Probabilistic Circuits for Autonomous Learning: A Simulation Study

1. Introduction

Machine learning, inference, and many other emerging applications (Schuman et al., 2017) make use of stochastic neural networks comprising (1) a binary stochastic neuron (BSN) (Ackley et al., 1985; Neal, 1992) and (2) a synapse that constructs the inputs I_i to the ith BSN from the outputs m_j of all other BSNs.

The output m_i of the ith BSN fluctuates between +1 and −1 with a probability controlled by its input

$\begin{array}{ll}{m}_i(t+{\tau }_N)=\text{sgn }\left[tanh\left(I_i(t)\right)-r\right]& (1)\end{array}$

where r represents a random number in the range [−1, +1], and τ_N is the time it takes for a neuron to provide a stochastic output m_i in accordance with a new input I_i¹.

Usually the synaptic function, I_i({m}) is linear and is defined by a set of weights W_ij such that

$\begin{array}{ll}{I}_i(t+{\tau }_S)={\displaystyle \sum _j}{W}_{ij}{m}_j(t)& (2)\end{array}$

where τ_S is the time it takes to recompute the inputs {I} everytime the outputs {m} change. Typically Equations (1), (2) are implemented in software, often with special accelerators for the synaptic function using GPU/TPUs (Schmidhuber, 2015; Jouppi, 2016).

The time constants τ_N and τ_S are not important when Equations (1) and (2) are implemented on a digital computer using a clock to ensure that neurons are updated sequentially and the synapse is updated between any two updates. But they play an important role in clockless operation of autonomous hardware that makes use of the natural physics of specific systems to implement Equations (1) and (2) approximately. A key advantage of using BSNs is that Equation (1) can be implemented compactly using stochastic magnetic tunnel junctions (MTJs) as shown in Camsari et al. (2017a,b), while resistive or capacitive crossbars can implement Equation (2) (Hassan et al., 2019a). It has been shown that such hardware implementations can operate autonomously without clocks, if the BSN operates slower than the synapse, that is, if τ_N >> τ_S shown by Sutton et al. (2019).

Stochastic neural networks defined by Equations (1) and (2) can be used for inference whereby the weights W_ij are designed such that the system has a very high probability of visiting configurations defined by {m} = {v}_n, where {v}_n represents a specified set of patterns. However, the most challenging and time-consuming part of implementing a neural network is not the inference function, but the learning required to determine the correct weights W_ij for a given application. This is commonly done using powerful cloud-based processors and there is great interest in accelerating the learning process and making it more energy efficient so that it can become a routine part of mobile and edge computing.

In this paper we present a new approach to the problem of fast and efficient learning that makes no use of digital computing at all. Instead it makes use of the natural physics of a fully autonomous probabilistic circuit composed of standard electronic components like resistors, capacitors, and transistors along with stochastic MTJs.

We focus on a fully visible Boltzmann machine (FVBM), a form of stochastic recurrent neural network, for which the most common learning algorithm is based on the gradient ascent approach to optimize the maximum likelihood function (Carreira-Perpinan and Hinton, 2005; Koller and Friedman, 2009). We use a slightly simplified version of this approach, whereby the weights are changed incrementally according to

$\begin{array}{l}{W}_{ij}(t+\Delta t)=W_{ij}(t)+ϵ\left[v_i{v}_j-m_i{m}_j-\lambda W_{ij}(t)\right]\end{array}$

where ϵ is the learning parameter and λ is the regularization parameter (Ng, 2004). The term v_iv_j is the correlation between the ith and the jth entry of the training vector {v}_n. The term m_im_j corresponds to the sampled correlation taken from the model's distribution. The advantage of this network topology is that the learning rule is local since it only requires information of the two neurons i and j connected by weight W_ij. In addition, the learning rule can tolerate stochasticity for example in the form of sampling noise which makes it an attractive algorithm to use for hardware machine learning (Carreira-Perpinan and Hinton, 2005; Fischer and Igel, 2014; Ernoult et al., 2019).

