Probably Correct Optimal Stable Matching under Two-Sided Uncertainty

We study a sequential learning problem for stable matchings in two-sided markets where preferences on both sides are initially unknown.

We focus on a centralized setting where an algorithm matches agents at each time step and receives noisy rewards that reflect the preferences of the matched agents, following a semi-bandit feedback structure.

We adopt a pure exploration perspective, aiming to efficiently identify the optimal stable matching with high probability.

Our work extends prior results by handling \emph{two-sided uncertainty} and by exploiting \emph{partial preference} information.

A central ingredient is the notion of \textbf{pervasive stable matching}, which enables the identification of optimal stable matchings under partial preferences.

We propose elimination-based algorithms whose stopping criteria exploit the structure of the learned partial preferences, and provide a refined sample-complexity analysis.

Beyond pure exploration, we extend our approach to regret minimization and establish regret bounds with respect to the \emph{optimal} stable matching that avoid dependence on the minimum reward gap $\Delta_{\min}$.