# A glimpse into Feynman’s contributions to the debate on the foundations of
quantum mechanics

M. Di Mauro  Dipartimento di Matematica, Università di Salerno, Via Giovanni
Paolo II,
Fisciano (SA), 84084, Italy
E-mail<EMAIL_ADDRESS>S. Esposito∗ and A. Naddeo∗∗ INFN, Sezione di
Napoli, C. U. Monte S. Angelo, Via Cinthia,
Napoli, 80125, Italy
∗E-mail<EMAIL_ADDRESS>
∗∗E-mail<EMAIL_ADDRESS>
###### Abstract

The broad debate on foundational issues in quantum mechanics, which took place
at the famous 1957 Chapel Hill conference on The Role of Gravitation in
Physics, is here critically analyzed with an emphasis on Richard Feynman’s
contributions. One of the most debated questions at Chapel Hill was whether
the gravitational field had to be quantized and its possible role in wave
function collapse. Feynman’s arguments in favor of the quantization of the
gravitational field, based essentially on a series of gedanken experiments,
are here discussed. Then the related problem of the wave function collapse,
for which Feynman hints to decoherence as a possible solution, is discussed.
Finally, another topic is analyzed, concerning the role of the observer in a
closed Universe. In this respect, Feynman’s many-worlds characterization of
Everett’s approach at Chapel Hill is discussed, together with later
contributions of his, including a kind of Schrödinger’s cat paradox, which are
scattered throughout the 1962-63 Lectures on Gravitation. Philosophical
implications of Feynman’s ideas in relation to foundational issues are also
discussed.

###### keywords:

Gedanken experiment; Wave function collapse; Many-worlds.

##  1 Introduction: the Chapel Hill conference

Richard Feynman’s most famous contribution in quantum theory is undoubtedly
his celebrated path integral approach [1]. His contributions to the debate on
interpretational issues are instead much less well known [2, 3]. The first
such contributions took place in the wide discussions which characterized the
1957 Chapel Hill conference [4], whose title was: _The Role of Gravitation in
Physics_. The Chapel Hill conference, which played a key role in triggering
the so called _Renaissance_ of general relativity [5] was of capital
importance in establishing the future research lines in the field. Broadly,
the main tracks were[6]: classical gravity, quantum gravity, and the classical
and quantum theory of measurement (as a link between the previous two topics).
In particular, foundational quantum issues were widely discussed by
researchers attending the conference. The main motivation for addressing them
came from the fundamental question of whether the gravitational field had to
be quantized, like other fundamental fields, or not. As its title declares,
the ambitious goal of the conference was the merging of general relativity
with the rest of physics and, especially, with the world of elementary
particle physics, which was ruled by quantum mechanics, hence it was (and it
still is, of course) very important to understand physics in regimes where
both gravity and quantum mechanics become important. This was clearly stated
by Peter G. Bergmann in the opening session of the second half of the
conference, which was in fact focused on problems within the quantum domain
(Ref. 4, p. 165):

> Physical nature is an organic whole, and various parts of physical theory
> must not be expected to endure in “peaceful coexistence.” An attempt should
> be made to force separate branches of theory together to see if they can be
> made to merge, and if they cannot be united, to try to understand why they
> clash. Furthermore, a study should be made of the extent to which arguments
> based on the uncertainty principle force one to the conclusion that the
> gravitational field must be subject to quantum laws: (a) Can quantized
> elementary particles serve as sources for a classical field? (b) If the
> metric is unquantized, would this not in principle allow a precise
> determination of both the positions and velocities of the Schwarzschild
> singularities of these particles?

The answer to these questions was expected to lead to a novel perspective on
the ordinary notions of space and time, by introducing uncertainty relations
for quantities such as distances and volumes. Physicists hoped that, as a
result, the divergences which plagued quantum field theory would get
suppressed by the gravitational field. Along with the formal discussion on the
main approaches to the quantization of the gravitational field (namely the
canonical, the functional integral and the covariant perturbative approaches),
which would have been developed in the following decades, a lot of time was
devoted to conceptual issues, especially in discussion sessions. A broad
debate arose on the problem of quantum measurement, the main conceptual
question being:

> What are the limitations imposed by the quantum theory on the measurements
> of space-time distances and curvature? (Ref. 4, p. 167)

or, equivalently

> What are the quantum limitations imposed on the measurement of the
> gravitational mass of a material body, and, in particular, can the principle
> of equivalence be extended to elementary particles? (Ref. 4, p. 167)

As the editors of the written records of the conference[4] emphasized, an
answer to the above questions could not be simply found in dimensional
arguments, since the Planck mass does not set a lower limit to the mass of a
particle whose gravitational field can be measured. Indeed, a simple argument
shows that the gravitational field of _any_ mass can in principle be measured,
thanks to the “long tail” of the Newtonian force law (Ref. 4, pp. 167-8).

The great part of the discussions of interest to us here developed in Section
VIII of the conference (Ref. 4, pp. 243-60), where the focus was mainly on the
contradictions eventually arising in the logical structure of quantum theory
if gravity quantization is not assumed, but in fact many even more
foundamental issues for quantum physics in general were touched as well.
Feynman greatly contributed to these discussions, proposing several gedanken
experiments in order to argue in favor of the necessity of gravitational
quantization, and hinting to decoherence as a viable solution to the problem
of wave function collapse. His contributions in fact triggered a wide debate
on the quantum measurement problem and on the existence and meaning of
macroscopic quantum superpositions. Further, in the subsequent session of the
conference, which was the closing one (Ref. 4, pp. 263-278), he gave a “many-
worlds” characterization of Hugh Everett III’s relative state interpretation
of quantum mechanics [7], which had just been described for the first time by
Everett’s advisor, John A. Wheeler. It is interesting to notice that, in fact,
the Chapel Hill conference was one of the few places where Feynman was
directly involved in discussions about the foundations and interpretation of
quantum mechanics [2].

The Chapel Hill discussions played a pivotal role in triggering subsequent
research on foundational issues, mainly on the quantum measurement problem and
on the various possible interpretations of quantum mechanics, with an emphasis
on the consequences of Everett’s one. In this respect it is worth mentioning
further contributions by Feynman, who despite not being often involved with
foundational quantum issues, nonetheless kept thinking deeply on them also in
the following years, especially when working on gravity, as hinted for example
in a letter written in 1961 to Viktor Weisskopf [8]:

> How can we experimentally verify that these waves are quantized? Maybe they
> are not. Maybe gravity is a way that quantum mechanics fails at large
> distances.

as well as in several suggestions and considerations scattered throughout his
1962-63 graduate lectures on gravitation [9].

In this paper, we historically and critically analyze Feynman’s contributions
to the debate about the foundations of quantum mechanics and gravity, with an
emphasis on the Chapel Hill discussions, which constitute the main source
about his thoughts on the matter. In particular, Section 2 deals with the
problem of the quantization of the gravitational field while in Sections 3 and
4 the focus is on wave function collapse following a measurement and on
Everett’s relative state interpretation of quantum mechanics. Finally our
concluding remarks are summarized in Section 5.

##  2 Quantization of the gravitational field: gedanken experiments

In this Section we report on Feynman’s arguments in favor of the quantization
of the gravitational field. As a matter of fact, Feynman believed that nature
cannot be half classical and half quantum and shared Bergmann’s general ideas,
as recalled by himself in some concluding remarks made at the end of the
conference:

> The questions raised in the last three days have to do with the relation of
> gravity to the rest of physics. We have gravity - electrodynamics - quantum
> theory - nuclear physics - strange particles. The problem of physics is to
> put them all together. The original problem after the discovery of gravity
> was to put gravity and electrodynamics together since that was essentially
> all that was known. Therefore, we had the unified field theories. After
> quantum theory one tries to quantize gravity (Ref. 4, p. 272).

In particular, for Feynman gravity, like the other fundamental interaction
that we experience at the macroscopic, classical level, i.e. electromagnetism,
has a quantum foundation (see e.g. Refs. 10, 11, 12). Coherently with this, he
developed an approach to quantum gravity (whose first hints were given at
Chapel Hill, cf. Ref. 4, pp. 271-276) which was characterized by being fully
quantum from the beginning [9, 13].

