Asymptotics

Multi-robot task and motion planning (MR-TAMP) requires jointly reasoning about discrete task decisions and continuous collision-free motions of multiple interacting robots. Although asymptotically optimal algorithms have been developed for task and motion planning, extending these guarantees to the multi-robot setting introduces an important challenge: different task transitions may involve different subsets of robots and therefore impose constraints of different dimensions on the composite configuration space. Consequently, an asymptotically optimal planner must not only optimize motion within each task mode, but also ensure sufficient exploration of the different types of transitions connecting them. We characterize this transition structure and establish sufficient conditions for global asymptotic optimality in MR-TAMP, requiring persistent coverage of relevant transitions and asymptotically improving motion planning within connected feasible regions. Based on these conditions, we develop an efficient asymptotically optimal MR-TAMP algorithm that combines evolving individual-robot roadmaps with implicit tensor-product search, avoiding explicit construction of the composite roadmap. The planner further employs conditional transition sampling, lazy collision checking, and mode- and solution-level guidance to improve finite-time planning efficiency while retaining persistent exploration. The resulting framework provides asymptotic optimality guarantees for multi-robot manipulation while efficiently exploiting the structure of individual-robot motion planning.

We study the large-depth behavior of residual networks whose weights are correlated across layers at initialization. Our results confirm and extend a conjecture of Marion et al. [2025], according to which correlated initializations should interpolate continuously between the Brownian stochastic differential equation arising from independent initialization and the ordinary differential equation arising from perfectly correlated initialization. When the initialization is obtained from the application of a feature function to a stationary Gaussian sequence with regularly varying correlation, we prove that there exists a unique critical scaling such that the infinite-depth limit is the solution of a Young differential equation driven by a Hermite process. Hermite processes reduce to the fractional Brownian motion if the feature function generating the initialization has Hermite rank one, which is the case for the identity function, for example. We show that the critical scaling and asymptotic limit are uniquely determined by the decay of correlations together with the Hermite rank of the feature function. Consequently, the correlation structure and Hermite rank of the initialization represent meaningful hyperparameters in the asymptotic regime. By contrast, under finite-variance iid initialization, the asymptotic driver is universally Brownian up to normalization regardless of the choice of distribution. Our proofs rely on a collection of novel results establishing a robust stability theory for Young differential equations in Banach spaces.

Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.

Prior work on LLM behavior under anomalous conditions asks whether a model notices anomalies. We ask a narrower question: once a model sits in a workflow with a low, controllable failure rate, does its explanatory engagement - length, specificity, self-reported confidence - change as failure grows asymptotically rarer? We built a local, zero-cost harness on three open-weight models (qwen3:8b, llama3.1:8b, mistral:7b) running a repeated tool-call task where one call fails at probability p, swept across eight rates from 0.2 to 0.0001, under five elicitation conditions from immediate prompting to none. We hypothesized a rise in engagement as failures grew rarer, then a collapse near a detectability threshold. Pooled across conditions this appeared false: length fell in a flat, monotonic pattern. Splitting by condition overturned that. Under immediate_forced, where the model must explain every failure instantly, the predicted rise is confirmed but followed by a plateau, not a collapse: length peaks at 28.4 words at p=0.05, settles to 17.4-19.0 words at the rarest rates, and confidence rises unevenly from about 53% to the 70s-90s. Under grouped_runs, explanation batched to run-end, no collapse appears. Under passive_unprompted, aggregate magnitude is a floor artifact, but a recovered logging gap revealed real, model-specific self-monitoring: llama3.1:8b volunteers structured confidence reports unprompted, sometimes eroding its own confidence as trials accumulate; the other two do so only once, as boilerplate. Elicitation structure is a first-class moderator of collapse observability. A companion guaranteed-failure run (72 cells, backfilling rates where random sampling gave zero real failures) shows models differ in whether they recognize an anomaly, distinct from engagement once recognized. Limitation: discrete rate points cannot capture behavior between them, a direction for future work.

