arXiv subjects

Ankur Moitra

Publications and source records attributed to Ankur Moitra.

At least 37 records · Page 2Linked to original sources

arXiv2025

High-Temperature Gibbs States are Unentangled and Efficiently Preparable

We show that thermal states of local Hamiltonians are separable above a constant temperature. Specifically, for a local Hamiltonian $H$ on a graph with degree $\mathfrak{d}$, its Gibbs state at inverse temperature $β$, denoted by $ρ= e^{-βH}/ \operatorname{tr}(e^{-βH})$, is a classical distribution over product states for all $β< 1/(c\mathfrak{d})$, where $c$ is a constant. This proof of sudden death of thermal entanglement resolves the fundamental question of whether many-body systems can exhibit entanglement at high temperature. Moreover, we show that we can efficiently sample from the distribution over product states. In particular, for any $β< 1/( c \mathfrak{d}^2)$, we can prepare a state $\varepsilon$-close to $ρ$ in trace distance with a depth-one quantum circuit and $\operatorname{poly}(n, 1/\varepsilon)$ classical overhead.

quant-ph↗

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

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 $n^{Θ(k)}$ 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

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

Fast sampling of satisfying assignments from random $k$-SAT with applications to connectivity

We give a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $α=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(Θ(k))}$. Here $n$ is the number of variables and $m$ is the number of clauses. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $α\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting. Our main technical contribution is a $o_k(\log n )$ bound of the sum of influences in the $k$-SAT model which turns out to be robust against the presence of high-degree variables. This allows us to apply the spectral independence framework and obtain fast mixing results of a uniform-block Glauber dynamics on a carefully selected subset of the variables. The final key ingredient in our method is to take advantage of the sparsity of logarithmic-sized connected sets and the expansion properties of the random formula, and establish relevant connectivity properties of the set of satisfying assignments that enable the fast simulation of this Glauber dynamics. Our results also allow us to conclude that, with high probability, a random $k$-CNF formula with density at most $2^{0.227 k}$ has a giant component of solutions that are connected in a graph where solutions are adjacent if they have Hamming distance $O_k(\log n)$. We are also able to deduce looseness results for random $k$-CNFs in the same regime.

cs.DS↗

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

On Learning Parities with Dependent Noise

In this expository note we show that the learning parities with noise (LPN) assumption is robust to weak dependencies in the noise distribution of small batches of samples. This provides a partial converse to the linearization technique of [AG11]. The material in this note is drawn from a recent work by the authors [GMR24], where the robustness guarantee was a key component in a cryptographic separation between reinforcement learning and supervised learning.

cs.CR↗

arXiv2024

Exploration is Harder than Prediction: Cryptographically Separating Reinforcement Learning from Supervised Learning

Supervised learning is often computationally easy in practice. But to what extent does this mean that other modes of learning, such as reinforcement learning (RL), ought to be computationally easy by extension? In this work we show the first cryptographic separation between RL and supervised learning, by exhibiting a class of block MDPs and associated decoding functions where reward-free exploration is provably computationally harder than the associated regression problem. We also show that there is no computationally efficient algorithm for reward-directed RL in block MDPs, even when given access to an oracle for this regression problem. It is known that being able to perform regression in block MDPs is necessary for finding a good policy; our results suggest that it is not sufficient. Our separation lower bound uses a new robustness property of the Learning Parities with Noise (LPN) hardness assumption, which is crucial in handling the dependent nature of RL data. We argue that separations and oracle lower bounds, such as ours, are a more meaningful way to prove hardness of learning because the constructions better reflect the practical reality that supervised learning by itself is often not the computational bottleneck.

cs.LG↗

arXiv2024

Strong spatial mixing for colorings on trees and its algorithmic applications

Strong spatial mixing (SSM) is an important quantitative notion of correlation decay for Gibbs distributions arising in statistical physics, probability theory, and theoretical computer science. A longstanding conjecture is that the uniform distribution on proper $q$-colorings on a $Δ$-regular tree exhibits SSM whenever $q \ge Δ+1$. Moreover, it is widely believed that as long as SSM holds on bounded-degree trees with $q$ colors, one would obtain an efficient sampler for $q$-colorings on all bounded-degree graphs via simple Markov chain algorithms. It is surprising that such a basic question is still open, even on trees, but then again it also highlights how much we still have to learn about random colorings. In this paper, we show the following: (1) For any $Δ\ge 3$, SSM holds for random $q$-colorings on trees of maximum degree $Δ$ whenever $q \ge Δ+ 3$. Thus we almost fully resolve the aforementioned conjecture. Our result substantially improves upon the previously best bound which requires $q \ge 1.59Δ+γ^*$ for an absolute constant $γ^* > 0$. (2) For any $Δ\ge 3$ and girth $g = Ω_Δ(1)$, we establish optimal mixing of the Glauber dynamics for $q$-colorings on graphs of maximum degree $Δ$ and girth $g$ whenever $q \ge Δ+3$. Our approach is based on a new general reduction from spectral independence on large-girth graphs to SSM on trees that is of independent interest. Using the same techniques, we also prove near-optimal bounds on weak spatial mixing (WSM), a closely-related notion to SSM, for the antiferromagnetic Potts model on trees.

cs.DS↗

arXiv2023

Classification Under Misspecification: Halfspaces, Generalized Linear Models, and Connections to Evolvability

