Privacy-preserving authenticated key exchange (PPAKE) is a cryptographic protocol that enables two users to exchange a session key while protecting users' privacy (i.e., hiding the user's identity) against the machine-in-the-middle adversary. To hide user identities, PPAKE messages are broadcast to the network, increasing communication complexity. In ASIACRYPT2022, Lyu et al. introduced a concept of robustness to reduce communication complexity. Roughly, robust PPAKE allows receivers to decide whether it is the intended user by processing the first message with its long-term secret key. As a result, only the intended user replies to the first message, and thus, messages in the network are reduced. However, if a user's secret key is leaked, an adversary can also use it to determine whether the past first message was intended for the user, and thus, the PPAKE scheme of Lyu et al. does not have full forward privacy. Lyu et al. leave an open problem of constructing a PPAKE scheme with robustness and full forward privacy. In this work, we solve this problem by introducing a new framework called key updatable PPAKE (kuPPAKE). In kuPPAKE schemes, a long-term secret key is updated so that the updated key does not work for past messages. Therefore, robustness no longer conflicts with full forward privacy. We propose a generic construction of a 2-round kuPPAKE and show a concrete scheme in the standard model from DH-style assumptions over bilinear groups.