Lightweight proxy models enable rapid experimentation without repeatedly training frontier-scale systems, but their small kernels often leave modern accelerators underutilized. Conventional training compounds this inefficiency by scheduling the forward and backward passes as disjoint phases, so spare capacity in one cannot be filled by work from the other. We reinterpret the detachment mechanism of Forward-Forward (FF) as a scheduling primitive: given a local objective, detaching a block's output removes downstream gradient dependencies, making its backward pass ready when its forward pass finishes. ERASE launches each detached subgraph's backward pass early on a separate CUDA stream, overlapping it with subsequent forward work. Execution trace on a lightweight transformer demonstrates this overlap and its limit: a kernel that saturates the device leaves no capacity for concurrency. On a large-scale click-through-rate model, detaching six dense subarchitectures improves training throughput by up to $9.51\%$ while keeping normalized entropy close to the baseline.
Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from model-reality gap. On-device gradient computation could resolve this issue, and a handful of theoretical and experimental studies have proposed ways to achieve it. Yet a unifying theory identifying when a physical system can compute the gradient of its own performance has been missing. Here we develop such a unification, based on the adjoint method: we identify sufficient conditions under which the adjoint field required for formally exact gradients can be generated on the same hardware that performs the computation. Linear and nonlinear systems obey fundamentally different conditions: for linear systems damping or gain is admissible provided reciprocity is preserved. For nonlinear trajectory systems the sufficient conditions are reciprocity of the linearized system and the existence of a time-reversal mirror. Algorithmically, the nonlinear case requires infinitesimal nudging, whereas linear systems admit a finite-amplitude experiment. We recover Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. We further show that reciprocity is only the simplest instance of a more general intertwining condition, which extends exact on-device gradient computation to a class of non-Hermitian, non-reciprocal systems. Further generalizations include time-dependent parameters, Onsager-reciprocal dynamics and nonlinear, PT-symmetric Schrödinger equations. Our work provides a unified theoretical basis for formally exact physical learning algorithms and a template for constructing them across a range of physical systems.
Biologically plausible learning models aim to explain how neural circuits can implement effective learning under the constraints of real neurons. Although significant progress has been made, a major remaining challenge is that existing models often allow neurons or synapses to represent mixed-sign values, both positive and negative, in violation of a basic aspect of cortical circuitry -- Dale's constraint: biological neurons are either excitatory or inhibitory, but not both, and synapses cannot change sign. In this work, we address this discrepancy by introducing a biologically motivated neural architecture in which both neural activations and learning signals are represented by non-negative activity, and synapses have fixed sign, while still supporting backpropagation-like learning. Our approach uses two complementary interacting non-negative channels to represent positive and negative contributions, inspired by evidence of on-off representations in the brain. These channels are implemented through a simple neural circuit motif, which is repeated throughout the network in both bottom-up and top-down pathways. Combined with a local Hebbian learning rule, the resulting model propagates learning signals and updates weights using only local interactions between neurons. We show theoretically that our learning scheme can exactly recover the backpropagation update despite relying solely on non-negative error signals. Empirically, beyond satisfying stronger biological constraints, the on-off architecture learns efficient representations, yielding substantial gains over comparable vanilla networks on the Tiny ImageNet benchmark. These results demonstrate that effective learning can emerge from biologically plausible mechanisms without requiring mixed-sign signals, providing a step toward more realistic models of neural computation.
Group Relative Policy Optimization (GRPO) is a powerful reinforcement learning algorithm for aligning generative models with human preferences. While successful in large language models~\cite{shao2024deepseekmathpushinglimitsmathematical}, its extension to diffusion and flow matching models introduces a severe computational bottleneck: gradients must be back-propagated through the high-capacity DiT backbone at \emph{every} timestep of the sampling trajectory, making high-resolution text-to-image (T2I) training prohibitively expensive. Training-free DiT inference acceleration methods (e.g., $Δ$-DiT, ScalingCache) exploit the fact that DiT hidden states and velocity predictions vary \emph{smoothly and nearly linearly} along the trajectory. We ask whether the same linearity can reduce the backward-pass cost of DiT RL training, and answer affirmatively with \textbf{JAGG} (\textbf{J}acobian-\textbf{A}ggregated \textbf{G}roup \textbf{G}radient), which reduces full transformer backward passes from $W$ to $2$ per group of $W$ consecutive steps. JAGG approximates intermediate-step Jacobians via $t$-weighted interpolation of the endpoint Jacobians, then aggregates per-step upstream signals into two composite gradients applied through a single joint backward pass. We prove this interpolation is \emph{exact} when the velocity is linear in $(z,t)$, and a cosine-similarity routing rule (\texttt{jagg\_frac}) deploys JAGG only where the assumption holds. Experiments on T2I benchmarks show JAGG delivers $\sim$2$\times$ backward speedup with negligible quality degradation. The code for this work can be accessed through https://github.com/SchumiDing/JAGG.
