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The advent of deep learning algorithms for mobile devices and sensors has led to a dramatic expansion in the availability and number of systems trained on a wide range of machine learning tasks, creating a host of opportunities and challenges in the realm of transfer learning. Currently, most transfer learning methods require some kind of control over the systems learned, either by enforcing constraints dur- ing the source training, or through the use of a joint optimization objective between tasks that requires all data be co-located for training. However, for practical, pri- vacy, or other reasons, in a variety of applications we may have no control over the individual source task training, nor access to source training samples. Instead we only have access to features pre-trained on such data as the output of “black-boxes.” For such scenarios, we consider the multi-source learning problem of training a classifier using an ensemble of pre-trained neural networks for a set of classes that have not been observed by any of the source networks, and for which we have very few training samples. We show that by using these distributed networks as feature extractors, we can train an effective classifier in a computationally-efficient manner using tools from (nonlinear) maximal correlation analysis. In particular, we develop a method we refer to as maximal correlation weighting (MCW) to build the required target classifier from an appropriate weighting of the feature functions from the source networks. We illustrate the effectiveness of the resulting classi- fier on datasets derived from the CIFAR-100, Stanford Dogs, and Tiny ImageNet datasets, and, in addition, use the methodology to characterize the relative value of different source tasks in learning a target task. 

In this study we use latent class analysis, distractor analysis, and qualitative analysis of cognitive interviews of student responses to questions on an algebra concept inventory, in order to generate theories about how students’ selections of specific answer choices may reflect different stages or types of algebraic conceptual understanding. Our analysis reveals three groups of students in elementary algebra courses, which we label as “mostly random guessing”, “some procedural fluency with key misconceptions”, and “procedural fluency with emergent conceptual understanding”. Student responses also revealed high rates of misconceptions that stem from misuse or misunderstanding of procedures, and whose prevalence often correlates with higher levels of procedural fluency.