Stochastic Gradient Descent

We study the long-time behavior of Wasserstein gradient flows for interaction energies $$\mathsf E[μ] = \frac12\iint_{M\times M}K(x,y)\,\mathrm dμ(x)\,\mathrm dμ(y)$$ on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies $$\mathsf E[μ_t]-\mathsf E_{\min}=o(t^{-1}).$$ If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $δ>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-δ}$ along a sequence of times tending to infinity. Moreover, for any $δ>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-δ}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

Quasars are luminous objects in the universe that exhibit stochastic brightness variations encoding information about the supermassive black holes powering them, and modeling these variations from ground-based survey data time series, known as light curves, is a statistical challenge. This paper reviews how stochastic differential equations (SDEs) have been adapted with neural network parameterizations to overcome this challenge in history. We create the Continuous-Delayed-Memory Stochastic Gradient Descent which depend on the past state of the discrete iteration process. We performed the simulation on some 2-dimensional landscape and observed some wider-exploration and more precise convergent behavior compared to Vanilla SGD by adjusting hyperparameters. Besides, we proposed a reinforcement learning structure with continuous time policy gradients for exploratory policies without solving HJB PDE, and we show that its optimality conditions recover the Gibbs policy of previous works.

Scaling deep learning faces critical bottlenecks: data exhaustion, exponential training costs, and resource concentration. Model merging combines pre-trained checkpoints without gradient descent, offering orders-of-magnitude savings versus retraining. Combining independently trained vision models is difficult when their architectures and parameter shapes differ. Existing weight-space merging methods generally assume aligned, shape-compatible checkpoints, whereas a Vision Transformer (ViT) and a state-space model (SSM) implement token mixing with different operators. We study a hybrid Heterogeneous merging setting that retains both architectures while aligning parameter groups by semantic role. Our proposed Riemannian--Lorentz Parameter Fusion (RLPF) method projects aligned groups to common coordinates, lifts selected coordinates to the Lorentz hyperboloid model of hyperbolic space, computes a regularized geodesic barycenter, and decodes the result into the two branches. A learned gate then combines branch logits for each input. Component groups use fixed curvature values, with normalization parameters treated as Euclidean. In the results available in this manuscript, the fine-tuned system obtains 82.37% on CIFAR-10, 75.04% on Oxford-IIIT Pet, and 78.58% top-1 accuracy on ImageNet-1K; the corresponding best-parent accuracies are 76.54%, 71.42%, and 76.42%. On ImageNet-1K, the reported pre-fine-tuning initialization reaches 77.80%. These results support further study of geometry-aware heterogeneous fusion, but not a training-free single-checkpoint merge: RLPF is a two-branch hybrid whose gate and reported final models are trained.

The test error of a model plotted against its number of parameters $d$ falls, peaks when the model can just fit the training data, and falls again, exhibiting the double descent phenomenon. We explain the phenomenon with statistical mechanics. The training trajectory of a stochastic gradient-based method is a particle wandering over the energy landscape of the training loss at an induced temperature $T$, and a run that has equilibrated visits every parameter vector of a given training loss equally often, the fundamental postulate of statistical mechanics, with probability given by the Boltzmann distribution. Because training starts at an initial point and has only finite time to diffuse, it carries an effective weight decay, which makes every parameter a quadratic degree of freedom. The equipartition theorem then distributes the energy among the $d$ degrees of freedom in shares of $T/2$, so at a fixed training loss adding parameters lowers the temperature and drives the Boltzmann distribution toward the stationary path. Finally, adding parameters can only lower the $L^2$ norm of the stationary path, so a solution sampled at fixed loss is less likely to be large with increasing $d$, effectively increasing weight regularization.

In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+ξ_j$ or by a known physical constraint $Du^*=v$. We consider the setting where $D$ is a linear differential operator and analyze a physics-informed kernel estimator $\hat u$ combining $n$ value observations and $m$ differential observations. In this context, we ask how much can differential information improve predictions, and how does this improvement depend quantitatively on $n$, $m$, and $D$. We prove finite-sample bounds, supported by numerical simulations, revealing a two-regime structure for the prediction error. When $m$ is limited, the rate depends jointly on $n$ and $m$; when $m$ exceeds a problem-dependent threshold, the rate saturates and matches the oracle rate obtained when the perfect constraint $D \hat u = Du^*$ is imposed. Examples are discussed for Sobolev spaces which are reproducing kernel Hilbert spaces and include partial Laplacian constraints on the torus and gradient observations on bounded domains. These examples illustrate the range of possible learning rate improvements --- from the standard nonparametric $n^{-1/4}$ to the parametric rate $n^{-1/2}$. Finally, we derive physically consistent rates in a stronger norm that jointly controls the errors in $\hat u$ and $D\hat u$.

Stochastic subspace methods have gained popularity as gradient descent based techniques for large scale optimisation problems, especially in distributed settings. In this paper, we introduce the technique of "persistence of memory" to greatly extend and improve the random subspace methods. To this end, we leverage a vector that is only weakly correlated with the gradient in order to provide a guiding structure to the generative process of the random subspace along which the descent is going to take place. This guidance vector may be fixed for a large number of iterations, only to be refreshed at wide intervals (on whose size we can provide guarantees in terms of problem parameters). In important machine learning settings, such as optimisation problems embodying sparsity or a minibatch structure, we show that the guidance vector can be obtained in an effective and computationally inexpensive manner by leveraging the structured properties of the problem. En route, we establish to our knowledge the first theoretical analysis of classical SSD methods for sparse functions. In a local neighbourhood of the optimum, we demonstrate an alignment phenomenon of our gradient estimates with a low-lying eigenvector of the Hessian, allowing a once-for-all computation of the guidance vector which renders the method computationally favourable even in scenarios with unstructured objectives.

