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Title: Learning to solve the credit assignment problem

Paper content:
Backpropagation is driving today's artificial neural networks (ANNs).
However, despite extensive research, it remains unclear if the brain implements this algorithm.
Among neuroscientists, reinforcement learning (RL) algorithms are often seen as a realistic alternative: neurons can randomly introduce change, and use unspecific feedback signals to observe their effect on the cost and thus approximate their gradient.
However, the convergence rate of such learning scales poorly with the number of involved neurons.
Here we propose a hybrid learning approach.
Each neuron uses an RL-type strategy to learn how to approximate the gradients that backpropagation would provide.
We provide proof that our approach converges to the true gradient for certain classes of networks.
In both feedforward and convolutional networks, we empirically show that our approach learns to approximate the gradient, and can match the performance of gradient-based learning.
Learning feedback weights provides a biologically plausible mechanism of achieving good performance, without the need for precise, pre-specified learning rules.
It is unknown how the brain solves the credit assignment problem when learning: how does each neuron know its role in a positive (or negative) outcome, and thus know how to change its activity to perform better next time?
This is a challenge for models of learning in the brain.
Biologically plausible solutions to credit assignment include those based on reinforcement learning (RL) algorithms and reward-modulated STDP (Bouvier et al., 2016; Fiete et al., 2007; Legenstein et al., 2010; Miconi, 2017) .
In these approaches a globally distributed reward signal provides feedback to all neurons in a network.
Essentially, changes in rewards from a baseline, or expected, level are correlated with noise in neural activity, allowing a stochastic approximation of the gradient to be computed.
However these methods have not been demonstrated to operate at scale.
For instance, variance in the REINFORCE estimator (Williams, 1992) scales with the number of units in the network (Rezende et al., 2014) .
This drives the hypothesis that learning in the brain must rely on additional structures beyond a global reward signal.
In artificial neural networks (ANNs), credit assignment is performed with gradient-based methods computed through backpropagation (Rumelhart et al., 1986; Werbos, 1982; Linnainmaa, 1976) .
This is significantly more efficient than RL-based algorithms, with ANNs now matching or surpassing human-level performance in a number of domains (Mnih et al., 2015; Silver et al., 2017; LeCun et al., 2015; He et al., 2015; Haenssle et al., 2018; Russakovsky et al., 2015) .
However there are well known problems with implementing backpropagation in biologically realistic neural networks.
One problem is known as weight transport (Grossberg, 1987) : an exact implementation of backpropagation requires a feedback structure with the same weights as the feedforward network to communicate gradients.
Such a symmetric feedback structure has not been observed in biological neural circuits.
Despite such issues, backpropagation is the only method known to solve supervised and reinforcement learning problems at scale.
Thus modifications or approximations to backpropagation that are more plausible have been the focus of significant recent attention (Scellier & Bengio, 2016; Lillicrap et al., 2016; Lee et al., 2015; Lansdell & Kording, 2018) .
These efforts do show some ways forward.
Synthetic gradients demonstrate that learning can be based on approximate gradients, and need not be temporally locked (Jaderberg et al., 2016; Czar-necki et al., 2017b) .
In small feedforward networks, somewhat surprisingly, fixed random feedback matrices in fact suffice for learning (Lillicrap et al., 2016 ) (a phenomenon known as feedback alignment).
But still issues remain: feedback alignment does not work in CNNs, very deep networks, or networks with tight bottleneck layers.
Regardless, these results show that rough approximations of a gradient signal can be used to learn; even relatively inefficient methods of approximating the gradient may be good enough.
On this basis, here we propose an RL algorithm to train a feedback system to enable learning.
Recent work has explored similar ideas, but not with the explicit goal of approximating backpropagation (Miconi, 2017; Miconi et al., 2018; Song et al., 2017) .
RL-based methods like REINFORCE may be inefficient when used as a base learner, but they may be sufficient when used to train a system that itself instructs a base learner.
We propose to use REINFORCE-style perturbation approach to train feedback signals to approximate what would have been provided by backpropagation.
This sort of two-learner system, where one network helps the other learn more efficiently, may in fact align well with cortical neuron physiology.
For instance, the dendritic trees of pyramidal neurons consist of an apical and basal component.
Such a setup has been shown to support supervised learning in feedforward networks (Guergiuev et al., 2017; Kording & Konig, 2001) .
Similarly, climbing fibers and Purkinje cells may define a learner/teacher system in the cerebellum (Marr, 1969) .
These components allow for independent integration of two different signals, and may thus provide a realistic solution to the credit assignment problem.
Thus we implement a network that learns to use feedback signals trained with reinforcement learning via a global reward signal.
We mathematically analyze the model, and compare its capabilities to other methods for learning in ANNs.
We prove consistency of the estimator in particular cases, extending the theory of synthetic gradient-like approaches (Jaderberg et al., 2016; Czarnecki et al., 2017b; Werbos, 1992; Schmidhuber, 1990) .
We demonstrate that our model learns as well as regular backpropagation in small models, overcomes the limitations of feedback alignment on more complicated feedforward networks, and can be used in convolutional networks.
Thus, by combining local and global feedback signals, this method points to more plausible ways the brain could solve the credit assignment problem.
Here we implement a perturbation-based synthetic gradient method to train neural networks.
We show that this hybrid approach can be used in both fully connected and convolutional networks.
By removing the symmetric feedforward/feedback weight requirement imposed by backpropagation, this approach is a step towards more biologically-plausible deep learning.
