Stochastic Gradient Descent

The effectiveness of active learning hinges on the choice of the acquisition criterion by which a learning algorithm selects potentially informative data points whose label is subsequently queried. This paper proposes a novel gradient-based acquisition criterion, derived from a generalization bound introduced by Luo et al. (2022). This criterion can be applied in lieu of uncertainty measures in uncertainty sampling, or incorporated into diversity-based methods that consider the spread of sampled points in addition to the uncertainty of their labels. We provide a theoretical justification of the proposed acquisition criterion, and demonstrate its effectiveness in an empirical evaluation.

The efficacy of deep neural networks is heavily reliant on the design of non-linear activation functions, yet existing approaches often struggle to balance optimization stability with computational efficiency. While piecewise linear functions offer inference speed, they suffer from optimization instability due to non-differentiability at the origin, whereas smooth counterparts typically incur significant computational overhead through their reliance on transcendental operations. To address these limitations, this paper proposes a general smoothing framework based on constructive approximation theory and introduces the Bernstein Linear Unit (BerLU). This novel activation function utilizes Bernstein polynomials to construct a differentiable quadratic transition region that effectively eliminates singularities while maintaining a piecewise linear structure. Theoretical analysis demonstrates that the proposed method guarantees strictly continuous differentiability and a non-expansive Lipschitz constant of one, which ensures stable gradient propagation and prevents the gradient explosion problems common in deep architectures. Comprehensive empirical evaluations across representative Vision Transformer and Convolutional Neural Network architectures confirm that this approach consistently outperforms state-of-the-art baselines on standard image classification benchmarks while delivering superior computational and memory efficiency.

Adaptive optimizers such as Adam and non-adaptive methods like SGD exhibit distinct generalization capabilities across different architectures. Prior tunable optimizers attempt to bridge this gap by strictly interpolating between SGD and Adam, effectively confining adaptivity within the 0-to-1 bound. However, this restricted interpolation is fundamentally insufficient: we reveal that optimal adaptivity often requires extrapolation, such as negative adaptivity for classical CNNs and adaptivity of at least one ($γ\geq 1$) for Transformers. Extrapolating adaptivity theoretically violates the strict non-decreasing pre-conditioner assumption, often leading to divergence in existing methods. To break this barrier, we propose Anon, an optimizer that achieves fully continuous adaptivity extrapolation across the entire real-number spectrum. To guarantee provable stability in these out-of-bound regimes, we introduce Incremental Delay Update (IDU), a novel mechanism that bypasses hard max-tracking strategies. We theoretically establish Anon's convergence in both convex and non-convex settings. Empirically, by exploring previously unreachable adaptivity landscapes, Anon demonstrates highly competitive and scalable performance among state-of-the-art element-wise optimizers on representative image classification, diffusion, and large language modeling tasks.

Asynchronous stochastic gradient descent (SGD) enables scalable distributed training but suffers from gradient staleness. Existing mitigation strategies, such as delay-adaptive learning rates and staleness-aware filtering, typically attenuate or discard delayed gradients, introducing systematic bias: updates from simpler or faster-to-process samples are overrepresented, while gradients from more complex samples are delayed or suppressed. In contrast, prior approaches to data-dependent delays rely on a Lipschitz assumption that yields suboptimal rates or leave the smooth, convex case unaddressed. We propose a momentum-based asynchronous framework designed to preserve information from delayed gradients while mitigating the effects of staleness. We establish the first optimal convergence rates for data-dependent delays in both convex and non-convex smooth setups, providing a new result for asynchronous optimization under standard assumptions. Additionally, we derive robust learning-rate schedules that simplify hyperparameter tuning in practice.

Hard thresholds, quantization, and discrete routing can produce training losses with flat regions and jumps, where ordinary gradients vanish or are undefined. We introduce PolyStep, a forward-only optimizer that evaluates rotated polytope probes and moves parameter blocks along weighted averages of the probe directions. We derive the weights from one-sided entropic transport and use its uncoupled softmax solution in our primary experiments. Our analysis explains when variation among probe costs produces motion and when that motion decreases the loss. On a regular simplex, nonconstant costs always give a nonzero direction. For monotone ridge losses, the softmax update cannot increase the loss at any positive temperature; a perturbation bound gives sufficient conditions for descent near curved jumps. For bounded measurable losses, we randomize the probe radii and identify an exact smoothing whose gradient equals the expected linear cost-weighted direction up to scale. This identity yields a stationarity bound for an idealized fixed-temperature variant: under regularity and sampling assumptions stronger than those met by our trained configurations, the bound has an $O(T^{-1/2})$ term and a persistent bias floor. We evaluate the practical method on networks with hard operations, discrete optimization, and policy search. On MNIST with hard-threshold spiking neurons, PolyStep reaches $93.0 \pm 0.2\%$, compared with $79.6 \pm 5.2\%$ for the best-tuned gradient-free baseline at matched evaluations. These gains come with a query cost proportional to the search dimension per fresh step, which limits the number of updates available at a fixed budget.

We introduce a finite-size gradient-transport framework for real language-model training, based on five observables $(D,z,β,δ,v_{\mathrm{rel}})$ that separate cascade size, duration, absolute transport, and intensive transport efficiency. We analyze direct raw-gradient measurements from Pico-LM across four scales and 125 aligned steps, together with a five-scale Pythia companion dataset built from 153 aligned checkpoint-difference update fields. The same algebraic closure holds in both families, and both share a near-unity cascade-size backbone, but they occupy distinct transport regimes: Pico-LM shows positive duration scaling and negative intensive-efficiency scaling, whereas Pythia remains near the $D=1$ baseline with only weak positive efficiency scale dependence. Randomized-field controls give nearly matched null floors in the intensive and duration channels, indicating that the contrast reflects different real departures from a shared null skeleton rather than different null calibrations. The families also differ in stepwise power-law compressibility: Pico-LM retains clean duration and efficiency power laws, whereas Pythia preserves the size backbone but shows weaker one-slope compressibility in those channels. External performance associations are correspondingly channel-level, carried mainly by $v_{\mathrm{rel}}$ and normalized cascade duration, while $D(t)$ acts as a shared size backbone without a significant exponent-level performance association. These results support a reusable transport measurement framework without claiming a universal fixed point or a first-principles derivation of neural scaling laws.

Stochastic gradient methods are central to large-scale learning, yet their generalization theory typically relies on independent sampling assumptions. In many practical applications, data are generated by Markov chains and learning is performed in a decentralized manner, which introduces significant analytical challenges. In this work, we investigate the stability and generalization of decentralized stochastic gradient descent (SGD) and stochastic gradient descent ascent (SGDA) under Markov chain sampling. Leveraging a stability-based framework, we characterize how Markovian dependence and decentralized communication jointly influence generalization behavior. Our analysis captures the effects of network topology, Markov chain mixing properties, and primal-dual dynamics. We establish non-asymptotic generalization bounds for both algorithms, extending existing results on Markov stochastic gradient methods to decentralized and minimax settings.

