Gradient Flow

We study the convergence of Wasserstein-Fisher-Rao (WFR) gradient flows for sampling from probability distributions known up to a normalisation constant. By combining Wasserstein transport with Fisher-Rao birth-death dynamics, WFR flows balance exploration and selection. These flows have been recognised as a promising mechanism to accelerate convergence beyond Langevin dynamics. We show that for a class of strongly log-concave target distributions satisfying additional curvature conditions, WFR flows preserve strong log-concavity, in contrast to Wasserstein flows which enjoy this property only in the Gaussian setting. Exploiting this result, we derive explicit non-asymptotic convergence rates for the symmetrised Kullback-Leibler divergence, without requiring a warm-start as required in current estimates. In particular, we show that the convergence rate decomposes additively into Wasserstein and Fisher-Rao contributions, thereby confirming a recent conjecture within this setting. These results provide refined convergence guarantees and further develop the theoretical foundations of WFR gradient flows for sampling and Bayesian inference.

Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

In data-driven training, multivariate time-series forecasting is usually optimized with a scalar loss averaged over samples, variables, and horizons. This averaging is convenient, but the optimizer sees only the aggregated gradient, which does not reveal whether the variable-wise contributions align or oppose one another. To quantify how often this disagreement arises, we measure the variable-wise gradients directly and find that 30.6% of their pairwise cosine similarities are negative on average across seven datasets. However, conflict and harm are not the same thing. Under shared training 35 of the 64 variables do worse than a full-input single-target oracle, and the harmed fraction is not reliably predicted by how often gradients conflict. We propose Per-Variable Surgery (PV-Surgery), an optimizer-side training strategy for backbones with cache-compatible layers. One backward pass builds variable-wise gradient proxies from output-side signals and keeps the pointwise forecasting loss. Reliability-aware selection targets layers whose proxy sums closely approximate their shared-gradient slices. Conditional pooling forms anchor and conflict pools without dropping variables. Common-direction surgery aligns variable or pooled gradients with their normalized mean and restores input norms to avoid reweighting. In experiments across five backbones, seven datasets, and four horizons, PV-Surgery lowers MSE by 3.61% and MAE by 2.93% on average. For multivariate forecasting, this indicates that the variable-wise structure hidden by mean-loss training is a usable optimization signal.

3D Gaussian Splatting has become a de facto scene representation for novel view synthesis, yet robustly learning 3D Gaussian primitives from visual input remains challenging. Standard optimization relies on gradient-based updates, but a common issue is the gradient vanishing phenomenon: a pixel far from a Gaussian primitive often has diminishing gradient magnitudes to influence primitive attributes, resulting in suboptimal scene reconstruction. In this paper, we propose a method to address gradient vanishing with a piecewise truncated gradient formulation that improves the optimization stability and robustness to initializations. We show that our method consistently improves 3D Gaussian Splatting with random and COLMAP initializations while being generalizable across static and dynamic Gaussian Splatting. As a by-product, we also examine the limitations of current benchmarks for dynamic scenes, and introduce a novel dataset for benchmarking dynamic Gaussian Splatting using synthetic 3D scenes. We demonstrate the effectiveness of our method in both static and dynamic settings for the public benchmarks and our proposed dataset.

Flow Matching has emerged as a leading framework for generative modeling, powering state-of-the-art systems such as FLUX and Stable Diffusion 3.5. However, the iterative nature of its ODE-based sampling process creates a fundamental efficiency bottleneck: the quality of generated samples is highly sensitive to the choice of step-sizes, and current models typically require 20 to 30 steps for good quality. In this work, we propose two lightweight, training-free algorithms, CAT-OV and CAT-OT that adapt step-sizes at inference time based on a novel connection between Flow Matching sampling and gradient flow. Our algorithms are computed efficiently by not requiring additional neural function evaluations. Specifically, CAT-OT estimates curvature over time via a finite-difference approximation of the time-derivative of the vector field, while CAT-OV approximates curvature over the state space via a gradient of the vector field. Under suitable conditions, both methods have truncation error bounds of constant order. Empirically, CAT-OV and CAT-OT outperform existing step-size heuristics in image quality metrics across four text- to-image Flow Matching models, reducing the number of generation steps required to reach comparable quality by up to 40%.

The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as $f = g + \varepsilon h$; in the limit $\varepsilon \to 0$, the dynamics of gradient descent with learning rate $η$ converge to the gradient flow of $h$ constrained to the minimizers of $g$ of sharpness at most $2/η$. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in $\varepsilon$. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of $g$-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in $\varepsilon$: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.

Gradient flow is widely used as a continuous-time surrogate for gradient descent, but state convergence does not imply convergence of differentiated training maps in nonsmooth networks. We characterize the fixed-horizon, vanishing-step limit of exact automatic differentiation through hard-ReLU gradient descent. Under a stable finite itinerary of separated, same-direction transverse activation events, gradient-descent states converge at first order to the corresponding piecewise-smooth gradient flow, while the exact derivative of every nonresonant discrete program converges to an event-free regional propagator. The true flow derivative instead interleaves classical saltation matrices that encode event-time sensitivity. For globally convex objectives, any strict activation event prevents complete cancellation of these missing transfers. Moreover, minimal globally 1-strongly convex residual-ReLU risks can realize arbitrarily large reciprocal sensitivity gaps, subject to an explicit transversality-scale tradeoff, and a coupled strongly convex construction yields an open set on which the largest initialization-gradient coordinate is reversed. In a controlled 17-parameter ReLU MLP, state and regional-AD errors vanish under mesh refinement while AD-to-flow errors remain between 0.18 and 0.39; an event-aware corrected product restores convergence. Resolved smoothing likewise recovers the flow sensitivity when the transition layer is sufficiently resolved. These results show that the gradient-flow limit of hard-ReLU training need not remain valid after differentiation.

Physical learning lets a trainable material or network use its own physical response to carry error signals, reducing the need for a separately programmed backward computation. We ask what determines whether such a system follows conventional gradient descent or evolves along a genuinely different learning trajectory. Our canonical model is a directed layered transport network in which every node redistributes a fixed amount of flow, so learning preserves positivity and total mass. In this model, conservation constrains only the allowable learning directions. Within the matched response class studied here, adjoint matching gives the physical output response a symmetric form. Non-negative mode-wise feedback then produces a reciprocal closed-loop response and a reweighted gradient flow. Adding an antisymmetric boundary component makes the closed-loop response rotational: the learning path can turn while the error driving that update still decreases at that moment. Turning is not automatically beneficial. Its finite-step effect is set by local curvature, and its accumulated effect also depends on step selection and on the new states visited along the path. Numerical consistency checks reproduce the exact response structure, predict the sign of the local effect across new network families, and show how trajectory drift can negate a local advantage. These results separate the roles of conservation, reciprocity, and nonreciprocity in physical learning.

