arXiv subjects

Ankur Moitra

Publications and source records attributed to Ankur Moitra.

At least 19 recordsLinked to original sources

arXiv2026

Honeycombs and Sums of Hermitian Matrices, Revisited

We give a new proof of the celebrated theorem of Knutson and Tao that the spectra of triples $A, B, A+B$ of Hermitian matrices exactly correspond to positions of boundary rays of honeycombs. Most importantly, our proof gives new insights into why honeycombs are related to Hermitian matrices in the first place. Our proof is axiomatic: We distill four essential properties shared by honeycombs and spectra of Hermitian triples, and show that any two objects sharing these four properties must be equivalent. In this way, we argue that honeycombs are `model organisms' for Hermitian triples: they are families of objects satisfying the same defining properties, but in more obvious ways.

math.CO↗

arXiv2026

Learning Gaussian Graphical Models from a Glauber Trajectory Without Mixing

We study the task of learning the structure of a $d$-sparse Gaussian graphical model on $n$ variables from a single trajectory of Glauber dynamics. Beyond algorithmic considerations, many applications present temporally correlated observations rather than i.i.d.\ samples. In the classical i.i.d.\ setting, under comparably general sparsity and minimum edge-strength assumptions, sublinear-in-$n$ sample guarantees are known, but achieving them in polynomial-time remains open. Motivated in part by this gap, we give a polynomial-time algorithm that recovers the conditional-independence graph from a single Glauber trajectory, with a trajectory-length guarantee that does not depend on the mixing time. Technically, our algorithm has three components. First, we estimate the conditional variances and rescale the trajectory to reduce to the unit-diagonal case, without changing the underlying graph. Second, we design a local edge test that extracts adjacency information from short update windows by isolating pairwise influence. Third, we aggregate these local statistics using a robust median-based estimator, and prove accuracy despite temporal dependence arising from a single trajectory.

cs.LG↗

arXiv2026

The Power of Test-Time Training for Approximate Sampling

Efficiently sampling from a complex probability distribution is a fundamental problem which has become increasingly pertinent in recent years with the rise of generative AI, as sophisticated sampling procedures from LLMs have been proposed to solve challenging reasoning problems. The efficacy of such sampling algorithms is limited, however, by the relationship between the LLM and the particular sampling task at hand, which has motivated the framework of test-time training (TTT). TTT works by updating a model's weights in response to partial generations and reward feedback received at inference time, thus adapting to the particular problem. In this work, we propose a formalization for TTT as the problem of producing a sample from a given probability measure $\mu^\star$ belonging to a known class ${F}$ of distributions, given an oracle $\hat \mu$ which yields approximate density estimates for $\mu^\star$. This is closely related to the problem of reducing sampling to approximate counting studied in seminal works of Jerrum, Valiant & Vazirani (1986) and Jerrum & Sinclair (1989): namely, when ${F}$ is the class of all distributions, it coincides exactly with the aforementioned counting-to-sampling reduction. In this paper, we first show a quadratic lower bound on the query complexity of sampling from $\mu^\star$ given query access to $\hat \mu$ (for sufficiently large classes ${F}$), thus showing that the random walk approach proposed by Jerrum & Sinclair (1989) and refined by Hayes & Sinclair (2010), is optimal. This answers an open question posed by Hayes & Sinclair. We then show that this lower bound can be circumvented if the size of ${F}$ is bounded appropriately. As we discuss, this latter result can be viewed as an abstraction of TTT, and thus represents a starting point for the development of a principled theoretical framework for TTT.

cs.DS↗

arXiv2026

The tractability landscape of diffusion alignment: regularization, rewards, and computational primitives

Inference-time reward alignment asks how to turn a pre-trained diffusion model with base law $p$ into a sampler that favors a reward $r$ while remaining close to $p$. Since there is no canonical distributional distance for this closeness constraint, different choices lead to different "reward-aligned" laws and, just as importantly, different algorithmic problems. We develop a primitive-based approach to reward alignment: rather than assuming arbitrary reward-aligned laws can be sampled, we ask which simple algorithmic primitives suffice to implement alignment for non-trivial reward classes. If closeness is measured in KL distance, the target law is $q(x) \propto p(x) \exp(\lambda^{-1}r(x))$. For this setting, we show that linear exponential tilts of the form $q(x)\propto p(x)\exp(\langle \theta, x \rangle)$ -- which according to recent work [MRR26] can be efficiently sampled from -- are a sufficient primitive for aligning to a very broad class of convex low-dimensional rewards. If closeness is measured in Wasserstein distance, the corresponding primitive is a proximal transport oracle: given $x$, solve $\mbox{argmax}_y \{r(y)- \lambda c(x,y)\}$. This oracle can be efficiently implemented for concave or low-dimensional Lipschitz rewards $r(x)=f(Ax)$. Together, these results illustrate that the choice of distribution distance for alignment affects the computational primitive and the tractable reward class.