We present a non-asymptotic theory of generalization in deep learning where the empirical neural tangent kernel partitions the output space. In directions corresponding to signal, error dissipates rapidly; in the vast orthogonal dimensions corresponding to noise, the kernel's near-zero eigenvalues trap residual error in a test-invisible reservoir. Within the signal channel, minibatch SGD ensures that coherent population signal accumulates via fast linear drift, while idiosyncratic memorization is suppressed into a slow, diffusive random walk. We prove generalization survives even when the kernel evolves $\mathcal{O}(1)$ in operator norm, the full feature-learning regime. This theory naturally explains disparate phenomena in deep learning theory, such as benign overfitting, double descent, implicit bias, and grokking. Lastly, we derive an exact population-risk objective from a single training run with no validation data, for any architecture, loss, or optimizer, and prove that it measures precisely the noise in the signal channel. This objective reduces in practice to an SNR preconditioner on top of Adam, adding one state vector at no extra cost; it accelerates grokking by $5 \times$, suppresses memorization in PINNs and implicit neural representations, and improves DPO fine-tuning under noisy preferences while staying $3 \times$ closer to the reference policy.

For a risk-averse finite-horizon Markov Decision Problem, we introduce a special class of Markov coherent risk measures, called mini-batch measures. We also define the class of multipattern risk-averse problems that generalizes the class of linear systems. We use both concepts in a feature-based $Q$-learning method with multipattern $Q$-factor approximation and we prove a high-probability regret bound of $\mathcal{O}\big(H^2 N^H \sqrt{ K}\big)$, where $H$ is the horizon, $N$ is the mini-batch size, and $K$ is the number of episodes. We also propose an economical version of the $Q$-learning method that streamlines the policy evaluation (backward) step. The theoretical results are illustrated on a stochastic assignment problem and a short-horizon multi-armed bandit problem.

Both Dimensionality Reduction (DR) and Graph Drawing (GD) aim to visualize abstract, non-linear structures, yet rely on different optimization paradigms. This contrast is evident in Multidimensional Scaling (MDS), which typically depends on the SMACOF algorithm despite graph drawing results showing that simpler stochastic optimization schemes can be more effective for the same objective. We bridge these domains by adapting Stochastic Gradient Descent (SGD) techniques from graph drawing to vector data embedding. We present a scikit-learn compatible estimator that minimizes global stress through local pairwise updates, improving upon the existing implementation. Experiments on standard high-dimensional benchmarks show that our stochastic solver converges substantially faster than SMACOF while achieving comparable or lower stress.

Contrastive vision-language models like CLIP exhibit remarkable zero-shot generalization. However, prompt tuning remains highly sensitive to label noise, as mislabeled samples generate disproportionately large gradients that can overwhelm pre-trained priors. We argue that because CLIP already provides a near-optimal initialization, adaptation should be inherently conservative, particularly against the extreme gradient updates common in noisy settings. To this end, we propose Double-Softmax Prompt Tuning (DSPT), a hyperparameter-free method for intrinsic gradient suppression. By applying a sequential probabilistic normalization, DSPT induces a self-adaptive saturation zone that suppresses gradients from high-error noisy samples while maintaining informative updates. We also provide both theoretical analysis and empirical evidence about how this mechanism achieves adaptive suppression. This design transforms ``gradient vanishing'', traditionally a training bottleneck, into a principled noise-filtering shield for label-noise prompt tuning. Extensive experiments confirm that this simple, drop-in design achieves state-of-the-art robustness across various noisy benchmarks, outperforming methods with complex architectures and handcrafted hyperparameters.

In financial predictions, the performance of machine learning models is often assessed by Rank IC, which is the Spearman rank correlation between the model predictions and the realized asset returns. Despite its wide adoption, most existing models are trained using regression losses or ranking objectives that may not align with Rank IC. We propose LambdaRankIC, a novel learning-to-rank approach that directly optimizes Rank IC. We circumvent the non-differentiability of the ranking operator by deriving the closed-form expression for the lambda gradients induced by the pairwise rank swaps, which enables efficient gradient-based optimization within the LambdaRank framework. We implement LambdaRankIC as a custom objective in XGBoost. Theoretically, we show that our approach optimizes an upper bound on Rank IC. We evaluate the proposed approach on both simulated and real-world financial data. In simulation studies, LambdaRankIC accurately recovers the true ranking structure in noiseless settings and consistently outperforms regression-based and NDCG-oriented ranking methods under low signal-to-noise ratios and heavy-tailed noise regimes. In empirical experiments using real market data, LambdaRankIC achieves the best out-of-sample performance on evaluation metrics commonly used in finance, including Rank IC, ICIR, monthly return, and Sharpe ratio. These results show that directly optimizing Rank IC can yield substantial improvements over conventional learning objectives in financial predictions when the full-order ranking quality is the primary goal.

Federated learning using mobile and Internet of Things devices requires not only the ability to handle heterogeneity of clients' data distributions but also high adaptability to varying communication environments. We propose FedHAW (Federated Learning with Hypergradient-based update of Aggregation Weights) that implements online updates of aggregation weights. FedHAW updates the aggregation weights by using hypergradient, the gradient of the objective function with respect to the weights, which can be calculated with low computational overhead. Simulation results show that the proposed method possesses high generalization performance in heterogeneous environments and high robustness to communication errors.

We study high-probability (HP) convergence guarantees in decentralized stochastic optimization, where multiple agents collaborate to jointly train a model over a network. Existing HP results in decentralized settings almost exclusively focus on the Decentralized Stochastic Gradient Descent ($\mathtt{DSGD}$) algorithm, which requires strong assumptions, such as bounded data heterogeneity, or strong convexity of each agent's cost. This is contrary to the mean-squared error (MSE) results, where methods incorporating bias-correction techniques are known to converge under relaxed assumptions and achieve better practical performance. In this paper we provide the first step toward bridging the gap, by studying HP convergence of $\mathtt{DSGD}$ incorporating the gradient tracking technique, in the presence of noise satisfying a relaxed sub-Gaussian condition. We show that the resulting method, dubbed $\mathtt{GT-DSGD}$, achieves order-optimal HP convergence rates for both non-convex and Polyak-Łojasiewicz costs, of order $\mathcal{O}\Big(\frac{\log(1/δ)}{\sqrt{nT}}\Big)$ and $\mathcal{O}\Big(\frac{\log(1/δ)}{nT}\Big)$, respectively, where $n$ is the number of agents, $T$ is the time horizon and $δ\in (0,1)$ is the confidence parameter. Our results establish that $\mathtt{GT-DSGD}$ converges in the HP sense under the same conditions on the cost as in the MSE sense, while achieving comparable transient times. To the best of our knowledge, these are the first HP guarantees for decentralized optimization methods incorporating bias-correction. Numerical experiments on real and synthetic data verify our theoretical findings, underlining the superior performance of $\mathtt{GT-DSGD}$ and highlighting that the benefits of incorporating bias-correction are also maintained in the HP sense.

We consider the problem of Cost-Aware Learning, where sampling different components of a finite-sum objective incurs different costs. The objective is to reach a target error while minimizing the total cost. We propose Cost-Aware SGD, which uses a distribution based on gradient norms and costs to sample components. We provide a thorough analysis of this algorithm, including cost-improvement bounds over baselines, a characterization of distribution proxy sub-optimality, and a lower bound. We apply our theoretical insights to reinforcement learning with language models, where the computational cost of sequence-level policy gradients varies with length. We find that the advantage magnitude serves as a high-fidelity proxy for gradient norms, and use this to introduce Cost-Aware GRPO. Empirical results on 1.5B, 4B, and 8B LLMs demonstrate that this algorithm significantly reduces the tokens used in policy optimization while matching or exceeding baseline accuracy.

