Published as a conference paper at ICLR 2018
ON THE INFORMATION BOTTLENECK
THEORY OF DEEP LEARNING
Andrew M. Saxe, Yamini Bansal, Joel Dapello, Madhu Advani
Harvard University
{asaxe,madvani}@fas.harvard.edu,{ybansal,dapello}@g.harvard.edu
Artemy Kolchinsky, Brendan D. Tracey
Santa Fe Institute
{artemyk,tracey.brendan}@gmail.com
David D. Cox
Harvard University
MIT-IBM Watson AI Lab
davidcox@fas.harvard.edu
david.d.cox@ibm.com
ABSTRACT
The practical successes of deep neural networks have not been matched by theoret-
ical progress that satisfyingly explains their behavior. In this work, we study the
information bottleneck (IB) theory of deep learning, which makes three specific
claims: first, that deep networks undergo two distinct phases consisting of an
initial fitting phase and a subsequent compression phase; second, that the compres-
sion phase is causally related to the excellent generalization performance of deep
networks; and third, that the compression phase occurs due to the diffusion-like
behavior of stochastic gradient descent. Here we show that none of these claims
hold true in the general case. Through a combination of analytical results and
simulation, we demonstrate that the information plane trajectory is predominantly
a function of the neural nonlinearity employed: double-sided saturating nonlineari-
ties like tanh yield a compression phase as neural activations enter the saturation
regime, but linear activation functions and single-sided saturating nonlinearities
like the widely used ReLU in fact do not. Moreover, we find that there is no evident
causal connection between compression and generalization: networks that do not
compress are still capable of generalization, and vice versa. Next, we show that
the compression phase, when it exists, does not arise from stochasticity in training
by demonstrating that we can replicate the IB findings using full batch gradient
descent rather than stochastic gradient descent. Finally, we show that when an
input domain consists of a subset of task-relevant and task-irrelevant information,
hidden representations do compress the task-irrelevant information, although the
overall information about the input may monotonically increase with training time,
and that this compression happens concurrently with the fitting process rather than
during a subsequent compression period.
1 I NTRODUCTION
Deep neural networks (Schmidhuber, 2015; LeCun et al., 2015) are the tool of choice for real-world
tasks ranging from visual object recognition (Krizhevsky et al., 2012), to unsupervised learning
(Goodfellow et al., 2014; Lotter et al., 2016) and reinforcement learning (Silver et al., 2016). These
practical successes have spawned many attempts to explain the performance of deep learning systems
(Kadmon & Sompolinsky, 2016), mostly in terms of the properties and dynamics of the optimization
problem in the space of weights (Saxe et al., 2014; Choromanska et al., 2015; Advani & Saxe, 2017),
or the classes of functions that can be efficiently represented by deep networks (Montufar et al.,
2014; Poggio et al., 2017). This paper analyzes a recent inventive proposal to study the dynamics of
learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby,
2017). In this view, deep learning is a question of representation learning: each layer of a deep neural
network can be seen as a set of summary statistics which contain some but not all of the information
present in the input, while retaining as much information about the target output as possible. The
1

Published as a conference paper at ICLR 2018
amount of information in a hidden layer regarding the input and output can then be measured over the
course of learning, yielding a picture of the optimization process in the information plane. Crucially,
this method holds the promise to serve as a general analysis that can be used to compare different
architectures, using the common currency of mutual information. Moreover, the elegant information
bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target
output information that any representation can achieve (Tishby et al., 1999). The IB bound thus
serves as a method-agnostic ideal to which different architectures and algorithms may be compared.
A preliminary empirical exploration of these ideas in deep neural networks has yielded striking
findings (Shwartz-Ziv & Tishby, 2017). Most saliently, trajectories in the information plane appear to
consist of two distinct phases: an initial “fitting” phase where mutual information between the hidden
layers and both the input and output increases, and a subsequent “compression” phase where mutual
information between the hidden layers and the input decreases. It has been hypothesized that this
compression phase is responsible for the excellent generalization performance of deep networks, and
further, that this compression phase occurs due to the random diffusion-like behavior of stochastic
gradient descent.
Here we study these phenomena using a combination of analytical methods and simulation. In Section
2, we show that the compression observed by Shwartz-Ziv & Tishby (2017) arises primarily due to
the double-saturating tanh activation function used. Using simple models, we elucidate the effect of
neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation
function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We
discuss how this compression via nonlinearity is related to the assumption of binning or noise in the
hidden layer representation. To better understand the dynamics of learning in the information plane,
in Section 3 we study deep linear networks in a tractable setting where the mutual information can be
calculated exactly. We find that deep linear networks do not compress over the course of training for
the setting we examine. Further, we show a dissociation between generalization and compression.
In Section 4, we investigate whether stochasticity in the training process causes compression in the
information plane. We train networks with full batch gradient descent, and compare the results to those
obtained with stochastic gradient descent. We find comparable compression in both cases, indicating
that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover,
we show that the two phases of SGD occur even in networks that do not compress, demonstrating
that the phases are not causally related to compression. These results may seem difficult to reconcile
with the intuition that compression can be necessary to attain good performance: if some input
channels primarily convey noise, good generalization requires excluding them. Therefore, in Section
5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show
that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during
training, though the overall information with the input increases. However, instead of a secondary
compression phase, this task-irrelevant information is compressed at the same time that the task-
relevant information is boosted. Our results highlight the importance of noise assumptions in applying
information theoretic analyses to deep learning systems, and put in doubt the generality of the IB
theory of deep learning as an explanation of generalization performance in deep architectures.
2 C OMPRESSION AND NEURAL NONLINEARITIES
The starting point for our analysis is the observation that changing the activation function can
markedly change the trajectory of a network in the information plane. In Figure 1A, we show
our replication of the result reported by Shwartz-Ziv & Tishby (2017) for networks with the tanh
nonlinearity.1 This replication was performed with the code supplied by the authors of Shwartz-Ziv
& Tishby (2017), and closely follows the experimental setup described therein. Briefly, a neural
network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic
gradient descent to produce a binary classification from a 12-dimensional input. In our replication we
used 256 randomly selected samples per batch. The mutual information of the network layers with
respect to the input and output variables is calculated by binning the neuron’stanh output activations
into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then
used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true
output labels. In line with prior work (Shwartz-Ziv & Tishby, 2017), the dynamics in Fig. 1 show a
1Code for our results is available at https://github.com/artemyk/ibsgd/tree/iclr2018
2

Published as a conference paper at ICLR 2018
Figure 1: Information plane dynamics and neural nonlinearities. (A) Replication of Shwartz-Ziv &
Tishby (2017) for a network with tanh nonlinearities (except for the final classification layer which
contains two sigmoidal neurons). The x-axis plots information between each layer and the input,
while the y-axis plots information between each layer and the output. The color scale indicates
training time in epochs. Each of the six layers produces a curve in the information plane with the
input layer at far right, output layer at the far left. Different layers at the same epoch are connected
by fine lines. (B) Information plane dynamics with ReLU nonlinearities (except for the final layer
of 2 sigmoidal neurons). Here no compression phase is visible in the ReLU layers. For learning
curves of both networks, see Appendix A. (C) Information plane dynamics for a tanh network of
size 784− 1024− 20− 20− 20− 10 trained on MNIST, estimated using the non-parametric kernel
density mutual information estimator of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017),
no compression is observed except in the final classification layer with sigmoidal neurons. See
Appendix B for the KDE MI method applied to the original Tishby dataset; additional results using
a second popular nonparametric k-NN-based method (Kraskov et al., 2004); and results for other
neural nonlinearities.
3

