The aim of this paper is to present a novel physics-based framework for the identification of dynamical systems, in which the physical and structural insights are reflected directly into a backpropagation-based learning algorithm. The main result is a method to compute in closed form the gradient of a multi-step loss function, while enforcing physical properties and constraints. The derived algorithm has been exploited to identify the unknown inertia matrix of a space debris, and the results show the reliability of the method in capturing the physical adherence of the estimated parameters.

Deep learning has become a commonplace solution to numerous modern problems, occupying a central spot of today's technological and social attention. The recipe of its power is the combining effect of unprecedented large dimensions and learning process based on gradient backpropagation [LeCun et al., 1998]. In particular, thanks to the simplicity of neuron model that is decomposed into a weighted linear sum followed by a non-linear activation function, the gradient of weights is solely determined by their respective input without involving cross-parameter dependency. Hence, in terms of computational process, gradient backpropagation is automated by gradient chain rule and only requires a buffering of the forward input data. However, deep learning is computationally intensive.

Iterative differential approximation methods that rely upon backpropagation have enabled the optimization of neural networks; however, at present, they remain computationally expensive, especially when training models at scale. In this paper, we propose a computationally efficient alternative for optimizing neural networks that can both reduce the costs of scaling neural networks and provide high-efficiency optimizations for low-resource applications. We derive an explicit solution to a simple feed-forward language model (LM) by mathematically analyzing its gradients. This solution generalizes from single-layer LMs to the class of all single-layer feed-forward softmax-activated neural models trained on positive-valued features, as is demonstrated by our extension of this solution application to MNIST digit classification. For both LM and digit classifiers, we find computationally that explicit solutions perform near-optimality in experiments showing that 1) iterative optimization only marginally improves the explicit solution parameters and 2) randomly initialized parameters iteratively optimize towards the explicit solution. We also preliminarily apply the explicit solution locally by layer in multi-layer networks and discuss how the solution's computational savings increase with model complexity -- for both single- and mult-layer applications of the explicit solution, we emphasize that the optima achieved cannot be reached by backpropagation alone, i.e., better optima appear discoverable only after explicit solutions are applied. Finally, we discuss the solution's computational savings alongside its impact on model interpretability and suggest future directions for the derivation of explicit solutions to complex- and multi-layer architectures.

The Backpropagation algorithm has often been criticised for its lack of biological realism. In an attempt to find a more biologically plausible alternative, the recently introduced Forward-Forward algorithm replaces the forward and backward passes of Backpropagation with two forward passes. In this work, we show that the internal representations obtained by the Forward-Forward algorithm can organise into category-specific ensembles exhibiting high sparsity - i.e. composed of an extremely low number of active units. This situation is reminiscent of what has been observed in cortical sensory areas, where neuronal ensembles are suggested to serve as the functional building blocks for perception and action. Interestingly, while this sparse pattern does not typically arise in models trained with standard Backpropagation, it can emerge in networks trained with Backpropagation on the same objective proposed for the Forward-Forward algorithm. These results suggest that the learning procedure proposed by Forward-Forward may be superior to Backpropagation in modelling learning in the cortex, even when a backward pass is used.

Spiking neural networks (SNNs) are a viable alternative to conventional artificial neural networks when resource efficiency and computational complexity are of importance. A major advantage of SNNs is their binary information transfer through spike trains which eliminates multiplication operations. The training of SNNs has, however, been a challenge, since neuron models are non-differentiable and traditional gradient-based backpropagation algorithms cannot be applied directly. Furthermore, spike-timing-dependent plasticity (STDP), albeit being a spike-based learning rule, updates weights locally and does not optimize for the output error of the network. We present desire backpropagation, a method to derive the desired spike activity of all neurons, including the hidden ones, from the output error. By incorporating this desire value into the local STDP weight update, we can efficiently capture the neuron dynamics while minimizing the global error and attaining a high classification accuracy. That makes desire backpropagation a spike-based supervised learning rule. We trained three-layer networks to classify MNIST and Fashion-MNIST images and reached an accuracy of 98.41% and 87.56%, respectively. In addition, by eliminating a multiplication during the backward pass, we reduce computational complexity and balance arithmetic resources between forward and backward pass, making desire backpropagation a candidate for training on low-resource devices.