For our autonomous operation we replace the equation above with its continuous time version (τ_L: learning time constant)

$\begin{array}{ll}\frac{d{W}_{ij}}{dt}=\frac{v_i{v}_j-m_i{m}_j-\lambda W_{ij}}{{\tau }_L}& (3)\end{array}$

which we translate into an RC circuit by associating W_ij with the voltage on a capacitor C driven by a voltage source (V_v,ij − V_m,ij) with a series resistance R (Figure 1):

$\begin{array}{ll}C\frac{d{V}_{ij}}{dt}=\frac{V_{v,ij}-V_{m,ij}-V_{ij}}{R}& (4)\end{array}$

with v_iv_j = V_v,ij/(V_DD/2) and m_im_j = V_m,ij/(V_DD/2). From Figure 1 and comparing Equations (3), (4) it is easy to see how the weights and the learning and regularization parameters are mapped into circuit elements: W_ij = A_vV_ij/V₀, λ = V₀/(A_vV_DD/2), and τ_L = λRC where A_v is the voltage gain of OP3 in Figure 1 and V₀ is the reference voltage of the BSN. For proper operation the learning time scale τ_L has to be much larger than the neuron time τ_N to be able to collect enough statistics throughout the learning process.

FIGURE 1

Figure 1. Clockless learning circuit designed to emulate Equations (1)–(3) autonomously.

A key element of this approach is the representation of the weights W with voltages rather than with programmable resistances for which memristors and other technologies are still in development (Li et al., 2018b). By contrast the charging of capacitors is a textbook phenomenon, allowing us to design a learning circuit that can be built today with established technology. The idea of using capacitor voltages to represent weights in neural networks has been presented by several authors for different network topologies in analog learning circuits (Schneider and Card, 1993; Card et al., 1994; Kim et al., 2017; Sung et al., 2018). The use of capacitors has the advantage of having a high level of linearity and symmetry for the weight updates during the training process (Li et al., 2018a).

In section 2, we will describe such a learning circuit that emulates Equations (1)–(3). The training images or patterns {v}_n are fed in as electrical signals into the input terminals, and the synaptic weights W_ij can then be read out in the form of voltages from the output terminals. Alternatively the values can be stored in a non-volatile memory from which they can subsequently be read and used for inference. In section 3, we will present SPICE simulations demonstrating the operation of this autonomous learning circuit.

2. Methods

The autonomous learning circuit has three parts where each part represents one of the three Equations (1)–(3). On the left hand side of Figure 1, the training data is fed into the circuit by supplying a voltage V_v,ij which is given by the ith entry of the bipolar training vector v_i multiplied by the jth entry of the training vector v_j and scaled by the supply voltage V_DD/2. The training vectors can be fed in sequentially or as an average of all training vectors. The weight voltage V_ij across capacitor C follows Equation (4) where V_v,ij is compared to voltage V_m,ij which represents correlation of the outputs of BSNs m_i and m_j. Voltage V_m,ij is computed in the circuit by using an XNOR gate that is connected to the output of BSN i and BSN j. The synapse in the center of the circuit connects weight voltages to neurons according to Equation (2). Voltage V_ij has to be multiplied by 1 or −1 depending on the current value of m_j. This is accomplished by using a switch which connects either the positive or the negative node of V_ij to the operational amplifiers OP1 and OP2. Here, OP1 accumulates all negative contributions and OP2 accumulates all positive contributions of the synaptic function. The differential amplifier OP3 takes the difference between the output voltages of OP2 and OP1 and amplifies the voltage by amplification factor A_v. This voltage conversion is used to control the voltage level of V_ij in relation to the input voltage of each BSN. The voltage level at the input of the BSN is fixed by the reference voltage of the BSN which is V₀. However, the voltage level of V_ij can be adjusted and utilized to adjust the regularization parameter λ in the learning rule (Equation 3). The functionality of the BSN is described by Equation (1) where the dimensionless input is given by I_i(t) = V_i,in(t)/V₀. This relates the voltage V_ij to the dimensionless weight by W_ij = A_vV_ij/V₀. The hardware implementation of the BSN uses a stochastic MTJ in series with a transistor as presented by Camsari et al. (2017b). Due to thermal fluctuations of the low-barrier magnet (LBM) of the MTJ the output voltage of the MTJ fluctuates randomly but with the right statistics given by Equation (1). The time dynamics of the LBM can be obtained by solving the stochastic Landau-Lifshitz-Gilbert (LLG) equation. Due to the fast thermal fluctuations of the LBM in the MTJ, Equation (1) can be evaluated on a subnanosecond timescale leading to fast generation of samples (Hassan et al., 2019b; Kaiser et al., 2019b).