Published as a conference paper at ICLR 2018
transition between an initial fitting phase, during which information about the input increases, and a
subsequent compression phase, during which information about the input decreases.
We then modified the code to train deep networks using rectified linear activation functions (f(x) =
max(0,x )). While the activities of tanh networks are bounded in the range [−1, 1], ReLU networks
have potentially unbounded positive activities. To calculate mutual information, we first trained the
ReLU networks, next identified their largest activity value over the course of training, and finally
chose 100 evenly spaced bins between the minimum and maximum activity values to discretize
the hidden layer activity. The resulting information plane dynamics are shown in Fig. 1B. The
mutual information with the input monotonically increases in all ReLU layers, with no apparent
compression phase. To see whether our results were an artifact of the small network size, toy dataset,
or simple binning-based mutual information estimator we employed, we also trained larger networks
on the MNIST dataset and computed mutual information using a state-of-the-art nonparametric
kernel density estimator which assumes hidden activity is distributed as a mixture of Gaussians
(see Appendix B for details). Fig. C-D show that, again, tanh networks compressed but ReLU
networks did not. Appendix B shows that similar results also obtain with the popular nonparametric
k-nearest-neighbor estimator of Kraskov et al. (2004), and for other neural nonlinearities. Thus, the
choice of nonlinearity substantively affects the dynamics in the information plane.
To understand the impact of neural nonlinearity on the mutual information dynamics, we develop
a minimal model that exhibits this phenomenon. In particular, consider the simple three neuron
network shown in Fig. 2A. We assume a scalar Gaussian input distributionX∼N (0, 1), which is
fed through the scalar first layer weightw1, and passed through a neural nonlinearityf(·), yielding
the hidden unit activityh =f(w1X). To calculate the mutual information with the input, this hidden
unit activity is then binned yielding the new discrete variableT = bin(h) (for instance, into 30 evenly
spaced bins from -1 to 1 for the tanh nonlinearity). This binning process is depicted in Fig. 2B. In
this simple setting, the mutual informationI(T ;X) between the binned hidden layer activityT and
the inputX can be calculated exactly. In particular,
I(T ;X) = H(T )−H(T|X) (1)
= H(T ) (2)
= −
N∑
i=1
pi logpi (3)
whereH(·) denotes entropy, and we have used the fact thatH(T|X) = 0 sinceT is a deterministic
function ofX. Here the probabilitiespi =P (h≥bi andh<b i+1) are simply the probability that
an inputX produces a hidden unit activity that lands in bini, defined by lower and upper bin limits
bi andbi+1 respectively. This probability can be calculated exactly for monotonic nonlinearitiesf(·)
using the cumulative density ofX,
pi =P (X≥f −1(bi)/w1 andX <f −1(bi+1)/w1), (4)
wheref −1(·) is the inverse function off(·).
As shown in Fig. 2C-D, as a function of the weight w1, mutual information with the input first
increases and then decreases for thetanh nonlinearity, but always increases for the ReLU nonlinearity.
Intuitively, for small weightsw1≈ 0, neural activities lie near zero on the approximately linear part
of the tanh function. Therefore f(w1X)≈ w1X, yielding a rescaled Gaussian with information
that grows with the size of the weights. However for very large weightsw1→∞ , the tanh hidden
unit nearly always saturates, yielding a discrete variable that concentrates in just two bins. This
is more or less a coin flip, containing mutual information with the input of approximately 1 bit.
Hence the distribution ofT collapses to a much lower entropy distribution, yielding compression for
large weight values. With the ReLU nonlinearity, half of the inputs are negative and land in the bin
containing a hidden activity of zero. The other half are Gaussian distributed, and thus have entropy
that increases with the size of the weight.
Hence double-saturating nonlinearities can lead to compression of information about the input, as
hidden units enter their saturation regime, due to the binning procedure used to calculate mutual
information. The crux of the issue is that the actual I(h;X) is infinite, unless the network itself
adds noise to the hidden layers. In particular, without added noise, the transformation fromX to the
continuous hidden activityh is deterministic and the mutual informationI(h;X) would generally be
4

Published as a conference paper at ICLR 2018
-5 0 5
-1
-0.5
0
0.5
1
0 2 4 6 8 10
w1
0
1
2
3
4
5I(X;T)
0 2 4 6 8 10
w1
0
0.5
1
1.5
2
2.5I(X;T)
𝑥 ℎ
𝑓(⋅)
𝑦(
𝑤*
Net input (𝑤*𝑥)
Hidden activity ℎ
Continuous activity
Bin borders
A B
C D
Tanhnonlinearity ReLUnonlinearity
Figure 2: Nonlinear compression in a minimal model. (A) A three neuron nonlinear network which
receives Gaussian inputs x, multiplies by weight w1, and maps through neural nonlinearity f(·)
to produce hidden unit activityh. (B) The continuous activity h is binned into a discrete variable
T for the purpose of calculating mutual information. Blue: continuous tanh nonlinear activation
function. Grey: Bin borders for 30 bins evenly spaced between -1 and 1. Because of the saturation
in the sigmoid, a wide range of large magnitude net input values map to the same bin. (C) Mutual
information with the input as a function of weight size w1 for a tanh nonlinearity. Information
increases for small w1 and then decreases for large w1 as all inputs land in one of the two bins
corresponding to the saturation regions. (D) Mutual information with the input for the ReLU
nonlinearity increases without bound. Half of all inputs land in the bin corresponding to zero activity,
while the other half have information that scales with the size of the weights.
infinite (see Appendix C for extended discussion). Networks that include noise in their processing
(e.g., Kolchinsky et al. (2017)) can have finiteI(T ;X). Otherwise, to obtain a finite MI, one must
compute mutual information as though there were binning or added noise in the activations. But
this binning/noise is not actually a part of the operation of the network, and is therefore somewhat
arbitrary (different binning schemes can result in different mutual information with the input, as
shown in Fig. 14 of Appendix C).
We note that the binning procedure can be viewed as implicitly adding noise to the hidden layer
activity: a range of X values map to a single bin, such that the mapping between X andT is no
longer perfectly invertible (Laughlin, 1981). The binning procedure is therefore crucial to obtaining
a finite MI value, and corresponds approximately to a model where noise enters the system after
the calculation ofh, that is,T =h +ϵ, whereϵ is noise of fixed variance independent fromh and
X. This approach is common in information theoretic analyses of deterministic systems, and can
serve as a measure of the complexity of a system’s representation (see Sec 2.4 of Shwartz-Ziv &
Tishby (2017)). However, neither binning nor noise is present in the networks that Shwartz-Ziv &
Tishby (2017) considered, nor the ones in Fig. 2, either during training or testing. It therefore remains
unclear whether robustness of a representation to this sort of noise in fact influences generalization
performance in deep learning systems.
Furthermore, the addition of noise means that different architectures may no longer be compared in a
common currency of mutual information: the binning/noise structure is arbitrary, and architectures
that implement an identical input-output map can nevertheless have different robustness to noise added
in their internal representation. For instance, Appendix C describes a family of linear networks that
compute exactly the same input-output map and therefore generalize identically, but yield different
mutual information with respect to the input. Finally, we note that approaches which view the weights
obtained from the training process as the random variables of interest may sidestep this issue (Achille
& Soatto, 2017).
5

