A Deep Dive into the Computational Fidelity of High Variability Low Energy Barrier Magnet Technology for Accelerating Optimization and Bayesian Problems ^††thanks: [email protected],[email protected]

Low energy barrier magnet (LBM) technology, which utilizes nanomagnets with barrier height in the order of thermal energy, has recently been proposed as a potential candidate for hardware accelerators for probabilistic computing and stochastic sampling [1, 2]. These accelerators may be broadly considered as hardware Markov chain Monte Carlo implementation that utilizes the built-in stochasticity provided by the dynamics of the LBM, which results in highly compact devices with true stochasticity, as compared to linear feedback-shift register (LFSR) based pseudo-random number generators (pRNGs) [3]. The magnetization component $m_{z}$ of the LBM randomly fluctuates between two stable states ($\uparrow,~{}\downarrow$) under the influence of the thermal noise, and the probability of getting any one of the two stable states can be driven via an external current [4]. There are a handful of applications ranging from probabilistic computing to machine learning and artificial intelligence that leverage the intrinsic stochastic nature of LBMs [4, 5, 6, 7, 8]. The prototype hardware building blocks are the binary stochastic neurons (BSNs), popularly known as “p-bits” with programmable weights in a recurrent configuration. An illustrative example of a dual-stacked feedback cross-bar structure is shown in Fig. 1(a). The synaptic weights or the “program” is loaded in memristors located at the cross-points of the core cross-bar structure, whereas the neurons are at the peripheries. Using a dual cross-bar structure, it is possible to build recurrent networks, including a Restricted Boltzmann Machine [RBM, Fig. 1(b)], an example application area of this accelerator. The RBM is embedded in the computing fabric by enabling certain neurons and synaptic connections while disabling the rest.

Although BSN-based non-Boolean probabilistic applications are inherently more error resilient than conventional nanomagnet switches used for deterministic Boolean memory and logic applications, the computational reliability of these accelerators that employ LBMs as their hardware RNG, needs to be carefully assessed. Recently, several studies have discussed the impact of geometric, structural, and process variation from device-to-device that can create ignorable to high variability in the characteristics of LBMs depending on the degree of variation [9, 10, 11], however, the resulting impact of these “non-idealities” on the computational networks is still largely not understood.

In this letter, we discuss the issues of variability in the context of circuits and networks built from LBM-based BSN devices. We categorize the variability into a few broad classes, namely shifting and scaling of the device characteristics from the ideal as expected from the mathematical model, and the variability of the barrier heights for two broad classes of algorithms that can be solved using p-bits, such as energy minimization-based optimization algorithm (EMOA) and probabilistic graphical algorithm (PGA). EMOA includes problems such as Ising model and RBMs, which seek to define a problem in terms of a thermodynamically definable “energy-landscape” with the embedding of the desired optimal result in the ground/vacuum energy, while PGA includes Bayesian decision diagrams, which do not have an inherent notion of energy and thermodynamics. In terms of network connectivity (using the spectral theorem of linear systems) [12], this implies that the EMOA networks have symmetric or undirected connections, resulting in eigenstates that are real-valued and reachable via real-space computation, whereas PGA networks are asymmetric or directed, resulting in non-real or complex eigenstates not reachable via real-space computation.

We estimate the mean absolute error (MAE) to quantify the performance deviation from the ideal devices. We find the MAE shows a sub-linear saturation for EMOA, while in the PGA, the error grows linearly to super-linearly. Moreover, the networks are found to be more prone to shifting variability than scaling. Additionally, for EMOA, larger networks are less affected by the variability, while for PGA, the trend is the opposite. Our findings may provide a potential path forward toward designing reliable LBM-based hardware accelerators.

Thin film magnets used in magnetic random access memory (MRAM) technology exhibit a double potential well corresponding to the two easy points [Fig. 1(f)]. The height of the barrier determines the expected state retention time using the Arrhenius relation given by:

In the above equation, $U(=\mu_{0}M_{s}H_{k}\Omega/2)$ is the energy barrier, where the symbols respectively stand for permeability of free space, saturation magnetization, magnetic anisotropy field strength, and volume. For a conventional storage class memory, $U$ is set to $40-60~{}k_{B}T$ for the free layer of an MTJ, which yields a decade-long state retention time $\tau$ depending on the $\tau_{0}$, the inverse of attempt frequency that ranges from $0.1-1~{}ns$ [13]. However, if the magnet is ultra-scaled by reducing the volume $\Omega$ or its profile is made circular, which reduces the $H_{k}$ by removing the shape anisotropy, the retention time can be scaled down to near $\tau_{0}$ [14]. In this case, the free layer’s magnetization vector fluctuates between the two easy points under the influence of the thermal noise, which is able to “kick” the magnetization over the barrier with ease, at near $\mathrm{GHz}$ frequencies. MTJ structure allows this fluctuation to be translated into an equivalent fluctuation in the resistance of the device, which can be used for building useful devices that can harvest true randomness from the environment.