The availability and consistency of online services remain vulnerable due to Distributed Denial of Service (DDoS) attacks. These attacks are evolving by adopting more complex strategies to evade traditional network security systems. Despite the effectiveness of machine learning models in detecting DDoS traffic, targeted adversarial attacks can degrade their classification accuracy. This work proposes a robust detection framework that integrates generative adversarial modelling with advanced machine learning models. We trained Random Forests, Deep Neural Ensembles, and Transformer-based models using the CICDDoS2019 dataset to establish the frameworks baseline performance. To enhance the models defensive capacity, we generated synthetic adversarial flows that simulate potential evasion attempts and adversarial traffic using a Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP). Then, we combined the generated traffic with benign and malicious traffic to construct hybrid datasets to train the models to learn more generalizable decision boundaries. The experimental results indicate that the proposed methodology significantly enhances detection accuracy and resilience, especially against unseen adversarial traffic. We also tested the designed framework using real-world generated traffic, which demonstrates its capability in practical settings. The scalable and efficient solution against adversarial DDoS attacks, introduced in this work, paves the way towards more resilient and adaptive network defense systems that combine generative adversarial augmentation with recent advances in learning models.

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.

We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of $O(L^2D_Y\barΔ_0\varepsilon^{-3})$, where $L$ is the gradient Lipschitz constant, $D_Y$ bounds the diameter of the dual feasible set, and $\barΔ_0$ is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to $O(L^2D_YΔ_φ\varepsilon^{-3})$, up to an additive lower-order cost, where $Δ_φ:=φ(x_0)-\inf_xφ(x)$. We further establish a lower bound of $Ω(L^2D_YΔ_φ\varepsilon^{-3})$ for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves $\mathcal{O}\!(L^{3/2}D_Y^{1/2}Δ_φ\varepsilon^{-5/2})$ gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves $O\!(\sqrtκ\,LΔ_φ\varepsilon^{-2})$ leading complexity for both stationarity criteria, where $κ=L/μ$ is the dual condition number, up to an additive initialization cost. The $\varepsilon^{-2}$ accuracy dependence is optimal under fixed regularity and initialization bounds.

Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.

The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices. Euclidean, Bures-Wasserstein and affine-invariant metrics are common choices, but their relative effectiveness depends on the objective. We introduce a two-parameter family defined by $X^{p}LX^{q}+X^{q}LX^{p}=U$, solved for $L$ at each tangent vector $U$, that contains all three as exact members, at $(0,0)$, $(1,0)$ and $(1,1)$, and extends past them. We treat the choice of member as a particular way of preconditioning for a given problem. To this end, we analyze the conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that depends on $(p,q)$ only through the exponent $r=p+q$. When the Euclidean Hessian is a pure power that mixes no eigendirections, the member $p=q=r/2$ attains that bound, and a closed-form criterion identifies the other members that do. We discuss ways to tune $r$ for a given problem. Experiments on real covariance data confirm the predicted conditioning and the benefit of tuning $r$. A task covariance example shows a further gain from tuning the shape.

High-capacity associative memories based on Kernel Logistic Regression (KLR) exhibit a "Ridge of Optimization" characterized by extreme stability and a highly skewed weight spectrum. However, the dynamical process by which learning converges to this critical regime has remained unclear. This paper provides a geometric analysis of the learning trajectories on the statistical manifold of a KLR-trained Hopfield network. By comparing the paths of Gradient Descent (GD) and Natural Gradient Descent (NGD), we elucidate the mechanisms governing the optimization process. Our analysis reveals that learning on the Ridge proceeds in two distinct phases. We show that the extreme curvature of the Ridge causes standard GD to follow a highly oscillatory, non-geodesic path. In stark contrast, NGD explicitly corrects for this geometry, following the ideal geodesic path and completely overcoming the instabilities faced by GD. We demonstrate experimentally that NGD not only converges significantly faster but also achieves a solution with superior generalization performance. These results establish that the highly structured geometry of the Ridge is optimally suited for information-geometric optimization, providing a new perspective on the interplay between learning dynamics and emergent representation geometry.

Warped Gaussian processes (GPs) handle non-Gaussian observations by mapping them into a latent standard GP via a parametric transformation called warping. Existing streaming variants, however, either optimize the warping parameters periodically or sacrifice analytical tractability for a higher model capacity. To bridge this gap, we show that the gradient of the instantaneous negative log-likelihood of a warped GP admits an exact recursive computation. Based on this result, we propose a novel online method for warped GPs that jointly updates the latent GP moments and optimizes the warping parameters.

Federated stochastic bilevel optimization has been actively studied in recent years due to its widespread applications in machine learning. However, most existing federated stochastic bilevel optimization algorithms require the computation of second-order Hessian and Jacobian matrices, which leads to longer running times in practice. To address these challenges, we propose a novel federated stochastic variance-reduced bilevel gradient descent algorithm that relies solely on first-order oracles. Specifically, our approach does not require the computation of second-order Hessian and Jacobian matrices, significantly reducing running time. Furthermore, we introduce a novel learning rate mechanism, i.e., a constant single-timescale learning rate, to coordinate the update of different variables. We also present a new strategy to establish the convergence rate of our algorithm. Finally, the extensive experimental results confirm the efficacy of our proposed algorithm.