The initialization of deep neural networks determines whether information and gradients can propagate across depth, yet a unified theory connecting these properties to learning dynamics remains elusive. Combining mean-field theory and random matrix theory, we establish a direct link between correlation propagation and the Neural Tangent Kernel (NTK) that governs learning in the sequential limit of infinitely wide, infinitely deep networks. Correlation propagation to infinite depth is possible only at a single, critical point in the weight-bias variance plane. At this point, we leverage the algebraic decay of the end-to-end Jacobian with depth to prove that the NTK becomes exactly proportional to the output correlation at infinite depth, tying together information propagation and learning dynamics. We further show that orthogonal initialization suppresses the leading finite-size corrections present under Gaussian initialization, clarifying the respective roles of the two initialization ensembles in this limit. These theoretical predictions are validated quantitatively on finite-width, finite-depth networks. Together, these results demonstrate that orthogonal initialization and criticality are required to control the asymptotic dynamics of deep learning.

For a Markov kernel $T$ with an invariant probability measure $π$, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of $T^{n}_{x}$ and $π$ for every starting point $x$: 1.) asymptotic absolute continuity: the singular mass sing$(T^{n}_{x}\midπ)$ tends to $0$; 2.) asymptotic domination of the target: the singular mass sing$(π\mid T^{n}_{x})$ tends to $0$, as $n \to \infty$. This criterion, on countably generated measurable spaces, is both sufficient and necessary for the Markov chain convergence. A density version of this criterion is verified on general measurable spaces in three cases: (i) $T$ has a positive transition density wrt $π$; (ii) $T$ consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis--Hastings algorithm; (iii) the transition density is positive only after a finite number of steps that may depend on the starting point $x$. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In most results, the state space is a general measurable space, which carries no structure beyond a $σ$-algebra. Countable generation is only assumed where the density-free form of the criterion is stated. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion, which is both sufficient and necessary.

We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-δ)$ quantile over all fixed collections of design vectors and all target parameters: $$d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}δ\right).$$ This is a nonasymptotic analogue of the Wilks $χ^2_d$ phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension $d=2$ is sharply of order $$\log\log\log n+\log\left(\frac{1}δ\right).$$ The worst case quantile in dimension $d=1$ is of order $\log(1/δ)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/δ)$, we prove the sharp bound $$d+\log\left(\frac{1}δ\right).$$ Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $δ$.

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $κ\sqrt{d}/\varepsilon$, where $κ$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence---or even, counterintuitively, divergence---in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.

We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth $σ\rightarrow \infty$. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size $O \left (\fracρσ \right )^2$, where $ρ$ is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that $ρ$ provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.

Dropout and Random Gradient Masking (RaM) are two training techniques used to improve performance in deep learning. Both techniques inject randomness into the training dynamics, but in significantly different ways: dropout applies random masks to the activations in the forward pass, whereas RaM leaves the forward pass unchanged and instead masks the gradients. In particular, the noise induced by RaM in the parameter updates is unbiased, so standard explanations for the effectiveness of dropout, such as the penalization effect or the prevention of co-adaptation between neurons, do not apply to RaM. In this work, we show that the difference between the two methods disappears for ResNets in the large depth and width asymptotics: in the complete feature learning regime, they both converge to the same large-scale limiting dynamics. This asymptotic equivalence holds for several variants of dropout and RaM, including layerwise dropout as used in stochastic-depth ResNets, albeit at slower quantitative rates. In fact, we also show that several of these variants collapse to the same limit asymptotically.

The paper develops the approach to the runtime analysis of evolutionary algorithms on the basis of limit theorems from probability theory. We consider the family of Jump$_k$ benchmark functions, defined on the search space of binary strings of length $n$, parametrized by the integer $k$, which have a plateau of multiple local optima at the Hamming distance $k$ from a unique global optimum. In this work, we consider the genetic algorithm $(1+(λ,λ)) GA$ from (Doerr, Doerr and Ebel, 2015) with tunable parameters of the mutation rate $p$, crossover bias $c$, and two intermediate population sizes $λ_M$ and $λ_C$. We study the time it escapes from the plateau of local optima and reaches the global optimum in the case of Jump$_k$ fitness function and tighten the upper bounds on the expected escape time, known from the work of Antipov, Doerr and Karavaev (2022). The obtained bounds also apply to a wider range of algorithmic parameters. The main result of this work applies to the case when $k\to \infty$ as $n \to \infty.$ The case of finite $k$ is investigated quite simply and considered tangentially.

