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# Equilibrium Point Learning

*Proceedings of the 15th Asian Conference on Machine Learning*, PMLR 222:74-89, 2024.

#### Abstract

We present a novel approach, Equilibrium Point Learning (EPL), for training the deep equilibrium model (DEQ). In this method, the equilibrium point of the DEQ serves as the learnable parameters. Notably, the DEQ parameters encapsulate the learning algorithm itself and remain fixed. Consequently, by exploring the parameter space, we can discover a more efficient learning algorithm without relying on conventional techniques such as backpropagation or Q-learning. In this paper, we adopt an evolutionary approach inspired by biological neurons to evolve the DEQ model parameters. Initially, we examine the physical dynamics of neurons at the molecular level and translate them into a dynamical system representation. Subsequently, we formulate a deep implicit layer that is mathematically proven to possess an equilibrium point. The energy function of the implicit layer is defined using a quadratic form augmented with entropy and momentum terms. Given the resemblance between the dynamics of the deep implicit layer and the principles of physics and chemistry, it can effectively capture the biomodel of systems biology and the neural model of spiking neural networks (SNNs). This equivalence enables us to define the implicit layer of the DEQ, allowing for seamless integration with existing artificial neural networks (ANNs). Finally, we employ HyperNEAT to evolve the parameters of the dynamical system. Through our experiments, we observe a consistent improvement in learning efficiency, with each successive generation exhibiting a 0.2% increase in learning speed per generation.