# The Curious Case of Convex Neural Networks

Sarath Sivaprasad  1,2 Work done while at IIIT Hyderabad Ankur Singh1,3 Naresh
Manwani1 Vineet Gandhi1
1KCIS  IIIT Hyderabad  2TCS Research  Pune  3IIT Kanpur
<EMAIL_ADDRESS><EMAIL_ADDRESS>{naresh.manwani,
<EMAIL_ADDRESS>
###### Abstract

This paper investigates a constrained formulation of neural networks where the
output is a convex function of the input. We show that the convexity
constraints can be enforced on both fully connected and convolutional layers,
making them applicable to most architectures. The convexity constraints
include restricting the weights (for all but the first layer) to be non-
negative and using a non-decreasing convex activation function. Albeit simple,
these constraints have profound implications on the generalization abilities
of the network. We draw three valuable insights: (a) Input Output Convex
Neural Networks (IOC-NNs) self regularize and significantly reduce the problem
of overfitting; (b) Although heavily constrained, they outperform the base
multi layer perceptrons and achieve similar performance as compared to base
convolutional architectures and (c) IOC-NNs show robustness to noise in train
labels. We demonstrate the efficacy of the proposed idea using thorough
experiments and ablation studies on six commonly used image classification
datasets with three different neural network architectures.

##  1 Introduction

Deep Neural Networks use multiple layers to extract higher-level features from
the raw input progressively. The ability to automatically learn features at
multiple levels of abstractions makes them a powerful machine learning system
that can learn complex relationships between input and output. Seminal work by
Zhang et al. [31] investigates the expressive power of neural networks on
finite sample sizes. They show that even when trained on completely random
labeling of the true data, neural networks achieve zero training error,
increasing training time and effort by only a constant factor. Such potential
of brute force memorization makes it challenging to explain the generalization
ability of deep neural networks. They further illustrate that the phenomena of
neural network fitting on random labeling of training data is largely
unaffected by explicit regularization (such as weight decay, dropout, and data
augmentation). They suggest that explicit regularization may improve
generalization performance but is neither necessary nor by itself sufficient
for controlling generalization error. Moreover, recent works show that
generalization (and test) error in neural networks reduces as we increase the
number of parameters [24, 23], which contradicts the traditional wisdom that
overparameterization leads to overfitting. These observations have given rise
to a branch of research that focuses on explaining the neural network’s
generalization error rather than just looking at their test performance [25].

|  |  |
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(a) True Label Experiment | (b) Random Label Experiment
|  |  |
(c) IOC-AllConv (50% noise) | (d) AllConv (50% noise)
Figure 1:  Training of AllConv and IOC-AllConv on CIFAR-10 dataset. (a) Loss
curve while training with true labels. AllConv starts overfitting after few
epochs. IOC-AllConv does not exhibit overfitting, and the test loss nicely
follows the training loss. (b) Accuracy plots while training with randomized
labels (labels were randomized for all the training images). If sufficiently
trained, even a simple network like MLP achieves 100% training accuracy and
gives around 10% test accuracy. IOC-MLP resists any learning on the randomized
data and gives 0% generalization gap. (c) and (d) Loss and accuracy plots on
CIFAR-10 data when trained with 50% labels randomized in the training set.

We propose a principled and reliable alternative that tries to affirmatively
resolve the concerns raised in [31]. More specifically, we investigate a novel
constrained family of neural networks called Input Output Convex Neural
Networks (IOC-NNs), which learn a convex function between input and output.
Convexity in machine learning typically refers to convexity in terms of the
parameters w.r.t to the loss [3], which is not the case in our work. We use an
IOC prefix to indicate the Input Output Convexity explicitly. Amos et al. [1]
have previously explored the idea of Input Output convexity; however, their
experiments limit to Partially Input Convex Neural Networks (PICNNs), where
the output is convex w.r.t some of the inputs. They deem fully convex networks
_unnecessary_ in their studied setting of structured prediction, _highly
restricted_ on the allowable class of models, _highly limited_ , even failing
to do simple identity mapping without additional skip (pass-through)
connections. Hence, they do not present even a single experiment on fully
convex networks.

We wake this sleeping giant up and thoroughly investigate fully convex
networks (outputs are convex w.r.t to all the inputs) on the task of multi-
class classification. Each class in multi-class classification is represented
as a convex function, and the resulting decision boundaries are formed as an
$\operatorname*{argmax}$ of convex functions. Being able to train IOC with NN-
like capacity, we, for the first time, discover the beautiful underlying
properties, especially in terms of generalization abilities and robustness to
label noise. We investigate IOC-NNs on six commonly used image classification
benchmarks and pose them as a preferred alternative over the non-convex
architectures. Our experiments suggest that IOC-NNs avoid fitting over the
noisy part of the data, in contrast to the typical neural network behavior.
Previous work shows that [2] neural networks tend to learn simpler hypotheses
first. Our experiments show that IOC-NNs tend to hold on to the simpler
hypothesis even in the presence of noise, without overfitting in most
settings.

A motivating example is illustrated in Figure 1, where we train an All
Convolutional network (AllConv) [29] and its convex counterpart IOC-AllConv on
the CIFAR-10 dataset. AllConv starts overfitting the train data after a few
epochs (Figure 1(a)). In contrast, IOC-AllConv shows no signs of overfitting
and flattens at the end (the test loss values pleasantly follow the training
curve). Such an observation is consistent across all our experiments on IOC-
NNs across different datasets and architectures, suggesting that IOC-NNs have
lesser reliance on explicit regularization like early stopping. Figure 1(b)
presents the accuracy plots for the randomized test where we train Multi-Layer
Perceptron (MLP) and IOC-MLP on a copy of the data where the true labels were
replaced by random labels. MLP achieves 100% accuracy on the train set and
gives a random chance performance on the test set (observations are coherent
with [31]). IOC-MLP resists any learning and gives random chance performance
(10% accuracy) on both train and test sets. As MLP achieves zero training
error, the test error is the same as generalization error, i.e., 90% (the
performance of random guessing on CIFAR10). In contrast, the IOC-MLP has a
near 0% generalization error. We further present experiment with 50% noisy
labels Figure 1(c). The neural network training profile concurs with the
observation of Krueger et al. [18], where the network learns a simpler
hypothesis first and then starts memorizing. On the other hand, IOC-NN
converges to the simpler hypothesis, showing strong resistance to fit the
noise labels.

