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Backward propagation of errors (backpropagation) is a method to minimize objective functions (e.g., loss functions) of deep neural networks by identifying optimal sets of weights and biases. Imposing constraints on weight precision is often required to alleviate prohibitive workloads on hardware. Despite the remarkable success of backpropagation, the algorithm itself is not capable of considering such constraints unless additional algorithms are applied simultaneously. To address this issue, we propose the constrained backpropagation (CBP) algorithm based on a pseudo-Lagrange multiplier method to obtain the optimal set of weights that satisfy a given set of constraints. The defining characteristic of the proposed CBP algorithm is the utilization of a Lagrangian function (loss function plus constraint function) as its objective function. We considered various types of constraints--binary, ternary, one-bit shift, and two-bit shift weight constraints. As a post-training method, CBP applied to AlexNet, ResNet-18, ResNet-50, and GoogLeNet on ImageNet, which were pre-trained using the conventional backpropagation. For all cases, the proposed algorithm outperforms the state-of-the-art methods on ImageNet, e.g., 66.6%, 74.4%, and 64.0% top-1 accuracy for ResNet-18, ResNet-50, and GoogLeNet with binary weights, respectively. This highlights CBP as a learning algorithm to address diverse constraints with the minimal performance loss by employing appropriate constraint functions., Comment: Accepted. NeurIPS 2021. The code is available at https://github.com/dooseokjeong/CBP

The brain is the perfect place to look for inspiration to develop more efficient neural networks. The inner workings of our synapses and neurons provide a glimpse at what the future of deep learning might look like. This paper serves as a tutorial and perspective showing how to apply the lessons learnt from several decades of research in deep learning, gradient descent, backpropagation and neuroscience to biologically plausible spiking neural neural networks. We also explore the delicate interplay between encoding data as spikes and the learning process; the challenges and solutions of applying gradient-based learning to spiking neural networks (SNNs); the subtle link between temporal backpropagation and spike timing dependent plasticity, and how deep learning might move towards biologically plausible online learning. Some ideas are well accepted and commonly used amongst the neuromorphic engineering community, while others are presented or justified for the first time here. The fields of deep learning and spiking neural networks evolve very rapidly. We endeavour to treat this document as a 'dynamic' manuscript that will continue to be updated as the common practices in training SNNs also change. A series of companion interactive tutorials complementary to this paper using our Python package, snnTorch, are also made available. See https://snntorch.readthedocs.io/en/latest/tutorials/index.html .

We propose a model for synaptic plasticity based on a calcium signaling cascade. The model simplifies the full signaling pathways from a calcium influx to the phosphorylation (potentiation) and dephosphorylation (depression) of glutamate receptors that are gated by fictive C1 and C2 catalysts, respectively. This model is based on tangible chemical reactions, including fictive catalysts, for long-term plasticity rather than the conceptual theories commonplace in various models, such as preset thresholds of calcium concentration. Our simplified model successfully reproduced the experimental synaptic plasticity induced by different protocols such as (i) a synchronous pairing protocol and (ii) correlated presynaptic and postsynaptic action potentials (APs). Further, the ocular dominance plasticity (or the experimental verification of the celebrated Bienenstock--Cooper--Munro theory) was reproduced by two model synapses that compete by means of back-propagating APs (bAPs). The key to this competition is synapse-specific bAPs with reference to bAP-boosting on the physiological grounds., Comment: 42 pages, 7 figures, Accepted by Neural Networks

Spiking neural networks (SNN) as time-dependent hypotheses consisting of spiking nodes (neurons) and directed edges (synapses) are believed to offer unique solutions to reward prediction tasks and the related feedback that are classified as reinforcement learning. Generally, temporal difference (TD) learning renders it possible to optimize a model network to predict the delayed reward in an ad hoc manner. Neuromorphic SNNs--networks built using dedicated hardware--particularly leverage such TD learning for not only reward prediction but also temporal sequence prediction in a physical time domain. In this tutorial, such learning in a physical time domain is referred to as temporal learning to distinguish it from conventional TD learning-based methods that generally involve algorithmic (rather than physical) time. This tutorial addresses neuromorphic SNNs for temporal learning from the scratch. It first concerns general characteristics of SNNs including spiking neurons and information coding schemes and then moves on to temporal learning including its general concept, feasible algorithms, and their association with neurophysiological learning rules that have intensively been enriched for the last few decades., Comment: 40 pages, 10 figures

