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Emerging research in computer graphics, inverse problems, and machine learning requires us to differentiate and optimize parametric discontinuities. These discontinuities appear in object boundaries, occlusion, contact, and sudden change over time. In many domains, such as rendering and physics simulation, we differentiate the parameters of models that are expressed as integrals over discontinuous functions. Ignoring the discontinuities during differentiation often has a significant impact on the optimization process. Previous approaches either apply specialized hand-derived solutions, smooth out the discontinuities, or rely on incorrect automatic differentiation. We propose a systematic approach to differentiating integrals with discontinuous integrands, by developing a new differentiable programming language. We introduce integration as a language primitive and account for the Dirac delta contribution from differentiating parametric discontinuities in the integrand. We formally define the language semantics and prove the correctness and closure under the differentiation, allowing the generation of gradients and higher-order derivatives. We also build a system, Teg, implementing these semantics. Our approach is widely applicable to a variety of tasks, including image stylization, fitting shader parameters, trajectory optimization, and optimizing physical designs.
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Designing Discontinuities
Discontinuities can be fairly arbitrary but also cause a significant impact on outcomes in social systems. Indeed, their arbitrariness is why they have been used to infer causal relationships among variables in numerous settings. Regression discontinuity from econometrics assumes the existence of a discontinuous variable that splits the population into distinct partitions to estimate causal effects. Here we consider the design of partitions for a given discontinuous variable to optimize a certain effect. To do so, we propose a quantization-theoretic approach to optimize the effect of interest, first learning the causal effect size of a given discontinuous variable and then applying dynamic programming for optimal quantization design of discontinuities that balance the gain and loss in the effect size. We also develop a computationally-efficient reinforcement learning algorithm for the dynamic programming formulation of optimal quantization. We demonstrate our approach by designing optimal time zone borders for counterfactuals of social capital.
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- Award ID(s):
- 2033900
- PAR ID:
- 10440937
- Date Published:
- Journal Name:
- Neural Compression Workshop (ICML 2023)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
