# On foundations for deductive mathematics

Frank Quinn

(Date: June 2024)

##  1\. abstract

This article was motivated by the discovery of a potential new foundation for
mainstream mathematics. The goals are to clarify the relationships between
primitives, foundations, and deductive practice; to understand how to
determine what is, or isn’t, a foundation; and get clues as to how a
foundation can be optimized for effective human use. For this we turn to
history and professional practice of the subject. We have no asperations to
Philosophy.

The first section gives a short abstract discussion, focusing on the
significance of consistency. The next briefly describes foundations, explicit
and implicit, at a few key periods in mathematical history. We see, for
example, that at the primitive level human intuitions are essential, but can
be problematic. We also see that traditional axiomatic set theories, Zermillo-
Fraenkel-Choice (ZFC) in particular, are not quite consistent with mainstream
practice. The final section sketches the proposed new foundation and gives the
basic argument that it is uniquely qualified to be considered the foundation
of mainstream deductive mathematics. The “coherent limit axiom” characterizes
the new theory among ZFC-like theories. This axiom plays a role in recursion,
but is implicitly assumed in mainstream work so does not provide new leverage
there. In principle it should settle set-theory questions such as the
continuum hypothesis.

##  2\. Abstractions

“Foundation” refers to a body of deductive practice, here mainstream deductive
mathematics. Historically practice comes first. Foundations are a way to
analyze the body of practice, particularly its consistency.

  1. (1)

A primitive is an object, hypotheses, or method of argument that is postulated
rather than deduced from something else.

  2. (2)

A primitive foundation is a collection of primitives that is complete in the
sense that no other primitives are needed. Note that the methods of argument
(the logic) is part of the foundation.

  3. (3)

A foundation is consistent if every chain of deductions that produces a
contradiction can be shown to have a logical error.

###  2.1. About consistency

The operational view of consistency is “complete reliability”, in the sense
that the outcome of an error-free argument will never be contradicted by the
outcome of any other such argument. Moreover, the outcome will not introduce
errors if used as an ingredient in further arguments. Arguments by
contradiction for example, assume something, deduce a contradiction, and
conclude that the thing assumed must be false. Success is sensitively
dependent on _everything else_ used in the argument being completely reliable,
and that the logic itself does not introduce errors.

Consistency provides an internal criterion for correctness. In practice,
however, it is far more important as a criterion for _in_ correctness. In
reasonably ambitious research programs, 95 percent or more of attempted proofs
turn out to have flaws. Sometimes, with luck and persistence, these flaws can
be corrected, and the attempts converge to a valid argument. Sometimes years
of effort have to be abandoned as hopeless, or outright wrong. The payoff for
this painful process is closure: even outrageously counterintuitive
conclusions are accepted when the arguments are carefully checked and found to
be error-free.

#### Empirical

Gödel has shown that foundations generally cannot prove their own consistency.
However, we can _test_ consistency: make lots of deductions, and verify that
any contradictions are accounted for by logical errors. Sometimes
inconsistencies do not appear until work reaches a certain level of
sophistication. Another wrinkle is that a foundation can be technically
inconsistent but consistent _in practice_ in the sense that methodology has
evolved so as to steer people away from the inconsistencies. The main point is
that empirical consistency is a property of a methodology rather than of a
foundation. Explicit foundations are largely retrofits that record successful
experience and narrow down potential sources of error.

#### Global

Evidence for consistency applies to the development as a whole, not just
individual results. In other sciences, and in mathematics when trying out new
methodologies, experiment can give evidence that individual conclusions are
reliable, within limits and independently of most other conclusions.
Foundational consistency checks are global in that they support consistency of
the whole development. This is the flip side of the fact that a single genuine
failure would throw doubt on the whole development.

###  2.2. Definitions and axioms vs primitives

We clarify usage in the mathematical community.

Definitions are located inside a developed context, and usually consist of
axioms that the things being defined should satisfy. Properties of these
things are to be inferred from the axioms and more-basic ingredients. In
particular, while intuition and intent can guide the formation of a definition
they have no logical force. Indeed, first attempts at definitions typically
fail to have intended properties. Major standard definitions are often the
result of years of work to fix failures and consolidate partial successes, and
should be thought of as distilled hard-won wisdom. They are certainly not
arbitrary constructs.

Primitive objects, hypotheses, etc. are starting points for the logical
context, _not_ defined in terms of lower-level objects. Primitive hypotheses,
in particular, are not axioms in the contemporary sense. Since properties
cannot be inferred from lower-level material, they depend on heuristic ideas,
syntax specifications, and examples.

For example, Euclid’s description of points as “things without width or
length” is not a definition in the contemporary sense because “width” and
“length” have not been given precise meaning. It succeeds as a primitive
because this, filtered through physical experience, some pictures, and
examples of usage, reliably evokes consistent understandings. Polygons are
‘defined’ because they are described in terms of primitives.

We emphasize that consistency at the primitive level depends heavily on
different users extracting functionally equivalent understandings. This has
often been a problem.

###  2.3. Deeper is better

In practice, definitions are often easier to learn and use with precision than
are complicated primitives. For example, attempts to use the real numbers as a
primitive were problematic. Now the reals are defined in terms of natural
numbers, and natural numbers are defined using set theory. This approach has
made full-precision understanding of the reals faster and more routine.

Curiously, another benefit of working several levels above the primitives is
that it can filter out misunderstandings or ambiguities about the primitives.
For instance, the traditional axiomatic approach to set theory takes ‘sets’ as
primitives, and the primitive hypotheses proscribing their behavior are quite
complicated. But these do not have to be understood in detail to get a
completely solid description of the real numbers. The new approach to set
theory has a similar advantage: the primitives (object generators) are several
levels below sets, so sets are _defined_ and their properties proved rather
than hypothesized. This gives a more accessible, and slightly sharper,
understanding of set theory.

###  2.4. Other views

There is a vast literature, mostly Philosophical, about the nature and role of
foundations. It is far from consistent, and while there are interesting
insights, cf. [Maddy], some of it is pretty silly. It does not seem to be
useful to try to sort through it here.

##  3\. Sketchy history

We draw lessons from a few key historical deveopments, mainly to provide
perspective on what happened in the early twentieth century. The accounts are
ideosyncratic, brief, and superficial, but sufficient for the purposes here.
For a bit more detail I have found the essays in ‘The Stanford Encyclopedia of
Philosophy’ to be helpful, for example [Bagaria] and [Ferreirós].

###  3.1. Ancient Greece

The first relatively explicit use of a foundation was in the geometry and
number theory of ancient Greece. The primary significance at the time seems to
have been its use as a metaphor for order in the natural world, not for its
“real-world” applications. Everyone could see, and be impressed by, genuine
consistency coming from logic and innate understanding, as opposed to the
chaotic array of beliefs due to cultural bias or religious or philosophical
doctrine. The practical applications are pretty weak. To some extent this can
be seen as a quantifier issue: for metaphors it is sufficient that the theory
have _some_ striking outcomes. Practical use requires ways to describe
outcomes for _all_ problems in appropriate settings. For example, clever
geometric tricks can determine the area between a line and an arc of a circle
in a few special cases. Calculus gives a description of _every_ such area, and
the appearence of transcendental functions in the solution shows that very few
cases can be solved with ancient techniques. Nonetheless many people find the
classical tricks more meaningful than routine but effective calculus.

#### Meaningful vs effective

We expand on this last remark. It has been known for over fifty years that
humans have an innate physics, and it is essentially Aristotlean. People
therefore find Aristotlean physics comfortable and meaningful, and it was
accepted for centuries even though it fails very simple consistency tests. In
general, humans seem not to expect consistency, either internal or with
reality, of the things we find meaningful.