Feynman’s arguments in favor of the quantization of gravity were totally
different from the usual dimensional arguments, which rely on the assumption
that gravity has to dominate over all other interactions at the Planck scale.
According to him quantum effects involving gravity should be subject to
probing without invoking such high scales. He provided further evidence to
support this claim by putting forward thought experiments designed to show
that, if quantum mechanics is required to hold for objects massive enough to
produce a detectable gravitational field (the opposite would require a
modification of quantum mechanics), then the gravitational field has to be
quantized, in order to avoid contradictions. This view supported the hope that
quantization of the metric would help taming the divergences present in
quantum field theory and, in this way, it was relevant also for the theory of
elementary particles. Let us thus retrace the discussion in more detail.

Section VIII began with Thomas Gold stating that, in the absence of phenomena
not explainable by classical gravity, the only way to argue in favor of
quantization of gravity is the presence of logical contradictions, of which he
was not convinced (Ref. 4, pp. 243-244). Bryce S. DeWitt pointed out that
difficulties may arise in the presence of quantized matter fields depending on
the choice of the quantum expectation value of the stress-energy tensor as the
source of the gravitational field (Ref. 4, p. 244). Indeed a measurement may
change this expectation value, which implies a change of the gravitational
field itself. The classical theory of gravitation is valid because
fluctuations are negligible at the scale where gravitational effects become
sizeable. At this point Feynman put forward his first thought experiment,
which was a variant of the two-slit diffraction experiment, in which a mass
indicator has been put behind the two-slit wall. Within a space-time region
whose linear dimensions are of order $L$ in space and ${L}/{c}$ in time, the
uncertainty on the gravitational potential (divided by $c^{2}$ so that it is
dimensionless and thus homogeneous to a metric) is in general $\Delta
g=\sqrt{\frac{hG}{c^{3}L^{2}}}=\frac{L_{P}}{L}$, where
$L_{P}=\sqrt{\frac{hG}{c^{3}}}$ is the Planck length.111This follows from an
argument given by Wheeler in Ref. 4, pp. 179-180, involving a path integral
for the gravitational field. The order of magnitude of the potential generated
by a mass $M$ within the spatial part of the considered region is
$g=\frac{MG}{Lc^{2}}$, which implies $\Delta g=\frac{\Delta MG}{Lc^{2}}$. A
comparison with the previous general expression gives a mass uncertainty
$\Delta M=\frac{c^{2}L_{P}}{G}\approx 10^{-5}$ grams, if the time of
observations is less than ${L}/{c}$. On the other side, by allowing an
infinite time, $M$ would be obtained with infinite accuracy. This option,
according to Feynman, could not take place for a mass $M$ fed into the two-
slit apparatus, so that the apparatus will not be able to uncover the
difficulty unless $M$ is at least of order $10^{-5}$ grams. Thus he concluded
that

> Either gravity must be quantized because a logical difficulty would arise if
> one did the experiment with a mass of order $10^{-4}$ grams, or else […]
> quantum mechanics fails with masses as big as $10^{-5}$ grams (Ref. 4, p.
> 245).

Subsequent discussions moved on more formal matters such as the meaning of the
equivalence principle in quantum gravity, with Feynman contrasting Helmut
Salecker’s suggestion of a possible violation of the equivalence principle in
the quantum realm. Then a further remark by Salecker himself brought the
participants’ attention back to the topical issue, the necessity of
gravitational quantization. In particular, as explained by an Editor’s note
(see Ref. 4, p. 249), Salecker hinted to the possibility to build up an
action-at-a-distance theory of gravitation in whole analogy to the
electromagnetic case, with charged quantized particles acting as a source of a
unquantized Coulomb field. At this stage Frederik J. Belinfante proposed the
quantization of both the static part and the transverse part of the
gravitational field (the last one describing gravitational radiation) as a way
to circumvent difficulties related to the choice of the expectation value of
the stress energy tensor as the source of the gravitational field. As noticed
by Zeh [2], Belinfante’s proposal reflects his ideas on the ontology of
quantum mechanics, which point toward an epistemic rather than ontic
interpretation of the wave function:

> “There are two quantities which are involved in the description of any
> quantized physical system. One of them gives information about the general
> dynamical behavior of the system, and is represented by a certain operator
> (or operators). The other gives information about our knowledge of the
> system; it is the state vector […] the state vector can undergo a sudden
> change if one makes an experiment on the system. The laws of nature
> therefore unfold continuously only as long as the observer does not bring
> extra knowledge of his own into the picture.” (Ref. 4, p. 250).

Belinfante’s description used the Heisenberg picture, at variance with Feynman
who reasoned in terms of wave functions as dynamical objects. Indeed,
according to Belinfante “the wave function […] must change for reasons beyond
the system’s physical dynamics. He does not refer to ensembles of wave
functions or a density matrix in order to represent incomplete knowledge”
(Ref. 2, p. 65).

Feynman promptly replied with his second gedanken experiment, a Stern-Gerlach
experiment with a gravitational apparatus. More in detail, he considered a
spin-$1/2$ particle going through a Stern-Gerlach apparatus and then crossing
the first or the second of two counters (denoted as $1$ and $2$,
respectively), each one connected by means of a rod to an indicator. He took
the indicator as a little ball with a diameter of $1$ cm, going up or down
depending on the position of the object, at counter $1$ or $2$, respectively.
Quantum mechanics, as underlined by Feynman, provides in principle an
amplitude for the ball up and an amplitude for the ball down222Zeh [2] points
out how Feynman’s description of the measurement process is very close to the
standard measurement and registration device proposal by John von Neumann
[14].. However, the ball is chosen to be macroscopic, so as to be able to
produce a detectable gravitational field, which in turn can move a probe ball.
In other words a channel between the object and the observer is established
via the gravitational field. This ideal experiment led Feynman to infer the
following conclusion about gravity quantization:

> Therefore, there must be an amplitude for the gravitational field, provided
> that the amplification necessary to reach a mass which can produce a
> gravitational field big enough to serve as a link in the chain does not
> destroy the possibility of keeping quantum mechanics all the way. There is a
> bare possibility (which I shouldn’t mention!) that quantum mechanics fails
> and becomes classical again when the amplification gets far enough, because
> of some minimum amplification which you can get across such a chain. But
> aside from that possibility, if you believe in quantum mechanics up to any
> level then you have to believe in gravitational quantization in order to
> describe this experiment (Ref. 4, p. 251).

A subsequent answer to a question by Hermann Bondi333“What is the difference
between this and people playing dice, so that the ball goes one way or the
other according to whether they throw a six or not?”(Ref. 4, p. 252). further
highlighted, according to Zeh (Ref. 2 p. 67), Feynman’s position against the
epistemic interpretation of the wave function:

> I don’t really have to measure whether the particle is here or there. I can
> do something else: I can put an inverse Stern-Gerlach experiment on and
> bring the beams back together again. And if I do it with great precision,
> then I arrive at a situation which is not derivable simply from the
> information that there is a 50 percent probability of being here and a 50
> percent probability of being there. In other words, the situation at this
> stage is not 50-50 that the die is up or down, but _there is an amplitude_
> that it is up and an amplitude that it is down – a complex amplitude – and
> as long as it is still possible to put those amplitudes together for
> interference you have to keep quantum mechanics in the picture. (Ref. 4, p.
> 252, our emphasis)

##  3 Wave function collapse

The aim of this Section is to highlight Feynman’s ideas about the quantum
measurement problem. Indeed, from the last sentence of the last quote, one can
infer a clear reference to the problem of wave function collapse, as well as a
hint to decoherence as a possible solution. A further suggestion can be found
in a subsequent remark:

> Well, it’s a question of what goes on at the level where the ball flips one
> way or the other. In the amplifying apparatus there’s already an uncertainty
> - loss of electrons in the amplifier, noise, etc. - so that by this stage
> the information is completely determined. Then it’s a die argument. You
> might argue this way: Somewhere in your apparatus this idea of amplitude has
> been lost. You don’t need it any more, so you drop it. The wave packet would
> be reduced (or something). Even though you don’t know where it’s reduced,
> it’s reduced. And then you can’t do an experiment which distinguishes
> _interfering_ alternatives from just plain odds (like with dice). (Ref. 4,
> p. 252, our emphasis)