Pretraining followed by fine-tuning introduces a compute-allocation problem: under a fixed training budget, compute spent improving the upstream objective reduces the compute available for downstream adaptation. Despite its practical importance, this trade-off is not yet well understood theoretically, even in simple models. In this paper, we cast this allocation as a compute-split problem under a two-stage pretrain--fine-tune procedure with fixed total optimisation budget, using regularised least squares trained by gradient descent as a tractable setting. We characterise the optimal split under data-dependent evaluation geometries induced by the fine-tuning problem. Our results show that the allocation depends on how pretraining directions affect fine-tuning predictions and how fine-tuning shifts are seen through downstream data geometry. In particular, the relevant quantities are determined by prediction-relevant spectral components of the pretraining and fine-tuning empirical covariances. Technically, the analysis relies on a basis-invariant, eigenspace-level spectral decomposition, together with perturbative control of the non-commuting pretraining and fine-tuning dynamics.

Stopgrads are widely used in training machine learning models, but stopgrads can alter the gradient, stationary points and convergence guarantees of the original objective, which can make stopgrad training theoretically ungrounded. We introduce a stopgrad regression principle, which identifies a general template for stopgrad objectives with a closed-form characterization of stationary points and their uniqueness, unifying stopgrad objectives for flow maps, reinforcement learning, and diffusion samplers. We provide theoretical grounding for optimizing stopgrad flow map objectives by showing their unique stationary point is the true flow map, and showing positive convergence results for Eulerian and Lagrangian objectives, including MeanFlow and improved MeanFlow. Remarkably, we show that under functional semi-gradient flow, the learned flow map has a closed-form expression composing the initial flow map and the true flow map. We additionally use our stopgrad regression principle to propose modified stopgrad placements for flow map objectives which reduce training memory by 2x.

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.

Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice. In contrast, we study nonconvex stochastic optimization under the joint relaxation of these assumptions: objectives with $(L,s)$-H"older continuous gradients, $s\in(0,1]$, and gradient noise satisfying only a bounded $\alpha$-th moment condition for $\alpha\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $\alpha\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and H"older smoothness simultaneously. Secondly, we analyze $\delta$-regularized gradient clipping ($\delta$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(\alpha-1)/[(1+s)(2\alpha-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $\alpha\ge1+s$ while in the very heavy-tailed regime $\alpha<1+s$, it has a convergence rate $O(T^{-2s(\alpha-1)/[(\alpha-1)+s(2\alpha-1)]})$ --- the first convergence guarantee in this regime for any stochastic gradient based method.

Deep neural networks are expressive enough to contain worst-case targets that can be evaluated in polynomial time but cannot be learned in polynomial time by gradient descent. For practical tasks they nonetheless learn well, raising the question of what non-generic structure of real-world targets enables this. Existing surrogate models cannot pose this question because they either lack hard-to-learn targets entirely (deep linear networks) or cannot evaluate such targets efficiently (kernel methods, infinite-width limits). We study tree tensor networks (TTNs), a model class that generalizes deep linear networks and Tucker decompositions. We show they embed arbitrary read-once Boolean formulas, and thus contain polynomial-size targets that cannot be learned by gradient descent in polynomial time under the same mechanism as neural networks. Despite this, we prove that their loss landscapes are conditionally benign for every realizable target: every local minimum that is minimum-norm is global. Thus, surprisingly, bad local minima are not what distinguishes between typical and worst-case problems in TTNs. Instead, learning difficulty in TTNs can arise from high-order degenerate saddle points, which we show are caused by rank-deficiency. This is explored through a case study of the parity function, illustrating the potential for TTNs to relate landscape geometry to computational hardness.

We investigate the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in any fixed Euclidean dimension. The potential is $\mu$-strongly convex and $L$-smooth, with an unknown mode in the ball of radius $\mu^{-1/2}$ about the origin. We have access to unbiased stochastic oracles with the variance at most $\sigma^2$. For every $\sigma^2\ge0$ and total variation (TV) accuracy $0<\varepsilon\le1/10$, we prove that the tight complexity of sampling a distribution within $\epsilon$-TV distance from the target distribution is $$N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)+ \frac{\sigma^2}{\mu\epsilon}\right),$$ where $\kappa:=\frac L\mu$ is the condition number. Note that this complexity bound is simultaneously tight for the condition number $\kappa$ and accuracy $\epsilon$. Besides, our tight complexity bound is adaptive to noiseless setting $\sigma=0$, which is $N^\star_{\text{TV}}=\Theta\!\left(\log(1+\kappa)\right)$.

We study loss-based filtering for finite-sum optimization with a subset of corrupted component functions whose gradients may be highly unreliable. Motivated by minimum-loss-based SGD (min-$k$-loss) and quantile-based methods for corrupted linear systems, we propose and analyze a general loss-filtering framework -- Quantile-$k$-Loss SGD (Q$k$L-SGD) -- that samples $k$ component losses at each iteration and updates using an index chosen uniformly from the lower empirical $q$-quantile. We prove linear convergence of this family of methods under standard convexity assumptions, requiring the sample size to scale with the number of corruptions and a subset strong-convexity threshold. For the cases when large enough sampling is impossible or undesirable, we give a complementary small-sample probabilistic analysis that covers any sample size $k$ and the convergence behavior depends on the probability of selecting an outlier and on the curvature of the selected good step. Experiments on polynomial regression, regularized logistic regression, and regularized hinge loss show that intermediate quantiles often outperform both standard SGD and min-$k$-loss SGD. In particular, min-$k$ often stalls by repeatedly selecting nearly solved components, while intermediate quantiles retain robustness and produce more informative updates.