Input Output Convexity shows a promising paradigm, as any feed-forward network
can be re-worked into its convex counterpart by choosing a non-decreasing (and
convex) activation function and restricting its weights to be non-negative
(for all but the first layer). Our experiments suggest that activation
functions that allow negative outputs (like leaky ReLU or ELU) are more suited
for the task as they help retain negative values flowing to subsequent layers
in the network. We show that IOC-MLPs outperforms traditional MLPs in terms of
test accuracy on five of the six studied datasets and IOC-NNs almost recover
the performance of the base network in case of convolutional networks. In
almost all studied scenarios, IOC networks achieve multi-fold improvements in
terms of generalization error over unconstrained Neural Networks. Overall, our
work makes the following contributions:

  * •

We bring to light the little known idea of Input Output Convexity in neural
networks. We propose a revised formulation to efficiently train IOC-NNs,
retaining adequate capacity (with changes like using ELU, increasing nodes in
the first layer, whitening transform at the input, etc.). To the best of our
knowledge, we for the first time explore a usable form of IOC-NNs, and shows
that they can be trained with NN like capacity.

  * •

Through a set of intuitive experiments, we detail its internal functioning,
especially in terms of its self regularization properties and decision
boundaries. We show that how sufficiently complex decision boundaries can be
learned using an $\operatorname*{argmax}$ over a set of convex functions
(where each class is represented by a single convex function). We further
propose a framework to learn the ensemble of IOC-NNs.

  * •

With a comprehensive set of quantitative and qualitative experiments, we
demonstrate IOC-NN’s outstanding generalization abilities. IOC-MLPs achieve
near zero generalization error in all the studied datasets and a negative
generalization error (test accuracy is higher than train accuracy) in a couple
of them, even at convergence. Such never seen behaviour opens up a promising
avenue for more future explorations.

  * •

We explore the robustness of IOC-NNs to label noise and find that it strongly
resists fitting the random labels. Even while training, IOC-NNs show no signs
of fitting on noisy data and efficiently learns patterns from non noisy data.
Our findings ignites explorations towards tighter generalization bounds for
neural networks.

##  2 Related Work

#### Simple Convex models:

Our work relates to parameter estimation on models that are guaranteed to be
convex by its construction. For regression problems, Magnani and Boyd [20]
study the problem of fitting a convex piecewise linear function to a given set
of data points. For classification problems, this traditionally translates to
polyhedral classifiers. A polyhedral classifier can be described as an
intersection of a finite number of hyperplanes. There have been several
attempts to address the problem of learning polyhedral classifiers [21, 15].
However, these algorithms require the number of hyperplanes as an input, which
is a major constraint. Furthermore, these classifiers do not give completely
smooth boundaries (at the intersection of hyperplanes). As another major
limitation, these classifiers cannot model the boundaries in which each class
is distributed over the union of non-intersecting convex regions (e.g., XOR
problem). The proposed IOC-NN (even with a single hidden layer) supersedes
this direction of work.

#### Convex Neural Networks:

Amos et al. [1] mentions the possibility of fully convex networks, however,
does not present any experiments with it. The focus of their work is to
achieve structured predictions using partially convex network (using convexity
w.r.t to some of the inputs). They propose a specific architecture called
FICNN which is fully convex and has fully connected layers with skip
connections. The skip connections are a must because their architecture cannot
even achieve identity mapping without them. In contrast, our work can take any
given architecture and derive its convex counterpart (we use the IOC suffix to
suggest model agnostic nature of our work). The work by Kent et al. [16]
analyze the links between polynomial functions and input convex neural
networks to understand the trade-offs between model expressiveness and ease of
optimization. Chen et al. [7, 8] explore the use of input convex neural
network in a variety of control applications like voltage regulation. The
literature on input convex neural networks has been limited to niche tailored
scenarios. Two key highlights of our work are: (a) to use activations that
allow the flow of negative values (like ELU, leaky ReLU, etc.), which enables
a richer representation (retaining fundamental properties like identity
mapping which are not achievable using ReLU) and (b) to bring a more in-depth
perspective on the functioning of convex networks and the resulting decision
boundaries. Consequently, we present IOC-NNs as a preferred option over the
base architectures, especially in terms of generalization abilities, using
experiments on mainstream image classification benchmarks.

#### Generalization in Deep Neural Nets:

Conventional machine learning wisdom says that overparameterization leads to
poor generalization performance owing to overfitting. Counter-intuitively,
empirical evidence shows that neural networks give better generalization with
an increased number of parameters even without any explicit regularization
[26]. Explaining how neural networks generalize despite being
overparameterized is an important question in deep learning [26, 23].

Neyshabur et al. [24] study different complexity measures and capacity bounds
based on the number of parameters, VC dimension, Rademacher complexity etc.,
and conclude that these bounds fail to explain the generalization behavior of
neural networks on overparameterization. Neyshabur et al. [25] suggest that
restricting the hypothesis class gives a generalization bound that decreases
with an increase in the number of parameters. Their experiments show that
restricting the spectral norm of the hidden layer leads to tighter
generalization bounds.

