Backpropagation

This paper provides a self-contained proof of the rank of the pathlifting Jacobian of a DAG ReLU network by performing an induction on the network's number of hidden nodes. In fact, the induction is elementary, and the key recipe is to consider the skeleton matrix of the network, a sparse matrix encoding the network paths, and transform the representation of one of its hidden neurons into an output node. The proof relies on intermediate propositions which link the pathlifting, its Jacobian, the network parameters, and its skeleton matrix, which, on top of permitting to conclude on the rank of the pathlifting Jacobian, also provide a way to compute it without backpropagation and whose computation cost is super efficient in practice compare to usual backpropagation. The paper is provided with a Python module that implements the different propositions of the paper for feed forward networks and is used to experimentally quantifies the computational gain of computing the pathlifting Jacobian with the proposed theory.

Recurrent models provide a natural path to long-context modeling, yet models trained with backpropagation through time (BPTT) often fail beyond their training horizon. Classical analyses emphasize gradients that vanish or explode along temporal paths. However, dense per-token losses can still train a shared recurrent rule despite severe decay, showing that decay alone does not determine whether learning fails. We instead study state credit: the signal through which future losses reach earlier recurrent states before contributing to parameter updates. Accordingly, we intervene directly on state credit and propose Credit Stabilization through Time (CST). During backward propagation, CST locally rescales the state-credit signal to stabilize its norm without rotating the component being corrected, while leaving the forward computation unchanged. Because controlled synthetic tasks and real data exhibit different credit dynamics, we specialize CST to each regime. In both settings, CST improves performance beyond the training horizon, with gains observed at up to 128x the training length.

Full-parameter fine-tuning of large language models has substantial memory costs because backpropagation stores activations and gradients. Zeroth-order optimization avoids this by estimating update directions from loss evaluations, but existing methods require tuning a sensitive learning rate for each model and task. We propose the momentum $p$-dimensional subspace trust-region method (MpSub). At each iteration, MpSub searches within a $p$-dimensional subspace: one direction preserves historical momentum from the most recent accepted step, while the remaining directions explore via fresh random sampling. The subspace gradient is estimated by central differences, a trial step is computed from a linear trust-region model, and the trust-region radius adapts according to the agreement between predicted and observed loss reduction, eliminating the learning rate. For LLM fine-tuning, evaluations within an iteration share a minibatch, and directions are regenerated in place from seeds, using forward passes alone. For smooth deterministic objectives under unorthogonalized Gaussian directions, we bound the finite-difference error, quantify gradient energy captured by the subspace, and prove that $\lim_{k\to\infty} \|\nabla f(x_k)\|_2 = 0$ almost surely under a safeguarded radius update. Under a matched budget of 8,400 training-objective forward passes, we fine-tune OPT-125M and OPT-350M on CommitmentBank. With the same preset parameters at both model sizes, MpSub attains mean test accuracies of 0.673 and 0.690 over three seeds, matching tuned MeZO (0.685) without any learning-rate search.

Conventional neural networks learn predominantly through affine transformations followed by nonlinear activations, while elapsed time is often treated as an auxiliary feature or assumed to be uniformly sampled. This paper introduces Wave Function Backpropagation (WFB), a wave-parameterized learning formulation in which neural responses are represented by learnable amplitude, wavenumber, angular frequency, and phase. The formulation associates an observed state with its temporal interval Delta t through the phase of a differentiable spatiotemporal wave. We derive standard WFB gradients and a spatial-curvature correction based on the Laplacian of the wave response. WFB is instantiated in a deliberately feed-forward trajectory predictor to provide a controlled proof of concept; sequence learning is outside the scope of the present evaluation. With motion features, STD-WFB using real intervals reduces average displacement error (ADE) by 20.4% relative to the original FFN baseline. In a new position-only evaluation that removes temporal leakage through precomputed velocity and acceleration, real-interval WFB reduces ADE by 10.4% relative to the original FFN and remains competitive with parameter-matched ReLU controls, obtaining 2.1% lower mean ADE than the matched FFN with explicit Delta t. Shuffled-interval WFB attains the lowest mean ADE, indicating that the present evidence supports the effectiveness of the wave representation but does not attribute the gain to interval alignment. These results establish WFB as a viable structured feed-forward learning formulation and define a clear basis for subsequent architectural studies.

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.

Representational similarity analysis (RSA) is increasingly used to ask which learning rules give convolutional networks brain-like representations. Because biologically plausible rules such as feedback alignment, predictive coding and STDP do not scale, studies that include them train small networks on small images (typically 32x32 CIFAR) and then compare them to brain responses modeled at much higher resolution. We find that a common result in this setting, that untrained or locally trained networks rival or beat backpropagation at early visual cortex, depends strongly on the resolution at which the network is evaluated. The V1 gap between an untrained network and a backpropagation-trained one widens from -0.001 +/- 0.007 at the 32px training resolution to +0.044 +/- 0.006 at 224px, growing monotonically across six resolutions (n=5 seeds). It holds in human fMRI and, directionally, in single-seed macaque electrophysiology, along the training trajectory, and for an ImageNet ResNet-50 and a Swin-Tiny transformer trained at 224px. Four candidate mechanisms are tested and none accounts for it: train/eval resolution matching, low-level Gabor and pixel structure, the normalization state of the untrained baseline, and convergence of the pooled descriptor toward a global brightness statistic; three are excluded by interventions holding the convolutional weights bit-identical. A fifth experiment locates the effect: capping image detail at the training resolution while letting the pooled positions grow 12-fold removes about 90% of it, so the dependence is carried by image detail rather than by pooling. Separately, a single scalar luminance value per image reaches rho = 0.075 against the V1 RDM, essentially matching the untrained network's 0.076, which bounds what this style of comparison can resolve. The one learning effect that holds across resolution is backprop above untrained, at LOC.

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.

Ensuring safety and policy compliance in text-to-image diffusion models remains a critical challenge, as benign or adversarial prompts can often elicit prohibited content, e.g. nudity and protected intellectual property. While training-based unlearning methods are effective, they are computationally expensive and prone to catastrophic interference with general capabilities. Conversely, existing test-time defenses are primarily prompt-centric, relying on modifying textual descriptions only, and overlook the visual signals for detection. In this paper, we propose to leverage the intermediate clean image estimated during the generation process and employ a sparse margin objective to detect prohibited concepts. When a violation is detected, we immediately intervene by optimizing a structured low-rank residual in the text-conditioning space via truncated backpropagation. This design allows weight-preserving detection, keeps non-violating inference latency nearly unchanged as the maximum budget increases, and offers flexibility in safety performance via test-time scaling. Extensive experiments on Stable Diffusion v1.4 and v3.5 across nudity removal, IP protection, and style erasure demonstrate superior performance across suppression, fidelity and preservation compared to prior weight-preserving baselines, providing a scalable and flexible solution for safe generative deployment.

Parameter-efficient post-training reduces the number of trainable parameters, but still requires repeated end-to-end backpropagation through the frozen backbone. Every adaptation step therefore needs backward-capable hardware and must store or recompute activations. We ask whether this repeated backward chain can be replaced by a one-time calibration. We introduce Local Credit Assignment (LoCA), a two-stage method for small-shift adaptation. One probe backward pass fits a low-rank map at each transformer block from the final prediction error to a local hidden-state correction. LoCA then reuses these maps to form blockwise regression targets from forward activations and fits low-rank adapters with closed-form ridge solves. No further backbone backward pass is required. We evaluate LoCA on five discriminative benchmarks with Qwen2.5 models from 0.5B to 14B. In 16 of 25 reported task--scale comparisons, LoCA yields lower evaluation cross-entropy than the corresponding LoRA run. Its measured full-run GPU peak, including calibration, is 26--29% lower than LoRA's. After calibration, its CPU steady-state memory is 36--52% lower and its per-pass time is 43--48% lower. A shared scale-normalized candidate set is reused across all tested Qwen2.5 sizes and on SmolLM2-1.7B. LoCA thus amortizes global credit assignment into one calibration and enables later forward-only tuning when repeated backpropagation is impractical. The code associated with this paper is available \href{https://github.com/Xia12121/LoCA}{here}.