cs.LG↗

arXiv2026

Learning $\mathsf{AC}^0$ Under Graphical Models

In a landmark result, Linial, Mansour and Nisan (J. ACM 1993) gave a quasipolynomial-time algorithm for learning constant-depth circuits given labeled i.i.d. samples under the uniform distribution. Their work has had a deep and lasting legacy in computational learning theory, in particular introducing the $\textit{low-degree algorithm}$. However, an important critique of many results and techniques in the area is the reliance on product structure, which is unlikely to hold in realistic settings. Obtaining similar learning guarantees for more natural correlated distributions has been a longstanding challenge in the field. In particular, we give quasipolynomial-time algorithms for learning $\mathsf{AC}^0$ substantially beyond the product setting, when the inputs come from any graphical model with polynomial growth that exhibits strong spatial mixing. The main technical challenge is in giving a workaround to Fourier analysis, which we do by showing how new sampling algorithms allow us to transfer statements about low-degree polynomial approximation under the uniform setting to graphical models. Our approach is general enough to extend to other well-studied function classes, like monotone functions and halfspaces.

cs.LG↗

arXiv2026

Steering diffusion models with quadratic rewards: a fine-grained analysis

Inference-time algorithms are an emerging paradigm in which pre-trained models are used as subroutines to solve downstream tasks. Such algorithms have been proposed for tasks ranging from inverse problems and guided image generation to reasoning. However, the methods currently deployed in practice are heuristics with a variety of failure modes -- and we have very little understanding of when these heuristics can be efficiently improved. In this paper, we consider the task of sampling from a reward-tilted diffusion model -- that is, sampling from $p^{\star}(x) \propto p(x) \exp(r(x))$ -- given a reward function $r$ and pre-trained diffusion oracle for $p$. We provide a fine-grained analysis of the computational tractability of this task for quadratic rewards $r(x) = x^\top A x + b^\top x$. We show that linear-reward tilts are always efficiently sampleable -- a simple result that seems to have gone unnoticed in the literature. We use this as a building block, along with a conceptually new ingredient -- the Hubbard-Stratonovich transform -- to provide an efficient algorithm for sampling from low-rank positive-definite quadratic tilts, i.e. $r(x) = x^\top A x$ where $A$ is positive-definite and of rank $O(1)$. For negative-definite tilts, i.e. $r(x) = - x^\top A x$ where $A$ is positive-definite, we prove that the problem is intractable even if $A$ is of rank 1 (albeit with exponentially-large entries).

cs.LG↗

arXiv2026

Subliminal Effects in Your Data: A General Mechanism via Log-Linearity

Training modern large language models (LLMs) has become a veritable smorgasbord of algorithms and datasets designed to elicit particular behaviors, making it critical to develop techniques to understand the effects of datasets on the model's properties. This is exacerbated by recent experiments that show datasets can transmit signals that are not directly observable from individual datapoints, posing a conceptual challenge for dataset-centric understandings of LLM training and suggesting a missing fundamental account of such phenomena. Towards understanding such effects, inspired by recent work on the linear structure of LLMs, we uncover a general mechanism through which hidden subtexts can arise in generic datasets. We introduce Logit-Linear-Selection (LLS), a method that prescribes how to select subsets of a generic preference dataset to elicit a wide range of hidden effects. We apply LLS to discover subsets of real-world datasets so that models trained on them exhibit behaviors ranging from having specific preferences, to responding to prompts in a different language not present in the dataset, to taking on a different persona. Crucially, the effect persists for the selected subset, across models with varying architectures, supporting its generality and universality.