The lesson for geometry is that Euclidean geometry is in large part an
exploration of human visual perception and physical experience. The fact that
it accurately reflects the structure of the plane is because our visual
perception does, not because it is a direct exploration of the plane. Much of
its power as a metaphor comes from this coincidence: it is comfortable because
it matches our perceptions, and effective because in this case perception
accurately reflects reality. Another lesson from Aristotlean physics is that
outside of some areas in geometry our physical intuitions do not match reality
very well. Approaching mathematics through direct perception seldom leads to
effective internalization, but is attractive and comfortable enough that it
remains a strong theme in elementary education.

###  3.2. Europe in the 1600s

The next major shift began in the early 1600s when Galileo, Kepler, and others
showed that mathematics could be directly powerful in the world, not just a
metaphor. Mathematical development became driven by the needs of physical
models, and methodology evolved rapidly. For instance, there is a conflict
between ratios and negative numbers that had inhibited use of negatives. This
was resolved by replacing ratios with fractions. Compact algebraic notation
replaced earlier discursive formulations. The ancient Greeks were justly proud
of their resolution of “multitudes” into natural numbers, and units indicating
the things being counted, but the analogous resolution of “magnitudes” into
real numbers and units was finally accomplished only in the 1600s. By the time
Newton, Leibnitz, and their contemporaries were active, the power of
application-oriented mathematics was firmly established and core methodologies
were taking shape.

Infinitesimals were often part of the foundational intuitions of the real
numbers in this period. They worked quite well for a time, but eventually
became problematic and had to be replaced by limits. We explain how this fits
into the analysis here. Consider the truncated polynomial ring
$\mathbb{R}[\delta]/(\delta^{2}=0)$. The implicit presumption was that a
function $f$ of a real variable should extend to this ring, and
$f[x+y\delta]=f[x]+g[x]y\delta$ where $g$ is the derivative of $f$. This is
true if $f$ is given by a power series: simply plug the extended variables
into the series. Infinitesimals thus worked as long as it was harmless to
assume functions were piecewise analytic. But limits of analytic functions
need not even be piecewise continuous, and continuous limits can be nowhere
differentiable. When limits became important (eg. for Fourier and Laplace
series) infinitesimals became untenable. The methodology had to evolve from
one in which differentiability was assumed, to a more primitive approach in
which differentiability had to be defined and proved.

In the 1960s Abraham Robinson showed infinitesimals can be used consistently
if arguments are restricted to first-order logic. But this first-order logic
constraint is strongly inconsistent with standard practice. This is a
potential special-purpose tool, not an alternative general approach.

###  3.3. Mid to late 1800s

In this period it was still generally felt that real numbers should be
understood through intuitions from physical experience, though, of course,
this should not include earlier intuitions about infinitesimals. This became
problematic as goals became more ambitious. People without sufficiently
accurate intuitions had unreliable outcomes, and were reduced to appeals to
authority (“Gauss did it so it must be ok”) or were shut out of the game. From
a foundational point of view, “intuitive reals” became unsatisfactory as a
primitive because it was not uniformly understood by different users. This is
a bit different from problems with infinitesimals, which in the larger context
are inconsistent even when correctly understood.

In the mid 1800s Dedekind rigorously described the reals in terms of natural
numbers. This made them reliably accessible but didn’t fully fix the
difficulties, because the natural numbers had varying interpretations. Some
intuitively-attractive ones lacked logical force. This was addressed by
defining natural numbers using early versions of set theory. For human use,
sets seemed to be effective starting points. By 1880 Dederkind was using set-
theoretic methods and terminology, and the idea that mathematics could have a
foundation in a theory of sets began to have traction; cf. [Ferreirós].

By 1900, foundation-oriented mathematicians were having good success with what
we now know as naïve set theory. Coming from another direction, Frege, Cantor,
and others clarified intuitive notions about set theory, and showed that it
had its own substance.

###  3.4. Early 1900s

In 1902 Russell publicized his paradox showing naïve set theory to be
inconsistent. Reactions were quite different in the mainstream and set-theory
communities, and resulted in them drifting apart.

#### Mainstream community

By 1900 quite a few mathematicians had found naïve set theory to be effective
and reliable, and many leading mathematicians were satisfied that all
mainstream work could be derived from it. The response to Russell’s paradox
was to add the caution “don’t say ‘set of all sets’ ”. This did not effect
actual practice since nobody saw a need to do that anyway.

By the 1920s much of the younger generation had embraced the set-theory
foundation and the rigor it enables, and were extremely successful with it.
The main set-theory concern of the period was that the axiom of Choice seemed
too good to be true, not that there would be more problems with naïve set
theory.

#### Set-theory community

The paradox seriously set back the study of sets as a subject. After a few
more unsuccessful attempts they gave up trying to define sets directly, and
instead began abstracting key properties (of naïve set theory) to use as
axioms.

In the next phase of the development, the notion of “function” seemed
problematic. A naïve function $X\to Y$ is “an assignment of an element of $Y$
to each element of $X$”. This does not qualify as a definition because
“assignment” is neither defined nor declared as a primitive. In fact this
description of ‘function’ works well as a primitive, but the community choose
not to follow this route. Instead a single function, the ‘membership’ operator
was taken as a primitive. Specifically, an implementation (or “model”) of the
axioms is a pair $(U,\in)$, where $U$ is a ‘universe’ of possible elements,
and the membership operator is a binary pairing $\in\colon U\times
U\to\textsf{yes}/\textsf{no}$. In general, subcollections of $U$ correspond to
binary functions $U\to\textsf{yes}/\textsf{no}$. The _sets_ of the theory
correspond to binary functions of the form $(\\#\in x)$ for some $x\in U$.
Note that elements are used to parameterize sets. One source of confusion is
that this parameterization is sometimes described as an identification:
“elements of sets are again sets”. This phrasing may be a hangover from 19th
century philosophy. More general functions are then defined in terms of sets.
Specifically, a function $X\to Y$ is a correspondence whose graph is a sub
_set_ of $X\times Y$. The standard forms of the axioms up to this point are
mainly due to Zermillo.

A problem with the Zermillo axioms is that they do not ensure there are enough
functions to transact the basic business of set theory. Experimentation,
drawing on a strong legacy of formal logic, revealed that functions obtained
from set operations by first-order logic are sufficient to give workable
rigorous set theories. Standard formulations of the logical hypotheses are
mainly due to Fraenkel. Many variations have been considered, but after about
1920 the Zermillo-Fraenkel (ZF) axioms were widely accepted as standard. We
will see, however, that while this solved a problem in set theory, it
disconnected formal set theory from the mainstream.

###  3.5. Mid 1900s to early 2000s

Early in this period the axiom of Choice was accepted as well-tested, and
included in standard formulations of both communities. Zermillo-Fraenkel-
Choice became the “gold standard” in the set-theory community, and naïve-with-
choice (as always, with the set-of-sets constraint) remained the implicit
standard for the mainstream.

#### Mainstream community

Naïve set theory—as used in standard practice—enabled highly rigorous
arguments of a depth unimaginable in the previous century. The consistency of
naïve set theory—again, as used in standard practice—is extremely well tested.
It is possibly our most solidly established empirical conclusion. There are
some drawbacks, but there is no reason to accept any loss of functionality to
address these drawbacks.

One drawback is that category theory does not fit comfortably in standard set
theories. Categories were introduced essentially as bookkeeping devices to
encode large-scale structures in algebraic topology and homological algebra.
For these purposes ignoring set-theory mismatches seemed harmless. By the late
1900s delicate work with higher-order category theory and point-set properties
of universal spaces needed more precision. There were proposals to use
categories as a substitute for set theory, cf. [Marquis], [Maddy].
Grothendieck developed a variation, topos theory, initially as a setting for
vigorous abstract algebraic geometry. As it matured this was also promoted as
a substitute for set theory. Later Voevodsky proposed using his “univalent
foundation” as a setting for mathematics, citing its effectiveness for
computer implementation.