Long-horizon video editing agents receive final-product feedback only after many interdependent decisions. Yet editing quality is subjective, admits multiple valid solutions, and is not meaningfully calibrated across heterogeneous requests, making a global scalar objective both ambiguous and temporally uninformative. Our key observation is that fixing the request, materials, and production constraints converts this subjective objective into an ordinal comparison among directly comparable alternatives. We introduce Group-Relative Preference Backpropagation (GRPB), which transforms same-task rankings into zero-sum advantages and redistributes them as bounded credit over semantic editing segments. A lagged allocator and guarded transmission prevent current judgments or unreliable estimates from directly shaping the same rollout group. We manually construct a project-disjoint, horizon-stratified suite of realistic editing tasks for training and controlled evaluation. Across matched baselines, credit interventions, external benchmarking, and blinded human evaluation, GRPB improves both editing behavior and rendered products. The resulting 9B Crayotter model surpasses several proprietary systems on AgenticVBench, supporting task-local preference reduction as a practical approach to learning from subjective, delayed outcomes. Code and all supporting materials are publicly available at https://github.com/idwts/Crayotter.

A Noise-modulated Neural Network (NNN) learns and infers only in the presence of noise, treating noise as a computational resource rather than a disturbance. The noise lets it learn efficiently by backpropagation while transmitting spike-like signals, but backpropagation needs a reverse path through transposed weights, the weight transport problem, which undermines biological and neuromorphic plausibility. Forward-only alternatives typically substitute a different objective or fixed random feedback, sacrificing stability and accuracy. We show that backpropagation itself can be reconstructed in the NNN from forward-pass statistics alone: a weight mirror estimates each weight matrix from the covariance between a previous-layer unit's output and the next-layer unit's input, and combining it with local differential estimation inside the units propagates the output error recursively along the computational graph, with no transposed-weight readout and no backward data path. The resulting gradient is empirically near-unbiased, and with local per-weight Adam updates it matches the final accuracy of backpropagation on simple regression tasks. With uniformly distributed noise, the local operations reduce to polynomials and comparators, making the whole system, learning rule included, well suited to digital circuits. Thus, in the NNN, noise is a resource not only for inference but also for reconstructing backpropagation.

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding governing physical laws into deep neural networks. However, their reliance on computationally expensive gradient-based optimization and deep architectures often results in slow training, high computational cost, and limited scalability. In this work, we propose a novel physics-informed broad learning system (PI-BLS), the first physics-informed learning framework based on broad RdNNs. The proposed formulation embeds the governing differential operator and the associated initial and boundary constraints directly into a linear output-layer optimization problem, thereby replacing nonlinear gradient-based training with a deterministic least-squares solution obtained via the pseudoinverse. Consequently, the entire learning process is reduced to a single linear optimization stage while preserving the underlying physical constraints. As a result, PI-BLS offers an efficient learning paradigm for a physics-informed learning framework for solving PDEs that eliminates iterative backpropagation while preserving the underlying physical constraints. Experimental results on representative forward PDE benchmarks demonstrate that PI-BLS achieves competitive and often superior performance with reduced training time and model parameters compared with conventional PINNs.

We consider the problem of learning high-dimensional semi-global feedback controllers under hard safety constraints enforced by control barrier functions (CBFs). Incorporating CBFs into end-to-end policy training requires embedding a quadratic-program-based safety filter as an optimization layer, but computational and differentiation bottlenecks have largely restricted prior approaches to low-dimensional systems, typically with at most 16 state dimensions. We address this limitation by combining operator splitting with the recently developed Jacobian-Free Backpropagation (JFB) method to enable scalable end-to-end training while preserving hard safety guarantees through the CBF safety filter. We justify this training methodology theoretically using nonsmooth analysis techniques and demonstrate its effectiveness on high-dimensional multi-agent nonlinear control problems with state and control dimensions up to 1200 and 400, respectively.

In the \emph{latent posterior model} of transformer behavior, the next-token distribution arises from a posterior over latent predictive models conditioned on the context, mixed to generate continuations. We exploit this model in settings where it is exact, namely Bayes-filtered transformers (BFTs) meta-learned on sequences from a hierarchical prior, to introduce \textbf{Posterior Prefix Tuning (PPT)}, a new method for \emph{eliciting} behavior from a transformer: given a utility function on continuations, find a prompt under which the transformer generates continuations of high expected utility. For a BFT, the elicitation objective factors through the latent posterior, and the gradient of this objective can be estimated from samples of the prior alone. PPT optimizes the parameters of a distribution over hard prompts: it draws prior samples once from the BFT via predictive Monte Carlo (PMC), then estimates the gradient by importance sampling against them. The optimization performs no transformer forward passes and no backpropagation through the transformer, and the prior samples are utility-independent, so a single set of samples drives elicitation against any number of utilities at negligible marginal cost. We validate PPT on Beta--Bernoulli and reinforced urn BFTs across three utility families (reverse cross-entropy, frequency matching, Dyck validity).

Direct feedback alignment (DFA) trains hidden layers with fixed random projections of the output error, avoiding the transposed-weight backward pass of backpropagation (BP). We study a failure mode of DFA training that is distinct from feedback quality: the local weight update is calculated by an outer product, so anisotropy can enter through either its presynaptic-activity factor or its local-error factor. Our analyses with controlled synthetic regimes isolate the first failure mode and show an approximately 40-percentage-point activity-conditioning gain when high-variance directions contain task-irrelevant nuisance. Three clean confirmations isolate a different regime: error conditioning improves raw DFA by 1.77--7.53 percentage points, and combining independently selected activity and error factors adds 0.40--0.90 points over activity conditioning. The signs hold for tanh/one-vs-rest MNIST and preregistered Fashion-MNIST, and replicate on eight fresh seeds in a ReLU/softmax MNIST model. This factorization yields a symmetric block-local family of normalized DFA (nDFA): activity nDFA right-preconditions by an inverse activity second moment, error nDFA left-preconditions by an inverse local-error second moment, and K-nDFA applies both factors with separately tuned damping. A linearized post-alignment calculation gives an exact input-side spectral identity and a Kronecker-factor motivation for the two-sided rule, whereas norm matching rules out a scalar step-size explanation. The error factor is fragile when under-damped, BatchNorm is a strong activity-side alternative, and convnet gains remain partial. We therefore frame conditioned DFA as a factor-level study of when local outer-product rules fail, not as a general replacement for BP or a solution to all-layer convolutional credit assignment.

On-device model adaptation is essential to enable lifelong personalization on resource-constrained hardware, but compute, power, and memory limitations of such devices make end-to-end backpropagation impractical for modern deep neural networks. This work proposes a heterogeneous adaptation pipeline that repurposes a commercial edge AI inference accelerator, Hailo-8L, for frozen-backbone feature extraction during on-device training. The computational graph is partitioned so that the pre-trained backbone is quantized to INT8 and run on the accelerator, while only a lightweight FP32 classification head is fine-tuned on the host CPU, enabling frequent, energy-efficient in-field updates with most weights remaining fixed. Across multiple architectures and datasets, this pipeline achieves up to 15.4x faster wall-clock training time compared to a Raspberry Pi 5 CPU baseline, offers competitive throughput in favorable settings, and consistently reduces energy per sample. Post-training quantization restoration is shown to be crucial for preserving the quality of accelerator-generated features and mitigating accuracy loss in quantization-sensitive architectures. Overall, the results demonstrate a practical approach to efficient on-device adaptation using inference-oriented edge accelerators. The implementation is available at https://github.com/MatPiech/accelerator-training.