The impact of a given training point on a statistical model is classically measured through its leave-one-out influence, which quantifies the effect of its removal from the training set on the model accuracy. While the statistics of leave-one-out influences are well understood in the low-dimensional, large sample limit $n\to \infty, d=O(1)$, they become more intricate in high dimensions, as the influence of a given sample develops non-trivial dependencies on all other training samples. For convex M-estimation under Gaussian design, in the high-dimensional limit $n\asymp d$, we show that the distribution of the influences across the training set converges to a limiting measure which we sharply characterize. Building on these results, we provide evidence that influential samples tend to lie close to the decision boundary, thereby making contact with a standard data selection heuristic in active learning.

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(η^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $τ=ηk$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(η)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(η^2)$; a bare fixed-$β$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$β$ momentum, and 4.00 for SGD; matching the memory clock to $τ$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

Long-range learning is hard for recurrent networks trained with stochastic gradient descent, because the influence of a past input fades with the lag $\ell$, and if it fades too fast the dependence cannot be learned from finite data. This fade is captured by an envelope $f(\ell)$. An exponential fade makes the data needed to learn a lag-$\ell$ dependence grow exponentially, putting long horizons out of reach; a power-law fade keeps the cost polynomial. We show that the asymptotic decay class of $f(\ell)$ is not fixed by the architecture. Instead, it emerges from the coupling between the state dynamics and parameter dynamics, settling into either a collapsed regime (fast, exponential forgetting) or an extended, anti-collapsed regime (slow, power-law forgetting). The intuition is a competition within these coupled dynamics. Training drives the network's effective time scales toward short ones, while rare, heavy-tailed fluctuations of the learning dynamics push a few of them to very long values. The extended regime survives only when these heavy-tailed pushes are strong enough to balance the pull. We make this mathematically precise with a coarse-grained stochastic process and prove exactly when the extended regime exists. A single exponent, the spectral exponent~$β$, then governs both the spread of time scales and how slowly the network forgets. Realizing the regime in practice needs one more ingredient: the joint action of the architecture and the optimizer must be able to hold such a broad spread. A network whose capacity to generate broad time-scale spectra is severely constrained still collapses, even when supplied with strong heavy-tailed forcing. Heavy-tailed fluctuations thus act not as noise to be suppressed, but as the mechanism that sustains long-range learning.

We present AO-ARC, an anytime multi-robot motion planning (MRMP) method that achieves initial solution times on par with state-of-the-art MRMP feasibility solvers while converging faster and more reliably than existing anytime MRMP methods as the number of robots increases. AO-ARC adapts the AO-x meta-algorithm for converting feasibility solvers into anytime algorithms by iteratively calling the original ARC method on bounded MRMP instances under a makespan cost metric. This exploits the adaptive (de)coupling of ARC while maintaining the consistent cost bound across robot (de)compositions needed for AO-x. We provide theoretical analysis proving the asymptotic optimality properties of AO- ARC and conduct empirical evaluation on a set of 2D scenarios with different levels of coordination complexity and a 3D manipulator scenario representative of real-world applications.

We develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces. Under mild conditions on the loss we establish existence and measurability of the estimator, covering a wide range of convex and non-convex losses, including bounded robust losses. We further prove sharp rates of convergence with an explicit bias-variance decomposition governed by a novel complexity measure. We show that the variance is independent of misspecification, while the bias depends on a source condition parameter known in the learning literature. For tensor product Sobolev spaces we obtain new rates that connect to spaces of functions with dominating mixed smoothness, substantially extending existing results and explaining why these estimators circumvent the curse of dimensionality. Our methodology, combining elements from both functional analysis and empirical process theory, allows for an asymptotic linearisation of the objective function that avoids both closed-form solutions and global Lipschitz assumptions, and may be of independent interest. The estimators are implemented in C++ and theory is supported by numerical experiments.