According to Feynman, wave packet reduction occurs somewhere in his
experimental apparatus thanks to the amplifying mechanism, so that a huge
amount of amplification (via the macroscopic gravitational field of the ball)
may be effective in changing amplitudes to probabilities. Then he wondered
about the possibility to devise an experiment able in principle to avoid the
wave packet reduction due to the amplification process. Subsequent criticism
by Leon Rosenfeld and Bondi further stimulated Feynman’s intuition. So he was
led to envisage a sort of quantum interference in his experiment, driven by
the gravitational interaction between macroscopic balls, which could be
described by means of a quantum field with suitable amplitudes taking a value
or another value, or to propagate here and there. But, as Bondi suggested, any
irreversible element should be removed, such as for instance the possibility
of gravitational links to radiate. Probably, this is another point in which a
further hint to the role of decoherence in destroying quantum interference can
be recognized, even if in Bondi’s and Feynman’s words a reference is made only
to classical irreversibility [2]. In fact the meaning of decoherence, its
origin and its role in smearing out phase relations as well as in triggering
the transition to classicality, were still unclear in the late 1950s444For a
historical and research account on decoherence, see Ref. 15 and references
therein. The discussion went on with Feynman arguing that quantum interference
might eventually take place with a mass of macroscopic size, say about
$10^{-5}$ gram or $1$ gram, and hinting to the possible role of gravity in
destroying quantum superpositions. In his words:

> There would be a new principle! It would be fundamental! The principle would
> be: – roughly: Any piece of equipment able to amplify by such and such a
> factor ($10^{-5}$ grams or whatever it is) necessarily must be of such a
> nature that it is irreversible. It might be true! But at least it would be
> fundamental because it would be a new principle. There are two
> possibilities. Either this principle – this missing principle – is right, or
> you can amplify to any level and still maintain interference, in which case
> it’s absolutely imperative that the gravitational field be quantized… I
> believe! or there’s another possibility which I have’t thought of (Ref. 4,
> pp. 254-255, emphasis in original).

The same ideas would have been pursued later by Feynman in his Lectures on
Gravitation[9], where he considered “philosophical problems in quantizing
macroscopic objects” and hinted to the possibility of a failure of quantum
mechanics induced by gravity:

> I would like to suggest that it is possible that quantum mechanics fails at
> large distances and for large objects. Now, mind you, I do not say that I
> think that quantum mechanics does fail at large distances, I only say that
> it is not inconsistent with what we do know. If this failure of quantum
> mechanics is connected with gravity, we might speculatively expect this to
> happen for masses such that $\frac{GM^{2}}{\hbar c}=1$, of $M$ near
> $10^{-5}$ grams, which corresponds to some $10^{18}$ particles (Ref. 9, pp.
> 12-13).

Within the same set of lectures (Ref. 9, p. 14), the possibility is recognized
that amplitudes may reduce to probabilities for a sufficiently complex object,
thanks to a smearing effect on the evolution of the phases of all parts of the
object. Such a smearing effect could have a gravitational origin. A similar
idea is expressed in the end of the letter to Weisskopf [8], as already
mentioned in the Introduction. This shows how Feynman was open-minded with
respect to all possibilities, despite his strong belief in the quantum nature
of reality.

The debate on gravity quantization here highlighted and, in particular,
Feynman’s deep insights, triggered subsequent research on the possibility of a
gravity-induced collapse of wave function, as a viable solution to the
measurement problem in quantum mechanics [16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. The idea is attractive since gravity is ubiquitous in nature, and
gravitational effects depend on the size of objects, so it has been greatly
developed in the subsequent years up to the present day. Along this line of
thinking, Roger Penrose suggested that a conflict emerges when a balanced
superposition of two separate wave packets representing two different position
of a massive object is considered [22, 23]. In his words (see Ref. 26, p.
475): “My own point of view is that as soon as a significant amount of space-
time curvature is introduced, the rules of quantum linear superposition must
fail”. Clearly the conflict is the result of putting together the general
covariance of general relativity and the quantum mechanical superposition
principle. By assuming in each space-time the validity of the notions of
stationarity and energy, and by taking the difference between the identified
time-translation operators as a measure of the ill-definiteness of the final
superposition’s energy, the decay time for the balanced superposition of two
mass distributions can be estimated:

$\displaystyle t_{D}=\frac{\hbar}{\Delta E_{grav}}.$ (1)

Here $\Delta E_{grav}$ is the gravitational self-energy of the difference
between the mass distributions of each of the two locations of the object.
This means that massive superpositions would immediately undergo collapse. The
same idea led Penrose to introduce the so called Schrödinger-Newton equation
[24], which governs a peculiar non-linear evolution of the center-of-mass wave
function. Its soliton-like dynamics is the result of a competition between a
partial shrinking due to the non-linear term and a partial spreading due to
the usual dynamical term.

More recently, other collapse models(called dynamical collapse models) have
been put forward [27, 28, 29, 30], where the collapse of the wave function is
induced by a different mechanism, i.e. the interaction with a random source
such as an external noise source.

From the experimental side, a lot of proposals have been made as well, mainly
aimed to explore a parameter range where both quantum mechanics and gravity
are significant. For instance it has been possible to create a quantum
superposition state with complex organic molecules with masses of the order
$m=10^{-22}$ kg [31, 32]. Recent experimental proposals involve matter-wave
interferometers [33, 34, 35], quantum optomechanics [36, 37, 38, 39] and
magnetomechanics [40]. Today a major challenge is, on one hand, to design
viable experiments at the interface between the quantum and classical worlds,
while on the other hand to reveal and discriminate gravity-induced decoherence
from environmental decoherence in such experiments. However, despite such a
huge theoretical and experimental effort, a definite answer to the quantum
measurement problem as well as to the problem of the emergence of classical
from the quantum world is still lacking.

##  4 The role of the observer in a closed Universe

In this Section we deal with the issue of the role of the observer in a closed
Universe, as emerged from discussions carried out in the closing session of
Chapel Hill conference. Here Wheeler gave the first public presentation of
Everett’s relative-state interpretation of quantum mechanics [7, 41], as
follows:

> General relativity, however, includes the space as an integral part of the
> physics and it is impossible to get outside of space to observe the physics.
> Another important thought is that the concept of eigenstates of the total
> energy is meaningless for a closed Universe. However, there exists the
> proposal that there is one “universal wave function”. This function has
> already been discussed by Everett, and it might be easier to look for this
> “universal wave function” than to look for all the propagators (Ref. 4, p.
> 270).

Feynman promptly replied by characterizing of Everett’s approach as “many-
worlds”:

> The concept of a “universal wave function” has serious conceptual
> difficulties. This is so since this function must contain amplitudes for all
> possible worlds depending on all quantum-mechanical possibilities in the
> past and thus one is forced to believe in the equal reality of an infinity
> of possible worlds (Ref. 4, p. 270).

The same idea will be presented some years later in the Lectures on
Gravitation (Ref. 9, pp. 13-14), where Feynman also discussed the role of the
observer in quantum mechanics. The Schrödinger’s cat paradox allowed him to
illustrate the difference between the results of a measurement carried out by
an external as well as an internal observer. While the external observer
describes his results by an amplitude, with the system collapsing into a well-
defined final state after the measurement, according to the internal observer
the results of the same measurement are given by a probability. Clearly the
absence of an external observer leads to a paradox, which is much more
effective when considering the whole Universe as described by a complete wave
function. This Universe wave function is governed by a Schrödinger equation
and implies the presence of an infinite number of amplitudes, which bifurcate
from each atomic event. In other words, according to Feynman, the Universe is
constantly spitting into an infinite number of branches, as a result of the
interactions between its components. Here the key observation is that
interactions play the role of measurements. As a consequence, an inside
observer knows which branch the world has taken, so that he can follow the
track of his past. Feynman’s conclusion brings into play a conceptual problem:

> Now, the philosophical question before us is, when we make an observation of
> our track in the past, does the result of our observation become real in the
> same sense that the final state would be defined if an outside observer were
> to make the observation (Ref. 9, p. 14)?

A further discussion of the meaning of the wave function of the Universe is
carried out later in the Lectures on Gravitation (Ref. 9, pp. 21-22). Here
Feynman restated his “many-worlds” characterization of Everett’s approach with
reference to a “cat paradox on a large scale”, from which our world eventually
could be obtained by a “reduction of the wave packet”. Concerning this
reduction, he wondered about the relation between Everett’s approach and
collapse mechanisms of whatever origin. Perhaps it may be relevant to quote a
comment by John P. Preskill, within a talk about the Feynman legacy, given at
the APS April Meeting (cf. Ref. 42, slide 29):

> When pressed, Feynman would support the Everett viewpoint, that all
> phenomena (including measurement) are encompassed by unitary evolution
> alone. According to Gell-Mann, both he and Feynman already held this view by
> the early 1960s, without being aware of Everett’s work555We do not agree
> that Feynman was not aware of Everett’s work in the 1960s, since he
> commented on it at Chapel Hill in 1957.. However, in 1981 Feynman says of
> the many-worlds picture: “It’s possible, but I am not very happy with it”.

