Guided exploration with expert demonstrations improves data efficiency for reinforcement learning, but current algorithms often overuse expert information. We propose a novel algorithm to speed up Q-learning with the help of a limited amount of imperfect expert demonstrations. The algorithm avoids excessive reliance on expert data by relaxing the optimal expert assumption and gradually reducing the usage of uninformative expert data. Experimentally, we evaluate our approach on a sparse-reward chain environment and six more complicated Atari games with delayed rewards. We can achieve better results with the proposed methods than Deep Q-learning from Demonstrations (Hester et al., 2017) in most environments.