Stochastic gradient descent (SGD) is the primary workhorse for large-scale optimization. While the average behavior of its iterates, typically characterized by mean-squared error bounds, is well-understood, obtaining high-probability guarantees for the last iterate remains challenging. Prior approaches to this problem have either imposed restrictive assumptions (such as bounded domains or gradients) or relied on complex proofs involving auxiliary sequences. In this work, we propose Batched SGD, a simple variant that partitions online samples into epochs and performs a single update per epoch using a refined, low-variance gradient estimate. Our main contribution demonstrates that this batching mechanism enables a surprisingly simple high-probability analysis that avoids both restrictive assumptions and auxiliary sequences. Under standard smoothness and norm-sub-Gaussian noise assumptions, we establish near-optimal rates for both strongly convex and non-convex objectives. Furthermore, we show that our batching idea extends naturally to federated learning (FL). We provide the first high-probability guarantees for FL, achieving logarithmic communication complexity, linear speedup in the number of agents, and resilience to data heterogeneity.

Finite element stress fields often exhibit strong local non-smoothness, where stress concentrations near holes, notches, and loading regions induce sharp spatial gradients and high-frequency graph components. Although graph neural networks naturally operate on irregular finite element meshes, conventional message passing is inherently smoothing and progressively attenuates such high-frequency information. Unitary propagation alleviates this problem by preserving spectral magnitudes, but typically relies on matrix functions and high-order approximations with $O(Ked)$ propagation complexity. We propose LiftGCN, an efficient spectrally stable graph network based on Joukowski spectral lifting. LiftGCN maps the real spectrum of a normalized graph operator onto the unit circle through the Joukowski relation and realizes the resulting spectral transformation as a simple second-order recurrence, avoiding matrix exponentials, eigendecomposition, and high-order polynomial truncation. We show that the linear Joukowski backbone has unit-modulus characteristic roots and admits an energy-preserving structure under a positive-definite metric, preventing exponential attenuation of graph-frequency components with depth. Each layer requires only one sparse neighborhood aggregation, yielding $O(ed)$ propagation complexity, while lightweight local nonlinear residuals provide expressive feature transformations. Experiments on finite element stress prediction demonstrate that LiftGCN achieves competitive overall accuracy while improving reconstruction of stress concentrations and local high-gradient structures with substantially reduced computational cost. Our code is available at https://github.com/ChenZeng001/LiftGCN.

Stochastic gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of stochastic gradients by Gaussian noise, providing a powerful tool for understanding its dynamics and long-time behavior. We investigate whether an analogous approximation principle holds for optimization over probability measures, where the objective is a functional defined on the Wasserstein space P2. The nonlinear geometry and infinite-dimensional nature of P2 prevent a direct extension of the classical Euclidean theory. Using Lions differentiability, we lift the problem to a linear Hilbert space, where higher-order differential calculus becomes available. We then construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. By exploiting this moment matching through higher-order Taylor expansions, we show that the Gaussian approximation captures the SGD dynamics with second-order weak accuracy. Our result provides a rigorous foundation for replacing sample-driven randomness by analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

Stochastic gradient descent (SGD) with gradient clipping and additive noise has become a standard technique for training machine learning models, particularly in applications requiring robustness or privacy guarantees. However, clipping introduces a bias in stochastic gradients, while additive noise introduces additional variance, making the long-run behaviour of individual optimization trajectories difficult to characterize. In this work, we prove that SGD with clipping and additive Gaussian noise (SGD-CN) converges almost surely (a.s.) under smoothness and uniformly bounded stochastic-gradient noise assumptions, provided the step sizes satisfy some standard decaying conditions. Our analysis extends to momentum variants such as the stochastic heavy ball and Nesterov's accelerated gradient, where we show that careful energy constructions yield similar guarantees. These results provide stronger theoretical foundations for understanding the pathwise behaviour of clipped stochastic gradient methods and suggest that, despite the bias and noise introduced by clipping and perturbation, the algorithm remains stable in both convex and nonconvex regimes.

We introduce AdamX, a first-order optimizer that incorporates cosine similarity as an adaptive mechanism for controlling update magnitudes. The proposed method is scalable, model-agnostic, and straightforward to integrate into existing training pipelines. We further introduce a variance rectification scheme that promotes smoother optimization during the early stages of training. Overall, we provide empirical evidence that AdamX achieves competitive convergence rates across a range of benchmark datasets and architectures. Performance is evaluated in terms of the number of epochs required to reach predefined performance thresholds under a fixed hyperparameter budget. Code and Experiments available at: https://github.com/FranciscoCaldas/adamX.

In this paper, we investigate the generalization performance of distributed gradient descent algorithms in a reproducing kernel Hilbert space under a robust loss function $l_σ$. By exploiting the spectral characterization of gradient descent together with the intrinsic properties of robust loss functions, we establish optimal learning rates for the distributed kernel-based robust gradient descent (DKRGD) algorithm with an appropriately chosen scale parameter $σ$. The proposed parameter choice of $σ$ simultaneously alleviates the saturation phenomenon and guarantees statistical robustness. A key technical contribution is a novel error analysis that provides substantially sharper bounds for products of operators, thereby significantly relaxing existing restrictions on the maximum number of local machines while retaining optimal learning rates. Finally, we develop a communication-efficient strategy that further improves the convergence performance of DKRGD.

Federated learning shares model updates rather than raw data, yet these updates can be inverted to reconstruct the clients' training data. Analytic reconstruction attacks, which invert a gradient in closed form, degrade as the batch grows: prior single-round attacks recover only about half of a batch of size $100$ even when the attacker fully controls the network parameters, and known upper bounds limit what any such method can recover. We establish a connection between gradient inversion and the theory of erasure-correcting codes, and use it to construct attacks that exceed these bounds. Our attacks recover batches exactly, together with every sample's label, from a single FedSGD round, and certify each recovery without ground-truth data. On eight image and tabular benchmarks they outperform prior single-round attacks by a wide margin. Even a passive attacker who only observes an honestly trained network recovers $94$--$100\%$ of ImageNet batches at sizes up to $128$, more than prior single-round attacks achieve even with active manipulation of the model, and in the active setting more than $90\%$ is recovered at batch sizes of several hundred. These results show that the privacy leakage of federated learning has been underestimated.