The above discussion implies that a hypothetical neural network that can fit
any hypothesis will have a worse generalization than the practical neural
networks which span a restricted hypothesis class. Inspired by this idea, we
propose a principled way of restricting the hypothesis class of neural
networks (by convexity constraints) that improves their generalization ability
in practice. In the previous efforts to train fully input output convex
networks, they were shown to have a limited capacity compared to its neural
network counterpart [1, 3], making their generalization capabilities
ineffective in practice. To our knowledge, we for the first time present a
method to formulate and efficiently train IOC-NNs opening an avenue to explore
their generalization ability.

##  3 Input Output Convex Networks

We first consider the case of an MLP with $k$ hidden layers. The output of
$i^{th}$ neuron in the $l^{th}$ hidden layer will be denoted as $h_{i}^{(l)}$.
For an input $\mathbf{x}=(x_{1},\ldots,x_{d})$, $h_{i}^{(l)}$ is defined as:

$h_{i}^{(l)}=\phi(\sum_{j}w_{ij}^{(l)}h_{j}^{l-1}+b_{i}^{(l)}),$ (1)

where, $h_{j}^{(0)}=x_{j}$ ($j=1\ldots d$) and $h_{j}^{(k+1)}=y_{j}$ ($j^{th}$
output). The first hidden layer represents an affine mapping of input and
preserves the convexity (i.e. each neuron in $h^{(1)}$ is convex function of
input). The subsequent layers are a weighted sum of neurons from the previous
layer followed by an activation function. The final output $\mathbf{y}$ is
convex with respect to the input $\mathbf{x}$ by ensuring two conditions: (a)
$w_{ij}^{(2:k+1)}\geq 0$ and (b) $\phi$ is convex and a non-decreasing
function. The proof follows from the operator properties [5] that the non-
negative sum of convex functions is convex and the composition $f(g(x))$ is
convex if $g$ is convex and $f$ is convex and non-decreasing.

A similar intuition follows for convolutional architectures as well, where
each neuron in the next layer is a weighted sum of the previous layer.
Convexity can be assured by restricting filter weights to be non-negative and
using a convex and non-decreasing activation function. Filter weights in the
first convolutional layer can take negative values, as they only represent an
affine mapping of the input. The maxpool operation also preserves convexity
since point-wise maximum of convex functions is convex [5]. Also, the skip
connection does not violate Input Output Convexity, since the input to each
layer is still a non-negative weighted sum of convex functions.

We use an ELU activation to allow negative values; this is a minor but a key
change from the previous efforts that rely on ReLU activation. For instance,
with non-negativity constraints on weights ($w_{ij}^{(2:k+1)}\geq 0$), ReLU
activations restrict the allowable use of hidden units that mirror the
identity mapping. Previous works rely on passthrough/skip connections to
address [1] this concern. The use of ELU enables identity mapping and allows
us to use the convex counterparts of existing networks without any
architectural changes.

###  3.1 Convexity as Self Regularizer

We define self regularization as the property in which the network itself has
some functional constraints. Inducing convexity can be viewed as a self
regularization technique. For example, consider a quadratic classifier in
$\mathbb{R}^{2}$ of the form
$f(x_{1},x_{2})=w_{1}x_{1}^{2}+w_{2}x_{2}^{2}+w_{3}x_{1}x_{2}+w_{4}x_{1}+w_{5}x_{2}+w_{0}$.
If we want the function $f$ to be convex, then it is required that the network
imposes following constraints on the parameters, $w_{1}\geq 0,\;w_{2}\geq
0,\;-2\sqrt{w_{1}w_{2}}\leq w_{3}\leq 2\sqrt{w_{1}w_{2}}$, which essentially
means that we are restricting the hypothesis space.

Similar inferences can be drawn by taking the example of polyhedral
classifiers. Polyhedral classifiers are a special class of Mixture of Experts
(MoE) network [13, 27]. VC-dimension of a polyhedral classifier in
$d$-dimension formed by the intersection of $m$ hyperplanes is upper bounded
by $2(d+1)m\log(3m)$ [30]. On the other hand, VC-dimension of a standard
mixture of $m$ binary experts in $d$-dimension is $O(m^{4}d^{2})$ [14]. Thus,
by imposing convexity, the VC-dimension becomes linear with the data dimension
$d$ and $m\log(m)$ with the number of experts. This is a huge reduction in the
overall representation capacity compared to the standard mixture of binary
experts.

Furthermore, adding non-negativity constraints alone can lead to
regularization. For example, the VC dimension of a sign constrained linear
classifier in $\mathbb{R}^{d}$ reduces from $d+1$ to $d$ [6, 19]. The proposed
IOC-NN uses a combination of sign constraints and restrictions on the family
of activation functions for inducing convexity. The representation capacity of
the resulting network reduces, and therefore, regularization comes into
effect. This effectively helps in improving generalization and controlling
overfitting, as clearly observed in our empirical studies (Section 4.1).

|  |  |
---|---|---|---
(a) | (b) | (c) | (d)

Figure 2:  Decision boundaries of different networks trained for two class
classification. (a) Original data: one class shown by blue and the other
orange. (b) Decision boundary learnt using MLP. (c) Decision boundary learnt
using IOC-MLP with single node in the output layer. (d) Decision boundary
learnt using IOC-MLP with two nodes in the output layer (ground truth as one
hot vectors)

.

|  |
---|---|---
(a) | (b)
Figure 3:  (a) Using two simple 1-D functions we illustrate that
$\operatorname*{argmax}$ of two convex functions can result into non-convex
decision boundaries. (b) Two convex functions whose $\operatorname*{argmax}$
results into the decision boundaries shown in Figure 2(d). The same plot is
shown from two different viewpoints.

