Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations
Anonymous Author(s)
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Abstract
Equilibrium propagation (EP) is an alternative to backpropagation (BP) that allowsthe training of deep neural networks with local learning rules.
It thus providesa compelling framework for training neuromorphic systems and understandinglearning in neurobiology.
However, EP requires infinitesimal teaching signals,thereby limiting its applicability in noisy physical systems.
Moreover, the algorithmrequires separate temporal phases and has not been applied to large-scale problems.
Here we address these issues by extending EP to holomorphic networks.
Weshow analytically that this extension naturally leads to exact gradients even forfinite-amplitude teaching signals.
Importantly, the gradient can be computed asthe first Fourier coefficient from finite neuronal activity oscillations in continuoustime without requiring separate phases.
Further, we demonstrate in numericalsimulations that our approach permits robust estimation of gradients in the presenceof noise and that deeper models benefit from the finite teaching signals.
Finally,we establish the first benchmark for EP on the ImageNet 32 × 32 dataset and showthat it matches the performance of an equivalent network trained with BP.
Ourwork provides analytical insights that enable scaling EP to large-scale problemsand establishes a formal framework for how oscillations could support learning inbiological and neuromorphic systems.
Introduction
The backpropagation (BP) of error algorithm [1] underpins the ability of state-of-the-art deep neuralnetworks to learn useful representations from structured data such as speech, vision and text [2].
BP stands out as the most successful algorithm to solve the credit assignment problem in artificialneural networks [3, 4], which can be defined by the following question: How should a synapticconnection be modified in order to improve the global performance of the network to perform a task,as measured by some objective function?
This is a difficult question since individual synapse mayhave a complicated influence on downstream processing.
BP solves credit assignment through thechain rule of differentiation [1].
Although BP is efficiently implemented in software, it is difficult toconceive how BP could plausibly be implemented in biological systems.
The problematic aspects are
BP’s use of symmetric connections and the need for two separate phases: A nonlinear forward pass that propagates neuronal activity and a linear backward pass that carries signed gradient signals [3].
These two types of processing are also inconvenient for training physical neural networks, since both should be handled by the same circuit, and explicitly propagating errors does not harness the device neuromorphic hardware
Despite its implausibility, representations learned with BP match representations of in-vivo data [7] better than networks trained with purely biologically-motivated learning rules such as STDP [8, 9].
This discrepancy raises the question as towhether and how neural dynamics could implement gradient-based credit assignment, and whether itcould be as effective as BP to learn useful representations [3].
Equilibrium propagation (EP) [10] is an alternative algorithm for performing credit assignment indynamical systems that converge to a fixed point, such as energy-based models [11].
EP also proceedsin two phases:
In the first phase the dynamical system is presented with static input data until theunits settle into an equilibrium or fixed point.
We refer to this state as the free equilibrium.
In asecond phase, a teaching signal slightly nudges designated output units towards a target value untilthe dynamics settle into a second equilibrium that is called the nudged equilibrium.
EP estimates lossgradients by comparing the neuronal activity between the two equilibria.
EP is appealing because theresulting learning rule is spatially local when the energy function consists of two-body interactions,as for instance in continuous Hopfield networks [11].
Furthermore EP provably approximates thetrue gradient in the limit of vanishing nudging [10].
More generally, the implicit differentiationcarried out by EP makes it suitable for meta learning [12], where explicitly backpropagating errorsthrough an inner optimization loop becomes prohibitive due to the high memory requirement ofstoring intermediate time steps for regular automatic differentiation.
Nevertheless, classic EP [10, 13, 14] has several limitations.
First, EP estimates only approach theactual loss gradient in the limit of a vanishing nudging or teaching signal.
This requirement makesit impractical for noisy neuromorphic systems where noise can confound small amplitude teachingsignals and also unrealistic as a model for learning in the brain where feedback strongly modulatesprocessing.