Figure 1 just shows the hardware implementation of one weight and one BSN. The size of the whole circuit depends on the size of the training vector N. For every entry of the training vector one BSN is needed. The number of weights which is the number of RC-circuits is given by N(N − 1)/2 where every connection between BSNs is assumed to be reciprocal. To learn biases another N RC-circuits are needed.

The learning process is captured by Equations (3) and (4). The whole learning process has similarity with the software implementation of persistent contrastive divergence (PCD) (Tieleman, 2008) since the circuit takes samples from the model's distribution (V_m,ij) and compares it to the target distribution (V_v,ij) without reinitializing the Markov Chain after a weight update. During the learning process voltage V_ij reaches a constant average value where $\frac{d{V}_{ij}}{dt}\approx 0$. This voltage V_ij = V_ij,learned corresponds to the learned weight.

For inference the capacitor C is replaced by a voltage source of voltage V_ij,learned. Consequently, the autonomous circuit will compute the desired functionality given by the training vectors. In general, training and inference have to be performed on identical hardware in order to learn around variations (see Supplementary Material for more details). It is important to note that in inference mode this circuit can be used for optimization by performing electrical annealing. This is done by increasing all weight voltages V_ij by the same factor over time. In this way the ground state of a Hamiltonian like the Ising Hamiltionian can be found (Sutton et al., 2017; Camsari et al., 2019).

3. Results

In this section the autonomous learning circuit in Figure 1 is simulated in SPICE. We show how the proposed circuit can be used for both inference and learning. As examples, we demonstrate the learning on a full adder (FA) and on 5 × 3 digit images. The BSN models are simulated in the framework developed by Camsari et al. (2015). For all SPICE simulations the following parameters are used for the stochastic MTJ in the BSN implementation: Saturation magnetization M_S = 1,100 emu/cc, LBM diameter D = 22 nm, LBM thickness l = 2 nm, TMR = 110%, damping coefficient α = 0.01, temperature T = 300 K and demagnetization field H_D = 4π M_S with V = (D/2)²πl. For the transistors, 14 nm HP-FinFET Predictive Technology Models (PTM)² are used with fin number fin = 1 for the inverters and fin = 2 for XNOR-gates. Ideal operational amplifiers and switches are used in the synapse. The characteristic time of the BSNs τ_N is in the order of 100 ps (Hassan et al., 2019b) and much larger than the time it takes for the synaptic connections, namely the resistors and operational amplifiers, to propagate BSN outputs to neighboring inputs. It has to be noted that in principle other hardware implementations of the synapse for computing Equation (2) could be utilized as long as the condition τ_N ≫ τ_S is satisfied.

