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Inferring programs which generate 2D and 3D shapes is important for reverse engineering, editing, and more. Training models to perform this task is complicated because paired (shape, program) data is not readily available for many domains, making exact supervised learning infeasible. However, it is possible to get paired data by compromising the accuracy of either the assigned program labels or the shape distribution. Wake-sleep methods use samples from a generative model of shape programs to approximate the distribution of real shapes. In self-training, shapes are passed through a recognition model, which predicts programs that are treated as "pseudo-labels" for those shapes. Related to these approaches, we introduce a novel self-training variant unique to program inference, where program pseudo-labels are paired with their executed output shapes, avoiding label mismatch at the cost of an approximate shape distribution. We propose to group these regimes under a single conceptual framework, where training is performed with maximum likelihood updates sourced from either Pseudo-Labels or an Approximate Distribution (PLAD). We evaluate these techniques on multiple 2D and 3D shape program inference domains. Compared with policy gradient reinforcement learning, we show that PLAD techniques infer more accurate shape programs and converge significantly faster. Finally, we propose to combine updates from different PLAD methods within the training of a single model, and find that this approach outperforms any individual technique.