By reaching comparable performance to backpropagation on MNIST, the method is able to solve larger problems than perturbation-only methods (Xie & Seung, 2004; Fiete et al., 2007; Werfel et al., 2005) .
By working in cases that feedback alignment fails, the method can provide learning without weight transport in a more diverse set of network architectures.
We thus believe the idea of integrating both local and global feedback signals is a promising direction towards biologically plausible learning algorithms.
Of course, the method does not solve all issues with implementing gradient-based learning in a biologically plausible manner.
For instance, in the current implementation, the forward and the backwards passes are locked.
Here we just focus on the weight transport problem.
A current drawback is that the method does not reach state-of-the-art performance on more challenging datasets like CIFAR.
We focused on demonstrating that it is advantageous to learn feedback weights, when compared with fixed weights, and successfully did so in a number of cases.
However, we did not use any additional data augmentation and regularization methods often employed to reach state-of-theart performance.
Thus fully characterizing the performance of this method remains important future work.
However the method does has a number of computational advantages.
First, without weight transport the method has better data-movement performance (Crafton et al., 2019; Akrout et al., 2019) , meaning it may be more efficiently implemented than backpropagation on specialized hardware.
Second, by relying on random perturbations to measure gradients, the method does not rely on the environment to provide gradients (compared with e.g. Czarnecki et al. (2017a) ; Jaderberg et al. (2016) ).
Our theoretical results are somewhat similar to that of Alain & Bengio (2015) , who demonstrate that a denoising autoencoder converges to the unperturbed solution as Gaussian noise goes to zero.
However our results apply to subgaussian noise more generally.
While previous research has provided some insight and theory for how feedback alignment works (Lillicrap et al., 2016; Ororbia et al., 2018; Moskovitz et al., 2018; Bartunov et al., 2018; Baldi et al., 2018 ) the effect remains somewhat mysterious, and not applicable in some network architectures.
Recent studies have shown that some of these weaknesses can be addressed by instead imposing sign congruent feedforward and feedback matrices (Xiao et al., 2018 ).
Yet what mechanism may produce congruence in biological networks is unknown.
Here we show that the shortcomings of feedback alignment can be addressed in another way: the system can learn to adjust weights as needed to provide a useful error signal.
Our work is closely related to Akrout et al. (2019) , which also uses perturbations to learn feedback weights.
However our approach does not divide learning into two phases, and training of the feedback weights does not occur in a layer-wise fashion, assuming only one layer is noisy at a time, which is a strong assumption.
Here instead we focus on combining global and local learning signals.
Here we tested our method in an idealized setting.
However the method is consistent with neurobiology in two important ways.
First, it involves separate learning of feedforward and feedback weights.
This is possible in cortical networks, where complex feedback connections exist between layers (Lacefield et al., 2019; Richards & Lillicrap, 2019) and pyramidal cells have apical and basal compartments that allow for separate integration of feedback and feedforward signals (Guerguiev et al., 2017; Körding & König, 2001) .
A recent finding that apical dendrites receive reward information is particularly interesting (Lacefield et al., 2019) .
Models like Guerguiev et al. (2017) show how the ideas in this paper may be implemented in spiking neural networks.
We believe such models can be augmented with a perturbation-based rule like ours to provide a better learning system.
The second feature is that perturbations are used to learn the feedback weights.
How can a neuron measure these perturbations?
There are many plausible mechanisms (Seung, 2003; Xie & Seung, 2004; Fiete et al., 2007) .
For instance, birdsong learning uses empiric synapses from area LMAN (Fiete et al., 2007) , others proposed it is approximated (Legenstein et al., 2010; Hoerzer et al., 2014) , or neurons could use a learning rule that does not require knowing the noise (Lansdell & Kording, 2018) .
Further, our model involves the subtraction of a baseline loss to reduce the variance of the estimator.
This does not affect the expected value of the estimator -technically the baseline could be removed or replaced with an approximation (Legenstein et al., 2010; Loewenstein & Seung, 2006) .
Thus both separation of feedforward and feedback systems and perturbation-based estimators can be implemented by neurons.
As RL-based methods do not scale by themselves, and exact gradient signals are infeasible, the brain may well use a feedback system trained through reinforcement signals to usefully approximate gradients.
There is a large space of plausible learning rules that can learn to use feedback signals in order to more efficiently learn, and these promise to inform both models of learning in the brain and learning algorithms in artificial networks.
Here we take an early step in this direction.
It is worth making the following points on each of the assumptions:
• A1.
In the paper we assume ξ is Gaussian.
Here we prove the more general result of convergence for any subgaussian random variable.
• A2.
In practice this may be a fairly restrictive assumption, since it precludes using relu nonlinearities.
Other common choices, such as hyperbolic tangent and sigmoid non-linearities with an analytic cost function do satisfy this assumption, however.
• A3.
It is hard to establish general conditions under whichẽ i (ẽ i ) T will be full rank.
While it may be a reasonable assumption in some cases.
Extensions of Theorem 2 to a non-linear network may be possible.
However, the method of proof used here is not immediately applicable because the continuous mapping theorem can not be applied in such a straightforward fashion as in Equation (15).
In the non-linear case the resulting sums over all observations are neither independent or identically distributed, which makes applying any law of large numbers complicated.

assistant: text: Perturbations can be used to train feedback weights to learn in fully connected and convolutional neural networks