Historically, Everett’s proposal [7] was the first attempt to go beyond the
Copenhagen interpretation in order to apply quantum mechanics to the Universe
as a whole. To this end, the well known separation of the world into
“observer” and “observed” has to be superseded by promoting the observer to be
part of the system, while the usual quantum rules are still effective in
measuring, recording or doing whatever operation. Quantum fluctuations of
space-time in the very early Universe have also to be properly taken into
account. On the other hand, this proposal lacks an adequate description of the
origin of the quasi-classical realm as well as a clear explanation of the
meaning of the branching of the wave function.

Everett’s work has been further developed by many authors [43, 44, 45, 46, 47,
48] and its more recent generalization is known as decoherent histories
approach to quantum mechanics of closed systems. A characterizing feature of
this formulation is that neither observers nor their measurements play a
prominent role. Furthermore the so called retrodiction, namely the ability to
construct a history of the evolution of the Universe towards its actual state
by using today’s data and an initial quantum state, is allowed. The process of
prediction requires to select out decoherent sets of histories of the Universe
as a closed system, while decoherence in this context plays the same role of a
measurement within the Copenhagen interpretation. Decoherence is a much more
observer-independent concept and gives a clear meaning to Everett’s branches,
the main issue being to identify mechanisms responsible for it. In particular,
it has been shown that decoherence is frequent in the Universe for histories
of variables such as the center of mass positions of massive bodies [49].

##  5 Concluding remarks

In this paper we have retraced Feynman’s thoughts on foundational issues in
quantum mechanics. Such ideas were rarely expressed by him, who mainly linked
his name to his masterful application of thus theory, but they clearly emerged
whenever he focused on the problem the interface of quantum mechanics and
gravity, with the ensuing great conceptual questions. These problems, such as
that of the collapse of the wave function, of macroscopic superpositions and
of observers which are parts of the observed system itself, are of course not
limited to the gravitational realm, but lie at the very heart of quantum
physics. Despite much progress having been achieved since Feynman’s times,
part of which had been anticipated by him and other participants to the Chapel
Hill conference, much work still needs to be done, before his most famous
remark[50] “I think I can say that nobody understands Quantum Mechanics” can
be considered to be no longer true.

## References

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  * [3] M. Di Mauro, S. Esposito and A. Naddeo, A road map for Feynman’s adventures in the land of gravitation, Eur. Phys. H 46, 22 (2021).
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# Federated Learning via Lattice Joint Source-Channel Coding

Seyed Mohammad Azimi-Abarghouyi  KTH Royal Institute of Technology
Stockholm, Sweden
Email<EMAIL_ADDRESS>Lav R. Varshney  University of Illinois Urbana-Champaign
Urbana, IL, USA
Email<EMAIL_ADDRESS>
###### Abstract

This paper introduces a universal federated learning framework that enables
over-the-air computation via digital communications, using a new joint source-
channel coding scheme. Without relying on channel state information at
devices, this scheme employs lattice codes to both quantize model parameters
and exploit interference from the devices. A novel two-layer receiver
structure at the server is designed to reliably decode an integer combination
of the quantized model parameters as a lattice point for the purpose of
aggregation. Numerical experiments validate the effectiveness of the proposed
scheme. Even with the challenges posed by channel conditions and device
heterogeneity, the proposed scheme markedly surpasses other over-the-air FL
strategies.

###### Index Terms:

Federated learning, over-the-air computation, joint source-channel coding,
lattice codes, digital communications

##  I Introduction

Given the rise of advanced wireless edge devices, federated learning (FL)
addresses challenges like data privacy, latency, and bandwidth by enabling
local machine learning processing, ensuring data remains on-device [2]. In
IoT-dense wireless settings with network constraints, FL faces communication
issues. While FL methods using orthogonal multiple access transmission prevent
interference, they result in latency and require more resources [3]. To
address these challenges, a source coding scheme based on lattice codes has
been used in [4, 5] to quantize model parameters in FL. In [5], privacy
enhancement is also considered. This use of lattice-based quantization is
based on the principles of subtractive dithered lattice quantization [6],
which is grounded in information-theoretic arguments. The method in [4] offers
a more precise finite-bit representation for FL than traditional scalar
quantization methods. In a sense, it uses functional quantization [7].

Leveraging interference from simultaneous edge device transmissions, over-the-
air computation [8, 9] offers another strategy. Over-the-air FL, built on
this, manages FL aggregation amidst interference, making it more efficient,
resource-conservative, and faster than FL using orthogonal transmission.
However, previous research on over-the-air FL has assumed analog modulation,
where the transmitter can freely shape the carrier waveform by opting for any
real number I/Q coefficients [10, 11, 12, 13, 14]. This assumption may not be
possible for existing wireless devices, as they come with digital modulation
chips that may not support arbitrary modulation schemes. Moreover, following a
power control approach, past work requires perfect channel state information
at the transmitter (CSIT) for all devices to determine their transmission
powers to counteract wireless channel effects during the aggregation process.
In this approach, there exist physical constraints on both instantaneous and
average transmission power. Consequently, a device experiencing poor channel
conditions may either be unable to engage in the learning process or be
required to utilize exceptionally high power [10, 11, 12, 14, 13]. Past work
also requires perfect synchronization among the transmitters. These
prerequisites lead to a substantial overhead for channel estimation training
and feedback before any transmission, causing increased delays and reduced
spectral efficiency.

The special case of over-the-air FL using BPSK modulation is studied in [14].
While significantly reducing implementation cost and resource requirements,
one-bit quantization can result in high latency and performance gap with the
non-quantized case [15]. Furthermore, other challenges related to channel
estimation and power control continue to persist.

Most wireless systems use a constant power for blind transmission, not
directly adjusting for the channel. Beyond not requiring CSIT and power
control, this approach can offer a range of advantages. Firstly, it enables
the maintenance of average energy for signal transmission, no matter how the
channel varies. Additionally, it helps prevent the dynamic range of the
transmitted signal from enlarging, making hardware implementations
significantly simpler and reducing costs. Lastly, approaches that adjust for
the channel at the transmitter can run into channel estimation errors, causing
the values to be multiplied by unpredictable gains when received [16].
Considering these, [18] proposes a blind over-the-air FL strategy. However, by
suggesting a particular receiver equalization, this method calls for a large
number of receiver antennas, and the impact of wireless fading diminishes as
the antenna count grows without bound. On the other hand, [16, 17] investigate
blind over-the-air FL that does not compensate for the adverse effects of
fading.

In this work, we propose a joint source-channel coding scheme that
incorporates novel transmission and aggregation strategies using lattice codes
with adjustable quantization levels. Developing a novel coding base, our aim
is to enhance the resilience of over-the-air FL against noise and
interference, thus ensuring the realization of desired learning outcomes. This
particularly ensures effective performance in challenging channel conditions,
even with a limited number of antennas available at the server. Notably our
blind approach does not rely on any prior knowledge or CSIT. It is inspired by
past research [19, 20, 21, 22] which indicates advantages of nested lattice
coding based modulation in compute-and-forward relaying as a channel coding
scheme, enhancing bit-rate throughput at relays. The key contributions are as
follows.

Compute-Update Scheme: We provide an end-to-end real-valued model parameter
transmission framework for FL, named compute-update FL— FedCPU. On the
transmitter side, we develop lattice codes for quantization of the model
parameters. Our quantization approach is based on normalization and dithering.
On the server side, we decode a single integer combination of transmitted
quantized model parameters as a lattice point, which after processing yields a
new form of aggregation. The reason for using integer combinations is because
the aggregation of model parameters in FL has an additive structure.
Additionally, lattice Voronoi region over the decoded integer combination
protects against the decoding error arising from interference and noise. This
level of protection is absent in analog modulation schemes, given their
continuous transmitted values [10, 11, 12, 14, 13, 16, 17, 18]. Also, the
adjustable aggregation method we propose marks a significant shift from the
traditional practice in FL of employing fixed, predefined weights, which are
either equal or based on the size of local datasets. This conventional
approach, recommended for ideal, error-free communication conditions as
outlined in the seminal FL paper [2], has been consistently used in over-the-
air FL research [10, 11, 2, 3, 12, 14, 13, 16, 17, 18, 15, 4, 5]. However,
this persists despite the fact that over-the-air FL deals with imperfect
communication scenarios, plagued by interference and noise. In response to
this, our method strategically tailor the aggregation weights to address the
wide range of decoding errors unique to each integer combination in our model.

