arXiv subjects

Elchanan Mossel

Publications and source records attributed to Elchanan Mossel.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotics for the harmonic descent chain and applications to critical beta-splitting trees

Motivated by the connection to a probabilistic model of phylogenetic trees introduced by Aldous, we study the recursive sequence governed by the rule $x_n = \sum_{i=1}^{n-1} \frac{1}{h_{n-1}(n-i)} x_i$ where $h_{n-1} = \sum_{j=1}^{n-1} 1/j$, known as the harmonic descent chain. While it is known that this sequence converges to an explicit limit $x$, not much is known about the rate of convergence. We first show that a class of recursive sequences including the above are decreasing and use this to bound the rate of convergence. Moreover, for the harmonic descent chain we prove the asymptotic $x_n - x = n^{-γ_* + o(1)}$ for an implicit exponent $γ_*$. As a consequence, we deduce central limit theorems for various statistics of the critical beta-splitting random tree. This answers a number of questions of Aldous, Janson, and Pittel.

math.PR↗

arXiv2026

Extremal subtrees of critical beta-splitting trees

We determine the most and least likely shapes for an instance of the critical beta-splitting tree via a connection to data compression and Huffman's minimum redundancy codes. This allows us to answer combinatorial questions about the distribution of clades posed by Aldous and Janson, stated as problem 7 in arXiv:2303.02529.

math.CO↗

arXiv2026

Consensus times for monotone aggregation dynamics

We study an asynchronous consensus dynamics on $N$ agents: at each step a uniformly chosen agent replaces its state by $f(Y_1,\dots,Y_r)$, where $f$ is a fixed monotone aggregation rule and $Y_1,\dots,Y_r$ are the states of $r$ agents sampled uniformly with replacement. Let $T$ be the first time at which all agents agree. Let $f:\{0,1\}^r\to\{0,1\}$ be monotone and non-constant. The expected consensus time is governed by the two \emph{endpoint degrees} $D_0(f)=\#\{i:f(e_i)=1\}$ and $D_1(f)=\#\{i:f(\mathbf{1}-e_i)=0\}$, where $e_i$ is the $i$th standard basis vector and $\mathbf{1}$ the all-ones vector. If $D_0(f)\ne 1$ and $D_1(f)\ne 1$, then $E[T]=Θ\bigl(N(1+\log d_f)\bigr)$, where $d_f$ counts the agents that must change state before the nearest attracting consensus is reached, so that $E[T]=O(N\log N)$ uniformly over initial states. If $f$ is a dictator, $E[T]$ is given by a voter-model formula and equals $Θ\bigl(N^2 Ent(p_0)\bigr)$ with absolute constants, uniformly over the initial state, where $Ent$ is the binary entropy and $p_0$ the initial fraction of agents in state $1$. Otherwise exactly one of $D_0(f),D_1(f)$ equals $1$ and the other equals $0$. The worst-case expected consensus time is then $Θ(N^{2-1/m})$, where $m\ge 2$ is the least number of coordinates that force the value $1$ in the {\em residual rule} of $f$, defined in the paper (of its dual, when $D_1(f)=1$).

math.PR↗

arXiv2026

Stochastic block models with many communities and the Kesten--Stigum bound

We study the inference of communities in stochastic block models with a growing number of communities. For block models with $n$ vertices and a fixed number of communities $q$, it was predicted in Decelle et al. (2011) that there are computationally efficient algorithms for recovering the communities above the Kesten--Stigum (KS) bound and that efficient recovery is impossible below the KS bound. This conjecture has since stimulated a lot of interest, with the achievability side proven in a line of research that culminated in the work of Abbe and Sandon (2018). Conversely, recent work by Sohn and Wein (2025) provides evidence for the hardness part using the low-degree paradigm. In this paper we investigate community recovery in the regime $q=q_n \to \infty$ as $n\to\infty$ where no such predictions exist. We show that efficient inference of communities remains possible above the KS bound. Furthermore, we show that recovery of block models is low-degree hard below the KS bound when the number of communities satisfies $q\ll \sqrt{n}$. Perhaps surprisingly, we find that when $q \gg \sqrt{n}$, there is an efficient algorithm based on non-backtracking walks for recovery even below the KS bound. We identify a new threshold and ask if it is the threshold for efficient recovery in this regime. Finally, we show that detection is easy and identify (up to a constant) the information-theoretic threshold for community recovery as the number of communities $q$ diverges. Our low-degree hardness results also naturally have consequences for graphon estimation, improving results of Luo and Gao (2024).

