Barcodes as Summary of Loss Function Topology
This provides insights into loss surface topology for neural network researchers, but it is incremental as it builds on existing topological methods.
The authors tackled the problem of understanding neural networks' loss surfaces by applying topological data analysis, specifically using barcodes of Morse complexes, and found that barcodes of local minima are concentrated in a small lower range of loss values and decrease with increased network depth and width.
We propose to study neural networks' loss surfaces by methods of topological data analysis. We suggest to apply barcodes of Morse complexes to explore topology of loss surfaces. An algorithm for calculations of the loss function's barcodes of local minima is described. We have conducted experiments for calculating barcodes of local minima for benchmark functions and for loss surfaces of small neural networks. Our experiments confirm our two principal observations for neural networks' loss surfaces. First, the barcodes of local minima are located in a small lower part of the range of values of neural networks' loss function. Secondly, increase of the neural network's depth and width lowers the barcodes of local minima. This has some natural implications for the neural network's learning and for its generalization properties.