These proposals reflect a misapprehension about foundations. We have stressed
that the core job of a foundation is to give a distilled starting point that
identifies key consistency checks, and to encode the results of such checks.
To be effective a foundation should be minimal and as simple as possible. The
proposals above are quite elaborate, and designed to display mathematical
structure or make it more mechanically accessible. These are good things for
applications but, as we saw with the real numbers, working directly with high-
level concepts is likely to cause problems. The situation seems to be that for
generations, set theory has provided such a rock-solid base for mainstream
practice that we have forgotten what it is like to have to worry about the
reliability of our tools. The proposals implicitly assume consistency will
follow from our experience with set theory. Indeed many of them implicitly
depend on a background set theory, so implicitly presume consistency to be
inherited from that.

The foundation described in the next section begins several levels below set
theory. The lower-level environment seems to give an effective setting for
categories.

#### Set-theory community

The ZFC axioms were found to be under-determined, and in 1963 Paul Cohen
introduced Forcing as a systematic way to construct new models. A flowering of
the subject ensued, with many thousands of pages devoted to showing a wide
variety of hypotheses are consistent with, or independent of, the basic
axioms. Deep results were found in other directions, PCF theory and large
cardinal axioms to mention only two. See [Jech] for an extensive introduction.

However, these theories do not provide a foundation for mainstream practice.
Sufficiency of naïve set theory was asserted around 1900, and this assertion
does _not_ apply to theories that require first-order logic. The general
problem is described by the “coherent limit axiom of [Quinn 1] but we
illustrate it in an important special case.Recall that every real number in
the unit interval has a base-2 decimal expansion that assigns either 0 or 1 to
each natural number. This assignment can also be thought of as specifying a
subcollection of $\mathbb{N}$. The restriction of the assignment to any finite
interval $[0,n]$ of natural numbers is a function in any ZFC theory.
Equivalently, the intersection of the subcollection with $[0,n]$ is a subset
in any ZFC theory. The full assignment is the limit of these restrictions, or
equivalently the subcollection is the union of the intersections. However,
there are ZFC models in which some of these limits are not functions in the
model, or equivalently the union of the finite subsets is not a set in the
model. The corresponding real numbers are missing from such theories. To
insure a particular real number is in a particular model we would have to
employ either special information about the model, or first-order logic. This
is an issue even for elementary calculus. But in the mainstream nobody
addresses this. The conclusion is that ZFC is too restrictive to describe
standard mathematical practice, even at basic levels.

The insufficiency of ZFC does not seem to be a deep point. The fact that it
went generally unremarked may be another instance of meaningfulness
overshadowing concern for effectiveness.

##  4\. Object generators

We sketch the proposed new foundation; see [Quinn 1] for details, and [Quinn
2] for a guide to routine use.

###  4.1. Generators

The full story begins several levels below set theory, with “object
generators” and their morphisms. The intuition is that object generators
“generate objects”, essentially like the ‘object’ primitive in category
theory. If $G$ is an object generator then $X\in\in G$ means “$X$ is a an
object in $G$”, and generators are defined using the syntax “$X\in\in G$ means
($\cdots$)”.

Essentially nothing else is included. Unlike sets, generators provide no way
of identifying their own outputs, and no way of knowing if two outputs are the
same. Further, while expected uses are for mathematical objects, for instance
$X\in\in\textbf{ groups }$ means “$X$ is a set with a group structure”,
nothing prevents silly examples, for instance $X\in\in G$ means “$X$ is a
Tuesday in the month of May, 1917”. Rather than try to prevent such things we
filter them out at a later stage. Note that few requirements means few
opportunities for contradiction.

“Morphisms” of generators are essentially the same as “functions” in naïve set
theory: a morphism $F\colon G\to H$ is an assignment of an object $F[X]\in\in
H$ to every object $X\in\in G$.

###  4.2. Logic

The native logic of object generators is non-binary in the sense that we might
_assert_ that $X,Y$ are the same, but in general there is no (yes, no)-valued
function that can detect this. This logic is unfamiliar and somewhat
complicated, and the first step toward set theory is to establish a sub-
context that does use binary logic.

The key ingredients are binary functions: functions to an object generator
with exactly two objects. We denote this by $\\{\textsf{yes},\textsf{no}\\}$,
or $\\{0,1\\}$. One of the primitive hypotheses (axioms) asserts the existence
of a two-object generator. Standard binary logic applies to such functions.
Incidently, these functions have no allowance for time dependence, so they
cannot detect things in the physical world. This is how silly examples of
object generators get filtered out.

###  4.3. Domains and sets

A logical domain is an object generator, say $A$, with a binary pairing
$A\times A\to\\{\textsf{yes},\textsf{no}\\}$ that returns ‘yes’ if the two
elements are the same. The name ‘logical domain’ is supposed to suggest that
these are natural settings for binary logic.

Quantification is the final ingredient. If $A$ is a logical domain then the
object generator whose objects are binary functions on $A$ is called the
“powerset” of $A$, and is denoted by $\textsf{P}[A]$. The domain supports
quantification if there is a binary function
$\textsf{P}[A]\to\\{\textsf{yes},\textsf{no}\\}$ that returns ‘yes’ if and
only if the input is the empty (always-‘no’) function. In standard logic the
empty-detecting operator is given by $f\mapsto\forall x\in
A,(f[x]=\textsf{no})$. The point is that if this one quantification expression
is implemented by a binary function, then all quantifications over $A$ are.

Finally, sets (_relaxed_ sets, if distinctions are necessary) are defined to
be logical domains that support quantification.

###  4.4. Hypotheses

There are four primitive hypotheses. Three are standard: there is a 2-element
set; the axiom of Choice; and the natural numbers support quantification. To
clarify this last, the first two hypotheses suffice to construct the natural
numbers, but do not imply that it supports quantification. This is equivalent
to the standard Axiom of Infinity in ZFC. The final hypothesis asserts that if
a logical domain supports quantification then so does its powerset.

Note that ZFC has many more, and more complex, axioms.

###  4.5. Comparisons

We compare this theory, naïve set theory, ZFC axiomatic set theories, and
standard mathematical practice. First, the “=” operator built into the logic
of naïve and ZFC allows comparison of any two elements of any two sets. In
relaxed set theory “=” is only defined for elements of a single set.
Mathematical practice is modeled on naïve set theory so in principle a global
“=” is available but, as far as the author can tell, it is never used.

Functions are primitives in relaxed and naïve set theory; are described in the
same way; and both are consistent with standard practice. In ZFC only the
equality and membership functions are primitive. Functions obtained by first-
order logical expressions are also functions in the theory. Beyond this,
functions are defined to be correspondences whose graphs are subsets. As
explained in the previous section, this is not consistent with standard
practice.

Finally, relaxed sets satisfy versions of the ZFC axioms. This is made precise
in [Quinn 1] with the construction of an object that satisfies the ZFC-1
axioms, where “-1” means “ignore first-order logical constraints”. As above,
this makes sense because functions are taken as primitives. This construction
turns out to be maximal so, in particular, all models of ZFC uniquely embed in
this as transitive sub-models. The “coherent limit axiom”, a generalization of
the real-number problem described at the end of the last section, is shown in
[Quinn 1] to characterize this maximal theory. This axiom is routinely assumed
in standard practice so standard practice, by default, takes place in relaxed
set theory. We note that identifying the ‘missing axiom’ (Gödel’s phrase) does
not lead to new mainstream methodology, but rather emphasizes the extra work
that would be required to stay in a smaller ZFC model.

The conclusion is that the relaxed set theory defined in [Quinn 1] is uniquely
qualified to be a foundation consistent with mainstream mathematical practice.

