SHAPIQ (2024)

SHAPIQ (2024)#

SHAPIQ is a Python library designed for computing and analyzing Shapley Interactions (SIs) of any order, which can reveal synergies and redundancies between entities in machine learning tasks. It provides a general interface for state-of-the-art SI algorithms, an explanation API for interpreting model predictions using SIs, and tools for benchmarking and visualizing these interactions across various machine learning domains, offering a more comprehensive understanding of feature relationships than traditional Shapley Values.

Cooperative Game with Three Players#

Consider a cooperative game with three players. So, set \(N = \{1, 2, 3\}\). And let the value function \( v(S) \) define the payout for each possible coalition \(S \subseteq N\). So the possible combinations for the game are as follows:

  • Order 0 interactions: \( \{\} \)

  • Order 1 interactions: \( \{1\}, \{2\}, \{3\} \)

  • Order 2 interactions: \( \{1, 2\}, \{2, 3\}, \{1, 3\} \)

  • Order 3 interactions: \( \{1, 2, 3\} \)

With SHAPIQ, we are interested in the calculation of interactions like these: \( \{1, 2\}, \{2, 3\}, \{1, 3\}, \{1, 2, 3\} \).

Interaction Index Representation#

Interaction Index is represented by:

\[ \Phi^{II}(S) = \sum_{T\subseteq N \setminus S }^{} p^{|S|}_{|T|}(n) \Delta_{S}(T) \]

Here,

\[ p^{|S|}_{|T|}(n) = \frac{1}{(n-s+1) (\binom{n-s}{t})} \]
\[ \Delta_{S}(T) = \textnormal{Discrete derivative for } S \subseteq N \textnormal{ in the presence of } T \subseteq N \setminus S \]

Interaction Index Calculation for \( \{1, 2\} \)#

\[ \Phi^{II}(\{1, 2\}) = \sum_{T\subseteq \{1, 2, 3\} \setminus \{1, 2\} }^{} \frac{1}{(3-2+1)(\binom{3-2}{t})} \Delta_{\{1, 2\}}(T) \]

Here, the possible cases of \(T\) are \(\{\}\) and \(\{3\}\).

So,

\[ \Phi^{II}(\{1, 2\}) = \frac{1}{(3-2+1)(\binom{3-2}{0})} \Delta_{\{1, 2\}}(\{\}) + \frac{1}{(3-2+1)(\binom{3-2}{1})} \Delta_{\{1, 2\}}(\{3\}) \]

Discrete Derivative Calculation#

\[ \Delta_{\{1, 2\}}(\{\}) = \sum_{L\subseteq S }^{} (-1)^{|S|-|L|} v(\{T \cup L\}) \]
\[ \Delta_{\{1, 2\}}(\{\}) = \sum_{L\subseteq S }^{} (-1)^{2-|L|} v(\{\{\} \cup L\}) \]

Possible values for \(L\) are \(\{\}, \{1\}, \{2\}, \{1, 2\}\).

\[ \Delta_{\{1, 2\}}(\{\}) = v(\{\}) - v(\{1\}) - v(\{2\}) + v(\{1, 2\}) \]

For \( \Delta_{\{1, 2\}}(\{3\}) \):

\[ \Delta_{\{1, 2\}}(\{3\}) = \sum_{L\subseteq S }^{} (-1)^{|S|-|L|} v(\{T \cup L\}) \]
\[ \Delta_{\{1, 2\}}(\{3\}) = v(\{3\}) - v(\{1, 3\}) - v(\{2, 3\}) + v(\{1, 2, 3\}) \]

Final Interaction Index Formula#

\[ \Phi^{II}(\{1, 2\}) = \frac{1}{2} \left( v(\{\}) - v(\{1\}) - v(\{2\}) + v(\{1, 2\}) \right) + \frac{1}{2} \left( v(\{3\}) - v(\{1, 3\}) - v(\{2, 3\}) + v(\{1, 2, 3\}) \right) \]

Value Function Definition#

Now, we need to get the values from the value function for the terms: \( v(\{\}), v(\{1\}), v(\{2\}), v(\{3\}), v(\{1, 2\}), v(\{2, 3\}), v(\{1, 3\}), v(\{1, 2, 3\}) \).

Here, \(v\) is the value function representing the worth of a coalition (e.g., the model’s prediction with a subset of features).

  • \(v(\{\})\): The value for the empty coalition, meaning it holds the value when no features are added. This is the model’s base value or the prediction without any feature information.

  • For any other coalition \(v(\{S\})\), it is obtained by evaluating the model (or its conditional expectation) on the input where only the features in \(S\) are present, and other features are handled (e.g., by marginalization or conditioning).

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