arXiv subjects

Elchanan Mossel

Publications and source records attributed to Elchanan Mossel.

At least 37 records · Page 2Linked to original sources

arXiv2026

A Theory of Online Learning with Autoregressive Chain-of-Thought Reasoning

Autoregressive generation lies at the heart of the mechanism of large language models. It can be viewed as the repeated application of a next-token generator: starting from an input string (prompt), the generator is applied for $M$ steps, and the last generated token is taken as the final output. [Joshi et al., 2025] proposed a PAC model for studying the learnability of the input-output maps arising from this process. We develop an online analogue of this framework, focusing on the mistake bound of learning the final output induced by an unknown next-token generator. We distinguish between two forms of feedback. In the End-to-End model, after each round the learner observes only the final token produced after $M$ autoregressive steps. In the Chain-of-Thought model, the learner is additionally shown the entire $M$-step trajectory. Our goal is to understand how the optimal mistake bound depends on the generation horizon $M$, and to what extent observing intermediate tokens can reduce this dependence. Our main results show that the online theory of autoregressive learning exhibits a qualitative picture analogous to the statistical one found by [Hanneke et al., 2026], but with a different scale of dependence on the generation horizon. In the End-to-End model, we prove a taxonomy of possible mistake-bound growth rates in the generation horizon $M$: essentially any rate between constant and logarithmic can arise. We further show that this logarithmic ceiling is unavoidable. In the Chain-of-Thought model, we show that access to the full generated trajectory eliminates the dependence on $M$ altogether. We also analyze autoregressive linear threshold classes, and prove optimal mistake bounds, as well as a new lower bound for the statistical setting. Along the way, our results resolve several questions left open by [Joshi et al., 2025].

cs.LG↗

arXiv2026

Some Theoretical Limitations of t-SNE

t-SNE has gained popularity as a dimension reduction technique, especially for visualizing data. It is well-known that all dimension reduction techniques may lose important features of the data. We provide a mathematical framework for understanding this loss for t-SNE by establishing a number of results in different scenarios showing how important features of data are lost by using t-SNE.

cs.LG↗

arXiv2026

Reconstructing the Geometry of Random Geometric Graphs

Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work, we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the manifold assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in $\mathbb{R}^N$. Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.

cs.LG↗

arXiv2026

Denoising distances beyond the volumetric barrier

We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise distance estimation has been fundamentally constrained by the volumetric barrier, namely the natural sample-spacing scale $n^{-1/d}$ coming from the fact that a generic point of the manifold typically lies at distance of order $n^{-1/d}$ from the nearest sampled point. In this paper, we introduce a novel approach, Orthogonal Ring Distance Estimation Routine (ORDER), which achieves a pointwise distance estimation precision of order $n^{-2/(d+5)}$ up to polylogarithmic factors in $n$ in polynomial time. This strictly beats the volumetric barrier for dimensions $d > 5$. As a consequence of obtaining pointwise precision better than $n^{-1/d}$, we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order $n^{-1/d}$. This matches the Wasserstein convergence rate of empirical measures, demonstrating that our reconstructed graph metric is asymptotically as good as having access to the full pairwise distance matrix of the sampled points. Our results are proven in a very general setting which includes general models of noisy pairwise distances, sparse random geometric graphs, and unknown connection probability functions.

stat.ML↗

arXiv2026

Sharp Threshold for the Convergence of Nonstationary Averaging

We study non-stationary averaging processes, where each term of a sequence is a weighted average of previous terms, namely $a_{n+1} = \sum_{j=1}^n p_n(j) a_j$. Our results extend classical theory in two distinct regimes. First, we prove a sharp threshold for convergence in the regime where the weights are bounded between two envelopes $(\log n)^{-α} \le np_n(\cdot) \leq (\log n)^β$. We show that the sequence necessarily converges when $α+ β/ 2 \leq 1$, while $α+ β/ 2 > 1$ the convergence can fail. Second, we study complementary fixed shape regime, when $p_n$ is obtained by a fixed limiting density on $(0,1)$. We show that under mild regularity assumptions, the sequence converges.

