This paper investigates the transmission design in the reconfigurable-intelligent-surface (RIS)-assisted downlink system. The channel state information (CSI) is usually difficult to be estimated at the base station (BS) when the RIS is not equipped with radio frequency chains. In this paper, we propose a downlink transmission framework with unknown CSI via Bayesian optimization. Since the CSI is not available at the BS, we treat the unknown objective function as the black-box function and take the beamformer, the phase shift, and the receiving filter as the input. Then the objective function is decomposed as the sum of low-dimension subfunctions to reduce the complexity. By re-expressing the power constraint of the BS in spherical coordinates, the original constraint problem is converted into an equivalent unconstrained problem. The users estimate the sum MSE of the training symbols as the objective value and feed it back to the BS. We assume a Gaussian prior of the feedback samples and the next query point is updated by minimizing the constructed acquisition function. Furthermore, this framework can also be applied to the power transfer system and fairness problems. Simulation results validate the effectiveness of the proposed transmission scheme in the downlink data transmission and power transfer.