Logistic regression uses θ(wᵀx) to estimate P(+1|x), yet the data contains only circles and crosses. This lecture derives the cross-entropy error ln(1 + exp(−y wᵀx)) from maximum likelihood and uses the chain rule to show that the gradient is a θ-weighted average of −yₙxₙ. Setting the gradient to zero gives a non-linear equation of w with no closed-form solution, so the lecture switches to fixed learning rate gradient descent, w ← w − η∇E_in(w), and walks to the bottom of the valley one step at a time.
The sum of AdaBoost's weights in each round is exactly the sum of the exponential error exp(−ys). From that, this lecture derives that the gₜ found by the base algorithm is an approximate functional gradient direction, and the vote αₜ is the best step size along that direction. Swap the exponential error for squared error and the same derivation becomes 'in each round, run regression on the residuals yₙ − sₙ, then set the step size with one-variable linear regression'; with a weakened C&RT as the base model, that is GBDT.
Lecture 2 breaks supervised learning into three pieces: a hypothesis h, the least-squares cost J(θ), and an algorithm that drives J down. Gradient descent sweeps all n examples on every step; minibatch SGD updates after looking at a small batch, and pays for it by bouncing around near the optimum. Linear regression also has a closed-form solution, θ = (XᵀX)⁻¹Xᵀy, provided XᵀX is invertible.
L2 gave us a score function and a loss. L3 answers how to find a good W. The first half covers regularization: add λR(W) next to the data loss so the model does not fit the training data too well. The second half is a lineage of optimizers. SGD zigzags in narrow valleys, Momentum builds up velocity, RMSProp scales each dimension's step, Adam combines the two and adds bias correction, and AdamW moves weight decay outside the moment estimates. The slides close with practical advice: Adam(W) is a good default in many cases, and SGD+Momentum can do better but needs more tuning of the learning rate and schedule.
CS189 Spring 2026 covers gradient descent in two lectures. Lec 13 derives the learning-rate limit and the condition number from the Hessian's eigenvalues, then moves through momentum, learning-rate schedules, AdaGrad/RMSProp/Adam, and mini-batch SGD. In Lec 15, Dimakis walks through the same material again, starting from a gradient computed by hand on a small data table. Both slide decks, the recordings, Lec 13's handwritten notes, and Discussions 6 and 7 (with solutions) are all publicly accessible.
CS181 Spring 2026 HW2 (due Feb 27) has four problems worth 90 points: train 10 logistic models on planet observations to see bias and variance, derive the MLE of a generative classifier, implement five classifiers on 27 loan applicants, and watch ridge reshape the loss surface under SGD, momentum, and Adam. The core skill is separating what one model's probability says from how much 10 models disagree.
A model uses loss to know how wrong it is and gradients to know which direction to adjust. Gradient descent repeats three things: compute loss, compute gradients, update parameters. The learning rate controls step size — too large and you overshoot, too small and training takes forever.
Lecture 8 moves from a one-dimensional parabola to vector gradients and compares batch GD, SGD, and mini-batches; the learning rate determines whether updates converge, oscillate, or diverge.