Transmitter-Receiver Architecture: To accomplish the aggregation, we propose a
two-layer receiver architecture at the server side that incorporates both the
real and imaginary parts of the signal. The initial layer incorporates an
equalization vector, which is subsequently optimized to minimize the decoding
error. In the subsequent layer, a normalizing factor is introduced and
optimized to minimize the quantization error.

System Insights: Our experimental results highlight the efficacy of FedCPU in
addressing the challenges posed by the lack of CSIT and the presence of only a
limited number of antennas at the server. Shrinking the lattice Voronoi
regions reveals a balance between increased decoding error and diminished
quantization error. Notably, FedCPU showcases markedly superior learning
accuracy compared to other over-the-air schemes, due to its distinctive
aggregation technique and finely-tuned receiver design. The learning results
closely follow those expected in an ideal, error-free orthogonal transmission.

##  II System Model

###  II-A Setup

There are $K$ devices and a single server as the basic setup for FL systems,
see Fig. 1. All the devices are single-antenna units, but the server has $M$
antennas. The downlink channels from the server to the devices are considered
error-free. The uplink channel from each device $k$ to the $m$th antenna of
the server at communication round $t$ is represented by
${h}_{mk,t}\in{\mathbb{C}}$. All devices are considered to have the same
transmission power constraint $P$. However, by appropriately adjusting the
channel coefficients, we can integrate asymmetric power constraints. Let the
entire channel matrix $\mathbf{H}_{t}^{\text{c}}\in{\mathbb{C}}^{M\times K}$
be

$\displaystyle\small\mathbf{H}_{t}^{\text{c}}=\begin{bmatrix}{h}_{11,t}&\cdots&{h}_{1K,t}\\\
\vdots&\ddots&\vdots\\\ {h}_{M1,t}&\cdots&{h}_{MK,t}\\\ \end{bmatrix}.$ (1)

The server is the only node that knows $\mathbf{H}_{t}^{\text{c}}$ as the
channel state information at the receiver (CSIR) after the transmission,
whereas the devices have no knowledge of channels.

###  II-B Learning Algorithm

A frequently adopted FL algorithm named FedAvg [2] is elucidated as follows.
For a specific round $t$ within the set $\\{0,\ldots,T-1\\}$, where $T$ is the
total number of rounds, every device $k$ initiates by refining its individual
learning model over $\tau$ local training epochs. Each of these epochs
utilizes a mini-batch $\boldsymbol{\xi}_{k}^{i}$ with size of $B$ selected at
random from the local dataset $\mathcal{D}_{k}$. This process is further
detailed as $\mathbf{w}_{k,t,i+1}={\mathbf{w}_{k,t,i}}-\mu_{t}\nabla
F_{k}(\mathbf{w}_{k,t,i},\boldsymbol{\xi}_{k}^{i}),\forall
i\in\left\\{0,\ldots,\tau-1\right\\}$, where $\mu_{t}$ is the learning rate at
round $t$, and $\nabla F_{k}(\mathbf{w}_{k,t,i},\boldsymbol{\xi}_{k}^{i})$ is
the local gradient computed at the local model parameter vector
$\mathbf{w}_{k,t,i}$. Then, each device $k$ uploads the local model update
$\Delta\mathbf{w}_{k,t}={\mathbf{w}_{k,t,\tau}}-{\mathbf{w}_{k,t,0}}$ to the
server for aggregation. For ideal aggregation, the global model update can be
achieved by averaging the model updates from all devices, giving them equal
weights, as

$\displaystyle\Delta\mathbf{w}_{\text{G},t+1}=\mathbf{w}_{\text{G},t+1}-\mathbf{w}_{\text{G},t}=\frac{1}{K}\sum_{k=1}^{K}\Delta\mathbf{w}_{k,t}.$
(2)

Subsequently, the server broadcasts the acquired global model
$\mathbf{w}_{\text{G},t+1}$ to all devices. Using this model, each device $k$
sets its starting state for the upcoming round as
$\mathbf{w}_{k,t+1,0}=\mathbf{w}_{\text{G},t+1}$.

###  II-C Lattice Preliminaries

###### Definition 1

A lattice $\Lambda$ is a discrete subgroup of $\mathbb{R}^{s\times 1}$ and can
be expressed as a linear transformation of integer vectors as
$\Lambda=\left\\{\mathbf{G}\mathbf{s}:\mathbf{s}\in\mathbb{Z}^{s\times
1}\right\\}$, where $\mathbf{G}\in\mathbb{R}^{s\times s}$ is the lattice
generator matrix. In $\Lambda$, any linear integer combination of lattice
points is itself a lattice point. Lattice codebooks can be formed by combining
a regular (i.e., low-dimensional lattice) constellation (e.g., PAM, QAM) with
a linear code (e.g., LDPC code) or via more intricate nested lattice
constructions [23].

###### Definition 2

A lattice quantizer is a map $Q_{\Lambda}:\mathbb{R}^{s\times 1}\to\Lambda$
that quantizes a point to its nearest lattice point based on Euclidean
distance as
$Q_{\Lambda}(\mathbf{x})=\arg\min_{\lambda\in\Lambda}\|\mathbf{x}-\lambda\|^{2}$.

###### Definition 3

Fundamental Voronoi region $\mathcal{V}$ is the set of all points that are
quantized to the origin as
$\mathcal{V}=\left\\{\mathbf{x}\in\mathbb{R}^{s\times
1}:Q_{\Lambda}(\mathbf{x})=\mathbf{0}\right\\}$.

###### Definition 4

The second moment of a lattice is the per-dimension second moment of a uniform
distribution over the foundational Voronoi region $\cal V$ as
$\sigma_{\text{q}}^{2}=\frac{\int_{\mathcal{V}}\|\mathbf{x}\|^{2}\mathrm{d}\mathbf{x}}{s\int_{\mathcal{V}}\mathrm{d}\mathbf{x}}$.

##  III FedCPU: Compute-Update Scheme

FedCPU consists of two primary components: the transmission scheme employed by
the devices and the aggregation scheme used by the server. The FedCPU develops
a new version of the aggregation vector. This version features adjustable
aggregation weights through integer coefficients, making use of the lattice
structure and the additive properties of wireless multiple-access channels.
Unlike over-the-air FL schemes in [10, 11, 12, 14, 13] that depend on CSIT,
FedCPU operates with constant power and does not rely on prior information.
This sets FedCPU apart from these schemes that use power control, imposing
average and maximum power constraints, and necessitate device selection. With
FedCPU, every device participates in the learning process based on an
appropriate aggregation weight tailored to its communication conditions, free
from the mentioned constraints. Additionally, though Sec. II.B discusses a
specific algorithm, FedCPU is not limited to it. For simplicity, the iteration
index is omitted here.

###  III-A Transmission Scheme

Based on a lattice $\Lambda$, the transmission preparation procedure carried
out by each device $k$ is as follows.

1) The model update parameters undergo normalization to achieve zero mean and
unit variance. The normalized model update is obtained as
${\widehat{\Delta\mathbf{w}}_{k}}=\frac{{\Delta{\mathbf{w}}_{k}}-\vartheta_{k}\mathbf{1}}{\sigma_{k}}$,
where $\mathbf{1}$ is the all-one vector, and $\vartheta_{k}$ and
${\sigma_{k}}$ denote the mean and standard deviation of the $s$ entries of
the model update vector given by
$\vartheta_{k}=\frac{1}{s}\sum_{i=1}^{s}\delta w_{k,i},\
\sigma_{k}^{2}=\frac{1}{s}\sum_{i=1}^{s}(\delta w_{k,i}-\vartheta_{k})^{2}$,
where $\delta w_{k,i}$ is the $i$-th entry of the vector
$\Delta\mathbf{w}_{k}$. Each device $k$ shares the two scalars
$(\vartheta_{k},\sigma_{k})$ error-free to the server via an orthogonal
feedback channel.

