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<!--
Copyright 2018 The Distill Template Authors
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<!doctype html>
<head>
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<title>The Probabilistic Fault Tolerance of Neural Networks in the Continuous Limit</title>
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<div class="cover">
<h3>The Probabilistic<br /> Fault Tolerance of<br /> Neural
Networks in the<br /> Continuous Limit<br />
<font size="5">a short introduction to the paper</font></h3>
<div class="hint unselectable">scroll down</div>
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<div class="l-page-outset">
<h4>This page introduces our paper in a graphical manner.
The main goal is to explain the positioning and the content of the paper in an efficient way.</h4>
<div class="marketing">
<div class="row vertical-align featurette">
<div class="col-sm-7 featurette-text">
<span><h2 class="featurette-heading">Fault tolerance</h2>
<p class="lead">
Fault Tolerance is a property of systems to maintain
their functionlity even when its components crash.
For neural networks, this means that the output is preserved even when some neurons crash <d-cite key="torres2017fault"></d-cite>.
</p>
</span>
</div>
<div class="col-sm-5">
<img class="featurette-image img-responsive center-block"
src="dropout.png">
<d-footnote>Picture is taken from https://towardsdatascience.com/coding-neural-network-dropout-3095632d25ce</d-footnote>
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<hr />
<div class="row vertical-align featurette">
<div class="col-sm-5">
<img class="featurette-image img-responsive center-block"
src="brain.jpg">
<d-footnote>Picture is taken from https://medium.com/the-spike/yes-the-brain-is-a-computer-11f630cad736</d-footnote>
</div>
<div class="col-sm-7 featurette-text">
<span><h2 class="featurette-heading">Fault tolerance in biological brains</h2>
<p class="lead">
Biological systems are known to be fault-tolerant <d-cite key="navlakha2015distributed"></d-cite>.
Neurons are unreliable, and, therefore, any single neuron cannot be crucial to the computation.
</p>
</span>
</div>
</div>
<hr />
<div class="row vertical-align featurette">
<div class="col-sm-7 featurette-text">
<span><h2 class="featurette-heading">Fault tolerance in neuromorphic hardware</h2>
<p class="lead">
Neuromorphic hardware is an emergent computing paradigm <d-cite key="neuromorphicIBM"></d-cite>.
It promises to speed up neural network computations drastically.
However, the hardware is faulty, and thus neurons crash randomly.
</p>
</span>
</div>
<div class="col-sm-5">
<img class="featurette-image img-responsive center-block"
src="neuromorphic.png">
<d-footnote>Picture is taken from https://www.researchgate.net/figure/Comparison-between-Brains-Computing-System-with-Conventional-Von-Neumann-Computing_fig1_316727654</d-footnote>
</div>
</div>
<hr />
<div class="row vertical-align featurette">
<div class="col-sm-5">
<img class="featurette-image img-responsive center-block"
src="timeline.png">
<d-footnote>Picture is taken from http://virgipla.wixsite.com/travellers/single-post/2017/03/07/111-Timeline-of-the-ancient-Greece</d-footnote>
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<div class="col-sm-7 featurette-text">
<span><h2 class="featurette-heading">Research on fault tolerance</h2>
<p class="lead">
The problem of fault tolerance in neural networks consists of defending the performance of the network
against random crashes of individual neurons.
The problem was well-studied in the 90s
for small networks <d-cite key="torres2017fault"></d-cite>.
However, advances in neuromorphic hardware ask for research in that direction for deep networks.
</p>
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<hr />
<div class="row vertical-align featurette">
<div class="col-sm-6 col-lg-6 col-md-6 featurette-text">
<span><h2 class="featurette-heading">Definitions</h2>
<p class="lead">
We quantify fault tolerance of a neural network as the error in the output: $$\Delta=\hat{y}-y$$ where $$\hat{y}$$ is the output of the network with crashed neurons, and $$y$$ is the original (correct) output.
