The Forward-Forward (FF) algorithm provides a bottom-up alternative to backpropagation (BP) for training neural networks, relying on a layer-wise "goodness" function to guide learning. Existing goodness functions, inspired by energy-based learning (EBL), are typically defined as the sum of squared post-synaptic activations, neglecting the correlations between neurons. In this work, we propose a novel goodness function termed dimensionality compression that uses the effective dimensionality (ED) of fluctuating neural responses to incorporate second-order statistical structure. Our objective minimizes ED for clamped inputs when noise is considered while maximizing it across the sample distribution, promoting structured representations without the need to prepare negative samples. We demonstrate that this formulation achieves competitive performance compared to other non-BP methods. Moreover, we show that noise plays a constructive role that can enhance generalization and improve inference when predictions are derived from the mean of squared outputs, which is equivalent to making predictions based on the energy term. Our findings contribute to the development of more biologically plausible learning algorithms and suggest a natural fit for neuromorphic computing, where stochasticity is a computational resource rather than a nuisance. The code is available at https://github.com/ZhichaoZhu/StochasticForwardForward

翻译：前向-前向（FF）算法为训练神经网络提供了一种自下而上替代反向传播（BP）的方法，其依赖逐层的“优度”函数来指导学习。受基于能量的学习（EBL）启发，现有优度函数通常定义为突触后激活值的平方和，忽略了神经元间的相关性。本文提出一种称为维度压缩的新型优度函数，该函数利用波动神经响应的有效维度（ED）来纳入二阶统计结构。我们的目标是在考虑噪声时最小化钳制输入下的ED，同时在整个样本分布上最大化该值，从而无需准备负样本即可促进结构化表征的形成。实验证明，该公式相比其他非BP方法取得了具有竞争力的性能。此外，我们揭示了噪声能发挥建设性作用：当预测基于平方输出的均值（等价于基于能量项进行预测）时，噪声可增强泛化能力并改进推理性能。我们的研究推动了更具生物合理性的学习算法的发展，并表明该方法天然适用于神经形态计算——其中随机性是一种计算资源而非干扰因素。代码发布于 https://github.com/ZhichaoZhu/StochasticForwardForward