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Title: Pooling Is Neither Necessary nor Sufficient for Appropriate Deformation Stability in CNNs

Paper content:
Many of our core assumptions about how neural networks operate remain empirically untested.
One common assumption is that convolutional neural networks need to be stable to small translations and deformations to solve image recognition tasks.
For many years, this stability was baked into CNN architectures by incorporating interleaved pooling layers.
Recently, however, interleaved pooling has largely been abandoned.
This raises a number of questions: Are our intuitions about deformation stability right at all?
Is it important?
Is pooling necessary for deformation invariance?
If not, how is deformation invariance achieved in its absence?
In this work, we rigorously test these questions, and find that deformation stability in convolutional networks is more nuanced than it first appears: (1) Deformation invariance is not a binary property, but rather that different tasks require different degrees of deformation stability at different layers.
(2) Deformation stability is not a fixed property of a network and is heavily adjusted over the course of training, largely through the smoothness of the convolutional filters.
(3) Interleaved pooling layers are neither necessary nor sufficient for achieving the optimal form of deformation stability for natural image classification.
(4) Pooling confers \emph{too much} deformation stability for image classification at initialization, and during training, networks have to learn to \emph{counteract} this inductive bias.
Together, these findings provide new insights into the role of interleaved pooling and deformation invariance in CNNs, and demonstrate the importance of rigorous empirical testing of even our most basic assumptions about the working of neural networks.
Within deep learning, a variety of intuitions have been assumed to be common knowledge without empirical verification, leading to recent active debate BID22 BID13 BID25 .
Nevertheless, many of these core ideas have informed the structure of broad classes of models, with little attempt to rigorously test these assumptions.In this paper, we seek to address this issue by undertaking a careful, empirical study of one of the foundational intuitions informing convolutional neural networks (CNNs) for visual object recognition: the need to make these models stable to small translations and deformations in the input images.
This intuition runs as follows: much of the variability in the visual domain comes from slight changes in view, object position, rotation, size, and non-rigid deformations of (e.g.) organic objects; representations which are invariant to such transformations would (presumably) lead to better performance.
This idea is arguably one of the core principles initially responsible for the architectural choices of convolutional filters and interleaved pooling BID15 , as well as the deployment of parametric data augmentation strategies during training BID28 .
Yet, despite the widespread impact of this idea, the relationship between visual object recognition and deformation stability has not been thoroughly tested, and we do not actually know how modern CNNs realize deformation stability, if they even do at all.Moreover, for many years, the very success of CNNs on visual object recognition tasks was thought to depend on the interleaved pooling layers that purportedly rendered these models insensitive to small translations and deformations.
However, despite this reasoning, recent models have largely abandoned interleaved pooling layers, achieving similar or greater success without them ; .These
observations raise several critical questions. Is deformation
stability necessary for visual object recognition? If so, how is
it achieved in the absence of pooling layers? What role does
interleaved pooling play when it is present?Here, we seek to
answer these questions by building a broad class of image deformations, and comparing CNNs' responses to original and deformed images. While this class
of deformations is an artificial one, it is rich and parametrically controllable, includes many commonly used image transformations (including affine transforms: translations, shears, and rotations, and thin-plate spline transforms, among others) and it provides a useful model for probing how CNNs might respond to natural image deformations. We use these to
study CNNs with and without pooling layers, and how their representations change with depth and over the course of training. Our contributions
are as follows:• Networks without pooling are sensitive to deformation at initialization, but ultimately learn representations that are stable to deformation.• The inductive bias
provided by pooling is too strong at initialization, and deformation stability in these networks decrease over the course of training.• The pattern of deformation
stability across layers for trained networks with and without pooling converges to a similar structure.• Networks both with and without
pooling implement and modulate deformation stability largely through the smoothness of learned filters.More broadly, this work demonstrates that our intuitions as to why neural networks work can often be inaccurate, no matter how reasonable they may seem, and require thorough empirical and theoretical validation.
In this work, we have rigorously tested a variety of properties associated with deformation stability.
We demonstrated that while pooling confers deformation stability at initialization, it does not determine the pattern of deformation stability across layers.
This final pattern is consistent across network architectures, both with and without pooling.
Moreover, the inductive bias conferred by pooling is in fact too strong for ImageNet and CIFAR-10 classification; this therefore has to be counteracted during training.
We also found that filter smoothness contributes significantly to achieving deformation stability in CNNs.
Finally, these patterns remain a function of the task being learned: the joint distribution of inputs and outputs is important in determining the level of learned deformation stability.Together, these results provide new insights into the necessity and origins of deformation stability.
They also provide an instructive example of how simple properties of learned weights can be investigated to shed light on the inner workings of deep neural networks.One limitation of this work is that we only focused on deformations sampled from a particular distribution.
We also only measured average sensitivity over these deformations.
In future work, it would be informative to explore similar questions but with the worst case deformations found via maximization of the deformation sensitivity BID5 ; BID11 .Finally
, our work compares only two points in time: the beginning and the end of training. There
remain open questions about how these characteristics change over the course of training. For example
, when do filters become smooth? Is this a
statistical regularity that a network learns early in training, or does filter smoothness continue to change even as network performance begins to asymptote? Does this
differ across layers and architectures? Is the trajectory
toward smooth filters and deformation stability monotone, or are there periods of training where filters become smoother and then periods when the filter smoothness decreases? Future work will
be required to answer all of these questions.For the ImageNet experiments, we used networks with block structure 2x64, 2x128, 3x256, 3x512, 3x512. For the CIFAR10
experiments, we used networks with block structure 2x32, 2x64, 2x128, 2x256.We compared networks with the following downsampling layers in our CIFAR10 experiments: Subsample: Keep top left corner of each 2x2 block. Max-pool: Standard
max-pooling layer. Average-pool: Standard
average-pooling layer. Strided: we replace the
max pooling layer with a convolutional layer with kernels of size 2x2 and stride 2x2. Strided-ReLU: we replace
the max pooling layer with a convolutional layer with kernels of size 2x2 and stride 2x2. The convolutional layer
is followed by batch-norm and ReLU nonlinearity. For our ImageNet experiments
, we compared only Max-pool and Strided-ReLU due to computational considerations.To rule out variability due to random factors in the experiment (initial random weights, order in which data is presented), we repeated all experiments 5 times for each setting. The error bands in the plots
correspond to 2 standard deviations estimated across these 5 experiments.

assistant: text: We find that pooling alone does not determine deformation stability in CNNs and that filter smoothness plays an important role in determining stability.