Supervise learning!

Some Notational Conventions

• \(n\): number of training examples
• \(m\): number of features
• \(x\): input feature(s)
• \(y\): output*/*target feature
• \(\theta\): parameters
• \(h_{\theta}\qty(x)\): the predictor function

And so, a tuple \(\qty(x,y)\) is a particular training example. We will use the parentheses notation to denote samples, so \(\qty(x^{(i)}, y^{(i)})\) as the ith example of training. We typically use \(h\qty(x)\) as the predictor, parameters are \(\theta_{j}\).

New Concepts

• Linear Regression
  • least-squares error
  • gradient descent
    • gradient descent for least-squares error
    • variants
      • summing over dataset: batch gradient descent
      • pick one sample and run it: stochastic gradient descent
      • pick some samples and run them: mini-batch gradient descenmini-bach gradient descet
• a primer on Vector Calculus
  • trace
• Normal Equation

Key Sequence

Review even more! Linear Regression, give some intuition, discuss logistic regression and give an optimization method for it.

Notation

Recall the notation:

• \(\qty(x^{(i)}, y^{(i)})\), ith example
• \(x^{(i)} \in \mathbb{R}^{m+1}\), where \(x_0^{(i)}, \forall i = 1\)
• \(y^{(i)} \in \mathbb{R}\)

\(n\) — number of examples; \(m\) — number of features

New Concepts

• Locally-Weighted Regression
• logistic regression
• Newton’s Method
• parametricity of learning algorithms
  • Non-Parametric Learning Algorithms
  • Parametric Learning Algorithm

Important Results / Claims

• probabilistic intuition for least-squares error

Questions

Interesting Factoids

Scratch

Homo Sapiens are animals that depend on ecosystems. Our social system depends on ecosystems.

A primer on Vector Calculus.

trace

constituents

for square \(A \in \mathbb{R}^{m\times m}\), we write:

requirements

\(\text{tr}\qty(A) = \sum_{i}^{} A_{ii}\) is the sum of the diagonals

additional information

properties of traces

\begin{equation} \text{tr}\qty(AB) = \text{tr}\qty(BA) \end{equation}

\begin{equation} \text{tr}\qty(ABC) = \text{tr}\qty(CAB) \end{equation}

\begin{equation} \nabla_{A} \qty [\text{tr}\qty(AB)] = B^{T} \end{equation}

As a function of \(n\), what is the runtime of the “worst” input?

• pros: very strong guarantee
• cons: too strong of an upper bound, may have better algorithm for Beyond Worst-Case Analysis