On the Sample Complexity of Learning Graphical Games
This work addresses the sample efficiency challenge for learning game structures in multi-agent systems, providing theoretical guarantees for researchers in algorithmic game theory and machine learning.
The paper tackles the problem of learning graphical games from behavioral data without payoff information, deriving both sufficient and necessary sample complexity bounds for recovering pure-strategy Nash equilibria. For sparse graphs, the bounds are O(k n log^2 n) and Ω(k n log^2 n), and for dense graphs, O(n^2 log n) and Ω(n^2 log n).
We analyze the sample complexity of learning graphical games from purely behavioral data. We assume that we can only observe the players' joint actions and not their payoffs. We analyze the sufficient and necessary number of samples for the correct recovery of the set of pure-strategy Nash equilibria (PSNE) of the true game. Our analysis focuses on directed graphs with $n$ nodes and at most $k$ parents per node. Sparse graphs correspond to ${k \in O(1)}$ with respect to $n$, while dense graphs correspond to ${k \in O(n)}$. By using VC dimension arguments, we show that if the number of samples is greater than ${O(k n \log^2{n})}$ for sparse graphs or ${O(n^2 \log{n})}$ for dense graphs, then maximum likelihood estimation correctly recovers the PSNE with high probability. By using information-theoretic arguments, we show that if the number of samples is less than ${Ω(k n \log^2{n})}$ for sparse graphs or ${Ω(n^2 \log{n})}$ for dense graphs, then any conceivable method fails to recover the PSNE with arbitrary probability.