## References

  * [Bagaria] Joan Bagaria, _Set Theory_ , The Stanford Encyclopedia of Philosophy (Winter 2023 Edition), Edward N. Zalta and Uri Nodelman (eds.), URL = https://plato.stanford.edu/archives/win2023/entries/set-theory/ .
  * [Ferreirós] José Ferreirós, _The Early Development of Set Theory_ , The Stanford Encyclopedia of Philosophy (Summer 2023 Edition), Edward N. Zalta and Uri Nodelman (eds.), URL = https://plato.stanford.edu/archives/sum2023/entries/settheory-early/ .
  * [Marquis] Jean-Pierre Marquis, ”Category Theory”, The Stanford Encyclopedia of Philosophy (Fall 2023 Edition), Edward N. Zalta and Uri Nodelman (eds.), URL = https://plato.stanford.edu/archives/fall2023/entries/category-theory/ .
  * [Jech] Thomas J. Jech: _Set theory – 3rd Millennium ed, revised and expanded._ Springer monographs in mathematics (2002) ISBN 3-540-44085-2
  * [Maddy] Penelope Maddy _What do we want a foundation to do? Comparing set-theoretic, category-theoretic, and univalent approaches_ Springer nature 2019, ebook ISBN 978-3-030-15655-8
  * [Quinn 1] Frank Quinn, _Object generators, relaxed sets, and a foundation for mathematics_ preprint 2023, http://arXiv.org2110.01489
  * [Quinn 2] Frank Quinn, _Object generators, categories, and everyday set theory_ preprint 2023, http://arxiv.org/abs/2210.07280


∎

11institutetext: Longling Zhang, Boshen Shen, Shan Xi, and Yi Wu
22institutetext: School of Data Science and Technology, Heilongjiang
University, Harbin, China, 150080
22email:
20185759@s.hlju.edu.cn,20194983@s.hlju.edu.cn,2191831<EMAIL_ADDRESS>33institutetext: Ahmed Barnawi 44institutetext: King Abdul Aziz University,
Riyadh 11543, Saudi Arabia
44email<EMAIL_ADDRESS>55institutetext: Neeraj Kumar 66institutetext:
Thapar Institute of Engineering and Technology Pariala India
66email<EMAIL_ADDRESS>
# FedDPGAN: Federated Differentially Private Generative Adversarial Networks
Framework for the Detection of COVID-19 Pneumonia

Longling Zhang$\dagger$ Bochen Shen$\dagger$ Ahmed Barnawi  Shan Xi  Neeraj
Kumar** Yi Wu* $\dagger$Equal contributions*First corresponding author**Second
corresponding author

(Received: date / Accepted: date)

###### Abstract

Existing deep learning technologies generally learn the features of chest
X-ray data generated by Generative Adversarial Networks (GAN) to diagnose
COVID-19 pneumonia. However, the above methods have a critical challenge: data
privacy. GAN will leak the semantic information of the training data which can
be used to reconstruct the training samples by attackers, thereby this method
will leak the privacy of the patient. Furthermore, for this reason that is the
limitation of the training data sample, different hospitals jointly train the
model through data sharing, which will also cause the privacy leakage.  To
solve this problem, we adopt the Federated Learning (FL) framework which is a
new technique being used to protect the data privacy. Under the FL framework
and Differentially Private thinking, we propose a Federated Differentially
Private Generative Adversarial Network (FedDPGAN) to detect COVID-19 pneumonia
for sustainable smart cities. Specifically, we use DPGAN to privately generate
diverse patient data in which differential privacy technology is introduced to
make sure the privacy protection of the semantic information of training
dataset. Furthermore, we leverage FL to allow hospitals to collaboratively
train COVID-19 models without sharing the original data. Under Independent and
Identically Distributed (IID) and non-IID settings, The evaluation of the
proposed model is on three types of chest X-ray (CXR) images dataset
(COVID-19, normal, and normal pneumonia). A large number of the truthful
reports make the verification of our model can effectively diagnose COVID-19
without compromising privacy.

###### Keywords:

Generative Adversarial Networks Federated Learning Differential Privacy
COVID-19 Privacy Protection.

##  1 Introduction

COVID-19 is a highly contagious infectious disease which is caused by Severe
Acute Respiratory Syndrome Coronavirus-2 (SARS-CoV-2) in which it and its
variants are still spreading worldwide (Cao 2020). As COVID-19 is raging
around the world, hospitals lack sufficient staff to diagnose suspected
patients with COVID-19 disease. To improve the efficiency of diagnosing
COVID-19, researchers generally develop deep learning systems to diagnose
patients’ chest X-ray images (Liang et al. 2020, Ting et al. 2020, Chang
2018). Dhiman et al. (2021) proposed the J48 decision three model in order to
discover the COVID-19 samples and gained the conclusion that the method is
useful. Although deep learning technologies have greatly improved the
efficiency of diagnosis, these technologies base on a great amount of
annotated instances and labeled data, which is hard for hospitals to find
enough training samples (Wang et al. 2020, Abdel-Basset et al. 2021a, 2020,
b). Furthermore, due to the privacy of medical image dataset, hospitals find
it hard to gather enough samples. Therefore, the above data availability
issues have brought many severe challenges to such a diagnostic method.

Inspired by the Generative Adversarial Networks (GANs) techniques (Goodfellow
et al. 2014), researchers generally use GAN to generate diverse training data
to solve the data availability issues. For example, Waheed et al. (2020)
proposed a GAN model to generate CXR images by developing an ACGAN model. Bao
et al. (2020) proposed a model named COVID-GAN to predict the impact of
COVID-19 epidemic. However, the above-mentioned method using GAN as a
generative model will leak the patient’s private information. Since the GAN-
based model remembers the semantic information of the distribution of a big
deal number of training samples, it is easy for a hacker to use reverse
engineering to reconstruct the patient’s private data (Jordon et al. 2018, Liu
et al. 2019, Xu et al. 2019, Chen et al. 2020a). For instance, Gu et al.
(2020) used reverse engineering technique to obtain the hidden encoding of the
real image combining the corresponding feature maps of the middle layer
generated in the generator and then reconstructed picture similar to the real
image. Therefore, we need to seek a way to design a data generative model that
does not reveal privacy.

Furthermore, researchers also utilize data sharing strategies to solve data
availability issues (Cosgriff et al. 2020). Moorthy et al. (2020) designed a
data sharing strategy to enable the hospital to have enough labeled data to
train deep learning models. Data sharing methods have been used by hospitals
to expand the scale of training samples until the issuance of the General Data
Protection Regulation (GDPR) (Voigt and Von dem Bussche 2017). The reason why
we can no longer use data sharing methods is that the GDPR stipulates that
organizations do not allow arbitrary sharing of user data, because this will
leak user privacy. In particular, COVID-19 data is medical data, which is very
sensitive to patients. Therefore, we need to seek novel learning methods to
avoid data sharing that would leak privacy and violate the law.

First, since Differential Privacy (DP) technology is generally used in privacy
protection, previous works (Xie et al. 2018, Liu et al. 2019, Xu et al. 2019,
Jordon et al. 2018, Hitaj et al. 2017) focused on using DP technology to
alleviate the problem of GAN leaking privacy. For example, Xie et al. (2018)
and Liu et al. (2019) proposed Differentially Private GAN (DPGAN) to protect
the user privacy by leveraging $(\varepsilon,\delta)$-DP technique. However,
such DPGAN-based models are only suitable for centralized learning rather than
distributed learning. Second, due to GDPR’s restrictions on data sharing
strategies, data exists between hospitals in the form of “islands” (Liu et al.
2020e, Li et al. 2020), which inspired researchers to develop a privacy-
persevering distributed machine learning paradigm, i.e., Federated Learning
(McMahan et al. 2017). In this context, references (Li et al. 2019, Ge et al.
2020, Sheller et al. 2020, Sui et al. 2020) applied FL in medical fields to
develop some privacy-persevering applications such as Medical Imaging (Li et
al. 2019), Medical Relation Extraction (Sui et al. 2020), and Medical Named
Entity Recognition (Ge et al. 2020).

