这篇论文把神经网络解空间的OGP理论从零温度推广到了有限温度，发现允许一点训练误差就能让算法在原来计算困难的区域找到好解，而且泛化也不差。
论文将零温度下的重叠间隙性质(OGP)扩展到有限温度，允许训练误差存在并统计惩罚。首先证明零温度平衡测度中占主导的冻结核1步复制对称破缺解在任意有限温度下依然存活。然后基于决策边界附近单模式吉布斯权重的平滑性，导出准则判断有限温度松弛何时消除冻结。进一步将OGP推广到有限温度，显示算法可访问的稠密有限能量区域在OGP临界密度α_OGP之上存在，直到随训练误差ε增长的阈值α_OGP(ε)。在师生设置中，这些宽有限能量区域仍保持良好泛化，且有限能量消息传递算法表明热噪声能在恢复教师和零温度解均困难的密度区域实现有效泛化。
On the robustness of noisy solutions in non-convex neural networks
Optimization in non-convex neural network models is strongly influenced by the geometry of the solution space: sparse, isolated, point-like clusters are typically algorithmically inaccessible, whereas wide and flat regions can be found efficiently despite being relatively rare. At zero temperature this picture has been formalized in binary perceptrons through the overlap gap property (OGP), which limits algorithmic access to configurations with zero training error above a critical constraint density $α_{\rm OGP}$. Here we extend this description to finite temperature, where a positive training error is allowed and statistically penalized. We first show that the frozen one-step replica-symmetry-breaking solution, dominating the zero temperature equilibrium measure, survives at any finite temperature. We furthermore derive a general criterion, based on the smoothness of the single-pattern Gibbs weight near the decision boundary, that determines when a finite-temperature relaxation of the loss removes freezing. We then extend the OGP construction to finite temperature and show that dense, algorithmically accessible regions of finite-energy configurations persist beyond $α_{\rm OGP}$, up to a threshold $α_{\rm OGP}(ε)$ that grows with the allowed training error $ε$. Finally, in the teacher-student setting, we show that these wide, finite-energy regions still retain good generalization. Using a finite energy message-passing algorithm, we demonstrate numerically that thermal noise enables effective generalization in the regime of constraint densities where both recovering the teacher and finding a zero temperature solution are computationally hard.