# Glassy dynamics in deep neural networks: A structural comparison

Max Kerr Winter  Liesbeth M. C. Janssen  Eindhoven University of Technology

###### Abstract

Deep Neural Networks (DNNs) share important similarities with structural
glasses. Both have many degrees of freedom, and their dynamics are governed by
a high-dimensional, non-convex landscape representing either the loss or
energy, respectively. Furthermore, both experience gradient descent dynamics
subject to noise. In this work we investigate, by performing quantitative
measurements on realistic networks trained on the MNIST and CIFAR-10 datasets,
the extent to which this qualitative similarity gives rise to glass-like
dynamics in neural networks. We demonstrate the existence of a Topology
Trivialisation Transition as well as the previously studied under-to-
overparameterised transition analogous to jamming. By training DNNs with
overdamped Langevin dynamics in the resulting disordered phases, we do not
observe diverging relaxation times at non-zero temperature, nor do we observe
any caging effects, in contrast to glass phenomenology. However, the weight
overlap function follows a power law in time, with exponent $\approx-0.5$, in
agreement with the Mode-Coupling Theory of structural glasses. In addition,
the DNN dynamics obey a form of time-temperature superposition. Finally,
dynamic heterogeneity and ageing are observed at low temperatures. These
results highlight important and surprising points of both difference and
agreement between the behaviour of DNNs and structural glasses.

††preprint: APS/123-QED

##  I Introduction

Deep neural networks (DNNs) have experienced an explosion of popularity in the
last decade as they have proven to be highly effective at tackling a wide
range of problems across the natural sciences and engineering. Consequently,
there is significant interest from the statistical physics community in
understanding the fundamental mechanisms by which neural networks train and
generalise [1, 2, 3, 4, 5]. An important strand of this work is the similarity
between DNNs and glasses. The link between spin glasses and neural networks
has received considerable attention since the 1980s, with an emphasis on the
mapping between the Hopfield network and Ising-like systems [6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17]. More recently, the authors of [18] identified
both points of similarity and difference between the mean square displacement
of a spin glass and a variety of feed forward networks, trained with
Stochastic Gradient Descent (SGD). However, the possible analogy between
neural networks and structural glasses, which exhibit a richer phenomenology
than spin glasses, is still largely unexplored. Moreover, since modern deep
learning comprises a wide variety of (typically feed forward, as opposed to
Hopfield) architectures, one may also wonder whether this analogy might be
extended to feed forward DNNs with more complex interactions between
parameters.

Structural glasses exhibit a wide variety of typical phenomena (growing and
ultimately diverging relaxation times, caging, dynamic heterogeneity,
violation of the Stokes-Einstein relation, ageing) that have been the subject
of intense research for many decades, resulting in a rich body of theory of
the kind that is currently lacking in deep learning [19, 20, 21, 22].
Structural glasses are a potentially valuable analogy for DNNs as they involve
complicated interaction potentials between the constituent particles. This is
reminiscent of the situation in DNNs where parameters also interact via highly
non-linear, indirect couplings. Furthermore, recent studies have made a
quantitative comparison between the jamming transition (a phenomenon closely
related to the structural glass transition [23]) and an under-to-
overparameterised transition (UOT) in DNNs [24, 25, 26, 27]. Large networks
are able to fit their training data exactly, resulting in zero loss
(equivalent to an unjammed state with zero interaction energy), whereas small
networks have a non-zero loss similar to the positive interaction energy of
jammed particles. Much like the energy landscape of an unjammed system, the
loss landscape in the long-time limit of overparameterised networks has been
shown to consist of a large number of flat directions [18]. These recent
results raise the intriguing possibility, explored in this work, of whether
similar connections can be drawn between the dynamics of glass forming
particle systems and DNNs.

As well as the analogy with jamming, results derived from high-dimensional
random energy landscapes [28] are also relevant to the statistical physics of
deep learning. Random energy landscapes, which are in some respects highly
idealised models of glasses, have been widely applied in physics [29, 30, 31]
and are now increasingly being used to study DNNs and related machine learning
systems [32, 33, 34]. Of particular relevance to the current work is the
Topology Trivialisation Transition (TTT), where the energy landscape of a
system switches from a single minimum to many minima on decreasing a quadratic
energy penalty. The existence of TTT-like transitions has been demonstrated in
high-dimensional elastic models subject to random fields [35, 36], in large
ecosystems [37], and in random neural networks [32, 13]. These diverse systems
all share a common structure with feed forward DNNs subject to regularisation
in that their dynamics balance a disordered term with a quadratic potential.
It has remained unexplored, to the best of the authors’ knowledge, whether
regularised feed forward DNNs undergo a TTT (and in fact, we find that they
do.)

The goal of this work is to quantitatively assess the similarities between
DNNs and structural glass-forming materials by performing comparisons across a
wide range of numerical experiments typical of glassy formers. Do DNNs undergo
glass-like transitions? Do they exhibit the typical glassy behaviours of
diverging relaxation times, caging, violation of the Stokes-Einstein relation,
dynamic heterogeneity, and ageing? Answering these questions is an important
first step in guiding the application of the theory of structural glasses to
the field of deep learning.

##  II  Numerical setup

In this work we use fully connected feedforward networks with no biases, only
weights, so that all parameters are of the same type. The DNNs are denoted
$f(\mathbf{x},\mathbf{W})$ for weight vector $\mathbf{W}$, of length $N$, and
input data $\mathbf{x}$. The activation functions are ReLU,
$\sigma(z)=\text{max}(0,z)$. We consider a problem of supervised binary
classification, with a training set
$\mathcal{T}=\\{\mathbf{x}_{i},y_{i}\\}_{i=1}^{P_{D}}$, of size $P_{D}$,
consisting of input examples $\mathbf{x}$ of dimension $d$, and corresponding
labels $y\in\\{+1,-1\\}$.

Our primary dataset is a preprocessed version of MNIST [38], consisting of
$P_{D}=6\times 10^{4}$ images of hand-drawn digits from 0–9, with
corresponding labels. We binarise the classification problem by assigning a
label of +1 to all even digits, and -1 to all odd digits. Furthermore, we use
Principal Component Analysis (PCA) to reduce the dimension of the input data
to $d=10$ [39], in order to decrease the number of weights in the first layer.
Hence all networks have an input layer of width 10 and an output of width 1.
The networks have a fixed depth of 6 hidden layers, all with a constant width.
This width is varied to alter the size of the networks, resulting in a total
number of weights between 4830 (a width of 30) and 34,522 (a width of 82). We
also repeat our analysis with the CIFAR-10 dataset (see Appendix E).

The choice of using the full MNIST and CIFAR-10 datasets, which consequently
require large DNNs to reach the overparameterised state, was made to maximise
the realism of our investigation, thus ensuring that our numerical experiments
reflect the true complexity of modern deep learning applications. However,
this complexity comes with a significant computational cost. Compounding this
problem, glassy measurements often require the networks to be trained for a
long time, and averaged over many repeats. Approximately ten thousand networks
were involved in this study, many of which were trained for over a hundred CPU
hours.

As our loss function for the training of our DNNs, we use the quadratic hinge
loss,

$\displaystyle\mathcal{L}_{\text{data}}=\frac{1}{P_{D}}\sum_{i=1}^{P_{D}}\ell(f(\mathbf{x}_{i},\mathbf{W}),y_{i}),$
(1)

where $\ell(f,y_{i})=\frac{1}{2}\text{max}(0,\Delta(y_{i},f))^{2}$, and
$\Delta(y,f)=1-yf$. This loss was chosen following the example of [24], as it
is similar to the energy of a system of particles interacting via a finite-
range harmonic potential. When the particles are separated beyond the cut-off
distance, all pairwise interactions are 0, and the total energy is 0. When the
particles are close together they repel each other and the energy is a sum of
quadratic terms. Analogously, when all data is fit exactly,
$\ell(f(\mathbf{x}_{i},\mathbf{W}),y_{i})=0,\>\forall i$, hence
$\mathcal{L}_{\text{data}}=0$, whereas when this is not the case, the loss is
an average of quadratic terms. However, the loss is quadratic in
$f(\mathbf{x},\mathbf{W})$, a highly non-linear function of the weights
(which, in our analogy to structural glasses, may be interpreted as the
equivalent of particle positions, i.e. the degrees of freedom.)