Synthetic data has become a promising way to scale model training beyond limited human-generated data but it may also induce strong model collapse (Dohmatob et al., 2024), where any fixed fraction of synthetic data prevents model performance from improving under data scaling, leaving a non-vanishing excess risk floor. In this paper, we study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift. We establish finite-sample risk bounds for mixed and two-stage training, separating standard bias and variance from source-mismatch effects, namely fluctuation and persistent drift under mixing and filtered initialization bias under two-stage. These bounds reveal a sharp contrast: mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage, showing that collapse is not inevitable under a simple data curriculum. Under a random sketch model, we further obtain scaling laws for both protocols, with tight results for mixed training in the optimization-saturated regime. These laws show that larger models may amplify synthetic-induced degradation under mixing, and quantify how high-quality synthetic pretraining may reduce bias in two-stage training. Finally, we establish an exact finite-sample necessary-and-sufficient condition for two-stage training to strictly outperform real-only training under the same real-data budget and identical real-stage updates. Overall, our results highlight that synthetic data is neither inherently harmful nor beneficial; its effect depends critically on both its quality and the training protocol used to incorporate it.

Many learning systems return a single point estimate even when admissible outputs form disconnected or non-convex sets. Under squared loss, an ambiguous conditional distribution can therefore have a Bayes-optimal conditional mean that is invalid. We formalize this failure as conditional mean collapse and introduce Settling, an equilibrium-based inference operator that separates proposal generation, consistency evaluation, and test-time equilibrium selection. The operator treats a mean-seeking proposal as an initialization and refines it toward a locally stable configuration; conditional on initialization, refinement is deterministic. We establish exact-gradient descent, local convergence, and an inexact-gradient robustness condition relevant to learned consistency critics. In a reproducible 100-context geometric diagnostic, the mean-seeking baseline succeeds in 0/100 contexts, stochastic denoising in 100/100, and Settling in 99/100 while producing substantially lower trajectory roughness. A 1,200-run sensitivity study yields 97-100% success across obstacle-jitter ranges up to 0.20 and 94-100% across one-time initialization perturbations from 0.05 to 0.50. Cross-domain panels remain mechanism illustrations; learned high-dimensional validation remains an open empirical test.

Online gradient descent is usually studied through external regret, where the learner competes with fixed alternatives. Recent work shows that first-order methods control richer action-dependent deviations. We ask for a geometric characterization of the deviations with respect to which online gradient descent, mirror descent, and follow-the-regularized-leader (FTRL) achieve no regret. We identify exactness as the common principle. Exactness means that the relevant displacement field is generated by a scalar potential, or equivalently that the associated one-form is exact in the geometry used by the algorithm. This geometry depends on the algorithm. For gradient descent it is Euclidean geometry, for mirror descent it is the geometry induced by the regularizer, and for FTRL it is the cumulative dual state. Under mild regularity conditions, exactness yields sublinear regret, while nonzero circulation provides the complementary obstruction and leads to linear regret. This gives a unified geometric framework for understanding the deviation classes controlled by these algorithms and reveals that different first-order methods can control genuinely different classes of deviations. These deviation classes have direct consequences for learning, particularly in games. We study the equilibrium notions induced by exact-form deviations and introduce conservative correlated equilibrium, reflecting both the conservative geometry of the underlying displacement fields and the restricted family of deviations available to the players. We characterize its relation to correlated equilibrium, determine when the resulting equilibrium notions coincide and when they separate, and show how these relationships depend on the geometry and the learning algorithm. Overall, this work gives a unified geometric account of what first-order online learning algorithms are no-regret with respect to, beyond fixed comparators.

Dimensional attention in learning is often implemented as a globally shared attention vector, where each stimulus dimension corresponds to a single scalar. These scalars are learned by models through gradient-descent on error, where predictive features acquire more salience. We show that under multi-outcome learning, where models predict more than one outcome, this shared vector becomes unstable; it collapses to its bounds and prevents the models from learning meaningful attentional tunings for learning and generalization. We address this by introducing an outcome-indexed attentional matrix that converts globally shared attentional tuning into an outcome-indexed representation. We present an analysis of the unstable shared vectors and derive the conditions under which it holds. Empirically, three synthetic experiments benchmark the proposed attention matrices and show that they converge to meaningful representations, something shared attention vectors fail to do. These results suggest that outcome-indexed attentional matrices are a general fix for gradient-based attentional processes, which improves models of learning under multi-outcome conditions.

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.

We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-α$ and uniformly over the problem class, is achievable if and only if $$\sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2} < \infty.$$ The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.

Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about latent variables in the data-generating process. The resulting inequality decomposes the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different latent variables are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.

In nonconvex optimization problems arising in geometric machine learning, data augmentation is commonly used to promote invariance by averaging empirical losses over transformations of the data. Computing the fully augmented objective, however, requires access to every element of the transformation group $G$, which may be prohibitively expensive when $G$ is large or accessible only through sampling. We study whether full augmentation can instead be approximated using a small, fixed sample of transformations acquired before optimization and reused thereafter. Under suitable regularity conditions, we show that, with probability at least $1-δ$, gradient descent (GD) on the resulting sparsely augmented objective returns an $\varepsilon$-stationary point of the fully augmented objective using $\mathcal{O}\bigl((\log |G|+\log(1/δ))/\varepsilon^2\bigr)$ group-transformation-oracle queries. By comparison, standard group stochastic gradient descent (group-SGD), which samples a fresh transformation at every iteration, uses $\mathcal{O}(1/\varepsilon^4)$ transformation queries. Therefore, gradient descent with fixed sparse augmentation requires fewer transformation queries than both GD applied to the fully augmented objective and group-SGD. Our proof techniques, which may be of independent interest, establish a uniform approximation of the full group-averaged gradient field by a random group average using spectral properties of group-induced operators and tools from representation theory.