Moreover, the mechanisms by which biological circuits could satisfy the requirement forseparate phases remains elusive.
Finally, while EP can train deep networks on CIFAR-10 [14], it hasremained an open question whether it can be scaled up to larger and more complex tasks [4].
In this article, we show that by extending EP with holomorphic network dynamics it naturallyestimates exact gradients for finite teaching signals.
Mathematically, the exact gradients are encodedas a Fourier coefficient of adiabatic neural oscillations.
This finding suggest a natural way ofestimating the gradients online through suitable synaptic filtering operations which dispenses with theneed for separate phases, in a similar spirit to Baldi and Pineda [15] for contrastive Hebbian learning[16].
Our main contributions are the following:
• We develop the theory of holomorphic EP (hEP) and prove that this allows computing exactgradients locally at synapses from finite teaching signal amplitudes of adiabatic oscillations.• We numerically quantify the accuracy of our estimate and show that it outperforms classic
EP, especially in the presence of substrate noise and in deep neural networks.
• We demonstrate learning with an always-on oscillating teaching signal, thereby alleviating
the need for separated phases.
• Finally, we show that hEP achieves the same performance as BP in deep convolutional
neural networks (CNNs) trained on CIFAR-10/100 [17], and ImageNet 32 × 32 [18].
Background and previous work
Equilibrium propagation (EP).
EP [10] allows training convergent dynamical systems to optimizea loss function.
We denote neuronal unit activity by the vector s , and the learnable parameters suchas weights and biases by θ (Fig. 1a).
The system’s dynamics are given as the gradient of a scalarenergy function E ( θ , s ) :
[Equation]
As a consequence, EP can train any energy-based models, e.g., Hopfield networks [11] to performclassification [10, 13, 14].
In classic EP, training proceeds in two phases.
First, a subset of unitsare clamped to the input x and the system goes to a ‘free’ fixed point denoted by s ∗ 0 .
Second, theloss function ℓ ( θ , s , y ) , with target y , is scaled with a small positive nudging factor β and added tothe energy function E which yields the total energy F ( θ , s , β, y ) :
=
E + βℓ
( θ , s , y ) .
This added teaching signal causes the system to reach a second equilibrium s ∗ β , again by minimizing the totalenergy F .
Although we write that ℓ takes all units s as argument, in practice typically only outputunits which encode the target label and thus serve as inputs for teaching signals are considered(Fig. 1b).
The learning objective of the system is to optimize the loss function ℓ at the free fixed point,which is defined by L ( θ , x , y ) :
= ℓ ( θ , s ∗ 0 , y ) .
Scellier and Bengio [10] showed that:
[Equation]
This result requires F to be twice continuously differentiable and assumes that one can apply theimplicit function theorem to the equilibrium equation
∂ s F ( θ , s ∗ 0 , 0)
=
(the y argument is hereafteromitted for clarity), so that β (cid:55)→ s ∗ β is a continuously differentiable map [10].
In practice, theleft-hand side of Eq.
(2) is estimated by finite differences [10, 13, 14, 19].
The appeal of EP forbiological plausibility [3, 20] and neuromorphic hardware [15, 21–23] arises from the fact that (i) thesystem only needs to propagate neural activities (Fig. 1a) and (ii) in layered neural networks witha Hopfield energy [11] (Fig. 1b), the left-hand side of Eq.
(2) can be computed by a Hebbian-likelearning rule as the product of pre- and postsynaptic activity.
In summary, EP implicitly propagateserror signals through differences of neuronal activity during the two phases, whereas BP propagateserror gradients explicitly [3, 20].
However, Eq.
(2) only holds in the limit β → 0 where activitydifferences vanish, which can pose a problem in the presence of noise or when activity differencesvanish with network depth, which is related to the problem of vanishing gradients.
In the following,we introduce holomorphic EP (hEP) which avoids these issues by estimating exact gradients withfinite β , and thus from finite amplitude teaching signals.