We also use L2-regularisation, a common method to avoid overfitting [40]. The
resulting loss function is

$\displaystyle\mathcal{L}=\mathcal{L}_{\text{data}}+\frac{\lambda}{2}|\mathbf{W}|^{2},$
(2)

where $\lambda>0$ is a control parameter. When studying dynamic variables, we
train the DNNs with overdamped Langevin dynamics (also known as Stochastic
Gradient Langevin Dynamics, SGLD, in the deep learning literature),

$\displaystyle\partial_{t}w_{i}=-\mu\frac{\partial\mathcal{L}}{\partial
w_{i}}+\xi_{i},$ (3)

where $w_{i}$ is a weight, $\mu$ is the learning rate (set to $10^{-3}$ unless
otherwise stated), and the noise term obeys $\langle\xi_{i}\rangle=0$, and
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=(2k_{B}T\mu/P_{D})\delta_{ij}\delta(t-t^{\prime})$,
for temperature $T$ and Boltzmann constant $k_{B}$. For the remainder of the
paper we work in natural units where $k_{B}=1$. In contrast to physical
systems a factor of $1/P_{D}$, the size of the dataset, appears in the noise
correlation due to $\mathcal{L}$ being intensive with respect to the dataset
size. See Appendix A for a more detailed discussion. Equation (2) highlights
the similarity between DNNs and the disordered elastic model of [28], which
undergoes a TTT. The Hamiltonian of the elastic system is
$\mathcal{H}(x_{i})=V(x_{i})+\frac{\eta}{2}|\mathbf{x}|^{2}$, where $V$ is a
random potential, and the second term describes the elasticity of the medium.
This is also similar to the Hamiltonian of spherical spin glasses [41]. The
question we wish to address is, in essence, how much glassy behaviour remains
when we substitute $\mathcal{H}$ for $\mathcal{L}$.

We will now consider various aspects of glassy dynamics, each of which
requires a different protocol to potentially probe it in DNNs. We will discuss
both the protocol and the results, starting with the phase diagram in order to
identify the transition lines. We will then address relaxation dynamics in the
resulting disordered phases, time-temperature superposition, caging and the
mean square displacement, the Stokes-Einstein relation, non-Gaussianity,
dynamic heterogeneity, and finally end with aging. Concluding comments are
given in Sec. IV.

##  III  Results

###  III.1 Phase diagram

We first seek to establish a phase diagram to identify two relevant transition
lines for our DNNs. The first is a transition in the DNN loss as a function of
the ratio of network size, $N$, to dataset size, $P_{D}$, from
underparameterised ($\mathcal{L}>0$) to overparameterised ($\mathcal{L}\approx
0$). This transition is analogous to the jamming transition of interacting
particles, and was discovered by the authors of [24]. The second transition,
the Topology Trivialisation Transition of [28], separates a regime with a
single energy minimum, from a disordered energy landscape with many minima. As
discussed below, we find the same under-to-overparameterised/jamming
transition, and show that an analogous TTT exists in DNNs as a function of the
regularisation strength, $\lambda$.

In order to construct the phase diagram, we train DNNs with Stochastic
Gradient Descent in the $(N,\lambda)$ plane, for 7000 epochs with a batch size
of 100. We then perform 50 epochs of gradient descent at the original learning
rate, and a further 50 epochs of gradient descent after reducing the learning
rate by a factor of 10. Convergence was checked manually on a representative
subset of DNNs. As such we are sampling (close to) the minima of the
landscape. Finally we average over 10 DNNs with different, random initial
conditions at each point in the $(N,\lambda)$ plane, resulting in the phase
diagram shown in Fig. 1(a). We use SGD instead of Langevin dynamics as we are
studying static properties of the landscape, not the dynamics, and SGD is much
faster than Eq. (3).

The UOT is visible on the $\lambda=0$ line, whereas the TTT occurs in the
vertical direction. The transitions are shown separately in Fig. 1(b) (UOT)
and (c) (TTT). Figure 1(b) uses the same networks as on the $\lambda=0$ line
from Fig. 1(a), and shows that achieving the $\mathcal{L}\approx 0$ state is
by no means guaranteed. It is strongly dependent on the initial conditions and
training trajectory through weight space, even in the overparameterised phase,
in agreement with [24]. On either side of the UOT we have two disordered
phases, underparameterised, and overparameterised, illustrated schematically
in Fig. 1(a). The key difference between these phases is the presence of flat
plateaus at zero loss in the overparameterised phase. The UOT does not exist
when $\lambda>0$ as the addition of the regularisation term destroys the
plateaus. This is similar to the case where the addition of a spherical
constraint makes a system of equations unsolvable [42]. Figure 1(c) uses 20
DNNs of width 30, each with different initial conditions, to demonstrate the
TTT. Each DNN undergoes a sharp TTT, but the position of the transition
varies, resulting in an average with some width. In this transition region a
small number of initial conditions find a long-lived, high loss minimum,
resulting in a kink in the average curve. The TTT separates a trivial phase,
where the landscape is essentially a quadratic bowl (high $\lambda$), from a
disordered phase (low $\lambda$).

It is possible to make a heuristic argument about how regularisation affects
the number of minima in the loss landscape. Turning points in $\mathcal{L}$
can be classified by the eigenvalues of the Hessian,
$H_{ij}=\partial^{2}\mathcal{L}/\partial w_{i}\partial w_{j}$, for weights
$w_{i}$ and $w_{j}$. A turning point with only positive eigenvalues is a
minimum, only negative eigenvalues a maximum, and anything in between a saddle
point. $H$ is essentially a random matrix, with some distribution of
eigenvalues. Adding L2 regularisation, as in Eq. (2), transforms the Hessian
to $H_{ij}=\partial^{2}\mathcal{L}_{\text{data}}/\partial w_{i}\partial
w_{j}+\lambda\delta_{ij}$, with corresponding characteristic polynomial,
$\text{det}(H-(\sigma-\lambda)I)=0$, for eigenvalues $\sigma$, and identity
matrix, $I$. Hence the regularisation term, $\lambda$, shifts the eigenvalue
spectrum towards positive values, and makes minima more likely.

Figure 1: Glass-like transitions in neural networks. (a) The final training
loss (colour bar) in the $(N/P_{D},\lambda)$ plane, where $N$ is the number of
weights, $P_{D}$ the number of training data, and $\lambda$ the regularisation
strength. Schematic illustrations of the loss landscape in the resulting four
phases are shown around the edge. At high $\lambda$ the quadratic L2
regularisation term of Eq. (2) dominates and the loss is a simple well. As
$\lambda$ decreases the loss landscape undergoes a TTT and develops many
random minima. On the $\lambda=0$ line, at low $N/P_{D}$ the network is
underparameterised, corresponding to another disordered landscape. Finally, at
high $N/P_{D}$ the network is over parameterised and has a loss landscape
combining disordered regions with flat plateaus at zero loss. (b) The final
training loss either side of the UOT. Each black point is a different initial
condition. (c) The final training loss on either side of the TTT. The
translucent lines correspond to different initial conditions, and the blue
line is their average.

