This paper presents an Early Exit Neural Network (EENN) architecture, which enables budgeted classification by dynamically selecting the most relevant exit point for each input sample of a dataset to achieve the best performance while adhering to a pre-defined computational budget. The key contribution of this work is a novel method that jointly learns the classifier model and the sample exiting policy, in contrast to prior approaches that treated these components separately. Specifically, the paper introduces a bi-level optimization framework that simultaneously optimizes the cross-entropy loss of the classifier and the probabilities of each sample exiting at different stages of the network. This joint learning approach allows the classifier parameters and the sample-dependent exiting policy to be mutually optimized, leading to improved classification accuracy under computational constraints. The proposed EENN method is evaluated on three computer vision benchmarks - CIFAR-10, CIFAR-100, and ImageNet - and demonstrates state-of-the-art results in budgeted classification compared to existing early exit strategies. The code for this work will be made publicly available upon acceptance of the paper.