Daily Papers

This work tackles the fundamental challenges in Federated Learning (FL) posed by arbitrary client participation and data heterogeneity, prevalent characteristics in practical FL settings. It is well-established that popular FedAvg-style algorithms struggle with exact convergence and can suffer from slow convergence rates since a decaying learning rate is required to mitigate these scenarios. To address these issues, we introduce the concept of stochastic matrix and the corresponding time-varying graphs as a novel modeling tool to accurately capture the dynamics of arbitrary client participation and the local update procedure. Leveraging this approach, we offer a fresh decentralized perspective on designing FL algorithms and present FOCUS, Federated Optimization with Exact Convergence via Push-pull Strategy, a provably convergent algorithm designed to effectively overcome the previously mentioned two challenges. More specifically, we provide a rigorous proof demonstrating that FOCUS achieves exact convergence with a linear rate regardless of the arbitrary client participation, establishing it as the first work to demonstrate this significant result.

This paper rigorously shows how over-parameterization changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where M^* in R^{n times n} is a positive semi-definite unknown matrix of rank r ll n, and one uses a symmetric parameterization XX^top to learn M^*. Here X in R^{n times k} with k > r is the factor matrix. We give a novel Omega (1/T^2) lower bound of randomly initialized GD for the over-parameterized case (k >r) where T is the number of iterations. This is in stark contrast to the exact-parameterization scenario (k=r) where the convergence rate is exp (-Omega (T)). Next, we study asymmetric setting where M^* in R^{n_1 times n_2} is the unknown matrix of rank r ll min{n_1,n_2}, and one uses an asymmetric parameterization FG^top to learn M^* where F in R^{n_1 times k} and G in R^{n_2 times k}. Building on prior work, we give a global exact convergence result of randomly initialized GD for the exact-parameterization case (k=r) with an exp (-Omega(T)) rate. Furthermore, we give the first global exact convergence result for the over-parameterization case (k>r) with an exp(-Omega(alpha^2 T)) rate where alpha is the initialization scale. This linear convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from Omega (1/T^2) to linear convergence. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of alpha, recovering the rate in the exact-parameterization case.

Autoregressive language models execute Transformer layers sequentially, creating a latency bottleneck that is not removed by conventional tensor or pipeline parallelism. We study whether this layerwise dependency can be relaxed by treating the hidden-state trace across layers as the solution of a nonlinear residual equation and solving it with parallel Newton-style updates. While this view is principled, exact Newton corrections require expensive Jacobian-vector products and naive fixed-point iterations are unstable on trained Transformers. We introduce Structured Newton Layer Parallelism (SNLP), a training and inference framework that replaces exact layer Jacobians with cheap architecture-induced surrogate dynamics. In residual Transformers, this yields Identity Newton (IDN), where the correction reduces to a prefix-sum-like update; in mHC-style architectures, HC Newton (HCN) uses the model's residual mixing matrix. We further introduce SNLP-aware regularization, which trains models to make one or a few structured Newton iterations accurately approximate the sequential forward. Experiments on nanochat-scale Transformers show that SNLP regularization improves layer-parallel compatibility and can also improve standard sequential perplexity, reducing baseline PPL by 4.7%-23.4%. At inference time, SNLP combined with layer fusion and chunkwise decomposition achieves practical wall-clock speedups: on a 0.5B Nanochat model, it reaches 2.3x speedup while still improving PPL by 6.1%. These results suggest that layer-parallel inference is not merely a numerical approximation to sequential execution, but can act as a useful solver-induced inference bias. We also characterize limitations: off-the-shelf pretrained models are less amenable to this procedure, and exact convergence recovers the sequential computation rather than providing monotonic inference-time scaling.

Recently, diffusion models have achieved significant advances in vision, text, and robotics. However, they still face slow generation speeds due to sequential denoising processes. To address this, a parallel sampling method based on Picard iteration was introduced, effectively reducing sequential steps while ensuring exact convergence to the original output. Nonetheless, Picard iteration does not guarantee faster convergence, which can still result in slow generation in practice. In this work, we propose a new parallelization scheme, the Picard Consistency Model (PCM), which significantly reduces the number of generation steps in Picard iteration. Inspired by the consistency model, PCM is directly trained to predict the fixed-point solution, or the final output, at any stage of the convergence trajectory. Additionally, we introduce a new concept called model switching, which addresses PCM's limitations and ensures exact convergence. Extensive experiments demonstrate that PCM achieves up to a 2.71x speedup over sequential sampling and a 1.77x speedup over Picard iteration across various tasks, including image generation and robotic control.

Objective perturbation adds a random linear term to a regularized empirical risk and releases the exact perturbed minimizer. We study the finite computation obtained by releasing the N-th iterate of deterministic gradient descent on wmapsto F(w;S)+langle z,wrangle, where zsimmathcal N(0,σ^2I_d) is drawn once before optimization. For strongly convex and smooth objectives with Lipschitz Hessian, we prove an explicit condition under which the map zmapsto w_N is a C^1-diffeomorphism on the bounded domains used in the privacy argument, with a quantitative lower bound on the smallest singular value of its Jacobian. This permits a direct change-of-variables analysis of the finite iterate. For generalized linear models, the resulting privacy-profile bound has no explicit ambient-dimension factor once the iteration condition holds, and its finite-iteration correction decreases geometrically. By letting the free truncation parameter grow slowly with N, we recover the corresponding exact-minimizer certificate in the limit. We also bound the expected excess empirical risk by dσ^2/(2μ) plus a geometrically decreasing optimization term, and transfer the result to population risk without an additional multiplicative condition-number factor in the leading statistical terms.

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes--Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos--Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within n updates for an ntimes n matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.

Increasing the number of parameters in language models is a common strategy to enhance their performance. However, smaller language models remain valuable due to their lower operational costs. Despite their advantages, smaller models frequently underperform compared to their larger counterparts, even when provided with equivalent data and computational resources. Specifically, their performance tends to degrade in the late pretraining phase. This is anecdotally attributed to their reduced representational capacity. Yet, the exact causes of this performance degradation remain unclear. We use the Pythia model suite to analyse the training dynamics that underlie this phenomenon. Across different model sizes, we investigate the convergence of the Attention and MLP activations to their final state and examine how the effective rank of their parameters influences this process. We find that nearly all layers in larger models stabilise early in training - within the first 20% - whereas layers in smaller models exhibit slower and less stable convergence, especially when their parameters have lower effective rank. By linking the convergence of layers' activations to their parameters' effective rank, our analyses can guide future work to address inefficiencies in the learning dynamics of small models.

Variational inference is becoming more and more popular for approximating intractable posterior distributions in Bayesian statistics and machine learning. Meanwhile, a few recent works have provided theoretical justification and new insights on deep neural networks for estimating smooth functions in usual settings such as nonparametric regression. In this paper, we show that variational inference for sparse deep learning retains the same generalization properties than exact Bayesian inference. In particular, we highlight the connection between estimation and approximation theories via the classical bias-variance trade-off and show that it leads to near-minimax rates of convergence for H\"older smooth functions. Additionally, we show that the model selection framework over the neural network architecture via ELBO maximization does not overfit and adaptively achieves the optimal rate of convergence.

We investigate the convergence rates and data sample sizes required for training a machine learning model using a stochastic gradient descent (SGD) algorithm, where data points are sampled based on either their loss value or uncertainty value. These training methods are particularly relevant for active learning and data subset selection problems. For SGD with a constant step size update, we present convergence results for linear classifiers and linearly separable datasets using squared hinge loss and similar training loss functions. Additionally, we extend our analysis to more general classifiers and datasets, considering a wide range of loss-based sampling strategies and smooth convex training loss functions. We propose a novel algorithm called Adaptive-Weight Sampling (AWS) that utilizes SGD with an adaptive step size that achieves stochastic Polyak's step size in expectation. We establish convergence rate results for AWS for smooth convex training loss functions. Our numerical experiments demonstrate the efficiency of AWS on various datasets by using either exact or estimated loss values.

We analyze Muon as originally proposed and used in practice -- using the momentum orthogonalization with a few Newton-Schulz steps. The prior theoretical results replace this key step in Muon with an exact SVD-based polar factor. We prove that Muon with Newton-Schulz converges to a stationary point at the same rate as the SVD-polar idealization, up to a constant factor for a given number q of Newton-Schulz steps. We further analyze this constant factor and prove that it converges to 1 doubly exponentially in q and improves with the degree of the polynomial used in Newton-Schulz for approximating the orthogonalization direction. We also prove that Muon removes the typical square-root-of-rank loss compared to its vector-based counterpart, SGD with momentum. Our results explain why Muon with a few low-degree Newton-Schulz steps matches exact-polar (SVD) behavior at a much faster wall-clock time and explain how much momentum matrix orthogonalization via Newton-Schulz benefits over the vector-based optimizer. Overall, our theory justifies the practical Newton-Schulz design of Muon, narrowing its practice-theory gap.

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

We present EGN, a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges to an epsilon-stationary point at a linear rate. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, and SGN optimizers across various supervised and reinforcement learning tasks.

Despite the widespread practical success of deep learning methods, our theoretical understanding of the dynamics of learning in deep neural networks remains quite sparse. We attempt to bridge the gap between the theory and practice of deep learning by systematically analyzing learning dynamics for the restricted case of deep linear neural networks. Despite the linearity of their input-output map, such networks have nonlinear gradient descent dynamics on weights that change with the addition of each new hidden layer. We show that deep linear networks exhibit nonlinear learning phenomena similar to those seen in simulations of nonlinear networks, including long plateaus followed by rapid transitions to lower error solutions, and faster convergence from greedy unsupervised pretraining initial conditions than from random initial conditions. We provide an analytical description of these phenomena by finding new exact solutions to the nonlinear dynamics of deep learning. Our theoretical analysis also reveals the surprising finding that as the depth of a network approaches infinity, learning speed can nevertheless remain finite: for a special class of initial conditions on the weights, very deep networks incur only a finite, depth independent, delay in learning speed relative to shallow networks. We show that, under certain conditions on the training data, unsupervised pretraining can find this special class of initial conditions, while scaled random Gaussian initializations cannot. We further exhibit a new class of random orthogonal initial conditions on weights that, like unsupervised pre-training, enjoys depth independent learning times. We further show that these initial conditions also lead to faithful propagation of gradients even in deep nonlinear networks, as long as they operate in a special regime known as the edge of chaos.

Despite the rapid development of safety alignment techniques for LLMs, defending against multi-turn jailbreaks is still a challenging task. In this paper, we conduct a comprehensive comparison, revealing that some existing defense methods can improve the robustness of LLMs against multi-turn jailbreaks but compromise usability, i.e., reducing general capabilities or causing the over-refusal problem. From the perspective of mechanism interpretability of LLMs, we discover that these methods fail to establish a boundary that exactly distinguishes safe and harmful feature representations. Therefore, boundary-safe representations close to harmful representations are inevitably disrupted, leading to a decline in usability. To address this issue, we propose X-Boundary to push harmful representations away from boundary-safe representations and obtain an exact distinction boundary. In this way, harmful representations can be precisely erased without disrupting safe ones. Experimental results show that X-Boundary achieves state-of-the-art defense performance against multi-turn jailbreaks, while reducing the over-refusal rate by about 20% and maintaining nearly complete general capability. Furthermore, we theoretically prove and empirically verify that X-Boundary can accelerate the convergence process during training. Please see our code at: https://github.com/AI45Lab/X-Boundary.