math.PR↗

arXiv2026

Stochastic Autoregressive Learning

Motivated by LLMs, which generate outputs by iteratively sampling from next-token distributions, we introduce a PAC-learning model for binary stochastic autoregressive learning. This generalizes the deterministic autoregressive learning framework of Joshi et al., COLT 2025. In our model, one fixed generator assigns a Bernoulli next-token distribution to every prompt string. Starting from an input prompt, a token is sampled and appended to the prompt; the same generator is then applied again to this expanded prompt; this procedure is repeated for $M$ steps. Three forms of supervision are considered: base one-step samples, chain-of-thought (CoT) samples that reveal full random trajectories of length $M$, and end-to-end (e2e) samples that reveal only the final token of length $M$ trajectories. For a generator class, we study the minimum number of samples $m_{base}(\varepsilon),m_{CoT}(\varepsilon), m_{e2e}(\varepsilon)$, resp., required to learn the one-step probabilities in the base model, and the final-token probability in the CoT and e2e models, under squared loss error~$\varepsilon$. We show that stochastic autoregressive learning fundamentally differs from the deterministic theory. At scale $\varepsilon$, there is no universal comparison between the three learning tasks: both $m_{CoT}/m_{base}$ and $m_{e2e}/m_{CoT}$ can be made simultaneously arbitrarily larger than $M/\varepsilon$, the natural analogue for the existing deterministic results. Nevertheless, after altering scales, for every class, CoT learning at scale $\varepsilon$ is upper-bounded by base learning at scale $\varepsilon/M^2$, whereas e2e learning at scale $\varepsilon$ is upper-bounded, up to logarithmic factors, by $(M/\varepsilon) m_{CoT}(Θ(\varepsilon))$. These dependencies and scales are essentially tight. We complement these bounds by studying dimension $d$ logistic functions in our model.

cs.LG↗

arXiv2026

Recovering Assignments with One-Sided Noise

We study the query complexity of recovering a planted assignment from a random constraint-satisfaction instance with one-sided noise. We consider the following 1-CNF recovery problem: an unknown binary string with $n/2$ ones and $n/2$ zeros is queried at individual variables. A query to a $1$-variable returns "$1$" with probability $p$ and "$0$" otherwise, while a $0$-variable always returns "$0$" (each query is a fresh noisy draw). The goal is to recover the binary string with probability at least $1 - δ$. While the naive counting argument may suggest a query complexity of $\log_2 \binom{n}{n/2}=Θ(n)$, we show that the query complexity is $(1+o(1))c(p) \frac{n}{2} \left( \log_2 n + \log_2(1/δ)\right)$, where $c(p) = \tfrac{1}{-\log_2(1-p)}$. We then study planted $k$-CNF satisfaction with one-sided noise. Each $k$-set containing a $1$-variable is included as a clause independently with probability $p$, and an algorithm may ask whether any given $k$-set is a clause. Unlike the $1$-CNF case, a clause-existence query is one-shot: each $k$-set either is or is not a clause, so repeating yields no new information. The model is one-sided because an observed clause certifies that at least one queried variable is assigned 1, whereas its absence does not certify all are assigned 0. The goal is to recover the planted assignment with probability at least $1 - δ$. The counting baseline is $Θ(n)$, yet we prove a query complexity of $(1+o(1))\,c(p,k)\, \frac{n}{2}\left( \log_2 n + \log_2(1/δ)\right)$, where $c(p,k) = \tfrac{1}{k(-\log_2(1-p))}$. These bounds are for adaptive algorithms. We also prove bounds for nonadaptive algorithms, showing that for fixed $p$, adaptivity gives a factor $\exp(Θ(k))$ improvement. Our results also imply lower bounds for noisy sorting of $\{0,1\}$-valued strings, and we study a variant of the model with negations.

