# A Higher-Order Ising Model with Gradient-Free Update

## TL;DR

This paper introduces an extended Ising model that can store and converge to desired patterns without gradient-based training, enabling faster and more flexible solutions.

## Contribution

The paper proposes a higher-order Ising model with a gradient-free update method, extending beyond the limitations of traditional second-order models.

## Key findings

- The extended Ising model allows for explicit closed-form update formulas without gradient evaluation.
- The model converges to a local minimum in finite steps without requiring network training.
- The new model can handle desired patterns where the Hebbian rule fails in conventional Ising models.

## Abstract

The Ising model is able to memorize some patterns or solutions as stable states. An Ising network may automatically converge to a pre-stored solution for a random input. However, in many cases, the Ising model cannot perform this task. The gap is that for a set of desired patterns, one may not be able to construct an Ising model such that the desired patterns are the stable solutions of the Ising model. The Ising model has limited power, because its energy function is limited to a second-order polynomial. Our research outline is as follows. This paper extends the conventional Ising model so that it has wider applications, where the Hebbian rule no longer works. The extended model does not have a limit on the order of the energy function. The extended Ising is defined by combining all desired patterns in a product. Our findings are that the extended Ising model has explicit closed-from update formulas, which do not require the evaluation of gradients. Thus, no network training is necessary. The update algorithm takes finite steps to reach a local minimum.