###  III.2 Relaxation with overdamped Langevin dynamics

We now turn to the time-dependent dynamics of our DNNs to explore potential
analogies with the relaxation dynamics of structural glassformers. There are
four regions of interest in our phase diagram, on either side of each
transition: the regularised phase ($0<\lambda<\lambda^{\ast}$); the quadratic
phase ($\lambda>\lambda^{\ast}$); the underparameterised phase ($\lambda=0$,
$N<N^{\ast}$); and the overparameterised phase ($\lambda=0$, $N>N^{\ast}$). In
practice, the quadratic phase is trivial as networks rapidly reach steady
state at the bottom of the quadratic well ($w_{i}\approx 0$ $\forall i$),
hence we do not discuss this phase further. As can be seen from Fig. 1(a),
$\lambda^{\ast}\approx 0.03$, with some $N$ dependence, and $N^{\ast}\approx
0.18$.

We first measure the dynamics of a network by defining the overlap correlation
function,

$\displaystyle
Q(t,t_{w})=\left\langle\frac{1}{N}\sum_{i=1}^{N}\left[1-\Theta\left(|w_{i}(t_{w}+t)-w_{i}(t_{w})|-\epsilon\right)\right]\right\rangle,$
(4)

where $t_{w}$ is the waiting time between the system being prepared and the
start of the measurement of $Q$, $\Theta$ is the Heaviside function,
$\epsilon>0$ is a sensitivity parameter, and $\langle\cdot\rangle$ denotes
averaging over both initial conditions and noise realisations. $Q(0,t_{w})=1$,
and as the weights of a network decorrelate from their initial value $Q$
decreases. For physical systems there will typically be a relevant length
scale (e.g. the particle diameter) that determines $\epsilon$. No such scale
exists for DNN weights, and so the choice of $\epsilon$ is somewhat arbitrary.
We define $\epsilon$ with respect to a new control parameter, $\chi$, and the
standard deviation of weights at the end of training, $\sigma$, such that
$\epsilon=\sigma/\chi$. A more in-depth discussion of our choice of
$\epsilon$, and the effect of varying it, is presented in Appendix B.

We perform a set of numerical experiments designed to be similar to the quench
and anneal protocol used to study glasses. Networks are first initialised with
random weights drawn from a uniform distribution. Next, the networks are
trained with SGD, at batch size 100, to efficiently move them away from their
random initial position in the loss landscape, and into a local basin. The
networks are then trained for a further period using the overdamped Langevin
dynamics of Eq. (3), at a range of temperatures, $T$. After waiting some time,
$t_{w}$, to allow the networks to move beyond their transient response to the
new dynamics we start to measure $Q$. Measurements are made on approximately
50 independent initial conditions (some networks are lost as their training
dynamics diverge). This protocol is illustrated schematically in Fig. 2(a),
and a typical example is shown in Fig. A2. The decay of $Q$ at two
temperatures is shown in Fig. 2(b). As expected, networks decorrelate more
quickly at higher temperatures. We take as a typical decorrelation time
$\tau_{90}$, the time it takes for $Q$ to reach 90% of its initial value,
$Q(\tau_{90})=0.9$. We choose $\tau_{90}$ instead of the more typical e-fold
relaxation time, $\tau_{e}$, which probes the long time ($\alpha$-relaxation)
behaviour, as we do not observe a two stage relaxation process in any of our
experiments (e.g. see the left hand column of Fig. 3). Hence any choice of
threshold is somewhat arbitrary, and we use 90% for computational convenience
as it is reached quickly. For completeness we repeat our measurements with
$\tau_{e}$ and find no significant differences with $\tau_{90}$. These results
are presented in Appendix E.

Figure 2: (a) A schematic of the measurement procedure. Networks are generated
with random weights, and trained for an initial period with SGD. The networks
are then trained for a second period with overdamped Langevin dynamics.
Measurements start after a waiting time, $t_{w}$. (b) Examples of the overlap
function for networks in the high temperature, noise dominated regime (red,
lower curve, $T=10^{-4}$), and in the low temperature, landscape dominated
regime (blue, upper curve, $T=10^{-6}$). These measurements were made with
$t_{w}=5\times 10^{3}$, and using $\chi=400$. The analytic predictions of
$Q(t)$ for free Brownian particles are shown by the dashed black lines.

In Fig. 3(a), (b), and (c), $Q(t)$ is plotted for a range of temperatures in
the regularised (width=30, $\lambda=10^{-2}$), underparameterised (width=30,
$\lambda=0$), and overparameterised (width=72, $\lambda=0$) phases
respectively. The corresponding dependence of $\tau_{90}$ on the temperature,
$T$, is shown in (d), (e), and (f). Both the regularised and
underparameterised phases display two dynamical regimes. At high $T$,
decorrelation is dominated by thermal noise, and we find a power law
relationship, $\tau_{90}\sim T^{-1}$, which we call the noise-dominated
regime. This is what we would expect from an overdamped, memoryless system
with dynamics of the form $\dot{Q}=-\Omega Q$, where the frequency term
$\Omega\sim T$ (e.g. Eq. 28 in [43] if we neglect the memory kernel).
Interestingly, a $1/T$ relationship has also been observed for the
decorrelation of the loss of DNNs trained on random data, and using the cross
entropy loss function [44]. At low $T$, the decorrelation process becomes
temperature independent, and instead is dominated by the first term of Eq.
(3), the deterministic gradients of the loss landscape. The mean value of
$\tau_{90}$ at $T=0$ is shown with the horizontal black line. Consequently,
the value $T^{*}$ where plots (d), (e), and (f) exhibit a kink is the point at
which the networks become trapped in the basins of the loss landscape. We call
this low $T$ behaviour the landscape-dominated regime. In the
overparameterised phase only the noise-dominated regime, $\tau_{90}\sim
T^{-1}$, exists. We suggest that this is because the landscape consists of
flat plateaus at $\mathcal{L}\approx 0$ where there are no gradients and hence
the noise term of Eq. (3) always dominates. Furthermore, the observed
divergence of $\tau_{90}$ as $T\to 0$ is to be expected, as both terms in Eq.
(3) disappear.

Figure 3: (a) The overlap function in the regularised phase, measured with
$\chi=300$, at temperatures (from red to blue, left to right) $T=1$, $0.5$,
$0.1$, $0.05$, $0.01$, $0.005$, $0.001$, $5\times 10^{-4}$, $10^{-4}$,
$10^{-5}$, $10^{-6}$, $10^{-8}$, $10^{-10}$, $0$. (b) The overlap function in
the underparameterised phase, measured with $\chi=500$, at temperatures (from
red to blue, left to right) $T=1$, $0.5$, $0.1$, $0.05$, $0.01$, $0.005$,
$0.001$, $5\times 10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$, $0$. (c) The
overlap function in the overparameterised phase, measured with $\chi=6000$, at
temperatures (from red to blue, left to right) $T=1$, $0.5$, $0.1$, $0.05$,
$0.01$, $0.005$, $0.001$, $5\times 10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$,
$10^{-8}$, $10^{-10}$, $0$. (d), (e), (f) The dependence of $\tau_{90}$ on $T$
in the regularised, underparameterised, and overparameterised phases
respectively. All phases exhibit a noise-dominated regime where $\tau_{90}\sim
1/T$. The regularised and underparameterised phases also have a landscape-
dominated regime where $\tau_{90}$ becomes temperature independent below
$T^{\ast}$. All measurements were made with $t_{w}=2\times 10^{4}$.