We introduce CMP (Cognitive Memory Primitive), an architecture that represents inputs as sparse relational codes, stores them in a two-tier competitive memory, and learns entirely through local, gradient-free updates, with no backpropagation anywhere in the network. We use this architecture to test a specific hypothesis: that catastrophic forgetting, usually treated as a training-time defect to be patched with replay or regularization, is instead a structural consequence of how backpropagation assigns credit and that a learning rule that is local and sparse by construction should resist it without a patch. On a controlled domain-incremental protocol across 15 text domains, three-seed replicated, CMP's backward transfer is 15-19x better than a matched-size Transformer trained with online EWC, and the result survives a domain-order control (reported as a range, +0.24 to +0.44, rather than a single figure). We report this alongside a real, substantial accuracy gap versus the Transformer baseline, a null result on a recognized vision benchmark, and a diagnosed, unresolved failure attempting to combine this architecture with a separate mechanism that improves raw accuracy, disclosed because an honest negative result is more useful than an omitted one. The central claim is narrow and falsifiable: local, sparse, non-backpropagation learning measurably resists catastrophic forgetting better than backpropagation with its standard fix, under conditions we state precisely.

Many fine-grained recognition tasks contain hierarchical labels such as order, family and species. Although this supervision should be beneficial, jointly optimising all levels often leads to unstable training because coarse and fine classifiers impose inconsistent gradients on the shared backbone. This hierarchical gradient conflict prevents the model from learning a coherent coarse-to-fine representation. In this paper, we propose FlexiGrad, a simple and parameter-free method that regulates gradient interactions during backpropagation. FlexiGrad removes only the harmful conflicting component when tasks disagree and reinforces the shared direction when they partially agree through a smooth hierarchy-aware weighting function. This produces stable optimisation and preserves both global structure and fine-grained discriminative cues. FlexiGrad integrates into existing architectures without modification while improves multi-granularity accuracy on CUB-200-2011, FGVC-Aircraft and Stanford Cars. The code will be available at PRIS-CV/FlexiGrad.

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 makes training deep networks memory intensive because it must store intermediate activations. Forward-mode methods avoid this cost, but their gradient estimates become increasingly noisy as the number of trained parameters grows. We introduce Split Forward Gradient (Split-FG), which splits a network at an intermediate representation: it computes the output head gradient exactly and estimates only the trunk gradient with a Jacobian--vector product. This reduces estimator variance and requires no backward pass through the trunk, while retaining an Adam-style convergence guarantee. Our experiments reveal an important practical failure mode. On WikiText-103, naive forward-gradient training of the trunk performs worse than leaving a randomly initialized trunk frozen, likely because Adam updates every noisy, under-determined trunk coordinate too aggressively. Simply using a much smaller learning rate for the trunk reverses this result: a $16$M-parameter GPT-2-style model reaches validation perplexity $387$, compared with $668$ for the frozen-trunk control and $2{,}885$ for a matched pure forward-gradient baseline (backpropagation reaches $150$). Split-FG also produces the strongest backprop-free results on our tabular benchmarks and reaches $60.5\%$ on CIFAR-10 and $35.2\%$ on CIFAR-100 with a heavy-head design. It reduces peak memory by up to $35\%$ relative to matched backpropagation, although the performance gap widens as the forward-mode trunk grows.

Exhaustive site-by-site interventions on a neural network's computational graph -- activation-patching sweeps, circuit-discovery searches, systematic ablation studies -- mutate the graph at every candidate site, and their cost is dominated by recomputation after each mutation. On a reactive graph engine whose invalidation provably touches exactly the downstream cone of a mutated node, we give a complete cost accounting for such workloads. First, the aggregate speedup of an exhaustive sweep over independent full recomputations is not a universal constant: if per-layer weight varies regularly with depth at Karamata index q, the ratio converges to (q+2)/(q+1) when weight concentrates near the output and to q+2 near the input, recovering 2 only in the depth-uniform case; a wall-clock corollary predicts a ceiling of about 1.79, below 2, until interpreter overhead is compiled away. Second, we prove the exact cost of a sequence of persistent mutations, never undone between insertions: the interleaved cost exceeds the isolated sum by an exact overcount summed over comparable site pairs, with closed-form extremes over insertion orders, while batched application is order-independent and sub-additive, costing exactly the union of the sites' cones plus the fresh nodes. Third, we prove the exact mirror of forward locality for the backward pass, showing it collapses the aggregate speedup to 1 under backpropagation on architectures without long skip connections. Every identity is validated on NeuroDSL, a reactive graph engine in Julia: measured sweep ratios converge to the predicted limits under four cost profiles; the training-mode ratio collapses to 1 at the predicted rate; and all 18 per-graft sequential costs and the batched total match the closed forms at zero tolerance across three insertion orders.

Feature-attribution methods are central to explainable artificial intelligence. Their assumptions are expressed in several mathematical languages: cooperative-game values, path integrals, gradient operators, perturbation distributions, and backpropagation rules. This survey proposes a common framework for local additive feature attribution. It organizes Shapley, path-based, gradient/backpropagation, perturbation, and CAM-style methods around five specification choices: value function, reference, path, perturbation distribution, and conservation rule. It then compares these methods through an axiom-by-method matrix and links common failure modes, including baseline sensitivity, off-manifold perturbations, sanity-check failures, adversarial manipulation, and method disagreement, to the assumptions that produce them. Finally, the survey proposes a ten-item reporting checklist for studies that use local additive attributions. The central message is that attribution results are meaningful only relative to the mathematical assumptions under which they are defined, and that those assumptions should be reported.

Predictive Coding (PC) offers a biologically motivated alternative to backpropagation via local weight updates, yet routing error between layers still relies on an autograd Jacobian-transpose ($J^\top$) product - the last non-local operation in PC. We show that this dependency is largely avoidable. For any layer $f(x)=\mathrm{Act}(\mathrm{Norm}(L(x)))$ with frozen normalization statistics, the exact $J^\top$ factors into three locally available terms, $J^\top v = L^\top(s \odot σ'(z) \odot v)$, where $σ'$ is the activation derivative, $z$ is the pre-activation, and $s=γ/σ_{\mathrm{run}}$ is the normalization gain. Prior weight-feedback methods omitted both corrections; restoring them closes the transport gap for this layer class. Locality here holds up to three assumptions, which we state upfront: weight symmetry ($L^\top$ mirrors the forward operator, as assumed by all PC), a soft spectral-norm control that is not synapse-local, and a nearest-neighbour approximation for MaxPool. Substituting the identity into PC yields WF-Act-PC, which removes the autograd backward pass from error transport. On CIFAR-10/100 (50 epochs, 5 seeds), WF-Act-PC is the only PC method whose accuracy improves with depth, surpassing iPC - the strongest classical PC baseline - by 2.7-22.3 pp on CIFAR-10. With both methods tuned per architecture, it matches or exceeds a comparably-tuned backpropagation baseline on the deeper CIFAR-10 architectures (VGG-9: 93.57% vs. 92.43%; ResNet-18: 92.76% vs. 91.54%) and on the harder Tiny-ImageNet benchmark, while trailing tuned BP on the deeper CIFAR-100 VGG cells. Our WF-Act-PC implementation is publicly available at https://github.com/jlshen025/pcax

Interpreting a neural network requires understanding what its internal features extract from a particular input. Feature inversion seeks to express a selected feature in the input domain, but canonical iterative methods search for an input whose re-encoded representation matches the target. Because many inputs can satisfy this constraint, target matching alone does not specify the inverse associated with the sample that generated the feature. We formulate source-grounded feature inversion by conditioning the inverse on the source-local network geometry at the target-generating input. At each boundary of the computational DAG, backpropagation provides the correct reverse dependencies but transports an adjoint signal rather than an upstream-state estimate. We locally repair this signal with a closed-form matrix Wiener map from a mean-seed VJP to the upstream state, followed by a second Wiener map for the JVP forward-consistency residual, and compose the repaired states through the same DAG in one finite reverse pass. One calibrated zero-intercept map family supports new inputs, depths, channels, and channel groups across diverse CNN and Transformer architectures, tensor components, and visual distributions without query-specific optimisation. Matched target and source controls verify that each inverse depends on the selected feature and the local operators of the sample being explained, rather than a target-independent image template. Prediction-conditioned feature atlases align these visualisations with independent interventions on the corresponding internal features. Together, source-grounded feature inversion opens the model's hidden feature hierarchy to inspection at the level of individual layers and channels, linking what the network extracts from an input to the internal evidence that shapes its decision.

