The sum of subsets is the definition of addition upon two subsets.

Apparently, the unions of subsets are almost never subspaces (they don’t produce linearity?) Therefore, we like to work with sum of subsets more.

Remember this has arbitrarily many things!! as a part of the content. When defining, remember to open that possibility.

constituents

Sub-sets of \(V\) named \(U_1, U_2, \dots, U_{m}\)

requirements

The sum of subsets \(U_1, \dots, U_{m}\) is defined as:

\begin{equation} U_1, \dots, U_{m} = \{u_1+\dots+u_{m}: u_1\in U_1, \dots, u_{m} \in U_{m}\} \end{equation}

Consider “what’s the variable representing the sum of the result of 2 dice?”

\begin{equation} Y = \sum_{i=1}^{2} X \end{equation}

where \(X\) is a random variable representing the result of once dice.

Suppose \(v \in V\), and \(U \subset V\). Then, \(v+U\) is the subset (not a subspace, obviously):

\begin{equation} v + U = \{v+u : u \in U\} \end{equation}

Supervised learning (also known as behavioral cloning) if the agent is learning what to do in an observe-act cycle) is a type of decision making method.

constituents

• input space: \(\mathcal{X}\)
• output space: \(\mathcal{Y}\)
• hypothesis/model/prediction: \(h : \mathcal{X} \to \mathcal{Y}\)

requirements

Our ultimate goal is to learn a good model \(h\) from the training set:

• what “good” means is hard to define
• we generally want to use the model on new data, not just the training set

continuous \(\mathcal{Y}\) is then called a regression problem; discrete \(\mathcal{Y}\) is called a classification problem.