We show that, when private sector expectations are determined in line with adaptive learning, optimal policy responds persistently to cost-push shocks. The optimal response is stronger and more persistent, the higher is the initial level of perceived inflation persistence by the private sector. Such a sophisticated policy reduces inflation persistence and inflation volatility at little cost in terms of output gap volatility. Persistent responses to cost-push shocks and stability of inflation expectations resemble optimal policy under commitment and rational expectations. Nevertheless, it is clear that the mechanism at play is very different. In the case of commitment it relies on expectations of future policy actions affecting inflation expectations; in the case of sophisticated central banking it relies on the reduction in the estimated inflation persistence parameter based on inflation data generated by shocks and policy responses.