We study the training dynamics of multiclass logistic regression on high-dimensional Gaussian mixture models with a large number of classes and establish precise scaling laws governing the cross-entropy risk under gradient-based optimization. We show that learning proceeds sequentially across classes, from most to least frequent. When the class priors follow a power law distribution, the risk dynamics decompose into three phases: an initial plateau until the first class is learned, a power-law decay regime during which sequential learning occurs, and a final convergence regime. We then analyze how model capacity interacts with optimization under a fixed compute budget. When the effective dimension is restricted via projection onto leading principal components, the risk decomposes into a capacity term (a power law in the retained dimension) and an optimization term (a power law in training time). Optimizing this tradeoff yields a compute-optimal scaling law for logistic regression, with explicit prescriptions for model size and training time as functions of compute. These results extend theoretical scaling laws from linear regression to multiclass classification, while connecting to empirical scaling laws observed in large-scale neural networks.

Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training dynamics. In this paper, we develop a theoretical framework that characterizes how these factors jointly shape generalization performance throughout training. More precisely, we study a broad class of neural networks trained under the $\ell^2$ loss by gradient descent (GD) with weight decay, and prove the convergence of GD to a neighbourhood of the global minimizers of the empirical loss. By partitioning the space based on the input data, we then decompose the population error into data error, optimization error, and prediction variation error, and bound them separately. In particular, for the prediction variation error, which measures the oscillations of the learned function, we propose (local) approximate homogeneity and derive explicit cellwise and layerwise bounds for its evolution along the training trajectory. These bounds yield two important implications: a necessary condition of improved generalization explains differences in layerwise generalization behavior; a sufficient condition describes delayed generalization and provides a theoretical characterization of grokking.

This paper propose a robust decentralized federated distillation method that enables clients with heterogeneous models to collaborate through predictions on shared unlabeled public data. In the proposed method, each client first evaluates the received predictions in three modalities of class prediction, boundary decision, and prediction correlation. It then filters unreliable clients, assigns reliability-based weights to the retained clients, and constructs a teacher for each type of knowledge. Finally, the corresponding distillation gradients are validated using a supervised gradient computed from private data. Conflicting prediction and boundary gradients are removed, and conflicting relation gradients are suppressed before the final model update. We prove the convergence of the proposed method by showing stable local optimization for honest clients under Byzantine distillation. Particularly, we show that our method ensures a bounded Byzantine influence on both distillation gradients and individual client private gradients after cross-modality fusion, thereby enabling stable local optimization for honest clienunder Byzantine distillation. Extensive experiments on CIFAR-10 and CIFAR-100 demonstrate that the proposed method improves the prediction accuracy of heterogeneous models of clients under non-IID data and Byzantine attacks. As the booming demands of federated learning in decentralized environments such as edge computing and mission-oriented UAV collaborations, our method has a great potential for adoption of DFL in unreliable real-world scenarios where clients are exposed to receiver-specific Byzantine messages of malicious predictions.

Attention mechanisms are central to modern foundation models, yet their training dynamics remain poorly understood, especially when the attention matrices have extensive rank. In this work, we study attention-indexed models, a broad framework that can represent multi-layer and multi-head attention architectures. First, we show that, in a suitable high-dimensional limit, the population-loss landscape is characterized by a finite set of trace order parameters. In contrast, online stochastic gradient descent (SGD) is governed by an infinite hierarchy of matrix moments, which we show can be exponentially well-approximated by a finite truncated system. Second, this framework reveals that attention parameterization itself can act as an architectural implicit bias. Direct optimization of an attention matrix $S\in\mathbb{R}^{d\times d}$ can remain trapped in an uninformative state. Tied attention ($S=WW^\top$) induces an automatic symmetry-breaking mechanism and yields weak recovery in $Θ(d^2\log d)$ samples. For untied attention, $S=UV^\top$, we uncover a fast-slow mechanism: the pre-activation mean first evolves on a fast timescale, while the overlaps evolve on a slower one. Weak recovery on the $Θ(d^2\log d)$ scale occurs when the state selected by the fast dynamics breaks the initial symmetry.

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.

This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.

Decentralized, serverless learning increasingly connects devices running different architectures, where the standard tool, decentralized SGD, is undefined as models with different parameter counts cannot be averaged. Knowledge distillation (KD) exchanges soft predictions rather than weights and sidesteps this obstacle, yet convergence theory for fully decentralized, asynchronous peer-to-peer (P2P) KD is lacking. We provide one, relocating consensus from parameter space to function (output) space: a KD event is a geometric contraction operator in logit space on the peers' predictive distributions, which we analyse in the Hilbert space of predictions on a reference measure. Under standard smoothness/variance assumptions and two realizability assumptions, one bridging parameter SGD to the functional step and one controlling restricted task/KD alignment, the time-averaged functional stationarity and function-space disagreement converge at rate $O(1/(ηT))$ to an $O(η)+O(B_f^2)+O(ζ_f^2)$ neighbourhood. Here $B_f$ is the distance from the task optimum to the peers' reachable classes and $ζ_f$ measures persistent local-task heterogeneity. Across homogeneous, width-heterogeneous, and mixed-family networks of the experiments, KD contracts function disagreement by $40-61\times$, while isolated training does not. The sampled stationarity diagnostic has late transient exponents $0.99-1.90$ on the shared-skeleton main runs, and the four-point step-size sweep exhibits the predicted transient: neighbourhood tradeoff.