cs.LG↗

arXiv2025

Improved Pseudorandom Codes from Permuted Puzzles

Watermarks are an essential tool for identifying AI-generated content. Recently, Christ and Gunn (CRYPTO '24) introduced pseudorandom error-correcting codes (PRCs), which are equivalent to watermarks with strong robustness and quality guarantees. A PRC is a pseudorandom encryption scheme whose decryption algorithm tolerates a high rate of errors. Pseudorandomness ensures quality preservation of the watermark, and error tolerance of decryption translates to the watermark's ability to withstand modification of the content. In the short time since the introduction of PRCs, several works (NeurIPS '24, RANDOM '25, STOC '25) have proposed new constructions. Curiously, all of these constructions are vulnerable to quasipolynomial-time distinguishing attacks. Furthermore, all lack robustness to edits over a constant-sized alphabet, which is necessary for a meaningfully robust LLM watermark. Lastly, they lack robustness to adversaries who know the watermarking detection key. Until now, it was not clear whether any of these properties was achievable individually, let alone together. We construct pseudorandom codes that achieve all of the above: plausible subexponential pseudorandomness security, robustness to worst-case edits over a binary alphabet, and robustness against even computationally unbounded adversaries that have the detection key. Pseudorandomness rests on a new assumption that we formalize, the permuted codes conjecture, which states that a distribution of permuted noisy codewords is pseudorandom. We show that this conjecture is implied by the permuted puzzles conjecture used previously to construct doubly efficient private information retrieval. To give further evidence, we show that the conjecture holds against a broad class of simple distinguishers, including read-once branching programs.

cs.CR↗

arXiv2025

A Dobrushin condition for quantum Markov chains: Rapid mixing and conditional mutual information at high temperature

A central challenge in quantum physics is to understand the structural properties of many-body systems, both in equilibrium and out of equilibrium. For classical systems, we have a unified perspective which connects structural properties of systems at thermal equilibrium to the Markov chain dynamics that mix to them. We lack such a perspective for quantum systems: there is no framework to translate the quantitative convergence of the Markovian evolution into strong structural consequences. We develop a general framework that brings the breadth and flexibility of the classical theory to quantum Gibbs states at high temperature. At its core is a natural quantum analog of a Dobrushin condition; whenever this condition holds, a concise path-coupling argument proves rapid mixing for the corresponding Markovian evolution. The same machinery bridges dynamic and structural properties: rapid mixing yields exponential decay of conditional mutual information (CMI) without restrictions on the size of the probed subsystems, resolving a central question in the theory of open quantum systems. Our key technical insight is an optimal transport viewpoint which couples quantum dynamics to a linear differential equation, enabling precise control over how local deviations from equilibrium propagate to distant sites.

quant-ph↗

arXiv2025

Taming Imperfect Process Verifiers: A Sampling Perspective on Backtracking

Test-time algorithms that combine the generative power of language models with process verifiers that assess the quality of partial generations offer a promising lever for eliciting new reasoning capabilities, but the algorithmic design space and computational scaling properties of such approaches are still opaque, and their benefits are far from apparent when one accounts for the cost of learning a high-quality verifier. Our starting point is the observation that seemingly benign errors in a learned verifier can lead to catastrophic failures for standard decoding techniques due to error amplification during the course of generation. We then ask: can this be improved with more sophisticated decoding strategies? We introduce a new process-guided test-time sampling algorithm, VGB, which uses theoretically grounded backtracking to achieve provably better robustness to verifier errors. VGB interprets autoregressive generation as a random walk on a tree of partial generations, with transition probabilities guided by the process verifier and base model; crucially, backtracking occurs probabilistically. This process generalizes the seminal Sinclair-Jerrum random walk (Sinclair & Jerrum, 1989) from the literature on approximate counting and sampling in theoretical computer science, and a conceptual contribution of our work is to highlight parallels with this literature. Empirically, we demonstrate on both synthetic and real language modeling tasks that VGB outperforms baselines on a variety of metrics.