In spite of remarkable progress in machine learning techniques, the state-of-the-art machine learning algorithms often keep machines from real-time learning (online learning) due in part to computational complexity in parameter optimization. As an alternative, a learning algorithm to train a memory in real time is proposed, which is named as the Markov chain Hebbian learning algorithm. The algorithm pursues efficient memory use during training in that (i) the weight matrix has ternary elements (-1, 0, 1) and (ii) each update follows a Markov chain--the upcoming update does not need past weight memory. The algorithm was verified by two proof-of-concept tasks (handwritten digit recognition and multiplication table memorization) in which numbers were taken as symbols. Particularly, the latter bases multiplication arithmetic on memory, which may be analogous to humans' mental arithmetic. The memory-based multiplication arithmetic feasibly offers the basis of factorization, supporting novel insight into the arithmetic., Comment: 25 pages, 4 figures

Recent advances in predictive data analytics and ever growing digitalization and connectivity with explosive expansions in industrial and consumer Internet-of-Things (IoT) has raised significant concerns about security of people's identities and data. It has created close to ideal environment for adversaries in terms of the amount of data that could be used for modeling and also greater accessibility for side-channel analysis of security primitives and random number generators. Random number generators (RNGs) are at the core of most security applications. Therefore, a secure and trustworthy source of randomness is required to be found. Here, we present a differential circuit for harvesting one of the most stochastic phenomenon in solid-state physics, random telegraphic noise (RTN), that is designed to demonstrate significantly lower sensitivities to other sources of noises, radiation and temperature fluctuations. We use RTN in amorphous SrTiO3-based resistive memories to evaluate the proposed true random number generator (TRNG). Successful evaluation on conventional true randomness tests (NIST tests) has been shown. Robustness against using predictive machine learning and side-channel attacks have also been demonstrated in comparison with non-differential readouts methods.

A unique set of characteristics are packed in emerging nonvolatile reduction-oxidation (redox)-based resistive switching memories (ReRAMs) such as their underlying stochastic switching processes alongside their intrinsic highly nonlinear current-voltage characteristic, which in addition to known nano-fabrication process variation make them a promising candidate for the next generation of low-cost, low-power, tiny and secure Physically Unclonable Functions (PUFs). This paper takes advantage of this otherwise disadvantageous ReRAM feature using a combination of novel architectural and peripheral circuitry. We present a physical one-way function, nonlinear resistive Physical Unclonable Function (nrPUF), potentially applicable in variety of cyber-physical security applications given its performance characteristics. We experimentally verified performance of Valency Change Mechanism (VCM)-based ReRAM in nano-fabricated crossbar arrays across multiple dies and runs. In addition to a massive pool of Challenge-Response Pairs (CRPs), using a combination of experimental and simulation, our proposed PUF shows a reliability of 98.67%, a uniqueness of 49.85%, a diffuseness of 49.86%, a uniformity of 47.28%, and a bit-aliasing of 47.48%., Comment: 12 pages, 8 figures

We report the nanodiamond-derived onion-like carbon successfully applied as an electrocatalyst for [VO]2+/[VO2]+ redox flow battery, as drop-coated (in the as-synthesized state) on glassy carbon or carbon felt electrodes. We show that its reversibility and catalytic activity in its as-synthesized state was comparable to some of the best data in the literature which employed surface modifications. We clarified the origin of such excellent performances by physical/electrochemical analyses., Comment: 11 pages

We investigated electroforming-free bipolar resistive switching behavior in Pt/Ti/Al2O3/Al tunnel junctions where the Al2O3 tunnel barrier was naturally formed on Al in air. Various compliance current values for the junction's set switching successfully lead to various resistance values in its low resistance state, suggesting the possibility for multi-level-operation. A mechanism for the bipolar switching is qualitatively discussed in terms of the modulation of the tunnel barrier by the reactive Ti layer on top of the barrier.