###  3.2 IOC-NN Decision Boundaries

Consider a scenario of binary classification in 2D space as presented in
Figure 2(a). We train a three-layer MLP with a single output and a sigmoid
activation for the last layer. The network comfortably learns to separate the
two classes. The learned boundaries by the MLP are shown in Figure 2(b). We
then train an IOC-MLP with the same architecture. The learned boundary is
shown in Figure 2(c). IOC-MLP learns a single convex function as output w.r.t
the input and its contour at the value of 0.5 define the decision boundary.
The use of non-convex activation like sigmoid in the last layer does not
distort convexity of decision boundary (Appendix A)

We further explore IOC-MLP with a variant architecture where the ground truth
is presented as a one-hot vector (allowing two outputs). The network learns
two convex functions $f$ and $g$ representing each class, and their
$\operatorname*{argmax}$ defines the decision boundary. Thus, if
$g(\mathbf{x})-f(\mathbf{x})>0$, then $\mathbf{x}$ is assigned to class $C1$
and $C2$ otherwise. Therefore, it can learn non-convex decision boundaries as
shown in Figure 3. Please note that $g-f$ is no more convex unless
$g^{\prime\prime}-f^{\prime\prime}\geq 0$. In the considered problem of binary
classification in Figure 2, using one-hot output allows the network to learn
non-convex boundaries (Figure 2 (d)). The corresponding two output functions
(one for each class) are illustrated in Figure 3 (b). We can observe that both
the individual functions are convex; however, their arrangement is such that
the $\operatorname*{argmax}$ leads to a reasonably complex decision
boundary.This happens due to the fact that the sets
$S_{1}=\\{\mathbf{x}\;|\;g(\mathbf{x})-f(\mathbf{x})>0\\}$ and
$S_{2}=\\{\mathbf{x}\;|\;g(\mathbf{x})-f(\mathbf{x})\leq 0\\}$ can both be
non-convex (even though functions $f(.)$ and $g(.)$ are convex).

###  3.3 Ensemble of IOC-NN

We further explore the ensemble of IOC-NN for multi-class classification. We
explore two different ways to learn the ensembles:

  1. 1.

Mixture of IOC-NN Experts: Training a mixture of IOC-NNs and an additional
gating network [13]. The gating network can be non-convex and outputs a scalar
weight for each expert. The gating network and the multiple IOC-NNs (experts)
are trained in an Expectation-Maximization (EM) framework, i.e., training the
gating network and the experts iteratively.

  2. 2.

Boosting + Gating: In this setup, each IOC-NN is trained individually. The
first model is trained on the whole data, and the consecutive models are
trained with exaggerated data on the samples on which the previous model
performs poorly. For bootstrapping, we use a simple re-weighting mechanism as
in [10]. A gating network is then trained over the ensemble of IOC-NNs. The
weights of the individual networks are frozen while training the gating
network.

|  |
---|---|---
(a) | (b) | (c)
Figure 4: (a) Original Data. (b) Output of the gating network, each color
represents picking a particular expert. (c) Decision boundaries of the
individual IOC-MLPs. We mark the correspondences between each expert and the
segment for which it was selected. Notice how the V-shape is partitioned and
classified using two different IOC-MLPs.

We detail the idea of ensembles using a representative experiment for binary
classification on the data presented in Figure 4(a). We train a mixture of
$\mathbf{p}$ IOC-MLPs with a gating network using the EM algorithm. The gating
network is an MLP with a single hidden layer, the output of which is a
$\mathbf{p}$ dimensional vector. Each of the IOC-MLP is a three-layer MLP with
a single output. We keep a single output to ensure that each IOC-MLP learns a
convex decision boundary. The output of the gating network is illustrated in
Figure 4(b). A particular IOC-MLP was selected for each partition and led to
five partitions. The decision boundaries of individual IOC-MLPs are shown in
Figure 4(c). It is interesting to note that the MoE of binary IOC-MLPs
fractures the input space into sub-spaces where a convex boundary is
sufficient for classification.

##  4 Experiments

#### Dataset and Architectures:

To show the significance of enhanced performance of IOC-MLP over traditional
NN, we train them on six different datasets: MNIST, FMNIST, STL-10, SVHN,
CIFAR-10, and CIFAR-100. We use an MLP with three hidden layers and 800 nodes
in each layer. We use batch normalization between every layer, and it’s
activation in all hidden layers. ReLU and ELU are used as activations for NN
and IOC respectively, and softmax is used in the last layer. We use Adam
optimizer with an initial learning rate of 0.0001 and use validation accuracy
for early stopping.

We perform experiments that involve two additional architectures to extend the
comparative study between IOC and NN on CIFAR-10 and CIFAR-100 datasets. We
use a fully convolutional [29], and a densely connected architecture [12]. We
choose DenseNet with growth rate k=12, for our experiments. We term the convex
counterparts as IOC-AllConv, IOC-DenseNet, respectively, and compare against
their base neural network counterparts [12, 29]. In all comparative studies,
we follow the same training and augmentation strategy to train IOC-NNs, as
used by the aforementioned neural networks.

#### Training on duplicate free data:

The test sets of CIFAR-10 and CIFAR-100 datasets have 3.25% and 10% duplicate
images, respectively [4]. Neural networks show higher performance on these
datasets due to the bias created by this duplicate data (neural networks have
been shown to memorize the data). CIFAIR-10 and CIFAIR-100 datasets are
variants of CIFAR-10 and CIFAR-100 respectively, where all the duplicate
images in the test data are replaced with new images. Barz et al. [4] observed
that the performance of most neural architectures drops when trained and
tested on bias-free CIFAIR data. We train IOC-NN and their neural network
counterparts on CIFAIR-10 data with three different architectures: a fully
connected network (MLP), a fully convolutional network (AllConv) [29] and a
densely connected network (DenseNet) [12].

