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Title: Empirical Analysis of the Hessian of Over-Parametrized Neural Networks

Paper content:
We study the properties of common loss surfaces through their Hessian matrix.
In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk.
We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et.
al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers.
We believe that our observations have striking implications for non-convex optimization in high dimensions.
First, the *flatness* of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading.
And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create *large* connected components at the bottom of the landscape.
Second, the dependence of a small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs.
With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light on the geometry of high-dimensional and non-convex spaces in modern applications.
In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.
In this paper, we study the geometry of the loss surface of supervised learning problems through the lens of their second order properties.
To introduce the framework, suppose we are given data in the form of input-label pairs, D = {(x i , y i )} N i=1 where x ∈ R d and y ∈ R that are sampled i.i.d. from a possibly unknown distribution ν, a model that is parametrized by w ∈ R M ; so that the number of examples is N and the number of parameters of the system is M .
Suppose also that there is a predictor f (w,
x).
The supervised learning process aims to solve for w so that f (w,
x) ≈ y.
To make the '≈' precise, we use a non-negative loss function that measures how close the predictor is to the true label, (f (w,
x),
y).
We wish to find a parameter w * such that w * = arg min L(w) where, DISPLAYFORM0 (f (w, x i ), y i ).In
particular, one is curious about the relationship between L(w) andL(w) := d(ν). By
the law of large numbers, at a given point w, L w →L w almost surely as N → ∞ for fixed M . However
in modern applications, especially in deep learning, the number of parameters M is comparable to the number of examples N (if not much larger). And the
behaviour of the two quantities may be drastically different (for a recent analysis on provable estimates see BID18 ).A classical
algorithm to find w * is gradient descent (GD), in which the optimization process is carried out using the gradient of L. A new parameter is found iteratively by taking a step in the direction of the negative gradient whose size is scaled with a constant step size η that is chosen from line-search minimization. Two problems
emerge: (1) Gradient computation can be expensive, (2) Line-search can be expensive. More involved
algorithms, such as Newton-type methods, make use of second-order information BID19 . Under sufficient
regularity conditions we may observe: L(w + ∆w) ≈ L(w) + ∆w∇L(w) + ∆w T ∇ 2 L(w)∆w. A third problem
emerges beyond an even more expansive computational cost of the Hessian: (3) Most methods require the Hessian to be non-degenerate to a certain extent.When the gradients are computationally expensive, one can alternatively use its stochastic version (SGD) that replaces the above gradient with the gradient of averages of losses over subsets (such a subset will be called the mini-batch) of D (see BID5 for a classical reference). The benefit of
SGD on real-life time limits is obvious, and GD may be impractical for practical purposes in many problems. In any case, the
stochastic gradient can be seen as an approximation to the true gradient, and hence it is important to understand how the two directions are related to one another. Therefore, the discussion
around the geometry of the loss surface can be enlightening in the comparison of the two algorithms: Does SGD locate solutions of a different nature than GD? Do they follow different
paths? If so, which one is better
in terms of generalization performance?For the second problem of expensive
line-search, there are two classical solutions: using a small, constant step size, or scheduling the step size according to a certain rule. In practice, in the context of deep
learning, the values for both approaches are determined heuristically, by trial and error. More involved optimal step size choices
involve some kind of second-order information that can be obtained from the Hessian of the loss function BID24 . From a computational point of view, obtaining
the Hessian is extremely expensive, however obtaining some of its largest and smallest eigenvalues and eigenvectors are not that expensive. Is it enough to know only those eigenvalues and
eigenvectors that are large in magnitude? How do they change through training? Would such
a method work in SGD as well as it would
on GD?For the third problem, let's look at the Hessian a little
closer. A critical point is defined by w such that ||∇L(w)|| = 0
and the nature of it can be determined by looking at the signs of its Hessian matrix. If all eigenvalues are positive the point is called a local
minimum, if r of them are negative and the rest are positive, then it is called a saddle point with index r. At the critical point, the eigenvectors indicate the directions
in which the value of the function locally changes. Moreover, the changes are proportional to the corresponding -signed-eigenvalue
. Under sufficient regularity conditions, it is rather straightforward to show
that gradient-based methods converge to points where the gradient is zero. Recently BID15 showed that they indeed converge to minimizers. However, a significant
and untested assumption to establish these convergence results
is that the Hessian of the loss is non-degenerate. A relaxation of the above convergence to the case of non-isolated critical points can
be found in BID20 . What about the critical points of machine learning loss functions? Do they satisfy the
non-degeneracy assumptions? If they don't, can we still apply the
results of provable theorems to gain intuition?