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.

As the scale of large pre-trained models continues to grow, fine-tuning them under limited memory budgets has become increasingly challenging. Low-Rank Adaptation (LoRA), currently one of the most widely adopted parameter-efficient fine-tuning (PEFT) methods, mitigates this challenge by optimizing only low-rank adaptation matrices, thereby greatly reducing the number of trainable parameters. With the parameter overhead substantially reduced, the activations retained for backpropagation have emerged as the primary remaining memory bottleneck during LoRA fine-tuning. To address this, we propose CARE-LoRA, a data-aware Compressed Activation REconstruction framework. By exploiting the inherent projection structure of LoRA, CARE-LoRA replaces the full input activation with the low-rank compressed activation naturally produced by the LoRA branch. It further computes a lightweight reconstruction matrix during the forward pass with negligible additional computation cost, which is used during backpropagation to reconstruct the gradient signal, thereby keeping LoRA matrices fully trainable. Extensive experiments across diverse models and downstream tasks demonstrate that, while substantially reducing the overall memory footprint, CARE-LoRA achieves competitive or even superior performance compared with standard LoRA and representative LoRA variants. Our code is publicly available at https://github.com/fishandyu/CARE-LoRA .

Backpropagation (BP) dominates deep learning training, but its reliance on gradients brings inherent troubles -- vanishing and exploding gradients. The pursuit of gradient-free methods has long been a goal in the field of artificial intelligence. This paper shows that indeed the simplest Monte Carlo algorithm implemented on a single GPU -- randomly mutate a parameter, keep it if the loss decreases, otherwise retry -- can practically train deep networks. This gradient-free method does not even need common techniques such as batch normalization or residual connections to directly train sufficiently deep networks. More remarkably, its flexibility extends to several nontrivial scenarios: it enables pure pruning training, supports discrete weights, accommodates unconventional transfer functions such as Gaussian, and reveals the substantial redundancy of deep networks. We have demonstrated its feasibility on deep networks with more than 20 layers, single-hidden-layer wide networks with up to 16,384 hidden neurons, and even a simple Transformer architecture trained on both image classification (MNIST) and character-level language modeling (Tiny Shakespeare). This simple gradient-free method may offer a complementary perspective for understanding the self-organization and learning mechanisms of neural networks, and also provides an alternative route for building physically inspired deep learning systems.

The architecture of deep feedforward neural networks is ubiquitous in deep learning, either as a whole system or as a subnetwork of other architectures, and thus its mechanism is a key ingredient of the black box of neural networks. On the basis of the simplest two-layer ReLU network, this paper systematically studies the mechanism of deep feedforward ReLU networks with multiple hidden layers and successfully explains the training solution obtained by the back-propagation algorithm. The concept of a path, especially in terms of the relationships between paths, plays a central role in uncovering the mystery of the black box. It is shown that a unit of a deep ReLU network can form a piecewise linear manifold to divide the input space, instead of a hyperplane of the two-layer case. How to efficiently use the hidden-layer units to produce both linear functions and partitions of the input space is also a central problem. The principles of a two-layer ReLU network can be generalized to the deeper case to a large extent, such as multiple strict partial orders and continuity restriction. The combination of the basic and simple principles proposed can yield complicated instantiations including the training solutions, and in this sense the black box of deep feedforward ReLU networks is revealed.

We present K-ABENA (K-Adaptive Backpropagation with Error-based N-exclusion Algorithm), a selective gradient computation framework that reduces per-iteration training cost by excluding a fraction of low-loss ("minor") observations from the backward pass. Its canonical form (v3) combines a defensive-mixture sampling design over the minor set with Horvitz-Thompson inverse-probability reweighting, yielding a design-unbiased Horvitz-Thompson gradient estimator (Lemma 2) and whose self-normalized practical variant carries a bias of order O(1/m) with an explicit constant (Lemma 3). We prove an O(1/sqrt(T)) non-convex convergence guarantee for SGD under the estimator, with an additive term that quantifies the residual bias (Theorem 1). We further prove that uncompensated loss-based selection - a family that includes OHEM, SBP, and the two earlier K-ABENA variants - admits no stationary point at any minimizer where its selection bias is bounded away from zero (Proposition 2), and we quantify this failure empirically: at 0.17% class imbalance, uncompensated variants reach test AUC 0.53-0.62 versus 0.9998 for full-batch SGD, while the compensated estimator attains 0.9991 at identical 28.4% compute savings. On real datasets (Breast Cancer, Digits, Wine, Diabetes) the compensated estimator is statistically indistinguishable from full-batch SGD (paired permutation tests, p >= 0.5; Section 7) while saving 28-54% of per-epoch gradient computation. A biased "regularized mode" (the earlier half-domain variant) is retained as an option with a proven exact bias decomposition (Lemma 5) and quantified contraindications: it collapses to 0.386 accuracy under 40% label noise (baseline: 0.832) and to 0.53 AUC under extreme imbalance. Every advantage and every limitation reported in this paper is either proved or measured; all experiments are CPU-scale (NumPy/scikit-learn) and their scope is stated explicitly.

Vision-Language-Action (VLA) policies that execute fixed-length action chunks can exhibit multimodal bifurcation: a cross-chunk inconsistency in which adjacent chunks generated from independent Gaussian latents can converge to incompatible trajectory modes, producing abrupt discontinuities at chunk boundaries. Existing remedies either require backpropagation through the policy at each denoising step, rely on rejection sampling, or require retraining, each trading computational cost or task reliability for smoother transitions. We propose SEAM (Smooth Execution of Action-Chunked Motion), a training-free inference-time method for flow matching VLAs. SEAM exploits a simple synchronous-execution insight: after the robot consumes the executed prefix, the previous chunk's unexecuted tail is already available as an analytic consistency reference. Its core mechanism, Velocity-guided Loss Steering (VLS), derives a time-dependent target from this tail and applies a closed-form correction after each Euler step without backpropagating through the policy network. On LIBERO-10 with pi_0.5, SEAM reduces boundary jerk by 28%, reduces chunk transition discontinuity by 27%, preserves baseline-level task success, and keeps denoising-loop cost near the unguided baseline.

Test-time adaptation (TTA) mitigates domain shifts by using incoming test data to update a model on the fly. The majority of TTA methods require resource-intensive backpropagation (BP) for model updates, particularly demanding large memory sizes, which makes it infeasible to deploy them on resource-limited devices (e.g., edge devices). To address this issue, we integrate two different techniques, zeroth-order optimization (ZOO) and model merging, under the recently established cross-device collaborative TTA (CDC-TTA) framework, where the system is composed of a mixture of resource-abundant and resource-limited devices, and the model information (e.g., model weights obtained on each device) is shared across the devices. Our method is executable on resource-limited devices by introducing ZOO, which requires only forward processing and bypasses the resource-intensive BP optimization. Concurrently, to mitigate the high-dimensional optimization difficulty caused by the side effect of ZOO, we incorporate model merging of the shared multiple models and set the merge coefficients as the optimization objective, which successfully reduces the optimization dimension. In addition, to enhance the synergistic combination of ZOO and model merging, we propose a unique preprocessing strategy that trims intra-model non-influential weights and reduces the inter-model information redundancy. We empirically confirmed the effectiveness of our method using common corruption and style-transferred image benchmarks.