cs.DS↗

arXiv2026

Learning and Testing Convex Functions

We consider the problems of \emph{learning} and \emph{testing} real-valued convex functions over Gaussian space. Despite the extensive study of function convexity across mathematics, statistics, and computer science, its learnability and testability have largely been examined only in discrete or restricted settings -- typically with respect to the Hamming distance, which is ill-suited for real-valued functions. In contrast, we study these problems in high dimensions under the standard Gaussian measure, assuming sample access to the function and a mild smoothness condition, namely Lipschitzness. A smoothness assumption is natural and, in fact, necessary even in one dimension: without it, convexity cannot be inferred from finitely many samples. As our main results, we give: - Learning Convex Functions: An agnostic proper learning algorithm for Lipschitz convex functions that achieves error $\varepsilon$ using $n^{O(1/\varepsilon^2)}$ samples, together with a complementary lower bound of $n^{\mathrm{poly}(1/\varepsilon)}$ samples in the \emph{correlational statistical query (CSQ)} model. - Testing Convex Functions: A tolerant (two-sided) tester for convexity of Lipschitz functions with the same sample complexity (as a corollary of our learning result), and a one-sided tester (which never rejects convex functions) using $O(\sqrt{n}/\varepsilon)^n$ samples.

cs.DS↗

arXiv2026

Denoising Distances in Metric Measure Spaces

Recent work studied the problem of finding clusters and denoising pairwise distances from noisy distances of points sampled on a manifold. We study the same problems in more general metric measure spaces under a lower mass condition. We give an algorithm that extracts large localized clusters around every sampled point, which can be used to denoise distances, with near-linear running time in the dense regime for fixed target distance error $r$. When the target distance error $r$ is allowed to vanish as $n\to\infty$, we identify the sharp information-theoretic scale for achieving distance error $r$, suggesting a statistical-computational gap for high-accuracy denoising beyond the Riemannian setting.

cs.CG↗

arXiv2026

Multiplayer Games of War

A recent paper by Bhatia, Chin, Mani, and Mossel (2026) defined stochastic processes modeling the game of War for {\em two players} with $n$ cards. That paper showed that these models, assuming uniform random decks, are equivalent to the Gambler's Ruin problem and therefore have an expected termination time of $Θ(n^2)$. In this paper, we generalize these models to {\em any number of players} $m$. We prove that the game with $m$ players is equivalent to a simple sticky random walk on an $(m-1)$-simplex; therefore, the termination time is the same as the absorption time of the simple sticky random walk. Unlike Gambler's Ruin, this absorption time has not been analyzed before. We show that the absorption time of the walk and the termination time of the game are both $Θ(n^2)$ for any number of players.

math.PR↗

arXiv2026

Depth Lower Bounds for ReLU Networks with Binary Inputs

We study the role of depth in ReLU networks with discrete (Boolean) inputs and real-valued outputs, complementing two established lines of work. For Boolean inputs, striking depth separation results were proven for $\mathsf{AC}^0$ but with threshold ($\mathsf{TC}^0$) or ReLU gates depth separation is only established for depth two vs. three. On the other hand, for {\em real-valued} functions and ReLU networks, Telgarsky's (2016) constructed a simple one variable class of functions which establishes separation at higher depths. In this paper we are interested to establish an all-depths depth separation for ReLU networks on $\{0,1\}^n$. We do so by exhibiting an explicit family of functions computable exactly by a ReLU network of depth $n+1$ and constant width, such that any ReLU network of depth $d$ and width $w$ computing the function exactly must satisfy $w^d = Ω(2^n)$; in particular, no network of depth $d = o(n/\log n)$ can compute it with width polynomial in $n$. We note that our lower bound relies on \emph{exact, infinite-accuracy} computation as an exponential precision truncation of the output is computable by a polynomial-size $\mathsf{TC}^0$ circuit.

cs.CC↗

arXiv2026

Online Realizable Regression and Applications for ReLU Networks

Realizable online regression can behave very differently from online classification. Even without any margin or stochastic assumptions, realizability may enforce horizon-free (finite) cumulative loss under metric-like losses, even when the analogous classification problem has an infinite mistake bound. We study realizable online regression in the adversarial model under losses that satisfy an approximate triangle inequality (approximate pseudo-metrics). Recent work of Attias et al. shows that the minimax realizable cumulative loss is characterized by the scaled Littlestone/online dimension $\mathbb{D}_{\mathrm{onl}}$, but this quantity can be difficult to analyze. Our main technical contribution is a generic potential method that upper bounds $\mathbb{D}_{\mathrm{onl}}$ by a concrete Dudley-type entropy integral that depends only on covering numbers of the hypothesis class under the induced sup pseudo-metric. We define an \emph{entropy potential} $Φ(\mathcal{H})=\int_{0}^{diam(\mathcal{H})} \log N(\mathcal{H},\varepsilon)\,d\varepsilon$, where $N(\mathcal{H},\varepsilon)$ is the $\varepsilon$-covering number of $\mathcal{H}$, and show that for every $c$-approximate pseudo-metric loss, $\mathbb{D}_{\mathrm{onl}}(\mathcal{H})\le O(c)\,Φ(\mathcal{H})$. In particular, polynomial metric entropy implies $Φ(\mathcal{H})<\infty$ and hence a horizon-free realizable cumulative-loss bound with transparent dependence on effective dimension. We illustrate the method on two families. We prove a sharp $q$-vs.-$d$ dichotomy for realizable online learning (finite and efficiently achievable $Θ_{d,q}(L^d)$ total loss for $L$-Lipschitz regression iff $q>d$, otherwise infinite), and for bounded-norm $k$-ReLU networks separate regression (finite loss, even $\widetilde O(k^2)$, and $O(1)$ for one ReLU) from classification (impossible already for $k=2,d=1$).