Theoretical results
Our main contribution is to show that if F is holomorphic, i.e., differentiable in the sense of complexvariables (see Appendix A.1 for the definition), hEP computes the gradient of the objective function L for finite β , i.e., without requiring the vanishing nudging signals (cf.
Eq.
To accomplish thishEP requires a non-vanishing teaching signal that evolves ‘adiabatically’ in the complex plane withrespect to the dynamics of the system (Fig. 1b).
In other words, we require the dynamical system torelax to its equilibrium on a much shorter timescale than the timescale of the nudge.
Derivation of holomorphic EP.
To show that hEP yields an unbiased gradient through finiteadiabatic nudging, we use the same notation as in Section 2.
Specifically, we extend the theoryby Scellier and Bengio [10] to the complex case and to dynamical systems, or networks whosescalar function governing the dynamics is holomorphic.
In line with classic EP we assume that thedynamical system has a free fixed point as described above.
Lemma 1 (Holomorphic Equilibrium Propagation) .
Let F be a scalar function governing thedynamics, so that the holomorphic implicit function theorem can be applied to the fixed pointequation ∂ s F ( θ , s ∗ 0 , 0)
=
0 , then the gradient formula of equilibrium propagation (Eq.
Proof.
The proof is an extension of the one provided by [10] for the real nudging case.
Theholomorphic implicit function theorem ensures that there exists an open set U ∈ C including 0 suchthat the implicit map β ∈ U
(cid:55)→ s ∗ θ ,β is holomorphic on U .
In particular, the fixed point s ∗ θ ,β isdefined on U (see Fig.
2b for how this area looks for a toy example).
The proof proceeds in two steps.
First we show that the total derivatives of F with respect to θ and β can still be interchanged forcomplex variables by virtue of the Schwarz theorem.
Second we show that, at the fixed point, thetotal derivative of F with respect to β ( θ ) is still equal to the partial derivatives with respect to β ( θ ).
To that end, we apply the chain rule of complex differentiation in:
[Equation]
where s denotes the complex conjugate of s .
At equilibrium, the second term on the right handside cancels by definition of the fixed point, and the third term is zero because F is holomorphic,i.e., its derivative with respect to the conjugate variable is zero according to the Cauchy-Riemanncondition [24].
The same argument holds for the derivative with respect to θ .
Therefore, interchangingthe total derivatives of F with respect to β and θ , and replacing the inner total derivatives by thepartial derivatives, we obtain that the EP gradient formula (Eq.
(2)) still holds, but for complexdifferentiation (Appendix A.1):
[Equation]
which concludes the proof (a more detailed version is in Appendix A.2).
We can now evaluate the left hand side of Eq.
(4) using a Cauchy integral (Appendix A.1):
Theorem 1 (Exact gradient from finite teaching signals) .
Assuming that the conditions of Lemma 1are met and let | β | > 0 be the radius of a circular path around 0 in C contained in the openset U on which the fixed point s ∗ θ ,β is defined. Further assume that this path is parameterized byt ∈ [0 , T ] (cid:55)→ β ( t ) =
| β | e 2i πt/T , where i is the imaginary unit.
Then the loss gradient is given by:
[Equation]
The full proof is given in Appendix A.3. Theorem 1 guarantees that given holomorphic dynamics wecan dispense with the requirement of vanishing teaching signal | β | → 0 in the limit of ‘adiabatic’nudging which corresponds to integrating over infinitely many fixed points with a finite | β | .
Note,that complex-valued teaching signals β ∈ γ produce fixed points in the complex plane computedthrough the same equations as in the real case (see Appendix B).
In particular, multi-layered neuralnetworks (Fig. 1b) can be trained by using the continuous Hopfield dynamics [11].
The trainableparameters are the weights and biases θ = ( w ij , b i ) and the total energy function F is given by:
[Equation]
If the activation function σ is holomorphic, which is true in the case of sigmoid functions, the same F (Eq.