Deep learning (DL) methods dominate thyroid nodule segmentation in ultrasound (US) images, achieving high Dice scores but at the cost of millions of parameters, GPU-dependent training via backpropagation, and limited mathematical tractability. These limitations impede deployment in resource-constrained environments. In this paper, we propose MedSaab-US, a backpropagation-free segmentation framework grounded in the Green Learning paradigm. MedSaab-US extracts multi-scale spatial-frequency features by combining multi-level Discrete Wavelet Transform (DWT) with multi-scale channel-wise Saab (Subspace Approximation with Adjusted Bias) transforms at patch sizes of 5 x 5, 11 x 11, and 21 x 21 pixels. Label-Assisted Greedy (LAG) feature selection retains the most discriminative features, which are fed to an XGBoost classifier for pixel-wise prediction. The Saab transform parameters are determined analytically from data statistics, while XGBoost employs iterative greedy tree construction without requiring backpropagation. Evaluated on the TN3K dataset (2,879 training and 614 test images), MedSaab-US achieves a mean Dice coefficient of 0.4784 +/- 0.2190, precision of 0.5768, and recall of 0.5604, with a model footprint under 500K parameters and CPU-only inference in approximately 0.3 seconds per image. We present this result as an exploratory non-DL baseline for thyroid ultrasound segmentation and analyze the specific challenges posed by isoechoic nodules. An ablation study further quantifies the contribution of each pipeline component, including separate evaluations of LAG feature selection and training-set size.

Biological neural circuits obey Dale's principle: each neuron's synapses are uniformly excitatory or inhibitory. Artificial networks that respect this constraint must coordinate separate excitatory and inhibitory populations, fundamentally changing how credit is assigned during learning. Several biologically plausible learning rules avoid backpropagation's weight transport requirement, but it has been difficult to achieve strong performance under Dale's principle beyond MNIST. Error Diffusion (ED) was originally proposed in a dual-stream excitatory/inhibitory architecture, where learning is driven by routing global error signals to all layers without transporting transposed forward weights or relying on random feedback matrices. Whether such a rule can scale under Dale's principle across both supervised classification and reinforcement learning remains unknown. Here, we introduce modulo error routing to extend Error Diffusion beyond binary classification, and show that a dual-stream excitatory/inhibitory architecture trained with this method achieves 96.7% on MNIST and establishes a 61.7% baseline on CIFAR-10, demonstrating that representation learning is possible even when strictly enforcing Dale's principle. For the classification setting, we introduce three domain-specific innovations: layer-specific sigmoid widths, batch-centered class error signals, and asymmetric initialization, and ablation analysis reveals that their relative importance reverses between MNIST and CIFAR-10, exposing task-dependent credit-assignment bottlenecks invisible to single-benchmark evaluation. In reinforcement learning, we integrate ED with Proximal Policy Optimization (PPO) and evaluate it on continuous-control tasks in Google Brax and on Craftax, an open-ended exploration task. We show that ED-PPO achieves competitive performance relative to Direct Feedback Alignment, a backpropagation-free baseline.

We present a comparative study of CUDA optimization strategies applied to forward and backward propagation in a shallow neural network. Three stacked optimizations are evaluated: (1) tiled shared memory with bank-conflict elimination via +1-column padding, (2) pre-transposed weight matrices for coalesced global memory access, and (3) a fused MatMul+ReLU kernel that eliminates intermediate global-memory round-trips. Experiments on an NVIDIA Tesla T4 (CUDA 13.0) across three dataset sizes show that the fully optimized implementation achieves a 1.41x speedup over the baseline CUDA version on the large dataset (25,600 samples), reducing execution time from 21.0s to 14.8s. Results are compared against a sequential CPU baseline and an OpenMP parallel implementation, demonstrating the effectiveness of memory-access optimization in GPU-accelerated deep learning primitives.

Modern machine learning applications employ deep neural networks training with the error backpropagation algorithm. Although this algorithm is very effective, it lacks biological realism. For example, backpropagation requires symmetric connectivity, and a separate neural processing channel for error signals. Prior works have therefore proposed a number of more realistic alternatives for error backpropagation. However, most of them still suffer from demanding preassumptions that may be not fulfilled in the real brain, for example, they often still require either symmetric connectivity or two separate processing channels, and often require also special mathematical operations like subtractions or function inversions. Here I propose supervised counterstream learning in deep associative networks as a simpler approach that requires only recognition of errors during training, and then backpropagates correcting target activity through the same activity channel as used for forward propagation. For this, two activity waves are initiated at the same time in input and output layers and then traveling in opposite directions to meet in one of the hidden layers. By employing simple local Hebbian-type learning rules, the corresponding activity pattern sequences get linked bidirectionally, thereby decreasing error rates over time. Despite its simplicity and an incomplete hyperparameter optimzation, a high high test accuracy is achieved on the (binarized) MNIST data set that is comparable to more demanding architectures.

Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency. Identifying the system Hamiltonian from noisy observations of state variables is a challenging task. For simulations to faithfully reflect the long-term behavior of Hamiltonian systems, especially energy conservation, it is essential to use symplectic integrators, which preserve the system's geometric structure. This fidelity comes at a cost: implicit symplectic integrators are more computationally intensive and make backpropagation through the ODE solver non-trivial. However, by leveraging the fact that symplectic discretizations of the adjoint system yield the same sensitivities associated by backpropagation, we obtain an efficient method of training the Neural Network parameters. In our work, we explore this alternate method of HNN training under noisy observation of trajectories with our HNN model based on an implicit symplectic integrator. Computationally, a predictor-corrector based ODE solver and fixed point iteration help to mitigate the computational cost of the implicit timestepping, resulting in more efficient generation of gradient updates. We showcase the numerical advantage, in experiments, in system identification and energy preservation on a range of non-separable, chaotic systems and the efficient computation and memory complexity of our method. We also observe that the post-processing of the learned Hamiltonian using backward error analysis yields a modified Hamiltonian that is a more accurate approximation of the true Hamiltonian without the need to use more accurate discretizations of the flow map.

The rapid deployment of Vision-Language Models (VLMs) in dynamic environments necessitates the ability to learn continuously without forgetting. However, traditional continual learning (CL) settings often rely on white-box paradigms, which is increasingly invalidated by the shift toward cloud-hosted models. In this paper, we introduce Black-CL, a more realistic benchmark for VLMs that enforces three primary real-world challenges: weight and architecture inaccessibility, constrained computation, and task-agnostic inference. The learner can query only output embeddings or logits, with no gradient flow through or structural modification of the backbone. Current CL methodologies, which rely on backbone backpropagation or complex parameter expansion, are fundamentally incompatible with these constraints. Under this setting, we propose BETA, a simple yet effective baseline built on the key insight that solely optimizing textual prototypes can navigate the complexities of CL. BETA integrates three core components: Semantic Projection Accumulation (SPA) for incremental knowledge acquisition, Latent Distribution Replay (LDR) for anchoring the embedding space against catastrophic forgetting, and Test-Time Prototype Adaptation (TTPA) for dynamic, instance-aware boundary refinement. Extensive experiments across ten diverse datasets and various backbones demonstrate that BETA significantly outperforms existing black-box tuners. Remarkably, with only 0.05 M trainable parameters, a 180--3000$\times$ reduction compared to competitive methods, BETA achieves performance on par with or even exceeding white-box CL methods. We believe Black-CL and BETA provide a foundational framework for future advancements in continual learning and accelerates the transition of continual learning from academia to real-world systems.

Model-free reinforcement learning algorithms such as Proximal Policy Optimization (PPO) treat the environment as a black box, estimating policy gradients from sampled rewards; this process demands millions of interactions and relies on high-variance advantage estimates. When environment dynamics are differentiable, the return is an end-to-end differentiable function of the policy parameters, enabling exact gradient computation via backpropagation through simulation. We term this approach Analytic Policy Gradients (APG) and evaluate it against PPO on four continuous control tasks of increasing dynamical complexity: a one-dimensional point-mass target-reaching task, a 2D point-mass navigation task with obstacle avoidance, a 2D rigid-body T-block pushing task, and a 7-DOF Franka FR3 end-effector reaching task. Both algorithms share identical model architectures, observation normalization, and optimizer settings. To decouple sample efficiency from compute efficiency, we design a multi-axis evaluation protocol that records performance against environment steps and gradient steps. We report a segmented backpropagation scheme with MC and critic-based bootstrap modes that mitigates gradient degradation on long-horizon tasks, and present ablations over segment length and bootstrap strategy.