A Markov Logic Network (MLN) is a probabilistic relational model used in Statistical Relational Artificial Intelligence for defining a probability distribution on the set of possible worlds with domain $D$ for an arbitrary finite domain $D$. An MLN consists of soft constraints with associated weights which are nonnegative real numbers. In this study we consider a language speaking about a property $P(x)$ and a relation $R(x, y)$. We consider an MLN for which every Boolean combination of $P(x)$ and $R(x, y)$ is a soft constraint (with associated weight). Let $n$ denote the size (cardinality) of the domain. We show that, for every choice of weights, if the weights are scaled by $1/n$ then, for every first-order sentence $\varphi$, the probability that $\varphi$ holds tends to either 0 or 1 as $n \to \infty$; that is, a 0-1 law for first-order logic holds. Morover, the limit probability does {\em not} depend on the weights. If we instead use the standard semantics of MLNs, in the case of which the weights are {\em not} scaled, then the limit behaviour is more complicated and {\em depends} on the weights. With unscaled weights we get 7 qualitatively different cases which depend on the weights. In some cases we have a 0-1 law for first-order logic, in some cases not, but we may still have a convergence law. The influence of the weights on the asymptotic probability of a first-order sentence may be in the form of a sudden ``phase transition'' from one of the 7 cases to another. The presence of a convergence law has positive implications for inference on large domains.

We find the asymptotic ratio between the storage capacities when enforcing real pre-activations in a complex hypothesis class as opposed to complex ones in the same class. We use weights drawn from the complex Gaussian, which converge asymptotically in norm to the square root of dimension almost surely. Our methods depend on Gardner volume-type comparisons at critical capacity. Our proof relies on an application of the Harish-Chandra-Itzykson-Zuber (HCIZ) formula, nonstandard in literature. With the HCIZ formula, we may obtain a more robust approximation for the final asymptotic ratio. This strategy is applicable to our work specifically since we integrate over the unitary and orthogonal compact manifolds, facilitated via the Weyl integration formula and the Haar measure.

Chain-of-thought (CoT) reasoning has become a widely used mechanism for eliciting multi-step reasoning in large language models by generating intermediate reasoning steps at inference time. Yet the scaling behavior of generalization with CoT depth remains poorly understood. To address this question, we study a theoretically solvable model of CoT for in-context weight prediction in linear regression, where test-time reasoning is represented as an iterative refinement of the weight-parameter estimate. Using tools from random matrix theory under high-dimensional asymptotics, we derive an exact formula for the generalization error as a function of reasoning depth, pretraining data amount, and context length. Our analysis reveals a sharp phase transition separating exponential and polynomial improvement, saturation, and overthinking, and characterizes how the optimal reasoning depth scales. We further show that deeper reasoning is most effective with sufficiently rich pretraining and in-context information, whereas limited pretraining or context makes longer reasoning prone to error amplification or saturation. We also validate these predictions through experiments on fully learned linear attention and softmax attention models. Our results provide a unified theoretical account of how test-time CoT depth affects generalization.

We formulate data propagation through the Transformer, the machine learning architecture powering large language models, as a nonlinear control system on the space of probability measures. For the mean-field Transformer model with self-attention and affine feed-forward layers, we prove that Gaussian distributions remain exactly Gaussian along the induced flow. This invariance reduces the infinite-dimensional measure dynamics to a finite-dimensional bilinear control system governing the evolution of the mean and covariance, reformulates the expressive capacity of Transformers as a reachability problem for prescribed Gaussian moments, and reveals a novel connection with Riccati-type equations from classical filtering and control. For time-varying controls, we prove exact finite-time reachability of any target Gaussian distribution whose covariance matrix has the same rank as the initial one, this rank constraint being an intrinsic invariant of the dynamics. For time-invariant parameters, we derive explicit spectral conditions leading either to asymptotic stability toward positive-definite equilibria or to finite-time blow-up of the covariance. Numerical experiments complement the theory by showing that practical Transformers with Gaussian inputs remain close to moment-matched Gaussian distributions through early and intermediate layers, while Transformers with prescribed attention matrices reproduce the predicted covariance regimes: bounded evolution in stabilizing configurations and blow-up in destabilizing ones.

