可行集上随机平滑的几何性质
The Geometry of Randomized Smoothing on Feasible Sets
- Technische Universität Wien（维也纳工业大学）
- NeverBlink（NeverBlink公司）
- Universitat Pompeu Fabra（庞培法布拉大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究探讨随机平滑在可行集上的几何性质，将过滤预测器的认证分解为几何与认证问题，提出联合概率与条件Rényi界等方法，在图像分类等场景中实现更优的鲁棒性认证。
AI中文摘要：
随机平滑在噪声中心移动时，将固定输出事件的概率作为认证依据。可行性或置信度过滤仅在保留的候选集中报告标签概率，从而产生一个比率。该比率的分子是固定的高斯事件质量，而分母是保留概率，且可能随中心变化。将此比率代入普通平滑公式，因此可以认证包含决策边界的球体。我们将该问题分解为几何问题和认证问题。几何问题决定条件化何时保持高斯比较。凸保留集保持完整比较，而一般集合需要随着中心移动对保留分布进行几何控制。若无此类控制，条件概率无法推出任何正的通用半径。联合保留与标签概率始终为同一过滤预测器提供有效证书。均匀协方差界可将散度证书转移到保留分布，并且即使在高斯事件比较失败时也能产生更大的半径。两种方法均具有有限样本界。对于具有训练选择非凸过滤器的学习图像分类器，条件Rényi界在不增加额外模型评估的情况下，比联合质量界认证更多图像。发布的置信度过滤器在条件替换获得的半径内表现出验证的标签变化。自适应高斯组合的应用在路径能量界下覆盖了具有历史依赖中心偏移的因果有限时域执行。
英文摘要：
Randomized smoothing certifies the probability of a fixed output event as the center of Gaussian noise moves. Feasibility or confidence filtering reports label probabilities only among retained proposals, producing a ratio. Its numerator is a fixed Gaussian event mass, while its denominator is the probability of retention and can change with the center. Substituting this ratio into the ordinary smoothing formula can therefore certify a ball that contains a decision boundary. We separate the problem into a geometric question and a certification question. Geometry determines when conditioning preserves Gaussian comparisons. Convex retained sets preserve the full comparison, while general sets require geometric control of the retained law as the center moves. Without such control, conditional probabilities imply no positive universal radius. Joint retention-and-label probabilities always yield a valid certificate for the same filtered predictor. A uniform covariance bound transfers divergence certificates to the retained law and can yield larger radii even when the Gaussian event comparison fails. Both methods admit finite-sample bounds. For a learned image classifier with a training-selected nonconvex filter, conditional Rényi bounds certify more images than joint-mass bounds without additional model evaluations. A released confidence filter exhibits verified label changes inside radii obtained by conditional substitution. An application of adaptive Gaussian composition covers causal finite-horizon executions with history-dependent center shifts under a pathwise energy bound.