Published as a conference paper at ICLR 2018
TeacherA
B Student
𝑌"
𝑊$
𝑋
𝑊& … 𝑊(
ℎ$ ℎ& ℎ(*$ ℎ(
𝑊(+$
Depth 𝐷
…
𝑋
𝑌
𝜖
𝑊.
Train
Test
C
D
Figure 3: Generalization and information plane dynamics in deep linear networks. (A) A linear
teacher network generates a dataset by passing Gaussian inputsX through its weights and adding
noise. (B) A deep linear student network is trained on the dataset (here the network has 1 hidden
layer to allow comparison with Fig. 4A, see Supplementary Figure 18 for a deeper network). (C)
Training and testing error over time. (D) Information plane dynamics. No compression is observed.
Hence when a tanh network is initialized with small weights and over the course of training comes to
saturate its nonlinear units (as it must to compute most functions of practical interest, see discussion
in Appendix D), it will enter a compression period where mutual information decreases. Figures
16-17 of Appendix E show histograms of neural activity over the course of training, demonstrating
that activities in thetanh network enter the saturation regime during training. This nonlinearity-based
compression furnishes another explanation for the observation that training slows down as tanh
networks enter their compression phase (Shwartz-Ziv & Tishby, 2017): some fraction of inputs have
saturated the nonlinearities, reducing backpropagated error gradients.
3 I NFORMATION PLANE DYNAMICS IN DEEP LINEAR NETWORKS
The preceding section investigates the role of nonlinearity in the observed compression behavior,
tracing the source to double-saturating nonlinearities and the binning methodology used to calculate
mutual information. However, other mechanisms could lead to compression as well. Even without
nonlinearity, neurons could converge to highly correlated activations, or project out irrelevant direc-
tions of the input. These phenomena are not possible to observe in our simple three neuron minimal
model, as they require multiple inputs and hidden layer activities. To search for these mechanisms,
we turn to a tractable model system: deep linear neural networks (Baldi & Hornik (1989); Fukumizu
(1998); Saxe et al. (2014)). In particular, we exploit recent results on the generalization dynamics
in simple linear networks trained in a student-teacher setup (Seung et al., 1992; Advani & Saxe,
2017). In a student-teacher setting, one “student” neural network learns to approximate the output
of another “teacher” neural network. This setting is a way of generating a dataset with interesting
structure that nevertheless allows exact calculation of the generalization performance of the network,
exact calculation of the mutual information of the representation (without any binning procedure),
and, though we do not do so here, direct comparison to the IB bound which is already known for
linear Gaussian problems (Chechik et al., 2005).
We consider a scenario where a linear teacher neural network generates input and output examples
which are then fed to a deep linear student network to learn (Fig. 3A). Following the formulation
of (Advani & Saxe, 2017), we assume multivariate Gaussian inputsX∼N (0, 1
Ni
INi) and a scalar
outputY . The output is generated by the teacher network according to Y = W0X +ϵo, where
ϵo∼N (0,σ 2
o) represents aspects of the target function which cannot be represented by a neural
network (that is, the approximation error or bias in statistical learning theory), and the teacher weights
Wo are drawn independently fromN (0,σ 2
w). Here, the weights of the teacher define the rule to be
learned. The signal to noise ratio SNR =σ2
w/σ2
o determines the strength of the rule linking inputs to
6

Published as a conference paper at ICLR 2018
outputs relative to the inevitable approximation error. We emphasize that the “noise” added to the
teacher’s output is fundamentally different from the noise added for the purpose of calculating mutual
information:ϵo models the approximation error for the task–even the best possible neural network
may still make errors because the target function is not representable exactly as a neural network–and
is part of the construction of the dataset, not part of the analysis of the student network.
To train the student network, a dataset of P examples is generated using the teacher. The student
network is then trained to minimize the mean squared error between its output and the target output
using standard (batch or stochastic) gradient descent on this dataset. Here the student is a deep
linear neural network consisting of potentially many layers, but where the the activation function
of each neuron is simply f(u) = u. That is, a depth D deep linear network computes the output
ˆY =WD+1WD··· W2W1X. While linear activation functions stop the network from computing
complex nonlinear functions of the input, deep linear networks nevertheless show complicated
nonlinear learning trajectories (Saxe et al., 2014), the optimization problem remains nonconvex (Baldi
& Hornik, 1989), and the generalization dynamics can exhibit substantial overtraining (Fukumizu,
1998; Advani & Saxe, 2017).
Importantly, because of the simplified setting considered here, the true generalization error is easily
shown to be
Eg(t) =||Wo−Wtot(t)||2
F +σ2
o (5)
whereWtot(t) is the overall linear map implemented by the network at training epoch t (that is,
Wtot =WD+1WD··· W2W1).
Furthermore, the mutual information with the input and output may be calculated exactly, because
the distribution of the activity of any hidden layer is Gaussian. Let T be the activity of a specific
hidden layer, and let ¯W be the linear map from the input to this activity (that is, for layer l, ¯W =
Wl··· W2W1). Since T = ¯WX , the mutual information ofX andT calculated using differential
entropy is infinite. For the purpose of calculating the mutual information, therefore, we assume that
Gaussian noise is added to the hidden layer activity, T = ¯WX +ϵM I, with mean 0 and variance
σ2
M I = 1.0. This allows the analysis to apply to networks of any size, including overcomplete layers,
but as before we emphasize that we do not add this noise either during training or testing. With these
assumptions,T andX are jointly Gaussian and we have
I(T ;X) = log| ¯W ¯W T +σ2
M IINh|− log|σ2
M IINh| (6)
where|·| denotes the determinant of a matrix. Finally the mutual information with the output Y , also
jointly Gaussian, can be calculated similarly (see Eqns. (22)-(25) of Appendix G).
Fig. 3 shows example training and test dynamics over the course of learning in panel C, and the
information plane dynamics in panel D. Here the network has an input layer of 100 units, 1 hidden
layer of 100 units each and one output unit. The network was trained with batch gradient descent on a
dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network
behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it
learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting
we study here, generalization performance can be acceptable without any compression phase.
The results in (Advani & Saxe (2017)) show that, for the case of linear networks, overtraining is
worst when the number of inputs matches the number of training samples, and is reduced by making
the number of samples smaller or larger. Fig. 4 shows learning dynamics with the number of samples
matched to the size of the network. Here overfitting is substantial, and again no compression is
seen in the information plane. Comparing to the result in Fig. 3D, both networks exhibit similar
information dynamics with respect to the input (no compression), but yield different generalization
performance.
Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that
cause compression over the course of learning; and generalization behavior can be widely different
for networks with the same dynamics of information compression regarding the input. We note that,
in the setting considered here, all input dimensions have the same variance, and the weights of the
teacher are drawn independently. Because of this, there are no special directions in the input, and
each subspace of the input contains as much information as any other. It is possible that, in real
world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large
weights in the teacher). We have not investigated this possibility here.
7

Published as a conference paper at ICLR 2018
A B
Error
C D
Epochs
Figure 4: Overtraining and information plane dynamics. (A) Average training and test mean square
error for a deep linear network trained with SGD. Overtraining is substantial. Other parameters:Ni =
100, P = 100, Number of hidden units = 100, Batch size = 5 (B) Information plane dynamics. No
compression is observed, and information about the labels is lost during overtraining. (C) Average
train and test accuracy (% correct) for nonlineartanh networks exhibiting modest overfitting (N = 8).
(D) Information plane dynamics. Overfitting occurs despite continued compression.
Figure 5: Stochastic training and the information plane. (A)tanh network trained with SGD. (B)tanh
network trained with BGD. (C) ReLU network trained with SGD. (D) ReLU network trained with
BGD. Both random and non-random training procedures show similar information plane dynamics.
To see whether similar behavior arises in nonlinear networks, we trained tanh networks in the same
setting as Section 2, but with 30% of the data, which we found to lead to modest overtraining.
Fig. 4C-D shows the resulting train, test, and information plane dynamics. Here the tanh networks
show substantial compression, despite exhibiting overtraining. This establishes a dissociation between
behavior in the information plane and generalization dynamics: networks that compress may (Fig. 1A)
or may not (Fig. 4C-D) generalize well, and networks that do not compress may (Figs.1B, 3A-B) or
may not (Fig. 4A-B) generalize well.
4 C OMPRESSION IN BATCH GRADIENT DESCENT AND SGD
Next, we test a core theoretical claim of the information bottleneck theory of deep learning, namely
that randomness in stochastic gradient descent is responsible for the compression phase. In particular,
because the choice of input samples in SGD is random, the weights evolve in a stochastic way during
training.
Shwartz-Ziv & Tishby (2017) distinguish two phases of SGD optimization: in the first “drift” phase,
the mean of the gradients over training samples is large relative to the standard deviation of the
gradients; in the second “diffusion” phase, the mean becomes smaller than the standard deviation
of the gradients. The authors propose that compression should commence following the transition
8