#### Training IOC architectures:

We tried four variations for weight constraints to enforce convexity
constraints: clipping negative weights to zero, taking absolute of weights,
exponentiation of negative weights and shifting the weights after each
iteration. We use exponentiation strategy in all experiments, as it gave the
best results. We exponentiate the negative weights after every update. The IOC
constrained optimization algorithm differs only by a single step from the
traditional algorithms (Appendix B).

To conserve convexity in the batch-normalization layer, we also constrain the
gamma scaler with exponentiation. However, in practice we found that the IOC
networks retains all desirable properties without constraining the gamma
scalar. We make few additional modifications to facilitate the training of
IOC-NNs. Such changes do not affect the performance of the base neural
networks. We use ELU as an activation function instead of ReLU in IOC-NNs. We
apply the whitening transformation to the input so that it is zero-centered,
decorrelated, and spans over positive and negative values equally. We also
increase the number of nodes in the first layer (the only layer where
parameters can take negative values). We use a slower schedule for learning
rate decay than the base counterparts. The IOC-NNs have a softmax layer at the
last layer and are trained with cross-entropy loss (same as neural networks).

#### Training ensembles of binary experts:

We divide CIFAR-10 dataset into 2 classes, namely: ‘Animal’ (CIFAR-10 labels:
‘Bird’, ‘Cat’, ‘Deer’, ‘Dog’, ‘Frog’ and ‘Horse’) and ‘Not Animal’. We train
an ensemble of IOC-MLP, where each expert is a three-layer MLP with one output
(with sigmoid activation at the output node). The gating network in the EM
approach is a one layer MLP which takes an image as input and predicts the
weights by which the individual expert predictions get averaged. We report
test results of ensembles with each additional expert. This experiment
resembles the study shown in Figure 4.

#### Training Boosted ensembles:

The lower training accuracy of IOC-NNs makes them suitable for boosting (while
the training accuracy saturates in non-convex counterparts). For
bootstrapping, we use a simple re-weighting mechanism as in [10]. We train
three experts for each experiment. The gating network is a regular neural
network, which is a shallow version of the actual experts. We train an MLP
with only one hidden layer, a four-layer fully convolutional network, and a
DenseNet with two dense-blocks as the gate for the three respective
architectures. We report the accuracy of the ensemble trained in this fashion
as well as the accuracy if we would have used an oracle instead of the gating
network.

#### Partially randomized labeling:

Here, we investigate IOC-NN’s behavior in the presence of partial label noise.
We do a comparative study between IOC and neural networks using All-Conv
architecture, similar to the experiment performed by [31]. We use CIFAR-10
dataset and make them noisy by systematically randomizing the labels of a
selected percentage of training data. We report the performance of All-Conv,
and it’s IOC counterpart on 20, 40, 60, 80 and 100 percent noise in the train
data. We report train and test scores at peak performance (performance if we
had used early stopping) and at convergence (if loss goes below 0.001 or at
2000 epochs).

###  4.1 Results

#### IOC as a preferred alternative for Multi-Layer-Perceptrons:

MLP is most basic and earliest explored form of neural networks. We compare
the train and test scores of MLP and IOC-MLP in Table 1. With a sufficient
number of parameters, MLP (a basic NN architecture) perfectly fits the
training data. However, it fails to generalize well on the test data owing to
brute force memorization. The results in Table 1 indicate that IOC-MLP gives a
smaller generalization gap (the difference between train and test accuracies)
compared to MLP. The generalization gap even goes to negative values on three
of the datasets. MLP (being poorly optimized for parameter utilization) is one
of the architectures prone to overfitting the most, and IOC constraints help
retain test performance resisting the tendency to overfit. Obtaining negative
or almost zero generalization error even at convergence is a never seen
behaviour in deep networks and the results clearly suggest the profound
generalization abilities of Input Output Convexity, especially when applied to
fully connected networks.

| NN | IOC-NN
---|---|---
| train | test | gen. gap | train | test | gen. gap
MNIST | 99.34 | 99.16 | 0.19 | 98.77 | 99.25 | -0.48
FMNIST | 94.8 | 90.61 | 3.81 | 90.41 | 90.58 | -0.02
STL-10 | 81 | 52.32 | 28.68 | 62.3 | 54.55 | 7.75
SVHN | 91.76 | 86.19 | 5.57 | 81.18 | 86.37 | -5.19
CIFAR-10 | 97.99 | 63.83 | 34.16 | 73.27 | 69.89 | 3.38
CIFAR-100 | 84.6 | 32.68 | 51.92 | 46.9 | 41.08 | 5.82
Table 1: Table shows train accuracy, test accuracy and generalization gap for MLP and IOC-MLP on six different datasets. | CIFAR-10 | CIFAR-100
---|---|---
| NN | IOC-NN | NN | IOC-NN
| train | test | gen. gap | train | test | gen. gap | train | test | gen. gap | train | test | gen. gap
MLP | 99.17 | 63.83 | 35.34 | 73.27 | 69.89 | 3.3 | 84.6 | 32.68 | 51.9 | 46.9 | 41.08 | 5.8
AllConv | 99.31 | 92.8 | 6.5 | 93.2 | 90.6 | 2.6 | 97.87 | 69.5 | 28.4 | 67.07 | 65.08 | 1.9
DenseNet | 99.46 | 94.06 | 5.4 | 94.22 | 91.12 | 3.1 | 98.42 | 75.36 | 23.06 | 74.9 | 68.53 | 6.3
Table 2: Train accuracy, test accuracy and generalization gap of three neural architectures and their IOC counterparts | NN | IOC-NN
---|---|---
| peak | convergence | peak | convergence
| train | test | train | test | gen. gap | train | test | train | test | gen. gap
100 | 98.63 | 10.53 | 97.80 | 10.1 | 87.7 | 9.98 | 10.62 | 10.21 | 9.94 | 0.27
80 | 22.40 | 60.24 | 97.83 | 27.75 | 70.08 | 21.93 | 61.48 | 23.80 | 56.20 | -32.4
60 | 38.52 | 75.80 | 97.80 | 46.71 | 51.09 | 37.90 | 75.91 | 39.31 | 71.75 | -32.44
40 | 56.48 | 80.47 | 97.96 | 61.83 | 36.13 | 55.01 | 81.58 | 54.63 | 81.01 | -26.38
20 | 72.8 | 85.72 | 98.73 | 76.31 | 22.42 | 69.92 | 85.85 | 70.22 | 83.61 | -13.39
Table 3: Results for systematically randomized labels at peak and at convergence for both IOC-NN and NN. The IOC constraints bring huge improvements in generalization error and test accuracy at convergence. |  Base MLP |  Constrained MLP |  FICNN |  IOC-NN
---|---|---|---|---
train | 99.17 | 46.81 | 62.8 | 73.27
test | 63.83 | 27.36 | 53.07 | 69.89
gen-gap | 35.34 | 19.45 | 9.73 | 3.38
Table 4: Results comparing FICNN [1] with IOC-NN on CIFAR-10 using MLP
architecture. First column shows base MLP results. Second column presents
results with a convex MLP using ReLU activation. Third and final columns show
the accuracies of FICNN and IOC-NN, respectively.

