The Forward-Forward (FF) learning algorithm provides a bottom-up alternative to backpropagation (BP) for training neural networks, relying on a layer-wise goodness function with well-designed negative samples for contrastive learning. Existing goodness functions are typically defined as the sum of squared postsynaptic activations, neglecting correlated variability 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 noisy copies of individual inputs 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 output, which is equivalent to making predictions based on an 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 Forward-Forward (FF) algorithm offers a biologically plausible alternative to backpropagation, enabling neural networks to learn through local updates. However, FF's efficacy relies heavily on the definition of "goodness", which is a scalar measure of neural activity. While current implementations predominantly utilize a simple sum-of-squares metric, it remains unclear if this default choice is optimal. To address this, we benchmarked 21 distinct goodness functions across four standard image datasets (MNIST, FashionMNIST, CIFAR-10, STL-10), evaluating classification accuracy, energy consumption, and carbon footprint. We found that certain alternative goodness functions inspired from various domains significantly outperform the standard baseline. Specifically, \texttt{game\_theoretic\_local} achieved 97.15\% accuracy on MNIST, \texttt{softmax\_energy\_margin\_local} reached 82.84\% on FashionMNIST, and \texttt{triplet\_margin\_local} attained 37.69\% on STL-10. Furthermore, we observed substantial variability in computational efficiency, highlighting a critical trade-off between predictive performance and environmental cost. These findings demonstrate that the goodness function is a pivotal hyperparameter in FF design. We release our code on \href{https://github.com/aryashah2k/In-Search-of-Goodness}{Github} for reference and reproducibility.

The Forward-Forward (FF) learning algorithm provides a bottom-up alternative to backpropagation (BP) for training neural networks, relying on a layer-wise "goodness" function with well-designed negative samples for contrastive learning. Existing goodness functions are typically defined as the sum of squared postsynaptic activations, neglecting correlated variability 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 noisy copies of individual inputs 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 output, which is equivalent to making predictions based on an 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

The Forward-Forward (FF) Algorithm is a recently proposed learning procedure for neural networks that employs two forward passes instead of the traditional forward and backward passes used in backpropagation. However, FF remains largely confined to supervised settings, leaving a gap at domains where learning signals can be yielded more naturally such as RL. In this work, inspired by FF's goodness function using layer activity statistics, we introduce Action-conditioned Root mean squared Q-Functions (ARQ), a novel value estimation method that applies a goodness function and action conditioning for local RL using temporal difference learning. Despite its simplicity and biological grounding, our approach achieves superior performance compared to state-of-the-art local backprop-free RL methods in the MinAtar and the DeepMind Control Suite benchmarks, while also outperforming algorithms trained with backpropagation on most tasks. Code can be found at https://github.com/agentic-learning-ai-lab/arq.

The so-called Forward-Forward Algorithm (FFA) has recently gained momentum as an alternative to the conventional back-propagation algorithm for neural network learning, yielding competitive performance across various modeling tasks. By replacing the backward pass of gradient back-propagation with two contrastive forward passes, the FFA avoids several shortcomings undergone by its predecessor (e.g., vanishing/exploding gradient) by enabling layer-wise training heuristics. In classification tasks, this contrastive method has been proven to effectively create a latent sparse representation of the input data, ultimately favoring discriminability. However, FFA exhibits an inherent asymmetric gradient behavior due to an imbalanced loss function between positive and negative data, adversely impacting on the model's generalization capabilities and leading to an accuracy degradation. To address this issue, this work proposes the Symmetric Forward-Forward Algorithm (SFFA), a novel modification of the original FFA which partitions each layer into positive and negative neurons. This allows the local fitness function to be defined as the ratio between the activation of positive neurons and the overall layer activity, resulting in a symmetric loss landscape during the training phase. To evaluate the enhanced convergence of our method, we conduct several experiments using multiple image classification benchmarks, comparing the accuracy of models trained with SFFA to those trained with its FFA counterpart. As a byproduct of this reformulation, we explore the advantages of using a layer-wise training algorithm for Continual Learning (CL) tasks. The specialization of neurons and the sparsity of their activations induced by layer-wise training algorithms enable efficient CL strategies that incorporate new knowledge (classes) into the neural network, while preventing catastrophic forgetting of previously...

Forward-only learning algorithms have recently gained attention as alternatives to gradient backpropagation, replacing the backward step of this latter solver with an additional contrastive forward pass. Among these approaches, the so-called Forward-Forward Algorithm (FFA) has been shown to achieve competitive levels of performance in terms of generalization and complexity. Networks trained using FFA learn to contrastively maximize a layer-wise defined goodness score when presented with real data (denoted as positive samples) and to minimize it when processing synthetic data (corr. negative samples). However, this algorithm still faces weaknesses that negatively affect the model accuracy and training stability, primarily due to a gradient imbalance between positive and negative samples. To overcome this issue, in this work we propose a novel implementation of the FFA algorithm, denoted as Polar-FFA, which extends the original formulation by introducing a neural division (\emph{polarization}) between positive and negative instances. Neurons in each of these groups aim to maximize their goodness when presented with their respective data type, thereby creating a symmetric gradient behavior. To empirically gauge the improved learning capabilities of our proposed Polar-FFA, we perform several systematic experiments using different activation and goodness functions over image classification datasets. Our results demonstrate that Polar-FFA outperforms FFA in terms of accuracy and convergence speed. Furthermore, its lower reliance on hyperparameters reduces the need for hyperparameter tuning to guarantee optimal generalization capabilities, thereby allowing for a broader range of neural network configurations.