For the noise-dominated regime we can predict the decorrelation curve by
treating the weights as independent Brownian particles. The corresponding
probability density function of weight displacements $\Delta w$ obeys the
diffusion equation and has solution

$\displaystyle P(\Delta w,t)=\frac{\Theta(t)}{\sqrt{4\pi Dt}}e^{-\frac{\Delta
w^{2}}{4Dt}},$ (5)

with diffusion constant $D=\frac{\mu T}{P}$. The overlap is equivalent to
integrating $P(\Delta w)$ over the interval $[-\epsilon,\epsilon]$, resulting
in

$\displaystyle Q(t)=\int_{-\epsilon}^{\epsilon}P(\Delta w)d\Delta
w=\frac{2\Theta(t)}{\sqrt{\pi}}\text{erf}\left(\frac{\epsilon}{\sqrt{4Dt}}\right),$
(6)

where $\text{erf}(x)$ is the error function. This analytic curve is shown in
the dashed line of Fig. 2(b) where, for a network in the noise-dominated
regime, the prediction closely follows the DNN measurement at short and
intermediate times. In the landscape-dominated regime, the DNNs decorrelate
faster than predicted. Weights in a harmonic basin subjected to thermal noise
decorrelate faster the steeper the basin (see Appendix F for details), hence,
although we cannot perform analytic calculations with the true loss landscape,
it is to be expected that DNNs in the landscape-dominated regime decorrelate
faster than the prediction of Eq. (6).

###  III.3 The long-time limit of $Q(t)$

By considering the long-time behaviour of $Q$, we are able to make
quantitative comparisons with the theory of structural glasses. For this
comparison we would like to measure $Q_{\infty}=\lim_{t\to\infty}Q(t)$. An
approximate measurement of $Q_{\infty}$ can be performed in the regularised
phase by observing that the $Q(t)$ curves at different temperatures collapse
onto a master curve when time is rescaled in a temperature-dependent way,
$\hat{t}=t/\tau_{90}$, as shown in Fig. 4(a). This is similar to the time-
temperature superposition principle (TTSP) of glasses [45, 46, 47], however
here the collapse occurs at short and intermediate times, instead of long
times as found in standard TTSP [48]. The resulting master curve obeys a power
law $Q(t)\sim t^{-0.48}$. This is in close agreement with the prediction for
the intermediate scattering function, $\phi\sim t^{-0.5}$, from the F1
schematic version of Mode-Coupling Theory (MCT), a highly idealised model of
structural glasses [49, 50, 51]. This power law is also observed in the under
and overparameterised phases (see Appendix G).

Different temperatures leave the master curve and begin to plateau at
different times, with higher temperatures doing so at longer times and lower
values of $Q(t)$. Note that networks at $T<10^{-5}$ did not yet have time to
leave the master curve and so behave in a temperature-independent way. We take
the final value of each curve in Fig. 4(a) that leaves the master curve as an
approximate measurement of $Q_{\infty}$, while acknowledging that in all cases
this is an overestimate. Plotting $Q_{\infty}$ against temperature, as in Fig.
4(b), we find a power law, $Q_{\infty}\sim T^{-0.48}$. At first glance this
appears to be another point of quantitative agreement with MCT (specifically
the F2 version), which predicts a power-law scaling of the non-ergodicity
parameter (analogous to $Q_{\infty}$) with exponent $-0.5$ [52, 53]. However,
caution is required. The weights of a regularised network are constrained by
the quadratic term in the loss. As shown in Appendix G, for a landscape of the
form $\mathcal{L}=\frac{\lambda}{2}|\mathbf{W}|^{2}$, and the appropriate
parameter values from our DNNs, the resulting values of $Q_{\infty}$ are
similar to those measured in Fig. 4(b). As such, the $Q_{\infty}\sim
T^{-0.48}$ power law is a simple result of L2-regularisation, not a signature
of glassy behaviour.

A similar analysis cannot be performed for the over- or underparameterised
phases. $Q(t)$ in the overparameterised phase does not leave the master curve
during our experimental window. This is to be expected, as we believe it is
the effect of gradients in the landscape that breaks the TTSP in this system.
In the underparameterised phase $Q(t)$ does leave the master curve, but not in
a well controlled way, as shown in Appendix G.

Figure 4: (a) $Q(t)$ curves collapse onto a master curve for short and
intermediate times, when time is rescaled by $\tau_{90}$. Higher temperatures
(redder colours) leave the master curve at longer times, and lower values of
$Q(t)$. The measurements were made with the same parameters as Fig. 3 and the
displayed temperatures are $T=1$, $0.5$, $0.1$, $0.05$, $0.01$, $0.005$,
$0.001$, $5\times 10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-8}$,
$10^{-10}$, $0$. (b) The final value of the overlap, $Q(t_{f}=3\times 10^{5})$
vs $T$.

###  III.4 Caging

Caging is a fundamental aspect of the glass transition where relaxation is
temporarily arrested [54, 55, 56, 57, 58]. This process leads to a typical
signature in the mean square displacement (MSD) of glassy particles where at
short times the ballistic MSD goes as $t^{2}$, then there is an intermediate-
time plateau from the cage effect, followed by a long-time diffusive regime
which goes as $t^{1}$ (or $t^{1}$, plateau, $t^{1}$ in the case of Brownian
simulations [59]) [60]. The signature plateau of caging is not unique to the
MSD and can also be seen in many other correlation functions. In structural
glasses, the intuitive explanation of caging is where the motion of a particle
becomes constrained by steric repulsion with its neighbours [19].
Interestingly, temporarily arrested relaxation is also observed in spin
glasses, despite the lack of steric interactions [61, 62, 18]. This has been
explained by arguing that the dynamics of a spin glass exhibit a dramatic slow
down near the glass transition temperature as the system navigates saddle
points with very few unstable directions [63, 64]. In this picture, the glass
transition corresponds to a transition from a landscape dominated by saddles
($T>T_{g}$) to one dominated by minima ($T<T_{g}$) [65].

Once again, we ask the question: Does the typical caging behaviour of glasses
occur in DNNs? We define the weight MSD in an analogous way to particle
positions as

$\displaystyle\text{MSD}(t,t_{w})=\left\langle\frac{1}{N}\sum_{i=1}^{N}[w_{i}(t+t_{w})-w_{i}(t_{w})]^{2}\right\rangle.$
(7)

As shown in Fig. 5, we see no plateaus, and hence no evidence of caging. In
Fig. 5 (a), (b), and (c) the MSD is plotted for regularised,
underparameterised, and overparameterised networks respectively, subject to
overdamped Langevin dynamics. In all three cases, at high temperatures and
intermediate times the MSD demonstrates simple diffusive behaviour with a
slope of $1$. In the regularised and underparameterised phases the MSD curves
begin to collapse onto a single curve below $T^{\ast}$, i.e., the inflection
temperature from the decorrelation plots in Fig. 3. For comparison with the
decorrelation plots an MSD equivalent to $\epsilon^{2}$ is shown by the
horizontal line. Below $T^{\ast}$ the dynamics are dominated by the gradients
of the loss landscape, hence the MSD becomes temperature insensitive, and no
longer follows a simple power law in $t$.

Landscape effects are not present in the overparameterised phase as the
networks exist in the flat, $\mathcal{L}\approx 0$ plateaus. As such the
stochastic part of the dynamics always dominates, and the MSD is always
diffusive.

At low temperatures and short times there are oscillations in the MSD in the
regularised and underparameterised phases. This is a numerical artefact due to
gradients in the loss varying over length scales of a similar size to the
learning rate (i.e. step size) used for training. The oscillations disappear
when a smaller learning rate is used (not shown), however the long time
sections of the curves then become unobtainable.