Markov Decision Processes (MDPs) are a formal framework for modeling and solving sequential decision-making problems. In finite-time horizons such problems are relevant for instance for optimal stopping or specific supply chain problems, but also in the training of large language models. In contrast to infinite horizon MDPs optimal policies are not stationary, policies must be learned for every single epoch. In practice all parameters are often trained simultaneously, ignoring the inherent structure suggested by dynamic programming. This paper introduces a combination of dynamic programming and policy gradient called dynamic policy gradient, where the parameters are trained backwards in time. For the tabular softmax parametrisation we carry out the convergence analysis for simultaneous and dynamic policy gradient towards global optima, both in the exact and sampled gradient settings without regularisation. It turns out that the use of dynamic policy gradient training much better exploits the structure of finite-time problems which is reflected in improved convergence bounds.

The alignment of language models with human preferences is vital for their application in real-world tasks. The problem is formulated as optimizing the model's policy to maximize the expected reward that reflects human preferences with minimal deviation from the initial policy. While considered as a straightforward solution, reinforcement learning (RL) suffers from high variance in policy updates, which impedes efficient policy improvement. Recently, direct preference optimization (DPO) was proposed to directly optimize the policy from preference data. Though simple to implement, DPO is derived based on the optimal policy that is not assured to be achieved in practice, which undermines its convergence to the intended solution. In this paper, we propose efficient exact optimization (EXO) of the alignment objective. We prove that EXO is guaranteed to optimize in the same direction as the RL algorithms asymptotically for arbitary parametrization of the policy, while enables efficient optimization by circumventing the complexities associated with RL algorithms. We compare our method to DPO with both theoretical and empirical analyses, and further demonstrate the advantages of our method over existing approaches on realistic human preference data.

Alternating projections (AP) near a clean intersection of smooth manifolds can converge slowly when the Friedrichs angle is small. We develop the Fixed-Point Manifold Acceleration (FPMA) framework for {local convergence and polynomial acceleration of} smooth maps that fix a manifold pointwise. {Under suitable regularity and uniform normal spectral stability, FPMA establishes local R-linear convergence and a smooth limiting map inducing a local retraction. At fixed points, consistent polynomial acceleration with p(1)=1 preserves the selected limit's first-order response to normal initial perturbations whenever the transformed normal blocks are uniformly spectrally stable. Under sufficient smoothness, we give a curvature criterion for second-order induced retractions and show that stable consistent polynomial acceleration preserves both the quadratic retraction expansion and this criterion. {As a special case, we establish new local R-linear convergence guarantees for fixed-degree Chebyshev-accelerated exact AP and second-order induced retractions. For uniform Friedrichs angle θ_Fin(0,π/2), acceleration strictly improves the local convergence rate per AP evaluation. The large-degree limiting spectral factor at fixed θ_F predicts O(θ_F^{-1}) evaluations for fixed relative error reduction as θ_Fdownarrow0, versus O(θ_F^{-2}) for ordinary AP.}} {The framework also covers} inexact, relaxed, and generalized AP {under their respective regularity and stability conditions.}

We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney--Witten--Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For c_L=c_R=c and a=(c-1)/12, the construction realizes primary dimension gaps Δ_1=(1+κ)a for sufficiently small fixed κ>0, and Δ_1=a+δ for any fixed δge0, at every sufficiently large real c. Every nonvacuum primary satisfies h,bar hge(c-1)/24. In the fixed-δ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in 1/c. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein--Hawking entropy and its corrections at every fixed positive E/c, where E=Δ-c/12. This includes 0<E<c/12, where thermal AdS dominates the canonical ensemble.

In this paper, we develop a fully discrete Galerkin method for solving initial value fractional integro-differential equations(FIDEs). We consider Generalized Jacobi polynomials(GJPs) with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution.

Diffusion language models generate without a fixed left-to-right order, leaving token ordering as a central algorithmic choice. Existing systems mainly use random masking or confidence-driven ordering, which respectively suffer from train--test mismatch and myopic exploration. We introduce DPRM (Doob -transform Process Reward Model), a plug-in token-ordering module that keeps the host architecture, denoising objective and supervision unchanged, and modifies only the ordering policy. DPRM starts from confidence-driven ordering and gradually shifts to process-reward-guided ordering through online estimates. We characterize the exact DPRM policy as a reward-tilted Gibbs reveal law, prove convergence of its stagewise Soft-BoN approximation, show that the online bucketized controller tracks the exact DPRM score at empirical-Bernstein rates, and establish a sample-complexity advantage under tractable optimization assumptions. Across nine hosts covering language reasoning, test-time scaling, protein, single-cell, molecular, DNA, text-to-image generation, and VQA, DPRM order variants improve several language, DNA, and multimodal settings while also identifying boundary cases where confidence-only ordering or task-specific utilities are preferable. Code is available at: https://github.com/DakeBU/DPRM-DLLM

Proximal operators are ubiquitous in inverse problems, commonly appearing as part of algorithmic strategies to regularize problems that are otherwise ill-posed. Modern deep learning models have been brought to bear for these tasks too, as in the framework of plug-and-play or deep unrolling, where they loosely resemble proximal operators. Yet, something essential is lost in employing these purely data-driven approaches: there is no guarantee that a general deep network represents the proximal operator of any function, nor is there any characterization of the function for which the network might provide some approximate proximal. This not only makes guaranteeing convergence of iterative schemes challenging but, more fundamentally, complicates the analysis of what has been learned by these networks about their training data. Herein we provide a framework to develop learned proximal networks (LPN), prove that they provide exact proximal operators for a data-driven nonconvex regularizer, and show how a new training strategy, dubbed proximal matching, provably promotes the recovery of the log-prior of the true data distribution. Such LPN provide general, unsupervised, expressive proximal operators that can be used for general inverse problems with convergence guarantees. We illustrate our results in a series of cases of increasing complexity, demonstrating that these models not only result in state-of-the-art performance, but provide a window into the resulting priors learned from data.

We implement physics-informed neural networks (PINNs) to solve the time-independent Schr\"odinger equation for three canonical one-dimensional quantum potentials: an infinite square well, a finite square well, and a finite barrier. The PINN models incorporate trial wavefunctions that exactly satisfy boundary conditions (Dirichlet zeros at domain boundaries), and they optimize a loss functional combining the PDE residual with a normalization constraint. For the infinite well, the ground-state energy is known (E = pi^2 in dimensionless units) and held fixed in training, whereas for the finite well and barrier, the eigenenergy is treated as a trainable parameter. We use fully-connected neural networks with smooth activation functions to represent the wavefunction and demonstrate that PINNs can learn the ground-state eigenfunctions and eigenvalues for these quantum systems. The results show that the PINN-predicted wavefunctions closely match analytical solutions or expected behaviors, and the learned eigenenergies converge to known values. We present training logs and convergence of the energy parameter, as well as figures comparing the PINN solutions to exact results. The discussion addresses the performance of PINNs relative to traditional numerical methods, highlighting challenges such as convergence to the correct eigenvalue, sensitivity to initialization, and the difficulty of modeling discontinuous potentials. We also discuss the importance of the normalization term to resolve the scaling ambiguity of the wavefunction. Finally, we conclude that PINNs are a viable approach for quantum eigenvalue problems, and we outline future directions including extensions to higher-dimensional and time-dependent Schr\"odinger equations.

We study simplified bootstrap problems for probability distributions on the infinite line and the circle. We show that the rapid convergence of the bootstrap method for problems on the infinite line is related to the fact that the smallest eigenvalue of the positive matrices in the exact solution becomes exponentially small for large matrices, while the moments grow factorially. As a result, the positivity condition is very finely tuned. For problems on the circle we show instead that the entries of the positive matrix of Fourier modes of the distribution depend linearly on the initial data of the recursion, with factorially growing coefficients. By positivity, these matrix elements are bounded in absolute value by one, so the initial data must also be fine-tuned. Additionally, we find that we can largely bypass the semi-definite program (SDP) nature of the problem on a circle by recognizing that these Fourier modes must be asymptotically exponentially small. With a simple ansatz, which we call the shoestring bootstrap, we can efficiently identify an interior point of the set of allowed matrices with much higher precision than conventional SDP bounds permit. We apply this method to solving unitary matrix model integrals by numerically constructing the orthogonal polynomials associated with the circle distribution.

Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas-Rachford splitting, Davis-Yin splitting, and primal-dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.

Standard supervised fine-tuning (SFT) assigns the same explicit loss weight to every expert demonstration, regardless of the model's changing competence over training queries. Reinforcement learning (RL) based methods adapt update strength using model-generated rollouts, but often require substantially more sampling and can be unstable on hard tasks. We propose Online Self-Weighted Fine-Tuning (OSW-FT), a simple method that augments SFT with online, trajectory-level weighting. For each query, OSW-FT estimates the model's current success rate using a small number of inference-only rollouts and rescales the standard SFT loss accordingly. The optimization direction remains anchored to the expert trajectory, while the update magnitude adapts online. For binary-verifiable reasoning, we connect this weighting to SFT and RL at the gradient level, inspired by variance-reduction principles. The resulting estimator is unbiased for the exact OSW-FT surrogate update for any finite rollout count, and we analyze convergence with respect to the corresponding surrogate objective. Evaluated across Qwen3 series ranging from 0.6B to 4B on multiple challenging benchmarks (e.g., AIME), OSW-FT consistently improves over SFT on small-to-medium scale models. OSW-FT offers a favorable compute-performance trade-off as a practical approach for fine-tuning small-to-medium LLMs on binary-verifiable reasoning tasks with only 2 online rollouts.

LiDAR mapping is important yet challenging in self-driving and mobile robotics. To tackle such a global point cloud registration problem, DeepMapping converts the complex map estimation into a self-supervised training of simple deep networks. Despite its broad convergence range on small datasets, DeepMapping still cannot produce satisfactory results on large-scale datasets with thousands of frames. This is due to the lack of loop closures and exact cross-frame point correspondences, and the slow convergence of its global localization network. We propose DeepMapping2 by adding two novel techniques to address these issues: (1) organization of training batch based on map topology from loop closing, and (2) self-supervised local-to-global point consistency loss leveraging pairwise registration. Our experiments and ablation studies on public datasets (KITTI, NCLT, and Nebula) demonstrate the effectiveness of our method.

Hyper-Connections (HC) generalizes residual connections by introducing dynamic residual matrices that mix information across multiple residual streams, accelerating convergence in deep neural networks. However, unconstrained residual matrices can compromise training stability. To address this, DeepSeek's Manifold-Constrained Hyper-Connections (mHC) approximately projects these matrices onto the Birkhoff polytope via iterative Sinkhorn--Knopp (SK) normalization. We identify two limitations of this approach: (i) finite SK iterations do not guarantee exact doubly stochasticity, leaving an approximation gap that can accumulate through network depth and undermine stability; (ii) efficient SK implementation requires highly specialized CUDA kernels, raising engineering barriers and reducing portability. Motivated by the Birkhoff--von Neumann theorem, we propose mHC-lite, a simple reparameterization that explicitly constructs doubly stochastic matrices as convex combinations of permutation matrices. This approach guarantees exact doubly stochasticity by construction and can be implemented using only native matrix operations. Extensive experiments demonstrate that mHC-lite matches or exceeds mHC in performance while achieving higher training throughput with a naive implementation and eliminating the residual instabilities observed in both HC and mHC. The code is publicly available at https://github.com/FFTYYY/mhc-lite.