Furthermore, having the IOC constraints significantly boosts the test accuracy
on datasets where neural network gives a high generalization gap (Table 1).
This trend is clearly visible in Figure 5(b). For the CIFAR-10 dataset,
unconstrained MLP gives 34.16% generalization gap, while IOC-NN brings down
the generalization gap by more than ten folds and boosts the test performance
by about 6%. Even in scenarios where neural networks give a smaller
generalization gap (like MNIST and SVHN), IOC-NN marginally outperforms
regular NN and gives an advantage in generalization. Overall, the results in
Table 1 highlight that IOC constraints are extremely beneficial when training
Multi Layer Perceptrons for image classification, giving comprehensive
advantages in terms of generalization and test performance.

#### Better generalization:

We investigate the generalization capability of IOC-NN on other architectures.
The results of the base architectures and their convex counterparts on
CIFAR-10 and CIFAR-100 datasets are presented in Table 2. IOC-NN outperforms
base NN on MLP architecture and gives comparable test accuracies for
convolutional architectures. The train accuracies are saturated in the base
networks (reaching above 99% in most experiments). The lower train accuracy in
IOC-NNs suggests that there might still be room for improvement, possibly
through better design choices tailored for IOC-NNs. In Table 2, the difference
in train and test accuracy across all the architectures (generalization gap)
demonstrates the better generalization ability of IOC-NNs. The generalization
gap of base architectures is at least twofold more than IOC-NNs on the
CIFAR-100 dataset. For instance, the generalization error of IOC-AllConv on
CIFAR-100 is only 1.99%, in contrast to 28.4% in AllConv. The generalization
ability of IOC-NNs is further qualitatively reflected using the training and
validation loss profiles (e.g., Figure1(a)). We present a table showing the
confidence intervals of prediction across all three architectures with
repeated runs in Appendix C.

Table 5 shows the train and test performance of the three architectures on
CIFAR-10 dataset and the drop incurred when trained on CIFAIR-10. The drop in
test performance of IOC-NNs is smaller than the typical neural network. This
further strengthens the claim that IOC-NNs are not memorizing the training
data but learning a generic hypothesis.

#### Comparison with FICNN:

Table 4 shows the results of IOC-NN and FICNN [1] on CIFAR-10 data. For
comparison, we use a three layer MLP with 800 nodes in each layer, for both
IOC-NN and FICNN. FICNN uses a skip connection from input layer to each of the
intermittent layers. This enables each layer to learn identity mapping inspite
of non-negative constraint. The number of parameters in FICNN model is almost
twice compared to the base MLP and IOC models but still the test performance
drops by more than 10%. The results clearly shows that IOC-NN gives better
test accuracy and lower generalization gap compared to FICNN, while using the
same number of parameters as the base MLP architecture.

#### Robustness to random label noise:

Robustness of IOC-NNs on partial and fully randomized labels (Figure 1 (b, c,
and d)) is one of its key properties. We further investigate this property by
systematically randomizing increasing portion of labels. We report the results
of neural networks and their convex counterparts with percentage of label
noise varying from 20% to 100% in Table 3. The train performance of neural
networks at convergence is near 100% across all noise levels. It is
interesting to note that IOC-NN gives a large negative generalization gap,
where the train accuracy is almost equal to the percentage of true labels in
the data. This observation shows that IOC-NNs significantly resist learning
noise in labels as compared to neural networks. Both neural network and it’s
convex counterpart learns the simple hypothesis first. While IOC-NN holds on
to this, in later epochs, the neural network starts brute force memorization
of noisy labels. The observations are coherent with findings in [18, 28],
demonstrating neural network’s heavy reliance on early stopping. IOC-AllConv
outperforms test accuracy of AllConv + early stopping with a much-coveted
generalization behavior. It is clear from this experiment that IOC-NN performs
better in the presence of random label noise in the data in terms of test
accuracy both at peak and convergence.

| NN | IOC-NN
---|---|---
| C-10 | CIFAIR | Gap | C-10 | CIFAIR | Gap
MLP | 63.6 | 63.08 | 0.52 | 69.89 | 69.51 | 0.38
AllConv | 92.8 | 91.14 | 0.66 | 90.6 | 90.47 | 0.13
DenseNet | 94.06 | 93.28 | 0.78 | 91.12 | 90.73 | 0.39
Table 5: Results on CIFAIR-10 dataset  |  |
---|---|---
(a) | (b) |
Figure 5: (a) shows the test accuracy of IOC-MLP with increasing number of experts in the binary classification setting. Average performance of normal MLP is shown in red since it does not change with increase in number of experts. (b) The generalization gap of MLP plotted against the improvement gained by the IOC-MLP for the six different datasets (represented by every point on the plot). The performance gain with IOC constraints increase with the increase in generalization gap of MLP. | single expert | gate | oracle
---|---|---|---
MLP | 69.89 | 71.8 | 85.47
All-Conv | 90.6 | 92.83 | 96.3
DenseNet | 91.12 | 93.25 | 97.19
Table 6: Result for single expert, gated MoE and with oracle on CIFAR-10 for
three architectures