At long times and high temperatures the MSD becomes sub-diffusive in the
regularised phase. We can study the long-time limit further by using SGD
instead of overdamped Langevin dynamics. The use of batches in SGD drastically
speeds up training, however the price we pay is that batch noise is quite
dissimilar to thermal noise [66, 67], hence none of the curves are straight
lines. With this caveat in mind, the SGD MSD of regularised DNNs is shown in
Fig. 5(d). Localisation occurs at $\text{MSD}\approx 10^{-4}$, which is what
we would expect from the quadratic term in Eq. (2) (see Appendix H). As such,
it is unsurprising that the localisation effect is absent in both under and
overparameterised DNNs.

Figure 5: The MSD of (a) regularised, (b) underparameterised, and (c)
overparameterised networks. The dotted black line shows a gradient of 1, i.e.
normal diffusion. The solid black line is the MSD corresponding to the value
of $\epsilon$ used in Fig. 3. The measurement parameters are the same as those
in Fig. 3. The temperatures used in (a) are $T=1$, $0.5$, $0.1$, $0.05$,
$0.01$, $0.005$, $0.001$, $5\times 10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$,
$10^{-8}$, $10^{-10}$, $0$. In (b) they are $T=1$, $0.5$, $0.1$, $0.05$,
$0.01$, $0.005$, $0.001$, $5\times 10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$,
$0$. In (c) they are $T=1$, $0.5$, $0.1$, $0.05$, $0.01$, $0.005$, $0.001$,
$5\times 10^{-4}$, $10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-8}$, $10^{-10}$. In
all three cases the temperatures run from high (red, top curves) to low (blue,
bottom curves). Panel (d) shows the MSD in the regularised phase for networks
trained with SGD. The localisation of the networks at long times is visible.
The batch sizes used are $b=1$, $2$, $4$, $8$, $16$, $32$, $64$, $100$, $128$,
$300$, $600$, $1000$, $3000$, $6000$, $10000$, $60000$. Small batch sizes,
corresponding to high noise, are the high curves (red), and large batch sizes
are the low (blue) curves.

We hypothesise that the root cause of the lack of caging in DNNs is due to the
geometrical complexity of the basins of attraction in the loss landscape. The
energy landscape of a structural glass is made up of topologically simple,
distinct basins that are roughly spherical in shape as they are formed by the
repulsive interactions of the particles acting on length scales of
approximately one particle diameter. This situation is depicted schematically
in Fig. 6 (left hand panels). These basins have a typical associated length
scale, and the barriers separating them a typical height. Consequently the
signature plateau of caging is visible in a predictable way when the dynamics
are resolved on length scales at least as small as the cage size, and
measurements are made over timescales at least as long as the cage-escape
time. The situation with the DNN loss landscape is much more complicated.
There is no reason for the basins of attraction to have a simple structure,
and are likely to be geometrically complex (e.g. being much larger in some
dimensions than others) as well as containing topologically non-trivial level
sets. The basins have no typical length scale, hence no typical escape time.
The recent work of [68] demonstrating the existence of nearly flat paths
between distant loss minima, as well as that of [69] mapping the high degree
of connectivity between solutions of the spherical negative perceptron, also
suggest the basins have a highly extended shape. A schematic loss landscape
illustrating these complexities is shown in the right hand panels of Fig. 6,
and the real loss landscapes of DNNs are visualised in [70].

What about caging in spin glasses? Here the energy landscape is divided into a
high temperature region of saddle points, and a low temperature region of
minima, with caging occurring at saddles near the boundary as the number of
unstable modes gets very small. Figure 5 suggest this picture does not apply
to DNNs and raises the interesting question of whether the topology of the
loss and energy landscapes are different, or whether geometric features are
sufficient to explain the lack of caging.

Figure 6: Schematics of the energy and loss landscapes of a structural glass
and a DNN. The glass energy landscape has simple, approximately spherical
basins of a well defined size. In contrast, we expect the DNN loss landscape
to have extended basins with no associated length scale.

###  III.5 The Stokes-Einstein relation and Dynamic Heterogeneity

The Stokes-Einstein relation is the most famous example of a fluctuation-
dissipation theorem, and relates the diffusion constant of a colloidal tracer
particle, $D$, to the viscosity, $\eta$, of the surrounding fluid, according
to

$\displaystyle D\propto\frac{k_{B}T}{\eta},$ (8)

where the denominator comes from the Stokes drag equation, and the exact
constant of proportionality is system dependent [71]. In colloidal glasses
(which obey overdamped Langevin dynamics like Eq. (3)) the effective viscosity
of the fluid-colloid mixture, $\eta_{\text{eff}}$, is used instead of the bare
viscosity of the background fluid [72]. It is a well known signature of the
glass transition that the Stokes-Einstein relation begins to be violated as
the system enters the supercooled regime [73, 74, 75]. The cause of this
violation is thought to be associated with non-Gaussian fluctuations, as
measured by the non-Gaussian parameter, $\alpha_{2}$ [76], and the emergence
of dynamic heterogeneity (DH). DH is another typical glassy phenomenon,
whereby the system begins to separate into spatially localised populations of
mobile and non-mobile particles [75, 77, 78]. The mobile particles dominate
the diffusion, whereas the non-mobile particles dominate the viscosity,
resulting in the decoupling of these two processes, and the violation of Eq.
(8).

Figure 7: The non-Gaussian parameter, $\alpha_{2}$, deviates from 0 and hence
demonstrates non-Gaussian behaviour in the weight displacements of the
regularised (a), underparameterised (b), and overparameterised (c) phases. The
measurement parameters are the same as in Fig. 3. The temperatures used in (a)
are (going from red, lowest curves, to blue, highest curves) $T=0.1$, $0.01$,
$0.001$, $10^{-5}$, $10^{-6}$, $10^{-8}$, $10^{-10}$, $0$, in (b) they are
$T=0.1$, $0.01$, $0.001$, $10^{-5}$, $10^{-6}$, $0$, and in (c) they are
$T=0.1$, $0.01$, $0.001$, $10^{-5}$, $10^{-6}$, $10^{-8}$. Histograms of the
weight displacements are plotted in the right-hand column, for the regularised
(d), underparameterised (e), and overparameterised (f) phases. All phases
exhibit bimodal distributions at low temperatures. The temperatures (going
from red, right-most curves, to blue, left-most curves) in (d) are
$T=10^{-3}$, $10^{-5}$, $10^{-6}$, $10^{-8}$, in (e) they are $T=10^{-2}$,
$10^{-3}$, $10^{-5}$, $10^{-6}$, and in (f) they are $T=10^{-1}$, $10^{-3}$,
$10^{-6}$, $10^{-8}$.

To what extent does this phenomenology also occur in DNNs? The overdamped
Langevin dynamics of Eq. (3) result in a Fokker-Planck equation with diffusion
constant $D=\mu kT/P_{D}$ [79], as any Brownian process does. This expression
can be rewritten replacing the mobility, $\mu$, with an “effective” drag,
$\gamma=1/\mu$, resulting in the (trivial) recovery of the fluctuation-
dissipation theorem $D=kT/\gamma P_{D}$. However, there is no real
justification for introducing the concept of a viscosity like in Eq. (8), to a
system with no colloidal particle, and no background fluid. Despite this, we
can still look for deviations from Gaussian behaviour by measuring the non-
Gaussian parameter,

$\displaystyle\alpha_{2}(t)=\frac{\langle\Delta
w^{4}(t)\rangle}{3\langle\Delta w^{2}(t)\rangle^{2}}-1,$ (9)

where $\Delta w$ are the weight displacements [80]. $\alpha_{2}$ is 0 for
Gaussian fluctuations in $\Delta w$. As can be seen in Fig. 7 (left-hand
column), $\alpha_{2}$ does in fact deviate from 0 for our DNNs, with the
deviation starting at long times for low temperatures, and coming to dominate
the whole time window as the temperature decreases. This is to be expected as
it is the landscape term in Eq. (3) that causes the non-Gaussian behaviour.
The $\alpha_{2}$ curves are quite different from those measured in structural
glasses, where $\alpha_{2}$ falls again at long times as cage hopping restores
Brownian motion [81]. Our data does not reach long enough times to see if this
occurs in DNNs, however given the lack of caging we suggest it is unlikely.