Linear attention and state-space models offer constant-memory alternatives to softmax attention, but often struggle with in-context associative recall. The Delta Rule mitigates this by writing each token via one step of online gradient descent. However, its step size relies on a single scalar gate that ignores the feature-wise curvature of the inner objective. We propose Online Scaled DeltaNet (OSDN), which augments the scalar gate with a diagonal preconditioner updated online via hypergradient feedback. Crucially, this right-preconditioning is algebraically equivalent to a per-feature scaling of the write-side key. This equivalence allows OSDN to strictly preserve the hardware-friendly chunkwise parallel pipeline of DeltaNet without incurring high-dimensional state overhead. Theoretically, by exploiting the exact-quadratic structure of the inner regression loss, we establish super-geometric convergence against a right-Newton comparator and prove an algorithm-aligned token-local residual contraction bound. To handle non-stationary contexts, we further introduce Adaptive Preconditioner Forgetting (APF) to dynamically refresh stale calibration. Empirically, OSDN demonstrates strong performance across scales. At the 340M-parameter scale, OSDN improves JRT-style in-context recall by 32% over DeltaNet. Scaling to 1.3B parameters, it achieves a 39% reduction in the recall residual ratio while maintaining parity on general downstream tasks (e.g., perplexity and LongBench) -- demonstrating that our online-preconditioning mechanism effectively transfers and amplifies at the billion-parameter scale.

Evaluating the symbolic reasoning of large language models (LLMs) calls for geometry benchmarks that require multi-step proofs grounded in both text and diagrams. However, existing benchmarks are often limited in scale and rarely provide visually grounded multiple-choice questions, limiting reliable evaluation of complex reasoning. We introduce GeoChallenge, a dataset of 90K automatically generated multiple-choice geometry proof problems, each requiring multi-step reasoning over aligned textual descriptions and diagrams. GeoChallenge provides fine-grained complexity ratings and formal language annotations to enable controlled evaluation. Experiments on multiple advanced LLMs show a clear performance gap between models and humans (the best-performing model, GPT-5-nano, achieves 75.89 exact match vs. 94.74 for humans). Further analysis also reveals three common failure patterns of LLMs: (1) exact match failures under the multiple-choice setting; (2) weak visual reliance; and (3) overextended reasoning without convergence.

Computational sensing strategies often suffer from calibration errors in the physical implementation of their ideal sensing models. Such uncertainties are typically addressed by using multiple, accurately chosen training signals to recover the missing information on the sensing model, an approach that can be resource-consuming and cumbersome. Conversely, blind calibration does not employ any training signal, but corresponds to a bilinear inverse problem whose algorithmic solution is an open issue. We here address blind calibration as a non-convex problem for linear random sensing models, in which we aim to recover an unknown signal from its projections on sub-Gaussian random vectors, each subject to an unknown positive multiplicative factor (or gain). To solve this optimisation problem we resort to projected gradient descent starting from a suitable, carefully chosen initialisation point. An analysis of this algorithm allows us to show that it converges to the exact solution provided a sample complexity requirement is met, i.e., relating convergence to the amount of information collected during the sensing process. Interestingly, we show that this requirement grows linearly (up to log factors) in the number of unknowns of the problem. This sample complexity is found both in absence of prior information, as well as when subspace priors are available for both the signal and gains, allowing a further reduction of the number of observations required for our recovery guarantees to hold. Moreover, in the presence of noise we show how our descent algorithm yields a solution whose accuracy degrades gracefully with the amount of noise affecting the measurements. Finally, we present some numerical experiments in an imaging context, where our algorithm allows for a simple solution to blind calibration of the gains in a sensor array.

Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration iteration, but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluates the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.

We study the parity-perturbed Hofstadter Q-recursion $ widetilde Q(1)=widetilde Q(2)=1,qquad widetilde Q(n)=widetilde Q(n-widetilde Q(n-1)) +widetilde Q(n-widetilde Q(n-2))+(-1)^n, and the associated two-variable Dirichlet series Z_{widetilde Q}(s,t)=sum_{nge1}n^{-s}widetilde Q(n)^{-t}. The estimate \widetilde Q(n)=n/2+O(n/\log n) gives the exact domain of absolute convergence Re(s+t)>1. With w=s+t, we separate the universal term 2^tζ(w) and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns t=-1. For E(n)=2\widetilde Q(n)-n and A(X)=\sum_{n\le X}E(n), the binary-arch clock yields A(X)=Xlog_2X+XΩ!left(log_23X{32}right) +O!left(X{log X}right), where Ω is an explicit continuous periodic function. This continues the normalized correction to Rew>0 and yields a boundary resonance lattice: a double resonance at w=0 and simple resonances at 2πi m/\log2. After subtracting the full-slice order-X skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal n/\log n pointwise scale with an explicit signed constant. The negative-arch mass satisfies A_r=512{92π}16^r{sqrt r} left(1-13{16r}+O(r^{-2})right). We do not claim a full-slice continuation across Rew=0$.

Bilevel optimization recently has received tremendous attention due to its great success in solving important machine learning problems like meta learning, reinforcement learning, and hyperparameter optimization. Extending single-agent training on bilevel problems to the decentralized setting is a natural generalization, and there has been a flurry of work studying decentralized bilevel optimization algorithms. However, it remains unknown how to design the distributed algorithm with sample complexity and convergence rate comparable to SGD for stochastic optimization, and at the same time without directly computing the exact Hessian or Jacobian matrices. In this paper we propose such an algorithm. More specifically, we propose a novel decentralized stochastic bilevel optimization (DSBO) algorithm that only requires first order stochastic oracle, Hessian-vector product and Jacobian-vector product oracle. The sample complexity of our algorithm matches the currently best known results for DSBO, and the advantage of our algorithm is that it does not require estimating the full Hessian and Jacobian matrices, thereby having improved per-iteration complexity.

A variety of widely used optimization methods like SignSGD and Muon can be interpreted as instances of steepest descent under different norm-induced geometries. In this work, we study the implicit bias of mini-batch stochastic steepest descent in multi-class classification, characterizing how batch size, momentum, and variance reduction shape the limiting max-margin behavior and convergence rates under general entry-wise and Schatten-p norms. We show that, without momentum, worst-case convergence and successful classification can only be guaranteed with full-batch gradient. In contrast, momentum enables small-batch convergence to an approximate max-margin solution through a batch-momentum trade-off, though it slows convergence. This approach provides fully explicit, dimension-free rates that improve upon prior results. Moreover, we prove that variance reduction can recover the exact full-batch implicit bias for any batch size, albeit at a slower convergence rate. Finally, we further investigate the batch-size-one steepest descent without momentum, and reveal its convergence to a fundamentally different bias via a concrete data example, which reveals a key limitation of purely stochastic updates. Overall, our unified analysis clarifies when stochastic optimization aligns with full-batch behavior, and paves the way for perform deeper explorations of the training behavior of stochastic gradient steepest descent algorithms.

A selector that allocates work across opaque components must evaluate them without being told how they performed. We study the least communication that suffices. Each parent maintains an allocation vector over its children and updates it from outcomes by proportional redistribution. Each child reads the sign of its own allocation change, one bit per round it is selected, so no evaluation message crosses the parent-child boundary. (1) The update preserves the simplex with strict positivity. (2) Conditional on being selected, a child's sign is exactly the root outcome at every depth, a sufficient statistic. The conditioning is necessary. (3) Under interiority a single selector has a unique interior equilibrium, exact and closed-form for N=2, with almost sure convergence under decreasing steps. For general N an equi-ratio condition gives an explicit affine equilibrium. (4) The linearised mean flow has real, strictly negative tangent eigenvalues with explicit bounds for all N >= 2. (5) The mean flow converges globally. An explicit potential, up to normalisation the KL divergence between the failure and slack distributions, decreases at a rate equal to the variance of the equi-ratio statistic. Without interiority the rule drops exactly the children below the surviving support's threshold. (6) The update law composes along a single active path, since ancestors select a node's active rounds but do not alter a realised signal. No joint convergence theorem for a hierarchy whose levels adapt simultaneously is claimed. (7) The one-bit interface admits exactly four deterministic memoryless distortions, and at a single selector passing the bit on unchanged delivers strictly higher mean quality upward than negation or constant transmission. Illustrations on synthetic hierarchies up to 16,384 leaves and three real datasets (3.4 billion comparisons, no mismatches) are consistent with the theory.

Robust Markov Decision Processes (MDPs) and risk-sensitive MDPs are both powerful tools for making decisions in the presence of uncertainties. Previous efforts have aimed to establish their connections, revealing equivalences in specific formulations. This paper introduces a new formulation for risk-sensitive MDPs, which assesses risk in a slightly different manner compared to the classical Markov risk measure (Ruszczy\'nski 2010), and establishes its equivalence with a class of regularized robust MDP (RMDP) problems, including the standard RMDP as a special case. Leveraging this equivalence, we further derive the policy gradient theorem for both problems, proving gradient domination and global convergence of the exact policy gradient method under the tabular setting with direct parameterization. This forms a sharp contrast to the Markov risk measure, known to be potentially non-gradient-dominant (Huang et al. 2021). We also propose a sample-based offline learning algorithm, namely the robust fitted-Z iteration (RFZI), for a specific regularized RMDP problem with a KL-divergence regularization term (or equivalently the risk-sensitive MDP with an entropy risk measure). We showcase its streamlined design and less stringent assumptions due to the equivalence and analyze its sample complexity.

We study a capability the dominant paradigm in synthetic tabular data does not provide: exact satisfaction of a declared analytical outcome with no source data. Imitation methods (copulas, GANs, diffusion) learn a real distribution and sample from it, and are judged on fidelity to real data. A large, practical class of needs is different: generating data with no source data ("cold start") that reproduces a declared outcome (a revenue curve, a churn rate, a group share) across a relational schema. Off-the-shelf imitation tools offer no interface for such targets, and no sampler can hit an exact aggregate, because sampling has variance. On a real public dataset, off-the-shelf learned synthesizers trained on that very data miss the declared monthly aggregate by 74 to 86 percent; a per-period steelman cuts the miss to about 19 percent and still cannot reach 0; a closed-form generator reaches exactly 0. We name this task outcome-conformant synthesis, argue its evaluation axis is conformance rather than fidelity, and show the two axes are orthogonal. We contribute: (1) a formal account showing a widely-used family of exact-aggregate generators is exactly conditional-sum sampling of a Gamma population (via Lukacs' characterization), with closed-form exactness, a closed-form marginal CV, and scale-invariance; a controlled experiment maps the boundary, enforcing the exact aggregate costs at most 0.006 in 1-Wasserstein distance to an arbitrary external marginal, the rest being shape-family mismatch; (2) SpecBench, to our knowledge the first benchmark to measure conformance to analytical outcomes for cold-start relational synthesis; and (3) a closed-form, deterministic reference system. Exact aggregation alone is trivial; the contribution is conformance jointly with closed-form marginals, integrity, determinism, and zero source data. We concede fidelity to imitation where real data exists.