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%.

Generative data augmentation is widely used to mitigate class imbalance, yet its theoretical effect on downstream generalization remains poorly understood. In this work, we develop a statistical framework for conditional generative augmentation and analyze its impact on classification risk. We formalize augmentation as a distribution-mixing process and show that the resulting risk distortion is controlled by both the augmentation strength and the class-conditional Wasserstein discrepancy between real and generated distributions. We further derive a capacity-dependent generalization bound based on Rademacher complexity, revealing an explicit trade-off between hypothesis complexity, augmentation intensity, and generative fidelity. Empirically, we evaluate the framework on binary and multiclass imbalanced classification tasks using Conditional GAN and Conditional WGAN-GP augmentation. Across datasets, CWGAN-GP consistently achieves lower Wasserstein discrepancies than CGAN, indicating improved distributional fidelity. However, improved fidelity does not necessarily translate into superior classification performance, with classical oversampling methods often remaining competitive. These findings support the central theoretical prediction that augmentation reliability is governed by distributional approximation error rather than predictive performance alone. Overall, this work establishes generative augmentation as a distributional perturbation process whose reliability can be quantified through Wasserstein-based measures and supported by finite-sample generalization guarantees. The proposed framework provides a principled foundation for evaluating synthetic data quality beyond classification accuracy alone.

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.

The Levenberg-Marquardt (LM) algorithm is a well-known second-order method for rapid convergence and strong robustness when training small- to medium-sized neural networks (NNs). However, its computational and memory costs increase significantly as the number of parameters in an NN grows. To address this limitation, subspace methods have been proposed, such as the Krylov subspace LM (KSLM) and the hybrid subspace LM (HSLM), making second-order algorithms more efficient. In this work, we evaluate the subspace Levenberg-Marquardt algorithms for regression and classification tasks in neural networks. We compare the performance of subspace LM variants with the classical LM method, as well as other popular first-order algorithms, such as stochastic gradient descent (SGD) and Adam.

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.

By focusing on algorithmic stability as a means of establishing out-of-sample bounds, we provide a system-theoretic interpretation of generalization in learning-enabled dynamical systems arising in data-driven optimization and feedback control approximation. Given two neighboring datasets, we specifically model sample replacement as an exogenous disturbance acting on a sensitivity system, while the incremental behavior of the data-dependent operator is encoded through an integral quadratic constraint. By relying on dissipativity arguments, we establish a matrix inequality-based certificate and a uniform stability bound that separates the one-sample sensitivity of the learned operator, and an algorithm-dependent dynamical gain. The latter can then be optimized, offering a tractable tool for certifying and comparing generalization capabilities of learning dynamics. We show that our results recover classical ones for gradient descent, apply naturally to momentum-based methods such as heavy-ball and Nesterov acceleration, and extend to data-driven control.

Popular adaptive stochastic gradient descent (SGD) methods to train artificial intelligence (AI) systems include the RMSprop, the Adam, and the AdamW optimizers, where the adaptivity parts in Adam and AdamW basically just coincide with RMSprop. Such adaptive methods involve several hyperparameters including the regularization parameter $ε$ (which ensures that one does not divide by 0 and is often chosen to be very close to zero such as $10^{-8}$ in PyTorch by default) and the second moment decay parameter $β$ (which is often chosen to be very close to $1$ such as 0.99 (RMSprop) and 0.999 (Adam and AdamW) in PyTorch by default). Despite the high relevance of such methods, it remains an open research problem to provide error estimates for such methods with the error constants being not exploding but uniformly bounded with the respect to the hyperparameters, even in the situation of convex stochastic optimization problems. It is the key contribution of this work to essentially solve this problem for RMSprop. Specifically, we bound the expectation of the stopped evaluation of the objective function at the RMSprop process from above by the sum of an initialization term that decays exponentially in the training time, a stochastic approximation remainder of order $γ_n$, and a memory error of order $( 1 - β)^2$ with the error constants being uniformly controlled over all admissible choices of the step sizes, the second moment decay parameter $β$ and the regularization parameter $ε\in[0,1]$ (also covering $ε=0$). Our non-asymptotic error estimates hold not just for all sufficiently large n but hold for every gradient step $n=1,2,3,...$ with all error constants being explicitly specified. The key innovative new feature in the proof of our analysis are suitable inverse moment bounds for the second moment process in RMSprop.

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.

Vision-language-action (VLA) driving methods increasingly combine multi-trajectory imitation learning with group-relative policy optimization (GRPO), making trajectory selection critical to final performance. However, some high-scoring trajectories that improve imitation can degrade subsequent GRPO by inducing advantage estimates misaligned with the current policy's feasible behavior distribution, driving updates away from safe and compliant behaviors. To address this, we propose a novel framework that aligns multi-trajectory supervision with policy optimization. To address the policy gradient bias induced by infeasible noisy trajectories outside the feasible region, augmented trajectories are constrained to a neighboring manifold of the ground-truth feasible region, and a Pareto-optimality criterion is adopted in place of the conventional aggregate score, retaining only non-dominated candidates and thereby filtering out conflicting samples at the source. To ensure that expanded trajectory supervision is effectively absorbed during policy optimization, we introduce two complementary mechanisms: feasibility-first advantage assignment and dynamic distillation. The former adapts Pareto credit to the feasibility composition of each rollout group and guides fully infeasible groups toward safe references. The latter updates teacher trajectories across refinement rounds to continually transfer useful supervision. Together, they progressively translate the benefits of expanded supervision into policy improvement. On NAVSIM v1 and v2, our method achieves 91.4 PDMS and 89.1 EPDMS, respectively, under single-trajectory inference, and recovers 440 of 658 initially failed scenes, 11.1% higher than the original GRPO baseline.