3D Gaussian Splatting has emerged as a powerful scene representation for real-time novel-view synthesis. However, its standard adaptive density control relies on screen-space positional gradients, which do not distinguish between geometric misplacement and frequency aliasing, often leading to either over-blurred high-frequency textures or inefficient over-densification. We present a structure-aware densification framework. Our key insight is that the decision to subdivide a Gaussian should be driven by an explicit comparison between its projected screen-space extent and the local structure of the texture it seeks to represent. We introduce a multi-scale frequency analysis combining structure tensors with Laplacian scale space analysis to estimate the dominant frequency at each pixel, enabling robust supervision across varying texture scales. Based on this analysis, we define $η$, a per-Gaussian, per-axis frequency violation metric that indicates when a primitive may be under-resolving local texture details. Unlike methods that perform isotropic splitting (e.g., splitting each Gaussian into two smaller ones with uniform shape), our approach performs anisotropic splitting. For each axis with high $η$, we compute a split factor to better resolve the local frequency content. We further introduce a multiview consistency criterion that aggregates $η$ observations across multiple views. By performing densification early and faster, we skip the lengthy iterative densification phases required by baseline methods and achieve significantly faster convergence. Experiments on standard benchmarks demonstrate that our method also achieves superior reconstruction quality, particularly in high-frequency regions.

Fine-tuning pretrained models has become a standard approach to adapting pretrained knowledge to improve the accuracy on new sparse, imbalance datasets. However, issues arise when optimization falls into a collapsed state, where the model gets stuck, leading to degraded performance and unstable training. One possible reason for this is the cancellation of gradients across training examples. To address this problem, we propose a novel algorithm, dynamic scaled gradient descent (\mName), that directly modifies the gradients returned by training examples, specifically, scaling down the gradients of correctly classified examples using a dynamic scaler. This strategy offers both theoretical and empirical advantages in improving training stability. Experiments on a variety of benchmark datasets, spanning multiple tasks and large pretrained models, demonstrate that our method consistently reduces performance variance and surpasses the accuracy of existing approaches.

In this paper an attractor FCM is created, tested, and analyzed. This FCM is neither a hebbian based nor agentic, nor a hybrid; it rather is a gradient descent based, physics constrained, Jacobian version of an FCM. Moreover, this model has several quirks; it uses residual memory, back propagation through time, and a fixed point anchor that is recursively implemented to update its weights. The residuals update the recursive part without losing the system memory. The model's anchor enables it to converge in a fixed point for which back propagation through time unrolls it and ensures that the error minimization is for an accurate gradient. Furthermore, a new learning algorithm is utilized. The Newton's method finds the system's fixed point attractor and then gradient descend is adaptively changing the landscape; an adaptive term is used to directly manipulate the weights through the attractor dynamics. As the adaptive term changes, the descent through the landscape is constantly adjusting according to sigmoid saturation, and that prevents premature convergence to a local minimum. Lastly, the updates are filtered by causal mask that informs the network about the physics, respecting the initial expert based opinions, for which model reduces the error to the target in an efficient way.

Information-theoretic generalization bounds analyze stochastic optimization by relating expected generalization error to the mutual information between learned parameters and training data. Virtual perturbation analyses of SGD add auxiliary Gaussian noise only in the proof, making mutual information tractable while leaving the actual SGD trajectory unchanged. Existing bounds, however, typically require perturbation covariances to be fixed independently of the optimization history, limiting their ability to represent geometries induced by moving gradient statistics, preconditioners, curvature proxies, and other pathwise information. We introduce predictable history-adaptive virtual perturbations, where the perturbation covariance at each iteration may depend on the past real SGD history but not on current or future randomness. This predictability enables a conditional Gaussian relative-entropy argument and yields generalization bounds for SGD with adaptive virtual-noise geometry. The bounds replace fixed sensitivity and gradient-deviation terms with conditional adaptive counterparts, include an output-sensitivity penalty from accumulated perturbation covariance, and reduce the deviation term to a conditional variance only under conditional unbiasedness. Since adaptive covariances may be data-dependent, we separate local Gaussian smoothing from global reference-kernel comparison. The resulting bound includes a covariance-comparison cost measuring the KL price of using an admissible reference geometry different from the actual adaptive covariance. Fixed-noise-style bounds are recovered under admissible synchronization, such as deterministic, public, or prefix-observable covariance rules. The framework recovers fixed isotropic and geometry-aware bounds as special cases while extending virtual perturbation analysis to history-dependent SGD without modifying the algorithm.

Learning rate scheduling has evolved from the single global fixed rate of early SGD to sophisticated layer-wise adaptive strategies. We systematize this evolution into five generations: (Gen1) global fixed learning rates, (Gen2) global scheduling, (Gen3) parameter-level adaptation, (Gen4) layer-level differentiation, and (Gen5) joint layer-time scheduling. We trace the fundamental motivation behind each transition, showing how the shift from one-size-fits-all to tailoring by layer and time addresses the impossible trinity of transfer learning: lower layers require small updates to preserve general knowledge while higher layers need large updates to adapt to new tasks. Building on this taxonomy, we propose Discriminative Adaptive Layer Scaling (DALS), a unified framework that integrates phase-adaptive cosine scheduling, depth-aware Grokfast gradient filtering, and LARS-style trust ratios into a single coherent optimizer. We benchmark 18 strategies including three DALS variants across all five generations on five datasets: synthetic, CIFAR-10 (from scratch), RTE, TREC-6, and IMDb (fine-tuning). On synthetic, DALS achieves the best accuracy at 98.0%, while DALS-Fast reaches 90% in just 3 epochs. The cross-dataset analysis reveals striking regime-dependent patterns -- no single strategy wins across all regimes. Critically, STLR+Discriminative, the ULMFiT champion, catastrophically fails on from-scratch tasks (43.6% on TREC-6 from scratch vs. 96.8% with RAdam), confirming that directional decay biases are harmful without pretrained features. DALS avoids either extreme, achieving the best synthetic result while maintaining competitive fine-tuning performance.

Continual learning agents with finite capacity must balance acquiring new knowledge with retaining the old. This requires controlled forgetting of knowledge that is no longer needed, freeing up capacity to learn. Weight decay, viewed as a mechanism for forgetting, can serve this role by gradually discarding information stored in the weights. However, a fixed scalar weight decay drives this forgetting uniformly over time and uniformly across all parameters, even when some encode stable knowledge while others track rapidly changing targets. We introduce Forgetting through Adaptive Decay (FADE), which adapts per-parameter weight decay rates online via approximate meta-gradient descent. We derive FADE for the online linear setting and apply it to the final layer of neural networks. Our empirical analysis shows that FADE automatically discovers distinct decay rates for different parameters, complements step-size adaptation, and consistently improves over fixed weight decay across online tracking and streaming classification problems.

This paper presents a planning pipeline framework for locomotion in rope-assisted robots climbing vertical surfaces. The proposed framework is formulated as a bi-level optimization scheme that addresses a mixed-integer problem: selecting feasible terrain regions for landing while simultaneously optimizing the control inputs, namely rope tensions and leg forces, and landing location. The outer level of the optimization is solved using the Cross-Entropy Method, while the inner level relies on gradient-based nonlinear optimization to compute dynamically feasible motions. The approach is validated on a novel climbing robot platform, ALPINE, across a variety of challenging terrain configurations.