We estimate the conditional population-risk curve of a realized smooth nonconvex gradient flow from the training sample. Flow approximate leave-one-out (Flow-ALO) propagates a deletion response and evaluates omitted observations at approximate deleted paths. The risk-curve error decomposes into response approximation, exact-LOO fluctuation, and deletion-to-full risk transfer. On each fixed finite horizon, bounded centered training-loss gradients, a one-sided Hessian lower bound, locally Lipschitz Hessians, and a strict tube-closure condition yield an explicit $(n-1)^{-2}$ bound for the deletion-response error. Bounded evaluation-loss gradients transfer the deletion-response bound to the score without requiring the Hessian to be invertible. Direct first-order jackknife cancellation and exact-LOO concentration control deletion-to-full risk transfer and fluctuation, respectively, completing recovery of the conditional population-risk curve. For bounded smooth two-layer mean-field networks training both layers, the score-error bound is uniform in width.

We propose CVaR-penalized Generative Particle Algorithm (CVaR-GPA), a robust, tail-agnostic algorithm for fine-tuning generative models to learn heavy-tailed distributions and capture extreme events, requiring no prior knowledge or estimation of the target's tail characteristics. The method is the Wasserstein gradient flow of the Lipschitz-regularized Kullback-Leibler (KL) divergence penalized by a Conditional Value-at-Risk (CVaR) discrepancy term: the Lipschitz-regularized KL divergence enables robust learning under minimal assumptions on the target distribution, while the CVaR penalty restores the velocity that otherwise vanishes prematurely in the under-sampled tails. The penalized flow admits a bounded but non-Lipschitz velocity field. This departs from the Lipschitz transport maps of standard generators, which preserve the tail behavior of a light-tailed source, and enables transport toward heavier-tailed targets. To define this flow on empirical measures, we derive the first-variation subgradients of CVaR from its Rockafellar-Uryasev representation, valid precisely where the classical density-based formula fails. The particle algorithm CVaR-GPA fine-tunes the output samples of any pre-trained model, without access to its architecture, and runs on an adaptive time horizon set by a kinetic-energy stopping criterion rather than a preset depth. On synthetic isotropic and anisotropic Student-$t$ target distributions, Neal's funnel distribution, and the real-world high-dimensional Fama-French 25 portfolio dataset, CVaR-GPA dramatically improves global and tail accuracy on heavy-tailed targets over the pre-trained baseline.

Deep decoder-only Transformers often replace the original Post-Norm architecture with Pre-Norm variants because Post-Norm training is highly sensitive to warmup and learning rate under conventional initialization schemes. Although prior work has identified rank collapse and gradient vanishing as related symptoms, it remains poorly understood how causal attention creates high-similarity representations and why training dynamics fail to repair them. We give a two-stage analysis of Post-Norm rank collapse using token similarity as a scalar state variable. First, at initialization, causal attention acts approximately as a prefix-averaging operator that increases token similarity across depth, while the SwiGLU branch contributes only a smaller damping effect. Second, once training enters a high-similarity regime, growth of pre-normalization residual norms makes the RMSNorm backward factor contractive; under mild conditions, gradients to earlier layers decay geometrically. As a complementary result, we characterize the properties of a collapsed network: its best predictor is frequency distribution with relatively high loss floor, and gradients in collapsed layers vanish at frequency distribution. Experiments on 48-layer decoder-only Transformers trained on C4 dataset match the predicted initialization-time similarity growth and collapse-time gradient contraction, and show that collapsed runs stay near the predicted frequency loss. Together, these results distinguish the forward similarity amplification and backward repair incapacity in Post-Norm collapse, while also characterizing the behavior of collapsed networks.

Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum, "shared-scalar" Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A "spectral schedule" reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.

Existing theories of neural-network width characterize asymptotic limits, but provide limited guidance on whether an expansion direction identified from finite training data remains beneficial on unseen data. We study this problem for function-preserving residual expansion and introduce the effective alignment dimension, a measurable quantity describing the signal-noise geometry of activation gradients. By deriving the exact mean and variance of the inner product between independently estimated training and test gradients, we obtain a finite-sample upper bound on misalignment probability. The bound depends only on the effective alignment dimension and an effective sample size, requiring finite second moments and a nonzero population gradient, without covariance spectral assumptions or prescribed width-growth rates. We integrate this certificate into the train-test residual-expansion framework, yielding a high-probability condition for test-risk improvement. Experiments across width-controlled LLaMA-style Transformers, Pythia, and ResNet-20 show that wider models exhibit larger effective alignment dimensions and lower empirical misalignment. Direct residual interventions confirm that the alignment statistic predicts the sign and magnitude of held-out loss changes.

Transformers are known to have internal continuous symmetries that leave outputs invariant, while modifying quantization. GaugeQuant leverages this in-training by introducing a LogSumExp term to the loss that breaks the symmetries, thus selecting a basis that minimizes activation outliers. A stop-gradient operator ensures that only rotation matrices are updated, yielding the language modeling objective completely unaltered. Our requires no specific calibration data, no quantization simulation, and adds negligible training overhead. With the LLaMA-2 7B model under W4A4 quantization with group size 128, perplexity drops from 8.22 to 6.73, competing with post-training methods that require frozen models and calibration datasets. Under W4A16, perplexity drops from 11.16 to 5.45. Code is available at https://github.com/MPedraBento/gauge-quant.

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.

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

Reinforcement learning (RL) is traditionally concerned with learning a control policy for a fixed environment. In many engineering systems, however, the environment itself is alterable: physical or operational parameters can be tuned to shape the transition dynamics and costs experienced by the agent. This motivates jointly optimizing both the policy and the environment design parameters. To this end, we establish an Environment Parameter Gradient Theorem -- a formal expression for the gradient of the value function with respect to environment parameters. The key theoretical device is a generalized action-value function $Q_{π,ξ}(s,a,ζ)$, which comprises two copies of the environment parameters: $ζ$ governs the cost and transition dynamics at the current state--action pair, while $ξ$ governs the future rollouts. This decoupling yields a tractable closed-form gradient expression and is essential to the theorem's derivation. Building on this result, we develop a model-free algorithm that simultaneously learns the optimal policy and the environment parameters. We demonstrate the efficacy of our framework on a UAV network design problem, where the optimal UAV placement (environment parameters) and communication routes (governed by the policy) are learned jointly to minimize the total communication cost in the network.

Gradient estimation -- the task of computing the gradient of the expected value of a probabilistic program -- has diverse applications in scientific computing, but is notoriously difficult because of issues such as high-dimensional integration, discrete random choices, and complex stochastic dependencies. This article introduces gradient inference, a new approach to developing sound and efficient gradient estimators for probabilistic programs. Gradient inference rests on a formal reduction from a gradient estimation problem to a closely related probabilistic inference problem, whose solution can be differentiated to obtain a gradient estimator. This inference problem is obtained by applying two powerful statistical operations -- coupling and factorization -- to the input probabilistic program. Our reduction lets us leverage the rich toolkit of probabilistic inference algorithms to design novel gradient estimators that extend and improve upon existing methods. We introduce GradInf, a probabilistic programming system that facilitates the sound and automated implementation of gradient inference. GradInf is centered around programmable source-to-source transformations for coupling and factorizing higher-order probabilistic programs, whose soundness is proven in terms of a denotational semantics. Key to our development is the use of information-flow typing to allow random choices in a probabilistic program to be factored out and partially evaluated, which improves our ability to deploy sophisticated probabilistic inference algorithms. The resulting system offers practitioners a principled framework for designing gradient estimators. We apply GradInf to several challenging case studies, showing that it can express prominent gradient estimators from the literature and enables the construction of new state-of-the-art estimators that outperform the best existing baselines.

Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.

We study the implicit bias of noisy stochastic gradient descent in training wide two-layer ReLU networks for multivariate regression. In a mean-field regime, the training dynamics are approximated by a Wasserstein gradient flow that converges to a unique stationary measure. We characterize the structure of this stationary measure and the predictor it represents. We show that, despite the network being infinitely overparameterized, the learned predictor admits an effectively finite representation: the input weights and biases align along finitely many directions, leading to an effective width collapse. In particular, the solution function is continuous piecewise affine, with affine regions determined by the cells of a finite hyperplane arrangement. The number of learned directions, and hence hyperplanes, is bounded above by $2\mathcal{P}-1$, where $\mathcal{P}$ denotes the number of linear dichotomies realizable on the training inputs. We further establish a non-redundancy property of the learned representation by proving that each learned direction induces a unique ternary activation pattern on the training data. Consequently, the complexity of the learned predictor is governed by the combinatorial geometry of the training data.

Deep neural networks with repeated architectural blocks, such as transformers, often exhibit structured relationships across layers that emerge during training. Motivated by this observation, we introduce \emph{Depth-wise Gradient Augmentation}, a general optimization paradigm in which the update applied to each layer is obtained by transforming the collection of block-wise optimizer updates along the depth dimension. Within this framework, we study \emph{Gradient Smoothing}, a family of depth-wise smoothing methods, and instantiate it with a simple local \emph{Window Smoothing} operator. The resulting method operates directly on block-wise updates produced by arbitrary base optimizers (e.g., SGD, Adam, Muon), incurs minimal computational overhead, and is compatible with existing optimization pipelines. We evaluate Gradient Smoothing across a diverse set of architectures and training regimes, including language model pretraining, RL post-training of LLMs for reasoning, diffusion modeling, and image classification with Vision Transformers. Across these settings, Gradient Smoothing consistently improves optimization and generalization performance without modifying model architectures or training objectives. We further show that it promotes more structured representation evolution across depth, consistent with its interpretation as a structured depth-wise preconditioning method. Together, these results establish Depth-wise Gradient Augmentation as a promising framework for exploiting cross-depth structure in optimization and demonstrate Gradient Smoothing as a simple and broadly applicable instantiation.

Matrix factorization (i.e., problems of the form $\min_{\mathbf{P},\mathbf{Q}} \|\mathbf{M}^\star - \mathbf{P}^\top\mathbf{Q}\|_\mathrm{F}^2$) is a minimal learning problem that exhibits both nonlinear parameter dynamics and representation learning. In this setting, we study how parameter trajectories under the Muon optimizer differ from those of gradient descent. We identify three main dynamical differences: 1) Muon avoids the slow saddle-to-saddle dynamics from small initialization. Muon instead learns all the top modes of $\mathbf{M}^\star$ at the same rate, with the smaller modes converging first. 2) Muon remains stable even when the learning rate exceeds the critical threshold set by the local loss sharpness. This frees the learning rate from the condition number of the problem, enabling rapid convergence via exponential learning rate annealing. 3) Once the weights are aligned with each other and the target, Muon flow conserves the matrix quantity $\sqrt{\mathbf{P}^\top \mathbf{P}}-\sqrt{\mathbf{Q}^\top \mathbf{Q}}$, while gradient flow is known to conserve the matrix $\mathbf{P}^\top\mathbf{P} - \mathbf{Q}^\top\mathbf{Q}$. Despite having distinct conserved quantities, both optimizers find the so-called \textit{balanced} solution from vanishing initialization. When training from small random initialization, the weights spontaneously align early in training. We derive the alignment rates in simple settings and show that they predict the empirical alignment rates in general. Finally, we exploit structural properties of Muon to construct a learning rate schedule that achieves near-perfect alignment in only two optimization steps.

Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, computational cost, gradient accuracy, and numerical conditioning vary widely across solvers and are discoverable only by trial and error. We introduce Mosaic, an extensible benchmarking framework for differentiable PDE solvers that standardizes access to solver gradients. Each solver is packaged as a containerized component (Tesseract) exposing a uniform gradient API regardless of language or automatic differentiation (AD) strategy, enabling researchers to evaluate, compare, and build on non-trivial physical solvers. Our evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer demonstrates that the benchmark surfaces practically relevant differences: order-of-magnitude variation in computational cost and Jacobian conditioning, alongside structural incompatibilities that eliminate solvers from realistic tasks entirely. Despite this variation, all solvers that produce gradients converge to similar optima, indicating that the practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy alone. Mosaic is open-source and available at https://github.com/pasteurlabs/mosaic.

Vision-Language Large Models (VLLMs) trained on massive crawled corpora raise pressing copyright and data-provenance concerns. These concerns are particularly acute in healthcare, where patient medical images paired with clinical reports demand rigorous privacy safeguards. However, existing training data detection methods either fail in cross-modal scenarios or rely on superficial output signals with insufficient discriminative power. We introduce GradAudit, a gradient-based auditing framework that examines internal optimization dynamics rather than treating VLLMs as black boxes. Our approach builds on a key observation: model parameters converge to regions where gradients on training samples become stable and well-aligned, whereas gradients on non-training samples remain noisy and inconsistent. By analyzing these gradient signatures, GradAudit achieves strong separability and detects genuine image-text associations learned during training, not merely individual modality membership. Empirically, across both medical and general-domain datasets, GradAudit substantially outperforms state-of-the-art baselines in both pretraining and fine-tuning VLLMs. In a case study employing copyrighted content, we show that existing training data detection methods not only underestimate the extent of unauthorized data usage, but that this underestimation becomes more pronounced as models become more recent and more advanced.

The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.

Reverse-mode differentiation computes every weight gradient, writes it to memory, and only then lets the optimizer read it back. This two-phase schedule sets the memory ceiling of modern training: at the seam between the phases, every layer's gradient is live at once. We argue that this materialized gradient is an artifact of how differentiation is staged, not a quantity that learning requires -- and we eliminate it. FORGE folds the optimizer step into the backward pass and applies it one tile at a time, entirely in registers, so each gradient tile is consumed the instant it is produced and never becomes a tensor. The fusion changes only when the update happens, not what it computes: in full precision the fused step is provably exact -- the identical optimizer update, for every element-wise rule -- and that exactness survives tensor- and sequence-parallel sharding; in the bf16 and 8-bit regimes used in practice it is faithful rather than bit-identical, its deviation bounded and, for the weight store, rendered unbiased by stochastic rounding. Because each gradient tile is born and consumed in the same registers, it is never converted down to bf16 to be stored and read back; FORGE thus preserves the full-precision fidelity that both bf16 and 8-bit optimizers lose to that conversion. Nor is the method tied to one architecture or one optimizer: linear layers are ubiquitous, and FORGE reclaims the gradient memory of any of them under any element-wise rule. Empirically FORGE more than halves the memory of an optimizer step and, at the small batch sizes typical of fine-tuning and continued pretraining, runs about 1.5x faster; integrated into tensor-parallel Megatron-LM it fits 8B training at four times the micro-batch a standard optimizer allows on the same GPUs.

