Budget constrained optimal design of experiments is a well studied problem. Although the literature is very mature, not many strategies are available when these design problems appear in the context of sparse linear models commonly encountered in high dimensional machine learning. In this work, we study this budget constrained design where the underlying regression model involves a ℓ1-regularized linear function. We propose two novel strategies: the first is motivated geometrically whereas the second is algebraic in nature. We obtain tractable algorithms for this problem which also hold for a more general class of sparse linear models. We perform a detailed set of experiments, on benchmarks and a large neuroimaging study, showing that the proposed models are effective in practice. The latter experiment suggests that these ideas may play a small role in informing enrollment strategies for similar scientific studies in the future.