DNNs have gained widespread adoption in feature interaction recommendation models. However, there has been a longstanding debate on their roles. On one hand, some works claim that DNNs possess the ability to implicitly capture high-order feature interactions. Conversely, recent studies have highlighted the limitations of DNNs in effectively learning dot products, specifically second-order interactions, let alone higher-order interactions. In this paper, we present a novel perspective to understand the effectiveness of DNNs: their impact on the dimensional robustness of the representations. In particular, we conduct extensive experiments involving both parallel DNNs and stacked DNNs. Our evaluation encompasses an overall study of complete DNN on two feature interaction models, alongside a fine-grained ablation analysis of components within DNNs. Experimental results demonstrate that both parallel and stacked DNNs can effectively mitigate the dimensional collapse of embeddings. Furthermore, a gradient-based theoretical analysis, supported by empirical evidence, uncovers the underlying mechanisms of dimensional collapse.

We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(δ;\,\cdot)$ whose boundary-inward property induces a monotone state parameter $δ_t$, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-critical step sizes, the reduced dynamics exhibit period-2 behavior consistent with edge-of-stability phenomena. We further show that the scalar certificate is not an ad hoc algebraic construction: under structural axioms and a natural state-parameter normalization, it is uniquely determined by the monotonicity mechanism. Numerical experiments suggest that this state-dependent Lyapunov mechanism persists beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.

Graph neural networks such as ParticleNet and transformer based networks on point clouds such as ParticleTransformer achieve state-of-the-art performance on jet tagging benchmarks at the Large Hadron Collider, yet the physical reasoning behind their predictions remains opaque. We present different methods, i.e. perturbation-based (GNNExplainer), Shapley-value-based (GNNShap), and gradient-based (GRADCam); adapted to operate on LundNet's Lund-plane graph representation. Leveraging the fact that each node in the Lund plane corresponds to a physically meaningful parton splitting, we construct Monte Carlo truth explanation masks and introduce a physics-informed evaluation framework that goes beyond standard fidelity metrics. We perform the analysis in three transverse-momentum bins ($\mathrm{p_T} \in [500,700]$, $[800,1000]$, and the inclusive region $[500,1000]$ GeV), revealing how explanation quality and focus shift between non-perturbative and perturbative regimes. We further quantify the correlation between explainer-assigned node importance and classical jet substructure observables -- $N$-subjettiness ratios $τ_{21}$ and $τ_{32}$ and the energy correlation functions -- establishing the degree to which the model has learned known QCD features. We find that overall the weight assigned by explainability methods has a correlation with analytic observables, with expected shift across different phase space regimes, indicating that a trained neural network indeed learns some aspects of jet-substructure moments. Our open-source implementation enables reproducible explainability studies for graph-based jet taggers.

Deep learning models are used in critical applications, in which mistakes can have serious consequences. Therefore, it is crucial to understand how and why models generate predictions. This understanding provides useful information to check whether the model is learning the right patterns, detect biases in the data, improve model design, and build systems that can be trusted. This work proposes a new method for interpreting Convolutional Neural Networks in image classification tasks. The approach works by selecting the most representative feature maps that contribute to each prediction. To solve this combinatorial problem, we encode it into a quantum constrained optimization problem and propose to solve it using quantum annealing. We evaluate our method against the state-of-the-art explainable AI techniques, specifically GradCAM and GradCAM++, and observe an improved class disentanglement, i.e. the model's decision boundaries become more distinct and its reasoning more transparent. This demonstrates that our approach enhances the quality of explanations, making it easier to understand which features the model relies on for specific predictions. In addition, we study the computational behavior of the quantum annealing algorithm. Specifically, we analyze the minimum energy gap of the system during computation and the probability that the algorithm finds the correct solution. These analyses provide theoretical insight into why the method works effectively in practice.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

We propose a new regularized optimal transport (OT) formulation, termed sliced-regularized optimal transport (SROT). Unlike entropic OT (EOT), which regularizes the transport plan toward an independent coupling, SROT regularizes it toward a smoothened sliced OT (SOT) plan. To the best of our knowledge, SROT is the first approach to leverage a version of SOT plan as a reference to improve classical OT. We provide a formal definition of SROT, derive its dual formulation, and provide a post-Bayesian interpretation of SROT. We then develop a Sinkhorn-style algorithm for efficient computation, retaining the same scalability advantages as EOT. By incorporating a scalable SOT plan as a prior, SROT yields more accurate approximations of the exact OT plan than EOT under the same level of regularization. Moreover, the resulting transport plan improves upon the reference SOT plan itself. We further introduce the corresponding OT divergence induced by SROT, named SROT divergence, and analyze its topological and computational properties. Finally, we validate our approach through experiments on synthetic datasets and color transfer tasks, demonstrating that SROT is better than both EOT and SOT in approximating exact OT. Additional experiments on gradient flows further highlight the advantages of SROT divergence.

A levelset free but a hybrid image segmentation approach based on a modified version of the piece wise constant shape gradient of an Mumford Shah shape functional and a repulsive function is considered. The segmentation is performed a non-local shape based through an evolution of discrete curves driven by a non local shape based energy to segment images containing disjoint regions and multiple boundaries. This formulation has a novel additional component as a multivariable function dependent on a few sampled points of the curves that handles the occurrence of self intersection during boundary curves evolution. The method is applied to a few gray scale and color images, including images with nested structures and astronomical objects. The results indicate effective segmentation in complex scenarios with absolute control on the topology of the segments and self-intersections of the boundaries

Stochastic Gradient Descent (SGD) is a known stochastic iterative method popular for large-scale convex optimization problems due to its simple implementation and scalability. Some objectives, such as those found in complex-valued neural networks, benefit from updates like in SGD and Gradient Descent (GD) with a newly defined ``gradient'' that allows for complex parameters. This complex variant of the SGD/GD methods has already been proposed, but convergence guarantees without analyticity constraints have not yet been provided. We propose a variant of SGD (complex SGD) that allows for complex parameters, and we provide convergence guarantees under assumptions that parallel those from the real setting. Notably, these results extend to GD as well, and with the same set of assumptions, we confirm that some directional bias results extend from the real to the complex setting for kernel regression problems. We provide empirical results demonstrating the efficacy of the complex SGD in kernel regression problems utilizing complex reproducing kernel Hilbert spaces. In particular, we demonstrate we may recover superoscillation functions and Blaschke products from the Fock Space and Hardy Space, respectively, as the optimal functions for a particular choice of a loss function.

Current model training approaches incorporate user information directly into shared weights, making individual data removal computationally infeasible without retraining. This paper presents a three-layer architecture that decouples personal data from shared weights by combining a static base model, composable domain-expert LoRA adapters that shape behavior without imparting user data, and per-user proxy artefacts whose deletion constitutes deterministic unlearning. Evaluation on Phi-3.5-mini and Llama-3.1-8B confirms per-user differentiation in which personal data influences outputs while remaining isolated, verified by a return to baseline after proxy removal (KL divergence of approximately 0.21 nats, 82-89% verification pass rate) and near-zero cross-user contamination. Because user-specific information never enters shared weights, the architecture mitigates model inversion, membership inference, and training-data extraction against shared model components by construction. The approach converts machine unlearning from an intractable weight-editing problem into a deterministic deletion operation that preserves personalization alongside privacy-enhancing guarantees and is compatible with differentially private stochastic gradient descent (DP-SGD) for privacy-preserving shared model improvement.