Finally, we revisit the issue through the lens of the following question: What does overparametrization imply on the discussion around GD vs. SGD (or large batch vs small batch) especially for their generalization properties?
In this final section, we will argue that, contrary to what is believed in BID13 and BID7 the two algorithms do not have to be falling into different basins.As noted in the introduction, for a while the common sense explanation on why SGD works well (in fact better) than GD (or large batch methods) was that the non-convex landscape had local minima at high energies which would trap large-batch or full-batch methods.
Something that SGD with small batch shouldn't suffer due to the inherent noise in the algorithm.
However, there are various experiments that have been carried out in the past that show that, for reasonable large systems, this is not the case.
For instance, BID22 demonstrate that a two hidden layer fully connected network on MNIST can be trained by GD to reach at the same level of loss values as SGD 1 .
In fact, when the step size is fixed to the same value for both of the algorithms, they reach the same loss value at the same number of iterations.
The training accuracy for both algorithms are the same, and the gap between test accuracies diminish as the size of the network increase with GD falling ever so slightly behind.
It is also shown in BID13 that training accuracies for both large and small batch methods are comparably good.
Furthermore, BID26 demonstrates that training landscape is easy to optimize even when there is no clear notion of generalization.
Such observations are consistent with our observations: over-parametrization (due to the architecture of the model) leads to flatness at the bottom of the landscape which is easy to optimize.When we turn our attention to generalization, BID13 note that LB methods find a basin that is different than the one found by SB methods, and they are characterized by how wide the basin is.
As noted in FIG0 , indeed the large eigenvalues are larger in LB than in SB, but is it enough to justify that they are in different basins, especially given the fact that the number of flat directions are enormous.
One of the most striking implications of flatness may be the connected structure of the solution space.
We may wonder whether two given solutions can be connected by a continuous path of solutions.
This question has been explored in a recent work: in BID9 it is shown that for one hidden layer rectified neural networks the solution space is connected which is consistent with the flatness of the landscape.
The classical notion of basins of attractions may not be the suitable objects to study for neural networks.
Rather, we may look at the exploration of interiors of level sets of the landscape.
We may be tempted to speculate that such an exploration may indeed result in point that generalizes better.
However, the flat space itself is very high dimensional which comes with its own computational issues.The training curve can be seen as composed of two parts: (1) high gain part where the norm of the gradients are large, (2) noise of the gradients is larger relative to the size of the stochastic gradients (see BID25 for a recent reference).
We speculate that the first part is relatively easy and even a large batch method can locate a large level set that contains points that generalize better than what's initially found.
From a practical point of view, using larger batches with larger step sizes can, in fact, accelerate training.
An example of this can be found in BID10 , where training Imagenet with a minibatch size of 8192 can match small batch performance.
On a final note for further consideration, we remark that we used standard pre-processing and initialization methods that are commonly used in practice.
Fixing these two aspects, we modified the data, model, and algorithm in order to study their relative effects.
However, the effects of pre-processing and initialization on the Hessian is highly non-trivial and deserves a separate attention.
We have shown that the level of the singularity of the Hessian cannot be ignored from theoretical considerations.
Furthermore, we use the generalized Gauss-Newton decomposition of the Hessian to argue the cluster of zero eigenvalues are to be expected in practical applications.
This allows us to reconsider the division between initial fast decay and final slow progress of training.
We see that even large batch methods are able to get to the same basin where small batch methods go.
As opposed to the common intuition, the observed generalization gap between the two is not due to small batch finding a different, better, wider basin.
Instead, the two solutions appear to be in the same basin.
This lack of a barrier between solutions is demonstrated by finding paths between the two points that lie in the same level set.
To conclude, we propose a major shift in perspective on considerations of the energy landscape in deep learning problems.

assistant: text: The loss surface is *very* degenerate, and there are no barriers between large batch and small batch solutions.