Published as a conference paper at ICLR 2018
from a high to a low gradient signal-to-noise ratio (SNR), i.e., the onset of the diffusion phase. The
proposed mechanism behind this diffusion-driven compression is as follows. The authors state that
during the diffusion phase, the stochastic evolution of the weights can be described as a Fokker-Planck
equation under the constraint of small training error. Then, the stationary distribution over weights
for this process will have maximum entropy, again subject to the training error constraint. Finally,
the authors claim that weights drawn from this stationary distribution will maximize the entropy
of inputs given hidden layer activity,H(X|T ), subject to a training error constraint, and that this
training error constraint is equivalent to a constraint on the mutual information I(T ;Y ) for small
training error. Since the entropy of the input, H(X), is fixed, the result of the diffusion dynamics
will be to minimizeI(X;T ) :=H(X)−H(X|T ) for a given value ofI(T ;Y ) reached at the end
of the drift phase.
However, this explanation does not hold up to either theoretical or empirical investigation. Let
us assume that the diffusion phase does drive the distribution of weights to a maximum entropy
distribution subject to a training error constraint. Note that this distribution reflects stochasticity
of weights across different training runs. There is no general reason that a given set of weights
sampled from this distribution (i.e., the weight parameters found in one particular training run)
will maximizeH(X|T ), the entropy of inputs given hidden layer activity. In particular, H(X|T )
reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather
than uncertainty about any kind of distribution across different training runs.
We also show empirically that the stochasticity of the SGD is not necessary for compression. To do
so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which
learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from
the dataset and calculating the gradient of the error with respect to that single sample (the typical
procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset,
and updates weights using the gradient of the total error across all examples. Batch gradient descent
uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in
its updates.
We trained tanh and ReLU networks with SGD and BGD and compare their information plane
dynamics in Fig. 5 (see Appendix H for a linear network). We find largely consistent information
dynamics in both instances, with robust compression in tanh networks for both methods. Thus
randomness in the training process does not appear to contribute substantially to compression of
information about the input. This finding is consistent with the view presented in Section 2 that
compression arises predominantly from the double saturating nonlinearity.
Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between
compression and the transition from high to low gradient SNR. Fig. 20 of Appendix I shows the
gradient SNR over training, which in all cases shows a phase transition during learning. Hence
the gradient SNR transition is a general phenomenon, but is not causally related to compression.
Appendix I offers an extended discussion and shows gradient SNR transitions without compression
on the MNIST dataset and for linear networks.
5 S IMULTANEOUS FITTING AND COMPRESSION
Our finding that generalization can occur without compression may seem difficult to reconcile with
the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme,
if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To
study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the
inputX into a set of task-relevant inputs Xrel and a set of task-irrelevant inputs Xirrel, and alter
the teacher network so that the teacher’s weights to the task-irrelevant inputs are all zero. Hence the
inputsXirrel contribute only noise, while theXrel contain signal. We then calculate the information
plane dynamics for the whole layer, and for the task-relevant and task-irrelevant inputs separately.
Fig. 6 shows information plane dynamics for a deep linear neural network trained using SGD (5
samples/batch) on a task with 30 task-relevant inputs and 70 task-irrelevant inputs. While the overall
dynamics show no compression phase, the information specifically about the task-irrelevant subspace
does compress over the course of training. This compression process occurs at the same time as
the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the
9

Published as a conference paper at ICLR 2018
A B C
Figure 6: Simultaneous fitting and compression. (A) For a task with a large task-irrelevant subspace
in the input, a linear network shows no overall compression of information about the input. (B)
The information with the task-relevant subspace increases robustly over training. (C) However, the
information specifically about the task-irrelevant subspace does compress after initially growing as
the network is trained.
information with these inputs specifically will indeed be reduced; but overall mutual information
with the input in general may still increase.
6 D ISCUSSION
Our results suggest that compression dynamics in the information plane are not a general feature of
deep networks, but are critically influenced by the nonlinearities employed by the network. Double-
saturating nonlinearities lead to compression, if mutual information is estimated by binning activations
or by adding homoscedastic noise, while single-sided saturating nonlinearities like ReLUs do not
compress in general. Consistent with this view, we find that stochasticity in the training process does
not contribute to compression in the cases we investigate. Furthermore, we have found instances
where generalization performance does not clearly track information plane behavior, questioning the
causal link between compression and generalization. Hence information compression may parallel the
situation with sharp minima: although empirical evidence has shown a correlation with generalization
error in certain settings and architectures, further theoretical analysis has shown that sharp minima
can in fact generalize well (Dinh et al., 2017). We emphasize that compression still may occur within
a subset of the input dimensions if the task demands it. This compression, however, is interleaved
rather than in a secondary phase and may not be visible by information metrics that track the overall
information between a hidden layer and the input. Finally, we note that our results address the specific
claims of one scheme to link the information bottleneck principle with current practice in deep
networks. The information bottleneck principle itself is more general and may yet offer important
insights into deep networks (Achille & Soatto, 2017). Moreover, the information bottleneck principle
could yield fundamentally new training algorithms for networks that are inherently stochastic and
where compression is explicitly encouraged with appropriate regularization terms (Chalk et al., 2016;
Alemi et al., 2017; Kolchinsky et al., 2017).
ACKNOWLEDGMENTS
We thank Ariel Herbert-V oss for useful discussions. This work was supported by grant numbers
IIS 1409097 and CHE 1648973 from the US National Science Foundation, and by IARPA contract
#D16PC00002. Andrew Saxe and Madhu Advani thank the Swartz Program in Theoretical Theoretical
Neuroscience at Harvard University. Artemy Kolchinsky and Brendan Tracey would like to thank the
Santa Fe Institute for helping to support this research. Artemy Kolchinsky was supported by Grant
No. FQXi-RFP-1622 from the FQXi foundation and Grant No. CHE-1648973 from the US National
Science Foundation. Brendan Tracey was supported by AFOSR MURI on Multi-Information Sources
of Multi-Physics Systems under Award Number FA9550-15-1-0038.
REFERENCES
A. Achille and S. Soatto. On the Emergence of Invariance and Disentangling in Deep Representations.
arXiv preprint arXiv:1706.01350, 2017.
10

Published as a conference paper at ICLR 2018
M.S. Advani and A.M. Saxe. High-dimensional dynamics of generalization error in neural networks.
arXiv preprint arXiv:1710.03667, 2017.
A.A. Alemi, I. Fischer, J.V . Dillon, and K. Murphy. Deep variational information bottleneck. In
International Conference on Learning Representations, 2017.
P. Baldi and K. Hornik. Neural networks and principal component analysis: Learning from examples
without local minima. Neural Networks, 2:53–58, 1989.
P.L. Bartlett and S. Mendelson. Rademacher and Gaussian Complexities: Risk Bounds and Structural
Results. Journal of Machine Learning Research, 3:463–482, 2002.
M. Chalk, O. Marre, and G. Tkacik. Relevant sparse codes with variational information bottleneck.
In Advances in Neural Information Processing Systems, pp. 1957–1965, 2016.
G. Chechik, A. Globerson, N. Tishby, and Y . Weiss. Information bottleneck for gaussian variables.
Journal of Machine Learning Research, pp. 165–188, 2005.
J. Chee and P. Toulis. Convergence diagnostics for stochastic gradient descent with constant step size.
arXiv preprint arXiv:1710.06382, 2017.
A. Choromanska, M. Henaff, M. Mathieu, G. Arous, B., and Y . LeCun. The Loss Surfaces of
Multilayer Networks. In Proceedings of the 18th International Conference on Artificial Intelligence,
volume 38, 2015.
L. Dinh, R. Pascanu, S. Bengio, and Y . Bengio. Sharp Minima Can Generalize For Deep Nets. In
International Conference on Machine Learning, 2017.
K. Fukumizu. Effect of Batch Learning In Multilayer Neural Networks. In Proceedings of the 5th
International Conference on Neural Information Processing, pp. 67–70, 1998.
I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and
Y . Bengio. Generative Adversarial Nets.Advances in Neural Information Processing Systems, pp.
2672–2680, 2014.
J. Kadmon and H. Sompolinsky. Optimal Architectures in a Solvable Model of Deep Networks. In
Advances in Neural Information Processing Systems, 2016.
A. Kolchinsky and B.D. Tracey. Estimating mixture entropy with pairwise distances. Entropy, 19,
2017.
A. Kolchinsky, B.D. Tracey, and D.H. Wolpert. Nonlinear information bottleneck. arXiv preprint
arXiv:1705.02436, 2017.
A. Kraskov, H. Stögbauer, and P. Grassberger. Estimating mutual information.Physical Review E,
69:066138, 2004.
A. Krizhevsky, I. Sutskever, and G.E. Hinton. ImageNet classification with deep convolutional neural
networks. In Advances in Neural Information Processing Systems, pp. 1097–1105. 2012.
S. Laughlin. A simple coding procedure enhances a neuron’s information capacity. Zeitschrift für
Naturforschung c, 36:910–912, 1981.
Y . LeCun, Y . Bengio, and G.E. Hinton. Deep learning.Nature, 521:436–444, 2015.
W. Lotter, G. Kreiman, and D.D. Cox. Deep predictive coding networks for video prediction and
unsupervised learning. arXiv preprint arXiv:1605.08104, 2016.
G. Montufar, R. Pascanu, K. Cho, and Y . Bengio. On the Number of Linear Regions of Deep Neural
Networks. In Advances in Neural Information Processing Systems, 2014.
N. Murata. A statistical study of on-line learning. In On-line Learning in Neural Networks, pp. 63–92.
Cambridge University Press, 1998.
11

