In this paper, we propose an acceleration technique of the adaptive filtering scheme called adaptive proximal forward-backward splitting method. For accelerating the convergence rate, the proposed method includes a step to shift the current estimate in the direction of the difference between the current and previous estimates based on the Fast Iterative Shrinkage/Thresholding Algorithm (FISTA). The computational complexity for this additional step is fairly low compared to the overall complexity of the algorithm. As an example of the proposed method, we derive an acceleration of the composition of the Adaptively Weighted Soft-Thresholding (AWST) operator and the exponentially weighted adaptive parallel projection. AWST shrinks the estimated filter coefficients to zero for exploiting the sparsity of the system to be estimated and the exponentially weighted adaptive parallel projection algorithm realizes high accuracy by utilizing all available information at each iteration. This accelerated method improves the steady-state mismatch drastically with its convergence speed as fast as the proportionate affine projection algorithm.