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Title: Critical initialisation in continuous approximations of binary neural networks

Paper content:
The training of stochastic neural network models with binary ($\pm1$) weights and activations via continuous surrogate networks is investigated.
We derive, using mean field theory, a set of scalar equations describing how input signals propagate through surrogate networks.
The equations reveal that depending on the choice of surrogate model, the networks may or may not exhibit an order to chaos transition, and the presence of depth scales that limit the maximum trainable depth.
Specifically, in solving the equations for edge of chaos conditions, we show that surrogates derived using the Gaussian local reparameterisation trick have no critical initialisation, whereas a deterministic surrogates based on analytic Gaussian integration do.
The theory is applied to a range of binary neuron and weight design choices, such as different neuron noise models, allowing the categorisation of algorithms in terms of their behaviour at initialisation.
Moreover, we predict theoretically and confirm numerically, that common weight initialization schemes used in standard continuous networks, when applied to the mean values of the stochastic binary weights, yield poor training performance.
This study shows that, contrary to common intuition, the means of the stochastic binary weights should be initialised close to close to $\pm 1$ for deeper networks to be trainable.
The problem of learning with low-precision neural networks has seen renewed interest in recent years, in part due to the deployment of neural networks on low-power devices.
Currently, deep neural networks are trained and deployed on GPUs, without the memory or power constraints of such devices.
Binary neural networks are a promising solution to these problems.
If one is interested in addressing memory usage, the precision of the weights of the network should be reduced, with the binary case being the most extreme.
In order to address power consumption, networks with both binary weights and neurons can deliver significant gains in processing speed, even making it feasible to run the neural networks on CPUs Rastegari et al. (2016) .
Of course, introducing discrete variables creates challenges for optimisation, since the networks are not continuous and differentiable.
Recent work has opted to train binary neural networks directly via backpropagation on a differentiable surrogate network, thus leveraging automatic differentiation libraries and GPUs.
A key to this approach is in defining an appropriate differentiable surrogate network as an approximation to the discrete model.
A principled approach is to consider binary stochastic variables and use this stochasticity to "smooth out" the non-differentiable network.
This includes the cases when
(i) only weights, and
(ii) both weights and neurons are stochastic and binary.
In this work we study two classes of surrogates, both of which make use of the Gaussian central limit theorem (CLT) at the receptive fields of each neuron.
In either case, the surrogates are written as differentiable functions of the continuous means of stochastic binary weights, but with more complicated expressions than for standard continuous networks.
One approximation, based on analytic integration, yields a class of deterministic surrogates Soudry et al. (2014) .
The other approximation is based on the local reparameterisation trick (LRT) Kingma & Welling (2013) , which yields a class of stochastic surrogates Shayer et al. (2017) .
Previous works have relied on heuristics to deal with binary neurons Peters & Welling (2018) , or not backpropagated gradients correctly.
Moreover, none of these works considered the question of initialisation, potentially limiting performance.
The seminal papers of Saxe et al. (2013) , Poole et al. (2016) , Schoenholz et al. (2016) used a mean field formalism to explain the empirically well known impact of initialization on the dynamics of learning in standard networks.
From one perspective the formalism studies how signals propagate forward and backward in wide, random neural networks, by measuring how the variance and correlation of input signals evolve from layer to layer, knowing the distributions of the weights and biases of the network.
By studying these moments the authors in Schoenholz et al. (2016) were able to explain how heuristic initialization schemes avoid the "vanishing and exploding gradients problem" Glorot & Bengio (2010) , establishing that for neural networks of arbirary depth to be trainable they must be initialised at "criticality", which corresponds to initial correlation being preserved to any depth.
The paper makes three contributions.
The first contribution is the presentation of new algorithms, with a new derivation able to encompass both surrogates, and all choices of stochastic binary weights, or neurons.
The derivation is based on representing the stochastic neural network as a Markov chain, a simplifying and useful development.
As an example, using this representation we are easily able to extend the LRT to the case of stochastic binary neurons, which is new.
This was not possible in Shayer et al. (2017) , who only considered stochastic binary weights.
As a second example, the deterministic surrogate of Soudry et al. (2014) is easily derived, without the need for Bayesian message passing arguments.
Moreover, unlike Soudry et al. (2014) we correctly backpropagate through variance terms, as we discuss.
