Stepping back from a 10th-order polynomial to a 2nd-order one is the same as adding a constraint to the 10th-order problem. Once the constraint is softened to ‖w‖² ≤ C, the negative gradient of E_in at the optimal solution must be parallel to w, which gives w_REG = (ZᵀZ + λI)⁻¹Zᵀy and is the same as minimizing E_aug = E_in + (λ/N)wᵀw. That is weight decay. A larger λ means a smaller equivalent C and a smaller effective VC dimension d_EFF; switching to L1 gives sparse solutions instead.
This lecture splits test error into three terms: noise σ², squared bias, and variance. It then uses the closed-form solution of ridge regression to show how regularization moves each eigenvalue from λ to λ+ρ, trading a little bias for a large cut in variance. The second half covers double descent, ImageNet-V2, and choosing hyperparameters with a dev set, k-fold cross-validation, and successive halving.
Techniques Lecture 16 has no new model. It re-sorts the models of the first 15 lectures by three questions: where the features are hidden (kernel, aggregation, extraction, low-dimensional projection), how the error is brought down (gradients, equivalent problems, multiple steps), and how to hit the brakes when the model is too powerful (regularization, validation). This article works PCA plus AdaBoost-Stump on eight points from start to finish and checks the answer against each of the three questions; it closes by keeping the ReLU, Xavier/He initialization, momentum, RMSProp, and Adam material that the Fall 2024 on-campus course added.
A linear aggregation of perceptrons can implement AND and OR but not XOR; one more layer can, and that is the multi-layer perceptron. This lecture replaces the sign in the middle with tanh, puts linear regression at the output layer, and derives backpropagation from the chain rule: compute every layer's output x forward, then compute δ backwards from the output layer, and the gradient of each weight is the previous layer's x times this layer's δ. It ends with two troubles in training: E_in is non-convex, so the initial weights should be small and random; d_VC is roughly O(VD), so weight elimination and early stopping are needed against overfitting.
Over four lectures, CS189 Spring 2026 presents linear regression from three angles that meet in one formula: MLE under Gaussian noise is least squares; the least-squares solution is the orthogonal projection of y onto the column space of X; and when features are collinear or too many, ridge (the MAP estimate under a Gaussian prior) or lasso (a Laplace prior) pulls the solution back, with λ chosen on a validation set. Slides, videos, and Discussion 3–4 solutions are all publicly accessible.
Lecture 8 swaps MLE's argmax p(D|θ) for argmax p(θ|D). The prior p(θ) multiplies the likelihood, and after taking the negative log it becomes an extra term in the objective. A trick coin shows data overwhelming the prior, a Beta prior estimates a click rate, and Gaussian and Laplace priors on linear-regression weights turn into L2 and L1 regularization.
HW1 uses an 800,000-year ice-core temperature dataset across four problems: kNN and kernel regression, a geometric proof of least squares, basis-function regression, and a probabilistic derivation of ridge and LASSO, ending with a coordinate-descent LASSO implementation.
Lecture 10 uses φ(x) to let linear models express nonlinear functions, then controls the resulting overfitting with train/validation/test separation, L1/L2 regularization, and model selection.