#### Leverage IOC properties to train ensembles:

We train binary MoE on the modified two-class setting of CIFAR-10 as described
in Section 4. The result is shown in Figure 5 (a). Traditional neural network
gives a test accuracy of 89.63% with a generalization gap of 10%. Gated MoE of
NNs does not improve the test performance as we increase the number of
experts. In contrast, the performance of ensemble of IOC-NNs goes up with the
addition of each expert and moves closer to the performance of neural
networks. It is interesting to note that even in the higher dimensional space
(like CIFAR-10 images), the intuitions derived from Figure 4 holds. We also
note that gate fractures the space into p partitions (where p is the number of
experts). Moreover, in the binary case for a single expert, the generalization
gap is almost zero. This can be attributed to the convex hull like smooth
decision boundary that the network predicts in the binary setting with a
single output.

The results with the boosted ensembles of IOC-NNs are presented in Table 6.
The boosted ensemble improves the test accuracies of IOC-NNs, matching or
outperforming the base architectures. However, this performance gain comes at
the cost of increased generalization error (still lower than the base
architectures). In the boosted ensemble, the performance significantly
improves if the gating network is replaced by an oracle. This observation
suggests that there is a scope of improvement in model selection ability,
possibly by using a better gating architecture.

#### Confidence Calibration of IOC-NNs:

In a classification setting, given an input, the neural network predicts
probability-like scores towards each class. The class with the maximum score
is considered the predicted output, and the corresponding score to be the
confidence. The confidence and accuracy being correlated is a desirable
property, especially in high-risk applications like self-driving cars, medical
diagnoses, etc. However, many modern multi-class classification networks are
poorly calibrated, i.e., the probability values that they associate with the
class labels they predict overestimate the likelihoods of those class labels
being correct in the real world [11]. Recent works have explored methods to
improve the calibration of neural networks [11, 22].

We observe that adding IOC constraints improve calibration error on the base
NN architecture. We present the reliability diagrams [9] (presenting accuracy
as a function of confidence) of three neural architectures and their convex
counterparts in Figure 6. The sum of the difference between the blue bars and
the orange bars represents the Expected Calibration Error. IOC constraints
show improved calibration in all three architectures (with notable
improvements in the case of MLP and AllConv). Better calibration further
strengthens the case for IOC-NNs from the application perspective.

|  |  |  |  |
---|---|---|---|---|---
(a) MLP | (b) AllConv | (c) DenseNet
Figure 6:  These diagrams show expected sample accuracy as a function of
confidence [9]. The blue bar shows the confidence of the bin and the orange
bar shows the percentage correctness of prediction in that bin. If the model
is perfectly calibrated, the bars align to form identity function. Any
deviation from a perfect diagonal is a miscalibration.

##  5 Conclusions

We present a subclass of neural networks, where the output is a convex
function of the input. We show that with minimal constraints, existing neural
networks can be adopted to this subclass called Input Output Convex Neural
Networks. With a set of carefully chosen experiments, we unveil that IOC-NNs
show outstanding generalization ability and robustness to label noise while
retaining adequate capacity. We show that in scenarios where the neural
network gives a large generalization gap, IOC-NN can give better test
performance. An alternate interpretation of our work can be self
regularization (regularization through functional constraints). IOC-NN puts to
rest the concerns around brute force memorization of deep neural networks and
opens a promising horizon for the community to explore. We show that in the
case of Multi-Layer-Perceptrons, IOC constraints improve accuracy,
generalization, calibration, and robustness to noise, making an ideal
proposition from a deployment perspective. The improved generalization,
calibration, and robustness to noise are also observed in convolutional
architectures while retaining the accuracy. In future work, we plan to
investigate the use of IOC-NNs for recurrent architectures. Furthermore, we
plan to explore the interpretability aspects of IOC-NNs and study the effect
of convexity constraints on the generalization bounds.

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Appendix This document contains the supplementary material to support the main
text. The contents of this document are:

  * •

Appendix A: Using non-convex activations in the last layer of Input Output
Convex Neural Networks (IOC-NNs)

  * •

Appendix B: Optimization algorithm for training IOC-NNs

  * •

Appendix C: Confidence intervals of classification accuracies for neural
networks and IOC-NNs

  * •

Appendix D: Discussion on capacity of IOC-NNs

  * •

Appendix E: Partition of input space by MoEs of IOC-NNs

##  Appendix 0.A Using Non-Convex Activations to Facilitate Training

In a multi-class classification setting, the softmax function is widely used
to get a joint probability distribution over the output classes. This
facilitates training with categorical cross-entropy loss. The softmax layer
distorts input output convexity, but the decision boundary remains unchanged
even after applying softmax in the last layer. Each of the pre-softmax output
of IOC-NNs is convex with respect to the inputs. The classification decision
is determined by the order (rank) of these values, as this rank remains
unaffected with the application of the softmax function. At inference time, we
compute $\operatorname*{argmax}$ of convex functions (pre-softmax layer).