We study online inference and asymptotic covariance estimation for the stochastic gradient descent (SGD) algorithm. While classical methods (such as plug-in and batch-means estimators) are available, they either require inaccessible second-order (Hessian) information or suffer from slow convergence. To address these challenges, we propose a novel, fully online de-biased covariance estimator that eliminates the need for second-order derivatives while significantly improving estimation accuracy. Our method employs a bias-reduction technique to achieve a convergence rate of $n^{(α-1)/2} \sqrt{\log n}$, outperforming existing Hessian-free alternatives.

When training neural networks with full-batch gradient descent (GD) and step size $η$, the largest eigenvalue of the Hessian -- the sharpness $S(\boldsymbolθ)$ -- rises to $2/η$ and hovers there, a phenomenon termed the Edge of Stability (EoS). \citet{damian2023selfstab} showed that this behavior is explained by a self-stabilization mechanism driven by third-order structure of the loss, and that GD implicitly follows projected gradient descent (PGD) on the constraint $S(\boldsymbolθ)\leq 2/η$. For mini-batch stochastic gradient descent (SGD), the sharpness stabilizes below $2/η$, with the gap widening as the batch size decreases; yet no theoretical explanation exists for this suppression. We introduce stochastic self-stabilization, extending the self-stabilization framework to SGD. Our key insight is that gradient noise injects variance into the oscillatory dynamics along the top Hessian eigenvector, strengthening the cubic sharpness-reducing force and shifting the equilibrium below $2/η$. Following the approach of \citet{damian2023selfstab}, we define stochastic predicted dynamics relative to a moving projected gradient descent trajectory and prove a stochastic coupling theorem that bounds the deviation of SGD from these predictions. We derive a closed-form equilibrium sharpness gap: $ΔS = ηβσ_{\boldsymbol{u}}^{2}/(4α)$, where $α$ is the progressive sharpening rate, $β$ is the self-stabilization strength, and $σ_{ \boldsymbol{u}}^{2}$ is the gradient noise variance projected onto the top eigenvector. This formula predicts that smaller batch sizes yield flatter solutions and recovers GD when the batch equals the full dataset.

Full-batch gradient descent on neural networks drives the largest Hessian eigenvalue to the threshold $2/η$, where $η$ is the learning rate. This phenomenon, the Edge of Stability, has resisted a unified explanation: existing accounts establish self-regulation near the edge but do not explain why the trajectory is forced toward $2/η$ from arbitrary initialization. We introduce the edge coupling, a functional on consecutive iterate pairs whose coefficient is uniquely fixed by the gradient-descent update. Differencing its criticality condition yields a step recurrence with stability boundary $2/η$, and a second-order expansion yields a loss-change formula whose telescoping sum forces curvature toward $2/η$. The two formulas involve different Hessian averages, but the mean value theorem localizes each to the true Hessian at an interior point of the step segment, yielding exact forcing of the Hessian eigenvalue with no gap. Setting both gradients of the edge coupling to zero classifies fixed points and period-two orbits; near a fixed point, the problem reduces to a function of the half-amplitude alone, which determines which directions support period-two orbits and on which side of the critical learning rate they appear.

We consider the problem of sampling from a probability distribution $π$. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from $π$. We consider the effect of replacing $π$ with a sequence of moving targets $(π_t)_{t\ge0}$ defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases. We also consider popular time discretisations and explore their convergence properties. We show that in the Fisher--Rao case, replacing the target distribution with a geometric mixture of initial and target distribution never leads to a convergence speed up both in continuous time and in discrete time. Finally, we explore the gradient flow structure of tempered dynamics and derive novel adaptive tempering schedules.

Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.

This is our fourth work in the series on machine learning (ML) moment closure models for the radiative transfer equation (RTE). In the first three papers of this series, we considered the RTE in slab geometry in 1D1V (i.e. one dimension in physical space and one dimension in angular space), and introduced a gradient-based ML moment closure [1], then enforced the hyperbolicity through a symmetrizer [2], or together with physical characteristic speeds by learning the eigenvalues of the Jacobian matrix [3]. Here, we extend our framework to the RTE in 2D2V (i.e. two dimensions in physical space and two dimensions in angular space). The main idea is to preserve the leading part of the classical $P_N$ model and modify only the highest-order block row. By analyzing the structural properties of the $P_N$ model, we show that its coefficient matrices are symmetric and admit a block-tridiagonal structure. Then we use this property to introduce a block-diagonal symmetrizer for the ML moment model and derive explicit algebraic conditions on the closure blocks which guarantee the symmetrizable hyperbolicity of the resulting ML system. These conditions lead to a natural parametrization of the closure in terms of a symmetric positive definite matrix together with symmetric closure blocks, which can be learned from data while automatically enforcing symmetrizable hyperbolicity by construction. The numerical results show that the proposed framework improves upon the classical $P_N$ model while maintaining hyperbolicity.

This paper presents a model-based reinforcement learning (RL) framework for optimal closed-loop control of nonlinear robotic systems. The proposed approach learns linear lifted dynamics through Koopman operator theory and integrates the resulting model into an actor-critic architecture for policy optimization, where the policy represents a parameterized closed-loop controller. To reduce computational cost and mitigate model rollout errors, policy gradients are estimated using one-step predictions of the learned dynamics rather than multi-step propagation. This leads to an online mini-batch policy gradient framework that enables policy improvement from streamed interaction data. The proposed framework is evaluated on several simulated nonlinear control benchmarks and two real-world hardware platforms, including a Kinova Gen3 robotic arm and a Unitree Go1 quadruped. Experimental results demonstrate improved sample efficiency over model-free RL baselines, superior control performance relative to model-based RL baselines, and control performance comparable to classical model-based methods that rely on exact system dynamics.

Online federated learning (OFL) has emerged as a popular framework for decentralized decision-making over continuous data streams without compromising client privacy. However, the adversary model assumed in standard OFL typically precludes any potential benefits of parallelization. Further, it fails to adequately capture the different sources of statistical variation in OFL problems. In this paper, we extend the OFL paradigm by integrating a stochastically extended adversary (SEA). Under this framework, the loss function remains fixed across clients over time. However, the adversary dynamically and independently selects the data distribution for each client at each time. We propose the \algoOFL{} algorithm to solve this problem, which utilizes online stochastic gradient descent at the clients, along with periodic global aggregation via the server. We establish bounds on the global network regret over a time horizon $T$ for two classes of functions: (1) for smooth and convex losses, we prove an $\mathcal{O}(\sqrt{T})$ bound, and (2) for smooth and strongly convex losses, we prove an $\mathcal{O}(\log T)$ bound. Through careful analysis, we quantify the individual impact of both spatial (across clients) and temporal (over time) data heterogeneity on the regret bounds. Consequently, we identify a regime of mild temporal variation (relative to stochastic gradient variance), where the network regret improves with parallelization. Hence, in the SEA setting, our results improve the existing pessimistic worst-case results in online federated learning.