3.1. Learning Addition

As first training example, we use the probability distribution of a full adder. The FA has 5 nodes and 10 weights that have to be learned. In the case of the FA training, no biases are needed. The probability distribution of a full adder with bipolar variables is shown in Table 1. To learn this distribution the correlation terms v_iv_j in the learning rule have to be fed into the voltage node V_v,ij. The correlation is dependent on what training vector/truth table line is fed in. For the second line of the truth table for example v₁v₂ = −1 · −1 = 1 and v₁v₃ = −1 · 1 = −1 with A being the first node, B the second node and so on. In Figure 2B the correlation v₁v₅ is shown. For the sequential case the value of v₁v₅ is obtained by circling through all lines of the truth table where each training vector is shown for 1 ns. A and C_out in Table 1 only differ in the fourth and fifth line for which v₁v₅ = −1. For all other cases v₁v₅ = 1. The average of all lines is shown as red solid line. Figure 2A shows the weight voltage V_ij with i = 1 and j = 5 for FA learning and the first 1,000 ns of training. The following learning parameters have been used for the FA: τ_L = 62.5 ns where C = 1 nF and R = 5 kΩ, A_v = 10, and R_f = 1 MΩ. This choice of learning parameters ensures that τ_L ≫ τ_N. Due to the averaging effect of the RC-circuit both sequential and average feeding of the training vector result in similar learning behavior as long as the RC-constant is much larger than the timescale of sequential feeding. Figure 2C shows the enlarged version of Figure 2A. For the sequential feeding, voltage V_1,5 changes substantially every time v₁v₅ switches to −1.

TABLE 1

Table 1. Truth table of a full adder.

FIGURE 2

Figure 2. Feeding of training data into the circuit. (A) Weight voltage V_1,5 over time for sequential and average feeding in of the correlation between visible unit i and visible unit j for training a full adder. (B) Correlation v₁v₅ vs. time t. All eight lines of the truth table of a full adder are cycled through where every vector is shown for time T = 1 ns at a time. (C) Enlarged version of subfigure (A). For sequential feeding in of data, the voltage change in v₁v₅ directly affects V_1,5.

At the start of training all weight voltages are initialized to 0 V and the probability distribution is uniform. The training is performed for 5,500 ns. In Figure 3A the ideal probability distribution of the FA P_Ideal is shown together with the normalized histogram P_SPICE of the sampled BSN configurations taken from the last 500 ns of learning and compared to the ideal distribution P_Ideal. The training vector is fed in as an average. For P_SPICE the eight trained configurations of Table 1 are the dominant peaks. To monitor the training process, the Kullback-Leibner divergence between the trained and the ideal probability distribution KL(P_Ideal||P_SPICE(t)) is plotted as a function of training time t in Figure 3B where P_SPICE(t) is the normalized histogram taken over 500 ns. P_SPICE at t = 0 corresponds to the histogram taken from t = 0 to t = 500 ns. During training the KL divergence decreases over time until it reaches a constant value at about 0.1. It has to be noted that after the weight matrix is learned correctly for a fully visible Boltzmann machine, the KL divergence can be reduced further by increasing all weights uniformly by a factor I₀ which corresponds to inverse temperature of the Boltzmann machine (Aarts and Korst, 1989). Figure 3 shows that the probability distribution of a FA can be learned very fast with the proposed autonomous learning circuit. In addition, the learning performance is robust when components of the circuit are subject to variation. In the Supplementary Material, additional figures of the learning performance are shown when the diameter of the magnet and the resistances of the RC-circuits are subject to variation. The robustness against variations can be explained by the fact that the circuit can learn around variations. BSNs using LBMs under variations have also been analyzed by Abeed and Bandyopadhyay (2019) and Drobitch and Bandyopadhyay (2019).

FIGURE 3

Figure 3. Training of a full adder in SPICE. (A) Probability distribution of a trained full adder is compared to the ideal distribution with binary inputs A, B, C_in and outputs S and C_out. The training is performed for 5,500 ns. Blue bars are the probability distribution extracted from SPICE simulations by creating a histogram of the configurations of m over the last 500 ns of training. (B) Kullback–Leibler divergence between P_SPICE obtained by doing a moving average of 500 ns and the target distribution defined as $KL(P_{\text{Ideal}}\left|\right|P_{\text{SPICE}}(t))=\sum _{\text{m}}{P}_{\text{Ideal}}(\text{m})log(P_{\text{Ideal}}(\text{m})/P_{\text{Train}}(\text{m},t))$. Following parameters have been used in the simulations: C = 1 nF, R = 5 kΩ, R_F = 1 MΩ, A_v = 10, V₀ = 50 mV.

