Contrastive Hebbian Learning with Random Feedback Weights

Neural networks are commonly trained to make predictions through learning algorithms. Contrastive Hebbian learning, which is a powerful rule inspired by gradient backpropagation, is based on Hebb's rule and the contrastive divergence algorithm. It operates in two phases, the forward (or free) phase, where the data are fed to the network, and a backward (or clamped) phase, where the target signals are clamped to the output layer of the network and the feedback signals are transformed through the transpose synaptic weight matrices. This implies symmetries at the synaptic level, for which there is no evidence in the brain. In this work, we propose a new variant of the algorithm, called random contrastive Hebbian learning, which does not rely on any synaptic weights symmetries. Instead, it uses random matrices to transform the feedback signals during the clamped phase, and the neural dynamics are described by first order non-linear differential equations. The algorithm is experimentally verified by solving a Boolean logic task, classification tasks (handwritten digits and letters), and an autoencoding task. This article also shows how the parameters affect learning, especially the random matrices. We use the pseudospectra analysis to investigate further how random matrices impact the learning process. Finally, we discuss the biological plausibility of the proposed algorithm, and how it can give rise to better computational models for learning

What Drives Swap Spreads, Credit or Liquidity?

This paper investigates the determinants of swap spreads. Compared with previous work done in this area, such as the seminal paper by Duffie and Singleton (1997), the paper includes daily credit spreads data in the time series framework. The issue is whether “liquidity” or “credit” (or both) is the main determinant of swap spreads dynamics. Our results agree with the prevailing view among swap traders that swap spreads are mainly an indicator of “market liquidity”. However, the dynamics are influenced significantly by “credit” over longer horizons, although credit is not the main driving force. LIBOR rate dynamics seem to play a relatively minor role in this setting.

Disturbing Extremal Behavior of Spot Rate Dynamics

This paper presents a study of extreme interest rate movements in the U.S. Federal Funds market over almost a half century of daily observations from the mid 1950s through the end of 2000. We analyze the fluctuations of the maximal and minimal changes in short term interest rates and test the significance of time-varying paths followed by the mean and volatility of extremes. We formally determine the relevance of introducing trend and serial correlation in the mean, and of incorporating the level and GARCH effects in the volatility of extreme changes in the federal funds rate. The empirical findings indicate the existence of volatility clustering in the standard deviation of extremes, and a significantly positive relationship between the level and the volatility of extremes. The results point to the presence of an autoregressive process in the means of both local maxima and local minima values. The paper proposes a conditional extreme value approach to calculating value at risk by specifying the location and scale parameters of the generalized Pareto distribution as a function of past information. Based on the estimated VaR thresholds, the statistical theory of extremes is found to provide more accurate estimates of the rate of occurrence and the size of extreme observations.extreme value theory, volatility, interest rates, value at risk

On-chip Few-shot Learning with Surrogate Gradient Descent on a Neuromorphic Processor

Recent work suggests that synaptic plasticity dynamics in biological models of neurons and neuromorphic hardware are compatible with gradient-based learning (Neftci et al., 2019). Gradient-based learning requires iterating several times over a dataset, which is both time-consuming and constrains the training samples to be independently and identically distributed. This is incompatible with learning systems that do not have boundaries between training and inference, such as in neuromorphic hardware. One approach to overcome these constraints is transfer learning, where a portion of the network is pre-trained and mapped into hardware and the remaining portion is trained online. Transfer learning has the advantage that pre-training can be accelerated offline if the task domain is known, and few samples of each class are sufficient for learning the target task at reasonable accuracies. Here, we demonstrate on-line surrogate gradient few-shot learning on Intel's Loihi neuromorphic research processor using features pre-trained with spike-based gradient backpropagation-through-time. Our experimental results show that the Loihi chip can learn gestures online using a small number of shots and achieve results that are comparable to the models simulated on a conventional processor

Error-triggered Three-Factor Learning Dynamics for Crossbar Arrays

Recent breakthroughs suggest that local, approximate gradient descent learning is compatible with Spiking Neural Networks (SNNs). Although SNNs can be scalably implemented using neuromorphic VLSI, an architecture that can learn in-situ as accurately as conventional processors is still missing. Here, we propose a subthreshold circuit architecture designed through insights obtained from machine learning and computational neuroscience that could achieve such accuracy. Using a surrogate gradient learning framework, we derive local, error-triggered learning dynamics compatible with crossbar arrays and the temporal dynamics of SNNs. The derivation reveals that circuits used for inference and training dynamics can be shared, which simplifies the circuit and suppresses the effects of fabrication mismatch. We present SPICE simulations on XFAB 180nm process, as well as large-scale simulations of the spiking neural networks on event-based benchmarks, including a gesture recognition task. Our results show that the number of updates can be reduced hundred-fold compared to the standard rule while achieving performances that are on par with the state-of-the-art