In gradient-based learning, a step size chosen in parameter units does not produce a predictable per-step change in function output. This often leads to instability in the streaming setting (i.e., batch size=1), where stochasticity is not averaged out and update magnitudes can momentarily become arbitrarily big or small. Instead, we propose intentional updates: first specify the intended outcome of an update and then solve for the step size that approximately achieves it. This strategy has precedent in online supervised linear regression via Normalized Least Mean Squares algorithm, which selects a step size to yield a specified change in the function output proportional to the current error. We extend this principle to streaming deep reinforcement learning by defining appropriate intended outcomes: Intentional TD aims for a fixed fractional reduction of the TD error, and Intentional Policy Gradient aims for a bounded per-step change in the policy, limiting local KL divergence. We propose practical algorithms combining eligibility traces and diagonal scaling. Empirically, these methods yield state-of-the-art streaming performance, frequently performing on par with batch and replay-buffer approaches.

Linear attention mechanisms have emerged as promising alternatives to softmax attention, offering linear-time complexity during inference. Recent advances such as Gated DeltaNet (GDN) and Kimi Delta Attention (KDA) have demonstrated that the delta rule, an online gradient descent update, enables superior associative recall compared to simple additive updates. While KDA refined the coarse head-wise decay gate into channel-wise decay, the learning rate $β_t$ in the delta update remains a scalar, limiting the model's capacity for dimension-specific adaptation. We introduce FG$^2$-GDN, which replaces the scalar $β_t$ with a channel-wise vector analogous to the transition from SGD to per-coordinate adaptive optimizers such as AdaGrad and Adam. We further propose FG$^2$-GDN+, which decouples the scaling for keys and values, enabling independent control of erasure strength and write strength. Experiments on synthetic and real-world benchmarks show that FG$^2$-GDN and its variant improve associative recall and long-context understanding over GDN and KDA, with comparable computational efficiency.

Empirical studies of trained models often report a transient regime in which signal is detectable in a finite gradient descent time window before overfitting dominates. We provide an analytically tractable random-matrix model that reproduces this phenomenon for gradient flow in a linear teacher--student setting. In this framework, learning occurs when an isolated eigenvalue separates from a noisy bulk, before eventually disappearing in the overfitting regime. The key ingredient is anisotropy in the input covariance, which induces fast and slow directions in the learning dynamics. In a two-block covariance model, we derive the full time-dependent bulk spectrum of the symmetrized weight matrix through a $2\times 2$ Dyson equation, and we obtain an explicit outlier condition for a rank-one teacher via a rank-two determinant formula. This yields a transient Baik-Ben Arous-Péché (BBP) transition: depending on signal strength and covariance anisotropy, the teacher spike may never emerge, emerge and persist, or emerge only during an intermediate time interval before being reabsorbed into the bulk. We map the corresponding phase diagrams and validate the theory against finite-size simulations. Our results provide a minimal solvable mechanism for early stopping as a transient spectral effect driven by anisotropy and noise.

We prove a centre manifold theorem for a map along a manifold-with-boundary of fixed points, and provide an application to the study of gradient descent with large step size on two-layer matrix factorisation problems.

Enterprise SLM deployment faces epistemic asymmetry: small models cannot self-correct reasoning errors, while frontier LLMs incur prohibitive costs and data sovereignty risks at scale. We propose Semantic Gradient Descent (SGDe), a teacher-student framework that compiles agentic workflows into discrete execution plans--DAG topologies, system prompts, and deterministic code. The trailing e distinguishes this discrete, compilation-based approach from stochastic gradient descent. Operating in discrete semantic space, a frontier teacher generates natural-language critiques that serve as directional gradients to iteratively refine the SLM's workflow artefacts. We formalise SGDe under PAC learning, establishing sample-complexity bounds that enable convergence with as few as three training examples by leveraging the teacher as a statistical prior. On an adversarially synthesized GSM-Hard test set, compiled workflows achieve 91.3% accuracy at m=5 and 99.3% at m=3--a +26.3% to +34.3% absolute gain over state-of-the-art prompt optimisers. Within harness engineering, SGDe treats deterministic code placement (which subtasks to delegate to Python versus retain as LLM calls) as a trace-driven, per-node optimisation target, generalising static whole-problem offloading in PAL and PoT. The teacher compiles two deterministic structures: capability offloading (delegating subtasks to Python when the SLM is unreliable) and structural consensus (wrapping variance-sensitive steps in fan-out/fan-in subgraphs with deterministic voting).

Fine-tuning adapts pretrained networks to new objectives. Whether the resulting depth profile of representational change reflects an intrinsic property of the model or the magnitude of gradient flow has not been tested directly. We measure this profile across 240 fine-tuning runs spanning 15 models in four architecture families (encoder and decoder transformers, a state-space model, and an RNN) at scales from 125M to 6.9B parameters. Representational change concentrates in output-proximal layers in every standard-training run except one. We apply a per-layer control that equalizes $\|ΔW\|/\|W\|$ across layers after each optimizer step. Under this control, the profile persists in some conditions and collapses in others. At 125M--350M, sequential-block architectures (BERT, OPT, GPT-2) retain the slope across tested objectives while parallel-block architectures (Pythia, CodeGen) retain it only for causal-language-modeling objectives. This architectural distinction narrows at 1.3B--1.4B, where both block types show positive equal-step slopes for CausalLM. Under standard training, profile shape is described by two additional axes: steepness tracks a training-free objective distance at initialization, and profile width is dominated by architecture. We treat the locality gradient, the depthwise slope of representational change, as a composite phenomenon whose components are scale-dependent.

Analysis of Stochastic Gradient Descent (SGD) and its variants typically relies on the assumption of uniformly bounded variance, a condition that frequently fails in practical non-convex settings, such as neural network training, as well as in several elementary optimization settings. While several relaxations are explored in the literature, the Blum-Gladyshev (BG-0) condition, which permits the variance to grow quadratically with distance has recently been shown to be the weakest condition. However, the study of the oracle complexity of stochastic first-order non-convex optimization under BG-0 has remained underexplored. In this paper, we address this gap and establish information-theoretic lower bounds, proving that finding an $ε$-stationary point requires $Ω(ε^{-6})$ stochastic BG-0 oracle queries for smooth functions and $Ω(ε^{-4})$ queries under mean-square smoothness. These limits demonstrate an unavoidable degradation from classical bounded-variance complexities, i.e., $Ω(ε^{-4})$ and $Ω(ε^{-3})$ for smooth and mean-square smooth cases, respectively. To match these lower bounds, we consider Proximally Anchored STochastic Approximation (PASTA), a unified algorithmic framework that couples Halpern anchoring with Tikhonov regularization to dynamically mitigate the extra variance explosion term permitted by the BG-0 oracle. We prove that PASTA achieves minimax optimal complexities across numerous non-convex regimes, including standard smooth, mean-square smooth, weakly convex, star-convex, and Polyak-Lojasiewicz functions, entirely under an unbounded domain and unbounded stochastic gradients.