cs.LG↗

arXiv2026

Mathematical perspective on genetic algorithms with optimization guided operators

Recent work in ML applies genetic algorithms at inference time to iteratively improve solutions to optimization problems. The basic mutation and recombination operators involved are qualitatively different from those studied classically. Mutations are no longer random; an ML algorithm mutates a solution with the goal of improving an objective. Similarly, recombination is not based on random collages of parent solutions. Instead, it is an ML optimization-based operator whose goal is to synthesize improved solutions from its inputs. Thus, these mutation and recombination operators are more likely to improve the objective, but their computational cost is much higher. We introduce a general model of genetic algorithms and formulating optimization in this model as a query-complexity problem, using the language of reinforcement learning. We then study specialized models. We show that some optimization problems require generation, mutation, and recombination to be solved. We then obtain qualitatively tight algorithms for a family of problems within this framework that captures the nontrivial role of diversity in the solution pool, a key feature of practical ML genetic algorithms.

cs.NE↗

arXiv2026

LLMs, Reasoning and Plagiarism

Recent reports claim that Large Language Models (LLMs) derive new science and exhibit human-level general intelligence. Such claims are entangled with two different narratives about what LLMs do: one in which they are an engine of synthesis that genuinely reasons to new knowledge, and one in which they retrieve and re-emit the work of others without attribution. In the scientific setting these are best understood as a contrast between \emph{reasoning} and \emph{plagiarism}. Finding where the truth lies between these two narratives is very challenging, as central components of the model -- the training data and the interaction transcript -- remain opaque. Thus claims of LLM reasoning do not satisfy Popper's refutability principle. We propose guidelines for transparency and reproducibility that will allow reasoning claims to be studied using the scientific method. The dominance of the reasoning narrative, we suggest, is in practice encouraging plagiarism in the scientific literature; we discuss what might be done about it.

cs.CY↗

arXiv2026

A Hierarchical Language Model with Predictable Scaling Laws and Provable Benefits of Reasoning

We introduce a family of synthetic languages with hierarchical structure -- generated by a broadcast process on trees -- for which the role of context length and reasoning in autoregressive generation can be analyzed precisely. At the heart of our analytic approach is an \emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the \emph{Ising broadcast process} (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the \emph{coloring broadcast process} (a hard-constrained language) in the freezing regime, bounded-context autoregression produces sequences that, with high probability, are inconsistent with \emph{any} valid coloring of the underlying tree. Together these results imply an $Ω(n)$ lower bound on the context length required to faithfully sample length-$n$ sequences. In contrast, we prove that an autoregressive \emph{reasoning} model with only $Θ(\log n)$ working memory can sample exactly from the true language -- an exponential improvement. We confirm both the lower-bound predictions and the reasoning-based upper bound empirically with transformers trained on the synthetic language; the trained models track our asymptotic predictions quantitatively across a wide range of context sizes.

cs.LG↗

arXiv2026

The Benefits of Temporal Correlations: SGD Learns k-Juntas from Random Walks Efficiently

We study how temporal correlations in the data can make certain sparse learning problems efficiently learnable by gradient-based methods. Our focus is on Boolean k-juntas, a canonical sparse learning problem known to pose barriers for gradient-based methods under independent uniform samples. We show that this picture changes when the samples are generated by a lazy random walk on the hypercube. In this setting, the temporal dependencies can be exploited by a two-layer ReLU network trained using stylized-SGD with a temporal-difference loss, which compares target and predicted increments across consecutive samples. For every fixed k, the resulting sample complexity is essentially linear in the ambient dimension d. By contrast, we show that for large-batch gradient methods using standard convex pointwise losses, temporal correlations do not provide the same advantage.

cs.LG↗

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↗