math.PR↗

arXiv2026

Why ReLU? A Bit-Model Dichotomy for Deep Network Training

Theoretical analyses of Empirical Risk Minimization (ERM) are standardly framed within the Real-RAM model of computation. In this setting, training even simple neural networks is known to be $\exists \mathbb{R}$-complete -- a complexity class believed to be harder than NP, that characterizes the difficulty of solving systems of polynomial inequalities over the real numbers. However, this algebraic framework diverges from the reality of digital computation with finite-precision hardware. In this work, we analyze the theoretical complexity of ERM under a realistic bit-level model ($\mathsf{ERM}_{\text{bit}}$), where network parameters and inputs are constrained to be rational numbers with polynomially bounded bit-lengths. Under this model, we reveal a sharp dichotomy in tractability governed by the network's activation function. We prove that for deep networks with {\em any} polynomial activations with rational coefficients and degree at least $2$, the bit-complexity of training is severe: deciding $\mathsf{ERM}_{\text{bit}}$ is $\#P$-Hard, hence believed to be strictly harder than NP-complete problems. Furthermore, we show that determining the sign of a single partial derivative of the empirical loss function is intractable (unlikely in BPP), and deciding a specific bit in the gradient is $\#P$-Hard. This provides a complexity-theoretic perspective for the phenomenon of exploding and vanishing gradients. In contrast, we show that for piecewise-linear activations such as ReLU, the precision requirements remain manageable: $\mathsf{ERM}_{\text{bit}}$ is contained within NP (specifically NP-complete), and standard backpropagation runs in polynomial time. Our results demonstrate that finite-precision constraints are not merely implementation details but fundamental determinants of learnability.

cs.LG↗

arXiv2026

Detecting Mutual Excitations in Non-Stationary Hawkes Processes

We consider the problem of learning the network of mutual excitations (i.e., the dependency graph) in a non-stationary, multivariate Hawkes process. We consider a general setting where baseline rates at each node are time-varying and delay kernels are not shift-invariant. Our main results show that if the dependency graph of an $n$-variate Hawkes process is sparse (i.e., it has a maximum degree that is bounded with respect to $n$), our algorithm accurately reconstructs it from data after observing the Hawkes process for $T = \mathrm{polylog}(n)$ time, with high probability. Our algorithm is computationally efficient, and provably succeeds in learning dependencies even if only a subset of time series are observed and event times are not precisely known.

math.ST↗

arXiv2025

Monotonicity, Topology, and Convexity of Recurrence in Random Walks

We consider non-homogeneous random walks on the two-dimensional positive quadrant $\mathbb{N}^2$ and the one-dimensional slab $\{0,1,\dots,k\}\times\mathbb{N}$. In the 1960's the following question was asked for $\mathbb{N}^2$: is it true if such a random walk $X$ is recurrent and $Y$ is another random walk that at every point is more likely to go down and more likely to go left than $Y$, then $Y$ is also recurrent? We provide an example showing that the answer is negative. We also show, via a coupling argument, that if either the random walk $X$ or $Y$ is sufficiently homogeneous then the answer is in fact positive. In addition, we show using the Rayleigh monotonicity principle that the analogous question for random walks on trees is positive. These results show that the subset of parameter space that yields recurrent random walks possesses some geometric properties, in this case the structure of an order ideal. Motivated by this perspective, we consider the more symmetric setting of homogeneous random walks on finitely generated abelian groups, and ask when this subset possesses other geometric properties, namely various topological properties and convexity. We answer some of these questions: in particular, we show that this subset is closed, and under a symmetric support condition, show it is path-connected and additionally show it is convex if and only if its effective dimension is at most 2. We also show its complement is in some sense typically path-connected but not convex. We finally propose some related open problems.