Event-Driven Contrastive Divergence for Spiking Neuromorphic Systems

Restricted Boltzmann Machines (RBMs) and Deep Belief Networks have been demonstrated to perform efficiently in a variety of applications, such as dimensionality reduction, feature learning, and classification. Their implementation on neuromorphic hardware platforms emulating large-scale networks of spiking neurons can have significant advantages from the perspectives of scalability, power dissipation and real-time interfacing with the environment. However the traditional RBM architecture and the commonly used training algorithm known as Contrastive Divergence (CD) are based on discrete updates and exact arithmetics which do not directly map onto a dynamical neural substrate. Here, we present an event-driven variation of CD to train a RBM constructed with Integrate & Fire (I&F) neurons, that is constrained by the limitations of existing and near future neuromorphic hardware platforms. Our strategy is based on neural sampling, which allows us to synthesize a spiking neural network that samples from a target Boltzmann distribution. The recurrent activity of the network replaces the discrete steps of the CD algorithm, while Spike Time Dependent Plasticity (STDP) carries out the weight updates in an online, asynchronous fashion. We demonstrate our approach by training an RBM composed of leaky I&F neurons with STDP synapses to learn a generative model of the MNIST hand-written digit dataset, and by testing it in recognition, generation and cue integration tasks. Our results contribute to a machine learning-driven approach for synthesizing networks of spiking neurons capable of carrying out practical, high-level functionality.Comment: (Under review

Inherent Weight Normalization in Stochastic Neural Networks

Multiplicative stochasticity such as Dropout improves the robustness and generalizability of deep neural networks. Here, we further demonstrate that always-on multiplicative stochasticity combined with simple threshold neurons are sufficient operations for deep neural networks. We call such models Neural Sampling Machines (NSM). We find that the probability of activation of the NSM exhibits a self-normalizing property that mirrors Weight Normalization, a previously studied mechanism that fulfills many of the features of Batch Normalization in an online fashion. The normalization of activities during training speeds up convergence by preventing internal covariate shift caused by changes in the input distribution. The always-on stochasticity of the NSM confers the following advantages: the network is identical in the inference and learning phases, making the NSM suitable for online learning, it can exploit stochasticity inherent to a physical substrate such as analog non-volatile memories for in-memory computing, and it is suitable for Monte Carlo sampling, while requiring almost exclusively addition and comparison operations. We demonstrate NSMs on standard classification benchmarks (MNIST and CIFAR) and event-based classification benchmarks (N-MNIST and DVS Gestures). Our results show that NSMs perform comparably or better than conventional artificial neural networks with the same architecture

Stochastic Synapses Enable Efficient Brain-Inspired Learning Machines

Recent studies have shown that synaptic unreliability is a robust and sufficient mechanism for inducing the stochasticity observed in cortex. Here, we introduce Synaptic Sampling Machines, a class of neural network models that uses synaptic stochasticity as a means to Monte Carlo sampling and unsupervised learning. Similar to the original formulation of Boltzmann machines, these models can be viewed as a stochastic counterpart of Hopfield networks, but where stochasticity is induced by a random mask over the connections. Synaptic stochasticity plays the dual role of an efficient mechanism for sampling, and a regularizer during learning akin to DropConnect. A local synaptic plasticity rule implementing an event-driven form of contrastive divergence enables the learning of generative models in an on-line fashion. Synaptic sampling machines perform equally well using discrete-timed artificial units (as in Hopfield networks) or continuous-timed leaky integrate & fire neurons. The learned representations are remarkably sparse and robust to reductions in bit precision and synapse pruning: removal of more than 75% of the weakest connections followed by cursory re-learning causes a negligible performance loss on benchmark classification tasks. The spiking neuron-based synaptic sampling machines outperform existing spike-based unsupervised learners, while potentially offering substantial advantages in terms of power and complexity, and are thus promising models for on-line learning in brain-inspired hardware

Conversion of Artificial Recurrent Neural Networks to Spiking Neural Networks for Low-power Neuromorphic Hardware

In recent years the field of neuromorphic low-power systems that consume orders of magnitude less power gained significant momentum. However, their wider use is still hindered by the lack of algorithms that can harness the strengths of such architectures. While neuromorphic adaptations of representation learning algorithms are now emerging, efficient processing of temporal sequences or variable length-inputs remain difficult. Recurrent neural networks (RNN) are widely used in machine learning to solve a variety of sequence learning tasks. In this work we present a train-and-constrain methodology that enables the mapping of machine learned (Elman) RNNs on a substrate of spiking neurons, while being compatible with the capabilities of current and near-future neuromorphic systems. This "train-and-constrain" method consists of first training RNNs using backpropagation through time, then discretizing the weights and finally converting them to spiking RNNs by matching the responses of artificial neurons with those of the spiking neurons. We demonstrate our approach by mapping a natural language processing task (question classification), where we demonstrate the entire mapping process of the recurrent layer of the network on IBM's Neurosynaptic System "TrueNorth", a spike-based digital neuromorphic hardware architecture. TrueNorth imposes specific constraints on connectivity, neural and synaptic parameters. To satisfy these constraints, it was necessary to discretize the synaptic weights and neural activities to 16 levels, and to limit fan-in to 64 inputs. We find that short synaptic delays are sufficient to implement the dynamical (temporal) aspect of the RNN in the question classification task. The hardware-constrained model achieved 74% accuracy in question classification while using less than 0.025% of the cores on one TrueNorth chip, resulting in an estimated power consumption of ~17 uW