Our goal is to guarantee that the error $$\Delta$$ does not exceed $$\varepsilon$$ with a high probability $$1-\delta$$.
</p>
</span>
</div>
<div class="col-sm-6 col-lg-6 col-md-6">
<img class="img-responsive center-block"
src="tail_bound.png">
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<hr />
<div class="row vertical-align featurette">
<div class="col-sm-5">
<img class="featurette-image img-responsive center-block"
src="regul.png">
<d-footnote>Picture is taken from https://en.wikipedia.org/wiki/Regularization_(mathematics)</d-footnote>
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<span><h2 class="featurette-heading">Fault Tolerance and Dropout</h2>
<p class="lead">
Our problem is mathematically similar to the theoretical study of Dropout <d-cite key="hintonDropout"></d-cite>.
In fact, Dropout was invented in the context of Fault Tolerance research <d-cite key="kerlirzin"></d-cite>.
However, we are interested in the <b>average error in the output</b> $$\mathbb{E}\Delta$$.
In contrast, Dropout is a <b>regularization technique,</b>
and, thus, theoretical studies of it are concerned about generalization properties.
Thus, these research directions are <b>related, but fundamentally different</b>.
</p>
</span>
</div>
</div>
<hr />
<div class="row vertical-align featurette">
<div class="col-sm-6 col-md-6 featurette-text">
<span><h2 class="featurette-heading">Fault Tolerance and Adversarial Examples</h2>
<p class="lead">
Our problem is also related to the phenomenon of Adversarial Examples <d-cite key="goodfellow2015explaining"></d-cite>.
Indeed, if we consider faults in the input, then fault tolerance is concerned about
the classification outcome under the <b>average perturbation</b>
$$\mathbb E_{\delta x}y(x+\delta x)$$.
In contrast, Adversarial Examples are the <b>worst-case</b> perturbations
$$
\max_{\delta x}Loss(y(x+\delta x))
$$.
Thus, the problem we consider is related to adversarial examples, but, again, <b>quite different</b> from it.
</p>
</span>
</div>
<div class="col-sm-6 col-md-6">
<img class="featurette-image img-responsive center-block"
src="ae.png">
<d-footnote>Picture is taken from https://medium.com/@ml.at.berkeley/tricking-neural-networks-create-your-own-adversarial-examples-a61eb7620fd8</d-footnote>
</div>
</div>
<hr />
<div class="row vertical-align featurette">
<div class="col-sm-6 col-lg-6">
<img class="featurette-image img-responsive center-block"
src="continuous_net.png">
</div>
<div class="col-sm-6 col-lg-6 featurette-text">
<span><h2 class="featurette-heading">Main contribution</h2>
<p class="lead">
We derive a bound on $$\mathbb{E}\Delta$$ and <nobr>Var$$\Delta$$</nobr> in the case if number of neurons $$n\to\infty$$,
and if the network follows the Continuous Limit <d-cite key="le2007continuous,sonoda2017double"></d-cite>: nearby neurons compute similar functions<d-footnote>The figure shows activations of neurons at one layer as the width increases</d-footnote>.
The bound uses a Taylor expansion. The main technical difficulty is to bound the remainder, which we successfully do using our assumptions.
The bound is then analyzed qualitatively, and quantitatively in our experiments.
It is then used to give a probabilistic guarantee on fault tolerance, giving a <b>solution to the our problem.</b>
We show that our bound is the tight asymptotically, up to a constant factor.<d-footnote>In the Section 4 of the paper, paragraph "The best and the worst fault tolerance" – the best variance is O(p/n), and our bound from Theorem 1 is like this as well</d-footnote>
</p>
</span>
</div>
</div>
</div>
<center>
<h1><a href="https://openreview.net/forum?id=rkl_f6EFPS">
→See the full paper on OpenReview</a></h1>
</center>
</div>
</d-article>
<d-appendix>
<d-bibliography src="bibliography.bib"></d-bibliography>
</d-appendix>
</body>