We next test for the presence of DH by plotting the distribution of weight
motilities, $\Delta w=|w(t_{w}+\Delta t)-w(t_{w})|$, at a sequence of
decreasing $T$ values. For structural glasses, a bimodal distribution of
mobile and non-mobile particles emerges as $T$ approaches the glass
temperature [76]. As shown in Fig. 7 (right-hand column), a bimodal
distribution also emerges for DNNs in all three phases. The presence of DH in
DNNs is surprising, as DH in structural glasses is typically understood as a
result of spatially correlated movements of groups of particles interacting
via steric repulsion between neighbours [82, 77, 83]. In contrast, the weights
of a DNN interact by complicated, indirect couplings, and their neighbour
relations (or equivalently the topology of the space they inhabit) differ
considerably from particles packed in 3D Euclidean space.

###  III.6 Ageing

Ageing, the phenomenon where the behaviour of a system depends explicitly on
its age or the waiting time $t_{w}$ (e.g. the time following a quench) is
another hallmark of glassy systems. As $T$ decreases towards the glass
transition temperature, glass-forming systems fall out of equilibrium, and the
resulting broken ergodicity produces ageing effects. The effect of ageing on
$Q$ is shown for the regularised and underparameterised phases in Fig. 8 (a)
and (c) respectively, as well as the dependence of $\tau_{90}$ on $t_{w}$ in
those phases in (b) and (d). Both the regularised and underparameterised
regimes exhibit ageing in the landscape-dominated regime $(T<T^{\ast})$,
though the effect is stronger in the regularised phase. In both phases a
longer $t_{w}$ results in slower dynamics, as it does in glasses. At
$T>T^{\ast}$ the networks behave like an equilibrium system and there is no
ageing (not shown). The overparameterised phase does not have a transition
from noise-dominated to landscape-dominated dynamics as the networks exist in
flat plateaus. As such, overparameterised networks never exhibit ageing, as
demonstrated in Fig. A11. With the current data it is impossible to make a
quantitative statement about the ageing behaviour (e.g. the power law ageing
documented in glasses [84, 85, 86]), and varying $t_{w}$ over several more
orders of magnitude is computationally unfeasible.

Figure 8: The overlap function of networks as they decorrelate in (a) the
regularised phase, and (c) the underparameterised phase. The arrows denote
increasing $t_{w}$. In panels (b) and (d) $\tau_{90}$ is plotted against the
waiting time $t_{w}$ at $T=10^{-6}$ (regularised) and $T=10^{-4}$
(underparameterised). The featured waiting times in both phases are
$t_{w}\in\\{2\times 10^{4},5\times 10^{4},8\times 10^{4},10^{5},1.5\times
10^{5}\\}$. All other parameters are the same as in Fig. 3.

##  IV  Discussion

At first glance, glasses and DNNs share many important similarities, and yet
DNNs fail to exhibit an unambiguous glass phase. Despite the existence of two
glass-like transitions, the TTT and UOT, DNNs do not have diverging relaxation
times - the defining characteristic of a glass. This is despite the existence
of a critical temperature below which the dynamics of the DNNs fall out of
equilibrium, are dominated by the landscape, and exhibit non-Gaussian
fluctuations, DH, and ageing. We conclude that the disorder inherent in the
loss landscape of these DNNs is not sufficient to induce a dramatic slowdown
in the dynamics. In fact, our measurements suggest that the differences in the
detailed structure of the glass and DNN landscapes, in particular the lack of
a cage length in the latter, is enough to eliminate crucial glassy phenomena.
This latter point is particularly relevant to the wider field of deep learning
as this study was conducted with real image data, and using realistically
large networks (consisting of tens of thousands of weights), and as such we
expect the investigated loss landscapes to be representative of other “real-
life” loss landscapes.

The presence of DH in DNNs raises the intriguing question of whether there is
some “spatial” organisation of the mobile weights, as there is in structural
glasses. Such structure could have important practical applications in the
sub-field of DNN pruning. Pruning, i.e. reducing a network’s size with minimal
reduction in its performance, is an area of growing importance as the immense
size of modern networks makes them impractical for use on mobile phones or
smart devices, as well as consuming worrying amounts of energy [87, 88, 89].
The presence of two distinct populations of weights, coupled with the insights
of disordered physics, could lead to innovative new approaches to pruning.

The relationship between deep learning and structural glasses is complicated,
with both surprising points of agreement (e.g. DH) and disagreement (e.g. the
lack of diverging timescales). Looking beyond structural glasses, the wider
field of the physics of disordered systems potentially has a lot to offer as a
theoretical blueprint for deep learning. The Random Lorentz Gas and Anderson
Localisation both suggest themselves as useful, and as of yet unexplored,
physical analogies to DNNs as their complex landscapes result in a
localisation transition as opposed to the caging seen in structural glasses.
Furthermore, the link between the localisation transition of the RLG and
percolation theory is well known, as is the existence of a fractal void space
near the percolation transition [90, 91, 92, 93, 94]. Perhaps similar
phenomena can be discovered in the connectedness of plateaus in the
overparameterised phase. Exploring such topics further would be a highly novel
direction of research, and further enrich the exciting work currently being
done at the interface of deep learning and statistical physics.

##  V Acknowledgements

We warmly thank Ilian Pihlajamaa and Corentin C. L. Laudicina for their
detailed reading of this manuscript, and Thomas Voigtmann, Kees Storm, Wouter
Ellenbroek, and Sophie de Hont for their insightful discussions and advice. We
acknowledge the Dutch Research Council (NWO) for its financial support through
a START-UP grant.

##  Appendix A Temperature in networks with overdamped Langevin dynamics

We wish to construct the overdamped Langevin dynamics of a network to be as
similar as possible to that of a physical system. The degrees of freedom, the
weights, are analogous to the particle positions. The energy of a system is an
extrinsic quality, whereas the loss, being an average over the data, is
intrinsic. Consequently, the quantity $P_{D}\mathcal{L}$ plays the role of the
energy.

We can see this argument more clearly by considering the overdamped Langevin
equation for the weights of a DNN,

$\displaystyle\partial_{t}w_{i}=-\mu\partial_{w_{i}}\mathcal{L}+\xi_{i},$ (10)

where
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})=2D\delta_{ij}\delta(t-t^{\prime})$. At
this stage we do not specify how the diffusion constant, $D$, relates to
temperature. The probability distribution of weights that obey such an
overdamped Langevin equation, $\rho$, evolves according to the Fokker-Planck
equation [79],

$\displaystyle\partial_{t}\rho=-\nabla(\mu\rho\nabla\mathcal{L}-D\nabla\rho).$
(11)

The steady state of this equation is given by

$\displaystyle\frac{d\rho}{d\mathcal{L}}=-\frac{\mu}{D}\rho,$ (12)

with solution $\rho=\rho_{0}\exp\left(-\mu\mathcal{L}/D\right)$. In order to
mimic physics, we wish to choose the noise on Eq. (10) such that the steady
state solution of Eq. (11) is the Boltzmann distribution with “energy”
$U=P_{D}\mathcal{L}$. This results in

$\displaystyle
e^{-\frac{\mu\mathcal{L}}{D}}=e^{-\frac{U}{k_{B}T}}=e^{-\frac{P_{D}\mathcal{L}}{k_{B}T}},$
(13)

and hence we conclude that $D=\frac{\mu kT}{P_{D}}$, a version of the familiar
fluctuation dissipation relation but taking into account the intensive nature
of $\mathcal{L}$, and $\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=(2\mu
k_{B}T/P_{D})\delta_{ij}\delta(t-t^{\prime})$, as stated in the main body of
the text.