Data attribution and valuation are critical for understanding data-model synergy for Large Language Models (LLMs), yet existing gradient-based methods suffer from scalability challenges on LLMs. Inspired by human cognition, where decision making relies on a focused readout of relevant memories rather than replaying all pathways, we introduce RISE (Readout Influence Sketching Estimator). Instead of computing and indexing gradients across the entire LLM, RISE focuses on influence hotspots at the output layer, where influence signals concentrate, and the gradient admits a decomposed outer-product form. This enables a dual-channel representation combining a lexical residual channel (RH) and a semantic projected-error channel (GH). Applying CountSketch projections to these channels achieves strong compression while maintaining accurate attribution. Across the OLMo (1B-32B) and Pythia (14M-6.9B) families, RISE reduces index storage by up to 112$\times$ compared to RapidIn and scales to 32B parameters LLM, where gradient-based baselines such as RapidIn and ZO-Inf become memory-infeasible. We evaluate RISE on two paradigms: (1) retrospective attribution, retrieving influential training examples for specific predictions, and (2) prospective valuation, scoring candidate data utility zero-shot. We validate RISE on three tasks: Howdy backdoor data detection, Finance-Medical domain separation, and Brain Rot high-quality data selection. In a closed-loop Brain Rot study, continued pretraining on RISE-selected data yields consistent downstream improvements. Overall, RISE provides a practical and scalable primitive for influence analysis and training-data selection in modern large language models.

Test-Time Adaptation (TTA) for black-box models accessible only via APIs remains a largely unexplored challenge. Existing approaches such as post-hoc output refinement offer limited adaptive capacity, while Zeroth-Order Optimization (ZOO) enables input-space adaptation but faces high query costs and optimization challenges in the unsupervised TTA setting. We introduce BETA (Black-box Efficient Test-time Adaptation), a framework that addresses these limitations by employing a lightweight, local white-box steering model to create a tractable gradient pathway. Through a prediction harmonization technique combined with consistency regularization and prompt learning-oriented filtering, BETA enables stable adaptation with no additional API calls and negligible latency beyond standard inference. On ImageNet-C, BETA achieves a +7.1% accuracy gain on ViT-B/16 and +3.4% on CLIP, surpassing strong white-box and gray-box methods including TENT and TPT. On a commercial API, BETA achieves comparable performance to ZOO at 250x lower cost while maintaining real-time inference speed, establishing it as a practical and efficient solution for real-world black-box TTA.

We consider the problem of approximating a function by an element of a nonlinear manifold which admits a differentiable parametrization, typical examples being neural networks with differentiable activation functions or tensor networks. Natural gradient descent (NGD) for the optimization of a loss function can be seen as a preconditioned gradient descent where updates in the parameter space are driven by a functional perspective. In a spirit similar to Newton's method, a NGD step uses, instead of the Hessian, the Gram matrix of the generating system of the tangent space to the approximation manifold at the current iterate, with respect to a suitable metric. This corresponds to a locally optimal update in function space, following a projected gradient onto the tangent space to the manifold. Still, both gradient and natural gradient descent methods get stuck in local minima. Furthermore, when the model class is a nonlinear manifold or the loss function is not ideally conditioned (e.g., the KL-divergence for density estimation, or a norm of the residual of a partial differential equation in physics informed learning), even the natural gradient might yield non-optimal directions at each step. This work introduces a natural version of classical inertial dynamic methods like Heavy-Ball or Nesterov and show how it can improve the learning process when working with nonlinear model classes.

We consider the problem of minimizing the convergence time for decentralized federated learning (DFL) in wireless networks under broadcast communications, with focus on mixing matrix design. The mixing matrix is a critical hyperparameter for DFL that simultaneously controls the convergence rate across iterations and the communication demand per iteration, both strongly influencing the convergence time. Although the problem has been studied previously, existing solutions are mostly designed for decentralized parallel stochastic gradient descent (D-PSGD), which requires the mixing matrix to be symmetric and doubly stochastic. These constraints confine the activated communication graph to undirected (i.e., bidirected) graphs, which limits design flexibility. In contrast, we consider mixing matrix design for stochastic gradient push (SGP), which allows asymmetric mixing matrices and hence directed communication graphs. By analyzing how the convergence rate of SGP depends on the mixing matrices, we extract an objective function that explicitly depends on graph-theoretic parameters of the activated communication graph, based on which we develop an efficient design algorithm with performance guarantees. Our evaluations based on real data show that the proposed solution can notably reduce the convergence time compared to the state of the art without compromising the quality of the trained model.

Sign-based optimization algorithms, such as SignSGD, have garnered significant attention for their remarkable performance in distributed learning and training large foundation models. Despite their empirical superiority, SignSGD is known to diverge on non-smooth objectives, which are ubiquitous in modern machine learning due to ReLUs, max-pools, and mixture-of-experts. To overcome this fundamental limitation, we propose \textbf{StoSignSGD}, an algorithm that injects structural stochasticity into the sign operator while maintaining an unbiased update step. In the regime of (online) convex optimization, our theoretical analysis shows that StoSignSGD rigorously resolves the non-convergence issues of SignSGD, achieving a sharp convergence rate matching the lower bound. For the more challenging non-convex non-smooth optimization, we introduce generalized stationary measures that encompass prior definitions, proving that StoSignSGD improves upon the best-known complexity bounds by dimensional factors. Empirically, StoSignSGD exhibits robust stability and superior efficiency across diverse large language model (LLM) training regimes. Notably, in low-precision FP8 pretraining -- a setting where AdamW fails catastrophically -- StoSignSGD remains highly stable and yields a remarkable 1.44$\times$ to 2.14$\times$ speedup relative to established baselines. Furthermore, when fine-tuning 7B LLMs on mathematical reasoning tasks, StoSignSGD delivers substantial performance gains over both AdamW and SignSGD. Finally, to dissect the mechanisms driving its success, we develop a sign conversion framework capable of transforming any general optimizer into its unbiased, sign-based counterpart. Utilizing this framework, we deconstruct the core components of StoSignSGD and present a comprehensive ablation study to empirically validate our algorithmic design choices.

We present a geometric framework for Reinforcement Learning (RL) that views policies as maps into the Wasserstein space of action probabilities. First, we define a Riemannian structure induced by stationary distributions, proving its existence in a general context. We then define the tangent space of policies and characterize the geodesics, specifically addressing the measurability of vector fields mapped from the state space to the tangent space of probability measures over the action space. Next, we formulate a general RL optimization problem and construct a gradient flow using Otto's calculus. We compute the gradient and the Hessian of the energy, providing a formal second-order analysis. Finally, we illustrate the method with numerical examples for low-dimensional problems, computing the gradient directly from our theoretical formalism. For high-dimensional problems, we parameterize the policy using a neural network and optimize it based on an ergodic approximation of the cost.

This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain. The proposed method circumvents the curse-of-dimensionality that plagues traditional discretizations and the computational bottlenecks of gradient-based neural network solvers through a synergistic combination of techniques. First, a prescribed Gaussian envelope encodes the far-field decay of the wavefunction, enabling a space-time separation where the spatial approximation is handled by a frozen, single-hidden-layer neural network with data-driven sampled features. This yields a gradient-free formalism where spatial derivatives are analytically precomputed and time-dependence is evolved via reduced ODEs. Second, a stochastic-dimension sampler provides a conditionally unbiased estimate of the spatial operator by evaluating only a small subset of spatial dimensions at each time step, essentially reducing computational and memory costs. Discrete conservation laws are also enforced, ensuring long-term stability. Extensive numerical experiments on GPE in up to 1000 dimensions demonstrate that SD-FSNN achieves significantly higher accuracy and efficiency compared to state-of-the-art methods, including PINNs, randomized feature methods, and tensor-network approaches. The results confirm that SD-FSNN effectively mitigates the Kolmogorov $n$-width barrier for frozen-basis models on structured solution manifolds.

