Many real-world edge applications including object detection, robotics, and smart health are enabled by deploying deep neural networks (DNNs) on energy-constrained mobile platforms. In this article, we propose a novel approach to trade off energy and accuracy of inference at runtime using a design space called Learning Energy Accuracy Tradeoff Networks (LEANets). The key idea behind LEANets is to design classifiers of increasing complexity using pretrained DNNs to perform input-specific adaptive inference. The accuracy and energy consumption of the adaptive inference scheme depends on a set of thresholds, one for each classifier. To determine the set of threshold vectors to achieve different energy and accuracy tradeoffs, we propose a novel multiobjective optimization approach. We can select the appropriate threshold vector at runtime based on the desired tradeoff. We perform experiments on multiple pretrained DNNs including ConvNet, VGG-16, and MobileNet using diverse image classification datasets. Our results show that we get up to a 50% gain in energy for negligible loss in accuracy, and optimized LEANets achieve significantly better energy and accuracy tradeoff when compared to a state-of-the-art method referred to as Slimmable neural networks.