Published as a conference paper at ICLR 2018
B. Neyshabur, R. Tomioka, and N. Srebro. Norm-Based Capacity Control in Neural Networks. In
Proceedings of The 28th Conference on Learning Theory, volume 40, pp. 1–26, 2015.
H. Poggio, T.and Mhaskar, L. Rosasco, B. Miranda, and Q. Liao. Why and when can deep-but
not shallow-networks avoid the curse of dimensionality: A review. International Journal of
Automation and Computing, pp. 1–17, 2017.
A.M. Saxe, J.L. McClelland, and S. Ganguli. Exact solutions to the nonlinear dynamics of learning in
deep linear neural networks. In the International Conference on Learning Representations, 2014.
J. Schmidhuber. Deep Learning in Neural Networks: An Overview. Neural Networks, 61:85–117,
2015.
H.S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples.
Physical Review A, 45:6056–6091, 1992.
R. Shwartz-Ziv and N. Tishby. Opening the black box of deep neural networks via information.arXiv
preprint arXiv:1703.00810, 2017.
D. Silver, A. Huang, C.J. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser,
I. Antonoglou, V . Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner,
I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis. Mastering the
game of Go with deep neural networks and tree search. Nature, 529:484–489, 2016.
N. Tishby and N. Zaslavsky. Deep learning and the information bottleneck principle. In IEEE
Information Theory Workshop, 2015.
N. Tishby, F.C. Pereira, and W. Bialek. The information bottleneck method. Proceedings of the 37-th
Annual Allerton Conference on Communication, Control and Computing, pp. 368–377, 1999.
A L EARNING CURVES FOR tanh AND RELU NETWORKS
Supplementary Figure 7 shows the learning curves for tanh and ReLU networks depicted in Fig. 1.
Figure 7: Learning curves for (A) tanh neural network in 1 A and (B) ReLU neural network in 1 B.
Both networks show good generalization with regards to the test data.
B R OBUSTNESS OF FINDINGS TO MI ESTIMATION METHOD AND NEURAL
ACTIVATION FUNCTIONS
This Appendix investigates the generality of the finding that compression is not observed in neural
network layers with certain activation functions. Figure 1 of the main text shows example results
using a binning-based MI estimator and a nonparametric KDE estimator, for both thetanh and ReLU
activation functions. Here we describe the KDE MI estimator in detail, and present extended results
on other datasets. We also show results for other activation functions. Finally, we provide entropy
estimates based on another nonparametric estimator, the popular k-nearest neighbor approach of
Kraskov et al. (2004). Our findings consistently show that double-saturating nonlinearities can yield
compression, while single-sided nonlinearities do not.
12

Published as a conference paper at ICLR 2018
B.1 K ERNEL DENSITY ESTIMATION OF MI
The KDE approach of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) estimates the mutual
information between the input and the hidden layer activity by assuming that the hidden activity is
distributed as a mixture of Gaussians. This assumption is well-suited to the present setting under
the following interpretation: we take the input activity to be distributed as delta functions at each
example in the dataset, corresponding to a uniform distribution over these specific samples. In other
words, we assume that the empirical distribution of input samples is the true distribution. Next, the
hidden layer activityh is a deterministic function of the input. As mentioned in the main text and
discussed in more detail in Appendix C, without the assumption of noise, this would have infinite
mutual information with the input. We therefore assume for the purposes of analysis that Gaussian
noise of varianceσ2 is added, that is,T =h +ϵ whereϵ∼N (0,σ 2I). Under these assumptions,
the distribution ofT is genuinely a mixture of Gaussians, with a Gaussian centered on the hidden
activity corresponding to each input sample. We emphasize again that the noiseϵ is added solely for
the purposes of analysis, and is not present during training or testing the network. In this setting, an
upper bound for the mutual information with the input is (Kolchinsky & Tracey, 2017; Kolchinsky
et al., 2017)
I(T ;X)≤− 1
P
∑
i
log 1
P
∑
j
exp
(
−1
2
||hi−hj||2
2
σ2
)
(7)
whereP is the number of training samples andhi denotes the hidden activity vector in response to
input samplei. Similarly, the mutual information with respect to the output can be calculated as
I(T ;Y ) = H(T )−H(T|Y ) (8)
≤ − 1
P
∑
i
log 1
P
∑
j
exp
(
−1
2
‖hi−hj‖2
2
σ2
)
(9)
−
L∑
l
pl

− 1
Pl
∑
i,Yi=l
log 1
Pl
∑
j,Yj =l
exp
(
−1
2
‖hi−hj‖2
2
σ2
)
 (10)
whereL is the number of output labels,Pl denotes the number of data samples with output labell,
pl =Pl/P denotes the probability of output labell, and the sums overi,Y i =l indicate a sum over
all examples with output labell.
13

Published as a conference paper at ICLR 2018
Figure 8: Information plane dynamics for the network architecture and training dataset of Shwartz-
Ziv & Tishby (2017), estimated with the nonparametric KDE method of Kolchinsky & Tracey
(2017); Kolchinsky et al. (2017) and averaged over 50 repetitions. (A) tanh neural network layers
show compression. (B) ReLU neural network layers show no compression. (C) The soft-sign
activation function, a double-saturating nonlinearity that saturates more gently than tanh, shows
modest compression. (D) The soft-plus activation function, a smoothed version of the ReLU, exhibits
no compression. Hence double-saturating nonlinearities exhibit the compression effect while single-
saturating nonlinearities do not.
Figure 8A-B shows the result of applying this MI estimation method on the dataset and network
architecture of Shwartz-Ziv & Tishby (2017), with MI estimated on the full dataset and averaged
over 50 repetitions. Mutual information was estimated using data samples from the test set, and we
took the noise varianceσ2 = 0.1. These results look similar to the estimate derived from binning,
with compression in tanh networks but no compression in ReLU networks. Relative to the binning
estimate, it appears that compression is less pronounced in the KDE method.
Figure 1C-D of the main text shows the results of this estimation technique applied to a neural network
of size 784− 1024− 20− 20− 20− 10 on the MNIST handwritten digit classification dataset.
The network was trained using SGD with minibatches of size 128. As before, mutual information
was estimated using data samples from the test set, and we took the noise varianceσ2 = 0.1. The
smaller layer sizes in the top three hidden layers were selected to ensure the quality of the kernel
density estimator given the amount of data in the test set, since the estimates are more accurate for
smaller-dimensional data. Because of computational expense, the MNIST results are from a single
training run.
More detailed results for the MNIST dataset are provided in Figure 9 for the tanh activation function,
and in Figure 10 for the ReLU activation function. In these figures, the first row shows the evolution
of the cross entropy loss (on both training and testing data sets) during training. The second row
shows the mutual information between input and the activity of different hidden layers, using the
nonparametric KDE estimator described above. The blue region in the second row shows the range
of possible MI values, ranging from the upper bound described above (Eq. 10) to the following lower
14