Hard-label classification is usually trained with smooth surrogate losses, most prominently softmax cross-entropy. We isolate an asymptotic mechanism by which this mismatch between smooth surrogate and discrete labels produces power-law learning curves in an online teacher-student model. After subtracting the mean logit, the thermodynamic-limit dynamics close in centered variables: a growing centered student-teacher alignment $D$ and the residual student variance $Δ$. At late times, examples away from teacher decision boundaries are already classified confidently and contribute exponentially little. Only boundary layers of width $O(D^{-1})$ remain active, while the noise of fixed-learning-rate online gradient descent maintains a nonzero $Δ$. As a function of the training time $α$ the late-time solution yields a $α^{-1/3}$ power law not only for the test loss but also for the generalization error $ε_g$, i.e., one minus test accuracy. This is much slower than the $α^{-1}$ Bayes-optimal reference for the same model. We further show that learning-rate schedules can improve the generalization error towards a $ε_g \sim α^{-1/2}$ power law. Simulations support the predicted order parameter dynamics and learning curves. Controlled experiments with correlated Gaussian inputs and whitened pretrained features show that data structure can dominate transients. Therefore, our result is an asymptotic, complementary mechanism rather than an alternative to spectral explanations of neural scaling laws.

Autonomous robotic exploration in remote and extreme environments allows scientists to model complex transport phenomena and collective behaviors described by continuously deforming flow fields. Although these environments are naturally modeled as time-varying domains, most adaptive exploration methods assume static environments and fail to provide adequate coverage or satisfy any formal guarantees. This is especially the case in oceanography where autonomous underwater systems (UxS) have highly restrictive compute and payload requirements that necessitate path planning methods that yield robust data collection strategies in open-loop and underactuated settings. In this work, to address the aforementioned issues, we propose to formulate adaptive search as an ergodic coverage problem and investigate certifying coverage in the ergodic sense over evolving domains with flow-induced dynamics. We expand upon recent work demonstrating maximum mean discrepancy (MMD) as a functional ergodic metric, and derive a flow-adaptive formulation that explicitly accounts for domain evolution within the coverage objective. We show that this approach preserves ergodic coverage guarantees in ambient flows and enables effective exploration in under-actuated, and even open-loop planning settings by integrating environment dynamics. Experiments validate that our method generalizes to diverse spatiotemporal processes including ocean exploration, and tracking human and cattle movement. Physical experiments on aerial and legged robotic platforms validate our ability to obtain ergodic coverage in non-convex, flow-restricted environments while respecting robot dynamics.

Large language models demonstrate remarkable ability in factual recall, yet the fundamental limits of storing and retrieving input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their corresponding~$d$-dimensional targets via a single layer, requiring each mapped input to be well separated from all other targets. Unlike in supervised classification, this strict separation induces~$p$ constraints per association and produces strong correlations between constraints that make a direct characterisation of the storage capacity difficult. Here, we provide a precise characterisation of this capacity in the following way. We first introduce a decoupled model in which each input has its own independent set of competing outputs, and provide numerical and analytical evidence that this decoupled model is equivalent to the original model in terms of storage capacity, spectra of the learnt weights, and storage mechanism. Using tools from statistical physics, we show that the decoupled model can store up to $p_c \log p_c / d^2 = 1 / 2$ associations, and generalise the computation of $p_c$ to linear two-layer architectures. Our analysis also gives mechanistic insight into how the optimal solution improves over a naïve Hebbian learning rule: rather than boosting input-output alignments with broad fluctuations, the optimal solution raises the correct scores just above the extreme-value threshold set by the competing outputs. These findings give a sharp statistical-physics characterisation of factual storage in linear networks and provide a baseline for understanding the memory capacity of more realistic neural architectures.