Few-step distillation for video diffusion models has attracted significant attention, driven by the urgent demand for efficient deployment in real-world scenarios. However, Distribution Matching Distillation (DMD), a leading paradigm, tends to degrade under limited NFE budgets, manifesting in video generation as layout instability, oversaturation, and broken motion dynamics. We trace this failure to a structural limitation: standard DMD is an intra-sample distribution-matching objective with coordinate-wise gradients, and thus imposes no explicit constraint on the relational geometry across batch elements or temporal frames, leaving the underlying copula largely unregulated. Combined with the mode-seeking tendency of its reverse-KL objective, this absence of relational guidance makes DMD prone to collapsing into local optima in the few-step regime. Motivated by this insight, we propose Copula-aware DMD (CoDMD), a lightweight relational regularizer that reuses score estimates already produced by the frozen teacher and the online fake model to construct pairwise relation matrices across samples and frames. These are matched through a supplementary distributional objective that requires no additional networks, datasets, or sampling trajectories. On the Wan-2.1-T2V model series at 1.3B & 14B scales, CoDMD distills 50-step teachers into 4-step students, achieving an approximate 25$\times$ speed-up while attaining VBench scores of 84.46 & 84.87, outperforming prior trajectory-based (rCM 82.81 & 84.05) and distribution-based (DMD 83.38 & 83.81) methods.

The noise of Magnetic Resonance Imaging MRI poses challenges for Deep Learning DL when tumor boundaries are obscured tumor location and appearance are complex Therefore we develop BrainFusionNet that combines Convolutional Neural Networks CNNs Vision Transformers ViT and Gated Recurrent Units GRUs to extract spatial contextual and sequential features from MRI images for improved brain tumor classification Furthermore explainable AI such as SHAP LIME and GradCAM are integrated to visualise and highlight image regions that contribute to BrainFusionNets decisionmaking process The proposed BrainFusionNet model is evaluated on two publicly available MRI datasets Kfold validation suggests 98 accuracy on both datasets The model was compared with the six stateoftheart SOTA CNNs and transfer learning Among the SOTA CNNs DenseNet121 and VGG16 achieved the highest accuracy of 96 The novelty of BrainFusionNet is that the hybrid model effectively extracts local and global features from MRI images even in smallscale tumor regions and small tumor sizes The model has a balanced sequential CNN architecture to capture lowlevel and deeperlayer features a customized ViT that captures local features stabilizes gradient flow and reduces the risk of vanishing gradients during MRI image training The CNN and ViT outputs are fed into a GRU for final classification Furthermore we analyze pixel intensities to determine whether MRI image quality affects image classification Our findings are very novel in image interpretation as we found that the distribution of pixel intensities in MRI images affects DL performance

Understanding gradient descent dynamics is key to explaining the success of over-parameterized models, where implicit bias manifests through conservation laws in gradient flow. While such laws are well understood for linear and ReLU networks, they remain largely unexplored for modern architectures. This work develops a unified framework to characterize conservation laws for contemporary models, including feedforward networks with GELU, SiLU, and SwiGLU activations, multihead attention with sinusoidal and rotary positional encodings, and Mixture-of-Experts architectures under diverse gating designs. Our theoretical findings are supported by experiments that validate the predicted invariants.

This paper traces, with explicit numerical values, how PyTorch's automatic differentiation (AD) engine computes gradients for Physics-Informed Neural Network (PINN) training -- a setting that requires two levels of differentiation: computing the physics derivative $\hat{y}'(t)=d\hat{y}/dt$ through the network, and computing parameter gradients $\nabla_θL$ of a loss that itself depends on $\hat{y}'(t)$. Using a 1-3-3-1 multilayer perceptron and the initial value problem $y'(t)+y(t)=0$, $y(0)=1$, we trace the complete pipeline at every node: the computational graph built during the forward pass, the reverse-mode backward traversal that computes all 22 parameter gradients in a single pass, and the graph-on-graph mechanism by which \texttt{create_graph=True} enables correct differentiation through the physics-informed residual. Every adjoint value is verified against the hand derivations of Tahimi (2026), connecting the $P/Q$ sensitivity framework to the vector--Jacobian products used by PyTorch's autograd engine.

We study feed-forward ReLU networks with fixed readout and quadratic loss, and rewrite gradient descent as a collective dynamics of activation fields and conjugate fields on the training set. Working to first order in the learning rate inside a fixed activation chamber, we derive explicitly the one-, two- and three-hidden-layer cases, and then give the arbitrary-depth recursion. For one hidden layer the activation dynamics closes directly and the residual update is governed by the product of an input Gram matrix and a co-activation/backpropagation Gram matrix. For two hidden layers a conjugate field is required, but no nontrivial pullback Gram metric has yet appeared. For three hidden layers the first weight-induced pullback Gram metric enters the conjugate-field dynamics. At arbitrary depth, activation variations propagate forward through a recursive response operator $\cU_\ell^{αβ}$, while conjugate-field variations propagate backward through an effective transport operator $\cM_\ell^{αβ}$. Their contractions reconstruct a layerwise residual kernel $$K_{αβ}^{(L)}=\sum_{\ell=1}^{L}Q_{αβ}^{(\ell-1)}S_{αβ}^{(\ell)}.$$ The resulting description exposes a duality between push-forward and pullback transport across every layer cut, and identifies the first Gram metrics as the lowest nontrivial terms in a broader hierarchy of activation-conditioned transport operators. We deliberately stop at the level of collective fields, conjugate fields, residual kernels and cut-wise transport metrics, leaving the later tensorial geometric formulation outside the scope of this paper.

Theoretical studies of machine learning models commonly consider different limiting regimes in which the learning dynamics of gradient descent becomes theoretically tractable. It is, however, desirable to have a systematically obtained picture of all qualitatively different extreme learning regimes for a particular type of models. In this paper we propose such a picture for large weight-tied linear autoencoders characterized by input and latent dimensions, initialization magnitude, and training set size. This model is nonlinear in the weights and its gradient flow does not have a general theoretical solution. We show that at the level of the formal loss-expansion hierarchy, its extreme regimes are naturally associated with faces of a triangular prism. In particular, there are five basic extreme regimes associated with the 2-faces of the prism: (1) large-data, (2) small-data, (3) mean-field, (4) narrow-latent, and (5) free. For regimes (1,2,3,4), we derive explicit expressions for both train and population limiting loss evolutions under gradient flow, obtaining very good agreement with experimental results.

