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Title: Extending the Framework of Equilibrium Propagation to General Dynamics

Paper content:
The biological plausibility of the backpropagation algorithm has long been doubted by neuroscientists.
Two major reasons are that neurons would need to send two different types of signal in the forward and backward phases, and that pairs of neurons would need to communicate through symmetric bidirectional connections.
We present a simple two-phase learning procedure for fixed point recurrent networks that addresses both these issues.
In our model, neurons perform leaky integration and synaptic weights are updated through a local mechanism.
Our learning method extends the framework of Equilibrium Propagation to general dynamics, relaxing the requirement of an energy function.
As a consequence of this generalization, the algorithm does not compute the true gradient of the objective function,
but rather approximates it at a precision which is proven to be directly related to the degree of symmetry of the feedforward and feedback weights.
We show experimentally that the intrinsic properties of the system lead to alignment of the feedforward and feedback weights, and that our algorithm optimizes the objective function.
Deep learning BID18 is the de-facto standard in areas such as computer vision BID17 , speech recognition and machine translation BID3 .
These applications deal with different types of data and share little in common at first glance.
Remarkably, all these models typically rely on the same basic principle: optimization of objective functions using the backpropagation algorithm.
Hence the question: does the cortex in the brain implement a mechanism similar to backpropagation, which optimizes objective functions?The
backpropagation algorithm used to train neural networks requires a side network for the propagation of error derivatives, which is vastly seen as biologically implausible BID7 . One
hypothesis, first formulated by Hinton & McClelland (1988) , is that error signals in biological networks could be encoded in the temporal derivatives of the neural activity and propagated through the network via the neuronal dynamics itself, without the need for a side network. Neural
computation would correspond to both inference and error back-propagation. This work
also explores this idea.The framework of Equilibrium Propagation BID29 requires the network dynamics to be derived from an energy function, enabling computation of an exact gradient of an objective function. However,
in terms of biological realism, the requirement of symmetric weights between neurons arising from the energy function is not desirable. The work
presented here extends this framework to general dynamics, without the need for energy functions, gradient dynamics, or symmetric connections.Our approach is the following. We start
from classical models in neuroscience for the dynamics of the neuron's membrane voltage and for the synaptic plasticity (section 3). Unlike in
the Hopfield model BID16 , we do not assume pairs of neurons to have symmetric connections. We then describe
an algorithm for supervised learning based on these models (section 4) with minimal extra assumptions. Our model is based
on two phases: at prediction time, no synaptic changes occur, whereas a local update rule becomes effective when the targets are observed. The proposed update
mechanism is compatible with spike-timing-dependent plasticity , which supposedly governs synaptic changes in biological neural systems. Finally, we show that
the proposed algorithm has the desirable machine learning property of optimizing an objective function (section 5). We show this experimentally
( Figure 3 ) and we provide the beginning for a theoretical explanation.
Our model demonstrates that biologically plausible learning in neural networks can be achieved with relatively few assumptions.
As a key contribution, in contrast to energy-based approaches such as the Hopfield model, we do not impose any symmetry constraints on the neural connections.
Our algorithm assumes two phases, the difference between them being whether synaptic changes occur or not.
Although this assumption begs for an explanation, neurophysiological findings suggest that phase-dependent mechanisms are involved in learning and memory consolidation in biological systems.
Theta waves, for instance, generate neural oscillatory patterns that can modulate the learning rule or the computation carried out by the network BID26 .
Furthermore, synaptic plasticity, and neural dynamics in general, are known to be modulated by inhibitory neurons and dopamine release, depending on the presence or absence of a target signal.
BID10 ; BID27 .In
its general formulation (Appendix A), the work presented in this paper is an extension of the framework of BID29 to general dynamics. This
is achieved by relaxing the requirement of an energy function. This
generalization comes at the cost of not being able to compute the (true) gradient of the objective function but, rather a direction in the weight space which is related to it. Thereby
, precision of the approximation of the gradient is directly related to the "alignment" between feedforward and feedback weights. Even though
the exact underlying mechanism is not fully understood yet, we observe experimentally that during training the weights symmetrize to some extent, as has been observed previously in a variety of other settings BID19 BID31 BID2 . Our work shows
that optimization of an objective function can be achieved without ever computing the (true) gradient. More thorough
theoretical analysis needs to be carried out to understand and characterize the dynamics in the weight space that optimize objective functions. Naturally, the
set of all such dynamics is much larger than the tiny subset of gradient-based dynamics.Our framework provides a means of implementing learning in a variety of physical substrates, whose precise dynamics might not even be known exactly, but which simply have to be in the set of sup-ported dynamics. In particular,
this applies to analog electronic circuits, potentially leading to faster, more efficient, and more compact implementations.

assistant: text: We describe a biologically plausible learning algorithm for fixed point recurrent networks without tied weights