We gain inspiration in these above methods and propose the Federated
Differentially Private Generative Adversarial Network (FedDPGAN) model to
detect COVID-19 pneumonia without compromising patient privacy. In this model,
DPGAN is a key component of the proposed model and its function is by adding
Gaussian noise in training gradient that protects the training samples’
privacy. In particular, we introduce a federated learning framework and
developed a distributed DPGAN to enable different hospitals to train COVID-19
diagnostic models collaboratively without data sharing. Specifically, with the
help of FL and its aggregation mechanism, FedDPGAN can aggregate model
parameters from medical institutions in different geographical locations to
construct a global GAN model with well-preserved privacy. A large number of
truthful data studies that FedDPGAN model is better than the existing
centralized learning and FL-based models. The research contents and
contributions include as:

  * •

Unlike existing frameworks, we propose a Federated Differentially Private
Generative Adversarial Network framework, which enables different hospitals
can utilize the privacy-preserving data augmentation method, i.e., distributed
DPGAN model to generated high-quality training samples which relieves the
problem of lacking the training sample of COVID-19 then apply ResNet (He et
al. 2016) model in FL to achieve high-precision COVID-19 detection.

  * •

To address the data availability issues in detecting COVID-19, we design a
distributed DPGAN by leveraging FL framework. In particular, we find that
distributed DPGAN can alleviate the Non-independent and identically
distributed (non-IID) issue in FL. Specifically, with the help of FL and its
aggregation mechanism, FedDPGAN can construct a global and local data
augmentation model by aggregating model parameters from medical institutions
in different geographical locations to do different medical task.

  * •

We conduct extensive case studies on different pneumonia CXR images
demonstrate that the proposed model FedDPGAN is better than the existing
centralized learning and FL-based models. Specifically, our model shows that
the best centralized model by 1.52% and the FL model by 0.49% in IID
distribution. In non-IID data distribution, our model performance is 3.00%
higher than the best FL model.

The article organizational structure is as follows. Section II is literature
review about differential Private Generative Adversarial Networks and
Federated Learning. Section III Section IV presents the FedDPGAN algorithm. In
FedDPGAN, we applicate the FedAVG algorithm and differential private method
makes introduction particularly. The final result compares between Sections V
and V-F. The Section VI includes brief summary of the article.

##  2 Related Work

In this section, we summarize the advanced work in the Differentially Private
Generative Adversarial Networks (DPGAN) and Federated Learning (FL) fields.

###  2.1 Differentially Private Generative Adversarial Networks

How to combine deep learning technology with privacy protection technology is
an emerging research direction. For example, many researchers apply
differential privacy (DP) technologies to training model that ensures models
security. Abadi et al. (2016b) developed a privacy-preserving deep learning
model training paradigm by adding well-designed differential privacy noise
(i.e., Gaussian noise) when computing the model gradients. Voigt and Von dem
Bussche (2017) proposed a DP-based deep learning models to achieve privacy-
preserving disease classification application. In particular, Xie et al.
(2018) suggested using DPGAN to protect the user privacy by leveraging
$(\varepsilon,\delta)-\mathrm{DP}$ techniques. Inspired by the above work, the
current work focuses on using DPGAN to develop some medical-related
applications (Choi et al. 2017, Xie et al. 2018, Chang et al. 2020). Chang et
al. (2020) utilized DPGAN to develop a medical imaging application.

However, such DPGAN-based models are only suitable for centralized learning
rather than distributed learning.As a result, although researchers have solved
the above problems and proposed some privacy preserving methods to protect the
local model training, distributed DPGAN has not been studied yet. We propose
the distributed DPGAN that can be applied in distributed learning-based
applications.

###  2.2 Federated Learning

Federated Learning (FL) (McMahan et al. 2017) will establish a data protection
model, distributing dataset on each client machine, and aggregating locally-
computed updates for a globally model which helps the participating clients to
achieve experimental results similar to distributed data (Liu et al. 2020c,
b), while maintaining the privacy of the training data (Liu et al. 2020b).
Therefore, as a promising distributed machine learning framework for privacy
protection, FL has spawned many emerging applications such as Google Keyboard
(Hard et al. 2018), traffic flow prediction (Liu et al. 2020d), anomaly
detection (Liu et al. 2020a, Wu et al. 2019), medical imaging (Sheller et al.
2020), etc. In particular, medical institutions turn their attention to FL to
develop a collaborative learning paradigm for privacy protection, thereby
avoiding legal problems caused by data sharing. For example, Ge et al. (2020)
applied Medical Named Entity Recognition (NER) in FL by utilizing different
hospitals data to promote the hospitals training models. Chen et al. (2020b)
proposed FL-QSAR model contributing the performance in QSAR prediction by
collaborating among pharmaceutical institutions in drug discovery. However,
the non-IID problem in FL hinders the rapid development of FL in the medical
field.

To address this problem, many novel optimization algorithms are designed to
overcome the adverse effects of non-IID. Wang et al. (2020) designed a
mechanism based on deep Q learning to maximize the reward for overcoming this
problem by adopting the method of selecting a subset of devices during each
round of communication. However, such optimization algorithms are suitable for
mobile IoT and cannot be applied in the medical field. The reason is that this
algorithm requires complex client selection and training complex deep
Q-leaning models. In this paper, we find that GAN can address this problem by
generating diverse training samples for FL training. In short, solving the
non-IID problem in FL is the only way to apply FL in the medical field.

##  3 Preliminary

###  3.1 Differential Privacy

DP is widely used to maintain secure model and protects the training data,
hence it is a privacy protection technology. The classic definition of DP
provided below relies on the concept of so-called adjacent databases, that is,
databases that differ in only one element (or sample, as it is the case in
Machine Learning datasets). Therefore, the formal definition of DP is as
follows:

######  Definition 1 ($(\varepsilon,\delta)-DP$ (Dwork et al. 2014))

Any two adjacent datasets $b$ and $b^{\prime}$ are input, with the random
algorithm $\mathcal{K}$ and the subset of outputs $S$ hold that:

${\color[rgb]{0,0,0}\Pr[{\mathcal{K}}(b)\in
S]\leq{e^{\varepsilon}}\cdot\Pr\left[{{\cal M}\left({{b^{\prime}}}\right)\in
S}\right]+\delta,}$ (1)

where $\epsilon$ is the privacy budget and $\delta$ is the failure
probability. The smaller the $\delta$, the closer the distribution of the data
output by $\mathcal{K}$ in $b,b’$ datasets.

According to Definition 1, we can use DP to ensure the privacy of the semantic
information of training data. But DP can not be directly applied in deep
learning because DP is applicable to query functions in the database.
Therefore, researchers generally apply DP in deep learning by adding the well-
designed Gaussian noise that is in keeping with the definition of the
differential privacy. The formal definition of Gaussian noise mechanism is as
follows:

######  Definition 2 (Gaussian Noise Mechanism)

$S_{f}$ is a random function sensitivity for the two adjacent dataset $b,b’$,
$f(b)$ is the query function, and ${S_{f}^{2}\cdot{\sigma^{2}}}$ is the
variance of the Gaussian distribution. For each pair of adjacent inputs $b$
and $b^{\prime}$, the Gaussian noise mechanism can be expressed as follows:

${\color[rgb]{0,0,0}\mathcal{M}(b)\triangleq
f(b)+\mathcal{N}\left({0,S_{f}^{2}\cdot{\sigma^{2}}}\right),}$ (2)

where $\mathcal{N}\left(0,S_{f}^{2}\cdot\sigma^{2}\right)$ is the noise to
disturb the distribution in 0 and standard deviation ${S_{f}}\cdot\sigma$.
Then we give the definition of the sensitivity $S_{f}$ of the random function
$f$ as follows:

######  Definition 3 (Sensitivity (Mironov 2017))

The sensitivity of the random function $f$ is as follows:

${\color[rgb]{0,0,0}\Delta
f=\mathop{\max}\limits_{b,b^{\prime}}{\left\|{f(b)-f\left({{b^{\prime}}}\right)}\right\|_{2}},}$
(3)

where prioritize the two adjacent datasets $b,b^{\prime}$.