Published as a conference paper at ICLR 2018
bound (Kolchinsky & Tracey, 2017),
I(T ;Y ) ≥ − 1
P
∑
i
log 1
P
∑
j
exp
(
−1
2
‖hi−hj‖2
2
4σ2
)
(11)
−
L∑
l
pl

− 1
Pl
∑
i,Yi=l
log 1
Pl
∑
j,Yj =l
exp
(
−1
2
‖hi−hj‖2
2
4σ2
)
. (12)
The third row shows the mutual information between input and activity of different hidden layers,
estimated using the binning method (here, the activity of each neuron was discretized into bins of
size 0.5). For both the second and third rows, we also plot the entropy of the inputs, H(X), as a
dashed line. H(X) is an upper bound on the mutual information I(X;T ), and is computed using
the assumption of a uniform distribution over the 10,000 testing points in the MNIST dataset, giving
H(X) = log2 10000.
Finally, the fourth row visualizes the dynamics of the SGD updates during training. For each layer and
epoch, the green line shows the𝓁2 norm of the weights. We also compute the vector of mean updates
across SGD minibatches (this vector has one dimension for each weight parameter), as well as the
vector of the standard deviation of the updates across SGD minibatches. The𝓁2 norm of the mean
update vector is shown in blue, and the𝓁2 norm of the standard deviation vector is shown in orange.
The gradient SNR, computed as the ratio of the norm of the mean vector to the norm of the standard
deviation vector, is shown in red. For both the tanh and ReLU networks, the gradient SNR shows a
phase transition during training, and the norm of the weights in each layer increases. Importantly,
this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise
in SGD updates does not yield compression in this setting.
15

Published as a conference paper at ICLR 2018
100 101 102 103 104
Epoch
0.0
0.5
1.0
1.5
2.0
2.5Cross/uni00A0entropy/uni00A0loss
Loss
Train
Test
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A01/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A01/uni00A0Mutual/uni00A0Info/uni00A0(binned)
100 101 102 103 104
Epoch
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101
102 Layer/uni00A01/uni00A0SNR
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A02/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A02/uni00A0Mutual/uni00A0Info/uni00A0(binned)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101 Layer/uni00A02/uni00A0SNR
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A03/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A03/uni00A0Mutual/uni00A0Info/uni00A0(binned)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101 Layer/uni00A03/uni00A0SNR
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A04/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
H(X)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A04/uni00A0Mutual/uni00A0Info/uni00A0(binned)
H(X)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101 Layer/uni00A04/uni00A0SNR
Mean
Std
SNR
||W||
Figure 9: Detailed tanh activation function results on MNIST. Row 1: Loss over training. Row
2: Upper and lower bounds for the mutual information I(X;T ) between the input (X) and each
layer’s activity (T ), using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky
et al., 2017). Dotted line indicatesH(X) = log2 10000, the entropy of a uniform distribution over
10,000 testing samples. Row 3: Binning-based estimate of the mutual information I(X;T ), with
each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm
dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in
each layer increases.
16

Published as a conference paper at ICLR 2018
100 101 102 103 104
Epoch
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5Cross/uni00A0entropy/uni00A0loss
Loss
Train
Test
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A01/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A01/uni00A0Mutual/uni00A0Info/uni00A0(binned)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101
102 Layer/uni00A01/uni00A0SNR
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A02/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A02/uni00A0Mutual/uni00A0Info/uni00A0(binned)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101 Layer/uni00A02/uni00A0SNR
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A03/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A03/uni00A0Mutual/uni00A0Info/uni00A0(binned)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101 Layer/uni00A03/uni00A0SNR
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A04/uni00A0Mutual/uni00A0Info/uni00A0(KDE)
H(X)
100 101 102 103 104
Epoch
0
2
4
6
8
10
12
14
I(X; T)
Layer/uni00A04/uni00A0Mutual/uni00A0Info/uni00A0(binned)
H(X)
100 101 102 103 104
Epoch
10/uni00AD5
10/uni00AD4
10/uni00AD3
10/uni00AD2
10/uni00AD1
100
101 Layer/uni00A04/uni00A0SNR
Mean
Std
SNR
||W||
Figure 10: Detailed ReLU activation function results on MNIST. Row 1: Loss over training. Row
2: Upper and lower bounds for the mutual information I(X;T ) between the input (X) and each
layer’s activity (T ), using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky
et al., 2017). Dotted line indicatesH(X) = log2 10000, the entropy of a uniform distribution over
10,000 testing samples. Row 3: Binning-based estimate of the mutual information I(X;T ), with
each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm
dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in
each layer increases. Importantly, this phase transition occurs despite a lack of compression in the
ReLU network, indicating that noise in SGD updates does not yield compression in this setting.
17

Published as a conference paper at ICLR 2018
-5 0 5
Net input
-1
0
1
2
3
4
5
6
Activation
tanh
ReLU
Softsign
Softplus
Figure 11: Alternative activation functions.
B.2 O THER ACTIVATION FUNCTIONS
Next, in Fig. 8C-D, we show results from the kernel MI estimator from two additional nonlinear
activation functions, the softsign function
f(x) = x
1 +|x|,
and the softplus function
f(x) = ln(1 +ex).
These functions are plotted next to tanh and ReLU in Fig. 11. The softsign function is similar to
tanh but saturates more slowly, and yields less compression than tanh. The softplus function is a
smoothed version of the ReLU, and yields similar dynamics with no compression. Because softplus
never saturates fully to zero, it retains more information with respect to the input than ReLUs in
general.
B.3 K RASKOV ESTIMATOR
We additionally investigated the widely-used nonparametric MI estimator of Kraskov et al. (2004).
This estimator uses nearest neighbor distances between samples to compute an estimate of the
entropy of a continuous random variable. Here we focused for simplicity only on the compression
phenomenon in the mutual information between the input and hidden layer activity, leaving aside the
information with respect to the output (as this is not relevant to the compression phenomenon). Again,
without additional noise assumptions, the MI between the hidden representation and the input would
be infinite because the mapping is deterministic. Rather than make specific noise assumptions, we
instead use the Kraskov method to estimate the entropy of the hidden representationsT . Note that the
entropy ofT is the mutual information up to an unknown constant so long as the noise assumption is
homoscedastic, that is,T =h +Z where the random variableZ is independent ofX. To see this,
note that
I(T ;X) = H(T )−H(T|X) (13)
= H(T )−H(Z) (14)
= H(T )−c (15)
where the constantc =H(Z). Hence observing compression in the layer entropy H(T ) is enough to
establish that compression occurs in the mutual information.
The Kraskov estimate is given by
d
P
P∑
i=1
log(ri +ϵ) +d
2 log(π)− log Γ(d/2 + 1) +ψ(P )−ψ(k) (16)
whered is the dimension of the hidden representation,P is the number of samples,ri is the distance
to the k-th nearest neighbor of sample i,ϵ is a small constant for numerical stability, Γ(·) is the
18

