1. The problem: Explain Support Vector Machines (SVM) in machine learning. 2. Support Vector Machines are supervised learning models used for classification and regression tasks. 3. The main idea is to find a hyperplane that best separates data points of different classes with the maximum margin. 4. The formula for the decision boundary is $$\mathbf{w} \cdot \mathbf{x} + b = 0$$ where $\mathbf{w}$ is the weight vector, $\mathbf{x}$ is the input vector, and $b$ is the bias. 5. The margin is the distance between the hyperplane and the closest data points from each class, called support vectors. 6. Maximizing the margin can be formulated as the optimization problem: $$\min_{\mathbf{w},b} \frac{1}{2} ||\mathbf{w}||^2$$ subject to $$y_i (\mathbf{w} \cdot \mathbf{x}_i + b) \geq 1$$ for all training points $(\mathbf{x}_i, y_i)$. 7. This ensures the hyperplane separates the classes with the largest possible margin, improving generalization. 8. For non-linearly separable data, SVM uses kernel functions to map data into higher-dimensional spaces where a linear separator can be found. 9. Common kernels include linear, polynomial, and radial basis function (RBF). 10. In summary, SVM finds the optimal separating hyperplane by maximizing the margin between classes, using support vectors to define this margin, and kernel tricks to handle complex data.