  The problem of matching markets has been studied for a long time in the
literature due to its wide range of applications. Finding a stable matching is
a common equilibrium objective in this problem. Since market participants are
usually uncertain of their preferences, a rich line of recent works study the
online setting where one-side participants (players) learn their unknown
preferences from iterative interactions with the other side (arms). Most
previous works in this line are only able to derive theoretical guarantees for
player-pessimal stable regret, which is defined compared with the players'
least-preferred stable matching. However, under the pessimal stable matching,
players only obtain the least reward among all stable matchings. To maximize
players' profits, player-optimal stable matching would be the most desirable.
Though \citet{basu21beyond} successfully bring an upper bound for
player-optimal stable regret, their result can be exponentially large if
players' preference gap is small. Whether a polynomial guarantee for this
regret exists is a significant but still open problem. In this work, we provide
a new algorithm named explore-then-Gale-Shapley (ETGS) and show that the
optimal stable regret of each player can be upper bounded by $O(K\log
T/\Delta^2)$ where $K$ is the number of arms, $T$ is the horizon and $\Delta$
is the players' minimum preference gap among the first $N+1$-ranked arms. This
result significantly improves previous works which either have a weaker
player-pessimal stable matching objective or apply only to markets with special
assumptions. When the preferences of participants satisfy some special
conditions, our regret upper bound also matches the previously derived lower
bound.