Published as a conference paper at ICLR 2018
Gamma function, andψ(·) is the digamma function. Here the parameter ϵ prevents infinite terms
when the nearest neighbor distanceri = 0 for some sample. We tookϵ = 10−16.
Figure 12 shows the entropy over training for tanh and ReLU networks trained on the dataset of
and with the network architecture in Shwartz-Ziv & Tishby (2017), averaged over 50 repeats. In
these experiments, we usedk = 2. Compression would correspond to decreasing entropy over the
course of training, while a lack of compression would correspond to increasing entropy. Severaltanh
layers exhibit compression, while the ReLU layers do not. Hence qualitatively, the Kraskov estimator
returns similar results to the binning and KDE strategies.
A B
Figure 12: Entropy dynamics over training for the network architecture and training dataset of
Shwartz-Ziv & Tishby (2017), estimated with the nonparametric k-nearest-neighbor-based method of
Kraskov et al. (2004). Here the x-axis is epochs of training time, and the y-axis plots the entropy of
the hidden representation, as calculated using nearest-neighbor distances. Note that in this setting, if
T is considered to be the hidden activity plus independent noise, the entropy is equal to the mutual
information up to a constant (see derivation in text). Layers 0-4 correspond to the hidden layers of
size 10-7-5-4-3. (A) tanh neural network layers can show compression over the course of training.
(B) ReLU neural network layers show no compression.
C N OISE ASSUMPTIONS AND DISCRETE VS CONTINUOUS ENTROPY
A recurring theme in the results reported in this paper is the necessity of noise assumptions to yield a
nontrivial information theoretic analysis. Here we give an extended discussion of this phenomenon,
and of issues relating to discrete entropy as opposed to continuous (differential) entropy.
The activity of a neural network is often a continuous deterministic function of its input. That is, in
response to an inputX, a specific hidden layer might produce activityh =f(X) for some function
f. The mutual information betweenh andX is given by
I(h;X) = H(h)−H(h|X). (17)
Ifh were a discrete variable, then the entropy would be given by
H(h) =−
N∑
i=1
pi logpi (18)
wherepi is the probability of the discrete symboli, as mentioned in the main text. ThenH(h|X) = 0
because the mapping is deterministic and we haveI(h;X) =H(h).
Howeverh is typically continuous. The continuous entropy, defined for a continuous random variable
Z with densitypZ by analogy to Eqn. (18) as
H(Z) =−
∫
pZ(z) logpZ(z)dz, (19)
can be negative and possibly infinite. In particular, note that ifpZ is a delta function, thenH(Z) =
−∞. The mutual information between hidden layer activityh and the inputX for continuoush,X is
I(h;X) =H(h)−H(h|X). (20)
19

Published as a conference paper at ICLR 2018
NowH(h|X) =−∞ since given the inputX, the hidden activityh is distributed as a delta function
atf(X). The mutual information is thus generally infinite, so long as the hidden layer activity has
finite entropy (H(h) is finite).
Figure 13: Effect of binning strategy on minimal three neuron model. Mutual information for the
simple three neuron model shown in Fig. 2 with bin edges bi∈ tanh(linspace(−50, 50,N )). In
contrast to linear binning, the mutual information continues to increase as weights grow.
To yield a finite mutual information, some noise in the mapping is required such thatH(h|X) remains
finite. A common choice (and one adopted here for the linear network, the nonparametric kernel
density estimator, and the k-nearest neighbor estimator) is to analyze a new variable with additive
noise,T =h +Z, whereZ is a random variable independent ofX. ThenH(T|X) =H(Z) which
allows the overall informationI(T ;X) =H(T )−H(Z) to remain finite. This noise assumption is
not present in the actual neural networks either during training or testing, and is made solely for the
purpose of calculating the mutual information.
Another strategy is to partition the continuous variable h into a discrete variable T, for instance
by binning the values (the approach taken in Shwartz-Ziv & Tishby (2017)). This allows use
of the discrete entropy, which remains finite. Again, however, in practice the network does not
operate on the binned variablesT but on the continuous variablesh, and the binning is solely for
the purpose of calculating the mutual information. Moreover, there are many possible binning
strategies, which yield different discrete random variables, and different mutual information with
respect to the input. The choice of binning strategy is an assumption analogous to choosing a type
of noise to add to the representation in the continuous case: because there is in fact no binning in
the operation of the network, there is no clear choice for binning methodology. The strategy we use
in binning-based experiments reported here is the following: for bounded activations like the tanh
activation, we use evenly spaced bins between the minimum and maximum limits of the function. For
unbounded activations like ReLU, we first train the network completely; next identify the minimum
and maximum hidden activation over all units and all training epochs; and finally bin into equally
spaced bins between these minimum and maximum values. We note that this procedure places no
restriction on the magnitude that the unbounded activation function can take during training, and
yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger
than the maximum are never seen during training).
As an example of another binning strategy that can yield markedly different results, we consider
evenly spaced bins in a neuron’s net input, rather than its activity. That is, instead of evenly spaced
bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values
through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation
region as compared to the linear region. Figure 13 shows the results of applying this binning strategy
20

Published as a conference paper at ICLR 2018
Figure 14: Effect of binning strategy on information plane dynamics. Results for the same tanh
network and training regime as 1A, but with bin edgesbi∈ tanh(linspace(−50, 50,N )). Measured
with this binning structure, there is no compression in most layers.
to the minimal three neuron model with tanh activations. This binning scheme captures more
information as the weights of the network grow larger. Figure 14 shows information plane dynamics
for this binning structure. The tanh network no longer exhibits compression. (We note that the
broken DPI in this example is an artifact of performing binning only for analysis, as discussed below).
Any implementation of a neural network on digital hardware is ultimately of finite precision, and
hence is a binned, discrete representation. However, it is a very high resolution binning compared
to that used here or by Shwartz-Ziv & Tishby (2017): single precision would correspond to using
roughly 232 bins to discretize each hidden unit’s activity, as compared to the 30-100 used here. If the
binning is fine-grained enough that each input X yields a different binned activity patternh, then
H(h) = log(P ) whereP is the number of examples in the dataset, and there will be little to no
change in information during training. As an example, we show in Fig. 15 the result of binning at full
machine precision.
Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis.
First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual
information estimates. The DPI states that information can only be destroyed through successive
transformations, that is, ifX→h1→h2 form a Markov chain, thenI(X;h1)≥I(X;h2) (see, eg,
Tishby & Zaslavsky (2015)). Because noise is added only for the purpose of analysis, however, this
does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to
propagate through the network to higher layers. That is, if the transformation from hidden layer 1
to hidden layer 2 is h2 =f(h1) andT1 =h1 +Z1 is the hidden layer activity after adding noise,
then the DPI would hold for the variable ˜T2 = f(T1) +Z2 = f(h1 +Z1) +Z2, not the quantity
21

Published as a conference paper at ICLR 2018
I(
A B
I(Y;T)I(X;T)
I(X;T)
I(Y;T)
Figure 15: Effect of binning at full machine precision. (A) ReLU network. (B) tanh network.
Information in most layers stays pinned to log2(P ) = 12 . Compression is only observed in the
highest and smallest layers near the very end of training, when the saturation oftanh is strong enough
to saturate machine precision.
T2 = h2 +Z2 = f(h1) +Z2 used in the analysis. Said another way, the Markov chain for T2 is
X→h1→h2→T2, notX→h1→T1→T2, so the DPI states only thatI(X;h1)≥I(X;T2).
A second consequence of the noise assumption is the fact that the mutual information is no longer
invariant to invertible transformations of the hidden activity h. A potentially attractive feature
of a theory based on mutual information is that it can allow for comparisons between different
architectures: mutual information is invariant to any invertible transformation of the variables, so two
hidden representations could be very different in detail but yield identical mutual information with
respect to the input. However, once noise is added to a hidden representation, this is no longer the case:
the variableT =h +Z is not invariant to reparametrizations ofh. As a simple example, consider
a minimal linear network with scalar weights w1 andw2 that computes the output ˆy = w2w1X.
The hidden activity ish =w1X. Now consider the family of networks in which we scale downw1
and scale upw2 by a factorc⁄= 0, that is, these networks have weights ˜w1 =w1/c and ˜w2 =cw2,
yielding the exact same input-output map ˆy = ˜w2 ˜w1X =cw2(w1/c)X =w2w1X. Because they
compute the same function, they necessarily generalize identically. However after introducing the
noise assumption the mutual information is
I(T ;X) = log
(
w2
1/c2 +σ2
M I
)
− log
(
σ2
M I
)
(21)
where we have taken the setting in Section 3 in which X is normal Gaussian, and independent
Gaussian noise of varianceσ2
M I is added for the purpose of MI computation. Clearly, the mutual
information is now dependent on the scalingc of the internal layer, even though this is an invertible
linear transformation of the representation. Moreover, this shows that networks which generalize
identically can nevertheless have very different mutual information with respect to the input when it
is measured in this way.
D W EIGHT NORMS OVER TRAINING
Our argument relating neural saturation to compression in mutual information relies on the notion
that in typical training regimes, weights begin small and increase in size over the course of training.
We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin:
when initialized with small weights, the activity of a tanh network will be in this linear regime and
the network can only compute a linear function of its input. Hence a real world nonlinear task can
only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some
examples. This point can also be appreciated from norm-based capacity bounds on neural networks,
which show that, for instance, the Rademacher complexity of a neural network with small weights
must be low (Bartlett & Mendelson, 2002; Neyshabur et al., 2015). Finally, as an empirical matter,
the networks trained in this paper do in fact increase the norm of their weights over the course of
22