2) A dither vector $\mathbf{d}_{k}\in\mathcal{V}$ is generated independently
from other devices according to a random uniform distribution.

3) The dither is added to $\widehat{\Delta\mathbf{w}}_{k}$ and the nearest
codeword to the result is selected as

$\displaystyle{\overline{\Delta\mathbf{w}}_{k}}=Q_{\Lambda}(\widehat{\Delta\mathbf{w}}_{k}+\mathbf{d}_{k}).$
(3)

The dither addition results in the quantization error becoming uniform. For
the quantization error $\boldsymbol{\epsilon}_{k}$ defined as
$Q_{\Lambda}(\widehat{\Delta\mathbf{w}}_{k}+\mathbf{d}_{k})-\mathbf{d}_{k}=\widehat{\Delta\mathbf{w}_{k}}+\boldsymbol{\epsilon}_{k}$,
it holds that
$\mathbb{E}\left\\{\|\boldsymbol{\epsilon}_{k}\|^{2}\right\\}=\sigma_{\text{q}}^{2}$.

4) The device scales the resulting lattice point and transmits

$\displaystyle\mathbf{x}_{k}=\sqrt{\frac{P}{1+2\sigma_{\text{q}}^{2}}}{\overline{\Delta\mathbf{w}}_{k}},$
(4)

whereby we have
$\mathbb{E}\left\\{\|\mathbf{x}_{k}\|^{2}\right\\}=\frac{P}{1+2\sigma_{\text{q}}^{2}}\mathbb{E}\left\\{\|{\overline{\Delta\mathbf{w}}_{k}}\|^{2}\right\\}\leq\frac{P}{1+2\sigma_{\text{q}}^{2}}\left(\mathbb{E}\left\\{\|\widehat{\Delta\mathbf{w}}_{k}+\mathbf{d}_{k}\|^{2}\right\\}+\sigma_{\text{q}}^{2}\right)=\frac{P}{1+2\sigma_{\text{q}}^{2}}\times\left(\mathbb{E}\left\\{\|\widehat{\Delta\mathbf{w}}_{k}\|^{2}\right\\}+\mathbb{E}\left\\{\|\mathbf{d}_{k}\|^{2}\right\\}+\sigma_{\text{q}}^{2}\right)=P$,
which satisfies the power constraint at the devices.

The server knows all dither vectors due to shared randomness with the devices,
a common assumption in literature [4, 5, 6, 19, 20, 21, 22, 23]. Fig. 2
provides a schematic representation of the transmitter architecture.

Figure 1: Transmitter structure for device $k$.

###  III-B Aggregation Scheme

The baseband received signal at antenna $m$ of the server,
$\mathbf{y}_{m}\in\mathbb{C}^{{s}\times 1}$, is
$\mathbf{y}_{m}=\sum_{k=1}^{K}h_{mk}\mathbf{x}_{k}+\mathbf{z}_{m}$, where
$\mathbf{z}_{m}\in\mathbb{C}^{{s}\times 1}$ is complex Gaussian noise, with
each entry having a variance of $\sigma_{\text{z}}^{2}$. Thus, the real-valued
representation is as follows.

$\displaystyle\mathbf{Y}={\mathbf{H}}\mathbf{X}+\mathbf{Z},$ (5)

where
$\mathbf{Y}=\left[\mathfrak{Re}\left\\{\mathbf{y}_{1}\right\\},\ldots,\mathfrak{Re}\left\\{\mathbf{y}_{M}\right\\},\mathfrak{Im}\left\\{\mathbf{y}_{1}\right\\},\ldots,\mathfrak{Im}\left\\{\mathbf{y}_{M}\right\\}\right]^{\top}$,
${\mathbf{H}}=\begin{bmatrix}\mathfrak{Re}\left\\{\mathbf{H}^{\text{c}}\right\\}\\\
\mathfrak{Im}\left\\{\mathbf{H}^{\text{c}}\right\\}\end{bmatrix}$,
$\mathbf{X}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{K}]^{\top}$,
$\mathbf{Z}=[\mathfrak{Re}\left\\{\mathbf{z}_{1}\right\\},\ldots,\mathfrak{Re}\left\\{\mathbf{z}_{M}\right\\},\mathfrak{Im}\left\\{\mathbf{z}_{1}\right\\},\ldots,\mathfrak{Im}\left\\{\mathbf{z}_{M}\right\\}]^{\top}$.
Based on (5) and the transmission scheme in Sec. III.A, the receiver
architecture for the aggregation at the server is as follows.

1) An equalization vector $\mathbf{b}\in\mathbb{R}^{2M\times 1}$ is used to
obtain

$\displaystyle\mathbf{b}^{\top}\mathbf{Y}=\mathbf{b}^{\top}{\mathbf{H}}\mathbf{X}+\mathbf{b}^{\top}\mathbf{Z}.$
(6)

2) The decoder aims to recover a lattice point or equally a linear integer
combination of quantized model updates
$\sum_{k=1}^{K}a_{k}\overline{\Delta\mathbf{w}}_{k}^{\top}=\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}$
directly from $\mathbf{b}^{\top}\mathbf{Y}$, where
$\mathbf{a}=[a_{1},\ldots,a_{K}]^{\top}\in\mathbb{Z}^{K\times 1}$ denotes the
integer coefficient vector and
$\overline{\Delta\mathbf{W}}=[\overline{\Delta\mathbf{w}}_{1},\ldots,\overline{\Delta\mathbf{w}}_{K}]^{\top}$.

For this decoding, the result $\mathbf{b}^{\top}\mathbf{Y}$ is scaled and then
quantized to its nearest lattice point as

$\displaystyle
Q_{\Lambda}\Biggl{(}\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}\mathbf{b}^{\top}\mathbf{Y}\Biggr{)}=\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}.$
(7)

Figure 2: Receiver structure for the server.

We can rewrite
$\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}\mathbf{b}^{\top}\mathbf{Y}$ as
$\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}+\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}(\mathbf{b}^{\top}{\mathbf{H}}-\mathbf{a}^{\top})\mathbf{X}+\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}\mathbf{b}^{\top}\mathbf{Z}$.
Thus, the decoding error in recovering
$\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}$ is measured in terms of the
mean squared error (MSE), referred to as decoding MSE, as follows.

$\displaystyle\text{DMSE}(\mathbf{a})=\mathbb{E}\Biggl{\\{}\Biggl{\|}\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}(\mathbf{b}^{\top}{\mathbf{H}}-\mathbf{a}^{\top})\mathbf{X}+\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}\times$
$\displaystyle\mathbf{b}^{\top}\mathbf{Z}\Biggr{\|}^{2}\Biggr{\\}}=({1+2\sigma_{\text{q}}^{2}})\left(\|\mathbf{b}^{\top}{\mathbf{H}}-\mathbf{a}^{\top}\|^{2}+\frac{1}{\text{SNR}}\|\mathbf{b}\|^{2}\right),$
(8)

where the independence of $\mathbf{x}_{k}$ and $\mathbf{x}_{k^{\prime}}$,
$\forall k\neq k^{\prime}$, is assumed, due to the transmit normalization and
dithering. Independence is also a factor in other over-the-air FL schemes [11,
13]. Also, $\text{SNR}=\frac{P}{\sigma_{\text{z}}^{2}}$ denotes the signal-to-
noise ratio. The next theorem presents the vector $\mathbf{b}$ that minimizes
the decoding MSE.

