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Title: Barcodes as summary of objective functions' topology

Paper content:
We apply canonical forms of gradient complexes (barcodes) to explore neural networks loss surfaces.
We present an algorithm for calculations of the objective function's barcodes of minima.
Our experiments confirm two principal observations: (1) the barcodes of minima are located in a small lower part of the range of values of objective function and (2) increase of the neural network's depth brings down the minima's barcodes.
This has natural implications for the neural network learning and the ability to generalize.
The learning via finding minima of objective functions is the principal strategy underlying majority of learning algorithms.
For example, in Neural Network training, the objective function's input is model parameters (weights) and the objective function's output is the loss on training dataset.
The graph of the loss function, often called loss surface, typically has complex structure (e.g. see loss surface visualisations by Li et al. (2018) ): non-convexity, many local minima, flat regions, steep slopes.
These obstacles harm exploration of the loss surface and complicate searching for optimal network weights.
The optimization of modern neural networks is based on the gradient descent algorithm.
The global topological characteristics of the gradient vector field trajectories are captured by the Morse complex via decomposing the parameter space into cells of uniform flow, see Barannikov (1994) ; Le Roux et al. (2018) and references therein.
The invariants of Morse complex called "canonical forms"(or barcodes) constitute the fundamental summary of the topology of the gradient vector field flow.
The "canonical forms", or barcodes, in this context are decompositions of the change of topology of the sublevel sets of objective function into simple "birth-death" phenomena of topological feautures of different dimensions.
The calculation of the barcodes for different functions constitutes the essence of the topological data analysis.
The currently available software packages for the calculation of barcodes of functions, also called "sublevel persistence", are GUDHI, Dionysus, PHAT, and TDA package which incorporates all three previous packages B.T. Fasy et al. (2014) .
They are based on the algorithm, described in Barannikov (1994) , see also appendix and e.g. Bauer et al. (2014) and references therein.
This algorithm which has complexity of O(n 3 ).
These packages can currently handle calculations of barcodes for functions defined on a grid of up to 10 6 points, and in dimensions two and three.
Thus all current packages have the scalability issues.
We describe a new algorithm for computations of the barcodes of functions in lowest degree.
Our algorithm works with functions defined on randomly sampled or specifically chosen point clouds.
Point cloud based methods are known to work better than grid based methods in optimization related problems (Bergstra and Bengio (2012) ).
We also use the fact that the definition of the barcode of lowest degree can be reformulated in geometrical terms (see definition 1 in section 2).
The previously known algorithms were based on the more algebraic approach as in definition 3.
Our algorithm has complexity of O(n log(n)).
It was tested in dimensions up to 16 and with number of points of up to 10 8 .
In this work, we develop a methodology to describe the properties of the loss surface of the neural network via topological features of local minima.
We emphasize that the value of the objective function at the minimum can be viewed as only a part of its topological characteristic from the "canonical form" (barcode).
The second half can be described as the value of objective function at the index-one saddle, which can be naturally associated with each local minimum.
The difference between the values of objective function at the associated index-one saddle and at the local minimum is a topological invariant of the minimum.
For optimization algorithms this quantity measures, in particular, the obligatory penalty for moving from the given local minimum to a lower minimum.
The main contributions of the paper are as follows:
Applying the one-to-one correspondence between local minima and 1-saddles to exploration of loss surfaces.
For each local minimum p there is canonically defined 1-saddle q (see Section 2).
The 1-saddle associated with p can be described as follows.
The 1-saddle q is precisely the point where the connected component of the sublevel set Θ f ≤c = {θ ∈ Θ | f (θ) ≤ c} containing the minimum p merges with another connected component of the sublevel set whose minimum is lower.
This correspondence between the local minima and the 1-saddles, killing a connected component of Θ f ≤c , is one-to-one.
The segment [f
(p), f
(q)] is then the "canonical form" invariant attached to the minimum p.
The set of all such segments is the barcode ("canonical form") of minima invariant of f .
It is a robust topological invariant of objective function.
It is invariant in particular under the action of homeomorphisms of Θ.
Full "canonical form" invariants give a concise summary of the topology of objective function and of the global structure of its gradient flow.
Algorithm for calculations of the barcodes (canonical invariants) of minima.
We describe an algorithm for calculation of the canonical invariants of minima.
The algorithm works with function's values on a a randomly sampled or specifically chosen set of points.
The local minima give birth to clusters of points in sublevel sets.
The algorithm works by looking at neighbors of each point with lower value of the function and deciding if this point belongs to the existing clusters, gives birth to a new cluster (minimum), or merges two or more clusters (index one saddle).
A variant of the algorithm has complexity of O(n log(n)), where n is the cardinality of the set of points.
Calculations confirming observations on behaviour of neural networks loss functions barcodes.
We calculate the canonical invariants (barcodes) of minima for small fully-connected neural networks of up to three hidden layers and verify that all segments of minima's barcode belong to a small lower part of the total range of loss function's values and that with the increase in the neural network depth the minima's barcodes descend lower.
The usefulness of our approach and algorithms is clearly not limited to the optimization problems.
Our algorithm permits really fast computation of the canonical form invariants (persistence barcodes) of many functions which were not accessible until now.
These sublevel persistence barcodes have been successfully applied in different discipline, to mention just a few: cognitive science (M.
K. Chung and Kim (2009) ), cosmology (Sousbie et al. (2011) ), see e.g. Pun et al. (2018) and references therein.
Our viewpoint should also have applications in chemistry and material science where 1-saddle points on potential energy landscapes correspond to transition states and minima are stable states corresponding to different materials or protein foldings (see e.g. Dellago et al. (2003) , Oganov and Valle (2009) ).
The article is structured as follows.
First we describe three definitions of barcodes of minima.
After that our algorithm for their calculation is described.
In the last part we give examples of calculations, including the loss functions of simple neural nets.
In this work we have introduced a methodology for analysing the plots of functions, in particular, loss surfaces of neural networks.
The methodology is based on computing topological invariants called canonical forms or barcodes.
To compute barcodes we used a graph-based construction which approximates the function plot.
Then we apply the algorithm we developed to compute the barcodes of minima on the graph.
Our experimental results of computing barcodes for small neural networks lead to two principal observations.
First all barcodes sit in a tiny lower part of the total function's range.
Secondly, with increase of the depth of neural network the barcodes descend lower.
From the practical point of view, this means that gradient descent optimization cannot stuck in high local minima, and it is also not difficult to get from one local minimum to another (with smaller value) during learning.
The method we developed has several further research directions.
Although we tested the method on small neural networks, it is possible to apply it to large-scale modern neural networks such as convolutional networks (i.e. ResNet, VGG, AlexNet, U-Net, see Alom et al. (2018) ) for imageprocessing based tasks.
However, in this case the graph-based approximation we use requires wise choice of representative graph vertices, which is a hardcore in high-dimensional spaces (dense filling of area by points is computationally intractable).
Another direction is to study the connections between the barcode of local minima and the generalization properties of given minimum and of neural network.
There are clearly also connections, deserving further investigation, between the barcodes of minima and results concerning the rate of convergency during learning of neural networks.

assistant: text: We apply canonical forms of gradient complexes (barcodes) to explore neural networks loss surfaces.