Selectivity and Shape in the Design of Forward-Forward Goodness Functions

TL;DR

This paper explores the design space of goodness functions for the Forward-Forward algorithm, emphasizing shape sensitivity over energy, leading to improved performance across multiple datasets.

Contribution

It introduces shape-sensitive and selectivity-based goodness functions, demonstrating their effectiveness and robustness in training neural networks with the Forward-Forward algorithm.

Findings

Achieved 89.0% on Fashion-MNIST and 98.2% on MNIST, surpassing sum-of-squares by +32.6pp.

Shape-sensitive functions improve robustness to magnitude shifts.

Consistent performance gains across diverse datasets and activation functions.

Abstract

The Forward-Forward (FF) algorithm trains networks layer-by-layer using a local "goodness function," yet sum-of-squares (SoS) has remained the only choice studied. We systematically explore the goodness-function design space and identify a unifying principle: the goodness function must be sensitive to the shape of neural activity, not its total energy. This principle is motivated by the observation that deep network activations follow heavy-tailed distributions and that discriminative information is often concentrated in peak activities. We propose two complementary families: selective functions (top-k, entmax-weighted energy) that measure only peak activity, and shape-sensitive functions (excess kurtosis / "burstiness" and higher-order moments) that reward heavy-tailed distributions via scale-invariant statistics. Combined with separate label-feature forwarding (FFCL), controlled experiments across 13 goodness functions, 5 activations, 6 datasets, and three continuous sweeps, each tracing a characteristic inverted-U, yield 89.0% on Fashion-MNIST and 98.2+-0.1% on MNIST (4x2000), a +32.6pp gain over SoS, with consistent improvements across all benchmarks (+72pp USPS, +52pp SVHN). The scale-invariant nature of burstiness makes it particularly robust to magnitude shifts across layers and datasets.