The second contribution is the theoretical analysis of both classes of surrogate at initialisation, through the prism of signal propagation theory Poole et al. (2016) , Schoenholz et al. (2016) .
This analysis is achieved through novel derivations of the dynamic mean field equations, which hinges on the use of self-averaging arguments Mezard et al. (1987) .
The results of the theoretical study, which are supported by numerical simulations and experiment, establish that for a surrogate of arbitrary depth to be trainable, it must be randomly initialised at "criticality".
In practical terms, criticality corresponds to using initialisations that avoid the "vanishing and exploding gradients problem" Glorot & Bengio (2010) .
We establish the following key results:
• For networks with stochastic binary weights and neurons, the deterministic surrogate can achieve criticality, while the LRT cannot.
• For networks with stochastic binary weights and continuous neurons, the LRT surrogate can achieve criticality (no deterministic surrogate exists for this case)
In both cases, the critical initialisation corresponds to randomly initialising the means of the binary weights close to ±1, a counter intuitive result.
A third contribution is the consideration of the signal propagation properties of random binary networks, in the context of training a differentiable surrogate network.
We derive these results, which are partially known, and in order to inform our discussion of the experiments.
This paper provides insights into the dynamics and training of the class of binary neural network models.
To date, the initialisation of any binary neural network algorithm has not been studied, although the effect of quantization levels has been explored through this perspective Blumenfeld et al. (2019) .
Currently, the most popular surrogates are based on the so-called "Straight-Through" estimator Bengio et al. (2013) , which relies on heuristic definitions of derivatives in order to define a gradient.
However, this surrogate typically requires the use of batch normalization, and other heuristics.
The contributions in this paper may help shed light on what is holding back the more principled algorithms, by suggesting practical advice on how to initialise, and what to expect during training.
Paper outline: In section 2 we present the binary neural network algorithms considered.
In subsection 2.1 we define binary neural networks and subsection 2.2 their stochastic counterparts.
In subsection 2.3 we use these definitions to present new and existing surrogates in a coherent framework, using the Markov chain representation of a neural network to derive variants of both the deterministic surrogate, and the LRT-based surrogates.
We derive the LRT for the case of stochastic binary weights, and both LRT and deterministic surrogates for the case of stochastic binary weights and neurons.
In section 3 we derive the signal propagation equations for both the deterministic and stochastic LRT surrogates.
This includes deriving the explicit depth scales for trainability, and solving the equations to find the critical initialisations for each surrogate, if they exist.
In section 4 we present the numerical simulations of wide random networks, to validate the mean field description, and experimental results to test the trainability claims.
Finally in section 5 we summarize the key results, and provide a discussion of the insights they provide.
This first study of two classes of surrogate networks, and the derivation of their initialisation theories has yielded results of practical significance.
Based on the results of Section 3, in particular Claims 1-3, we can offer the following advice.
If a practitioner is interested in training networks with binary weights and neurons, one should use the deterministic surrogate, not the LRT surrogate, since the latter has no critical initialisation.
If a practitioner is interested in binary weights only,the LRT in this case does have a critical initialisation (and is the only choice from amongst these two classes of surrogate).
Furthermore, both networks are critically initialised when σ 2 b → 0 and by setting the means of the weights to ±1.
Interesting results were uncovered for the binary neural networks corresponding to the trained surrogate.
It was seen that during training, when evaluating the stochastic binary counterparts concurrently with the surrogate, the performance of binary networks is worse than the continuous model, especially as depth increases.
We reported that the stochastic binary network, with more samples, outperformed the deterministic binary network.
This makes sense since the objective optimised is the expectation over an ensemble of stochastic binary networks.
A study of random binary networks, included in the Appendices, and published recently Blumenfeld et al. (2019) for a different problem, showed that binary networks are always in a chaotic phase.
Of course, when evaluating any binary network which is trained via gradient descent on a given surrogate model, signals have different average behaviour through the corresponding binary network.
It makes sense that the closer one is to the early stages of the training process, the closer the signal propagation behaviour is to the randomly initialised case.
It is likely that as training progresses the behaviour of the binary counterparts approaches that of the trained surrogate.
Any such difference would not be observed for a heuristic surrogate as used in Courbariaux & Bengio (2016) or Rastegari et al. (2016) , which has no continuous forward propagation equations.

assistant: text: An in-depth exploration of stochastic binary networks, continuous surrogates, and their training dynamics, with insights on how to initialize weights for best performance.