math.PR↗

arXiv2025

Shotgun assembly of random regular graphs

Mossel and Ross (2019) introduce the shotgun assembly problem for random graphs: what radius $R$ ensures that the random graph $G$ can be uniquely recovered from its list of rooted $R$-neighborhoods, with high probability? Here we consider this question for random regular graphs of fixed degree $d\ge3$. A result of Bollobás (1982) implies efficient recovery at $R = (1 + ε) \frac12 \log_{d-1}n$ with high probability -- moreover, this recovery algorithm uses only a summary of the distances in each neighborhood. We show that using the full neighborhood structure gives a sharper bound \[ R = \frac{\log n + \log\log n}{2\log(d-1)} + O(1)\,, \] which we prove is tight up to the $O(1)$ term. One consequence of our proof is that if $G,H$ are independent graphs where $G$ follows the random regular law, then with high probability the graphs are non-isomorphic; furthermore, this can be efficiently certified by testing the $R$-neighborhood list of $H$ against the $R$-neighborhood of a single adversarially chosen vertex of $G$.

math.PR↗

arXiv2025

Two Agents, One Prompt, and Your Weight

We investigate a quantitative variant of the classic Two Doors logic puzzle, in which the answer space is no longer binary, for example when the goal is to recover a numerical fact (such as one's true weight) rather than choose between two doors. The puzzle retains the original structure: one agent always tells the truth, the other always lies. Our central contribution is to identify a class of self-referential prompts that successfully extract the correct quantitative answer under minimal assumptions. We also explore how well does \texttt{ChatGPT} does in reasoning for this problem which is just a little bit out of distribution.

math.LO↗

arXiv2025

Optimal Low degree hardness for Broadcasting on Trees

Broadcasting on trees is a fundamental model from statistical physics that plays an important role in information theory, noisy computation and phylogenetic reconstruction within computational biology and linguistics. While this model permits efficient linear-time algorithms for the inference of the root from the leaves, recent work suggests that non-trivial computational complexity may be required for inference. The inference of the root state can be performed using the celebrated Belief Propagation (BP) algorithm, which achieves Bayes-optimal performance. Although BP runs in linear time using real arithmetic operations, recent research indicates that it requires non-trivial computational complexity using more refined complexity measures. Moitra, Mossel, and Sandon demonstrated such complexity by constructing a Markov chain for which estimating the root better than random guessing (for typical inputs) is $NC^1$-complete. Kohler and Mossel constructed chains where, for trees with $N$ leaves, achieving better-than-random root recovery requires polynomials of degree $N^{Ω(1)}$. The papers above raised the question of whether such complexity bounds hold generally below the celebrated Kesten-Stigum bound. In a recent work, Huang and Mossel established a general degree lower bound of $Ω(\log N)$ below the Kesten-Stigum bound. Specifically, they proved that any function expressed as a linear combination of functions of at most $O(log N)$ leaves has vanishing correlation with the root. In this work, we get an exponential improvement of this lower bound by establishing an $N^{Ω(1)}$ degree lower bound, for any broadcast process in the whole regime below the Kesten-Stigum bound.

math.PR↗

arXiv2025

Reconstructing Riemannian Metrics From Random Geometric Graphs

Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a probability that depends on their distance. In recent work of Huang, Jiradilok, and Mossel~\cite{HJM24}, the authors study the problem of reconstructing an embedded manifold form a random geometric graph sampled from the manifold, where edge probabilities depend monotonically on the Euclidean distance between the embedded points. They show that, under mild regularity assumptions on the manifold, the sampling measure, and the connection probability function, it is possible to recover the pairwise Euclidean distances of the embedded sampled points up to a vanishing error as the number of vertices grows. In this work we consider a similar and arguably more natural problem where the metric is the Riemannian metric on the manifold. Again points are sampled from the manifold and a random graph is generated where the connection probability is monotone in the Riemannian distance. Perhaps surprisingly we obtain stronger results in this setup. Unlike the previous work that only considered dense graph we provide reconstruction algorithms from sparse graphs with average degree $n^{1/2}{\rm polylog}(n)$, where $n$ denotes the number of vertices. Our algorithm is also a more efficient algorithm for distance reconstruction with improved error bounds. The running times of the algorithm is $O(n^2\,{\rm polylog}(n))$ which up to polylog factor matches the size of the input graph. Our distance error also nearly matches the volumetric lower bounds for distance estimation.