Modern deep neural network architectures are trained via backpropagation, which requires errors to be sequentially propagated through all layers before parameters can be updated. This introduces two limitations: locking, where layer-wise updates are strictly interdependent and cannot proceed in parallel, and the weight transport problem, which requires symmetric forward and backward pathways for exact gradient computation. These constraints restrict parallelism, increase memory and communication overhead, and pose challenges for scalable learning. In this work, we propose Hierarchical Block-Local Learning (HBLL), a framework that decomposes deep neural networks into hierarchically linked blocks trained using local learning objectives derived from variational principles, eliminating the need for full end-to-end backpropagation while maintaining effective information propagation across the network. HBLL is the first algorithm that is able to train deep neural networks in $\mathcal{O}(\log N)$ parallel time complexity, where $N$ is the number of network layers. We show that HBLL implicitly defines a family of subnetworks corresponding to different hierarchical paths, enabling flexible inference with different effective numbers of layers. We evaluate HBLL on a set of challenging vision and language modeling tasks, achieving competitive performance. We also extend HBLL to recurrent sequence architectures, applying to settings that otherwise rely on backpropagation through time.

Low-Rank Adaptation (LoRA) enables efficient adaptation of large pre-trained models to downstream tasks by parameterizing weight updates with low-rank matrices. In this paper, we investigate the limitations of the LoRA parameterization from a geometric perspective. Specifically, we show that when a full fine-tuning gradient is backpropagated to the low-rank matrices, it undergoes anisotropic scaling driven by their singular values. We argue that this phenomenon is undesirable because it distorts the full fine-tuning gradient by skewing it toward dominant singular directions while suppressing others. Our analyses demonstrate that anisotropic gradient scaling reduces the effective rank of the low-rank matrices' gradients and results in suboptimal alignment between the full fine-tuning gradient and its low-rank approximation in LoRA, thereby exacerbating the gap to full fine-tuning. To address these limitations, we propose a new low-rank parameterization, SDS-LoRA, which structurally decouples singular values from the backward pass. Our method ensures that the full fine-tuning gradient backpropagates only through the orthonormal bases of the low-rank matrices' subspaces, independent of their scales. Convergence analysis demonstrates that while LoRA's convergence rate degrades with the condition number of the low-rank matrices, SDS-LoRA remains independent of it. Experimental results across natural language and vision benchmarks show that SDS-LoRA improves loss convergence and reduces the gap to full fine-tuning, significantly enhancing adaptation performance.

Deep learning models achieve state-of-the-art image classification but face deployment challenges due to computational costs and energy demands. We propose a lightweight training strategy that adapts normalization layers of the model to the new domain and decouples feature extraction from classifier optimization, reducing overhead by precomputing features only once. A redesigned classifier head with margin-based weighted loss further minimizes ambiguity without end-to-end backpropagation. Evaluated across four CNN architectures (ResNet18, ResNet50, MobileNet, DenseNet121), three Transformer models (ViT, Swin and DeiT) and three medical datasets (Brain Cancer MRI, BreakHis and PatchCamelyon), our approach significantly reduces the required training time with only a marginal accuracy trade-off, often matching or surpassing baseline performance. This efficiency translates to reducing CO2 by orders of magnitude, offering a practical and environmentally sustainable solution for resource-constrained clinical or prototyping environments.

Context adaptation automates prompt engineering in LLM-based systems by iteratively revising tunable prompts from task feedback, without modifying model weights. Extending this paradigm to multi-LLM agentic systems is crucial: existing methods suffer from inaccurate credit assignment and lack convergence guarantees. We propose \textbf{G}raph-based \textbf{T}arget \textbf{B}ack-\textbf{P}ropagation (GTBP), a context adaptation framework for agentic workflows modeled as directed acyclic graphs. GTBP propagates local target outputs backward through the workflow graph and uses target--output discrepancies to guide a stage-wise prompt update mechanism. Theoretically, we show that GTBP's stage-wise prompt updates become stable over iterations, and that a sufficiently capable LLM optimizer can decrease the overall objective. Empirically, GTBP consistently outperforms strong baselines across three benchmarks while maintaining comparable computational cost.

Learning in biological multilayer neuronal networks offers insights that extend beyond the classical weighted-sum neuron model commonly used in artificial neural networks. This article presents an accessible guide to a mechanistic neuronal network model that more accurately captures aspects of biological computation while enabling a simple yet powerful mechanism for learning in multilayer neural networks. The proposed approach supports efficient online streamed learning and provides a practical alternative to backpropagation. We demonstrate its potential in an image classification task, achieving competitive classification performance. The approach's simplicity, biological grounding, and broad applicability highlight a promising path toward algorithms that unify mechanistic neuron models and machine learning.

Masked diffusion language models (MDLMs) such as LLaDA now rival autoregressive (AR) LLMs, but every existing knowledge-editing and unlearning method (ROME, MEMIT, etc.) targets AR transformers and either makes assumptions that fail under iterative denoising, or requires gradient updates whose backward-pass activations cost tens of GB of extra VRAM and which collapse MDLMs at standard learning rates. We introduce TimeROME-DLM, the first training-free, gradient-free, inference-time knowledge-editing framework for MDLMs. It couples two components: a Temporal Indirect Effect (TIE) causal-tracing protocol that identifies, for each fact, the coordinate whose intervention most strongly drives the object prediction at later denoising steps; and a closed-form, low-rank residual edit memory that aggregates subject keys and target deltas across all forget facts and applies a single ridge-regularised update at that coordinate at every diffusion forward, with sparsification to limit utility spillover. Backbone weights stay frozen; only three hyperparameters (alpha, lambda, q) are tuned on a small validation split. On TOFU forget01 with TOFU-finetuned LLaDA-8B-Base, TimeROME-DLM cuts forget-set log-probability by roughly 83 nats. The same configuration transfers to LLaDA-8B-Instruct, Dream-7B, MMaDA-8B, DiffuLLaMA-7B, and LLaDA-MoE-1.4B. It keeps retain-set log-probability nearly flat (within ~1 nat at the utility-safe operating point) across 50 sequentially inserted facts, delivers a four- to fourteen-fold wall-clock speedup with zero additional VRAM over the strongest converged training-time baseline, and scales sub-linearly to 400 facts. TimeROME-DLM closes the locate-then-edit gap between AR LLMs and MDLMs at a fraction of the computational cost.

Modern conversational agents condition on an ever-growing dialogue history at each turn, incurring redundant attention and encoding costs that grow with conversation length. Naive truncation or summarization degrades fidelity, while existing context compressors lack cross-turn memory sharing or revision, causing information loss and compounding errors in long dialogues. We revisit the context compression under conversational dynamics and empirically present its fragility. To improve both efficiency and robustness, we introduce Context-Driven Incremental Compression (C-DIC), which treats a conversation as interleaved contextual threads and stores revisable per-thread compression states in a single, compact dialogue memory. At each turn, a lightweight retrieve, revise, and write-back loop shares information across turns and updates stale memories, stabilizing long-horizon behavior. In addition, we adapt truncated backpropagation-through-time (TBPTT) to our multi-turn setting, learning cross-turn dependencies without full-history backpropagation. Extensive experiments on long-form dialogue benchmarks demonstrate superior performance and efficiency of C-DIC; notably, C-DIC shows stable inference latency and perplexity over hundreds of dialogue turns, supporting a scalable path to high-quality dialogue modeling.

