The insect olfactory system, which includes the antennal lobe (AL), mushroom body (MB), and ancillary structures, is a relatively simple neural system capable of learning. Its structural features, which are widespread in biological neural systems, process olfactory stimuli through a cascade of networks where large dimension shifts occur from stage to stage and where sparsity and randomness play a critical role in coding. Learning is partly enabled by a neuromodulatory reward mechanism of octopamine stimulation of the AL, whose increased activity induces rewiring of the MB through Hebbian plasticity. Enforced sparsity in the MB focuses Hebbian growth on neurons that are the most important for the representation of the learned odor. Based upon current biophysical knowledge, we have constructed an end-to-end computational model of the Manduca sexta moth olfactory system which includes the interaction of the AL and MB under octopamine stimulation. Our model is able to robustly learn new odors, and our simulations of integrate-and-fire neurons match the statistical features of in-vivo firing rate data. From a biological perspective, the model provides a valuable tool for examining the role of neuromodulators, like octopamine, in learning, and gives insight into critical interactions between sparsity, Hebbian growth, and stimulation during learning. Our simulations also inform predictions about structural details of the olfactory system that are not currently well-characterized. From a machine learning perspective, the model yields bio-inspired mechanisms that are potentially useful in constructing neural nets for rapid learning from very few samples. These mechanisms include high-noise layers, sparse layers as noise filters, and a biologically-plausible optimization method to train the network based on octopamine stimulation, sparse layers, and Hebbian growth.