According to Definition 2 and Definition 3, it can be seen that the core
meaning of the parameter of sensitivity indicating the magnitude of noise is
to indicate the effect of deleting deleted records in the data set on the
query result. That is to say, the noise scale change of the Gaussian noise
mechanism is proportional to the sensitivity. In particular, when
$\delta\geqslant\frac{4}{5}\exp\left({-{{(\sigma\cdot\varepsilon)}^{2}}/2}\right)$
and $\varepsilon<1$, the random function $f$ satisfies the definition of
$(\varepsilon,\delta)$-DP after adding Gaussian noise.

###  3.2 Generative Adversarial Networks

Generative Adversarial Network (GAN) is an approach of the non-supervision
model. GAN contains two parts: Generator $N$ and Discriminator $M$. The
generator randomly takes samples which is from potential space (latent space)
and emulate the truthful traning data more and more. Input set of
discriminator network is truthful output data, that distinguish training data
from truthful samples as much as possible. Inspired by Game Theory, the
generative model $N$ make effort that capture distribution of the data, and
the model to discrimate $M$ tries to estimate the probability. They confront
and constantly adjust the parameters during training. The ultimate goal of GAN
is to make the discriminator unable to judge whether the output result of the
generator is true or fake. We mentioned above that the GAN optimization
problem is actually a game theory of $N$ and $M$, that is, a minimal-
maximization problem, so reference (Goodfellow et al. 2014) proposed an
important approach is to solve this problem:

######  Definition 4 (Optimal Generator)

For generator, it learn a distribution ${P_{g}}$ of dataset. The input data
distribution ${P_{z}(z)}$, the generator
$G\left(z;\theta_{g}\right):z\rightarrow x$, and the discriminator
$G\left(z;\theta_{g}\right):z\rightarrow x$. Therefore, the optimal generator
of a function can be expressed as follows:

${\color[rgb]{0,0,0}\begin{split}\min_{N}\max_{M}V(M,N)=&E_{x\sim
P_{\text{data }}(x)}[\log M(x)]+\\\ &E_{z\sim
P_{z}(z)}[\log(1-M(N(z)))].\end{split}}$ (4)

However, the above optimization model has the problems of vanishing gradient
and samples diversity. Therefore, researchers put forward an optimized GAN,
which solves the problem of gradient disappearance, which is defined as
follows:

######  Definition 5 (Optimize GAN)

$\prod\left(\mathrm{F}_{\gamma},\mathrm{F}_{g}\right)$ is the set of joint
distribution $\gamma$ include $\mathrm{F}_{\gamma}$ and $\mathrm{F}_{g}$ all
possible combinations. $\mathrm{F}_{\gamma}$ and $\mathrm{F}_{g}$ are the edge
distribution.

${\color[rgb]{0,0,0}W\left(\mathrm{~{}F}_{\gamma},\mathrm{F}_{g}\right)=\inf_{\gamma\in\prod\left(\mathrm{F}_{\gamma},\mathrm{F}_{g}\right)}E_{(a,b)\sim\gamma}[\|a-b\|]}$
(5)

###  3.3 Federated Learning

As a promising distributed machine learning framework for privacy protection,
Federated Learning protects users’ privacy data by keeping their local data
locally and only periodically exchanges updates with the server which reduces
their communication costs. The classic algorithm for optimizing federated
optimization problems is Federated Averaging (FedAvg) (McMahan et al. 2017).

In FL, we consider a server $\mathcal{S}$ and a subset of the clients
$\mathcal{K}$ participating in the training of a shared global model
$F(\cdot)$. We assume that each client holds an IID or non-IID datasets
${D_{k}}$. At the client side for data sample $x$, we let $\ell(\omega;x)$ be
the loss function, where $\omega\in{\mathbb{R}^{d}}$ denotes the model’s
trainable parameters. At the server side, we let
$\mathcal{L}(\omega)=E_{x\sim\mathcal{D}}[\ell(\omega;x)]$ be the loss
function and let the server to optimize the following objective function:

${\min_{\omega}}\mathcal{L}(\omega),{\text{ where
}}\mathcal{L}(\omega):=\sum\limits_{k=1}^{K}{{p_{k}}}{\mathcal{L}_{k}}(\omega),$
(6)

where $K$ represents the clients, $({p_{k}},\sum\limits_{k}{{p_{k}}}=1)$
indicates the relative influence of each client on the global model. In FL,
the training is conducted between the server and clients side in a $T$-round
communication rounds to minimize the above objective function following a
three step protocol:

  * •

(Step 1, Initialization): The $t$-th round of training, the server selects a
subset from clients $\mathcal{K}$ to participate in training and broadcasts
the initialized global model parameters $\omega^{t}$ to each client.

  * •

(Step 2, Local Training): Each client individually executes the local training
to obtain the model updates.  Specifically, the client trains the received
global model $\omega^{t}$ on the dataset $\mathcal{D}_{k}$ by using the local
optimizer, e.g., Stochastic Gradient Descent (SGD) and then uploads all
updates $\Delta\omega_{k}^{t}$ to the server.

  * •

(Step 3, Aggregation): After collecting all updates uploaded by
$\mathcal{K}_{t}$ clients, the central server uses the aggregation algorithm,
i.e., FedAvg Algorithm to obtain the new global model which serves as an
initial point for the next communication round by aggregate the model updates.

Repeat the above steps until the global model converges.

Figure 1: Overview of Federated Differentially Private Generative Adversarial
Network (FedDPGAN) framework.

##  4 PROPOSED FedDPGAN MODEL

We start with an introduction of medical DPGAN model. Then we introduce the
FedDPGAN to diagnosis COVID-19 CXR images from various medical platforms.
Specifically, FedDPGAN is a client-server architecture in which server shares
the global model and coordinates the client’s local privacy protection DPGAN
model with SGD optimizer.

###  4.1 Architecture of FedDPGAN

In this subsection, we present the approaches to protect the users’ private
data, including the basic DP noise mechanism and FedDPGAN algorithm. In our
framework, there is a central server $\mathcal{S}$ and a client set
$\mathcal{K}$ with their local dataset $\mathcal{D}_{k}$. Next, we introduce
in detail the functions of the components of the proposed model.

####  4.1.1 Distributed DPGAN

First, we present our distributed DPGAN mechanism. Since COVID-19 data is very
private, we need to protect the privacy of patients when we use GAN to
generate COVID-19 data. Therefore, following existing works (Liu et al. 2019,
Xie et al. 2018), we adopt the method of adding Gaussian noise to the training
gradient to ensure dataset security. Specifically, we adopt this way by adding
random noise in discriminator which interferes with original data
distribution, thereby protecting the privacy of the training data. Therefore,
according to Definitions 1–3 we have:

${g_{\sigma}}\leftarrow{g_{\sigma}}\cdot
min(1,\frac{C}{{\left\|{{g_{\sigma}}}\right\|}})+\mathcal{N}(0,\sigma_{n}^{2}c_{g}^{2}I),$
(7)

where $g_{\sigma}$ is the noisy gradient, $c_{g}$ is the sensitivity of the
gradient function and $C$ is the clipping threshold. The gradient as a random
variable approximately obeys Gaussian distribution, we have:

$f(x)=\frac{1}{{\sqrt{2\pi\sigma}}}\exp\left({-\frac{{{{(x-\mu)}^{2}}}}{{2{\sigma^{2}}}}}\right)\cdot\varepsilon,$
(8)

where $\mu$ is the mean of the random variable.

To make the above random function $f$ satisfy the definition of differential
privacy, we assume that $\Delta
f=\mathop{\max}\limits_{d,d^{\prime}\in\mathcal{D}}{\left\|{f(x)-f(x^{\prime})}\right\|_{2}}$
is the $L2$-Sensitivity and $\forall\delta\in(0,1)$, we have:

$\sigma>\frac{{\sqrt{2\ln(1.25/\delta)}\Delta f}}{\varepsilon}.$ (9)

When we add Gaussain noise
$\mathcal{N}\left(0,\sigma_{n}^{2}c_{g}^{2}I\right)$ to the gradient, If and
only if

${\sigma_{n}}=2q\sqrt{{n_{d}}\log\left({\frac{1}{\delta}}\right)}/\varepsilon,$
(10)

where sample rate is $q$. we can say that this function $f$ satisfies the
definition of DP. To prevent the gradient from exploding, we use gradient
clipping technology to make the gradient in the range of $(-c,c)$.