math.PR↗

arXiv2025

Comparison Theorems for the Mixing Times of Systematic and Random Scan Dynamics

A popular method for sampling from high-dimensional distributions is the \emph{Gibbs sampler}, which iteratively resamples sites from the conditional distribution of the desired measure given the values of the other coordinates. It is natural to ask to what extent does the order of site updates matter in the mixing time? Two natural choices are (i) standard, or \emph{random scan}, Glauber dynamics where the updated variable is chosen uniformly at random, and (ii) the \emph{systematic scan} dynamics where variables are updated in a fixed, cyclic order. We first show that for systems of dimension $n$, one round of the systematic scan dynamics has spectral gap at most a factor of order $n$ worse than the corresponding spectral gap of a single step of Glauber dynamics, tightening existing bounds in the literature by He, et al. [NeurIPS '16] and Chlebicka, Łatuszyński, and Miasodejow [Ann. Appl. Probab. '25]. The corresponding bound on mixing times is sharp even for simple spin systems by an explicit example of Roberts and Rosenthal [Int. J. Statist. Prob. '15]. We complement this with a converse statement: if all, or even just one scan order rapidly mixes, the Glauber dynamics has a polynomially related mixing time, resolving a question of Chlebicka, Łatuszyński, and Miasodejow.

math.PR↗

arXiv2025

Low-dimensional Functions are Efficiently Learnable under Randomly Biased Distributions

The problem of learning single index and multi index models has gained significant interest as a fundamental task in high-dimensional statistics. Many recent works have analysed gradient-based methods, particularly in the setting of isotropic data distributions, often in the context of neural network training. Such studies have uncovered precise characterisations of algorithmic sample complexity in terms of certain analytic properties of the target function, such as the leap, information, and generative exponents. These properties establish a quantitative separation between low and high complexity learning tasks. In this work, we show that high complexity cases are rare. Specifically, we prove that introducing a small random perturbation to the data distribution--via a random shift in the first moment--renders any Gaussian single index model as easy to learn as a linear function. We further extend this result to a class of multi index models, namely sparse Boolean functions, also known as Juntas.

cs.LG↗

arXiv2025

Better Models and Algorithms for Learning Ising Models from Dynamics

We study the problem of learning the structure and parameters of the Ising model, a fundamental model of high-dimensional data, when observing the evolution of an associated Markov chain. A recent line of work has studied the natural problem of learning when observing an evolution of the well-known Glauber dynamics [Bresler, Gamarnik, Shah, IEEE Trans. Inf. Theory 2018, Gaitonde, Mossel STOC 2024], which provides an arguably more realistic generative model than the classical i.i.d. setting. However, this prior work crucially assumes that all site update attempts are observed, \emph{even when this attempt does not change the configuration}: this strong observation model is seemingly essential for these approaches. While perhaps possible in restrictive contexts, this precludes applicability to most realistic settings where we can observe \emph{only} the stochastic evolution itself, a minimal and natural assumption for any process we might hope to learn from. However, designing algorithms that succeed in this more realistic setting has remained an open problem [Bresler, Gamarnik, Shah, IEEE Trans. Inf. Theory 2018, Gaitonde, Moitra, Mossel, STOC 2025]. In this work, we give the first algorithms that efficiently learn the Ising model in this much more natural observation model that only observes when the configuration changes. For Ising models with maximum degree $d$, our algorithm recovers the underlying dependency graph in time $\mathsf{poly}(d)\cdot n^2\log n$ and then the actual parameters in additional $\widetilde{O}(2^d n)$ time, which qualitatively matches the state-of-the-art even in the i.i.d. setting in a much weaker observation model. Our analysis holds more generally for a broader class of reversible, single-site Markov chains that also includes the popular Metropolis chain by leveraging more robust properties of reversible Markov chains.

