In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrix-tensor model. We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance from the signal/spike with the Kac-Rice formula, and locate trivialization of the landscape at large signal-to-noise ratios. We evaluate in a closed form the performance of a gradient flow algorithm using integro-differential PDEs as developed in physics of disordered systems for the Langevin dynamics. We analyze the performance of an approximate message passing algorithm estimating the maximum likelihood configuration via its state evolution. We conclude by comparing the above results: while we observe a drastic slow down of the gradient flow dynamics even in the region where the landscape is trivial, both the analyzed algorithms are shown to perform well even in the part of the region of parameters where spurious local minima are present.

本研究分析了梯度下降和其他算法在典型推理问题中对损失景观和性能之间的相互作用，并在spiked matrix-tensor模型上以负对数似然为损失函数来研究了大信噪比条件下的局部极小值数量及其所在位置。我们使用物理中无序系统的积分微分PDE分析了梯度流算法的性能，并分析了一种通过状态演化来估计极大似然配置的近似消息传递算法的性能。结果表明，尽管我们观察到景观的平凡化导致梯度流动力学明显放缓，但两种分析算法在存在虚假局部最小值的参数区域的部分区域中都表现良好。

Passed & Spurious: 基于下降算法的带尖峰矩阵-张量模型的局部极小点