###### Theorem 1

The optimal equalization vector for a given coefficient vector $\mathbf{a}$ is

$\displaystyle\mathbf{b}_{\text{opt}}^{\top}=\mathbf{a}^{\top}{\mathbf{H}}^{\top}\left(\frac{1}{\text{SNR}}\mathbf{I}+{\mathbf{H}}{\mathbf{H}}^{\top}\right)^{-1}.$
(9)

###### Proof:

Expanding $\frac{\text{DMSE}(\mathbf{a})}{{1+2\sigma_{\text{q}}^{2}}}$ as

$\displaystyle\|\mathbf{b}^{\top}{\mathbf{H}}-\mathbf{a}^{\top}\|^{2}+\frac{1}{\text{SNR}}\|\mathbf{b}\|^{2}=\left(\mathbf{b}^{\top}{\mathbf{H}}-\mathbf{a}^{\top}\right)\left({\mathbf{H}}^{\top}\mathbf{b}-\mathbf{a}\right)+$
$\displaystyle\frac{1}{\text{SNR}}\|\mathbf{b}\|^{2}=\mathbf{b}^{\top}{\mathbf{H}}{\mathbf{H}}^{\top}\mathbf{b}-2\mathbf{b}^{\top}{\mathbf{H}}\mathbf{a}+\mathbf{a}^{\top}\mathbf{a}+\frac{1}{\text{SNR}}\mathbf{b}^{\top}\mathbf{b}$
(10)

and taking derivative from the result with respect to $\mathbf{b}$, we obtain
$2{\mathbf{H}}{\mathbf{H}}^{\top}\mathbf{b}-2{\mathbf{H}}\mathbf{a}+\frac{2}{\text{SNR}}\mathbf{b}$,
which amounts to zero at (9). ∎

Substituting $\mathbf{b}_{\text{opt}}$ into (III-B), we obtain

$\displaystyle\text{DMSE}(\mathbf{a})=({1+2\sigma_{\text{q}}^{2}})\times$
$\displaystyle\mathbf{a}^{\top}\left[\mathbf{I}-{\mathbf{H}}^{\top}\left(\frac{1}{\text{SNR}}\mathbf{I}+{\mathbf{H}}{\mathbf{H}}^{\top}\right)^{-1}{\mathbf{H}}\right]\mathbf{a}.$
(11)

Applying the matrix inversion lemma [24], (III-B) can be expressed as

$\displaystyle\text{DMSE}(\mathbf{a})=({1+2\sigma_{\text{q}}^{2}})\mathbf{a}^{\top}\left(\mathbf{I}+{\text{SNR}}{\mathbf{H}}^{\top}{\mathbf{H}}\right)^{-1}\mathbf{a}.$
(12)

From the SNR and $\mathbf{H}$ as seen in (12), the DMSE arises due to
interference and noise impacting the decoding process.

The value of $\text{DMSE}(\mathbf{a})$ can be interpreted as the variance of
the effective noise in decoding
$\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}$. Therefore, the decoding error
probability, which represents the probability of decoding any other lattice
point besides $\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}$, is equated to
the event where the decoding noise lies outside
$\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}+\mathcal{V}$.

###### Remark 1

For a decoding noise within
$\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}+\mathcal{V}$, the system is
fully protected from interference and noise.

###### Remark 2

Unlike conventional digital communications, where a decoding error results in
an entirely incorrect message, in FedCPU a decoding error merely introduces an
additive estimation error in the subsequent equation (III-B). This estimation
error depends on the distance between the decoded lattice point and the target
lattice point. In essence, while our goal is to minimize the decoding error to
enhance system performance, it does not constitute a critical constraint or a
bottleneck for the system.

3) After decoding and removing the dithers, the aggregation vector at the
output of the receiver is estimated as

$\displaystyle\Delta\mathbf{w}_{\text{G}}^{\top}=\frac{Q_{\Lambda}\left(\sqrt{\frac{1+2\sigma_{\text{q}}^{2}}{P}}\mathbf{b}^{\top}\mathbf{Y}\right)-\mathbf{a}^{\top}{\mathbf{D}}}{\eta\mathbf{1}^{\top}\mathbf{a}}+\frac{1}{\mathbf{1}^{\top}\mathbf{a}}\sum_{k=1}^{K}a_{k}\vartheta_{k}\mathbf{1}^{\top},$
(13)

where $\eta$ is a normalizing factor, the summation
$\mathbf{1}^{\top}\mathbf{a}$ is to take the average for the aggregation, and
$\mathbf{D}=[{\mathbf{d}}_{1},\ldots,{\mathbf{d}}_{K}]^{\top}$. Due to the
transmit normalization, the estimation in (13) is unbiased, and one can
consider any vector $\mathbf{a}$ with non-negative integer coefficients except
the all-zero vector $\mathbf{0}$ for an aggregation. It differes from other
over-the-air schemes, e.g., [11, 3, 2, 10, 12, 14, 13, 16, 17, 18, 15, 4, 5],
which use predefined weights in FL, such as equal or proportional to local
dataset sizes.

We can rewrite (13) as

$\displaystyle\Delta\mathbf{w}_{\text{G}}^{\top}=\frac{1}{\mathbf{1}^{\top}\mathbf{a}}\sum_{k=1}^{K}a_{k}\Delta\mathbf{w}_{k}^{\top}+\frac{\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}-\mathbf{a}^{\top}\mathbf{D}}{\eta\mathbf{1}^{\top}\mathbf{a}}+$
$\displaystyle\frac{\sum_{k=1}^{K}a_{k}\left(\vartheta_{k}\mathbf{1}^{\top}-\Delta\mathbf{w}_{k}^{\top}\right)}{\mathbf{1}^{\top}\mathbf{a}}=\frac{1}{\mathbf{1}^{\top}\mathbf{a}}\sum_{k=1}^{K}a_{k}\Delta\mathbf{w}_{k}^{\top}+$
$\displaystyle\frac{\mathbf{a}^{\top}\overline{\Delta\mathbf{W}}-\mathbf{a}^{\top}\mathbf{D}}{\eta\mathbf{1}^{\top}\mathbf{a}}-\frac{\sum_{k=1}^{K}a_{k}\sigma_{k}\widehat{\Delta\mathbf{w}_{k}}^{\top}}{\mathbf{1}^{\top}\mathbf{a}}=\frac{1}{\mathbf{1}^{\top}\mathbf{a}}\sum_{k=1}^{K}a_{k}\Delta\mathbf{w}_{k}^{\top}$
$\displaystyle+\frac{\sum_{k=1}^{K}a_{k}\widehat{\Delta\mathbf{w}_{k}}^{\top}+\sum_{k=1}^{K}a_{k}\boldsymbol{\epsilon}_{k}^{\top}}{\eta\mathbf{1}^{\top}\mathbf{a}}-\frac{\sum_{k=1}^{K}a_{k}\sigma_{k}\widehat{\Delta\mathbf{w}_{k}}^{\top}}{\mathbf{1}^{\top}\mathbf{a}}$
(14)

In (III-B), the first term is the desirable aggregation and the remaining
terms are mainly due to the quantization errors. Thus, the overall error
against this estimation in terms of the MSE, referred to as quantization MSE,
is

$\displaystyle\text{QMSE}(\mathbf{a})=\mathbb{E}\Biggl{\\{}\Biggl{\|}\frac{\sum_{k=1}^{K}a_{k}\widehat{\Delta\mathbf{w}_{k}}+\sum_{k=1}^{K}a_{k}\boldsymbol{\epsilon}_{k}}{\eta\mathbf{1}^{\top}\mathbf{a}}-\frac{1}{\mathbf{1}^{\top}\mathbf{a}}\times$
$\displaystyle\sum_{k=1}^{K}a_{k}\sigma_{k}\widehat{\Delta\mathbf{w}_{k}}\Biggr{\|}^{2}\Biggr{\\}}=\frac{1}{(\mathbf{1}^{\top}\mathbf{a})^{2}}\Biggl{(}\mathbb{E}\Biggl{\\{}\Biggl{\|}\sum_{k=1}^{K}\left(\frac{a_{k}}{\eta}-a_{k}\sigma_{k}\right)\times$
$\displaystyle\widehat{\Delta\mathbf{w}_{k}}\Biggr{\|}^{2}\Biggr{\\}}+\frac{1}{\eta^{2}}\mathbb{E}\left\\{\left\|\sum_{k=1}^{K}a_{k}\boldsymbol{\epsilon}_{k}\right\|^{2}\right\\}\Biggr{)}=\frac{1}{(\mathbf{1}^{\top}\mathbf{a})^{2}}\times$
$\displaystyle\left(\left\|\left(\frac{1}{\eta}\mathbf{I}-\text{diag}(\boldsymbol{\sigma})\right)\mathbf{a}\right\|^{2}+\frac{1}{\eta^{2}}\|\mathbf{a}\|^{2}\sigma_{\text{q}}^{2}\right)=\frac{1}{(\mathbf{1}^{\top}\mathbf{a})^{2}}\times$
$\displaystyle\left(\mathbf{a}^{\top}\left(\frac{1}{\eta}\mathbf{I}-\text{diag}(\boldsymbol{\sigma})\right)^{2}\mathbf{a}+\frac{1}{\eta^{2}}\|\mathbf{a}\|^{2}\sigma_{\text{q}}^{2}\right),$
(15)

where $\boldsymbol{\sigma}=[\sigma_{1},\ldots,\sigma_{K}]^{\top}$. The factor
$\eta$ that minimizes the quantization MSE is presented in the next theorem.