Language models are accurate on most long-arithmetic problems, but their accuracy is estimated on samples and carries no guarantee for an unseen input; tool use is exact but moves the computation outside the network. We show that exactness at all input lengths becomes a finite, checkable property of a network when each of its steps reads only a bounded view of an external working memory. In Sketchpad, Qwen3.5-0.8B with a low-rank adapter executes arithmetic procedures on a rendered sheet. Each forward pass reads a three-cell view and emits one step: a digit, a carry or borrow, and the next operation. A fixed runtime moves the view and writes no digit of a result. We prove in Lean that the model is then exact on all inputs of any length if and only if it is exact on the finite set of views the runtime presents, that no finite test suffices when steps also read earlier steps, and that the guarantee tolerates bounded numerical perturbation and extends to compositions of procedures. All 2,446 views pass on the deployment device, a laptop GPU running the model in a web browser. The certified model performs signed addition and subtraction, long multiplication and division, square roots and greatest common divisors at arbitrary length. It solved 200 of 200 pre-registered problems in the browser, and 220 of 220 sums of up to 1,000 digits and products of up to 20 digits, of which GPT-6.1 Sol, the strongest of ten other models, solved 217. A symbolic runtime built on these procedures solves, expands and factors polynomials with every number computed by the model, and a multiplicative gate on the adapter recovers the base model exactly when the sheet is not in use.

We disprove a conjecture of Davis, Drusvyatskiy, and Jiang (2026) that subdifferential regularity can be removed from the generic guarantee of convergence to local minimizers for stochastic subgradient methods. We construct a globally Lipschitz, coercive, semialgebraic function in two dimensions for which the method converges to a nonminimizing Clarke critical point. For suitable power-law stepsizes, this convergence holds for open sets of initial points and linear tilts of the objective, uniformly over all perturbation sequences within a prescribed bound.

In this paper we study holomorphic properties of infinite dimensional spin factors. Among the infinite dimensional Banach spaces with homogeneous open unit balls, we show that the spin factors are natural outlier spaces in which to ask the question (as was proved in the early 1970s for Hilbert spaces): Do biholomorphic automorphisms g of the open unit ball B have fixed points in overline B? In this paper, for infinite dimensional spin factors, we provide reasonable conditions on g that allow us to explicitly construct fixed points of g lying on partial B. En route, we also prove that every spin factor has the density property. In another direction, we focus on (compact) holomorphic maps f:Brightarrow B, having no fixed point in B and examine the sequence of iterates (f^n). As (f^n) does not generally converge, we instead trace the target set T(f) of f, that is, the images of all accumulation points of (f^n)_n, for any topology finer than the topology of pointwise convergence on B. We prove for a spin factor that T(f) lies on the boundary of a single bidisc unique to f.

We investigate safe multi-agent reinforcement learning, where agents seek to collectively maximize an aggregate sum of local objectives while satisfying their own safety constraints. The objective and constraints are described by {\it general utilities}, i.e., nonlinear functions of the long-term state-action occupancy measure, which encompass broader decision-making goals such as risk, exploration, or imitations. The exponential growth of the state-action space size with the number of agents presents challenges for global observability, further exacerbated by the global coupling arising from agents' safety constraints. To tackle this issue, we propose a primal-dual method utilizing shadow reward and κ-hop neighbor truncation under a form of correlation decay property, where κ is the communication radius. In the exact setting, our algorithm converges to a first-order stationary point (FOSP) at the rate of Oleft(T^{-2/3}right). In the sample-based setting, we demonstrate that, with high probability, our algorithm requires mathcal{O}left(ε^{-3.5}right) samples to achieve an ε-FOSP with an approximation error of O(φ_0^{2κ}), where φ_0in (0,1). Finally, we demonstrate the effectiveness of our model through extensive numerical experiments.

There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. Examples include AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not otherwise compute or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in R^d with dge 2, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error Θ(dln(n/d)/n), and in the agnostic setting the deviation scales as τln(1/τ) at true error τ. In contrast, homogeneous halfspaces in R^2 exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error O(1/n). In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a lnln n overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.

Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions (that is, series whose terms are described and ordered in some way but which do not converge apriori) and, secondly, to study the convergence or summability of these formal solutions (the existence and uniqueness of actual solutions with the given asymptotic expansion in a certain domain). In this paper we deal only with the convergence of formal functional series having the form of an infinite sum of power functions with (complex, in general) power exponents and satisfying analytical functional equations of the following three types: a differential, q-difference or Mahler equation.

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

We give a computer-assisted counterexample to the open question, posed by Rudin, Schapire, and Daubechies in COLT 2012, of whether exhaustive AdaBoost always converges to a finite cycle. The construction is based on a block-product gadget whose two factors share an exact period-2 orbit for their 5-step branch maps, but whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio. This irrationality forces the burst-winner sequence to have an irrational asymptotic frequency, precluding eventual periodicity. All assertions are certified by exact rational arithmetic. This work was developed in collaboration with GPT-5.4 Pro and Claude Opus 4.6.

Optimization analyses for cross-entropy training rely on local Taylor models of the loss to predict whether a proposed step will decrease the objective. These surrogates are reliable only inside the Taylor convergence radius of the true loss along the update direction. That radius is set not by real-line curvature alone but by the nearest complex singularity. For cross-entropy, the softmax partition function F=sum_j exp(z_j) has complex zeros -- ``ghosts of softmax'' -- that induce logarithmic singularities in the loss and cap this radius. To make this geometry usable, we derive closed-form expressions under logit linearization along the proposed update direction. In the binary case, the exact radius is ρ^*=δ^2+ π^2/Δ_a. In the multiclass case, we obtain the lower bound ρ_a=π/Δ_a, where Δ_a=max_k a_k-min_k a_k is the spread of directional logit derivatives a_k=nabla z_kcdot v. This bound costs one Jacobian-vector product and reveals what makes a step fragile: samples that are both near a decision flip and highly sensitive to the proposed direction tighten the radius. The normalized step size r=τ/ρ_a separates safe from dangerous updates. Across six tested architectures and multiple step directions, no model fails for r<1, yet collapse appears once rge 1. Temperature scaling confirms the mechanism: normalizing by ρ_a shrinks the onset-threshold spread from standard deviation 0.992 to 0.164. A controller that enforces τleρ_a survives learning-rate spikes up to 10{,} 000times in our tests, where gradient clipping still collapses. Together, these results identify a geometric constraint on cross-entropy optimization that operates through Taylor convergence rather than Hessian curvature.

For a family of two-matrix models \[ 1{2} Tr(A^2+B^2) - g{4} Tr(A^4+B^4) - cases h{2} Tr( A BA B) \\ h{4} Tr( A BA B+ ABBA ) \\ h{2} Tr( A B BA ) cases \] with hermitian A and B, we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the (h,g)-plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.

Motivated by the open question of Bianchi, Hachem, and Schechtman (Bianchi et al., 2024, Remark 4), we show that in the non-weakly convex definable setting, omnidirectional noise with conditional fourth-moment control does not suffice for universal almost-sure avoidance of sharply repulsive critical points. We demonstrate this failure by constructing a globally Lipschitz, coercive, definable objective that is not weakly convex. For this objective and its sufficiently small linear perturbations, stochastic subgradient descent with independent noise uniformly distributed on a ball in R^2 converges with probability 1 to a nonminimal sharply repulsive critical point from every initial point in a specified ball. A two-cycle argument establishes uniform bounds on the rescaled iterates, yielding convergence to the sharply repulsive critical point. In contrast, for locally Lipschitz definable objectives, Gaussian noise ensures simultaneous almost-sure avoidance of any finite collection of sharply repulsive points. Through a construction of difference quotient functions, we show that convergence to any such point with positive probability would produce a stationary distribution with expected descent 0, whereas a global positive lower bound on the Fréchet subgradient norms of the limiting functions forces strictly positive expected descent under the same distribution. This contradiction proves avoidance, with the everywhere positive Gaussian density playing a key role.

Sharp bounds on two potential-outcome means need not combine into sharp bounds on their difference. We characterize exactly when composition is guaranteed across environments sharing a joint potential-outcome law. For binary treatment and outcome, fix strict selection-probability bounds and interior treatment propensities. Subtracting the full model's sharp mean intervals gives sharp average-treatment-effect bounds for every compatible observed law if and only if the absolute selection bounds coincide across environments. Under this alignment, the complete mean region is rectangular; its endpoints and attaining causal models require O(K) arithmetic operations for K environments. For every mismatch, failure occurs on a relatively open set of positive observed tables at the prescribed propensities and persists under small joint drift. We give computable perturbation certificates and an exact family whose gap is first order in the mismatch. A separate finite-confounding family shows when the lost joint information changes an identified causal sign. These results distinguish local marginal sharpness from joint attainability after combining environments.

In this paper we provide a new method to certify that a nearby polynomial system has a singular isolated root with a prescribed multiplicity structure. More precisely, given a polynomial system f =(f_1, ldots, f_N)in C[x_1, ldots, x_n]^N, we present a Newton iteration on an extended deflated system that locally converges, under regularity conditions, to a small deformation of f such that this deformed system has an exact singular root. The iteration simultaneously converges to the coordinates of the singular root and the coefficients of the so called inverse system that describes the multiplicity structure at the root. We use $alpha$-theory test to certify the quadratic convergence, and togive bounds on the size of the deformation and on the approximation error. The approach relies on an analysis of the punctual Hilbert scheme, for which we provide a new description. We show in particular that some of its strata can be rationally parametrized and exploit these parametrizations in the certification. We show in numerical experimentation how the approximate inverse system can be computed as a starting point of the Newton iterations and the fast numerical convergence to the singular root with its multiplicity structure, certified by our criteria.

The ability of an architecture to realize permutations is quite fundamental. For example, Large Language Models need to be able to correctly copy (and perhaps rearrange) parts of the input prompt into the output. Classical universal approximation theorems guarantee the existence of parameter configurations that solve this task but offer no insights into whether gradient-based algorithms can find them. In this paper, we address this gap by focusing on two-layer fully connected feed-forward neural networks and the task of learning permutations on nonzero binary inputs. We show that in the infinite width Neural Tangent Kernel (NTK) regime, an ensemble of such networks independently trained with gradient descent on only the k standard basis vectors out of 2^k - 1 possible inputs successfully learns any fixed permutation of length k with arbitrarily high probability. By analyzing the exact training dynamics, we prove that the network's output converges to a Gaussian process whose mean captures the ground truth permutation via sign-based features. We then demonstrate how averaging these runs (an "ensemble" method) and applying a simple rounding step yields an arbitrarily accurate prediction on any possible input unseen during training. Notably, the number of models needed to achieve exact learning with high probability (which we refer to as ensemble complexity) exhibits a linearithmic dependence on the input size k for a single test input and a quadratic dependence when considering all test inputs simultaneously.

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

Statistical inference for non-stationary data is hindered by the failure of classical central limit theorems (CLTs), not least because there is no fixed Gaussian limit to converge to. To resolve this, we introduce relative weak convergence, an extension of weak convergence that compares a statistic or process to a sequence of evolving processes. Relative weak convergence retains the essential consequences of classical weak convergence and coincides with it under stationarity. Crucially, it applies in general non-stationary settings where classical weak convergence fails. We establish concrete relative CLTs for random vectors and empirical processes, along with sequential, weighted, and bootstrap variants, that parallel the state-of-the-art in stationary settings. Our framework and results offer simple, plug-in replacements for classical CLTs whenever stationarity is untenable, as illustrated by applications in nonparametric trend estimation and hypothesis testing.

Developing general-purpose algorithms with global asymptotic convergence guarantees for variational inequalities (VIs) remains a longstanding challenge. This paper proposes a path-following framework for finite-dimensional VIs with arbitrary continuous mappings and polytope domains. The approach first transforms a general VI into a smooth VI on a simplex. Its key innovation is to formulate the resulting problem on a fiber bundle, called the fixed-point bundle, whose geometric structure enables the systematic integration of initialization, path-following, and singularity avoidance. Without imposing monotonicity or similar assumptions, the framework guarantees global linear convergence for nonsingular solutions and global linear convergence up to any prescribed precision for singular solutions. In numerical experiments, the algorithm attained a precision of 10^{-5} on all 6400 randomly generated instances, with dimensions up to 1024. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.

