Predicting dynamics using geometry & topology in random landscapes

In large random systems, certain behaviors are reliably predicted, like the energy density of the ground state. The long-time behavior of many physical and algorithmic dynamics is likewise predictable, through DMFT and related approaches. But can these behaviors be connected to static structures of the problem at hand, like its energy landscape? Recently, development of the Overlap Gap Property suggests that static topological properties can predict the performance of the best algorithms. Here, I will describe progress towards predicting the performance of the mediocre but simple algorithms we usually use. Virtually all approaches are doomed to find marginal (flat) minima, and I will show how to locate these. Then, I will describe some work on the topology of constant-energy slices of configuration space, which may explain the performance of gradient descent, the most mediocre algorithm of them all. ______________ JK-D, “Conditioning the complexity of random landscapes on marginal optima”, PRE 110, 064148 (2024) JK-D, “Algorithm-independent bounds on complex optimization through the statistics of marginal optima”, arXiv:2407.02092 [cond-mat.dis-nn] JK-D, “On the topology of solutions to random continuous constraint satisfaction problems”, arXiv:2409.12781 [cond-mat.dis-nn]