Backpropagation, the cornerstone of deep learning, is limited to computing gradients for continuous variables. This limitation poses challenges for problems involving discrete latent variables. To address this issue, we propose a novel approach to approximate the gradient of parameters involved in generating discrete latent variables. First, we examine the widely used Straight-Through (ST) heuristic and demonstrate that it works as a first-order approximation of the gradient. Guided by our findings, we propose ReinMax, which achieves second-order accuracy by integrating Heun's method, a second-order numerical method for solving ODEs. ReinMax does not require Hessian or other second-order derivatives, thus having negligible computation overheads. Extensive experimental results on various tasks demonstrate the superiority of ReinMax over the state of the art.

Despite considerable theoretical progress in the training of neural networks viewed as a multi-agent system of neurons, particularly concerning biological plausibility and decentralized training, their applicability to real-world problems remains limited due to scalability issues. In contrast, error-backpropagation has demonstrated its effectiveness for training deep networks in practice. In this study, we propose a local objective for neurons that, when pursued by neurons individually, align them to exhibit similarities to error-backpropagation in terms of efficiency and scalability during training. For this purpose, we examine a neural network comprising decentralized, self-interested neurons seeking to maximize their local objective -- attention from subsequent layer neurons -- and identify the optimal strategy for neurons. We also analyze the relationship between this strategy and backpropagation, establishing conditions under which the derived strategy is equivalent to error-backpropagation. Lastly, we demonstrate the learning capacity of these multi-agent neural networks through experiments on three datasets and showcase their superior performance relative to error-backpropagation in a catastrophic forgetting benchmark.

Text-to-image diffusion models have recently emerged at the forefront of image generation, powered by very large-scale unsupervised or weakly supervised text-to-image training datasets. Due to their unsupervised training, controlling their behavior in downstream tasks, such as maximizing human-perceived image quality, image-text alignment, or ethical image generation, is difficult. Recent works finetune diffusion models to downstream reward functions using vanilla reinforcement learning, notorious for the high variance of the gradient estimators. In this paper, we propose AlignProp, a method that aligns diffusion models to downstream reward functions using end-to-end backpropagation of the reward gradient through the denoising process. While naive implementation of such backpropagation would require prohibitive memory resources for storing the partial derivatives of modern text-to-image models, AlignProp finetunes low-rank adapter weight modules and uses gradient checkpointing, to render its memory usage viable. We test AlignProp in finetuning diffusion models to various objectives, such as image-text semantic alignment, aesthetics, compressibility and controllability of the number of objects present, as well as their combinations. We show AlignProp achieves higher rewards in fewer training steps than alternatives, while being conceptually simpler, making it a straightforward choice for optimizing diffusion models for differentiable reward functions of interest. Code and Visualization results are available at https://align-prop.github.io/.

One defining characteristic of Mixture-of-Expert (MoE) models is their capacity for conducting sparse computation via expert routing, leading to remarkable scalability. However, backpropagation, the cornerstone of deep learning, requires dense computation, thereby posting challenges in MoE gradient computations. Here, we introduce SparseMixer, a scalable gradient estimator that bridges the gap between backpropagation and sparse expert routing. Unlike typical MoE training which strategically neglects certain gradient terms for the sake of sparse computation and scalability, SparseMixer provides scalable gradient approximations for these terms, enabling reliable gradient estimation in MoE training. Grounded in a numerical ODE framework, SparseMixer harnesses the mid-point method, a second-order ODE solver, to deliver precise gradient approximations with negligible computational overhead. Applying SparseMixer to Switch Transformer on both pre-training and machine translation tasks, SparseMixer showcases considerable performance gain, accelerating training convergence up to 2 times.

The increasing scale of vision transformers (ViT) has made the efficient fine-tuning of these large models for specific needs a significant challenge in various applications. This issue originates from the computationally demanding matrix multiplications required during the backpropagation process through linear layers in ViT. In this paper, we tackle this problem by proposing a new Low-rank BackPropagation via Walsh-Hadamard Transformation (LBP-WHT) method. Intuitively, LBP-WHT projects the gradient into a low-rank space and carries out backpropagation. This approach substantially reduces the computation needed for adapting ViT, as matrix multiplication in the low-rank space is far less resource-intensive. We conduct extensive experiments with different models (ViT, hybrid convolution-ViT model) on multiple datasets to demonstrate the effectiveness of our method. For instance, when adapting an EfficientFormer-L1 model on CIFAR100, our LBP-WHT achieves 10.4% higher accuracy than the state-of-the-art baseline, while requiring 9 MFLOPs less computation. As the first work to accelerate ViT adaptation with low-rank backpropagation, our LBP-WHT method is complementary to many prior efforts and can be combined with them for better performance.