Deep Learning has revolutionized vision via convolutional neural networks (CNNs) and natural language processing via recurrent neural networks (RNNs). However, success stories of Deep Learning with standard feed-forward neural networks (FNNs) are rare. FNNs that perform well are typically shallow and, therefore cannot exploit many levels of abstract representations. We introduce self-normalizing neural networks (SNNs) to enable high-level abstract representations. While batch normalization requires explicit normalization, neuron activations of SNNs automatically converge towards zero mean and unit variance. The activation function of SNNs are "scaled exponential linear units" (SELUs), which induce self-normalizing properties. Using the Banach fixed-point theorem, we prove that activations close to zero mean and unit variance that are propagated through many network layers will converge towards zero mean and unit variance -- even under the presence of noise and perturbations. This convergence property of SNNs allows to (1) train deep networks with many layers, (2) employ strong regularization, and (3) to make learning highly robust. Furthermore, for activations not close to unit variance, we prove an upper and lower bound on the variance, thus, vanishing and exploding gradients are impossible. We compared SNNs on (a) 121 tasks from the UCI machine learning repository, on (b) drug discovery benchmarks, and on (c) astronomy tasks with standard FNNs and other machine learning methods such as random forests and support vector machines. SNNs significantly outperformed all competing FNN methods at 121 UCI tasks, outperformed all competing methods at the Tox21 dataset, and set a new record at an astronomy data set. The winning SNN architectures are often very deep. Implementations are available at: github.com/bioinf-jku/SNNs.

Deep learning has aroused extensive attention due to its great empirical success. The efficiency of the block coordinate descent (BCD) methods has been recently demonstrated in deep neural network (DNN) training. However, theoretical studies on their convergence properties are limited due to the highly nonconvex nature of DNN training. In this paper, we aim at providing a general methodology for provable convergence guarantees for this type of methods. In particular, for most of the commonly used DNN training models involving both two- and three-splitting schemes, we establish the global convergence to a critical point at a rate of {cal O}(1/k), where k is the number of iterations. The results extend to general loss functions which have Lipschitz continuous gradients and deep residual networks (ResNets). Our key development adds several new elements to the Kurdyka-{\L}ojasiewicz inequality framework that enables us to carry out the global convergence analysis of BCD in the general scenario of deep learning.

The ground-state energy, electron density, and related properties of ordinary matter can be computed efficiently when the exchange-correlation energy as a functional of the density is approximated semilocally. We propose the first meta-GGA (meta-generalized gradient approximation) that is fully constrained, obeying all 17 known exact constraints that a meta-GGA can. It is also exact or nearly exact for a set of appropriate norms, including rare-gas atoms and nonbonded interactions. This SCAN (strongly constrained and appropriately normed) meta-GGA achieves remarkable accuracy for systems where the exact exchange-correlation hole is localized near its electron, and especially for lattice constants and weak interactions.

Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis. By framing neural networks as operators with fixed points representing desired solutions, we develop a theoretical framework grounded in iterative methods for operator equations. Under defined conditions, we present convergence proofs based on fixed point theory. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Empirical assessments highlight that performing iterations through network operators improves performance. We also introduce an iterative graph neural network, PIGN, that further demonstrates benefits of iterations. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis, potentially guiding the design of future networks with clearer theoretical underpinnings and improved performance.

Large language model agents solve tasks by generating trajectories that interleave planning, tool calls, and intermediate results. Current evaluation metrics reduce such a trajectory to a binary success flag, compare it against a reference by exact matching, or delegate judgment to another language model. A success flag cannot distinguish a sound solution from one that succeeds by luck, and says nothing about why a failed run went wrong. Exact matching penalizes plans that are valid but reordered or decomposed differently from the reference. We reframe trajectory evaluation as a distance between the agent's execution graph and a set of valid solution graphs, and instantiate it via an unbalanced fused Gromov-Wasserstein transport problem over attributed dependency graphs. The resulting score, termed OTAP (Optimal Transport for Agentic Planning), is a pseudo-metric that is provably invariant to dependency-preserving reorderings and has bounded sensitivity to redundant steps. Its unbalanced marginals handle missing or hallucinated steps without forcing a match, and its soft coupling accommodates variation in plan granularity. On controlled perturbations and three public benchmarks, OTAP separates valid from invalid trajectories in a regime where semantics-only metrics score below chance. Its advantage tracks the fidelity of the dependency graph: largest where edges follow from operator semantics, smallest where they are inferred from free text. Where a formal verifier exists, strict surface metrics predict validity better than OTAP does, which places OTAP in open-ended domains where no verifier is available.

Bilevel optimization has been recently revisited for designing and analyzing algorithms in hyperparameter tuning and meta learning tasks. However, due to its nested structure, evaluating exact gradients for high-dimensional problems is computationally challenging. One heuristic to circumvent this difficulty is to use the approximate gradient given by performing truncated back-propagation through the iterative optimization procedure that solves the lower-level problem. Although promising empirical performance has been reported, its theoretical properties are still unclear. In this paper, we analyze the properties of this family of approximate gradients and establish sufficient conditions for convergence. We validate this on several hyperparameter tuning and meta learning tasks. We find that optimization with the approximate gradient computed using few-step back-propagation often performs comparably to optimization with the exact gradient, while requiring far less memory and half the computation time.

In an unbounded plane, straight lines are used extensively for mathematical analysis. They are tools of convenience. However, those with high slope values become unbounded at a faster rate than the independent variable. So, straight lines, in this work, are made to be bounded by introducing a parametric nonlinear term that is positive. The straight lines are transformed into bounded nonlinear curves that become unbounded at a much slower rate than the independent variable. This transforming equation can be expressed as a continued fraction of straight lines. The continued fraction is real-valued and converges to the solutions of the transforming equation. Following Euler's method, the continued fraction has been reduced into an infinite series. The usefulness of the bounding nature of continued fraction is demonstrated by solving the problem of image classification. Parameters estimated on the Fashion-MNIST dataset of greyscale images using continued fraction of regression lines have less variance, converge quickly and are more accurate than the linear counterpart. Moreover, this multi-dimensional parametric estimation problem can be expressed on xy- plane using the parameters of the continued fraction and patterns emerge on planar plots.

A common statistical task lies in showing asymptotic normality of certain statistics. In many of these situations, classical textbook results on weak convergence theory suffice for the problem at hand. However, there are quite some scenarios where stronger results are needed in order to establish an asymptotic normal approximation uniformly over a family of probability measures. In this note we collect some results in this direction. We restrict ourselves to weak convergence in mathbb R^d with continuous limit measures.

Consider a random uniform sample of n points in a compact region A of Euclidean d-space, d geq 2, with a smooth or (when d=2) polygonal boundary. Fix k bf N. Let T_{n,k} be the threshold r at which the geometric graph on these n vertices with distance parameter r becomes k-connected. We show that if d=2 then n (pi/|A|) T_{n,1}^2 - log n is asymptotically standard Gumbel. For (d,k) neq (2,1), it is n (theta_d/|A|) T_{n,k}^d - (2-2/d) log n - (4-2k-2/d) log log n that converges in distribution to a nondegenerate limit, where theta_d is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when (d,k)=(2,2) where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest k-nearest neighbour link U_{n,k} in the sample, and show T_{n,k}=U_{n,k} with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in A. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.

We give an improved theoretical analysis of score-based generative modeling. Under a score estimate with small L^2 error (averaged across timesteps), we provide efficient convergence guarantees for any data distribution with second-order moment, by either employing early stopping or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in reverse KL divergence in epsilon-accuracy can be done in tilde Oleft(d log (1/delta){epsilon}right) steps: 1) the variance-delta Gaussian perturbation of any data distribution; 2) data distributions with 1/delta-smooth score functions. Our analysis also provides a quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.

The full shadow of a complex polynomial P of degree dge2 is the closure of all zeros of (P^n)^{(m)}, nge1, 0le m<dn. We prove that it consists of a compact connected limiting core containing all roots and critical points of P, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as ntoinfty and the support of their limiting normalized root-counting measure. For m/ntoαin(0,d), we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.

For a real quadratic field Q(d), we study the norm-form energy N = S_ζ^2 - d cdot S_L^2, where S_ζ and S_L are Lorentzian-weighted zero sums with w(ρ) = 2/(1/4 + γ^2). We prove three main results. (1) Spacelike spectral data: N < 0 unconditionally for all squarefree d > 1, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting. (2) Effective density bound: at each verified truncation level M, dens{N > 0} leq 2|f_{S_L^{(M)}}|_infty cdot (W_1(ζ)/d + ε_M), established unconditionally via Jacobi--Anger resonance analysis. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice Λ_infty has finite rank (verified for M leq 20, where rank = 0), the sharp asymptotic dens{N > 0} = C(d)/d + o(1/d) holds. For d = 5, C(5) = 2,f_{S_L}(0)cdotE[|S_ζ|] = 0.1191; the constant depends on d through the zeros of L(s,χ_d), and C(d) = O(1/log d) as d to infty.

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors x_i, i=1,dots,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

A declared task can sometimes certify a reduced source before a downstream encoder, codebook, rate, distortion target, or optimizer is chosen. This paper studies when one such reduction preserves the complete downstream problem family, a property termed Task-Sufficient Contraction. The reduced source is fixed by the task before the later operating point is selected. An exact contraction allows the later problem to be solved on that source with the same result as if the full source had been retained. For a machine with a fixed set of possible actions and a fixed loss, the paper identifies a consumer-specific source by merging states only when every available action has the same regret in both. For finite action sets, replacing the richer source by this reduced source preserves the complete one-step rate-regret curve, even though the reduction is fixed before the distortion target is chosen. A second result gives an exact characterization for quadratic loss on affine feasible-action sets: the canonical reduced source is the projection onto the directions in which feasible actions can differ. Under a fixed energy budget, this becomes centered load, while retaining only the optimal water-filled action is too coarse. Earlier Information Bottleneck, semantic rate-distortion, and goal-oriented quantization results are then used to distinguish exact, architecture-conditioned, approximate, failed, and corrected contractions. The framework suggests a way for heterogeneous machines to exchange what a receiving task needs without first aligning their full internal representations.

In location estimation, we are given n samples from a known distribution f shifted by an unknown translation lambda, and want to estimate lambda as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cram\'er-Rao bound of error mathcal N(0, 1{nmathcal I}), where mathcal I is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of mathcal I_r, the Fisher information of the r-smoothed distribution. As n to infty, r to 0 at an explicit rate and this converges to the Cram\'er-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.