This paper investigates non-stationary online learning using the metric of interval regret, which requires an online algorithm to perform well over every time interval. We propose the first online learning algorithm that achieves an interval regret bound scaling with gradient variation, a fundamental measure of the cumulative change in online function gradients, which relates to various problem-dependent quantities and is closely connected to stochastic optimization and other problems. Our method employs a simple and efficient two-layer online ensemble structure that achieves strong theoretical guarantees. Specifically, it enjoys a regret bound that simultaneously adapts to various problem-dependent quantities while also preserving the minimax-optimal rate in the worst case. Moreover, recognizing the challenge of hyperparameter tuning, we introduce a Lipschitz- and smoothness-agnostic variant that automatically adapts to these potentially unknown constants. This is primarily enabled by a novel Lipschitz-adaptive meta algorithm, which may be of independent interest. Beyond interval regret, our method also yields broader implications: it provides versatile bounds for interval dynamic regret, a stronger measure that competes with changing comparators over any interval, and yields the first piecewise characterization for stochastic extended adversarial optimization. Theoretical findings are validated by experiments.

LLM-based search agents are trained predominantly with outcome-only reward, leaving the search process itself unsupervised. This signal degenerates on outcome-homogeneous groups where all sampled trajectories share the same correctness, yielding zero within-group advantage and no gradient. Existing process supervision either trains a costly verifier or generates per-query rubrics that are inconsistent across queries and discarded after one use. We propose ARBOR (Adaptive Rubric Buffer for Online Reward), a reusable process-reward framework that maintains a rubric memory shared across queries. Query-local drafts induced from contrastive trajectories are admitted, consolidated into cross-query common rubrics, and retired as the policy evolves. A small active subset of common rubrics scores trajectories via sparse pairwise judging, and the resulting scores are added to the base reward, providing process-level gradient even when outcome reward is uniform. ARBOR consistently outperforms GRPO and DAPO baselines on four multi-hop QA benchmarks, raising average LLM-judge accuracy by up to 4.2 points and converting up to 42% of otherwise-zero-gradient training groups into informative ones.

Understanding how structured internal structure emerges during neural network training is central to the study of deep learning. We investigate this phenomenon through the group composition task, where a two-layer neural network is trained to predict $g_1 \star g_2$ for elements of a finite group $G$. By lifting the projected gradient flow to the Fourier domain, we demonstrate that the training dynamics are governed by a Riemannian gradient ascent on a representation-theoretic energy functional. We prove that, under random initialization, this flow drives each neuron to converge almost surely toward a single irreducible representation, while the cross-layer Fourier coefficients achieve a rotational rank-one alignment. This framework provides a representation-theoretic account of feature learning and characterizes a novel low-rank compression phenomenon for matrix-valued group representations. Moreover, for Abelian groups, we provide a complete population-level description: random initialization promotes uniform diversification across nontrivial representations and induces Haar-uniform phases, jointly approximating the indicator via a majority-vote mechanism. We further prove that both phase alignment and representation competition emerge with exponential convergence rates.

On-policy distillation (OPD) is a widely used technique to transfer capabilities from capable teacher language models to the base student models, and can be formulated in a reinforcement learning style objective using student generated rollouts. Yet, despite the divergence reward being dependent on student model likelihood, existing works usually adopt a stop gradient design primarily for stability, which makes the resulting advantage estimation questionable. In this work, we provide a generic optimization framework based on f-divergence between the student and teacher, and mathematically revisit whether such design space is valid. We prove that general stop-gradient operation would lead to biased estimates of the reward objective and corresponding gradient for general divergence functions. We propose OPD+, the corrected version of OPD that demonstrates improved performance over the baseline KL approach and also supports the choice of various f-divergence. We validate our findings on mathematical reasoning and tool-use benchmarks.

Recent analyses of multi-pathway Deep Linear Networks use Gradient Flow to predict a "winner-takes-all" specialization in which path symmetry breaks and each feature concentrates in a single pathway. In this work, we show that discrete Gradient Descent (GD) with a large step size tells a different story. We prove that single-path solutions are sharp minima, whereas distributing signals across pathways reduces sharpness by a factor that decreases with both the number of pathways and depth. Consequently, while early training reproduces the depth-driven symmetry breaking predicted by GF, oscillations at the Edge of Stability subsequently override this tendency and drive the network into a re-balancing phase, where signals redistribute across pathways. Together, these results clarify how depth shapes pathway competition and explain why large-step GD favors shared representations rather than persistent single-pathway dominance.

Domain specialization can improve LLM behavior, but often weakens the general capabilities inherited from the original model. Recent Multi-Teacher On-Policy Distillation (MOPD) pipelines recover model capabilities by supervising student-generated trajectories with teacher feedback, but typically assume teacher-aligned prompt coverage, requiring prompts to match the teachers' training distributions. This assumption is difficult to satisfy when the general teacher is an open-source model whose post-training data are unknown. Instead of attempting to reconstruct this hidden distribution, we study general capability recovery with readily available proxy general prompts. We identify two failure modes of vanilla MOPD in this incomplete-coverage situation: recovery-preservation counteraction from mixing conflicting recovery and preservation gradients, and weak-signal flattening from uniformly averaging samples with unequal correction demand. We propose \textbf{Counteraction-Aware Multi-Teacher On-Policy Distillation} (\textbf{CaMOPD}), which addresses these issues with decoupled alternating training and gap-based sample selection. CaMOPD allocates dedicated updates to general recovery, periodically performs domain-preservation updates, and selects samples with larger averaged token-level teacher-student log-probability gaps to concentrate correction signals. Across role-play dialogue and medical reasoning QA scenarios, CaMOPD outperforms all baselines in general capability recovery while maintaining domain-specific behavior. Gradient coherence analyses further support the intended effect of CaMOPD in producing more coherent correction signals.

The residual update of a pre-norm Transformer layer admits an interpretation as one step of a first-order optimizer acting on a surrogate token energy, wherein the attention and MLP sublayers function as gradient oracles. Based on this observation, we build a family of optimizer-inspired Transformers (triple-momentum, Adam/AdamW, Muon, SOAP) and compare them under matched compute. In our main pretraining experiment, the triple-momentum TMMFormer achieves the lowest validation loss, outperforming the vanilla Transformer and prior architectural variants. A controlled ablation and supporting theory show that momentum, not preconditioning, is the main source of the gain. We further show that TMMFormer and other momentum-based designs reach flatter minima than the vanilla Transformer, which leads to less forgetting and better generalization.

Chain-of-thought (CoT) reasoning is useful for monitoring language models only when the reasoning trace faithfully reflects the computation that produces the final answer. However, models can rely on prompt-to-answer shortcuts that bypass the CoT, making the visible reasoning trace misleading even when it appears plausible. We study CoT faithfulness through a structural information-flow perspective: faithful reasoning should route answer-relevant information through the mediated path from prompt to CoT to answer, rather than through a direct prompt-to-answer shortcut. This perspective yields a task-agnostic framework based on three complementary properties, sufficiency, completeness, and necessity, which we instantiate with entropy-based, masked-KL, and gradient-based diagnostics. We show that these metrics recover externally judged faithfulness differences in hinted reasoning, and identify a low-entropy failure mode of KL-based diagnostics where gradient-based measures remain more stable. Building on this analysis, we introduce update-time interventions for verifier-based on-policy RL, including attention masking, backward-only gradient masking, CoT gradients, and adversarial perturbations of prompt representations. Across hinted arithmetic, reward-hackable code repair, and DAPO-Math models trained without hints but evaluated under wrong-hint injection, our interventions shift behavioral and structural indicators toward stronger CoT mediation. In particular, they make shortcut and reward-hacking behavior more transparent in the CoT and improve task-agnostic faithfulness metrics, while in some settings also reducing wrong-hint susceptibility. Our results suggest that controlling information flow during training is a practical route toward more faithful and monitorable CoT reasoning. Code is available at https://github.com/safety-research/faithful-cot.