Published as a conference paper at ICLR 2018
training, as shown by the green lines in Figure 20 for tanh and ReLU networks in the training setting
of Shwartz-Ziv & Tishby (2017); Figures 9 and 10 for the MNIST networks; and Figure 21 for a
linear network.
E H ISTOGRAMS OF NEURAL ACTIVATIONS
Supplementary Figures 16 and 17 show histograms of neural activities over the course of training in
tanh and ReLU networks respectively.
Figure 16: Histogram of neural activities in a tanh network during training. The final three layers
eventually saturate in the top and bottom bins corresponding to the saturation limits of the tanh
activation function, explaining the compression observed in tanh. x-axis: training time in epochs.
y-axis: Hidden activity bin values from lowest to highest. Colormap: density of hidden layer activities
across all input examples.
23

Published as a conference paper at ICLR 2018
Figure 17: Histogram of neural activities in a ReLU network during training. ReLU layers 1-5 have
a roughly constant fraction of activities at zero, corresponding to instances where the ReLU is off;
the nonzero activities disperse over the course of training without bound, yielding higher entropy
distributions. The sigmoid output layer 6 converges to its saturation limits, and is the only layer that
compresses during training (c.f. Fig. 1B). x-axis: training time in epochs. y-axis: Hidden activity
value. Colormap: density of hidden layer activities across all input examples.
F I NFORMATION PLANE DYNAMICS IN DEEPER LINEAR NETWORKS
Supplementary Figure 18 shows information plane dynamics for a deep neural network with five
hidden layers each containing 50 hidden units.
24

Published as a conference paper at ICLR 2018
Train
Test
A B
Figure 18: Information plane dynamics in a deep linear neural network. (A) Train and test error
during learning. (B) Information plane dynamics. No compression is visible.
G L INEAR MUTUAL INFORMATION CALCULATION
For the linear setting considered here, the mutual information between a hidden representationT and
the outputY may be calculated using the relations
H(Y ) = No
2 log(2πe) + 1
2 log|WoW T
o +σ2
oINo|, (22)
H(T ) = Nh
2 log(2πe) + 1
2 log| ¯W ¯W T +σ2
M IINh|, (23)
H(Y ;T ) = No +Nh
2 log(2πe) + 1
2 log
⏐⏐⏐⏐
¯W ¯W T +σ2
M IINh
¯WW T
o ,
Wo ¯W T WoW T
o +σ2
oINh
⏐⏐⏐⏐, (24)
I(Y ;T ) =H(Y ) +H(T )−H(Y ;T ). (25)
H S TOCHASTIC VS BATCH TRAINING
Figure 19 shows information plane dynamics for stochastic and batch gradient descent learning in a
linear network. Randomness in the training process does not dramatically alter the information plane
dynamics.
Figure 19: Effect of stochastic training in linear networks. (A) Information plane dynamics for
stochastic gradient descent in a linear network (same setting as Fig. 4). (B) Information plane
dynamics for batch gradient descent.
I G RADIENT SNR PHASE TRANSITION
The proposed mechanism of compression in Shwartz-Ziv & Tishby (2017) is noise arising from
stochastic gradient descent training. The results in Section 4 of the main text show that compression
25

Published as a conference paper at ICLR 2018
still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates
is not the cause of compression. Here we investigate a related claim, namely that during training,
networks switch between two phases. These phases are defined by the ratio of the mean of the gradient
to the standard deviation of the gradient across training examples, called the gradient signal-to-noise
ratio. In the first “drift” phase, the SNR is high, while in the second “diffusion” phase the SNR
is low. Shwartz-Ziv & Tishby (2017) hypothesize that the drift phase corresponds to movement
toward the minimum with no compression, while the diffusion phase corresponds to a constrained
diffusion in weight configurations that attain the optimal loss, during which representations compress.
However, two phases of gradient descent have been described more generally, sometimes known as
the transient and stochastic phases or search and convergence phases (Murata, 1998; Chee & Toulis,
2017), suggesting that these phases might not be related specifically to compression behavior.
In Fig. 20 we plot the gradient SNR over the course of training for the tanh and ReLU networks in
the standard setting of Shwartz-Ziv & Tishby (2017). In particular, for each layerl we calculate the
mean and standard deviation as
ml =
‖‖‖‖
⟨ ∂E
∂Wl
⟩‖‖‖‖
F
(26)
sl =
‖‖‖‖STD
( ∂E
∂Wl
)‖‖‖‖
F
(27)
where⟨·⟩ denotes the mean and STD (·) denotes the element-wise standard deviation across all
training samples, and‖·‖F denotes the Frobenius norm. The gradient SNR is then the ratioml/sl.
We additionally plot the norm of the weights‖Wl‖F over the course of training.
Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like
transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for
MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence
the two phase nature of gradient descent appears to hold across the settings that we examine here.
Crucially, this finding shows that the SNR transition is not related to the compression phenomenon
because ReLU networks, which show the gradient SNR phase transition, do not compress.
Finally, to show the generality of the two-phase gradient SNR behavior and its independence from
compression, we develop a minimal model of this phenomenon in a three neuron linear network. We
consider the student-teacher setting of Fig. 3 but withNi =Nh = 1, such that the input and hidden
layers have just a single neuron (as in the setting of Fig. 2). Here, with just a single hidden neuron,
clearly there can be no compression so long as the first layer weight increases over the course of
training. Figure 21AC shows that even in this simple setting, the SNR shows the phase transition but
the weight norm increases over training. Hence again, the two phases of the gradient are present even
though there is no compression. To intuitively understand the source of this behavior, note that the
weights are initialized to be small and hence early in learning all must be increased in magnitude,
yielding a consistent mean gradient. Once the network reaches the vicinity of the minimum, the mean
weight change across all samples by definition goes to zero. The standard deviation remains finite,
however, because on some specific examples error could be improved by increasing or decreasing the
weights–even though across the whole dataset the mean error has been minimized.
Hence overall, our results show that a two-phase structure in the gradient SNR occurs in all settings we
consider, even though compression occurs only in a subset. The gradient SNR behavior is therefore
not causally related to compression dynamics, consistent with the view that saturating nonlinearities
are the primary source of compression.
26

Published as a conference paper at ICLR 2018
Figure 20: Gradient SNR phase transition. (A) tanh networks trained in the standard setting of
Shwartz-Ziv & Tishby (2017) show a phase transition in every layer. (B) ReLU networks also show a
phase transition in every layer, despite exhibiting no compression.
Error
Error
B
C
A
D
Figure 21: Minimal model exhibiting gradient SNR phase transition. Here a three neuron linear
network (architecture 1− 1− 1) learns to approximate a teacher. Other parameters are teacher
SNR = 1, number of training samplesP = 100, learning rate.001. Left column: (A) The loss over
training with SGD (minibatch size 1). (C) The resulting gradient SNR dynamics. Right column:
(B) The loss over training with BGD. (D) The resulting gradient SNR dynamics averaging over all
training samples (not minibatches, see text).
27