####  4.1.2 FedDPGAN Framework

Following the client-server architecture in FL, the central server
$\mathcal{S}$ selects a random fraction $C$ of clients. Then the clients
aggregate a model broadcasting the model to each client. After global model is
initialized, the client uploads generator model’s parameter $\theta_{t+1}^{k}$
to the aggregator. Then the aggregator accumulates
$\sum_{k=1}^{k}\frac{N_{k}}{N}\theta_{t+1}^{k}$ getting the average value of
$\theta_{t+1}$ and updates to the global model.

$\theta_{t+1}\leftarrow\sum_{k=1}^{k}\frac{N_{k}}{N}\theta_{t+1}^{k}.$ (11)

The server coordinates multiple clients updating and shares a new global model
into the clients. The specific steps between clients and server can be
summarized in three steps:

  * •

(Step 1, FL Initialization): Firstly, central server picks the subset of the
clients. Then it broadcasts the initialized generator parameter $\theta_{t}$.

  * •

(Step 2, Distributed DPGAN Training): After the initialization, the selected
clients $K$ perform training iterations of SGD over their local data. We add
Gaussian noise $\mathcal{N}(0,\sigma_{n}^{2}c_{g}^{2}I)$ appropriately in a
bounded range $min(1,\frac{C}{\left\|g_{\sigma}\right\|})$ and then it will
automatically clipping parameters to add noise when the next time update. The
client updates the weight parameters $\omega$ and truncating in the range of
$(-c,c)$ after updating the weight parameters to optimize the discriminator.

  * •

(Step 3, FL Aggregation): The clients upload their model parameters
$\theta_{t+1}^{k}$ to the aggregator. The aggregator aggregates all model
parameters getting the average value of
$\sum_{k=1}^{k}\frac{N_{k}}{N}\theta_{t+1}^{k}$ and then updates to server to
create a global model that is used as an initialization point for the next
communication round.

Algorithm 1 Federated Differentially Private Generative Adversarial Network
(FedDPGAN) Algorithm.

0: $P$: The platform set; $\alpha$: The learning rate; $c$: The clipping
parameter; $m$: The mini-batch size; ${N_{d}}$: Discriminator iteration;
${N_{g}}$: Generator iteration; ${\sigma_{n}}$: Noise scale; ${P_{g}(Z)}$:
Noise prior; ${P_{data}(X)}$: Data generating distribution;

0: $\theta$: DP generator;

1: Server executes:

2: Initialize generator parameters $\theta_{0}$ and discriminator parameters
$\omega_{0}$;

3: for each client $k\in P$ in parallel do

4: $\theta_{t+1}^{k}\leftarrow$ ClientUpdate($\theta_{t}$, $\omega_{t},k$)

5: $\theta_{t+1}\leftarrow\sum_{k=1}^{k}\frac{N_{k}}{N}\theta_{t+1}^{k}$

6: end for

7: ClientUpdate($\theta_{t}$, $\omega_{t},k$):

8: for $t_{1}=1,...,N_{g}$ do

9: for $t_{2}=1,...,N_{d}$ do

10: A mini-batch $\left\\{Z_{1},...,Z_{m}\right\\}$ from $P_{g}(Z)$

11: A mini-batch $\left\\{X_{1},...,X_{m}\right\\}$ from $P_{data}(X)$

12: $g_{\omega}\leftarrow g_{\omega}$
min(1,$\frac{C}{\left\|g_{\omega}\right\|})+N(0,\sigma_{n}^{2}c_{g}^{2}I)$(adding
noise)

13: $\omega\leftarrow clip(\omega+\alpha\cdot SGD(\omega,g_{\omega}),-c,c)$

14: end for

15: $g_{\sigma}\leftarrow g_{\sigma}$
min(1,$\frac{C}{\left\|g_{\sigma}\right\|}$)

16: $\theta\leftarrow\theta-\alpha\cdot SGD(\theta,g_{\theta})$

17: end for

18: return $\theta$ to server

###  4.2 FedDPGAN-based COVID-19 Diagnosis Model

We represent the COVID-19 dataset characteristics and the COVID-19 diagnpsis
model. First, we use the publicly available COVID-19 dataset as a benchmark
dataset for evaluating the performance of the proposed model. This dataset
consists of chest X-ray images of patients. To this end, we need to apply
advanced Convolutional Neural Network (CNN) structure suitable for vision
tasks in FedDPGAN to achieve higher performance.

Therefore, in this paper, we use ResNet (He et al. 2016) model to diagnosis
COVID-19 by classifying the chest X-ray images. Specifically, ResNet is a
powerful emerging deep learning model that has attracted considerable
attention in recent years. ResNet adds a direct connection channel to the
network structure to quickly transfer the training gradient, which greatly
improves the efficiency of model training. Specifically, we get the gradient
after the lower layer network training the parameters, the gradient is direct
to transmit to the upper layer network parameters, that is, the original input
information is allowed to be directly transmitted to the upper layer. Also,
the correlation of gradients decays with the increase of layers. It has been
proved that RESNET can effectively reduce the attenuation of this correlation.
This feature enables ResNet to build a deeper network structure, which is
widely used in image classification and is suitable for our COVID-19 medical
image classification task.

##  5 EXPERIMENTAL RESULTS

We apply real-world CXR images that comprehensively evaluate the proposed
model. First, we give details of the experimental environment, datasets,
hyperparameters, and model details for this experiment. Second, we compare the
other baseline model like advanced centralized learning model and electronic
language-based model to determine the performance of our proposed model. Then,
we compare the model performance with other benchmark models under simulated
non-IID distributions. Finally, the influence of privacy parameters on model
performance is discussed.

###  5.1 Experimental Setting

####  5.1.1 Datasets

We evaluate the FedDPGAN on different pneumonia images dataset published by
Cohen et al. (2020b, a), where the dataset consists of normal lung images,
ordinary pneumonia images, and COVID-19 pneumonia images. Specifically, such a
dataset contains 2,000 normal images, 1,250 normal pneumonia images and 350
COVID-19 pneumonia images. As mentioned above, we can find that this data has
the problem of class imbalance, which is why we use the model DPGAN to
generate diverse data. We generate fake chest X-ray images through DPGAN model
and mix them into our dataset. More details can be seen in Fig.2. In addition,
we adjusted the image size about $28\times 28$ pixels that speeds up the
convergence of the model.

(a) Real image

(b) Fake image

Figure 2: Overview of the generated dataset.

####  5.1.2 Baseline Models and Aggregation Method

In this experiment, we use FedResNet, centralized ResNet (He et al. 2016),
Convolutional Neural Network (CNN) (Tajbakhsh et al. 2016), Multilayer
Perceptron (MLP) (Li et al. 2018), $K$ Nearest Neighbors (KNN) (Park and Lee
2018), and Support Vector Machines (SVM) (Morra et al. 2009) as our baseline
models that proves FedDPGAN performance. Note that we apply ResNet model in
our FedDPGAN framework.

Second, we use the FedAvg aggregation algorithm as the updated aggregation
algorithm in the proposed framework. The reason is that the training model
under the classic FedAvg aggregation algorithm performs well in various tasks.

####  5.1.3 Non-IID Setting

In the medical field, since the data of different hospitals are collected by
different types of collection equipment, the data between different hospitals
is non-IID. In this paper, to achieve non-IID data distribution, we assign two
types of data, i.e., normal chest images and general pneumonia images to most
clients and we put COVID-19 images into only a few clients. More details can
be seen in Fig. 3.

Figure 3: Overview of non-IID data allocation method.

