这篇论文用变分法啃下了结构化数据上感知器学习的难题，上下界只差参数优化顺序，数学上挺漂亮，推荐给喜欢理论推导的人。
这篇论文提出一种变分方法，研究在有限温度下基于高斯混合数据训练连续自旋感知器的模型。通过结合插值方法与对数凹性，作者导出了极限淬火压力的极小极大变分下界和上界。两个界仅在两个变分参数的优化顺序上不同，其余极值都由变分势的凹-凸结构控制。当两种优化可交换时，上下界匹配并确定模型解。该框架还给出了基态能量、训练损失和泛化误差的统一计算路径。
Variational Bounds for Perceptron Learning from Structured Data
We introduce a variational approach to a finite-temperature continuous-spin perceptron trained on a Gaussian mixture. The model allows for a broad class of concave utilities and log-concave separable prior measures on the spins. By combining the interpolation method with log-concavity and concentration estimates, we derive lower and upper minimax variational bounds for the limiting quenched pressure. Remarkably, the two bounds differ only in the order of optimization of two variational parameters, while all remaining extrema are controlled by the concave--convex structure of the variational potential. Whenever the two optimizations commute, the two bounds match and identify the solution of the model. The same potential yields the fixed-point equations as stationarity conditions and provides a unified route to the computation of the ground-state energy, training loss, and generalization error.