##  Appendix B Choosing $\epsilon$

The overlap function defined in Eq. (4) relies on an appropriate definition of
the parameter $\epsilon$. There is no natural length scale associated with the
weights of a DNN, hence we use the standard deviation of weights at the end of
training, $\sigma$, and define $\epsilon$ as

$\displaystyle\epsilon=\frac{\sigma}{\chi},$ (14)

where the value of the parameter $\chi>0$ can be chosen freely within some
reasonable bounds. We have verified experimentally that $\sigma$ varies by at
most roughly 1% between networks in the same phase, although it does vary more
between phases. Choosing a smaller $\chi$ slows down the decorrelation of
$Q(t)$, hence we are not so interested in a lower bound for $\chi$ stricter
than $\chi>0$. Choosing a larger $\chi$ speeds up decorrelation, which has
significant practical advantages in that we do not need to train our DNNs for
such a long time. What is the maximum value of $\chi$ we can reasonably
choose? The values in a Pytorch DNN are stored as single precision floating
point numbers with a resolution of approximately $1\times 10^{-7}$, i.e. the
next number after 1 is approximately $1+10^{-7}$. Consequently we place an
upper bound on $\chi$ such that $\epsilon>10^{-7}$.

How does the choice of $\epsilon$ affect our measurements of $Q(t)$, and in
particular the identification of two distinct regimes in the decorrelation
times of overdamped Langevin networks? A very large $\epsilon$ increases
$\tau_{90}$ beyond the maximum training time of our networks, whereas a very
small $\epsilon$ decreases $\tau_{90}$ below the resolution at which we save
data (which varies from every epoch at short times to every 100 epochs at long
times). Between these two extremes varying $\epsilon$ simply shifts the
$\tau_{90}$ vs $kT$ curve as shown in Fig. A1. In this figure we vary $\chi$
by a factor of 50 without changing the shape of the curve. The critical value
of $kT$ at which networks transition between the two dynamic regimes changes
by about an order of magnitude, so whenever we make quantitative comparisons
we do so at a fixed $\chi$.

In general the overlap function in spin glasses is not defined with respect to
a particular length scale, and yet avoids introducing a parameter like
$\epsilon$. This is because the spins are Ising-like binary variables, hence
simply taking an inner product like $\mathbf{s}_{1}\cdot\mathbf{s}_{2}$
results in fast decorrelation. The exception is the p-spin model, which uses
continuous spins. However, in this model the strict spherical constraint,
$|\mathbf{s}|=1$, also ensures fast decorrelation. In our case of DNN weights,
defining the overlap as the inner product of the weights results in very slow
decorrelation, and as such we turn to the definition in the main text.

Figure A1: The dependence of the decorrelation time on $\chi$. As $\chi$ is
increased the networks decorrelate more quickly, and the transition point
between noise and landscape-dominated dynamics occurs at lower temperatures.
However, the overall shape of the curves is unaffected.

##  Appendix C Measurement process

In Fig. A2 we demonstrate a typical training loss curve illustrating the
process of taking measurements. An initial period of SGD moves the network out
of its initial condition and into a local minimum. The dynamics then switch to
Eq. (3) (Langevin), there is an initial transient response to the new
dynamics, and after some time $t_{w}$, taken to be 20000 timesteps unless
otherwise stated, measurement starts. The choice of $t_{w}=20000$ was made
somewhat subjectively, balancing the fact that a longer $t_{w}$ reaches a
flatter loss curve, but we only have finite training time.

Figure A2: The training loss curve of a network, illustrating the measurement
process.

##  Appendix D Transitions with CIFAR10

The UOT is shown for CIFAR10 data in Fig. A3(a), and the TTT is shown in Fig.
A3(b).

Figure A3: (a) The UOT for CIFAR10 data. (b) The TTT for CIFAR10 data.

##  Appendix E Further decorrelation plots

We repeat the measurements of the decorrelation time as a function of
temperature shown in Fig. 3. We first plot the e-fold relaxation time,
$\tau_{e}$, as a function of $T$ for regularised, underparameterised, and
overparameterised networks, shown in Fig. A4. As with $\tau_{90}$, we see two
dynamical phases in regularised and underparameterised networks. At high
temperatures $\tau_{e}\sim 1/T$, and at low temperatures $\tau_{e}$ becomes
$T$ independent.

These findings are replicated by networks trained on the CIFAR-10 dataset
[95]. Following the same steps as with MNIST, we convert CIFAR-10 to a binary
classification problem by assigning labels of +1 to the categories ‘airplane’,
‘bird’, ‘deer’, ‘frog’, and ‘ship’, and assigning -1 to ‘automobile’, ‘cat’,
‘dog’, ‘horse’, and ‘truck’. We also use PCA to reduce the dimension of the
images to the first 10 PCA components. The $90\%$ relaxation times are shown
in Fig. A4.

Figure A4: Left-hand column: The e-fold decorrelation time vs temperature in
(a) regularised, (b) underparameterised, and (c) overparameterised networks
trained on MNIST. Right-hand column: The dependence of $\tau_{90}$ on
temperature in (d) regularised, (e) underparameterised, and (f)
overparameterised networks trained on CIFAR10.

##  Appendix F Calculation of $Q(t)$ for weights in a quadratic basin

Here we consider the case of DNNs trained with overdamped Langevin dynamics
within a loss landscape of a simple harmonic well,
$\mathcal{L}=\frac{1}{2}k\sum_{i=1}^{N}w_{i}^{2}$, with some “spring constant”
$k$. The resulting overdamped Langevin equation is

$\displaystyle\partial_{t}w_{i}=-\mu kw_{i}+\xi_{i},$ (15)

where the noise term obeys $\langle\xi(t)\rangle=0$,
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=2\mu
T\delta_{ij}\delta(t-t^{\prime})$. Note that here we do not define the noise
with a $1/P_{D}$ term as in the main text, as this loss function is
independent of the dataset size. Such an overdamped Langevin equation results
in the following Fokker-Planck equation for the weight probability
distribution,

$\displaystyle\partial_{t}P(w,t)=\partial_{w}\left(\mu kwP\right)+\mu
T\partial_{w}^{2}P,$ (16)

with solution

$\displaystyle P(w,t)=A\exp\left[\frac{-k\left(w-w(0)e^{-k\mu
t}\right)^{2}}{2T\left(1-e^{-2k\mu t}\right)}\right],$ (17)

where

$\displaystyle A=\left[\frac{k}{2\pi T\left(1-e^{-2\mu
kt}\right)}\right]^{\frac{1}{2}}.$ (18)

The overlap function is given by integrating over a region of size $2\epsilon$
around the initial condition $w(0)$,

$\displaystyle Q(t)$
$\displaystyle=\int_{w(0)-\epsilon}^{w(0)+\epsilon}P(w,t)dw,$ (19)

which evaluates to

$\displaystyle Q(t)=\frac{1}{2}\Bigg{\\{}$
$\displaystyle\text{erf}\left[\frac{w(0)\left(1-e^{-k\mu
t}\right)+\epsilon}{\sqrt{c}}\right]$
$\displaystyle-\text{erf}\left[\frac{w(0)\left(1-e^{-k\mu
t}\right)-\epsilon}{\sqrt{c}}\right]\Bigg{\\}},$ (20)

where

$\displaystyle\frac{1}{\sqrt{c}}=\left[\frac{k}{2T\left(1-e^{-2\mu
kt}\right)}\right]^{\frac{1}{2}}.$ (21)

The effect of varying the spring constant of the quadratic well is dependent
on $w(0)$. The case where $w(0)\gg\epsilon$, which is the relevant regime for
modelling landscape-dominated dynamics which have fallen out of equilibrium,
is shown in Fig. A5, where a higher $k$ results in faster decorrelation. In
Fig. A5, $w(0)=1$, $\epsilon=0.1$, $2T=1$, and $\mu=1$.

