Deep learning systems thrive on abundance of labeled training data but such data is not always available, calling for alternative methods of supervision. One such method is expectation regularization (XR) (Mann and McCallum, 2007), where models are trained based on expected label proportions. We propose a novel application of the XR framework for transfer learning between related tasks, where knowing the labels of task A provides an estimation of the label proportion of task B. We then use a model trained for A to label a large corpus, and use this corpus with an XR loss to train a model for task B. To make the XR framework applicable to large-scale deep-learning setups, we propose a stochastic batched approximation procedure. We demonstrate the approach on the task of Aspect-based Sentiment classification, where we effectively use a sentence-level sentiment predictor to train accurate aspect-based predictor. The method improves upon fully supervised neural system trained on aspect-level data, and is also cumulative with LM-based pretraining, as we demonstrate by improving a BERT-based Aspect-based Sentiment model.