约束端到端学习中的软径向投影
Soft-Radial Projection for Constrained End-to-End Learning
AI总结：
本文提出软径向投影方法，通过径向映射解决约束端到端学习中的梯度饱和问题，保证严格可行性并提升优化性能。
摘要：
将硬约束整合到深度学习中对于安全关键系统至关重要。然而，现有的构造层将预测投影到约束边界上面临根本性瓶颈：梯度饱和。通过将外部点压缩到低维表面，标准正交投影导致秩 deficient 的雅可比矩阵，这会消除与活动约束正交的梯度并阻碍优化。我们引入了软径向投影，这是一种可微重参数化层，通过从欧几里得空间到可行集内部的径向映射来绕过这一问题。这种构造保证了严格可行性，同时在几乎 everywhere 保留满秩雅可比矩阵，从而防止了基于边界的方法典型的优化停滞。我们理论证明了该架构保留了通用逼近性质，并实证显示在最先进的优化和投影基线之上改进了收敛行为和解的质量。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