This paper reports a step that had already been taken. Marcus's three components of an adequate cognitive architecture -- variables, recursive structure, individuals vs kinds -- left the neural substrate open; a companion paper answers it with VaCoAl, built on XOR-and-shift over GF(2). Those pillars are horizontal: how a mind represents the world. Minsky's Emotion Machine supplies the orthogonal, vertical dimension -- a mind reasoning about its own reasoning. The vertical step needed no new machinery. A representation exactly recoverable becomes introspection when what is recovered is the reasoner's own deliberative trace: Minsky's Reflective layer. Pillar 2 says nothing about whose structure is recovered; point the same unbind inward and Reflection follows. Only the argument changes. Capability: exact reversible binding at O(N), kept exact within a Frontier Size, yields a trace readable constituent by constituent: a self-report faithful by construction, not verbalization. Only exact-algebraic traces, not probabilistic ones, make the reflexive read faithful. Necessity: audit and valuation are one circuit at opposite settings of one parameter. Repair every collision and the trace is exactly auditable but no path outranks another; tolerate them and the CR2 decay that credits the direct route makes the read approximate. Position: we claim neither to have built the Reflective layer nor to surpass large language models; it supplies the auditable trace probabilistic deliberation cannot leave. The same substrate gives panalogy (content-addressable retrieval scored by CR) and credit assignment as counterfactual unbind-rebind, meeting Pearl's causal axis. Not claimed: Reflection's sufficient conditions (meta-control loop, context-tagging) are future work; Reflection is introspection, not evaluation; its reachability is argued, not shown; speed and power unbenchmarked.

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families {S_{i}}_{i in I} of contractions on a Hilbert space H, to commuting families {T_{i}}_{i in I} of contractive C_{0}-semigroups on L^{2}(prod_{i in I}T) otimes H. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for d in N with d geq 3 the existence of commuting families {T_{i}}_{i=1}^{d} of contractive C_{0}-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of R_{geq 0}^{d} of non-unitarily dilatable and non-unitarily approximable d-parameter contractive C_{0}-semigroups on separable infinite-dimensional Hilbert spaces for each d geq 3. Similar results are also developed for d-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, viz. that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of C_{0}-semigroups, which extends results of Eisner (2009--10).

The Hilfer fractional derivative generalizes and interpolates between the commonly used Riemann-Liouville and Caputo fractional derivative. In general, solutions to Hilfer fractional derivative initial value problems are singular for t downarrow t_0, providing a challenge in numerical approximation. In this work, we present a Bernstein splines method to obtain accurate approximations to solutions of nonlinear Hilfer fractional derivative initial value problems, including solutions with singular behavior. We provide explicit convergence requirements and asymptotic convergence rates to analytical solutions. Moreover, the method is efficiently implemented, using a vectorized approach and parallelization in polynomial order. Numerical experiments show empirical convergence corresponding to analytical results and a significantly higher accuracy compared to the commonly used fractional Adams-Bashforth-Moulton predictor-corrector method. Furthermore, our method is able to simulate nonlinear systems with Hilfer fractional derivatives, as demonstrated by the application to the fractional Van der Pol oscillator.

In an asynchronous federated learning framework, the server updates the global model once it receives an update from a client instead of waiting for all the updates to arrive as in the synchronous setting. This allows heterogeneous devices with varied computing power to train the local models without pausing, thereby speeding up the training process. However, it introduces the stale model problem, where the newly arrived update was calculated based on a set of stale weights that are older than the current global model, which may hurt the convergence of the model. In this paper, we present an asynchronous federated learning framework with a proposed adaptive weight aggregation algorithm, referred to as AsyncFedED. To the best of our knowledge this aggregation method is the first to take the staleness of the arrived gradients, measured by the Euclidean distance between the stale model and the current global model, and the number of local epochs that have been performed, into account. Assuming general non-convex loss functions, we prove the convergence of the proposed method theoretically. Numerical results validate the effectiveness of the proposed AsyncFedED in terms of the convergence rate and model accuracy compared to the existing methods for three considered tasks.

First-exit prediction for momentum SGD depends on sampling information that pointwise gradient laws omit. Strongly convex finite pools with identical pointwise laws and matched iid momentum dynamics can have different reshuffled exits; even analytic pools with uniformly close pointwise laws can have separated exit probabilities. For persistent sampling in one dimension, we establish an explicit O(h) mean-exit comparison with a finite-rate switching process under nonvanishing component velocities and step-size conditions; iid sampling instead averages to an ODE. Retaining realized minibatches and sampled curvature yields finite-window exit brackets beyond this regime. In 32 trained ridge-logistic runs, 29 meet prespecified targets with rounding-certified brackets; an enclosure control explains most of the width advantage. Exact-real certificates establish finite means at the original tested radii for three ResNet-18 checkpoints and give local finite-mean conditions for tanh and ViT networks; their upper bounds remain loose, and original-radius ViT mean guarantees remain unresolved despite 128 fresh continuation paths. A two-network replication gives 32/32 successful seeds in each of four finite-path persistence comparisons. Independent equal-length controls retain slower exits after refresh resets than without resets, but the refresh-versus-random ordering reverses with update length at both networks. These interventions do not causally decompose raw momentum SGD. Neural step-size scaling does not validate the switching limit.

In this paper, it is shown that if a sequence of resistance metric spaces equipped with measures converges with respect to the local Gromov-Hausdorff-vague topology, and certain non-explosion and metric-entropy conditions are satisfied, then the associated stochastic processes and their local times also converge. The metric-entropy condition can be checked by applying volume estimates of balls. Whilst similar results have been proved previously, the approach of this article is more widely applicable. Indeed, we recover various known conclusions for scaling limits of some deterministic self-similar fractal graphs, critical Galton-Watson trees, the critical Erdos-R\'enyi random graph and the configuration model (in the latter two cases, we prove for the first time the convergence of the models with respect to the resistance metric and also, for the configuration model, we overcome an error in the existing proof of local time convergence). Moreover, we derive new ones for scaling limits of uniform spanning trees and random recursive fractals. The metric-entropy condition also implies convergence of associated Gaussian processes.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

Harnessing parallelism in seemingly sequential models is a central challenge for modern machine learning. Several approaches have been proposed for evaluating sequential processes in parallel using iterative fixed-point methods, like Newton, Picard, and Jacobi iterations. In this work, we show that these methods can be understood within a common framework based on linear dynamical systems (LDSs), where different iteration schemes arise naturally as approximate linearizations of a nonlinear recursion. Moreover, we theoretically analyze the rates of convergence of these methods, and we verify the predictions of this theory with several case studies. This unifying framework highlights shared principles behind these techniques and clarifies when particular fixed-point methods are most likely to be effective. By bridging diverse algorithms through the language of LDSs, the framework provides a clearer theoretical foundation for parallelizing sequential models and points toward new opportunities for efficient and scalable computation.

The Platonic Representation Hypothesis (PRH) holds that as models scale, representations of heterogeneous networks converge toward a shared model of reality. We propose its sequel and boundary, the Capability Convergence Hypothesis (CCH): under a fixed per-token inference budget, representational convergence does not entail capability convergence. Capability instead converges toward a class, the access-complete hybrid: any architecture holding both a compressive O(1)-state channel and a scalable verbatim-index channel. We anchor it on a witness task, the Newton's-apple problem in an infinite stream, and name three resource walls: a Shannon wall barring any o(Nb)-state architecture, a horizon wall barring any fixed window, and a circuit wall barring fixed-depth attention-only composition (conditional on TC0 != NC1). Under an explicit separability assumption a hybrid crosses all three by paying each wall's price, so capability is strictly super-additive under composition. We separate what we prove from what we conjecture: the access-completeness principle rests on information-theoretic lower bounds and pre-registered experiments, while the field-level convergence trend is an economics-motivated conjecture. We report the first pre-registered small-scale tests under criteria frozen before the data: the predicted scissors gap is measured (exact-retrieval error 0.994 vs. 0.000 once a 64-scalar state gains one global-attention layer), the state-tracking bifurcation lands at the registered boundary, and a conjunction witness shows an irreducibly two-channel solution; one prediction failed with its direction reversed and is reported as such. Representational convergence is given freely by scale; capability convergence must be purchased by access structure.

We introduce a framework based on bilevel programming that unifies gradient-based hyperparameter optimization and meta-learning. We show that an approximate version of the bilevel problem can be solved by taking into explicit account the optimization dynamics for the inner objective. Depending on the specific setting, the outer variables take either the meaning of hyperparameters in a supervised learning problem or parameters of a meta-learner. We provide sufficient conditions under which solutions of the approximate problem converge to those of the exact problem. We instantiate our approach for meta-learning in the case of deep learning where representation layers are treated as hyperparameters shared across a set of training episodes. In experiments, we confirm our theoretical findings, present encouraging results for few-shot learning and contrast the bilevel approach against classical approaches for learning-to-learn.

Inference-time control steers a pretrained generative model towards a target distribution without retraining. We study tilted targets π_0propto G_0,p_0, where p_0 is the sampler output distribution and G_0 is an evaluable reweighting function. Existing approaches rely on sequential annealing with sequential Monte Carlo (SMC) or parallel annealing with replica exchange (RE). Sequential control is exact but needs large particle populations, whereas no exact parallel control method exists: existing RE corrections approximate an intractable time reversal and are biased. We propose Exact Non-equilibrium COntrol with Replica Exchange (ENCORE), the first exact parallel control method. Each replica stores its generation trajectory, so the upward move is a truncation and the intractable time reversal is never simulated. We prove target invariance and show that the resulting dynamics are those of non-equilibrium replica exchange with the exact time reversal as forward proposal. Under regularity conditions, our diffusion analysis shows that both sequential and parallel control become unstable under refinement of the time discretisation without guidance, whereas guided proposals remain stable and yield diagnostics for tuning the schedule and the computational budget. Across synthetic targets, Boltzmann sampling of biomolecules, and image generation, ENCORE achieves competitive accuracy and diversity, remains robust to sampler perturbations, and applies to distilled samplers where existing RE corrections are unavailable.

Non-cooperative game theory provides a robust framework for analyzing distributed resource allocation in multi-user wireless networks, with Iterative Water-Filling (IWF) emerging as a canonical solution for power control problems. Although classical fixed-point theorems guarantee the existence of a Nash Equilibrium (NE) under mild concavity and compactness conditions, the convergence of practical iterative algorithms to that equilibrium remains a challenging endeavor. This challenge intensifies under varying update schedules, interference regimes, and imperfections such as channel estimation errors or feedback delay. In this paper, we present an in-depth examination of IWF in multi-user systems under three different update schemes: (1) synchronous sequential updates, (2) synchronous simultaneous updates, and (3) totally asynchronous updates. We first formulate the water-filling operator in a multi-carrier environment, then recast the iterative process as a fixed-point problem. Using contraction mapping principles, we demonstrate sufficient conditions under which IWF converges to a unique NE and highlight how spectral radius constraints, diagonal dominance, and careful step-size selection are pivotal for guaranteeing convergence. We further discuss robustness to measurement noise, partial updates, and network scaling to emphasize the practical viability of these schemes. This comprehensive analysis unifies diverse threads in the literature while offering novel insights into asynchronous implementations. Our findings enable network designers to ascertain system parameters that foster both stable convergence and efficient spectrum usage.

Diffusion models are generative models that have recently demonstrated impressive performances in terms of sampling quality and density estimation in high dimensions. They rely on a forward continuous diffusion process and a backward continuous denoising process, which can be described by a time-dependent vector field and is used as a generative model. In the original formulation of the diffusion model, this vector field is assumed to be the score function (i.e. it is the gradient of the log-probability at a given time in the diffusion process). Curiously, on the practical side, most studies on diffusion models implement this vector field as a neural network function and do not constrain it be the gradient of some energy function (that is, most studies do not constrain the vector field to be conservative). Even though some studies investigated empirically whether such a constraint will lead to a performance gain, they lead to contradicting results and failed to provide analytical results. Here, we provide three analytical results regarding the extent of the modeling freedom of this vector field. {Firstly, we propose a novel decomposition of vector fields into a conservative component and an orthogonal component which satisfies a given (gauge) freedom. Secondly, from this orthogonal decomposition, we show that exact density estimation and exact sampling is achieved when the conservative component is exactly equals to the true score and therefore conservativity is neither necessary nor sufficient to obtain exact density estimation and exact sampling. Finally, we show that when it comes to inferring local information of the data manifold, constraining the vector field to be conservative is desirable.