Vision-language-action (VLA) models predict chunks of future actions from the current observation, an assumption that fails under partial observability, where decisions depend on information no longer visible. Existing memory-augmented VLAs simultaneously introduce recurrence, retrieval, compression modules, auxiliary objectives, hierarchical memory, or task-specific architectural changes, so the contribution of recurrence itself remains entangled with surrounding machinery. We present a controlled isolation study of recurrence in a strong pretrained VLA backbone. Our formulation augments the transformer with a small set of learnable memory tokens carried across timesteps and updated through self-attention, trained end to end with truncated backpropagation through time, with no auxiliary losses and no architectural changes. We instantiate this as $μ$VLA, a family of OpenVLA-OFT variants parameterized by memory width m, TBPTT length K, and the memory update rule (cross-step gradients or a detached EMA), so that recurrence is the only varying factor. On MIKASA-Robo, $μ$VLA improves average success rate on five training tasks from 0.42 to 0.84 at the strongest setting and reaches 0.23 on held-out tasks with the same memory structure versus 0.07 for the memoryless baseline. On tasks requiring different memory structure, performance remains near baseline. On LIBERO, the strongest recurrent variant achieves 96.2% average success, indicating no regression under full observability. We interpret these results as a calibration of the capability envelope of minimal in-backbone recurrence, identifying the regime in which it is sufficient and the regime where additional memory structure is required. Demos and videos can be found in https://avanturist322.github.io/mu-vla/.

There are two main Parameter-Efficient Fine-Tuning (PEFT) techniques for Large Language Models (LLMs). While Low-Rank Adaptation (LoRA) introduces additional weights between the LLM layers, Soft Prompting introduces additional fine-tuning-specific raw tokens to an LLM input. However, both require modification to the computational graphs of precompiled, preoptimized LLMs. As a result, neither is fully supported in high-throughput engines like vLLM. We propose fine-tuning with ART (Art-based Reinforcement Training). The method injects information into a frozen Multimodal Large Language Model (MLLM) by optimizing only its raw visual input, thus enabling the soft-token approach on pre-compiled computational graphs. It relies on backpropagation of gradients back into a plain pixel array and thus supports any fine-tuning objective. Moreover, the optimized visual input can be stylized as task-relevant computational artworks. The approach's effectiveness is confirmed for different sizes of a popular open Qwen architecture and for several textual benchmarks. Specifically, ART reaches accuracy competitive with LoRA across mathematics and structured-tool-use benchmarks.

Backpropagation (BP) is widely viewed as biologically implausible, in part because it requires feedback weights to be the transpose of forward weights for error propagation. Interestingly, when training a network with fixed random feedback weights to circumvent this issue, learning aligns the forward weights with the feedback weights, leading the backpropagated error signal to become an approximation of the standard gradient used by BP. This process, called Feedback Alignment (FA), occurs in MLPs and very shallow CNNs but does not scale well to deeper architectures. In this work, we first investigated differences between BP and FA models, trained on CIFAR10, specifically focusing on the effective rank of the signal. We found that the FA error has a considerably lower rank and hence is constrained to a lower-dimensional subspace compared to BP, limiting exploration of the parameter space. Motivated by this observation, we evaluated two mechanisms for increasing the effective dimensionality of FA: Muon, an optimiser that orthogonalises weight updates; and hidden activity normalisation, which promotes activation orthogonality. Across larger architectures and benchmarks, we find that these methods consistently improve over FA baselines, for example, on CIFAR100 with a Resnet-18, accuracy increases by 9 percentage points. Our results identify low-dimensional gradient dynamics as a key obstacle to scaling FA and suggest that inducing higher-dimensional update geometry is a promising route toward scaling alternatives to backpropagation.

Aligning text-to-image flow matching models with human preferences via direct reward backpropagation is sample-efficient but hampered by two well-known pathologies: activations cannot be stored across the full sampling trajectory at modern model scale, and chained Jacobian products across steps inflate the reward gradient as it travels back to early indices. Connector-based methods, such as LeapAlign, address these issues by replacing the full backward trajectory with a short pinned path, highlighting a useful decoupling between sampling and optimization. However, the quality of the resulting gradient depends on how accurately this short path approximates the full rollout, especially over long intervals. We propose FlowBP, a unified surrogate-trajectory framework that treats the backward trajectory itself as the design object. FlowBP keeps a no-gradient cached rollout for sampling, then builds a lightweight backward surrogate from cached and selectively re-forwarded velocities. This view separates four choices: the reward-model input, active set, integration weights, and bridge coupling, and recovers prior direct-gradient methods as particular settings. Within this framework, we instantiate three variants: FlowBP-Sparse uses sparse Euler reconstruction, FlowBP-Bridge adds controlled bridge coupling, and FlowBP-Lagrange raises the order of leap quadrature. All three bound memory by the active-set size and limit gradient chaining to at most one Jacobian factor. Across SD3.5-M, FLUX.1-dev, and FLUX.2-Klein-base on preference, quality, and compositional metrics, the three variants improve over direct-gradient baselines on most metrics.

Neurosymbolic AI combines neural networks with symbolic programs to create robust and explainable predictions. One such framework is NeurASP, which trains a neural network to predict concepts and reasons over them using rules written in answer set programming (ASP) to solve downstream tasks. Crucially, labels are only provided for the downstream prediction produced by the symbolic rules, not for the latent concepts themselves. Backpropagation through the non-differentiable ASP component requires expensive probability and gradient calculations, which has hindered scalability to more sophisticated tasks. In this paper, we address the current limitations of NeurASP by improving its computational performance through vectorization, batch processing and caching of intermediate computations during training. We compare computation speeds between the original and our new implementation of NeurASP and report speedups of multiple orders of magnitude for larger tasks. To this end, we propose a new dataset of difficult tasks involving playing cards, which we use to test the capabilities of NeurASP's enhanced learning function.

We recast predictive coding as continuous-time proximal gradient descent applied to a regularized maximum-a-posteriori (MAP) objective. We study first a single-level problem and then a multi-level hierarchy. For the single-level problem, we show that proximal gradient descent is precisely a leaky firing-rate network: the membrane leak, the effective recurrent matrix, the local synaptic drive, and the static nonlinearity all follow from one optimization principle, and the resulting circuit is the one proposed by Rao and Ballard. The prior selects the nonlinearity through its proximal operator, and the likelihood precision sets the gain on the observation. For the hierarchy, we show that a classical variable-splitting relaxation of the deep MAP problem yields hierarchical predictive coding as the interconnection of local and distributed solvers. In probabilistic modeling terms, this relaxation replaces the directed generative chain by an undirected Markov random field whose node potentials are the level-wise priors. Each level then applies its own activation function, namely the proximal operator of its prior.

Training recurrent neural networks (RNNs) requires assigning credit across long sequences of computations. Standard backpropagation through time (BPTT) addresses this problem poorly: it is sequential in time, limiting parallelism, and suffers from vanishing or exploding gradients, making long-range associations difficult to learn. We propose Supervised Memory Training (SMT), a method for training nonlinear RNNs that sidesteps recurrent credit propagation entirely by reducing RNN training to supervised learning on one-step memory transition labels $(m_t, x_{t+1}) \rightarrow m_{t+1}$. SMT acquires these memory labels by training a Transformer-based encoder on a predictive state objective--retaining only information from the past necessary to predict the future. By decoupling what to remember from how to update memory, SMT enables time-parallel RNN training with a stable $O(1)$ length gradient path between any two tokens--without ever unrolling the RNN. We find that SMT outperforms BPTT when pretraining various RNN architectures on tasks like language modeling and pixel sequence modeling. SMT enables nonlinear RNNs to better capture long-range dependencies and train in parallel, potentially unlocking the scaling of models that build temporal abstractions of past experience.