The Forward-Forward (FF) algorithm was recently proposed as a local learning method to address the limitations of backpropagation (BP), offering biological plausibility along with memory-efficient and highly parallelized computational benefits. However, it suffers from suboptimal performance and poor generalization, largely due to inadequate theoretical support and a lack of effective learning strategies. In this work, we reformulate FF using distance metric learning and propose a distance-forward algorithm (DF) to improve FF performance in supervised vision tasks while preserving its local computational properties, making it competitive for efficient on-chip learning. To achieve this, we reinterpret FF through the lens of centroid-based metric learning and develop a goodness-based N-pair margin loss to facilitate the learning of discriminative features. Furthermore, we integrate layer-collaboration local update strategies to reduce information loss caused by greedy local parameter updates. Our method surpasses existing FF models and other advanced local learning approaches, with accuracies of 99.7\% on MNIST, 88.2\% on CIFAR-10, 59\% on CIFAR-100, 95.9\% on SVHN, and 82.5\% on ImageNette, respectively. Moreover, it achieves comparable performance with less than 40\% memory cost compared to BP training, while exhibiting stronger robustness to multiple types of hardware-related noise, demonstrating its potential for online learning and energy-efficient computation on neuromorphic chips.

Growing economic, environmental, and social pressures require us to be efficient with limited resources (Aleksandrov and Walsh, 2020). Therefore, the fair division (Steinhaus, 1948) of limited resources among multiple parties/agents is needed to efficiently balance their competing interests in many real-life applications, e.g., Fisher market (Codenotti and Varadarajan, 2007; Vazirani, 2007), housing allocation (Benabbou et al., 2019), rent division (Edward Su, 1999; Gal et al., 2016), and many more (Demko and Hill, 1988). The fair division problem has been extensively studied in algorithmic game theory (Caragiannis et al., 2019; Codenotti and Varadarajan, 2007; Eisenberg and Gale, 1959; Vazirani, 2007) but focuses on the static setting where all information (items, agents, and their utility) is known in advance. However, fair division problems are often online (Aleksandrov and Walsh, 2020; Gao et al., 2021; Yamada et al., 2024), referred to as online fair division, where indivisible items arrive sequentially and each item needs to be irrevocably allocated to an agent. Existing algorithms for online fair division assume a small number of items with a sufficiently large number of copies (Yamada et al., 2024), which ensures a good utility estimation using the observed utilities for previous allocations for all item-agent pairs. The estimated utilities of item-agent pairs are used to allocate the item to an agent that maintains a desired balance between fairness (i.e., keeping the desired level of utilities across the agents) and efficiency (i.e., maximizing the total utility) (Sinclair et al., 2022). However, many real-life applications have a large number of items with only a few copies for each item. For example, consider an online food delivery platform that wants to recommend restaurants (agents) to its users (items) while balancing between fairly recommending restaurants to accommodate their competing interests (fairness) and maximizing its profit (efficiency).

Advances in neural computation have predominantly relied on the gradient backpropagation algorithm (BP). However, the recent shift towards non-stationary data modeling has highlighted the limitations of this heuristic, exposing that its adaptation capabilities are far from those seen in biological brains. Unlike BP, where weight updates are computed through a reverse error propagation path, Hebbian learning dynamics provide synaptic updates using only information within the layer itself. This has spurred interest in biologically plausible learning algorithms, hypothesized to overcome BP's shortcomings. In this context, Hinton recently introduced the Forward-Forward Algorithm (FFA), which employs local learning rules for each layer and has empirically proven its efficacy in multiple data modeling tasks. In this work we argue that when employing a squared Euclidean norm as a goodness function driving the local learning, the resulting FFA is equivalent to a neo-Hebbian Learning Rule. To verify this result, we compare the training behavior of FFA in analog networks with its Hebbian adaptation in spiking neural networks. Our experiments demonstrate that both versions of FFA produce similar accuracy and latent distributions. The findings herein reported provide empirical evidence linking biological learning rules with currently used training algorithms, thus paving the way towards extrapolating the positive outcomes from FFA to Hebbian learning rules. Simultaneously, our results imply that analog networks trained under FFA could be directly applied to neuromorphic computing, leading to reduced energy usage and increased computational speed.

We introduce a new approach in distributed deep learning, utilizing Geoffrey Hinton's Forward-Forward (FF) algorithm to speed up the training of neural networks in distributed computing environments. Unlike traditional methods that rely on forward and backward passes, the FF algorithm employs a dual forward pass strategy, significantly diverging from the conventional backpropagation process. This novel method aligns more closely with the human brain's processing mechanisms, potentially offering a more efficient and biologically plausible approach to neural network training. Our research explores different implementations of the FF algorithm in distributed settings, to explore its capacity for parallelization. While the original FF algorithm focused on its ability to match the performance of the backpropagation algorithm, the parallelism aims to reduce training times and resource consumption, thereby addressing the long training times associated with the training of deep neural networks. Our evaluation shows a 3.75 times speed up on MNIST dataset without compromising accuracy when training a four-layer network with four compute nodes. The integration of the FF algorithm into distributed deep learning represents a significant step forward in the field, potentially revolutionizing the way neural networks are trained in distributed environments.