Fast adversarial training (FAT) is beneficial for improving the adversarial robustness of neural networks. However, previous FAT work has encountered a significant issue known as catastrophic overfitting when dealing with large perturbation budgets, \ie the adversarial robustness of models declines to near zero during training. To address this, we analyze the training process of prior FAT work and observe that catastrophic overfitting is accompanied by the appearance of loss convergence outliers. Therefore, we argue a moderately smooth loss convergence process will be a stable FAT process that solves catastrophic overfitting. To obtain a smooth loss convergence process, we propose a novel oscillatory constraint (dubbed ConvergeSmooth) to limit the loss difference between adjacent epochs. The convergence stride of ConvergeSmooth is introduced to balance convergence and smoothing. Likewise, we design weight centralization without introducing additional hyperparameters other than the loss balance coefficient. Our proposed methods are attack-agnostic and thus can improve the training stability of various FAT techniques. Extensive experiments on popular datasets show that the proposed methods efficiently avoid catastrophic overfitting and outperform all previous FAT methods. Code is available at https://github.com/FAT-CS/ConvergeSmooth.

Guided sampling is a vital approach for applying diffusion models in real-world tasks that embeds human-defined guidance during the sampling procedure. This paper considers a general setting where the guidance is defined by an (unnormalized) energy function. The main challenge for this setting is that the intermediate guidance during the diffusion sampling procedure, which is jointly defined by the sampling distribution and the energy function, is unknown and is hard to estimate. To address this challenge, we propose an exact formulation of the intermediate guidance as well as a novel training objective named contrastive energy prediction (CEP) to learn the exact guidance. Our method is guaranteed to converge to the exact guidance under unlimited model capacity and data samples, while previous methods can not. We demonstrate the effectiveness of our method by applying it to offline reinforcement learning (RL). Extensive experiments on D4RL benchmarks demonstrate that our method outperforms existing state-of-the-art algorithms. We also provide some examples of applying CEP for image synthesis to demonstrate the scalability of CEP on high-dimensional data.

In modern machine learning, parallelization of training is an important strategy for increasing scale. Asynchronous stochastic gradient descent (ASGD), which maximizes the utilization of available hardware by avoiding waiting for slow workers. However, with constant step sizes, the convergence of ASGD is nonetheless affected negatively by slow workers due to large delays in updates. At the same time, it has been empirically observed in asynchronous training of deep learning models that gradient clipping "stabilizes" training. In this work, we provide a theoretical justification for this behavior, as we show that clipping removes the dependence of the maximum delay in the oracle complexity. We employ a sub-Weibull model of gradient noise which generalizes sub-Gaussian and sub-exponential distributions to more heavy-tailed distributions, motivated by empirical observations in deep learning. We show convergence in expectation, and the first time in asynchronous optimization, convergence with high probability.

Modern distributed training relies heavily on communication compression to reduce the communication overhead. In this work, we study algorithms employing a popular class of contractive compressors in order to reduce communication overhead. However, the naive implementation often leads to unstable convergence or even exponential divergence due to the compression bias. Error Compensation (EC) is an extremely popular mechanism to mitigate the aforementioned issues during the training of models enhanced by contractive compression operators. Compared to the effectiveness of EC in the data homogeneous regime, the understanding of the practicality and theoretical foundations of EC in the data heterogeneous regime is limited. Existing convergence analyses typically rely on strong assumptions such as bounded gradients, bounded data heterogeneity, or large batch accesses, which are often infeasible in modern machine learning applications. We resolve the majority of current issues by proposing EControl, a novel mechanism that can regulate error compensation by controlling the strength of the feedback signal. We prove fast convergence for EControl in standard strongly convex, general convex, and nonconvex settings without any additional assumptions on the problem or data heterogeneity. We conduct extensive numerical evaluations to illustrate the efficacy of our method and support our theoretical findings.

Let t(N) denote the largest number such that N! can be expressed as the product of N integers greater than or equal to t(N). The bound t(N)/N = 1/e-o(1) was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $ t(N){N} = 1{e} - c_0{log N} + Oleft( 1{log^{1+c} N} right) for an explicit constant c_0 = 0.30441901\dots and some absolute constant c>0, answering a question of Erdős and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of t(N) for wide ranges of N, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that t(N) \geq N/3 for N \geq 43632$, with the threshold shown to be best possible.

Models for learning probability distributions such as generative models and density estimators behave quite differently from models for learning functions. One example is found in the memorization phenomenon, namely the ultimate convergence to the empirical distribution, that occurs in generative adversarial networks (GANs). For this reason, the issue of generalization is more subtle than that for supervised learning. For the bias potential model, we show that dimension-independent generalization accuracy is achievable if early stopping is adopted, despite that in the long term, the model either memorizes the samples or diverges.

We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval [μ,L]. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian perturbations can differ. We prove that the sine-Jacobi recurrence of Part I is asymptotically optimal for this response norm within a factor of 1.000004, uniformly over 0<μ<L. Every terminal-exact recurrence satisfies A_Nge(2c_{cert}-o(1))N^{3/2}ε_N^star/(L-μ) with c_{cert}>1.14268353624, whereas the sine-Jacobi recurrence attains the constant 2c_{sin} with c_{sin}<1.14269218. The proof reduces the gain to the diagonal energy of a Jacobi Green matrix. A centering lift represents this matrix by an exponential kernel; packing its diagonal into a density on the line preserves the energy and controls the change in the Birman-Schwinger spectrum. A calibrated identity gives a lower bound using the ground parameter and one Fredholm determinant. Two further determinants and a trace observation improve the bound through a pointwise inequality on an explicit compact state region. One rational certificate verifies this inequality by interval arithmetic. A second certificate constructs a density satisfying the selected observations and both normalizations with energy below the sine value. Finally, no C^1 function of finitely many observations of the specified kinds gives a locally sharp lower bound at the sine density.

In this paper, we describe a phenomenon, which we named "super-convergence", where neural networks can be trained an order of magnitude faster than with standard training methods. The existence of super-convergence is relevant to understanding why deep networks generalize well. One of the key elements of super-convergence is training with one learning rate cycle and a large maximum learning rate. A primary insight that allows super-convergence training is that large learning rates regularize the training, hence requiring a reduction of all other forms of regularization in order to preserve an optimal regularization balance. We also derive a simplification of the Hessian Free optimization method to compute an estimate of the optimal learning rate. Experiments demonstrate super-convergence for Cifar-10/100, MNIST and Imagenet datasets, and resnet, wide-resnet, densenet, and inception architectures. In addition, we show that super-convergence provides a greater boost in performance relative to standard training when the amount of labeled training data is limited. The architectures and code to replicate the figures in this paper are available at github.com/lnsmith54/super-convergence. See http://www.fast.ai/2018/04/30/dawnbench-fastai/ for an application of super-convergence to win the DAWNBench challenge (see https://dawn.cs.stanford.edu/benchmark/).

We present a constructive lower bound of Ω(1/varepsilon) for Frank-Wolfe (FW) when both the objective and the constraint set are smooth and strongly convex, showing that the known uniform O(1/varepsilon) guarantees in this regime are tight. It is known that under additional assumptions on the position of the optimizer, FW can converge linearly. However, it remained unclear whether strong convexity of the set can yield rates uniformly faster than O(1/varepsilon), i.e., irrespective of the position of the optimizer. To investigate this question, we focus on a simple yet representative problem class: minimizing a strongly convex quadratic over the Euclidean unit ball, with the optimizer on the boundary. We analyze the dynamics of FW for this problem in detail and develop a novel computational approach to construct worst-case FW trajectories, which is of independent interest. Guided by these constructions, we develop an analytical proof establishing the lower bound.

Many-body perturbation theory is a powerful framework to calculate spectroscopic quantities, as well as the total energy. However, the field suffers from the fact that different flavors of approximations lead to a wide range of results. We present several in principle exact expansions and discuss how to use them consistently. We concentrate on the first two orders in the respective screened Coulomb interaction. Of special interest is a consistent expansion around the Kohn-Sham system. We illustrate the performance of the various choices using the symmetric Hubbard dimer over a wide range of correlation strength. In particular, the expansion based on exact Kohn-Sham ingredients stands out as a promising route towards good energies and good spectra.

In the present article we study diffuse interface models for two-phase biomembranes. We will do so by starting off with a diffuse interface model on R^n defined by two coupled phase fields u,v. The first phase field u is the diffuse approximation of the interior of the membrane; the second phase field v is the diffuse approximation of the two phases of the membrane. We prove a compactness result and a lower bound in the sense of Gamma-convergence for pairs of phase functions (u_varepsilon,v_varepsilon). As an application of this first result, we consider a diffuse approximation of a two-phase Willmore functional plus line tension energy.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function f_theta (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function f_theta follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

Even when massively overparameterized, deep neural networks show a remarkable ability to generalize. Research on this phenomenon has focused on generalization within distribution, via smooth interpolation. Yet in some settings neural networks also learn to extrapolate to data far beyond the bounds of the original training set, sometimes even allowing for infinite generalization, implying that an algorithm capable of solving the task has been learned. Here we undertake a case study of the learning dynamics of recurrent neural networks (RNNs) trained on the streaming parity task in order to develop an effective theory of algorithm development. The streaming parity task is a simple but nonlinear task defined on sequences up to arbitrary length. We show that, with sufficient finite training experience, RNNs exhibit a phase transition to perfect infinite generalization. Using an effective theory for the representational dynamics, we find an implicit representational merger effect which can be interpreted as the construction of a finite automaton that reproduces the task. Overall, our results disclose one mechanism by which neural networks can generalize infinitely from finite training experience.

Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.

Inertial Navigation Systems (INS) are algorithms that fuse inertial measurements of angular velocity and specific acceleration with supplementary sensors including GNSS and magnetometers to estimate the position, velocity and attitude, or extended pose, of a vehicle. The industry-standard extended Kalman filter (EKF) does not come with strong stability or robustness guarantees and can be subject to catastrophic failure. This paper exploits a Lie group symmetry of the INS dynamics to propose the first nonlinear observer for INS with error dynamics that are almost-globally asymptotically and locally exponentially stable, independently of the chosen gains. The observer is aided only by a GNSS measurement of position. As expected, the convergence guarantee depends on persistence of excitation of the vehicle's specific acceleration in the inertial frame. Simulation results demonstrate the observer's performance and its ability to converge from extreme errors in the initial state estimates.

We consider the distributionally robust optimization (DRO) problem with spectral risk-based uncertainty set and f-divergence penalty. This formulation includes common risk-sensitive learning objectives such as regularized condition value-at-risk (CVaR) and average top-k loss. We present Prospect, a stochastic gradient-based algorithm that only requires tuning a single learning rate hyperparameter, and prove that it enjoys linear convergence for smooth regularized losses. This contrasts with previous algorithms that either require tuning multiple hyperparameters or potentially fail to converge due to biased gradient estimates or inadequate regularization. Empirically, we show that Prospect can converge 2-3times faster than baselines such as stochastic gradient and stochastic saddle-point methods on distribution shift and fairness benchmarks spanning tabular, vision, and language domains.