Object detection is a safety-critical component of autonomous driving. It is essential to quantify the uncertainty in bounding-box predictions for safety assurance. Post hoc uncertainty quantification without retraining aligns with real-world deployment requirements; therefore, we employ the Laplace approximation. Because instance-level uncertainty is needed, linearized inference methods that require multiple backpropagations are not time-efficient, and sampling-based methods are not fully post hoc. We propose Monte-Carlo generalized linearized model (MC-GLM), which provides instance-level and approximately post hoc uncertainty quantification. The number of samples required in the Monte Carlo step is constant and independent of the number of output instances, so it can be parallelized. Experiments on the nuScenes dataset with the CenterPoint detector validate the effectiveness of our method, and the resulting uncertainties exhibit good quality.

The Forward-Forward (FF) algorithm offers a computationally efficient and biologically plausible alternative to backpropagation (BP) by training neural networks through purely local, layer-wise optimization. However, FF is inherently designed for classification via contrastive positive-negative sample pairs, and extending it to regression poses fundamental challenges: continuous target space lack natural "opposites" for contrastive learning, and the standard goodness function carries no information about target magnitude or ordering. We propose FFR (Forward-Forward for Regression), to our knowledge, the first framework to extend FF to real-world regression and demonstrate competitive performance across diverse real-world datasets. FFR introduces three key innovations: (1) an ordinal competitive goodness function that replaces contrastive pairs with competitive learning between partitioned neuron groups under distance-aware ordinal supervision; (2) a stratified ladder architecture where shallow layers learn coarse ordinal discrimination and deeper layers refine into fine-grained regression, with multi-scale feature aggregation for inter-layer collaboration; and (3) hierarchical prediction with uncertainty estimation, where multi-scale predictors jointly provide robust predictions and prediction confidence as a free-lunch. Extensive experimental results show FFR recovers on average 98.6% of BP's accuracy across five real-world regression benchmarks while reducing peak training memory to only 27% of BP's at depth 8 and 8% at depth 32, with per-iteration time around 72% of BP's, and substantially outperforms all BP-free competitors.

One-step text-to-image generators are attractive for deployment because they generate an image with a single forward pass, but preference finetuning them remains difficult: standard alignment methods often rely on policy likelihoods, denoising trajectories, differentiable reward gradients, or test-time optimization. We propose Drifting Preference Optimization (DrPO), an online preference-finetuning method for deterministic one-step generators. For each prompt, DrPO samples candidates from the current generator, ranks them with a target reward, and uses high- and low-scoring samples to synthesize a feature-space update direction. The update is a non-parametric dipole preference field plus a reference drift estimated from the frozen base generator, and is optimized through a detached feature-space regression target. The target reward is used only for ranking, so DrPO can train with large, black-box, or non-differentiable rewards while inference remains a single generator call. We evaluate DrPO on SD-Turbo and SDXL-Turbo with multiple target rewards and benchmarks, including HPSv3 and GenEval. DrPO improves alignment over reward-gradient-free one-step preference baselines and reduces HPSv3 training computation by $3.51\times$ under the matched effective-batch setting by removing reward-model backpropagation. Initial offline experiments suggest that sample-based gradient synthesis can also be used beyond online reward ranking.

We introduce Score Broadcast and Decorrelation (SBD), a principled framework for broadcast-based credit assignment for general families of differentiable losses. Error broadcast is a biologically plausible alternative to backpropagation that sends output information to hidden layers without weight transport. The Error Broadcast and Decorrelation (EBD) framework, recently introduced for the mean-squared-error (MSE) setting, grounded this mechanism in the stochastic orthogonality of optimal estimators, under which the optimal residual is orthogonal to functions of the input. We generalize that foundation by introducing an orthogonality principle between the output score (the gradient of loss with respect to the final-layer output) and hidden-layer activations, which holds whenever the optimal score has conditional mean zero. This single principle unifies broadcast-based credit assignment across the standard differentiable-loss families, including cross-entropy, Bregman divergences, proper scoring rules, and exponential-family negative log-likelihoods. The framework supplies a theoretical grounding for the three-factor learning rule under general losses, with the neuromodulatory factor derived as the broadcast loss score. We derive the cross-entropy case explicitly, characterize the admissible loss class, and introduce a score vector expansion technique that enriches the broadcast signal while preserving the orthogonality framework. Experiments on CIFAR-10 and Tiny ImageNet show that SBD substantially improves over existing broadcast approaches, with score vector expansion delivering further gains. Overall, this work identifies the loss score as the signal to broadcast, supplies the orthogonality theory and theoretical grounding for the three-factor learning rule from neuroscience, and shows how score vector expansion enriches the decorrelation directions of the resulting objective.

Pipeline parallelism is essential for large-scale model training, but existing asynchronous approaches often degrade convergence due to parameter mismatch between forward and backward passes. We propose Asynchronous Multi-Directional Pipeline parallelism (AMDP) to mitigate this issue while sustaining high utilization. AMDP limits the first stage of each pipeline to process at most two minibatches before backpropagation, bounding the number of parameter updates between forward and backward passes. To alleviate the resulting pipeline bubbles, AMDP launches multiple concurrent pipelines and adapts their number according to pipeline depth. In addition, AMDP accumulates gradients across minibatches and applies them in a single update, ensuring that only a bounded number of minibatches experience parameter mismatch, limited to within one optimization step. Experiments on GPT- and BERT-style models demonstrate that AMDP significantly accelerates training while preserving convergence.

Deep neural network training involves both forward propagation (from features through logits to loss) and backward propagation (from loss through gradients to parameter updates). While perturbations along the forward chain, including feature perturbation, logit perturbation, and label perturbation, have been extensively studied, the backward chain's gradient perturbation has received little systematic investigation. In this paper, we establish a unified framework for gradient perturbation, revealing that existing methods such as Sharpness-Aware Minimization (SAM), gradient clipping, and gradient noise injection can all be interpreted as imposing specific forms of gradient perturbation. Analogous to the recently proposed Logit Perturbation Learning (LPL), we conjecture that amplifying the gradient norm for a class acts as positive augmentation (enhancing learning), while dampening it acts as negative augmentation (suppressing overfitting). Based on these observations, we propose Learning to Perturb Gradients (LPG), which adaptively perturbs logit-level gradients at the class level to achieve category-aware training. We also establish theoretical connections between gradient perturbation bounds and generalization guarantees via PAC-Bayesian analysis. Experiments on balanced classification, long-tail classification, and noisy label learning demonstrate that LPG consistently outperforms existing methods and can be combined with them as a plug-in module.

Backpropagation is the core learning mechanism underlying deep learning. However, whether and how this algorithm is implemented in the brain remains highly debated. In particular, while forward activations of pretrained models reliably map onto the cortical hierarchy of visual processing, it is unknown whether backpropagated gradients exhibit a similar correspondence. Here, we address this question using functional magnetic resonance imaging (fMRI) and magnetoencephalography (MEG) recordings of human brain responses to natural images. For this, we extend standard encoding analyses of forward activations to map backpropagated gradients onto neural data. Focusing on a recent self-supervised vision model (DINOv3) and reproducing results on eight vision models, we find that backpropagated gradients can reliably predict both fMRI and MEG signals, specifically in higher-level visual cortex and for later latencies. However, the spatial and temporal organization of these backpropagated gradients in the brain diverges from the patterns expected under a biologically plausible backpropagation mechanism: specifically, both the order in which gradients are computed and their spatial organization diverge from the temporal and spatial hierarchies of the human brain. Together, these results suggest that, although deep networks and the brain may share similar representational content, they likely rely on fundamentally different mechanisms to learn those representations.

Backpropagation is the default learning rule for artificial neural networks and is often treated as the settled approach whenever differentiability is available. In this work, we revisit this convention through a theoretical lens of sample efficiency. We introduce a unified vectorized feedback framework for loss-based and reward-based learning on computational graphs, in which synthetic gradients emerge as a natural alternative to backpropagation. We characterize the conditions under which synthetic gradients can achieve a lower gradient-estimation mean squared error than backpropagation. We construct examples illustrating that this sample efficiency advantage can be arbitrarily large. Experiments on contextual bandits and reinforcement learning tasks demonstrate the potential of our theoretical findings.