cs.LG↗

arXiv2025

On Algorithmic Robustness of Corrupted Markov Chains

We study the algorithmic robustness of general finite Markov chains in terms of their stationary distributions to general, adversarial corruptions of the transition matrix. We show that for Markov chains admitting a spectral gap, variants of the \emph{PageRank} chain are robust in the sense that, given an \emph{arbitrary} corruption of the edges emanating from an $ε$-measure of the nodes, the PageRank distribution of the corrupted chain will be $\mathsf{poly}(\varepsilon)$ close in total variation to the original distribution under mild conditions on the restart distribution. Our work thus shows that PageRank serves as a simple regularizer against broad, realistic corruptions with algorithmic guarantees that are dimension-free and scale gracefully in terms of necessary and natural parameters.

math.PR↗

arXiv2025

Online Learning of Neural Networks

We study online learning of feedforward neural networks with the sign activation function that implement functions from the unit ball in $\mathbb{R}^d$ to a finite label set $\{1, \ldots, Y\}$. First, we characterize a margin condition that is sufficient and in some cases necessary for online learnability of a neural network: Every neuron in the first hidden layer classifies all instances with some margin $γ$ bounded away from zero. Quantitatively, we prove that for any net, the optimal mistake bound is at most approximately $\mathtt{TS}(d,γ)$, which is the $(d,γ)$-totally-separable-packing number, a more restricted variation of the standard $(d,γ)$-packing number. We complement this result by constructing a net on which any learner makes $\mathtt{TS}(d,γ)$ many mistakes. We also give a quantitative lower bound of approximately $\mathtt{TS}(d,γ) \geq \max\{1/(γ\sqrt{d})^d, d\}$ when $γ\geq 1/2$, implying that for some nets and input sequences every learner will err for $\exp(d)$ many times, and that a dimension-free mistake bound is almost always impossible. To remedy this inevitable dependence on $d$, it is natural to seek additional natural restrictions to be placed on the network, so that the dependence on $d$ is removed. We study two such restrictions. The first is the multi-index model, in which the function computed by the net depends only on $k \ll d$ orthonormal directions. We prove a mistake bound of approximately $(1.5/γ)^{k + 2}$ in this model. The second is the extended margin assumption. In this setting, we assume that all neurons (in all layers) in the network classify every ingoing input from previous layer with margin $γ$ bounded away from zero. In this model, we prove a mistake bound of approximately $(\log Y)/ γ^{O(L)}$, where L is the depth of the network.

stat.ML↗

arXiv2025

Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees

In this paper we continue to rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborová (2011) regarding the block model, in particular in the case of $q=3$ and $q=4$ communities. We prove that for $q=3$ and $q=4$ there is no computational-statistical gap if the average degree is above some constant by showing it is information theoretically impossible to detect below the Kesten-Stigum bound. The proof is based on showing that for the broadcast process on Galton-Watson trees, reconstruction is impossible for $q=3$ and $q=4$ if the average degree is sufficiently large. This improves on the result of Sly (2009), who proved similar results for regular trees for $q=3$. Our analysis of the critical case $q=4$ provides a detailed picture showing that the tightness of the Kesten-Stigum bound in the antiferromagnetic case depends on the average degree of the tree. We also prove that for $q\geq 5$, the Kestin-Stigum bound is not sharp. Our results prove conjectures of Decelle, Krzakala, Moore, Zdeborová (2011), Moore (2017), Abbe and Sandon (2018) and Ricci-Tersenghi, Semerjian, and Zdeborová (2019). Our proofs are based on a new general coupling of the tree and graph processes and on a refined analysis of the broadcast process on the tree.

math.PR↗