####  5.1.4 Hyperparameters

In this experiment, we set $K=100$ clients and place the equal size dataset in
each client. At each round of communication, we randomly select $C=10\%$ of
clients to participate in training and set the the local batch size $B=10$,
local epochs $E=5$, the learning rate of $\alpha=0.01$, and Gaussian noise
generator generates the noise which default $\sigma=0.0001$.

####  5.1.5 Implementation and Setup

The implementation of the model is under the TensorFlow 2.0(Abadi et al.
2016a), which is a powerful framework released by Google that can run on the
GPU for acceleration. PyTorch(Paszke et al. 2019) is an open-source ML toolkit
that hastens everything ranging from research prototyping to production
deployment. All of the experiments are conducted using PyTorch and TensorFlow
with Ubuntu 16.04. Experiments are conducted on a Linux Server with NVIDIA
GeForce RTX 2080TI GPU and an i7 9900K CPU.

Figure 4: Comparison of COVID-19 diagnostic performance between the proposed
model and the benchmark models.

###  5.2 Model Performance

We compare the model performance of the proposed FedDPGAN-based ResNet model
with that of FedResNet, DPGAN-based ResNet, ResNet, CNN, MLP, KNN, and SVM
models with the same simulation configuration. Among these seven competing
methods, DPGAN-based FedResNet and FedResNet are federated models and the rest
are centralized models. ResNet (He et al. 2016) has a good performance on
image classification tasks and become a widely used baseline model. The SVM,
KNN, and MLP models are popular ML models for general classification
applications (Hao Zhang et al. 2006).

Table. 1 indicates the accuracy of model and the compared model in diagnosing
COVID-19. From all of the results in the table, we can see that the proposed
model can not only protect privacy but also use data augmentation method to
improve performance. Fig. 4 shows that the performance of DPGAN-based ResNet
model is better than the best baseline method centralized ResNet by 0.84%,
which is 4.36% higher than the worst centralized baseline method SVM and is
better than the worst deep learning baseline method CNN which is 3.05% below.
The reason is that: (1) The large amount of data generated enables ResNet
models to learn more samples. (2) Model training with DPGAN makes ResNet more
generalizable.

In federated learning, our model can achieve the best model performance which
is 0.49% higher than the baseline method DPGAN-based ResNet model, as shown in
Fig.4. In a word, FedDPGAN-based ResNet model can achieve accurate without
compromising privacy.

Table 1: Comparison of COVID-19 diagnostic performance between the proposed model and the benchmark models under IID setting. Model | Accuracy | Data Augmentation | Privacy Protection
---|---|---|---
FedDPGAN-based ResNet | 94.45% | ✓ | ✓
FedResNet | 93.96% | × | ✓
DPGAN-based ResNet | 93.77% | ✓ | ✓
ResNet He et al. (2016) | 92.93% | × | ×
CNN Tajbakhsh et al. (2016) | 90.72% | × | ×
MLP Li et al. (2018) | 92.05% | × | ×
KNN Park and Lee (2018) | 92.78% | × | ×
SVM Morra et al. (2009) | 89.41% | × | ×

###  5.3 Performance of Federated Learning with Data Augmentation under IID
and non-IID Settings

In this part, we quest the influence of data augmentation methods in IID and
non-IID settings. First, we compare the performance of the FedResNet model
under IID and non-IID settings. Fig. 5 shows that the prediction error of
FedResNet model under the non-IID setting is 2.75% higher than under the IID
setting. Experimental results show that non-IID distribution will affect the
convergence performance of the model, resulting in the degradation of model.
The reason is that the distribution of non-IID data will affect the
convergence of the model, resulting in a decline in model performance.

Second, under non-IID settings, we make overall evaluation of FedResNet by
using data augmentation method and the FedResNet model without this method.
Fig.5 shows that the prediction error of FedDPGAN-based ResNet (using data
augmentation method) model is 3.00% lower than FedResNet (without using data
augmentation method) under non-IID setting. The reason is that data
augmentation methods can alleviate non-IID problems by generating diverse
data. Such a method can make the convergence of federated learning training
more stable.

Third, we compare the performance of FedDPGAN-based ResNet model under non-IID
setting and FedResNet model under IID setting. Fig. 6 shows that the
performance of the FedDPGAN-based ResNet model with non-IID setting is close
to the FedResNet with IID setting. Furthermore, Fig. 6 shows the prediction
error of the proposed models under IID and non-IID settings. In this case, our
model is superior to the centralized ResNet which without privacy protection
as shown in Fig. 7. In a word, our model is more suitable for real-world
medical application scenarios.

Figure 5: (a) Performance comparison of FedResNet model under IID and non-IID
settings; (b) Performance comparison between FedResNet model and FedDPGAN-
based ResNet model under non-IID settings.

Figure 6: (a) Performance comparison of FedResNet model under IID setting and
FedDPGAN-based ResNet model under non-IID settings; (b) Performance comparison
between FedDPGAN-based ResNet model under IID setting and under non-IID
settings. Figure 7: Performance comparison between ResNet model and FedDPGAN-
based ResNet model under non-IID settings.

###  5.4 Performance Comparison of Different Privacy Budgets

In this part, we assess the performance of the proposed model by setting
different privacy budgets. Since the privacy budget $\sigma$ indicates the
scale of Gaussian noise (Fredrikson et al. 2015), we need to explore the
relationship between the scale of Gaussian noise and the performance of the
proposed model. Specifically, we generate fake images of different quality by
adjusting privacy budgets $\sigma$ and then explore their performance. We
enhance the privacy protection ability of datasets through improving $\sigma$.
From Table 2, we can draw a conclusion that the smaller size of $\sigma$ the
higher model performance we will gain. The experimental results show that we
can adjust the privacy budget $\sigma$ to achieve a balance between
performance and privacy protection.

Table 2: Performance under the IID setting and non-IID setting of the proposed model under different privacy budgets $\sigma$. Model | Data distribution | $\sigma$ | Accuracy | Data Augmentation | Privacy Protection
---|---|---|---|---|---
FedDPGAN-based ResNet | non-IID | $10^{-4}$ | 94.11% | ✓ | ✓
| non-IID | $10^{-2}$ | 94.06% | ✓ | ✓
| non-IID | $1$ | 91.90% | ✓ | ✓
FedDPGAN-based ResNet | IID | $10^{-4}$ | 94.45% | ✓ | ✓
| IID | $10^{-2}$ | 93.81% | ✓ | ✓
| IID | $1$ | 94.01% | ✓ | ✓

##  6 RESULTS AND DISCUSSION

We propose the FedDPGAN model can be used in diagnosing COVID-19 under using
CXR images without compromising privacy. Such a model enables hospitals in
different geographic locations to collaboratively train a COVID-19 diagnostic
model without sharing data. Specifically, our method solves two serious
challenges currently encountered in diagnosing COVID-19: data availability and
data privacy. First, in this model, we design a distributed DPGAN model to
address data availability issue by generating COVID-19 image data. In
particular, we use $(\varepsilon,\delta)$-DP noise to protect the privacy of
GAN’ training gradient. Second, we introduce FL framework to protect patient’s
privacy and apply the ResNet model to diagnostic COVID-19. In the experiment
part, we test the performance of FedDPGAN model on COVID-19 chest X-ray image
datasets and compare it with centralized ResNet, CNN, MLP, KNN, and SVM
models. The results show that our method has the best model performance and
privacy protection ability compared with competing methods. Furthermore, the
experimental results indicate that the GAN component in the proposed model can
alleviate the non-IID problem in FL, which opens a window for the use of data
augmentation to solve the non-IID problem.

In the future, we will design a more realistic semi-supervised federated
learning system (Liu et al. 2020a) to solve the lack of data labeling and data
privacy issues in the medical field. Furthermore, we will explore how data
augmentation methods can improve the non-IID problem in FL, which motivates us
to design more efficient data augmentation methods to solve non-IID problem in
the future.

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