(a)

(b)

(c)

Figure 3: Test accuracy (a) for different $M$ values, (b) for different $\rho$
values, and (c) for different FL strategies.

###### Theorem 2

The optimal $\eta$ for a given $\mathbf{a}$ is

$\displaystyle\eta_{\text{opt}}=\frac{\left(1+\sigma_{\text{q}}^{2}\right)\|\mathbf{a}\|^{2}}{\mathbf{a}^{\top}\text{diag}(\boldsymbol{\sigma})\mathbf{a}}.$
(16)

###### Proof:

We can expand $(\mathbf{1}^{\top}\mathbf{a})^{2}\text{QMSE}(\mathbf{a})$ as

$\displaystyle\mathbf{a}^{\top}\left(\frac{1}{\eta}\mathbf{I}-\text{diag}(\boldsymbol{\sigma})\right)^{2}\mathbf{a}+\frac{1}{\eta^{2}}\|\mathbf{a}\|^{2}\sigma_{\text{q}}^{2}=$
$\displaystyle\mathbf{a}^{\top}\left(\frac{1}{\eta^{2}}\mathbf{I}-\frac{2}{\eta}\text{diag}(\boldsymbol{\sigma})+\text{diag}(\boldsymbol{\sigma}^{2})\right)\mathbf{a}+\frac{1}{\eta^{2}}\|\mathbf{a}\|^{2}\sigma_{\text{q}}^{2}.$
(17)

By taking derivative with respect to $\eta$ and equating the resulting
expression to zero, we obtain

$\displaystyle\mathbf{a}^{\top}\left(\frac{-2}{\eta^{3}}\mathbf{I}+\frac{2}{\eta^{2}}\text{diag}(\boldsymbol{\sigma})\right)\mathbf{a}-\frac{2}{\eta^{3}}\|\mathbf{a}\|^{2}\sigma_{\text{q}}^{2}=$
$\displaystyle\frac{1}{\eta}(1+\sigma_{\text{q}}^{2})\|\mathbf{a}\|^{2}-\mathbf{a}^{\top}\text{diag}(\boldsymbol{\sigma})\mathbf{a}=0,$
(18)

which leads to the final result. ∎

Substituting the optimal $\eta$, the QMSE is

$\displaystyle\text{QMSE}(\mathbf{a})=\frac{1}{(\mathbf{1}^{\top}\mathbf{a})^{2}}\left(-\frac{1}{\eta_{\text{opt}}}\mathbf{a}^{\top}\text{diag}(\boldsymbol{\sigma})\mathbf{a}+\mathbf{a}^{\top}\text{diag}(\boldsymbol{\sigma}^{2})\mathbf{a}\right)$
$\displaystyle=\frac{1}{(\mathbf{1}^{\top}\mathbf{a})^{2}}\left(\mathbf{a}^{\top}\text{diag}(\boldsymbol{\sigma}^{2})\mathbf{a}-\frac{\left(\mathbf{a}^{\top}\text{diag}(\boldsymbol{\sigma})\mathbf{a}\right)^{2}}{(1+\sigma_{\text{q}}^{2})\|\mathbf{a}\|^{2}}\right).$
(19)

Fig. 3 provides a schematic representation of the receiver architecture
designed for this aggregation process.

In FedCPU, the coefficient vector $\mathbf{a}$ can be chosen using any metric
to enhance learning performance. However, due to space limitations in
presenting a learning convergence analysis of FedCPU111A detailed convergence
analysis of FedCPU is presented in [1], along with a corresponding metric
which incorporates both DMSE and QMSE., we opt to minimize the decoding MSE as

$\displaystyle\arg\min_{\mathbf{a}\in\mathbb{Z}^{K\times
1}\backslash\mathbf{0}}\mathbf{a}^{\top}\left(\mathbf{I}+{\text{SNR}}{\mathbf{H}}^{\top}{\mathbf{H}}\right)^{-1}\mathbf{a}$
(20)

subject to $a_{k}\geq 0,\ \forall k$. This optimization problem is an integer
program that is NP-hard. Instead, we suggest the following suboptimal problem.

$\displaystyle\text{round-to-
integer}\left\\{\arg\min_{\mathbf{a}\in\mathbb{R}^{K\times
1}}\mathbf{a}^{\top}\left(\mathbf{I}+{\text{SNR}}{\mathbf{H}}^{\top}{\mathbf{H}}\right)^{-1}\mathbf{a}\right\\}$
(21)

subject to $a_{k}\geq 1,\ \forall k$. This problem is convex since the matrix
$\left(\mathbf{I}+{\text{SNR}}\mathbf{H}^{\top}\mathbf{H}\right)^{-1}$ is
positive definite.

##  IV Experimental Results

The task classifies standard MNIST images (parameters as $K=30$,
$\text{SNR}=10$, $\tau=3$, $\mu=0.01$, $B=100$, and $M=30$) using a CNN. The
CNN has two $3\times 3$ convolutional layers with ReLU (32 and 64 channels),
followed by $2\times 2$ max pooling. It concludes with a 128-unit fully
connected layer with ReLU and a softmax output. The lattice generator matrix
is given by
$\mathbf{G}=\text{diag}\left\\{\mathbf{G}_{2},\ldots,\mathbf{G}_{2}\right\\}$,
where $\mathbf{G}_{2}=\begin{bmatrix}0.25&0\\\ 0.125&0.25\\\ \end{bmatrix}$,
[4, 5]. Devices have non-i.i.d. datasets with samples from two classes,
varying in count per device. Performance, gauged by learning accuracy against
global iteration count $t$, is averaged over 20 samples, considering a
Gaussian channel with Rayleigh fading $\sim\exp(5)$ and uniform phase
distribution $\sim{\cal{U}}(0,2{\pi})$.

In Fig. 3.(a), the accuracy is shown for different numbers of antennas $M$ at
the server. The performance improves as $M$ increases because the decoding MSE
is decreased. However, for higher values of $M$, this performance improvement
tends to diminish. In Fig. 3.(b), we explore the implications of lattice
quantization by employing different lattice generator matrices as
$\rho\mathbf{G}$. As $\rho$ decreases, the lattice points become more densely
packed, leading to smaller Voronoi regions. Notably, while a reduced $\rho$
minimizes the quantization errors, it does not necessarily enhance performance
due to the rise in decoding error. This results in a balance of factors.

In Fig. 3.(c), comparisons are made between FedCPU and schemes from the
literature, serving as benchmarks. Among the evaluated benchmarks are ideal
UVeQFed [4] and BFEL [18] schemes. The ideal UVeQFed is the version of UVeQFed
when operating with ideal communications, given that UVeQFed does not delve
into the communication aspect of FL. It employs lattice quantization and
leverages orthogonal transmission to completely avoid interference, operating
under the premise of boundless communication resources. In contrast, the BFEL
scheme, which does not employ quantization, relies on multi-antenna processing
at the server to mitigate interference effects. Notice FedCPU significantly
outperforms BFEL scheme. The enhancement in performance is due to the use of
adaptive aggregation weights, a factor previously overlooked. Additionally, we
employ optimal equalization with a unique receiver architecture at the server,
different from the equalization proposed in [18]. Additionally, FedCPU,
subjected to interference and noise, nearly matches the performance of ideal
UVeQFed, which operates without interference and noise.

##  V Conclusions

In this paper, we introduced a universal federated learning scheme
incorporating lattice codes, pioneering a new over-the-air computation method.
The proposed scheme offers adjustable quantization, enabling distributed
learning through digital modulation. Additionally, it ensures resilience
against interference and noise through coding. In this scheme, with no need
for channel compensation at the transmitter end, an integer combination of
lattice quantized model parameters is reliably decoded and processed for
aggregation. In spite of channel conditions and data heterogeneity,
experimental findings showcased the superior learning accuracy of the proposed
scheme, outperforming existing over-the-air alternatives. Furthermore, even
with a limited number of antennas at the server, the proposed scheme can
nearly achieve the performance level anticipated in a scenario devoid of
errors.

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