In this lecture from Stanford’s CS229 Machine Learning course, the focus is on unsupervised learning algorithms, specifically K-Means clustering and Gaussian Mixture Models (GMM). The instructor begins by contrasting supervised learning, where labeled data guides the model, with unsupervised learning, where no labels are available, making the problem inherently more challenging. The goal in unsupervised learning is to uncover the underlying structure or clusters within the data, which requires stronger assumptions and often results in weaker guarantees compared to supervised methods. The lecture emphasizes the importance of understanding the modeling assumptions behind these algorithms rather than just their procedural steps.
The K-Means algorithm is introduced as an intuitive, iterative method for clustering data into K groups. The process involves initializing cluster centers randomly, assigning each data point to the nearest center, and then updating the centers to be the mean of their assigned points. This cycle repeats until the assignments stabilize. The instructor discusses practical considerations such as the algorithm’s convergence, the possibility of oscillations, and the fact that K-Means is NP-hard, meaning it may not find the global optimum. The importance of initialization is highlighted, with mention of the K-Means++ algorithm, which provides a smarter way to initialize centers to improve clustering results.
Transitioning to Gaussian Mixture Models, the lecture presents GMM as a probabilistic extension of K-Means that allows for soft assignments of points to clusters. Instead of hard cluster membership, each point has a probability of belonging to each cluster, modeled as a mixture of Gaussian distributions with different means and covariances. The instructor explains the generative model where data points are sampled from these Gaussian sources with unknown parameters, including the mixing proportions. The challenge is to estimate these parameters and the latent cluster assignments from the observed data, which is more complex than K-Means due to the probabilistic nature and the unknown cluster memberships.
The Expectation-Maximization (EM) algorithm is introduced as the method to fit GMMs. EM iteratively performs an E-step, where it estimates the probability that each data point belongs to each cluster given current parameters, and an M-step, where it updates the parameters (means, covariances, and mixing proportions) based on these probabilities. This soft assignment and parameter update process continues until convergence. The lecture details the mathematical formulation of these steps, emphasizing the use of Bayes’ rule to invert the generative model and compute the posterior probabilities. The instructor also notes that EM shares similarities with K-Means but incorporates the additional complexity of estimating covariance structures and mixing weights.
Finally, the lecture covers a theoretical detour into convexity and Jensen’s inequality, foundational concepts used to justify the EM algorithm’s convergence properties. Convexity is explained geometrically and algebraically, highlighting how Jensen’s inequality allows the construction of a surrogate function that lower-bounds the likelihood and is easier to optimize. This surrogate function is iteratively maximized in EM, ensuring that the likelihood increases with each iteration. The instructor concludes by previewing that the next lecture will delve deeper into the EM algorithm’s derivation and convergence analysis, reinforcing the conceptual understanding of these fundamental unsupervised learning techniques.