The widespread adoption of high-level deep learning libraries, while accelerating model development, has increasingly abstracted away the internal mechanics of neural networks, creating a gap between practical usage and fundamental understanding. To address this, the paper presents a self-contained neural network framework implemented entirely from scratch -- without relying on automatic differentiation or pre-built deep learning modules. The implementation encompasses all essential components, including multi-layer architectures, diverse activation functions, regularization techniques, and state-of-the-art optimizers. Beyond serving as a pedagogical instrument that demystifies forward/backward propagation, gradient dynamics, and optimization landscapes, the framework demonstrates robust performance when applied to a multi-class classification task, successfully validating its correctness, numerical stability, and generalization across varied configurations. The extensible design and clean modularity further position it as a reliable baseline for educational purposes and future research exploration.
Backpropagation is the computational engine of deep learning, yet its mathematical structure is typically treated as a procedural traversal of computational graphs. We present a global operator theory of the \emph{F-adjoint} framework, which reformulates the layerwise backward recursion of an $L$-depth feedforward network into a single linear system $(I-\cB)\Xs=\bG$, where $\bG$ is a source vector. We prove that the global backward operator $\cB$ is strictly block upper-triangular and nilpotent of index at most $L$. This nilpotency guarantees the exact termination of the Neumann series solution after at most $L$ terms, revealing classical backpropagation to be mathematically equivalent to block back-substitution on an upper bidiagonal system. We formalise \emph{F-symmetry} -- the condition in which the backward pass perfectly mirrors the forward pass -- identifying orthogonal weight matrices as canonical examples. Through worked numerical examples, we demonstrate how this operator perspective exposes the single-path collapse of strictly feedforward networks and its breakdown in residual architectures. Finally, we leverage this compositional structure to rigorously derive the mechanics of residual networks (gradient highways) and transfer learning (gradient truncation). This framework elevates backpropagation from an algorithmic recipe to a global nilpotent-operator formulation.
Thermodynamic computing devices based on the Ising model show great promise for low-power AI inference and edge computing, but scalable methods for training large models for such hardware remain limited. Prior theory shows that the time-averaged behavior of high-temperature Gibbs-sampled Ising systems can implement feed-forward neural inference. We turn this theoretical correspondence into a scalable and purely backpropagation-based algorithm for training deep convolutional networks for thermodynamic inference on Ising machine hardware. Our image classification models achieve accuracies of 94.9% on CIFAR-10 and 76.0% on CIFAR-100 under binary Gibbs sampling. We then develop and experimentally validate a mathematical theory relating inference cost to accuracy and controlling autocorrelation times. Subsequently, we calculate asymptotic results showing that inference cost is bounded by a well-controlled tradeoff with performance and exhibit algorithms for computing optimal inference schedules. Finally, we discuss implications for hardware development and the future of high-temperature thermodynamic AI models.
Despite the success of deep learning, training deep networks in biologically plausible and hardware-efficient ways remains an open challenge. Feedback alignment (FA) methods address this by replacing backpropagation's symmetric backward weights with fixed random matrices, but their effectiveness depends critically on whether they can be accurately evaluated. The standard evaluation relies on two quantities: task accuracy and cosine similarity between the method's credit signal and the backpropagation gradient. We show that this reporting pair is insufficient by identifying two independent failure modes, both silent under current reporting: (1) measurement degeneracy, where the BP reference gradient collapses to the numerical floor in terminal-LayerNorm residual architectures, rendering cosine uninterpretable; and (2) aggregation collapse, where the aggregate cosine masks layerwise heterogeneity that concentrates credit at one end of the network. To address these limitations, we propose a diagnostic evaluation protocol based on three checks -- scale stability, reference validity, and depth utility -- together with per-layer rather than aggregate cosine reporting. Across multiple architectures and methods, the standard reporting pair gives no signal of failure in any audited case, while our protocol identifies all failures with wide calibration margins. The two failure modes are causally independent: a per-block scale penalty alleviates Mode 1 (residual scale explosion driving reference collapse) without affecting Mode 2 (cosine ranking that contradicts every functional metric we measured). Identifying these silent failures prevents researchers from building on non-functional credit assignment and provides actionable guidance for developing FA methods that genuinely train deep layers.