We develop a unified treatment of credit assignment for RL training in multi-agent LLM systems. We show that observed reward alone cannot distinguish an agent that determines it from one that never affects it, and that standard shared-reward training performs exact gradient ascent on each agent's private utility rather than system performance. Moreover, we prove no single scalar per agent can consistently account for joint performance once agents interact. We thus develop the unique background-dependent notion of marginal contribution satisfying natural consistency requirements. From it we derive gradient-correct marginal contribution training signals, identify them from filtered feedback, and optimally allocate a budget of exact counterfactual evaluations against learned-signal error. Instantiated in GRPO, our signal improves routed GSM8K accuracy over winner-take-all training at no extra generation cost.

The critical damping condition of the damped harmonic oscillator model of SGD with momentum (Qian, 1999) yields a momentum schedule with no tuned hyperparameters: mu(t) = 1 - 2*sqrt(alpha(t)). Across five seeds on ResNet-18/CIFAR-10 (200-epoch cosine schedule) it reaches 90% test accuracy 2.34x faster than constant mu=0.9 (range 1.71-2.86x, 5/5 seeds, one-sided paired t-test p=4e-4), at the cost of a real final-accuracy deficit of 0.46 pp (5/5 seeds, p=0.009). A short-schedule control rules out a schedule-length artifact: compressed baselines either pay 0.5-0.9 pp of accuracy or stay slower to 90% at equal accuracy. A hybrid recipe -- critical-damping momentum until 90%, then constant mu=0.9 -- removes the deficit and keeps the speedup: 95.45 +/- 0.05% final accuracy at 2.4x faster progress to 90% (n=5). The speedup generalizes across architectures (VGG-16 without skip connections: 1.72x, n=3); on CIFAR-100 early gains persist (2-4x to mid-training thresholds) but the accuracy cost grows (-1.7 pp), compressing the accuracy-matched gain to 1.14x. We also report an exhaustive negative result on surgical layer selection. Version 2 of this paper claimed that gradient attribution on misclassified images selects which layers to retrain; running the identical correction protocol on all 35 combinations of 3-of-7 layer groups ranks the selected triple 11th of 35 (exact p=0.31) -- no better than random. What survives is weaker but real: combinations containing the top-ranked layer outperform the rest (+6.2 vs -1.4 mean net error reduction), and the bottom of the gradient-norm ranking reliably predicts the most harmful interventions (down to -20 net errors). Gradient attribution on errors is a harm-avoidance signal, not a selector of repair targets. We release the full 35-combination landscape as a baseline for layer-selection claims.

The adaptive moment estimation (Adam) optimizer proposed by Kingma & Ba (2014) is presumably the most popular stochastic gradient descent (SGD) optimization method for the training of deep neural networks (DNNs) in artificial intelligence (AI) systems. Despite its groundbreaking success in the training of AI systems, it still remains an open research problem to provide a complete error analysis of Adam, not only for optimizing DNNs but even when applied to strongly convex stochastic optimization problems (SOPs). Previous error analysis results for strongly convex SOPs in the literature provide conditional convergence analyses that rely on the assumption that Adam does not diverge to infinity but remains uniformly bounded. It is the key contribution of this work to establish uniform a priori bounds for Adam and, thereby, to provide -- for the first time -- an unconditional error analysis for Adam for a large class of strongly convex SOPs.

Counterfactual explanations (CE) explain model decisions by identifying input modifications that lead to different predictions. Most existing methods operate at the instance level. Distributional Counterfactual Explanations (DCE) extend this setting by optimizing an optimal transport objective that balances proximity to a factual input distribution and alignment to a target output distribution, with statistical certification via chance constrained bounds. However, DCE relies on gradient based optimization, while many real-world tabular pipelines are dominated by non-differentiable models. We propose DISCOVER, a model-agnostic solver for distributional counterfactual explanations. DISCOVER preserves the original DCE objective and certification while replacing gradient descent with a budgeted propose-and-select search paradigm. It exploits a sample-wise decomposition of the transport objective to compute per-row impact scores and enforce a top-k intervention budget, focusing edits on the most influential samples. To guide candidate generation without predictor gradients, DISCOVER introduces an OT-guided cone sampling primitive driven by input-side transport geometry. Experiments on multiple tabular datasets demonstrate strong joint alignment of input and output distributions, extending distributional counterfactual reasoning to modern black box learning pipelines. A code repository is available at: https://github.com/VALHALLA9/Discover.

Dataset distillation, a training-aware data compression technique, has recently attracted increasing attention as an effective tool for mitigating costs of optimization and data storage. However, progress remains largely empirical. Mechanisms underlying the extraction of task-relevant information from the training process and the efficient encoding of such information into synthetic data points remain elusive. In this paper, we theoretically analyze practical algorithms of dataset distillation applied to the gradient-based training of two-layer neural networks with width $L$. By focusing on a non-linear task structure called multi-index model, we prove that the low-dimensional structure of the problem is efficiently encoded into the resulting distilled data. This dataset reproduces a model with high generalization ability for a required memory complexity of \tildeΘ$$(r^2d+L), where $d$ and $r$ are the input and intrinsic dimensions of the task. To the best of our knowledge, this is one of the first theoretical works that include a specific task structure, leverage its intrinsic dimensionality to quantify the compression rate and study dataset distillation implemented solely via gradient-based algorithms.

This paper addresses the distributed stochastic minimax optimization problem subject to stochastic constraints. We propose a novel first-order Softmax-Weighted Switching Gradient method tailored for federated learning. Under full client participation, our algorithm achieves the standard $\tilde{\mathcal{O}}(ε^{-4})$ oracle complexity to satisfy a unified bound $ε$ for both the optimality gap and feasibility tolerance. We extend our theoretical analysis to the practical partial participation regime by quantifying client sampling noise through a stochastic superiority assumption. Furthermore, by relaxing standard boundedness assumptions on the objective functions, we establish a strictly tighter lower bound for the softmax hyperparameter. We provide a unified error decomposition and establish a sharp $\mathcal{O}(\log\frac{1}δ)$ high-probability convergence guarantee. Ultimately, our framework demonstrates that a single-loop primal-only switching mechanism provides a stable alternative for optimizing worst-case client performance, effectively bypassing the hyperparameter sensitivity and convergence oscillations often encountered in traditional primal-dual or penalty-based approaches. We verify the efficacy of our algorithm via experiment on the Neyman-Pearson (NP) classification, fair classification, and federated safe reinforcement learning tasks.

The Edge of Stability (EoS) is a phenomenon where the sharpness (largest eigenvalue) of the Hessian approaches and then hovers near the stability threshold $2/η$ during gradient descent (GD) with step size $η$. Despite (apparently) violating classical smoothness assumptions, EoS has been widely observed in deep learning, but its theoretical foundations remain incomplete. We provide an interpretation of EoS through the lens of Directional Smoothness [Mishkin et al., 2024]. This interpretation naturally extends to non-Euclidean norms, which we use to define generalized sharpness under an arbitrary norm. Our generalized sharpness measure includes previously studied vanilla GD and preconditioned GD as special cases, as well as methods for which EoS has not been studied, such as $\ell_{\infty}$-descent, Block CD, Spectral GD, and their normalized versions. Through experiments on neural networks, we show that non-Euclidean GD with our generalized sharpness also exhibits progressive sharpening followed by oscillations around or above the threshold $2/η$. Practically, our framework provides a geometry-aware spectral diagnostic that can be applied across a broad class of non-Euclidean gradient methods.