We develop a gradient flow on the space of probability measures defined on matrix-valued parameters induced by regularized Muon, an analytically smoothed version of the idealized Muon optimizer. The key observation is that the regularized orthogonalization map is the gradient of a smooth Fenchel-dual smoothing of the nuclear norm. This identifies the (regularized) Muon update as a mirror/prox step in the update variable, with momentum acting as the dual coordinate. We use this structure to lift Muon from a single matrix parameter to finite-particle probability objectives of the form $J(ρ)=R\left(\int F d ρ\right)$, a setting motivated by mean-field descriptions of neural-network training, and derive the inertial continuous-time limit. Using this structure, we derive the finite-particle continuous-time limit under the inertial scaling of step size and momentum, and then pass to a phase-space mean-field equation over probability laws on parameter-momentum pairs. The resulting flow can be shown to be a damped Hamiltonian probability dynamics whose kinetic energy is induced by the regularized Muon mirror potential. We prove an exact Hamiltonian dissipation identity, showing that the Hamiltonian energy decreases monotonically. While the target objective itself need not be monotone along the inertial Muon dynamics, under additional gradient-dominance, bounded-momentum, and curvature/alignment assumptions, we obtain continuous and discrete-time exponential convergence rates for the objective gap. We also study the well-posedness of the mean-field limit equation and establish propagation of chaos guarantees for the interacting particle system. Finally, we extend the formulation to Hilbert-valued feature maps on product matrix spaces, yielding a blockwise Muon probability flow applicable to smooth transformer mixture-of-experts models.

Segmentation is a fundamental task in computer vision, underpinning pixel-level scene understanding and serving as a cornerstone for applications ranging from autonomous perception to medical image analysis. For complex referring segmentation, recent methods pair large vision-language models with segmentation decoders: the former analyzes the image and prompt, while the latter predicts the target mask. Although reinforcement learning improves reasoning-intensive vision-language systems, trainable tools such as segmentation decoders are typically optimized separately with differentiable objectives, and the principled integration of such objectives into reinforcement learning remains underexplored. Thus, we introduce group relative tool optimization (GRTO), a mathematically grounded framework for jointly optimizing a policy with differentiable tool use. GRTO reuses group relative policy optimization (GRPO) rollouts to optimize the auxiliary tool objective, letting decoder gradients complement policy rewards. Further, we derive Bootstrapped-GRTO (B-GRTO), a pre-training method that cheaply bootstraps the tool, leading to faster convergence and superior performance. Across three challenging referring segmentation settings, B-GRTO results in substantial improvements over plain GRPO, matching or surpassing domain-specific state-of-the-art methods. This demonstrates the value of unifying reinforcement learning with differentiable auxiliary objectives for reasoning-intensive segmentation.

An accurate assessment of a model's complexity is crucial for topics such as interpretation, generalization, and model selection. However, most existing complexity measures either rely on heuristic assumptions or are computationally prohibitive. In this paper, we present a mathematically rigorous yet easy-to-compute measure of model complexity that is based on the similarities between the model gradients across inputs. It is thus well-defined for any parametric model, but also for kernel-based non-parametric models. We prove that our measure of complexity generalizes model-specific complexity measures such as polynomial degree (for polynomial regression), kernel length scale (for Matérn kernels), number of neighbors (for k-nearest neighbors), number of splits (for decision trees), and number of trees (for random forests). We also use our measure to obtain new insights into the double descent phenomenon for random Fourier features, random forests, neural networks, and gradient boosting.

We study the max-margin solutions reached by mirror flow in deep neural networks with homogeneous activation functions. Extending classical results on gradient flow, we derive a novel balance equation for mirror flow from convex duality, enabling a characterization of the horizon function governing the induced margin. We further establish max-margin characterizations together with convergence rates and norm growth estimates. Finally, we support our theory through experiments on synthetic datasets and standard vision tasks. Concretely, we show that: (1) distinct non-homogeneous mirror maps can induce the same max-margin solution; (2) convergence can be extremely slow, including exponentially slow regimes; and (3) although all considered mirror maps exhibit feature learning, they can produce markedly different representations, ranging from sparse to dense neuron activations. Together, these results provide a unified perspective on sparse and dense feature learning in homogeneous neural networks, highlighting how mirror maps shape both optimization dynamics and the geometry of the learned classifiers.

We develop a framework for analyzing parameter symmetries in deep ReLU networks and obtain a complete characterization of the generic parameter fibers for three-layer bottleneck architectures. Our approach provides explicit semi-algebraic descriptions of these fibers and yields a polynomial time algorithm for deciding functional equivalence of two parameters. The symmetries include discrete and continuous transformations arising from layer composition, and depend on whether deeper layers hide or preserve geometric structure from preceding layers. Finally, we show that some of these symmetries induce local conservation laws along gradient flow, while others do not.

Wide neural networks in the feature-learning regime drive modern deep learning, and yet they remain far less studied than their kernel-regime counterparts. We consider a critical yet under-explored difference between these two regimes: the regulariser and prior implied by gradient flow training. This canonical regularisation property is well-studied in kernel regime networks -- of all the infinite global minima, gradient flow selects exactly the vanishing ridge solution -- and underpins the celebrated NN-GP correspondence, precisely allowing the modelling of noise during training. However, we prove ridge regularisation biases gradient flow in feature-learning regime networks, even in the infinitesimal limit of vanishing regularisation. Over training, ridge distorts the inductive bias of the network, with a particular damage done to pretrained networks where the implicit prior is informative. We resolve this by axiomatising the canonical regulariser as a regime-agnostic function-space energy and lift, which uniquely identifies ridge in the kernel regime, and crucially generalises to the feature-learning regime. By studying the Riemannian geometry of feature-learning networks, we derive geodesic ridge from our framework, generalising ridge to the feature-learning regime. Correspondingly, we prove the canonical function-space prior is a Riemannian Gibbs Process, generalising the more familiar Gaussian Process. As a practical contribution, we propose arc ridge as a minimax-robust, scalable surrogate to geodesic ridge, revealing a deep relationship between early stopping and canonical regularisation across learning regimes. Finally, we demonstrate the consequences of our theory empirically on both image processing and NLP transfer-learning problems.