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↗

arXiv2024

Overcomplete Tensor Decomposition via Koszul-Young Flattenings

Motivated by connections between algebraic complexity lower bounds and tensor decompositions, we investigate Koszul-Young flattenings, which are the main ingredient in recent lower bounds for matrix multiplication. Based on this tool we give a new algorithm for decomposing an $n_1 \times n_2 \times n_3$ tensor as the sum of a minimal number of rank-1 terms, and certifying uniqueness of this decomposition. For $n_1 \le n_2 \le n_3$ with $n_1 \to \infty$ and $n_3/n_2 = O(1)$, our algorithm is guaranteed to succeed when the tensor rank is bounded by $r \le (1-\epsilon)(n_2 + n_3)$ for an arbitrary $\epsilon > 0$, provided the tensor components are generically chosen. For any fixed $\epsilon$, the runtime is polynomial in $n_3$. When $n_2 = n_3 = n$, our condition on the rank gives a factor-of-2 improvement over the classical simultaneous diagonalization algorithm, which requires $r \le n$, and also improves on the recent algorithm of Koiran (2024) which requires $r \le 4n/3$. It also improves on the PhD thesis of Persu (2018) which solves rank detection for $r \leq 3n/2$. We complement our upper bounds by showing limitations, in particular that no flattening of the style we consider can surpass rank $n_2 + n_3$. Furthermore, for $n \times n \times n$ tensors, we show that an even more general class of degree-$d$ polynomial flattenings cannot surpass rank $Cn$ for a constant $C = C(d)$. This suggests that for tensor decompositions, the case of generic components may be fundamentally harder than that of random components, where efficient decomposition is possible even in highly overcomplete settings.

cs.DS↗

arXiv2024

Model Stealing for Any Low-Rank Language Model

Model stealing, where a learner tries to recover an unknown model via carefully chosen queries, is a critical problem in machine learning, as it threatens the security of proprietary models and the privacy of data they are trained on. In recent years, there has been particular interest in stealing large language models (LLMs). In this paper, we aim to build a theoretical understanding of stealing language models by studying a simple and mathematically tractable setting. We study model stealing for Hidden Markov Models (HMMs), and more generally low-rank language models. We assume that the learner works in the conditional query model, introduced by Kakade, Krishnamurthy, Mahajan and Zhang. Our main result is an efficient algorithm in the conditional query model, for learning any low-rank distribution. In other words, our algorithm succeeds at stealing any language model whose output distribution is low-rank. This improves upon the previous result by Kakade, Krishnamurthy, Mahajan and Zhang, which also requires the unknown distribution to have high "fidelity", a property that holds only in restricted cases. There are two key insights behind our algorithm: First, we represent the conditional distributions at each timestep by constructing barycentric spanners among a collection of vectors of exponentially large dimension. Second, for sampling from our representation, we iteratively solve a sequence of convex optimization problems that involve projection in relative entropy to prevent compounding of errors over the length of the sequence. This is an interesting example where, at least theoretically, allowing a machine learning model to solve more complex problems at inference time can lead to drastic improvements in its performance.

cs.LG↗

arXiv2024

Towards characterizing the value of edge embeddings in Graph Neural Networks

Graph neural networks (GNNs) are the dominant approach to solving machine learning problems defined over graphs. Despite much theoretical and empirical work in recent years, our understanding of finer-grained aspects of architectural design for GNNs remains impoverished. In this paper, we consider the benefits of architectures that maintain and update edge embeddings. On the theoretical front, under a suitable computational abstraction for a layer in the model, as well as memory constraints on the embeddings, we show that there are natural tasks on graphical models for which architectures leveraging edge embeddings can be much shallower. Our techniques are inspired by results on time-space tradeoffs in theoretical computer science. Empirically, we show architectures that maintain edge embeddings almost always improve on their node-based counterparts -- frequently significantly so in topologies that have ``hub'' nodes.

cs.LG↗

arXiv2024

Bypassing the Noisy Parity Barrier: Learning Higher-Order Markov Random Fields from Dynamics

