Rescuing Points Trapped Inside Incorrect Decision Regions: A target for Evaluating and Improving Forward-Forward Learning

Forward-Forward (FF) training lets each layer learn from a local goodness criterion. In cumulative-goodness variants, later layers can inherit a task that earlier layers have partly separated. We formalize this as layer free-riding: under the softplus FF criterion, the class-discrimination gradient reaching block $d$ decays exponentially with the positive margin accumulated by earlier blocks (Theorem 3.1); a squared-hinge barrier reproduces the pathology. History-free and hardness-gated repairs raise deeper-layer separation by up to $5\times$ on CIFAR-100 and $45\times$ in an 8-block CIFAR-10 model, yet these and a depth-scaled auxiliary term move accuracy by under one percentage point among non-degenerate variants; on Tiny ImageNet, a harder cross-dataset check of the selected configuration, layer health likewise does not track accuracy. A theory-derived attenuation-compensated objective raises deepest-block separation $5.9\times$ at a consistent cost in the FF classifier's own accuracy that a probe trained on the frozen features recovers; in a smaller CIFAR-10 control, most of its separation gain comes from dropping the depth-scaled auxiliary term it replaces. Accuracy invariance is a property of the deployed sum-based scoring rule under redistribution (Proposition 3.2) and, empirically, of the structured trained readouts we test, not of what the network learns: retrained variants disagree on 13-26% of test predictions. Architecture and augmentation matter far more than the non-degenerate training rules studied. Cumulative free-riding is therefore a real and repairable pathology, but for the softplus and squared-hinge objectives, architectures, and datasets we study, it is not the dominant factor limiting accuracy.

Training neural networks has traditionally relied on backpropagation (BP), a gradient-based algorithm that, despite its widespread success, suffers from key limitations in both biological and hardware perspectives. These include backward error propagation by symmetric weights, non-local credit assignment, and frozen activity during backward passes. We propose Forward Target Propagation (FTP), a biologically plausible and computationally efficient alternative that replaces the backward pass with a second forward pass. FTP estimates layerwise targets using only feedforward computations, eliminating the need for symmetric feedback weights or learnable inverse functions, hence enabling modular and local learning. We evaluate FTP on fully connected networks, CNNs, and RNNs, demonstrating accuracies competitive with BP on MNIST, CIFAR10, and CIFAR100, as well as effective modeling of long-term dependencies in sequential tasks. Moreover, FTP outperforms BP under quantized low-precision and emerging hardware constraints while also demonstrating substantial efficiency gains over other biologically inspired methods such as target propagation variants and forward-only learning algorithms. With its minimal computational overhead, forward-only nature, and hardware compatibility, FTP provides a promising direction for energy-efficient on-device learning and neuromorphic computing.

The Forward-Forward Algorithm (FFA) replaces backpropagation (BP) with layer-wise local contrastive objectives, eliminating the backward pass and the need to retain intermediate activations, yet suffers a persistent performance gap with BP that worsens with depth. This paper diagnoses two structural failure modes: an optimization floor arising from concurrent local updates; and a geometric collapse of layer representations driven by the local update mechanism. On the optimization side, we prove that the FFA loss satisfies the Polyak--Lojasiewicz inequality at each layer; however, simultaneous layer updates induce inter-layer representation-distribution drift, so each layer optimizes against a moving input distribution and incurs an error floor. On the representational side, the pairwise similarity kernel of layer representations contracts exponentially toward rank one as depth increases, collapsing the diversity of per-layer error signals. This collapse bounds FFA's effective learning capacity, which measures the diversity of gradient information across layers, independently of depth, whereas BP's chain-rule signal preserves per-layer diversity, yielding a capacity that scales with depth.