Neural sampling machine with stochastic synapse allows brain-like learning and inference

Abstract

Many real-world mission-critical applications require continual online learning from noisy data and real-time decision making with a defined confidence level. Brain-inspired probabilistic models of neural network can explicitly handle the uncertainty in data and allow adaptive learning on the fly. However, their implementation in a compact, low-power hardware remains a challenge. In this work, we introduce a novel hardware fabric that can implement a new class of stochastic neural network called Neural Sampling Machine (NSM) by exploiting the stochasticity in the synaptic connections for approximate Bayesian inference. We experimentally demonstrate an in silico hybrid stochastic synapse by pairing a ferroelectric field-effect transistor (FeFET)-based analog weight cell with a two-terminal stochastic selector element. We show that the stochastic switching characteristic of the selector between the insulator and the metallic states resembles the multiplicative synaptic noise of the NSM. We perform network-level simulations to highlight the salient features offered by the stochastic NSM such as performing autonomous weight normalization for continual online learning and Bayesian inferencing. We show that the stochastic NSM can not only perform highly accurate image classification with 98.25% accuracy on standard MNIST dataset, but also estimate the uncertainty in prediction (measured in terms of the entropy of prediction) when the digits of the MNIST dataset are rotated. Building such a probabilistic hardware platform that can support neuroscience inspired models can enhance the learning and inference capability of the current artificial intelligence (AI).

Fig. 1. Overview of stochastic synapse.

a Synaptic stochasticity occurring at the molecular level in biological neural networks. The presynaptic neuronal spike causes the release of neurotransmitters at the synaptic release site with a probability around 0.1. b Schematic of a Neural Sampling Machine (NSM) incorporating a Bernoulli or “blank-out” multiplicative noise in the synapse. This acts as a continuous DropConnect mask on the synaptic weights such that a subset of the weights is continuously forced to be zero. c Illustration of an NSM implemented in a hardware using crossbar array architecture implementing compute-in-memory. The analog weight cell implemented using eNVMs are placed at each cross-point and are augmented with a stochastic selector element. This allows selectively sampling or reading the synaptic weights G_ij with some degree of uncertainty, based on random binary variables ξ_ij generated for each of the synapse. d Illustration of a scenario where an input voltage V_in,3 is applied to a row of the synaptic array with conductance states $\mathit{G}=\left\{G_1,G_2,G_3,G_4,\dots ,G_N\right\}$. Depending on the state of the selectors in the cross-points, an output weighted sum current ${\mathit{I}}_{\mathit{out}}=\left\{0,G_2{V}_{in,3},0,G_4{V}_{in,3},\dots ,0\right\}$ is generated which is exactly same as multiplying the weight sum of $w_{ij}{z}_j$ with a multiplicative noise ${\xi }_{ij}$. WL word line, BL bit line, SL source line, V_in input voltage, I_out output current, G conductance.

Fig. 2. FeFET-based analog synapse.

a Schematic of a stand-alone FeFET-based analog synapse. The channel conductance can be modulated by applying write pulses ±V_write to the gate of the FeFET while reading out the conductance state is achieved by applying a small read voltage V_read to the gate terminal. b Experimentally measured conductance modulation in a 500 nm × 500 nm high-K metal gate FeFET fabricated at 28 nm technology node. An amplitude modulation scheme is used where positive and negative write voltage pulses V_write of increasing amplitude from 2.8 V to 4 V and pulse widths of 1 μs are applied to modulate the conductance of the FeFET. c Measured continuous change in the conductance state of the FeFET upon applying multiple potentiation and depression pulses of varying amplitude. d The FeFET-based analog weight cell is modeled in the NSM by fitting the conductance update scheme for both potentiation and depression with the closed-form expression as shown in the figure. WL word line, BL bit line, SL source line, V_in input voltage, I_out output current, V_write write voltage, G conductance, LRS low resistance state, HRS high resistance state.

Fig. 3. Introducing multiplicative noise through stochastic selector.

a Schematic and TEM of a fabricated stack of [Ag/TiN/HfO₂/Pt] with 3 nm TiN and 4 nm HfO₂. b A stochastic synapse is realized by augmenting this stochastic selector in series with the FeFET-based analog weight cell. c Measured current-voltage characteristics showing abrupt electronic transition from insulating state to metallic state due to the formation of a continuous filament of Ag+ atoms bridge the top and bottom electrodes. A wide window of variation in the threshold voltage V_T that triggers the spontaneous formation of the Ag⁺ filament is observed. The stochasticity can be exploited by applying the input voltage V_in within the variation window of the V_T. d Measured threshold voltage V_T over multiple cycles. e Stochastically reading an LRS and an HRS of the FeFET through the stochastic selector. f Measured device-to-device variation across 17 selector devices. Error bar denotes standard deviation across the mean. g–i The stochasticity switching of the selector device is modeled using an Onrstein-Uhlenbeck (OU) Process. The model shows excellent agreement with the experimental data. WL word line, BL bit line, SL source line, V_in input voltage, I_out output current, LRS low resistance state, HRS high resistance state.

Fig. 4. Hardware NSM performing image classification and exhibiting self-normalization.

a Network architecture of the NSM consisting of an input layer, three hidden fully connected layers and an output layer. b Exact match witnessed between the measured switching probability of the stochastic selector device and theoretically predicted probability for a Bernoulli distribution, highlighting that our stochastic selector device can inject Bernoulli multiplicative noise. c Evolution of the test accuracy for the simulated hardware-NSM using the FeFET-based analog weight cell and the stochastic selector as a function of the epochs. d Comparison of the performance of the simulated hardware-NSM with a deterministic feedforward multilayer perceptron (MLP) and the theoretical NSM model with full precession synaptic weights and a Bernoulli multiplicative noise for the stochastic synapses. e Evolution of the weights of the third layer during learning for three different networks- an MLP without any regularization, an MLP with additional regularization added and the simulated hardware-NSM. f Evolution of the 15th, 50th and 85th percentiles of the input distributions to the last hidden layer during training for all the three networks. Overall, NSM exhibits a tighter distribution of the weights and activation concentrated around its mean, highlighting the inherent self-normalizing feature. MLP multilayer perceptron, NSM neural sampling machine, Q quantile.

Fig. 5. Bayesian inferencing and uncertainty in data comparison between simulated hardware-NSM and a conventional MLP network.

a, f Continuously rotated images of the digits 1 and 2 from the MNIST dataset, used for performing Bayesian inferencing. We perform 100 stochastic forward passes during the inference mode for each rotated image of digits 1 and 2 and record the distribution of the (b, g) softmax input and (c, h) softmax output for few representative output neurons. d, i Classification result produced by the NSM for each rotated image. e, j The uncertainty of the NSM associated with the prediction, calculated in terms of the entropy H = −Σp*log$\left(p\right)$, where p is the probability distribution of the prediction. When the NSM makes a correct prediction (classifying image 1 and 2 as belonging to class 1 and 2, respectively), the uncertainty measured in terms of the entropy remains 0. However, in the case of wrong predictions, the uncertainty associated with the prediction becomes large. MLP multilayer perceptron, NSM neural sampling machine.