Hard-thresholding is an important type of algorithm in machine learning that is used to solve $\ell_0$ constrained optimization problems. However, the true gradient of the objective function can be difficult to access in certain scenarios, which normally can be approximated by zeroth-order (ZO) methods. The SZOHT algorithm is the only algorithm tackling $\ell_0$ sparsity constraints with ZO gradients so far. Unfortunately, SZOHT has a notable limitation on the number of random directions % in ZO gradients due to the inherent conflict between the deviation of ZO gradients and the expansivity of the hard-thresholding operator. This paper approaches this problem by considering the role of variance and provides a new insight into variance reduction: mitigating the unique conflicts between ZO gradients and hard-thresholding. Under this perspective, we propose a generalized variance reduced ZO hard-thresholding algorithm as well as the generalized convergence analysis under standard assumptions. The theoretical results demonstrate the new algorithm eliminates the restrictions on the number of random directions, leading to improved convergence rates and broader applicability compared with SZOHT. Finally, we illustrate the utility of our method on a ridge regression problem as well as black-box adversarial attacks.

In recent years, transformer architectures have revolutionized the field of language processing, opening the door to previously unforeseen possibilities. However, from a theoretical point of view, the mathematical models proposed in the literature often lack direct contact with the actual architectures and depend on strong simplifying assumptions. In this paper, we reduce this gap by modelling the data flow in multi-headed transformer architectures as time-dependent gradient flows for a suitable interaction energy capturing the design of the attention mechanism. The explicit dependence on time allows us to consider different weights for each head and for each layer, without imposing constraints on the initialization method. Moreover, we prove that, under a suitable integrability assumption on the evolution of the weights, each element of the $ω$-limit set of the gradient flows is a stationary point of the interaction energy at a limiting weight distribution. Finally, we analyse the stability of the gradient flows considering perturbations of both the initial data and the weights. Specifically, on the one hand, we study the robustness of the proposed models with respect to noisy inputs, establishing a continuous dependence of the gradient flows on the initial data and uniqueness of the flows. On the other hand, we prove the $Γ$-convergence of the perturbed interaction energy to the unperturbed one, leading to the convergence of the corresponding gradient flows. We complement these theoretical results with numerical experiments that confirm the predicted energy-dissipation identity and clarify the asymptotic behavior of the dynamics in both the autonomous-like (Ornstein--Uhlenbeck) and the genuinely non-autonomous (oscillating-weights) regimes.

The Average Gradient Outer Product (AGOP) governs feature learning in neural networks: the Neural Feature Ansatz states that weight Gram matrices at each layer align with the corresponding AGOP matrices computed over the training distribution. We ask a complementary question: can this same quantity serve as a post-hoc attribution method for explaining individual predictions? We introduce AGOP-Weighted: a novel attribution method that multiplies the per-sample gradient by sqrt(diag(M) / max diag(M)), a training-distribution prior that suppresses gradient noise and amplifies consistently important pixels -- a combination not present in any prior attribution method. We formalise two companion variants -- AGOP-Local (per-sample gradient, equivalent to VanillaGrad) and AGOP-Global (diag(M) directly as a zero-cost saliency map) -- and implement an efficient training-time accumulation hook; AGOP-Global then requires zero inference cost (disk lookup) while AGOP-Weighted requires only a single gradient pass. We conduct the first rigorous comparison of AGOP attribution against Integrated Gradients (IG), SmoothGrad, GradCAM, and VanillaGrad across two benchmarks with pixel-level ground truth: (i) the synthetic XAI-TRIS benchmark (four classification scenarios, 8x8 images, CNN8by8) and (ii) the photorealistic CLEVR-XAI benchmark (ResNet-18 fine-tuned from ImageNet). AGOP-Weighted achieves 44% higher mIoU than IG on linear tasks; AGOP-Global achieves 7x higher mIoU than IG on multiplicative tasks (where IG falls below random) at zero inference cost. Both findings generalise to ResNet-18 on CLEVR-XAI (+18% and +37% respectively). We further show that GradCAM fails on small-resolution images due to spatial resolution collapse, and that diag(M) quality improves monotonically throughout training even after classification accuracy has plateaued.

We propose Sobolev-regularized Maximum Mean Discrepancy (SrMMD) gradient flow, a regularized variant of maximum mean discrepancy (MMD) gradient flow based on a gradient penalty on the witness function. The proposed regularization mitigates the non-convexity of the MMD objective and yields provable \emph{global} convergence guarantees in MMD in both continuous and discrete time. A more surprising appeal is that our convergence analysis does not rely on isoperimetric assumptions on the target distribution. Instead, it is based on a regularity condition on the difference between kernel mean embeddings. A key highlight of the proposed flow is that it is applicable in both sampling (from an unnormalized target distribution) -- using Stein kernels -- and generative modeling settings, unlike previous works, where a gradient flow is suitable for only generative modeling or sampling but not both. The effectiveness of the proposed flow is empirically verified on a broad range of tasks in both generative modelling and sampling.

Diffusion models and flow-based methods have shown impressive generative capability, especially for images, but their sampling is expensive because it requires many iterative updates. We introduce W-Flow, a framework for training a generator that transforms samples from a simple reference distribution into samples from a target data distribution in a single step. This is achieved in two steps: we first define an evolution from the reference distribution to the target distribution through a Wasserstein gradient flow that minimizes an energy functional; second, we train a static neural generator to compress this evolution into one-step generation. We instantiate the energy functional with the Sinkhorn divergence, which yields an efficient optimal-transport-based update rule that captures global distributional discrepancy and improves coverage of the target distribution. We further prove that the finite-sample training dynamics converge to the continuous-time distributional dynamics under suitable assumptions. Empirically, W-Flow sets a new state of the art for one-step ImageNet 256$\times$256 generation, achieving 1.29 FID, with improved mode coverage and domain transfer. Compared to multi-step diffusion models with similar FID scores, our method yields approximately 100$\times$ faster sampling. These results show that Wasserstein gradient flows provide a principled and effective foundation for fast and high-fidelity generative modeling.

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical documentation, these phenomena have resisted unified theoretical treatment: existing explanations are post-hoc, each fitted to a specific observation with whatever mathematics is at hand. We reverse this direction by deriving the mathematical structure that layerwise alignment inherently demands. Using geometric invariant theory, we prove that alignment geometry has a canonical closed, polystable stratum given by a flag variety, and that subspace intersection dimension is its unique reparameterization-invariant observable, establishing that subspace metrics are not empirical conventions but mathematical necessities. This unified framework yields two dynamical consequences: ridge regularization drives subspace alignment at an exponential rate set by weight decay, whereas nonlinear activations induce a commutator obstruction to exact basis alignment, generically present in nonlinear networks and absent in linear ones. Together these give a geometric explanation of the Level-2/3 hierarchy in Neural Collapse from first principles rather than post-hoc analysis. The commutator magnitude and head subspace overlap further serve as weight-space windows into internal alignment structure, requiring no forward passes. Experiments on multilayer perceptrons, residual networks, and pretrained language models support the proposed diagnostics and delineate their scope.

