This paper proposes a unilateral Adaptive update strategy based Interval Prediction (AIP) model for short-term load prediction, which is developed based on lower and upper bound estimation (LUBE) architecture. In traditional LUBE interval prediction model, the model training is usually trained by heuristic algorithms. In this paper, the model training is formulated as a bi-level optimization problem with the help of proposed unilateral adaptive update strategy and cost function. In lower-level problem, a simplified biased convex cost function is developed to supervise the learning direction of basic prediction engines. The basic prediction engine utilizes Gated Recurrent Unit (GRU) to extract features and Full Connected Neural Network (FNN) to generate interval boundary. In upper-level problem, a unilateral adaptive update strategy with unilateral coverage rate is put forward. It iteratively tunes hyper-parameters of cost function during training process. Comprehensive experiments based on residential load data are implemented and the proposed interval prediction model outperforms the tested state-of-the-art algorithms.