diff --git a/404.html b/404.html index ef70f696..7333aba6 100644 --- a/404.html +++ b/404.html @@ -215,7 +215,7 @@ document.addEventListener("astro:after-swap", () => init()); preloadTheme(); -
rezarezvan.com

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--- a/chalmers/dat038/dat038/index.html +++ b/chalmers/dat038/dat038/index.html @@ -215,7 +215,7 @@ document.addEventListener("astro:after-swap", () => init()); preloadTheme(); - +
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40 min read

Rezvan Explains: Classical Machine Learning

Table of Contents

+
36 min read

Rezvan Explains: Classical Machine Learning

Table of Contents

Rezvan Explains: Classical Machine Learning

Veni, disce mecum

As of writing this, last night as I was going to sleep, I was lying in bed and thought about the future. @@ -465,91 +465,6 @@

Generative Models

The class-conditional distribution $p(\mathbf{x} | y)$ is a bit more tricky. How do we model this distribution?

- - - @@ -1910,91 +1825,6 @@
Maximum Likelihood Estimate

In our example from Figure 1, if we plot the Gaussian estimates, we get the following plot,

- - - @@ -3643,91 +3473,6 @@
Bayes Optimal Classifier Summary
$$

- - - @@ -8260,91 +8005,6 @@
Logistic Regression

Our linear classifier will separate the feature space into 2 half-spaces

- - - @@ -9441,91 +9101,6 @@
Logistic Regression
But there’s a problem, to take a probabilistic approach we need a function to map the value of $f(\mathbf{x}) = \mathbf{w}^T \mathbf{x} + b$ to a probability value (between 0 and 1).

- - - @@ -10698,91 +10273,6 @@
Support Vector Machines (SVMs)

For now, let’s assume that our data is linearly separable.

- - - @@ -12232,91 +11722,6 @@
Maximum Margin Principle

Let us define a naive criterion, the maximum margin principle.

- - - @@ -13887,7 +13292,7 @@
Duality and KKT Conditions

$$ \begin{equation} \begin{aligned} -f_(\mathbf{x}^{\star}) = g(\mathbf{\lambda}^{\star}, \mathbf{\nu}^{\star}) & = \underset{\mathbf{x} \in \chi}{\inf} \left(f_0(\mathbf{x}) + \sum_{i=1}^{r} \lambda_i^{\star} f_i(\mathbf{x}) + \sum_{i=1}^{s} \nu_i^{\star} h_i(\mathbf{x})\right) \newline +f(\mathbf{x}^{\star}) = g(\mathbf{\lambda}^{\star}, \mathbf{\nu}^{\star}) & = \underset{\mathbf{x} \in \chi}{\inf} \left(f_0(\mathbf{x}) + \sum_{i=1}^{r} \lambda_i^{\star} f_i(\mathbf{x}) + \sum_{i=1}^{s} \nu_i^{\star} h_i(\mathbf{x})\right) \newline & \leq f_0(\mathbf{x}^{\star}) + \sum_{i=1}^{r} \lambda_i^{\star} f_i(\mathbf{x}^{\star}) + \sum_{i=1}^{s} \nu_i^{\star} h_i(\mathbf{x}^{\star}) \newline \end{aligned} \end{equation} @@ -13896,7 +13301,7 @@

Duality and KKT Conditions

$$ \begin{equation} \begin{aligned} -f_(\mathbf{x}^{\star}) = g(\mathbf{\lambda}^{\star}, \mathbf{\nu}^{\star}) & = \underset{\mathbf{x} \in \chi}{\inf} \left(f_0(\mathbf{x}) + \sum_{i=1}^{r} \lambda_i^{\star} f_i(\mathbf{x}) + \sum_{i=1}^{s} \nu_i^{\star} h_i(\mathbf{x})\right) \newline +f(\mathbf{x}^{\star}) = g(\mathbf{\lambda}^{\star}, \mathbf{\nu}^{\star}) & = \underset{\mathbf{x} \in \chi}{\inf} \left(f_0(\mathbf{x}) + \sum_{i=1}^{r} \lambda_i^{\star} f_i(\mathbf{x}) + \sum_{i=1}^{s} \nu_i^{\star} h_i(\mathbf{x})\right) \newline & \leq f_0(\mathbf{x}^{\star}) + \sum_{i=1}^{r} \lambda_i^{\star} f_i(\mathbf{x}^{\star}) + \sum_{i=1}^{s} \nu_i^{\star} h_i(\mathbf{x}^{\star}) \newline & \leq f_0(\mathbf{x}^{\star}) \end{aligned} @@ -13916,6 +13321,119 @@

Duality and KKT Conditions
$$

If strong duality holds and $\mathbf{x}^{\star}$ are optimal, then they must satisfy the KKT conditions.