Figure A5: The overlap function of Brownian weights in a quadratic well with
high ($k=100$) and low ($k=0.01$) spring constant.

##  Appendix G Further details of $\lim_{t\to\infty}Q(t)$

Although both the under and overparameterised phases also exhibit time-
temperature superposition, as shown in Fig. A6, their behaviour is markedly
different to the regularised phase. In the underparameterised phase, some
temperatures eventually leave the master curve, and some do not. Those that
do, do so in an irregular fashion, with individual curves leaving both above
and below the master curve, in a temperature non-monotonic manner. As such, we
cannot use the late time values of $Q(t)$ as an approximate measure of
$Q_{\infty}$.

As for the overparameterised phase, the individual temperatures never leave
the master curve. We believe this is because the breaking of the TTSP is due
to landscape effects in the dynamics, and overparameterised DNNs exist in a
gradient-less landscape.

Figure A6: Time-temperature superposition of $Q(t)$ in the underparameterised
(a) and overparameterised (b) phases.

An important question raised by the measurement of a power-law in Fig. 4(b)
is, could the large, quadratic basin of the regularisation term cause the
observed behaviour in $Q_{\infty}$? Starting with the results from the
previous section, modifying the temperature like $T\to T/P_{D}$ to replicate
the convention of the main text, and taking the long time limit of Eq. (20),
we get

$\displaystyle Q_{\infty}=\frac{1}{2}\Bigg{\\{}$
$\displaystyle\text{erf}\left[\sqrt{\frac{kP_{D}}{2T}}(w(0)+\epsilon)\right]$
$\displaystyle-\text{erf}\left[\sqrt{\frac{kP_{D}}{2T}}(w(0)-\epsilon)\right]\Bigg{\\}}.$
(22)

Setting $k$, the spring constant, to the regularisation parameter
$\lambda=0.01$, and using $w(0)$ and $\epsilon$ measured from the relevant
DNNs, we can plot $Q_{\infty}$ vs $T$, as shown in Fig. A7. We do indeed
recover the $T^{-0.5}$ power-law, and similar numerical values to those
measured in Fig. 4(b).

Figure A7: The analytic result of $Q_{\infty}$ vs $T$ for weights in a
quadratic bowl. The dashed red line is $T^{-0.5}$.

##  Appendix H MSD of weights in a quadratic well

Consider the case of weights evolving according to Eq. (3), in a quadratic
well, $\mathcal{L}=\frac{1}{2}k\sum_{i=1}^{N}w_{i}^{2}$. The Fokker-Planck
equation for the weights is

$\displaystyle\partial_{t}P=\partial_{w}(\mu kwP)+\frac{\mu
T}{P_{d}}\partial_{w}^{2}P,$ (23)

where $P$ is the probability density of weights. The steady state distribution
is

$\displaystyle P_{ss}=\sqrt{\frac{kP_{D}}{2\pi
T}}\exp\left(-\frac{kP_{D}w^{2}}{2T}\right).$ (24)

We can then calculate the long time limit MSD as

$\displaystyle\text{MSD}=\int_{-\infty}^{\infty}P_{ss}(w)w^{2}dw,$ (25)

resulting in

$\displaystyle\text{MSD}=\frac{T}{kP_{D}}.$ (26)

Subbing in $T=0.1$, $k=\lambda=10^{-2}$, and $P_{D}=6\times 10^{4}$, which are
the relevant values for the $T=0.1$ curve in Fig. 5(a), results in an MSD of
$1.6\times 10^{-4}$, close to the value at which the MSD plateaus in Fig.
5(d). So we conclude the weights are localised by the quadratic basin. This is
in agreement with our calculation of $Q_{\infty}$.

##  Appendix I Non-Gaussianity and Dynamic Heterogeneity in CIFAR-10

Using the DNNs trained on CIFAR-10 from Sec. E, we measure the non-Gaussian
parameter, $\alpha$, and the distribution of weight motilities at a range of
temperatures. These networks exhibit the same behaviour as the MNIST examples
in the main text. All three phases have $\alpha\neq 0$, starting at long times
for high temperatures, then occurring at all times for low temperatures. This
behaviour is less pronounced for the overparameterised phase. Note that the
pronounced oscillations up to $t=100$ are due to the fact that the network
states are saved at a higher frequency at shorter times. The weight motilities
follow bimodal distributions, i.e. demonstrate DH, at low temperatures for the
regularised and underparameterised, but not overparameterised, phases.

Figure A8: The non-Gaussian parameter, $\alpha$, and weight motility
distributions for DNNs trained on CIFAR-10. Non-Gaussian behaviour,
corresponding to $\alpha\neq 0$ occurs at long times for high temperatures,
and all times for low temperatures in the regularised (a), underparameterised
(b) and overparameterised (c) phases. The temperatures in (a) and (b), going
from blue (high curves) to red (low curves) are $T=10^{-10}$, $10^{-8}$,
$10^{-6}$, $10^{-5}$, $10^{-4}$, $10^{-3}$, $10^{-2}$, $10^{-1}$, $1$. The
temperatures in (c) are $T=10^{-10}$, $10^{-8}$, $10^{-6}$, $10^{-5}$,
$10^{-4}$, $10^{-3}$, $10^{-2}$, $1$. The regularised and underparameterised
networks also have bimodal weight motility distributions (panels (d) and (e)
respectively), unlike the overparameterised phase (f). The temperatures used
in (d), (e), and (f), going from blue (left curves) to red (right curves) are
$T=10^{-8}$, $10^{-6}$, $10^{-5}$, $10^{-3}$.

##  Appendix J Aging effects

Figure A9: The effect of waiting time on the $\tau_{90}$ vs $T$ curve in the
regularised phase. Figure A10: The effect of waiting time on the $\tau_{90}$
vs $T$ curve in the underparameterised phase.

The relaxation behaviour does not change significantly at different waiting
times. In Fig. A10 (a) and (b), $t_{w}$ is varied over an order of magnitude
for regularised and underparameterised networks respectively. In all cases,
$\tau_{90}$ displays the same behaviour of a noise-dominated regime with a
gradient $\approx-1$, and a temperature independent landscape-dominated
regime. In both phases, increasing $t_{w}$ slows down the decorrelation
process, without a noticeable effect on $T^{\ast}$. The ageing effect is
stronger in the regularised phase than the underparameterised phase. The
gradients are slightly less than 1 when $t_{w}\neq 20000$. This is due to a
data sampling effect. In general the network states are saved every 100
timesteps. However, for $t\in[20000,20100]$ the networks are saved every
timestep, and for $t\in[20100,21000]$ every 10 timesteps. This is to allow
high resolution measurements of short decorrelation times when $t_{w}=20000$,
as it is for the majority of our measurements. Consequently, the time
resolution of measurements at $t_{w}\neq 20000$ is limited to every 100
timesteps, which introduces some inaccuracy in the gradient measurements.

Figure A11: (a) The overlap function at various $t_{w}$, and (b) the (lack of)
dependence of $\tau_{90}$ on $t_{w}$, in the overparameterised phase.

The overparameterised phase does not undergo a transition from noise-dominated
to landscape-dominated dynamics, hence does not experience ageing. This is
demonstrated in Fig. A11.

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