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Title: Spike-based causal inference for weight alignment

Paper content:
In artificial neural networks trained with gradient descent, the weights used for processing stimuli are also used during backward passes to calculate gradients.
For the real brain to approximate gradients, gradient information would have to be propagated separately, such that one set of synaptic weights is used for processing and another set is used for backward passes.
This produces the so-called "weight transport problem" for biological models of learning, where the backward weights used to calculate gradients need to mirror the forward weights used to process stimuli.
This weight transport problem has been considered so hard that popular proposals for biological learning assume that the backward weights are simply random, as in the feedback alignment algorithm.
However, such random weights do not appear to work well for large networks.
Here we show how the discontinuity introduced in a spiking system can lead to a solution to this problem.
The resulting algorithm is a special case of an estimator used for causal inference in econometrics, regression discontinuity design.
We show empirically that this algorithm rapidly makes the backward weights approximate the forward weights.
As the backward weights become correct, this improves learning performance over feedback alignment on tasks such as Fashion-MNIST and CIFAR-10.
Our results demonstrate that a simple learning rule in a spiking network can allow neurons to produce the right backward connections and thus solve the weight transport problem.
Any learning system that makes small changes to its parameters will only improve if the changes are correlated to the gradient of the loss function.
Given that people and animals can also show clear behavioral improvements on specific tasks (Shadmehr et al., 2010) , however the brain determines its synaptic updates, on average, the changes in must also correlate with the gradients of some loss function related to the task (Raman et al., 2019) .
As such, the brain may have some way of calculating at least an estimator of gradients.
To-date, the bulk of models for how the brain may estimate gradients are framed in terms of setting up a system where there are both bottom-up, feedforward and top-down, feedback connections.
The feedback connections are used for propagating activity that can be used to estimate a gradient (Williams, 1992; Lillicrap et al., 2016; Akrout et al., 2019; Roelfsema & Ooyen, 2005; Lee et al., 2015; Scellier & Bengio, 2017; Sacramento et al., 2018) .
In all such models, the gradient estimator is less biased the more the feedback connections mirror the feedforward weights.
For example, in the REINFORCE algorithm (Williams, 1992) , and related algorithms like AGREL (Roelfsema & Ooyen, 2005) , learning is optimal when the feedforward and feedback connections are perfectly symmetric, such that for any two neurons i and j the synaptic weight from i to j equals the weight from j to i, e.g. W ji = W ij (Figure 1 ).
Some algorithms simply assume weight symmetry, such as Equilibrium Propagation (Scellier & Bengio, 2017) .
The requirement for synaptic weight symmetry is sometimes referred to as the "weight transport problem", since it seems to mandate that the values of the feedforward synaptic weights are somehow transported into the feedback weights, which is not biologically realistic (Crick, 1989-01-12; Grossberg, 1987) .
Solving the weight transport problem is crucial to biologically realistic gradient estimation algorithms (Lillicrap et al., 2016) , and is thus an important topic of study.
Several solutions to the weight transport problem have been proposed for biological models, including hard-wired sign symmetry (Moskovitz et al., 2018) , random fixed feedback weights (Lillicrap et al., 2016) , and learning to make the feedback weights symmetric (Lee et al., 2015; Sacramento et al., 2018; Akrout et al., 2019; Kolen & Pollack, 1994) .
Learning to make the weights symmetric is promising because it is both more biologically feasible than hard-wired sign symmetry (Moskovitz et al., 2018) and it leads to less bias in the gradient estimator (and thereby, better training results) than using fixed random feedback weights (Bartunov et al., 2018; Akrout et al., 2019) .
However, of the current proposals for learning weight symmetry some do not actually work well in practice (Bartunov et al., 2018) and others still rely on some biologically unrealistic assumptions, including scalar value activation functions (as opposed to all-or-none spikes) and separate error feedback pathways with one-to-one matching between processing neurons for the forward pass and error propagation neurons for the backward pass Akrout et al. (2019) ; Sacramento et al. (2018) .
Interestingly, learning weight symmetry is implicitly a causal inference problem-the feedback weights need to represent the causal influence of the upstream neuron on its downstream partners.
As such, we may look to the causal infererence literature to develop better, more biologically realistic algorithms for learning weight symmetry.
In econometrics, which focuses on quasi-experiments, researchers have developed various means of estimating causality without the need to actually randomize and control the variables in question Angrist & Pischke (2008); Marinescu et al. (2018) .
Among such quasi-experimental methods, regression discontinuity design (RDD) is particularly promising.
It uses the discontinuity introduced by a threshold to estimate causal effects.
For example, RDD can be used to estimate the causal impact of getting into a particular school (which is a discontinuous, all-or-none variable) on later earning power.
RDD is also potentially promising for estimating causal impact in biological neural networks, because real neurons communicate with discontinuous, all-or-none spikes.
Indeed, it has been shown that the RDD approach can produce unbiased estimators of causal effects in a system of spiking neurons Lansdell & Kording (2019) .
Given that learning weight symmetry is fundamentally a causal estimation problem, we hypothesized that RDD could be used to solve the weight transport problem in biologically realistic, spiking neural networks.
Here, we present a learning rule for feedback synaptic weights that is a special case of the RDD algorithm previously developed for spiking neural networks (Lansdell & Kording, 2019) .
Our algorithm takes advantage of a neuron's spiking discontinuity to infer the causal effect of its spiking on the activity of downstream neurons.
Since this causal effect is proportional to the feedforward synaptic weight between the two neurons, by estimating it, feedback synapses can align their weights to be symmetric with the reciprocal feedforward weights, thereby overcoming the weight transport problem.
We demonstrate that this leads to the reduction of a cost function which measures the weight symmetry (or the lack thereof), that it can lead to better weight symmetry in spiking neural networks than other algorithms for weight alignment (Akrout et al., 2019) and it leads to better learning in deep neural networks in comparison to the use of fixed feedback weights (Lillicrap et al., 2016) .
Altogether, these results demonstrate a novel algorithm for solving the weight transport problem that takes advantage of discontinuous spiking, and which could be used in future models of biologically plausible gradient estimation.
In order to understand how the brain learns complex tasks that require coordinated plasticity across many layers of synaptic connections, it is important to consider the weight transport problem.
Here, we presented an algorithm for updating feedback weights in a network of spiking neurons that takes advantage of the spiking discontinuity to estimate the causal effect between two neurons (Figure 2 ).
We showed that this algorithm enforces weight alignment (Figure 3 ), and identified a loss function, R self , that is minimized by our algorithm (Figure 4) .
Finally, we demonstrated that our algorithm allows deep neural networks to achieve better learning performance than feedback alignment on Fashion-MNIST and CIFAR-10 ( Figure 5 ).
These results demonstrate the potential power of RDD as a means for solving the weight transport problem in biologically plausible deep learning models.
One aspect of our algorithm that is still biologically implausible is that it does not adhere to Dale's principle, which states that a neuron performs the same action on all of its target cells (Strata & Harvey) .
This means that a neuron's outgoing connections cannot include both positive and negative weights.
However, even under this constraint, a neuron can have an excitatory effect on one downstream target and an inhibitory effect on another, by activating intermediary inhibitory interneurons.
Because our algorithm provides a causal estimate of one neuron's impact on another, theoretically, it could capture such polysynaptic effects.
Therefore, this algorithm is in theory compatible with Dale's principle.
Future work should test the effects of this algorithm when implemented in a network of neurons that are explicitly excitatory or inhibitory.
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assistant: text: We present a learning rule for feedback weights in a spiking neural network that addresses the weight transport problem.