We study the problem of Polyadic Prediction, where the input consists of an ordered tuple of objects, and the goal is to predict a measurement associated with them. Many tasks can be naturally framed as Polyadic Prediction problems. In drug discovery, for instance, it is important to estimate the treatment effect of a drug on various tissue-specific diseases, as it is expressed over the available genes. Thus, we essentially predict the expression value measurements for several (drug, gene, tissue) triads. To tackle Polyadic Prediction problems, we propose a general framework, called Polyadic Regression, predicting measurements associated with multiple objects. Our framework is inductive, in the sense of enabling predictions for new objects, unseen during training. Our model is expressive, exploring high-order, polyadic interactions in an efficient manner. An alternating Proximal Gradient Descent procedure is proposed to fit our model. We perform an extensive evaluation using real-world chemogenomics data, where we illustrate the superior performance of Polyadic Regression over the prior art. Our method achieves an increase of 0:06 and 0:1 in Spearman correlation between the predicted and the actual measurement vectors, for predicting missing polyadic data and predicting polyadic data for new drugs, respectively.