In this paper, we consider the inviscid limit problem to the higher dimensional incompressible Navier--Stokes equations in the whole space. It is shown in [Guo, Li, Yin: J. Funct. Anal., 276 (2019)] that given initial data u_0in B^{s}_{p,r} and for some T>0, the solutions of the Navier--Stokes equations converge strongly in L^infty_TB^{s}_{p,r} to the Euler equations as the viscosity parameter tends to zero. We furthermore prove the failure of the uniform (with respect to the initial data) B^{s}_{p,r} convergence in the inviscid limit of a family of solutions of the Navier-Stokes equations towards a solution of the Euler equations.

Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with ell_2-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new ell*-smoothness concept that measures the norm of Hessians in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish sharp convergence for stochastic mirror descent, matching state-of-the-art under classic smoothness. Our theory also extends to non-convex and composite optimization, which may shed light on practical usages of mirror descent, including pre-training and post-training of LLMs.

The Neural Arithmetic Logic Unit (NALU) is a neural network layer that can learn exact arithmetic operations between the elements of a hidden state. The goal of NALU is to learn perfect extrapolation, which requires learning the exact underlying logic of an unknown arithmetic problem. Evaluating the performance of the NALU is non-trivial as one arithmetic problem might have many solutions. As a consequence, single-instance MSE has been used to evaluate and compare performance between models. However, it can be hard to interpret what magnitude of MSE represents a correct solution and models sensitivity to initialization. We propose using a success-criterion to measure if and when a model converges. Using a success-criterion we can summarize success-rate over many initialization seeds and calculate confidence intervals. We contribute a generalized version of the previous arithmetic benchmark to measure models sensitivity under different conditions. This is, to our knowledge, the first extensive evaluation with respect to convergence of the NALU and its sub-units. Using a success-criterion to summarize 4800 experiments we find that consistently learning arithmetic extrapolation is challenging, in particular for multiplication.

Vision-language models encode continuous geometry that their text pathway fails to express: a 6,000-parameter linear probe extracts hand joint angles at 6.1 degrees MAE from frozen features, while the best text output achieves only 20.0 degrees -- a 3.3x bottleneck. LoRA fine-tuning (r=16, 2,000 images) narrows this gap to 6.5 degrees, providing evidence for a pathway-training deficit rather than a representational one. Training objective determines accuracy more than architecture: five encoders spanning self-supervised, contrastive, and hybrid paradigms converge to statistically equivalent accuracy (R^2 approximately 0.55, TOST-equivalent at delta=0.03) despite sharing as little as CKA=0.41 representational similarity -- functional convergence without representational convergence. Autoregressive generation damages geometric fidelity, but the damage originates in the generation process, not in language alignment: Qwen2.5-VL's LLM layers actually improve probe accuracy over its raw vision encoder. Layer-wise analysis reveals a universal mid-network accuracy peak across all architectures, with attention heads in layers 18-22 carrying disproportionate geometric signal. These findings enable a single frozen backbone to function as a multi-task geometric sensor through lightweight probes, without fine-tuning or text generation.

Recent work in probability-domain knowledge distillation has established axiomatic frameworks for temperature scaling, multi-teacher aggregation, and bias-variance trade-offs in single-stage settings. However, the mathematical behavior of recursive or multi-generation distillation remains poorly understood, with prior approaches relying primarily on empirical heuristics. In this work, we introduce an axiomatic and operator-theoretic framework for recursive meta-distillation, formalizing iterative knowledge distillation as a sequence of probability-distribution operators with explicit anchoring to base teachers. We define structural axioms for valid meta-teacher construction and prove the existence of non-trivial operator families satisfying these axioms without specifying particular algorithms or loss functions. Under mild realizability and convexity assumptions, we show that anchored recursive distillation induces contraction in KL divergence, yielding geometric convergence to base teacher distributions and a unique, globally attractive fixed point. The contribution is foundational rather than algorithmic: the framework characterizes when recursive distillation is mathematically well-posed and convergent rather than error-accumulating, independent of model architecture, optimization details, or specific operator instantiations. These results provide a theoretical basis for understanding stability, bias-variance behavior, and failure modes in iterative and multi-teacher distillation under capacity constraints.

Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency. For a fixed stepsize, the iterates typically admit a stationary distribution that is rarely tractable. Prior work shows that as the stepsize αdownarrow 0, the centered-and-scaled steady state converges weakly to a Gaussian random vector. However, for fixed α, this weak convergence offers no usable error bound for approximating the steady-state by its Gaussian limit. This paper provides explicit, non-asymptotic error bounds for fixed α. We first prove general-purpose theorems that bound the Wasserstein distance between the centered-scaled steady state and an appropriate Gaussian distribution, under regularity conditions for drift and moment conditions for noise. To ensure broad applicability, we cover both i.i.d. and Markovian noise models. We then instantiate these theorems for three representative SA settings: (1) stochastic gradient descent (SGD) for smooth strongly convex objectives, (2) linear SA, and (3) contractive nonlinear SA. We obtain dimension- and stepsize-dependent, explicit bounds in Wasserstein distance of order α^{1/2}log(1/α) for small α. Building on the Wasserstein approximation error, we further derive non-uniform Berry--Esseen-type tail bounds that compare the steady-state tail probability to Gaussian tails. We achieve an explicit error term that decays in both the deviation level and stepsize α. We adapt the same analysis for SGD beyond strongly convexity and study general convex objectives. We identify a non-Gaussian (Gibbs) limiting law under the correct scaling, which is validated numerically, and provide a corresponding pre-limit Wasserstein error bound.

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N^2) operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical order, with a O(N{log}N) complexity and requires the evaluation points to lie on a regular grid. The second one is based on the divide-and-conquer principle, with a O(Nlog(N)^{(d-1){vee}1}) complexity and requires the evaluation points to coincide with the input points. The two fast algorithms are described and detailed in the general d-dimensional case, and numerical experiments validate their speed and accuracy. Secondly, the paper establishes a direct connection between cumulative distribution functions and kernel density estimation (KDE) for a large class of kernels. This connection paves the way for fast exact algorithms for multivariate kernel density estimation and kernel regression. Numerical tests with the Laplacian kernel validate the speed and accuracy of the proposed algorithms. A broad range of large-scale multivariate density estimation, cumulative distribution estimation, survival function estimation and regression problems can benefit from the proposed numerical methods.

In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) X sim DP(αν_0), using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of αmapsto log E_{X sim DP(αν_0)}[exp( E_X[αf])], where E_X[f] = int f dX. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF logE_{Xsim DP(αν_0)}{exp(E_{X}{[f]})} for any α> 0. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence αKL(ν_0Vert cdot). This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.

We study the vanishing discount selection problem for critical Hamil\-ton--Jacobi equations posed on {mathbb R}^d with Hamiltonian G=H-V, where H is continuous, periodic in space, convex and superlinear in the momentum, and V is a compactly supported potential of constant sign. For Vleqslant0, we prove local uniform convergence of the discounted solutions to a distinguished critical solution, which we characterize in terms of Mather measures and of the solution selected by the unperturbed periodic problem. The proof relies on a decomposition of limiting discounted measures into a retained component and a periodic component accounting for the mass lost at infinity. For Vgeqslant0, we establish convergence under the additional assumption that the unperturbed selected solution has zero average with respect to every periodic Mather measure. We also give explicit examples in which the discounted solutions converge, but natural formulas for the candidate limit fail to identify the selected solution.

Offline reinforcement learning (offline RL) can benefit from nearby out-of-distribution (OOD) actions, but estimation errors at these actions may be amplified by bootstrapping. Existing regularization and local-generalization methods control either the admissible OOD region or the influence of generalized targets, often through separate mechanisms. We propose Convex Hull Neighborhood Smooth Dual Generalization (CSDG), which expresses the Bellman backup as an in-sample value target plus a CHN-local correction. This formulation makes the generalized contribution explicit and separates it from the in-sample reference path. The correction is obtained by smoothing in-sample-oriented and OOD-oriented candidates sampled at different perturbation radii. A mixture coefficient lambda scales its contribution to each backup, while the recursive discount remains gamma. Under boundedness and fixed perturbation kernels, we derive an exact one-step correction identity, a time-varying iterate bound, and a fixed-point bound that depends only on the branch discrepancy at the fixed point. We further characterize the implicit policies induced by the idealized operators and give a conditional non-degradation criterion. The practical algorithm approximates these quantities using asymmetric bounded noise and expectile regression, without exact support classification or an additional pessimistic OOD penalty. Experiments on Gym-MuJoCo and AntMaze show strong aggregate performance and stable value estimation. Code is available at: https://github.com/YOUNG-fnxm/CSDG

We study the problem of convergence to a stationary point in zero-sum games. We propose competitive gradient optimization (CGO ), a gradient-based method that incorporates the interactions between the two players in zero-sum games for optimization updates. We provide continuous-time analysis of CGO and its convergence properties while showing that in the continuous limit, CGO predecessors degenerate to their gradient descent ascent (GDA) variants. We provide a rate of convergence to stationary points and further propose a generalized class of alpha-coherent function for which we provide convergence analysis. We show that for strictly alpha-coherent functions, our algorithm convergences to a saddle point. Moreover, we propose optimistic CGO (OCGO), an optimistic variant, for which we show convergence rate to saddle points in alpha-coherent class of functions.

We present a SemiEmpirical Theory of Learning (SETOL) that explains the remarkable performance of State-Of-The-Art (SOTA) Neural Networks (NNs). We provide a formal explanation of the origin of the fundamental quantities in the phenomenological theory of Heavy-Tailed Self-Regularization (HTSR): the heavy-tailed power-law layer quality metrics, alpha and alpha-hat. In prior work, these metrics have been shown to predict trends in the test accuracies of pretrained SOTA NN models, importantly, without needing access to either testing or training data. Our SETOL uses techniques from statistical mechanics as well as advanced methods from random matrix theory and quantum chemistry. The derivation suggests new mathematical preconditions for ideal learning, including a new metric, ERG, which is equivalent to applying a single step of the Wilson Exact Renormalization Group. We test the assumptions and predictions of SETOL on a simple 3-layer multilayer perceptron (MLP), demonstrating excellent agreement with the key theoretical assumptions. For SOTA NN models, we show how to estimate the individual layer qualities of a trained NN by simply computing the empirical spectral density (ESD) of the layer weight matrices and plugging this ESD into our SETOL formulas. Notably, we examine the performance of the HTSR alpha and the SETOL ERG layer quality metrics, and find that they align remarkably well, both on our MLP and on SOTA NNs.

Follow the regularized leader FTRL is the premier algorithm for online optimization. However, despite decades of research on its convergence in constrained optimization -- and potential games in particular -- its behavior remained hitherto poorly understood. In this paper, we establish that FTRL can take exponential time to converge to a Nash equilibrium in two-player potential games for any (permutation-invariant) regularizer and potentially vanishing learning rate. By known equivalences, this translates to an exponential lower bound for certain mirror descent counterparts, most notably multiplicative weights update. On the positive side, we establish the potential property for FTRL and obtain an exponential upper bound exp(O_ε(1/ε^2)) for any no-regret dynamics executed in a lazy, alternating fashion, matching our lower bound up to factors in the exponent. Finally, in multi-player potential games, we show that fictitious play -- the extreme version of FTRL -- can take doubly exponential time to reach a Nash equilibrium. This constitutes an exponentially stronger lower bound for the foundational learning algorithm in games.