Why does full-pipeline FP4 training of large language models often diverge, even when forward activations and activation gradients remain stable? We address this question through a controlled study of MXFP4 quantization in transformer training, progressively enabling FP4 across forward propagation (Fprop), activation gradients (Dgrad), and weight gradients (Wgrad) while holding all other factors fixed. In full pretraining of Llama 3.1-8B on the C4 dataset, we observe that quantizing Wgrad is the primary driver of convergence degradation, whereas FP4 in Fprop and Dgrad alone introduces only modest additional token requirements. To interpret this behavior, we evaluate both structured and stochastic interventions under a controlled experimental setting. We find that stochastic rounding and randomized Hadamard rotations fail to stabilize training once Wgrad is quantized, whereas deterministic Hadamard rotations consistently restore stable optimization. These results suggest that FP4 training instability is driven by structured micro-scaling errors along sensitive gradient paths, rather than by insufficient stochasticity. We run experiments with native MXFP4 support on AMD Instinct MI355X GPUs, enabling controlled investigation of these effects without reliance on software emulation.

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.

Score Distillation Sampling (SDS) and its variants have been widely used for text-to-3D generation by distilling 2D image diffusion priors. However, the standard SDS objective is prone to severe mode collapse, frequently yielding over-smoothed and over-saturated results. Although recent advancements, such as Score Distillation via Inversion (SDI), mitigate these artifacts and produce visually sharper models, they ultimately fail to faithfully capture the full target distribution. In this work, we show that the bottleneck limiting the sampling capacity of SDI stems from its reliance on the posterior mean estimator, which is mathematically equivalent to a single-step Euler approximation of the deterministic reverse DDIM trajectory. To address this, we propose a naturally motivated extension termed Probability-Flow Distillation (PFD). We establish that PFD corresponds exactly to a Wasserstein gradient flow, thereby inducing principled distribution-matching dynamics. Finally, we show that PFD can synthesize 3D assets with fine-grained, high-fidelity details and achieve improved quality compared to existing methods.

The population dynamics of molecules, cells, and organisms are governed by a number of unknown forces. In the last decade, population dynamics have predominantly been modeled with Wasserstein gradient flows. However, since gradient flows minimize free energy, they fail to capture important dynamical properties, such as periodicity. In this work, we propose a change in perspective by considering dynamics that minimize a population-level action under a damped Wasserstein Lagrangian. By deriving the corresponding Hamiltonian equations of motion, we formalize Wasserstein Lagrangian Mechanics, a structured class of second-order dynamics that encompasses classical mechanics, quantum mechanics, and gradient flows. We then propose WLM as the first algorithm that learns these second-order dynamics from observed marginals, without specifying the Lagrangian. By directly learning the population mechanics, WLM can both forecast and interpolate unseen marginals, and outperforms existing gradient flow and flow matching methods across a wide range of dynamics, including vortex dynamics, embryonic development, and flocking.

We introduce a hybrid quantum-classical pipeline, based on neutral-atom reservoir computing, for medical image classification, focusing on the binary classification task of polyp detection. To deal effectively with the high dimensionality, we integrate a guided auto-encoder. This pipeline learns compact and discriminative representations of image data that are also well-suited for quantum reservoir computing. A key challenge in such systems is the non-differentiable nature of quantum measurements, which creates a 'gradient barrier' for standard training. We overcome this barrier by incorporating a differentiable surrogate model that emulates the quantum layer, enabling end-to-end backpropagation through the entire system. This guided training process is jointly optimized for classification accuracy and for faithful image recovery from the auto-encoder. The learned latent representations are encoded as pulse detuning parameters within a Rydberg Hamiltonian, and quantum embeddings are subsequently obtained through expectation values. These embeddings are then passed to a linear classifier. Our simulations show that this method outperforms some traditional approaches that use PCA or unguided autoencoders. We also conduct ablation studies to assess the impact of various quantum and training parameters, demonstrating the robustness and flexibility of our proposed pipeline for real-world medical imaging applications, even in the current NISQ era.

Incorporating prior data into online reinforcement learning accelerates training but typically forces a difficult trade-off between high computational costs and long, multi-stage training pipelines. While fixed-length stabilization phases are significantly more computationally efficient than static update schedules, they require task-dependent manual tuning, risking either the waste of prior knowledge or severe overfitting. To address this, we propose SOPE, an algorithm that uses an actor-aligned Off-Policy Policy Evaluation (OPE) signal as an automated early-stopping mechanism to dynamically control the length of offline training phases. By evaluating the critic on a held-out validation split under the current policy's action distribution, SOPE halts gradient updates exactly when out-of-distribution benefits saturate, eliminating the need for manual schedule tuning. Evaluated on 25 continuous control tasks from the Minari benchmark suite, SOPE improves baseline performance by up to 45.6% while reducing the required TFLOPs by up to 22x, thus balancing the tradeoff between sample and computational efficiency. These findings demonstrate that adaptive, evaluation-driven update schedules are more effective than relying on static, exhaustive update schedules.

The main XAI attribution methods for deep neural networks -- GradCAM, SHAP, LIME, Integrated Gradients -- operate on separate theoretical foundations and are not formally comparable. We present GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), a mathematical framework establishing a representation theory for attributions: every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM, but excludes nonlinear functionals such as standard GradCAM or attention maps. Seven formal theorems provide simultaneous guarantees absent in any individual method: (T1) necessary canonical form; (T2) exact completeness; (T3) Monte Carlo convergence O(1/sqrt(m))+O(1/k); (T4) exact Shapley Interaction Values; (T5) Hoeffding ANOVA decomposition; (T6) Sobol sensitivity generalization; (T7) multi-scale extension (MS-GRALIS) with minimum-variance weights. An algebraic appendix justifies the GRALIS-SIV correspondence via the Mobius transform without circularity. GRALIS satisfies 13.5/14 axiomatic properties vs. 2.5-6/14 for individual methods, including completeness, sensitivity, locality, order-k interactions and optimal multi-scale aggregation simultaneously. Preliminary validation on BreaKHis (1,187 histology images, DenseNet-121) reports deletion faithfulness AUC +0.015 (malignant), 96% class-conditional consistency, SAL = 0.762+/-0.109 and sparsity index 0.39. Extended comparison with baseline XAI methods is planned for a companion paper.

LLM post-training typically propagates task gradients through the full depth of the model. Although this end-to-end structure is simple and general, it couples task adaptation to full-depth activation storage, long-range backward dependencies and direct task-gradient access to pretrained representations. We argue that this full-depth backward coupling can be unnecessarily expensive and intrusive, particularly when post-training supervision is much narrower than pre-training. To this end, we propose \textbf{LoPT}: Local-Learning Post-Training, a simple post-training strategy that makes gradient reach an explicit design choice. LoPT places a single gradient boundary at the transformer midpoint: the second-half block learns from the task objective, while the first-half block is updated by a lightweight feature-reconstruction objective to preserve useful representations and maintain interface compatibility. LoPT shortens the task-induced backward path while limiting direct interference from narrow task gradients on early-layer representations. Extensive experiments demonstrate that LoPT achieves competitive performance with lower memory cost, higher training efficiency and better retention of pretrained capabilities. Our code is available at: https://github.com/HumyuShi/LoPT