Notation for valuation¶
Todo
Organize this page better and use its content consistently throughout the documentation.
The following notation is used throughout the documentation:
Let \(D = \{x_1, \ldots, x_n\}\) be a training set of \(n\) samples.
The utility function \(u:\mathcal{D} \rightarrow \mathbb{R}\) maps subsets of \(D\) to real numbers. In pyDVL, we typically call this mappin a score for consistency with sklearn, and reserve the term utility for the triple of dataset \(D\), model \(f\) and score \(u\), since they are used together to compute the value.
The value \(v\) of the \(i\)-th sample in dataset \(D\) wrt. utility \(u\) is denoted as \(v_u(x_i)\) or simply \(v(i)\).
For any \(S \subseteq D\), we denote by \(S_{-i}\) the set of samples in \(D\) excluding \(x_i\), and \(S_{+i}\) denotes the set \(S\) with \(x_i\) added.
The marginal utility of adding sample \(x_i\) to a subset \(S\) is denoted as \(\delta(i) := u(S_{+i}) - u(S)\).
The set \(D_{-i}^{(k)}\) contains all subsets of \(D\) of size \(k\) that do not include sample \(x_i\).
Created: 2023-12-21