(6)) can be evaluated with complex β , and we can apply Eq.
(5) to obtain:
[Equation]
where σ ∗ i ( t ) :
= σ ( s ∗ i,β ( t ) ) .
Therefore, assuming a T -periodic teaching signal (Fig. 1b), the gradientis proportional to the first exponential Fourier coefficient of the product of the oscillating activities.
Although this formulation assumes complex neuronal output, we show in Appendix A.4 that thegradient in Eq.
(7) can be expressed in terms of the real part or imaginary part only.
The complexteaching signal is therefore best thought of as a way to produce unbiased neuronal oscillations onthe nudging timescale.
In the next section, we numerically estimate the Fourier coefficient of Eq.
(5)with a fixed number of points N on the circle which we use to train networks in the subsequentexperiments and to compare it to the actual loss gradient computed with automatic differentiation.
Numerical estimation of the loss gradient as a Fourier coefficient.
Next, we explain how toestimate the gradient from the corresponding Fourier coefficient (Eq.
In practice, we use a
Riemann sum to compute the integral numerically.
We fix the nudging radius | β | > 0 such that thecircular path lies in the domain U in which equilibria exist as described in Theorem 1.
We sample thepath with N ≥ 2 nudging points { β k :=
| β | e 2i πk/N ; k ∈ [0 , ..., N − 1] } , and define the estimator:
[Equation]
We have that ˆ ∇ ( N ) −→ N [Equation] d L d θ , and the remaining bias term when using N points is:
[Equation]
where C p is the p -th derivative in 0 of the function β (cid:55)→ ∂ θ F ( θ , s ∗ β , β ) (see Appendix A.5 for theproof).
The bias term in Eq.
9 converges to zero with increasing N because it is a sub-sum ofthe ( N + 1) -th order remainder of the series expansion of β (cid:55)→ ∂ θ F ( θ , s ∗ β , β ) in 0.
The rate ofconvergence depends on the C p coefficients and the radius | β | .
In the case N = 2 , the estimate of
Eq.
(8) coincides with the ‘symmetric’ estimate of Laborieux et al.
However, the bias term onthe right hand side of Eq.
(9) is only valid when the dynamics are holomorphic.
Next, we illustratethe approach in a toy experiment, and list three practical improvements brought by hEP.
Experiments
In all the experiments, we used the discrete setting of convergent recurrent neural networks of [13],and the readout scheme of [14] for optimizing the cross-entropy loss function with EP (Appendix B).
All simulations were implemented in Jax [25] and Haiku [26] (Apache License 2.0).
The datasetswere obtained from the Tensorflow datasets library [27].
Our code is publicly available on GitHub[redacted].
The details of simulations and hyperparameters can be found in Appendix E and F.
Demonstration of holomorphic Equilibrium Propagation on a single data point.
To providethe first numerical validation of Theorem 1 while also allowing us to gain intuitions about dynamicsof individual neurons, we implemented a small-scale multi-layer perceptron (MLP) with layerdimensions 6-4-4-4, including input and output layers.
The activation function was a shifted sigmoid,which is holomorphic (Appendix B).
The network was fed with a single datapoint, namely a randomlysampled point from a Gaussian and a random one-hot target.
The dark blue region in Figure 2b showsthe map of complex β for which the network settles to a fixed point after 200 time steps.
We foundexperimentally that the area in the complex plane where stable fixed points exist strongly depends onthe activation function and the weight initialization (see Appendix D).
As β evolves on the circle ofradius 0.1 ( N = 24 ) hidden layer neurons settle into different equilibrium points in the complex plane(Fig. 2a).
We observed that while the teaching signal β ( t ) = | β | e 2i πt/T was purely sinusoidal, thenon-linearity of the network induces neural oscillations that are not purely sinusoidal.