To mitigate the time complexity of generative models, one-step generative models have recently emerged through direct mapping from noise to data in a single forward pass. However, the reward-guided fine-tuning method of one-step generative models remains largely unexplored. To address this, we consider one-step generators from an optimal transport view, investigating Wasserstein Gradient Flow (WGF) for modeling smooth and controlled distributional evolution in probability space. We then propose a novel reward-guided fine-tuning of a one-step generative model via WGF. We derive a practical training method that requires no reward gradients, thereby handling both non-differentiable and differentiable rewards. Moreover, our method provides smooth and stable reward-guided distributional updates while mitigating reward hacking and mode collapse. Experiments on 2D synthetic data, CIFAR-10, and ImageNet 256$\times$256 with diverse rewards, including JPEG (in)compressibility, class probability, Black-and-White and CLIP alignment, show that our method achieves better reward alignment compared to baselines.

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned landscape. Under a Calabi-Yau metric, specifically in a non-compact setting with a global potential so defined geometrically rather than invoking the topological requirements of the Calabi conjecture, a wedged nowhere-vanishing holomorphic form is the top exterior product of the Kähler form up to constants, yielding a constant determinant condition with respect to a background metric and ill-conditioned eigenvalues under nonuniform and almost low-rank assumptions. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist in a geometric analytic modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Algorithmic Information Dynamics (AID) studies systems by perturbing them and measuring changes in algorithmic complexity, but its usual estimator, the Block Decomposition Method, is piecewise constant, restricting the calculus to finite differences. We use $K^{\mathrm{CDM}}_{\mathrm{s}F}$, a certified, differentiable estimator, to bring the calculus into learning dynamics: grokking, where a complexity order parameter is known but has not been made to act. As a transient loss kick, the estimator becomes a controller that accelerates grokking in Levin's description-length--versus-time sense, within a data-dependent Occam boundary whose finite-size trend, $f_c\sim\ln p/p$, is consistent with a coupon-collector interpretation. Ablations show that a complexity gate matches a train-loss gate in rescuing failing seeds with $27\%$ less intervention; among the tested signals, only map complexity marks the transition's completion; the certified prior and the per-parameter $\nabla K$ attribution are both fungible (a uniform-prior sensor makes bit-identical gate decisions, and random supports match $\nabla K$-selected ones above a sparsity threshold); and direct field perturbation shows a nucleation-like response to the Occam field (no linear regime is resolved over the probed amplitudes, so these measurements do not justify a fluctuation--dissipation surrogate), with a finite-field response growing by orders of magnitude toward the phase-transition. These measurements account for the empirically tuned staircase: bang--bang pulses, stall-fired and released on yield, whose iteration plausibly builds the response it exploits. The kick transfers to sparse parity and to a transformer; a sustained weight-space loss fails. The algorithmic estimator's distinct contribution is timing (when to fire and when to release), not attribution.

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

Networked systems, from power grids to traffic networks and cloud clusters, carry loads across nodes with limited capacity. A node whose load exceeds its capacity fails and sheds its load onto its neighbors, which can trigger a system-wide cascade. We study how to allocate a fixed capacity budget across nodes to resist these cascades under local load redistribution. The problem is difficult because no optimal allocation is known, and the fail-or-survive objective is non-differentiable and piecewise constant, so exact and gradient-based optimization methods do not directly apply. We introduce TANGCO (Topology-Aware Neural Graph-Guided Capacity Optimization), which uses a graph neural network policy trained through the cascade simulator with policy-gradient learning and a heuristic anchor. We evaluate TANGCO on five synthetic graph families and five real networks spanning power, road, air, and Internet topologies. The learned policy improves on the best of four hand-designed heuristics in all 450 synthetic instances and in 40 of 45 real-network conditions, with robustness gains ranging from 1.6% to 246%. The learned policies transfer to unseen graphs within a family and partially across related topologies, and TANGCO$^{pre}$, pre-trained on synthetic graphs, matches per-network training on unseen real networks. Training scales near-linearly with graph size, and TANGCO$^{pre}$ allocates on a new network with no per-target training, matching the deployment cost of a hand-designed heuristic. Free-vector variants without the GNN, stay close to the heuristics, so the graph representation carries the gain beyond numerical search. Finally, analysis of the learned allocations identifies when local risk is sufficient, leads to an improved closed-form heuristic, and reveals the regimes where a topology-aware learned policy remains necessary.

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.

We establish convergence guarantees of gradient descent for general feedforward neural networks of arbitrary width or depth, with no special requirements on the initialization or dataset. We only assume that the activation functions are Lipschitz smooth, Lipschitz continuous, and linearly bounded--- properties that hold for linear, tanh, softplus, and sigmoid activation functions. For the loss function, we require that it is Lipschitz smooth in the model outputs, which is true for mean-squared error. The key theoretical insight is that the Lipschitz properties of the activation functions are partially preserved even through repeated compositions, leading to a novel generalized Lipschitz smoothness condition where the change in gradient is upper bounded by the change in the parameter space, multiplied by polynomial terms of the parameter norms at both endpoints. This type of condition holds for both the model function and the loss function, enabling a descent lemma where the loss decreases as long as the learning rate is small enough with respect to the parameter norms. By ensuring that the parameter norms do not grow too quickly to infinity, we prove that the minimum squared gradient norm converges to zero in $T$ iterations at rate $O(1/T^{1/L})$ for an $L$-layer neural network.