We consider the problem of learning graphical models, also known as Markov random fields (MRFs) from temporally correlated samples. As in many traditional statistical settings, fundamental results in the area all assume independent samples from the distribution. However, these samples generally will not directly correspond to more realistic observations from nature, which instead evolve according to some stochastic process. From the computational lens, even generating a single sample from the true MRF distribution is intractable unless $\mathsf{NP}=\mathsf{RP}$, and moreover, any algorithm to learn from i.i.d. samples requires prohibitive runtime due to hardness reductions to the parity with noise problem. These computational barriers for sampling and learning from the i.i.d. setting severely lessen the utility of these breakthrough results for this important task; however, dropping this assumption typically only introduces further algorithmic and statistical complexities. In this work, we surprisingly demonstrate that the direct trajectory data from a natural evolution of the MRF overcomes the fundamental computational lower bounds to efficient learning. In particular, we show that given a trajectory with $\widetilde{O}_k(n)$ site updates of an order $k$ MRF from the Glauber dynamics, a well-studied, natural stochastic process on graphical models, there is an algorithm that recovers the graph and the parameters in $\widetilde{O}_k(n^2)$ time. By contrast, all prior algorithms for learning order $k$ MRFs inherently suffer from $n^{\Theta(k)}$ runtime even in sparse instances due to the reductions to sparse parity with noise. Our results thus surprisingly show that this more realistic, but intuitively less tractable, model for MRFs actually leads to efficiency far beyond what is known and believed to be true in the traditional i.i.d. case.

cs.LG↗

arXiv2024

Linear Bellman Completeness Suffices for Efficient Online Reinforcement Learning with Few Actions

One of the most natural approaches to reinforcement learning (RL) with function approximation is value iteration, which inductively generates approximations to the optimal value function by solving a sequence of regression problems. To ensure the success of value iteration, it is typically assumed that Bellman completeness holds, which ensures that these regression problems are well-specified. We study the problem of learning an optimal policy under Bellman completeness in the online model of RL with linear function approximation. In the linear setting, while statistically efficient algorithms are known under Bellman completeness (e.g., Jiang et al. (2017); Zanette et al. (2020)), these algorithms all rely on the principle of global optimism which requires solving a nonconvex optimization problem. In particular, it has remained open as to whether computationally efficient algorithms exist. In this paper we give the first polynomial-time algorithm for RL under linear Bellman completeness when the number of actions is any constant.

cs.LG↗

arXiv2024

The Role of Inherent Bellman Error in Offline Reinforcement Learning with Linear Function Approximation

In this paper, we study the offline RL problem with linear function approximation. Our main structural assumption is that the MDP has low inherent Bellman error, which stipulates that linear value functions have linear Bellman backups with respect to the greedy policy. This assumption is natural in that it is essentially the minimal assumption required for value iteration to succeed. We give a computationally efficient algorithm which succeeds under a single-policy coverage condition on the dataset, namely which outputs a policy whose value is at least that of any policy which is well-covered by the dataset. Even in the setting when the inherent Bellman error is 0 (termed linear Bellman completeness), our algorithm yields the first known guarantee under single-policy coverage. In the setting of positive inherent Bellman error ${\varepsilon_{\mathrm{BE}}} > 0$, we show that the suboptimality error of our algorithm scales with $\sqrt{\varepsilon_{\mathrm{BE}}}$. Furthermore, we prove that the scaling of the suboptimality with $\sqrt{\varepsilon_{\mathrm{BE}}}$ cannot be improved for any algorithm. Our lower bound stands in contrast to many other settings in reinforcement learning with misspecification, where one can typically obtain performance that degrades linearly with the misspecification error.

cs.LG↗

arXiv2024

Edit Distance Robust Watermarks via Indexing Pseudorandom Codes

Motivated by the problem of detecting AI-generated text, we consider the problem of watermarking the output of language models with provable guarantees. We aim for watermarks which satisfy: (a) undetectability, a cryptographic notion introduced by Christ, Gunn & Zamir (2024) which stipulates that it is computationally hard to distinguish watermarked language model outputs from the model's actual output distribution; and (b) robustness to channels which introduce a constant fraction of adversarial insertions, substitutions, and deletions to the watermarked text. Earlier schemes could only handle stochastic substitutions and deletions, and thus we are aiming for a more natural and appealing robustness guarantee that holds with respect to edit distance. Our main result is a watermarking scheme which achieves both undetectability and robustness to edits when the alphabet size for the language model is allowed to grow as a polynomial in the security parameter. To derive such a scheme, we follow an approach introduced by Christ & Gunn (2024), which proceeds via first constructing pseudorandom codes satisfying undetectability and robustness properties analogous to those above; our key idea is to handle adversarial insertions and deletions by interpreting the symbols as indices into the codeword, which we call indexing pseudorandom codes. Additionally, our codes rely on weaker computational assumptions than used in previous work. Then we show that there is a generic transformation from such codes over large alphabets to watermarking schemes for arbitrary language models.

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