Nevertheless,the gradient is contained in the first mode of these non-linear oscillations (Fig. 2c).
To understand how the gradient is computed accurately when the magnitude of the teaching signal isincreased, we recorded the adiabatic product of activities σ ∗ i ( t ) σ ∗ j ( t ) for one pair of neurons (darkblue) over one teaching period and for increasing values of | β | (Fig. 2c).
In the case | β | = 0 .
001 ,the perturbation induced by the sinusoidal teaching signal is also purely sinusoidal, and the gradientmagnitude is simply the radius of the circle.
However, when | β | is increased to 0.01, the linearapproximation of the perturbation becomes less accurate, because higher powers of β becomesignificant in the series expansion around the free fixed point.
The gradient could still be wellapproximated by taking the mean of the two radii corresponding to real positive and negativeβ , as done by Laborieux et al.
Increasing | β | further to 0.1 and 0.5 yields an even moredeformed perturbation, but the gradient is still correctly contained in the first Fourier coefficient of [Figure] the perturbation.
hEP breaks down when β reaches amplitudes for which the corresponding path intersects with areas in which no stable equilibrium exists (cf.
Fig.
2b, light areas, and Fig. 2d).
However, as we will see in the next sections, the finite teaching amplitudes are beneficial when theneuronal dynamics are subject to noise and when training deep neural networks. [Figure]
fed with a minibatch of ten MNIST samples.
Holomorphic EP can estimate the gradient in continuous time.
Our theoretical findings allowus to revisit in a principled way an idea introduced by Baldi and Pineda [15] for a continuous-timeimplementation of contrastive Hebbian learning [16] and learning rules looking to maximize slowness[28–30].
In this context, the oscillating teaching signal is always-on, and the dynamical networkis governed by three mechanisms acting on distinct timescales.
The smallest timescale T dyn is thetypical time needed by the network to reach its fixed point.
The second timescale is the period of one teaching oscillation T osc , and the third timescale is the number of periods after which synapticplasticity occurs T plas .
The gradient can be estimated online by:
[Equation]
which converges to the gradient if T dyn ≪ T osc ≪ T plas (see Appendix A.6).
This expression issimilar to Eq.
(5.2) in [15].
However their teaching signal oscillates discretely between 0 and 1, andtherefore produces a biased estimate of the gradient.
To test the influence of the oscillation timescale
T osc on the online estimate of Eq.
(10), we compared the online estimation of the gradient over severalperiods between several values of T osc .
To this end, we used a MLP with two hidden layers with 256units each, which we fed with a minibatch of MNIST data [31].
We observed that the gradient couldbe accurately estimated in a few periods for high enough T osc (Fig. 3a, dark curves).
However, whenthe oscillations were too fast, a non-vanishing bias remained in the gradient estimates even for manyperiods (Fig. 3a, lighter curves).
This bias is in all likelihood due to the inability of the system toreach the fixed point (Fig. 3b).
Finally, we found that given appropriate period timings, hEP used inthe online setting can train a network on MNIST (Table 1).
Importantly, the online formulation ofhEP allows to dispense with the requirement of strictly separate learning phases by replacing themwith separate plasticity mechanisms acting on different timescales. [Table] With noise
[Table]
Finite size teaching oscillations provide robustness to noise.
To analyze hEP’s robustness tonoise, we injected a small-amplitude zero-mean Gaussian noise to each neuron in the network inaddition to the input from other neurons.
We then used a single minibatch from the MNIST datasetto compute gradient estimates using classic EP and hEP.
The latter was computed by using onerealisation using N = 15 points, whereas the classic EP estimate was computed using the free andnudged fixed points each averaged over ⌈ N/ 2 ⌉ to provide a fair comparison.
We found that for smallβ when noise amplitudes were comparable to the activity changes caused by teaching oscillations thegradient estimate diverged from the true gradient of the noise-free system (Fig. 3c).
