Learning Algorithms for Multiple Instance Regression

Multiple instance regression, introduced by Ray and Page [2001], is a generalisation of supervised regression in which the training data is available as a bag of feature-vectors (instances) and for each bag there is a bag-label which matches the label of one (unknown) primary instance from that bag. The goal is to compute a hypothesis regressor consistent with the underlying instance-labels. While most works on MIR focused on training models on such training data, computational learnability of MIR was only recently explored by Chauhan et al. [UAI 2024] who showed worst case intractability of properly learning *linear regressors* in MIR by showing a inapproximability bound. However, their work did not rule out efficient algorithms for this problem on natural distributions and randomly chosen labels. In this work we show that it is indeed possible to efficiently learn linear regressors in MIR when given access to random bags of uniformly randomly sampled primary instance chosen as the bag-label in which the feature vectors are independently sampled from Gaussian distributions. This is achieved by optimizing a certain bag-level loss which, via concentration bounds, yields a close approximation to the target linear regressor. Lastly, we show that the bag-level loss is also useful for learning general concepts (e.g. neural networks) in this setting: an optimizer of the loss on sampled bags is, w.h.p. a close approximation of a scaled version of the target function. We include experimental evaluations of our learning algorithms on synthetic and real-world datasets showing that our method outperforms the baseline MIR methods.