Soft-Radial Projection for Constrained End-to-End Learning
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911418510475264 |
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| author | Schneider, Philipp J. Kuhn, Daniel |
| author_facet | Schneider, Philipp J. Kuhn, Daniel |
| contents | Integrating hard constraints into deep learning is essential for safety-critical systems. Yet existing constructive layers that project predictions onto constraint boundaries face a fundamental bottleneck: gradient saturation. By collapsing exterior points onto lower-dimensional surfaces, standard orthogonal projections induce rank-deficient Jacobians, which nullify gradients orthogonal to active constraints and hinder optimization. We introduce Soft-Radial Projection, a differentiable reparameterization layer that circumvents this issue through a radial mapping from Euclidean space into the interior of the feasible set. This construction guarantees strict feasibility while preserving a full-rank Jacobian almost everywhere, thereby preventing the optimization stalls typical of boundary-based methods. We theoretically prove that the architecture retains the universal approximation property and empirically show improved convergence behavior and solution quality over state-of-the-art optimization- and projection-based baselines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Soft-Radial Projection for Constrained End-to-End Learning Schneider, Philipp J. Kuhn, Daniel Machine Learning Optimization and Control Computational Finance Integrating hard constraints into deep learning is essential for safety-critical systems. Yet existing constructive layers that project predictions onto constraint boundaries face a fundamental bottleneck: gradient saturation. By collapsing exterior points onto lower-dimensional surfaces, standard orthogonal projections induce rank-deficient Jacobians, which nullify gradients orthogonal to active constraints and hinder optimization. We introduce Soft-Radial Projection, a differentiable reparameterization layer that circumvents this issue through a radial mapping from Euclidean space into the interior of the feasible set. This construction guarantees strict feasibility while preserving a full-rank Jacobian almost everywhere, thereby preventing the optimization stalls typical of boundary-based methods. We theoretically prove that the architecture retains the universal approximation property and empirically show improved convergence behavior and solution quality over state-of-the-art optimization- and projection-based baselines. |
| title | Soft-Radial Projection for Constrained End-to-End Learning |
| topic | Machine Learning Optimization and Control Computational Finance |
| url | https://arxiv.org/abs/2602.03461 |