Analog In-memory Training on General Non-ideal Resistive Elements: The Impact of Response Functions

TL;DR

This work analyzes gradient-based training on analog in-memory computing hardware with generic non-ideal resistive response functions, showing that asymmetric updates induce an implicit penalty that biases toward a symmetric point and prevents exact convergence. It develops Residual Learning, a bilevel framework that introduces a residual array $P_k$ to align the algorithmic stationary point with the hardware's symmetric point, achieving provable convergence to a true optimum under zero-shift and extending to practical issues like limited granularity and IO noise. The paper further extends to Residual Learning v2 with digital buffering and thresholded transfer to combat read noise and discretization, and validates the theory with MNIST and CIFAR simulations using FCN, CNN, and ResNet architectures, where RL and RLv2 close the gap to Digital SGD while preserving AIMC efficiency. Overall, the approach provides a rigorous foundation for on-chip analog training with non-ideal devices and offers practical algorithms to mitigate hardware imperfections, supporting scalable energy-efficient learning on resistive crossbar arrays.

Abstract

As the economic and environmental costs of training and deploying large vision or language models increase dramatically, analog in-memory computing (AIMC) emerges as a promising energy-efficient solution. However, the training perspective, especially its training dynamic, is underexplored. In AIMC hardware, the trainable weights are represented by the conductance of resistive elements and updated using consecutive electrical pulses. While the conductance changes by a constant in response to each pulse, in reality, the change is scaled by asymmetric and non-linear response functions, leading to a non-ideal training dynamic. This paper provides a theoretical foundation for gradient-based training on AIMC hardware with non-ideal response functions. We demonstrate that asymmetric response functions negatively impact Analog SGD by imposing an implicit penalty on the objective. To overcome the issue, we propose Residual Learning algorithm, which provably converges exactly to a critical point by solving a bilevel optimization problem. We demonstrate that the proposed method can be extended to address other hardware imperfections, such as limited response granularity. As we know, it is the first paper to investigate the impact of a class of generic non-ideal response functions. The conclusion is supported by simulations validating our theoretical insights.

Figures (6)

• Figure 1: The weight's response curve. Positive and negative pulses are fired continuously on the left and right halves, respectively. One pulse is fired per cycle. Given $w$, the weight becomes $w^+$ or $w^-$ after one positive and negative pulse, respectively. The response factors $q_+(w)$ and $q_-(w)$ are approximately the slope of the curve at $w$, and $\Delta w_{\min}$ is the response granularity. (Left) Ideal response functions $q_+(w)\equiv q_-(w)\equiv 1$. Every point is a symmetric point. (Right) Asymmetric response functions $q_{+}(w) \ne q_{-}(w)$ almost everywhere expect for the symmetric point $w^\diamond$.
• Figure 2: Comparison of Analog SGD and Tiki-Taka under different parameter $c_{\texttt{Lin}}$. The error plateau in the order $10^{-5}$ comes from the limited response granularity $\Delta w_{\min} = 10^{-4}$.
• Figure 3: Examples of response functions from \ref{['assumption:response-factor']}; $w^\diamond$ is the symmetric point.
• Figure 4: Test accuracy of training on MNIST dataset under different $\tau$; (Left) FCN. (Right) CNN.
• Figure 5: The test accuracy of ResNet family models after 100 epochs trained by Residual Learning under different $\gamma$ in \ref{['recursion:HD-P']}; (Left) CIFAR10. (Right) CIFAR100.

Theorems & Definitions (12)

• proof : Proof of Theorem \ref{['theorem:pulse-update-error']}
• proof : Proof of Lemma \ref{['lemma:properties-weighted-norm']}
• proof : Proof of Lemma \ref{['lemma:lip-analog-update']}
• proof : Proof of Lemma \ref{['lemma:element-wise-product-error']}
• proof : Proof of Theorem \ref{['theorem:implicit-regularization-short']} and \ref{['theorem:implicit-regularization']}
• proof : Proof of Theorem \ref{['theorem:ASGD-convergence-noncvx']}
• proof : Proof of Theorem \ref{['theorem:TT-convergence-scvx']}
• proof : Proof of Lemma \ref{['lemma:TT-barW-descent']}
• proof : Proof of Lemma \ref{['lemma:TT-W-descent']}
• proof : Proof of Lemma \ref{['lemma:TT-P-descent']}