Abstract :^)

Here I implemented SVM from scratch (only numpy was used) and tried to train it with different data and hyperparameters. An optimization algorithm I use here is SMO (i.e. Sequential Minimal Optimization), which is commonly used with Support Vector Machine. Although my implementation is a simplified version of the algorithm.

P.S. The testing_smo.ipynb file was used to try different implementations for SVM, as well as the optimization techniques. For example, I started with a SGD optimization, so in the file you may see a learning rate variable (just a "leftlover"). Now it just repeats my main SMO script tho.

Advantages of SMO:

• Scalability: SMO is designed to work well with large datasets.
• Convergence: SMO has a proven convergence guarantee, which means that it will converge to the optimal solution given enough iterations.
• Flexibility: SMO can handle a wide range of kernel functions, including linear, polynomial, and RBF kernels.
• Memory efficiency: SMO only needs to store a small number of variables in memory, which makes it memory-efficient.

I used this paper to implement the algorithm. Here you can learn how it works from a mathematical point of view. Or, if you by some chance are interested in a full version of SMO, here you can read a paper about it.

A Brief Overview On How My Script Works

In this section I will quickly explain how to work with my SVM implementation.

SMO Class' Init Method

def __init__(X: ndarray, y: ndarray, kernel: str, c: float = 1, tol: float = 10e-4, sigma: float = 1)

• X - a 2D numpy array; the first dimention contains data points, the second one contains the features.
• y - a 1D numpy array; consists of the labels for every data point in X; those labels should be either 1 or -1 (only binary classification).
• kernel - a string; "linear" or "rbf" (i.e names of the kernel functions).
• C - regularization parameter (just read the paper).
• tol - numerical tolerance (just read the paper).
• sigma - a hyperparameter for the RBF kernel, isn't needed while using the linear kernel function.

Making Predictions

def predict(features: ndarray, labels: ndarray) -> tuple[ndarray, float]

• features - a 2D numpy array; the first dimention contains data points, the second one contains the features.
• labels - a 1D numpy array; consists of the labels for every data point in features; those labels should be either 1 or -1 (only binary classification).
• RETURNS - a tuple with the predictions and an accuracy respectively.

Training a Model

def fit() -> tuple[ndarray, float]

• BEHAVIOR - basically, performes only one "tuning step" through all the alphas; should be executed multiple times
• RETURNS - a tuple with the predictions and an accuracy respectively.

Getting Support Vectors

def get_support_vectors(zero: float = 10e-5) -> ndarray

• zero - a number that is being compared with all alphas as those that don't belong to support vectors may not converge to zero completely.
• RETURNS - a numpy array of the support vectors from the training dataset.

Experiments

The Titanic Dataset (titanic_experiments.ipynb)

The number of iterations is always 30, numerical tolerance is always the same: 10e-4. A random seed for numpy (to generate alphas) is 42. You can find the dataset on Kaggle.

№	Kernel	C	Gamma	Training Accuracy	Testing Accuracy
1	Linear	0.01	-	72.13%	61.03%
2	Linear	0.1	-	69.47%	56.19%
3	Linear	1	-	66.81%	56.49%
4	Linear	10	-	64.99%	56.19%
5	Linear	100	-	65.12%	56.19%
6	RBF	0.01	0.1	59.38%	61.63%
7	RBF	0.1	0.1	81.23%	60.72%
8	RBF	1	0.1	86.41%	61.33%
9	RBF	10	0.1	94.82%	67.37%
10	RBF	100	0.1	94.54%	60.73%
11	RBF	10	1	97.20%	53.78%
12	RBF	10	0.01	81.79%	77.34%

Blobs and Circles (blobs_circles_experiments.ipynb)

In this subsection I will visualize some data and the decision boundaries my model makes for the data. For this purpose I use two auto-generated datasets from the scikit-learn library: blobs and circles. And because it's better to always have the same data, I saved those as .csv files. P.S. All the datapoints within a blue area (the gutter) are the support vectors

Blobs

Used params:

• Numerical Tolerance = 0.001
• Iterations = 150
• Random Seed (for generating alphas) = 42
• Kernel = "Linear"

Plots	C
	0.01
	1
	100

Circles

Used params:

• Numerical Tolerance = 0.001
• Iterations = 25
• Random Seed (for generating alphas) = 42
• Kernel = "RBF"

Plots	C	Gamma
	0.01	0.1
	0.1	1
	100	0.1
	2	35