The increase in the deployed Internet-of-Things (IoT) devices has facilitated better functionality and connectivity across devices. Authentication of users on IoT devices plays a key role in the IoT networks to ensure security and integrity of the data. Multiple user authentication techniques including cryptographic and biometric approaches are introduced for authentication of users on these devices. Despite their effectiveness, these techniques incur large computational and communication overheads. In contrast, we propose a gait-based authentication, suitable for IoT devices in this work. Across multiple gait signals, we consider walking gait in this work, as it is unique to every individual and can be measured in an unobtrusive manner by utilizing the inertial sensors, which are inherently embedded in the IoT devices as well as smartphones. Given the limited resources available on IoT devices, we propose a lightweight authentication method that allows for early exit from the Neural Network (NN) in order to optimize the computational costs. A Deep Q-Learning Network (DQN) reinforcement learning method is further introduced to determine the exit dynamically during the authentication. The proposed method has been evaluated on the whuGAIT dataset. The proposed technique achieves more than 85% authentication accuracy with $6.94\times$ lower inference time and $5.9\times$ reduction in multiply-and-accumulate operations compared to ResNet50.