+
SVMs Continued
+

Let’s now write our hard-margin SVM as an optimization problem,

+

$$ +\begin{equation} +\begin{aligned} +\underset{\mathbf{w}, b}{\min} & \quad \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 \newline +\text{subject to} & \quad y^{(i)}(\mathbf{w}^T \mathbf{x}^{(i)} + b) \geq 1, \quad i = 1, \ldots, M +\end{aligned} +\end{equation} +$$

+

Note that our constraint is from the fact that rescaling $(\mathbf{w}, b)$ does not change the objective function. +Therefore, we set the margin to be $1$,

+

$$ +\begin{equation} +\underset{(\mathbf{x}, y) \in \mathcal{D}}{\min} y(\mathbf{w}^T \mathbf{x} + b) = 1 +\end{equation} +$$

+

For the soft-margin SVM, we introduced the slack variable $\xi_i \geq 0$ for each sample $(\mathbf{x}^{(i)}, y^{(i)})$,

+

$$ +\begin{equation} +\begin{aligned} +\underset{\mathbf{w}, b}{\min} & \quad \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 + C \sum_{i=1}^{M} \xi_i \newline +\text{subject to} & \quad y^{(i)}(\mathbf{w}^T \mathbf{x}^{(i)} + b) \geq 1 - \xi_i, \quad i = 1, \ldots, M \newline +& \quad \xi_i \geq 0, \quad i = 1, \ldots, M +\end{aligned} +\end{equation} +$$

+

We also introduce a (hyper)parameter $C$ as a penalty for violating the margin (too much slack). +Now that we have both the hard-margin and soft-margin SVMs, let’s derive their respective dual problems.

+
Deriving the Dual Problem
+

Let’s start with the hard-margin SVM first since it is simpler, but we will compare the two later.

+

We have the following optimization problem,

+

$$ +\begin{equation} +\begin{aligned} +\underset{\mathbf{w}, b}{\min} & \quad \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 \newline +\text{subject to} & \quad y^{(i)}(\mathbf{w}^T \mathbf{x}^{(i)} + b) \geq 1, \quad i = 1, \ldots, M +\end{aligned} +\end{equation} +$$

+

Firstly, we write the problem into standard form,

+

$$ +\begin{equation} +\begin{aligned} +\underset{\mathbf{w}, b}{\min} & \quad \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 \newline +\text{subject to} & \quad 1 - y^{(i)}(\mathbf{w}^T \mathbf{x}^{(i)} + b) \leq 0, \quad i = 1, \ldots, M +\end{aligned} +\end{equation} +$$

+

Then, we form the Lagrangian,

+

$$ +\begin{equation} +\begin{aligned} +L(\mathbf{w}, b, \mathbf{\lambda}) & = \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 + \sum_{i = 1}^M \lambda_i (1 - y^{(i)}(\mathbf{w}^T \mathbf{x}^{(i)} + b)) \newline +& = \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 + \sum_{i = 1}^M \lambda_i - \lambda_i y^{(i)} (\mathbf{w}^T \mathbf{x}^{(i)} + b) \newline +& = \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 + \sum_{i = 1}^M \lambda_i - \lambda_i y^{(i)} \mathbf{w}^T \mathbf{x}^{(i)} + \lambda_i y^{(i)} b \newline +& = \frac{1}{2} \Vert \mathbf{w} \Vert_2^2 - \sum_{i = 1}^M \lambda_i y^{(i)} \mathbf{w}^T \mathbf{x}^{(i)} + \sum_{i = 1}^M \lambda_i y^{(i)} b \sum_{i = 1}^M \lambda_i +\end{aligned} +\end{equation} +$$

+

The dual function is defined as, +$$ +g(\mathbf{\lambda}) = \underset{\mathbf{w}, b}{\max} \ L(\mathbf{w}, b, \mathbf{\lambda}). +$$

+

To maximize $L$ with respect to $\mathbf{w}$ and $b$, we take the partial derivative of $\mathbf{w}$ and $b$ of $L$ and set the expression to zero, respectively.