This paper investigates the autonomous synchronization problem for discrete-time heterogeneous multiagent systems. The synchronization problem is transformed into the asymptotic decoupling problem of stable modes in a class of discrete-time linear time-varying systems, for which we provide a sufficient condition. Leveraging this condition, synchronization conditions are established. The synchronization conditions are based on the average of the agents' initial dynamic matrices, without requiring the differences among these matrices to be small. This approach reduces the conservativeness of existing conditions and achieves a unification of both homogeneous and heterogeneous systems. Numerical simulation results are provided to support the theoretical findings.

An alternative generating function is proposed to enumerate row-convex polyominoes without internal holes on a discrete grid. The approach is based on integer partitions of the total area, where each partition corresponds to a sequence of row lengths, and the product of all permutations of the parts accounts for all possible horizontal alignments of consecutive rows. Summing over the products yields a formula for the total number of convex polyominoes of a given size. Numerical examples are provided for small areas, and the exact generating function is derived via a transfer series argument, establishing the asymptotic growth S(N) as A2^(N) cos(N*theta) + phi) with theta = arctan(sqrt(7)/3). The method establishes a direct connection between integer partitions and polyomino enumeration, offering a simple yet effective framework for both exact and asymptotic combinatorial analysis. Potential applications include shape priors in discrete image analysis, grid-based modeling, and combinatorial generation of convex structures.

Existing large-dimensional theory for spectral algorithms resolves either the optimally tuned point or the interpolation limit, but leaves the under-regularized regime unexplored. We study the learning curve and benign overfitting of spectral algorithms in the large-dimensional setting where the sample size and dimension are of comparable order, i.e., $n \asymp d^γ$ for some $γ>0$. We first consider inner-product kernels on the sphere $\mathbb{S}^{d-1}$ and establish a sharp asymptotic characterization of the excess risk across the full regularization path under various source conditions $s \geq 0$, where $s$ measures the relative smoothness of the regression function. Our results reveal that the learning curve is not simply U-shaped but instead consists of three distinct regimes: over-regularized, under-regularized, and interpolation regimes. This characterization allows us to fully capture the benign overfitting phenomenon, demonstrating that benign overfitting arises consistently across both the under-regularized and interpolation regimes whenever $s$ is positive but no larger than a critical threshold. We further show that, in the sufficiently regularized regime, the kernel learning curve is recovered by an associated sequence model. Finally, we extend the learning-curve analysis to large-dimensional KRR for a class of kernels on general domains in $\mathbb{R}^d$ whose low-degree eigenspaces satisfy spectral-scaling and hyper-contractivity conditions.

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

Multi--task learning seeks to improve the generalization error by leveraging the common information shared by multiple related tasks. One challenge in multi--task learning is identifying formulations capable of uncovering the common information shared between different but related tasks. This paper provides a precise asymptotic analysis of a popular multi--task formulation associated with misspecified perceptron learning models. The main contribution of this paper is to precisely determine the reasons behind the benefits gained from combining multiple related tasks. Specifically, we show that combining multiple tasks is asymptotically equivalent to a traditional formulation with additional regularization terms that help improve the generalization performance. Another contribution is to empirically study the impact of combining tasks on the generalization error. In particular, we empirically show that the combination of multiple tasks postpones the double descent phenomenon and can mitigate it asymptotically.

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-α}$ for the empirical covariance operator and the PCA excess risk, where $α>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic \enquote{free lunch} for PPI++, an adaptive form of PPI, showing that the \textit{asymptotic} variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds \textit{regardless of the quality of the pseudo-labels}. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a \enquote{no free lunch} result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, for instance, the correlation must be at least $1/\sqrt{n - 2}$ in order to see improvement. More broadly, by providing exact non-asymptotic expressions for the variance of PPI++ under sample splitting, we aim to empower practitioners to transparently reason about the benefits of PPI++ in specific applications. In experiments, we illustrate that our theoretical findings hold on real-world datasets.