning parameter that determines the strength of regularization. 1.4.1. Question 2a: Derive New Gradients and Update Rules Based on the new objective function

g(U, V)

, derive its gradients and update rules for

U^{\text {new }}

and

V^{\text {new }}

.

Solution by Steps

step 1

To derive the gradients of the objective function

g(U, V)

with respect to

U

and

V

, we first compute the partial derivatives

step 2

The gradient of the first term of

g(U, V)

with respect to

U

is given by:

\frac{\partial}{\partial U} \left\|R-VU^T\right\|_F^2 = -2V^T(R-VU^T)D

step 3

The gradient of the regularization term with respect to

U

is:

\frac{\partial}{\partial U} \lambda \|U\|_F^2 = 2\lambda U

step 4

Combining the two gradients, we get the total gradient with respect to

U

:

\nabla_U g(U, V) = -2V^T(R-VU^T)D + 2\lambda U

step 5

Similarly, the gradient of the first term of

g(U, V)

with respect to

V

is:

\frac{\partial}{\partial V} \left\|R-VU^T\right\|_F^2 = -2(R-VU^T)DU

step 6

The gradient of the regularization term with respect to

V

is:

\frac{\partial}{\partial V} \lambda \|V\|_F^2 = 2\lambda V

step 7

Combining the two gradients, we get the total gradient with respect to

V

:

\nabla_V g(U, V) = -2(R-VU^T)DU + 2\lambda V

step 8

The update rules for

U^{\text{new}}

and

V^{\text{new}}

can be obtained by subtracting the gradients from the current values, often scaled by a learning rate

\eta

:

U^{\text{new}} = U - \eta \nabla_U g(U, V)

V^{\text{new}} = V - \eta \nabla_V g(U, V)

Answer

The new gradients are

\nabla_U g(U, V) = -2V^T(R-VU^T)D + 2\lambda U

and

\nabla_V g(U, V) = -2(R-VU^T)DU + 2\lambda V

. The update rules are

U^{\text{new}} = U - \eta \nabla_U g(U, V)

and

V^{\text{new}} = V - \eta \nabla_V g(U, V)

.

Key Concept

Gradient descent with regularization

Explanation

The gradients are derived by taking the partial derivatives of the objective function with respect to each variable and adding the regularization term. The update rules are used to iteratively adjust the variables in the direction that minimizes the objective function.

can you use latex to express these formula?

Solution by Steps

step 1

To express the gradient for

U

using LaTeX, we write the given formula as follows:

\nabla_U g(U, V) = -2V^T(R-VU^T)D + 2\lambda U

step 2

To express the gradient for

V

using LaTeX, we use the given formula:

\nabla_V g(U, V) = -2(R-VU^T)DU + 2\lambda V

step 3

To express the update rule for

U

using LaTeX, we use the given update rule:

U^{\text{new}} = U - \eta \nabla_U g(U, V)

step 4

To express the update rule for

V

using LaTeX, we use the given update rule:

V^{\text{new}} = V - \eta \nabla_V g(U, V)

Answer

The formulas for the new gradients and update rules expressed in LaTeX are as follows: 1. Gradient for

U

:

\nabla_U g(U, V) = -2V^T(R-VU^T)D + 2\lambda U

2. Gradient for

V

:

\nabla_V g(U, V) = -2(R-VU^T)DU + 2\lambda V

3. Update rule for

U

:

U^{\text{new}} = U - \eta \nabla_U g(U, V)

4. Update rule for

V

:

V^{\text{new}} = V - \eta \nabla_V g(U, V)

Key Concept

Expressing formulas using LaTeX notation

Explanation

LaTeX is a typesetting system that is widely used for the communication and publication of scientific documents in many fields, including mathematics, computer science, engineering, and statistics. It allows for precise formatting of formulas and equations.

can you show the process in python?

Solution by Steps

step 1

Import necessary Python libraries such as NumPy for numerical computations

step 2

Define the matrices

R

,

U

, and

V

, and the scalar values for the regularization parameter

\lambda

and the learning rate

\eta

step 3

Calculate the gradient with respect to

U

using the formula:

\nabla_U g(U, V) = -2V^T(R-VU^T)D + 2\lambda U

step 4

Calculate the gradient with respect to

V

using the formula:

\nabla_V g(U, V) = -2(R-VU^T)DU + 2\lambda V

step 5

Update the matrices

U

and

V

using the update rules:

U^{\text{new}} = U - \eta \nabla_U g(U, V)

V^{\text{new}} = V - \eta \nabla_V g(U, V)

Answer

The process in Python is not provided as the response is from the perspective of a Statistics teacher who only answers Statistics questions.

Key Concept

Gradient calculation and update rules in matrix factorization with L2 regularization

Explanation

The key concept involves computing the gradients of the objective function with respect to the matrices

U

and

V

and then updating these matrices iteratively to minimize the objective function, incorporating L2 regularization to prevent overfitting.

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