$\footnotesize\operatorname*{argmax}(Y)=\operatorname*{argmax}(softmax(Y))$
(2)

A similar approach is taken for modelling independent probabilities of labels
in binary classification setting. Sigmoid activation is widely used to model
the probability $p(y=y_{n}|x)$. For IOC-NN, at inference, the predicted value
is $y=\sigma(f(x))$, where $f(x)$ is convex. Hence, decision boundary is:

$\footnotesize\sigma(f(x))=0.5,$ (3)

which is equivalent to

$\footnotesize f(x)=\sigma^{-1}(0.5)$ (4)

where, $f(x)$ is convex. Since $\sigma^{-1}(0.5)$ is a constant, the decision
boundary is convex.

##  Appendix 0.B Optimization Algorithm for Training IOC-NNs

The only architectural constraint in designing an IOC-NN is the choice of a
convex and non-decreasing activation function. Furthermore, any feed-forward
neural architecture can be trained as IOC-NN by adding two steps to the
optimization algorithm. For example, the constrained version of the vanilla
stochastic gradient descent algorithm is shown in Algorithm 1.

Input: Training data $S$, Labels $L$, learning rate $\eta$, constant
$\epsilon$, $\theta_{0}$ = ($w_{0}{{}^{n}_{ij}}$, $b_{0}{{}^{n}_{i}}$)

Output: $\Theta=(w^{*},b^{*})$

initialize: $\theta=\theta_{0}$;

while  _stopping criteria not met_ do

for  _i in 1:len(S) / batch_size_ do

sample$(x_{batch},y_{batch})\in(S,L)$;

$L\leftarrow Div(f_{\theta}(x_{batch}),y_{batch})$;

$w\leftarrow w-\eta(\frac{\partial}{\partial w}L)$;

$b\leftarrow b-\eta(\frac{\partial}{\partial b}L)$;

for  _layer $\in{2:k}$_ do

if  _$w <0$ _ then

$w\leftarrow e^{w-\epsilon}$;

end if

end for

end for

check stopping criteria

end while

Algorithm 1 Algorithm to train a k layer IOC-NN

$\epsilon$ is used to bring down the value of the updated weights post
exponentiation for negative values close to zero. Values near zero, post
exponentiation, will be close to one, and $\epsilon$ helps keep them close to
zero. We use $\epsilon=5$ across all experiments. To summarize, the only two
additional steps required to train an IOC-NN are condition (to check the sign
of updated weights) and exponentiation. We can develop IOC constrained
versions for all optimization algorithms that are used in training neural
networks.

##  Appendix 0.C Confidence Intervals of Classification Accuracies for Neural
Networks and IOC-NNs

We run MLP, AllConv and Densenet on CIFAR-10 data five times. We report the
accuracy in prediction and the confidence intervals obtained over five runs in
Table 7. We report the train and test accuracies along with the generalization
gap. The table confirms the reliability of the results in the main document.
The generalization gap of IOC-NNs are consistently lower than neural networks.
Also, compared to base MLP the convex counterpart gives higher test accuracy.

Table 7: . | NN | IOC-NN
---|---|---
| train | test | Gen. error | train | test | Gen. error
MLP |  97.99 $\pm$ 0.18 |  63.83 $\pm$ 0.12 | 34.1 |  73.27 $\pm$ 0.41 |  69.89 $\pm$ 0.21 | 3.38
All Conv |  99.12 $\pm$ 1.03 |  92.21 $\pm$ 0.18 | 7.09 |  92.61 $\pm$ 1.45 |  90.34 $\pm$ 0.20 | 2.27
DenseNet |  99.08 $\pm$ 0.75 |  93.86 $\pm$ 0.46 | 5.22 |  93.8 $\pm$ 0.83 |  90.43 $\pm$ 0.59 | 3.37

##  Appendix 0.D Discussion on Capacity of IOC-NNs

The decision boundaries of real-world classification problems are often not
convex. Each of the single outputs of IOC-NN is convex with respect to the
inputs by design. However, it can still learn arbitrarily complex decision
boundaries. Kripfganz et al.[17] show that we can represent any piecewise
linear function as a difference of two piecewise linear convex functions.
Using ReLU as an activation limits the function classes that convex networks
can learn. For instance, they cannot learn identity mapping [1]. A simple
architectural change of using ELU overcomes this issue and hence increases the
capacity of IOC-NNs. Results in the main text show empirical evidence of this
increase in capacity. Following this direction in the future, we would like to
explore the proof for universal approximation for IOC-NNs.

|  |  |  |
---|---|---|---|---
|  |  |  |
Figure 7: The figure shows the mean image of ten clusters within the class
’Automobile’. Each cluster corresponds to the expert picked by the gate. These
partitions signify the difference in the data points on which each of the
binary convex models gained expertise.

##  Appendix 0.E Partition of Input space by MoEs of IOC-NNs

Real-world data often lies in a sparse, high-dimensional space and neural
networks fit complex boundaries over them. These complex boundaries make the
classification results challenging to interpret. In section 3.2 of the main
text, we demonstrate on toy data, how an ensemble of binary IOC-NNs creates
meaningful partitions of input space. Gating network of MoEs of IOC-NNs
partition input space into mutually exclusive and exhaustive subspaces. The
partition chosen by the gate for each expert is a portion where a convex
boundary is enough to make a reasonable prediction. We train ten, three
layered single output IOC-MLPs using gated EM strategy on the two-class
setting of CIFAR-10 as explained in section 4: ’Training ensembles of binary
experts’. Figure 7 shows the mean image of ten partitions of class
’Automobile’. Each partition corresponds to one of the ten experts chosen by
the gate. Each of the IOC-NN has expertise on clusters with aspects that can
lead to some real-world interpretation. For instance, an expert is chosen by
the gate to classify images showing the front view of red cars from similar
images, while another expert is picked for side view images. Similar clusters
can be seen corresponding to different colors and orientations. In the future,
we would like to explore this property of MoE of binary IOC-NNs towards
improving the interpretability of decisions made by neural networks.