One such device is the “p-bit”, which is a binary stochastic neuron with a compact model given by:

In this device, the output swings between $-V_{DD}/2$ to $V_{DD}/2$ corresponding to $-1$ and $+1$ state labels of $m_{z}$, however, the ratio of these states is controllable by an input signal, which imposes a $\tanh$-like probability distribution. rnd is a uniform random distribution. The parameters $\beta$ and $\alpha$ represent the transfer gain of the unit and the relative contribution of the stochasticity to the characteristics, respectively. For large scale correlated networks, $V^{in}$ can be represented as:

where $j$ stands for the index over all input devices connected to the particular $i$-th device, $h$ is the bias vector, and $J$ is the synaptic matrix. Different functionalities correspond to different choices of $h$ and $J$. $\kappa$ is a coupling coefficient representing the inverse of the “temperature” of the system.

We implement the compact model of $p$-bit networks described by (2) and (3) in MATLAB according to the methodology discussed in Camsari [4]. The MATLAB model is a parameterized version of the compact modeling simulation performed in SPICE [4, 15]. In MATLAB implementation, we use $\alpha=1$, $\beta=1$ (for ideal case), $\kappa=0.8$, and $V_{DD}=2~{}V$ throughout the calculation unless otherwise specified.

We use computational networks constructed from p-bits of varying sizes. For EMOA, we use AND gate and full-adder having $J$ matrices sized $3\times 3$ and $14\times 14$, respectively [4]. We construct an arbitrary symmetric $J$ matrix of $50\times 50$ for a large network. For PGA, we use Bayesian networks (BNs) constructed from $8$, $20$, and $50$ p-bits ($J$ matrices are asymmetric in these cases). For EMOA, MAE is computed by taking the summation of the absolute difference between the output probability distribution of ideal and non-ideal cases, normalized by the number of LBMs in the network. However, for PGA, we calculate the normalized MAE from the difference in the correlation matrix ($\sigma(i,j)=\frac{1}{T}\int_{0}^{T}V_{i}^{out}V_{j}^{out}\,dt$) between the ideal and non-ideal cases. For both algorithms, we use $T=10^{6}$ simulation steps to get to the $V^{out}$. If the sample generation time is $2~{}ns$, this is equivalent to $2~{}ms$ of compute time. The average and standard deviation of the MAE are calculated from $N=100$ simulations.

LBM devices are hybrids of silicon CMOS, which is a highly mature technology, and spintronics/magnetics, which is a relatively new technology. While they have been successfully integrated into the context of high energy barrier storage class MRAM technology by several commercial vendors, its LBM variant comes with lithographic challenges that may require a long process of technological developments to perfect. These lithographic challenges mainly concern the quality of magnetic films and the precision control over their geometry. Abeed [2019a] studied the impact of geometrical irregularities such as dimples, holes, shape variance, etc. on the characteristic correlation times of LBMs and found that the distribution of correlation times can be large. These kinds of variations can have implications that are beyond the intrinsic behavior of the free-layer magnet of the MTJ itself.

In particular, two critical sets of variations are discussed next. Please note that these variations become relevant in the context of circuits and networks built from these devices.

In summary, we quantify the impact of non-idealities in computational networks built from LBM-based BSNs using two different techniques. In all the possible variances studied in this work, the error shows a sub-linear saturation at the extremal device variability points for EMOA, while in the PGA, the error grows linearly to super-linearly. We conjecture that this is because, in EMOA, the system tries to seek a single thermodynamically favorable fixed point in a finite phase space, which limits the growth of error, whereas, in PGA, there is no similar principle that can check the growth of the error. Additionally, running multiple samples of the same problem with different random seeds (thereby simulating the “real world”) helps in reducing the variance of the error, but not its mean value. This suggests that for a certain amount of device variability, the average error is fixed, which may be estimated or characterized beforehand, and the results are certified accordingly. These findings may provide critical design insights for building suitable LBM-based hardware accelerators.

This work is supported in part by the NSF I/UCRC on Multi-functional Integrated System Technology (MIST) Center; IIP-1439644, IIP-1439680, IIP-1738752, IIP-1939009, IIP-1939050, and IIP-1939012. We thank Kerem Yunus Camsari and Faiyaz Elahi Mullick for useful discussions. All the calculations are done using the computational resources from High-Performance Computing systems at the University of Virginia (Rivanna).