To some extentthis effect could be mitigated by choosing a finite teaching signal ( β ≫ 0 ) [12].
However, since βalso increases the bias for classic EP this creates a trade-off between choosing β either too small ortoo large.
Importantly, even for the optimal choice of β , classic EP did not accurately approximate thegradient of the noise-free system.
In contrast, hEP thanks to its robustness to finite teaching signalsdid provide an accurate estimate of the gradient despite the noise.
We verified that hEP is indeedmore robust to noise than classic EP when used to train the network (Table 1).
Thus, hEP combinedwith finite teaching amplitudes provides an effective way for training noisy computational substrates.
Holomorphic EP matches BP performance on large-scale vision benchmarks. To test hEP’sability to train deep neural networks, we first investigated the influence of the number of fixed points N and the teaching amplitude | β | on the approximating quality ˆ ∇ ( N ) of the loss gradient in aseven-layer VGG-like architecture ([32]; Fig. 4a,b).
We ensured holomorphic dynamics by usingsoftmax pooling layers [23] instead of the non-holomorphic max pooling, and by relying on sigmoidweighted linear units (dSiLU) [33] (see Appendix B) instead of standard ReLUs.
We first considereda minibatch of ten images from the ImageNet 32 × 32 dataset [18] and computed the gradient usinghEP as well as BPTT for reference. Here, our estimate (Eq.
(8)) was computed by first letting thenetwork settle to the free fixed ( β = 0 ), and then running the phases with complex β .
We found thatthe change to holomorphic dynamics already improved upon the gradient estimates used in previouswork [14, 19].
Moreover, we observed that for the last layer increasing N only extended the rangeof usable teaching magnitudes | β | , but did not improve the quality of the gradient estimate. This [Figure]
phenomenon can be understood from Eq.
(9), since higher N reduces the bias term considerably,which accommodates higher teaching magnitude | β | .
However, larger amplitudes tended to improvethe total gradient, particularly in deep layers where small teaching magnitudes were not enoughto produce sufficient error signals (see Appendix C for details).
Additionally, larger N were onlyrequired when using a higher teaching amplitude (Fig. 4b).
Finally, we tested how the gradientquality impacts the network training accuracy on CIFAR-10.
We observed that the non-holomorphic VGG was unable to reach low training error (Fig. 4c), which is consistent with the poor gradientquality (Fig 4a).
Changing to a holomorphic architecture with the same number of points resulted in a substantial improvement of training accuracy, which was further boosted when training with N consistent with our theory
[Table] Finally, we wondered whether hEP could train deep neural networks on large-scale datasets, whichhas remained an open problem for most if not all alternative algorithms to BP [4].
To this end,we trained a five-layer CNN based on the VGG architecture [32] on multiple vision benchmarksincluding CIFAR-10, CIFAR-100 [17], and the 32 × 32 pixel version of ImageNet [18], whichcontains 1.2 million data points and 1000 classes like the full ILSVRC dataset [34].
In all cases wefound that the validation accuracy reached by networks trained with BP and hEP using only two fixedpoints ( N = 2 ) were identical within their uncertainties (Table 2).
Note that the networks trainedwith BP were the feed-forward equivalent of the holomorphic networks used with EP, but with ReLUsinstead of dSiLU, which did not give satisfactory results.
Thus, hEP permits training deep CNNs on ImageNet 32 × 32 to comparable performance levels as standard BP.
Discussion
We have introduced hEP which extends classic EP by computing exact loss gradients throughintegration over finite size adiabatic neuronal oscillations caused by a teaching signal (Section 3).
Importantly, such integration can be accomplished online with purely local learning rules whichmakes it an exciting theoretical framework for studying learning in the brain where oscillations areubiquitously observed [35, 36].
In practice we found that numerically evaluating a small number of points during one oscillation cycle provides an excellent gradient approximation that outperformsclassic EP and thanks to the finite oscillation amplitude is robust to noise, which is an advantage fortraining neuromorphic hardware systems [21, 22, 37].
