Many probabilistic modeling problems in machine learning use gradient-based optimization in which the objective takes the form of an expectation. These problems can be challenging when the parameters to be optimized determine the probability distribution under which the expectation is being taken, as the na\"ive Monte Carlo procedure is not differentiable. Reparameterization gradients make it possible to efficiently perform optimization of these Monte Carlo objectives by transforming the expectation to be differentiable, but the approach is typically limited to distributions with simple forms and tractable normalization constants. Here we describe how to differentiate samples from slice sampling to compute \textit{slice sampling reparameterization gradients}, enabling a richer class of Monte Carlo objective functions to be optimized. Slice sampling is a Markov chain Monte Carlo algorithm for simulating samples from probability distributions; it only requires a density function that can be evaluated point-wise up to a normalization constant, making it applicable to a variety of inference problems and unnormalized models. Our approach is based on the observation that when the slice endpoints are known, the sampling path is a deterministic and differentiable function of the pseudo-random variables, since the algorithm is rejection-free. We evaluate the method on synthetic examples and apply it to a variety of applications with reparameterization of unnormalized probability distributions.
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Differentiable Spline Approximations
The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically require that the machine learning models be differentiable, limiting their applicability. Our goal in this paper is to use a new, principled approach to extend gradient-based optimization to functions well modeled by splines, which encompass a large family of piecewise polynomial models. We derive the form of the (weak) Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. Overall, we show that leveraging this redesigned Jacobian in the form of a differentiable" layer''in predictive models leads to improved performance in diverse applications such as image segmentation, 3D point cloud reconstruction, and finite element analysis. We also open-source the code at\url {https://github. com/idealab-isu/DSA}.
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- Award ID(s):
- 2053760
- NSF-PAR ID:
- 10344688
- Date Published:
- Journal Name:
- Advances in neural information processing systems
- Volume:
- 34
- ISSN:
- 1049-5258
- Page Range / eLocation ID:
- 20270-20282
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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