3.2. Learning Image Completion

As second example, the circuit is utilized to train 10 5 × 3 pixel digit images shown in Figure 4A. Here, 105 reciprocal weights and 15 biases have to be learned. The network is trained for 3,000 ns and the bipolar training data is fed in as average of the 10 v_iv_j terms for every digit. The same learning parameters as in the previous section are used here. In Figure 4B, the KL divergence is shown as a function of time between the SPICE histogram and the ideal probability distribution where the ideal distribution has 10 peaks with each peak being 10% for each digit. Most of the learning happens in the first 1,500 ns of training, however, the KL divergence still reduces slightly during the later parts of learning. After 3,000 ns the KL divergence reaches a value of around 0.5.

FIGURE 4

Figure 4. Training and testing of 5 × 3 digit images. (A) 5 × 3 digit images from 0 to 9. (B) Kullback Leibner divergence during training for 3,000 ns using the autonomous circuit. (C–E) Image completion: For inference, six unique pixels are clamped for every digit (as shown in C). (D,E) show the heatmap of BSN outputs during inference for running the circuit for 100 ns for (D) I₀ = 1 and (E) I₀ = 2.

For inference we replace the capacitor by a voltage source where every voltage is given by the previously learned voltage V_ij. The circuit is run for 10 instances where every instance has a unique clamping pattern of 6 pixels representing one of the 10 digits. The clamped inputs are shown in Figure 4C. The input of a clamped BSN is set to ±V_DD/2. Each instance is run for 100 ns and the outputs of the BSNs are monitored. The BSNs fluctuate between the configurations given by the learned probability distribution. In Figure 4D, the heat map of the output of the BSNs is shown. For every digit the most likely configuration is given by the trained digit image. To illustrate this point, the amount of BSN fluctuations is reduced by increasing the learned weight voltages by a factor of I₀ = 2. The circuit is again run in inference mode for 100 ns with the same clamping patterns. In Figure 4E the heatmap is shown. The circuit locks into the learned digit configuration. This shows that in inference mode the circuit can be utilized for image completion.

4. Discussion

In this paper we have presented a framework for mapping a continuous version of Boltzmann machine learning rule (Equation 3) to a clockless autonomous circuit. We have shown full SPICE simulations to demonstrate the feasibility of this circuit running without any digital component with the learning parameters set by circuit parameters. Due to the fast BSN operation, samples are drawn at subnanosecond speeds leading to fast learning, as such the learning speed should be at least multiple orders of magnitudes faster compared to other computing platforms (Adachi and Henderson, 2015; Korenkevych et al., 2016; Terenin et al., 2019). The advantage of this autonomous architecture is that it produces random numbers naturally and does not rely on pseudo random number generators like linear-feedback shift register (LFSRs) (which are for example used in Bojnordi and Ipek, 2016). These LFSRs have overhead and are not as compact and efficient as the hardware BSN used in this paper. As shown by Borders et al. (2019), typical LFSRs need about 10x more energy per flip and more than 100x more area than an MTJ-based BSN. Another advantage of this approach is that the interfacing with digital hardware only needs to be performed after the learning has been completed. Hence, no expensive analog-to-digital conversion has to be performed during learning. We believe this approach could be extended to other energy based machine learning algorithms like equilibrium propagation introduced by Scellier and Bengio (2017) to design autonomous circuits. Such standalone learning devices could be particularly of interest in the context of mobile and edge computing.

Data Availability Statement

The datasets generated for this study are available on request to the corresponding author.

Author Contributions

JK and SD wrote the paper. JK performed the simulations. RF helped setting up the simulations. KC developed the simulation modules for the BSN. All authors discussed the results and helped refine the manuscript.

Funding

This work was supported in part by ASCENT, one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA. KC gratefully acknowledges support from Center for Science of Information (CSoI), an NSF Science and Technology Center, under grant CCF-0939370.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

This manuscript has been released as a prepint at arXiv (Kaiser et al., 2019a).

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fncom.2020.00014/full#supplementary-material

Footnotes

1. ^Equation (1) can be written in binary notation with a unit step function and a sigmoid function.

2. ^http://ptm.asu.edu/

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