+

$$ +\begin{equation} +\begin{aligned} +\frac{\partial L(\mathbf{w}, b, \mathbf{\lambda})}{\partial \mathbf{w}} & = \frac{\partial}{\partial \mathbf{w}} \left[\frac{1}{2} \Vert \mathbf{w} \Vert_2^2 - \sum_{i = 1}^M \lambda_i y^{(i)} \mathbf{w}^T \mathbf{x}^{(i)} + \sum_{i = 1}^M \lambda_i y^{(i)} b \sum_{i = 1}^M \lambda_i \right] \newline +& = \mathbf{w} - \sum_{i = 1}^M \lambda_i y^{(i)} \mathbf{x}^{(i)} = 0. +\end{aligned} +\end{equation} +$$

+

$$ +\begin{equation} +\begin{aligned} +\frac{\partial L(\mathbf{w}, b, \mathbf{\lambda})}{\partial b} & = \frac{\partial}{\partial b} \left[\frac{1}{2} \Vert \mathbf{w} \Vert_2^2 - \sum_{i = 1}^M \lambda_i y^{(i)} \mathbf{w}^T \mathbf{x}^{(i)} + \sum_{i = 1}^M \lambda_i y^{(i)} b \sum_{i = 1}^M \lambda_i \right] \newline +& = \sum_{i = 1}^M \lambda_i y^{(i)} = 0. +\end{aligned} +\end{equation} +$$

+

Thus, we can solve for $\mathbf{w}$ from the first equation and use the second expression as a constraint.

+

$$ +\mathbf{w} = \sum_{i = 1}^M \lambda_i y^{(i)} \mathbf{x}^{(i)}. +$$

+

If we plug back these into our original Lagrangian,

+

$$ +\begin{aligned} +L(\mathbf{w}, b, \mathbf{\lambda}) & = \frac{1}{2} \left( \sum_{i = 1}^M \lambda_i y^{(i)} \mathbf{x}^{(i)} \right)^T \left( \sum_{j = 1}^M \lambda_j y^{(j)} \mathbf{x}^{(j)} \right) - \sum_{i = 1}^M \lambda_i y^{(i)} \left( \sum_{j = 1}^M \lambda_j y^{(j)} \mathbf{x}^{(j)} \right) ^T \mathbf{x}^{(i)} + \sum_{i = 1}^M \lambda_i y^{(i)} b \sum_{i = 1}^M \lambda_i \newline +& = \sum_{i = 1}^M \lambda_i + \frac{1}{2} \sum_{i = 1}^M \sum_{j = 1}^M y^{(i)} y^{(j)} \lambda_i \lambda_j (\mathbf{x}^{(i)})^T \mathbf{x}^{(j)} - \sum_{i = 1}^M \sum_{j = 1}^M y^{(i)} y^{(j)} \lambda_i \lambda_j (\mathbf{x}^{(i)})^T \mathbf{x}^{(j)} \newline +& = \sum_{i = 1}^M \lambda_i - \frac{1}{2} \sum_{i = 1}^M \sum_{j = 1}^M y^{(i)} y^{(j)} \lambda_i \lambda_j (\mathbf{x}^{(i)})^T \mathbf{x}^{(j)} +\end{aligned} +$$

+

So, our dual function is, +$$ +g(\alpha) = \sum_{i = 1}^M \alpha_i - \frac{1}{2} \sum_{i = 1}^M \sum_{j = 1}^M y^{(i)} y^{(j)} \alpha_i \alpha_j (\mathbf{x}^{(i)})^T \mathbf{x}^{(j)}. +$$

+

Combining this and our constraints, we get the following dual optimization problem,

+

$$ +\begin{equation} +\begin{aligned} +\underset{\alpha}{\max} & \quad \sum_{i = 1}^M \alpha_i - \frac{1}{2} \sum_{i = 1}^M \sum_{j = 1}^M y^{(i)} y^{(j)} \alpha_i \alpha_j (\mathbf{x}^{(i)})^T \mathbf{x}^{(j)}. \newline +\text{subject to} & \quad +\sum_{i = 1}^M \alpha_i y^{(i)} = 0, \alpha_i \geq 0, i = 1, \ldots, M. +\end{aligned} +\end{equation} +$$

+

Comparing to the dual form with soft margins. +We can see that the only difference lies in the constraint on $\alpha_i$. In the hard margin case, we do not have an upper-bound, this is due to the $\xi_i$ term (slackness term), we get the upper-bound $C$. With hard margins, we assume the data is perfectly separable, i.e., no mistakes in classification. +We can think of the hard margin case as a special case of the soft margin. If $C \to \infty$ we get the hard margin case.

+

NOTE: No closed form solution for SVMs, only quadratic programming/optimization problem.

+

Let’s see if linear regression is different, oh it is, look, a closed form solution.

+

However, both methods can be solved using gradient descent.

Gradient Descent
Generative VS. Discriminative Models

Discriminative - learn to directly discriminate the classes diff --git a/posts/uncertainty/index.html b/posts/uncertainty/index.html index f1ef55a2..9b2a5ea8 100644 --- a/posts/uncertainty/index.html +++ b/posts/uncertainty/index.html @@ -215,7 +215,7 @@ document.addEventListener("astro:after-swap", () => init()); preloadTheme(); - +

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