Additionally, the possibility of using finiteteaching signals is conducive for training deep CNNs, where infinitesimal teaching signals as used byclassic EP, may vanish (Section 4).
A body of previous work has attempted to reconcile BP with neurobiology [3, 38].
EP is most closelyrelated to classic theories of predictive coding (PC) [39–45] which similarly assumes convergentnetwork dynamics cast into an energy minimization problem.
PC further assumes that errors areencoded in neuronal dynamics, dedicated dendritic compartments, or separate temporal phases[20, 40, 46, 47].
In a similar vein, Target Propagation (TP) [48–50] assumes locally encoded errorsignals which in some cases are obtained by iterating approximate inverses, a property reminiscent of
EP [51–53] which comes with theoretical guarantees [54].
However, all previous EP studies and mostof the works above, with two notable exceptions [38, 50], were all limited to small-scale problems[4].
In contrast, we demonstrate in this article that hEP scales to ImageNet 32 × 32 .
While the ability to run hEP online makes it an appealing model for credit assignment in biologicalneural networks, this interpretation has several notable shortcomings.
First, hEP requires complexvalued neuronal outputs and a holomorphic dynamical system which precludes the use of max poolingand ReLUs and hampers a direct comparison to neurobiology.
However, we found that holomorphicalternatives exists which empirically yield comparable performance.
Moreover, complex outputs havea long-standing tradition in computational neuroscience where they appear in variations of Hopfieldnetworks [55–57], in the framework of theta neurons [58], and phasor networks [59] where they areused to describe oscillatory neuronal dynamics.
It is possible to interpret hEP within such frameworks. For instance, it is straightforward to interpret complex neuronal output as oscillating activity witha defined amplitude and relative phase to some reference signal accessible to the entire neuronalpopulation.
Such a signal could be provided by neuromodulators such as acetylcholine which hasbeen implicated in neural oscillations [60].
Within our framework, the oscillatory teaching signal thencorresponds to a slow phase precession between the neuronal activity and the reference. Importantly,such a mechanism implies a hierarchy of oscillation frequencies.
Such different oscillations areknown to exist in the brain, e.g., theta (4–8 Hz) and gamma (30–70 Hz), but their precise purposeremains elusive. While establishing formal circuit-level equivalences with hEP will require futurework, the principled link between oscillatory activity, learning and memory as developed in this articleseems promising.
Like other algorithms hEP requires symmetric synaptic connectivity between layerswhich seems biologically implausible.
While theoretical guarantees for exact gradient computationare lost without strict symmetry [61], it may not be required for learning [62–64].
Alternatively,symmetry may be acquired through plastic feedback connections [65, 66].
Although our approachneither uses time-varying input nor neuronal spiking dynamics, spiking extensions to EP have beenproposed [67, 68].
However, applying our theory to time-varying tasks requiring memory will requireadditional architectural modifications and theoretical concepts [69] and establishing links to presentspike-based approaches [38, 70–72].
Our work augments classic EP with desirable properties for potential neuromorphic applications,which promise power-efficient and equitable artificial intelligence (AI) at the edge and in IoT devices[73–75].
While current software implementation of EP are generally slow compared to backprop,its appeal lies in its potential for training physical networks on future neuromorphic mixed-signaldevices that are incompatible with backprop, but achieve settling times on the order of nanoseconds[21, 37].
As exciting as such developments are, they also risk negative societal impacts, e.g., throughmass surveillance or allowing AI systems with potentially discriminatory biases to permeate oureveryday lives further.
A transparent research strategy and taking into account ethical considerationsearly during product design will be essential to avoid such adverse outcomes.
On the upside, ourtheoretical work further consolidates EP as a conceptual framework for understanding the brain, afundamental requirement to inform future biomedical research targeted at nervous system disorders.
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