In this paper we revisit some classic problems on classification under misspecification. In particular, we study the problem of learning halfspaces under Massart noise with rate $η$. In a recent work, Diakonikolas, Goulekakis, and Tzamos resolved a long-standing problem by giving the first efficient algorithm for learning to accuracy $η+ ε$ for any $ε> 0$. However, their algorithm outputs a complicated hypothesis, which partitions space into $\text{poly}(d,1/ε)$ regions. Here we give a much simpler algorithm and in the process resolve a number of outstanding open questions: (1) We give the first proper learner for Massart halfspaces that achieves $η+ ε$. We also give improved bounds on the sample complexity achievable by polynomial time algorithms. (2) Based on (1), we develop a blackbox knowledge distillation procedure to convert an arbitrarily complex classifier to an equally good proper classifier. (3) By leveraging a simple but overlooked connection to evolvability, we show any SQ algorithm requires super-polynomially many queries to achieve $\mathsf{OPT} + ε$. Moreover we study generalized linear models where $\mathbb{E}[Y|\mathbf{X}] = σ(\langle \mathbf{w}^*, \mathbf{X}\rangle)$ for any odd, monotone, and Lipschitz function $σ$. This family includes the previously mentioned halfspace models as a special case, but is much richer and includes other fundamental models like logistic regression. We introduce a challenging new corruption model that generalizes Massart noise, and give a general algorithm for learning in this setting. Our algorithms are based on a small set of core recipes for learning to classify in the presence of misspecification. Finally we study our algorithm for learning halfspaces under Massart noise empirically and find that it exhibits some appealing fairness properties.

cs.LG↗

arXiv2023

Exploring and Learning in Sparse Linear MDPs without Computationally Intractable Oracles

The key assumption underlying linear Markov Decision Processes (MDPs) is that the learner has access to a known feature map $ϕ(x, a)$ that maps state-action pairs to $d$-dimensional vectors, and that the rewards and transitions are linear functions in this representation. But where do these features come from? In the absence of expert domain knowledge, a tempting strategy is to use the ``kitchen sink" approach and hope that the true features are included in a much larger set of potential features. In this paper we revisit linear MDPs from the perspective of feature selection. In a $k$-sparse linear MDP, there is an unknown subset $S \subset [d]$ of size $k$ containing all the relevant features, and the goal is to learn a near-optimal policy in only poly$(k,\log d)$ interactions with the environment. Our main result is the first polynomial-time algorithm for this problem. In contrast, earlier works either made prohibitively strong assumptions that obviated the need for exploration, or required solving computationally intractable optimization problems. Along the way we introduce the notion of an emulator: a succinct approximate representation of the transitions that suffices for computing certain Bellman backups. Since linear MDPs are a non-parametric model, it is not even obvious whether polynomial-sized emulators exist. We show that they do exist and can be computed efficiently via convex programming. As a corollary of our main result, we give an algorithm for learning a near-optimal policy in block MDPs whose decoding function is a low-depth decision tree; the algorithm runs in quasi-polynomial time and takes a polynomial number of samples. This can be seen as a reinforcement learning analogue of classic results in computational learning theory. Furthermore, it gives a natural model where improving the sample complexity via representation learning is computationally feasible.

cs.LG↗

arXiv2023

Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems

Recently Chen and Poor initiated the study of learning mixtures of linear dynamical systems. While linear dynamical systems already have wide-ranging applications in modeling time-series data, using mixture models can lead to a better fit or even a richer understanding of underlying subpopulations represented in the data. In this work we give a new approach to learning mixtures of linear dynamical systems that is based on tensor decompositions. As a result, our algorithm succeeds without strong separation conditions on the components, and can be used to compete with the Bayes optimal clustering of the trajectories. Moreover our algorithm works in the challenging partially-observed setting. Our starting point is the simple but powerful observation that the classic Ho-Kalman algorithm is a close relative of modern tensor decomposition methods for learning latent variable models. This gives us a playbook for how to extend it to work with more complicated generative models.

cs.LG↗

arXiv2023

Provable benefits of score matching

Score matching is an alternative to maximum likelihood (ML) for estimating a probability distribution parametrized up to a constant of proportionality. By fitting the ''score'' of the distribution, it sidesteps the need to compute this constant of proportionality (which is often intractable). While score matching and variants thereof are popular in practice, precise theoretical understanding of the benefits and tradeoffs with maximum likelihood -- both computational and statistical -- are not well understood. In this work, we give the first example of a natural exponential family of distributions such that the score matching loss is computationally efficient to optimize, and has a comparable statistical efficiency to ML, while the ML loss is intractable to optimize using a gradient-based method. The family consists of exponentials of polynomials of fixed degree, and our result can be viewed as a continuous analogue of recent developments in the discrete setting. Precisely, we show: (1) Designing a zeroth-order or first-order oracle for optimizing the maximum likelihood loss is NP-hard. (2) Maximum likelihood has a statistical efficiency polynomial in the ambient dimension and the radius of the parameters of the family. (3) Minimizing the score matching loss is both computationally and statistically efficient, with complexity polynomial in the ambient dimension.

cs.LG↗

arXiv2023

The Power of an Adversary in Glauber Dynamics

Glauber dynamics are a natural model of dynamics of dependent systems. While originally introduced in statistical physics, they have found important applications in the study of social networks, computer vision and other domains. In this work, we introduce a model of corrupted Glauber dynamics whereby instead of updating according to the prescribed conditional probabilities, some of the vertices and their updates are controlled by an adversary. We study the effect of such corruptions on global features of the system. Among the questions we study are: How many nodes need to be controlled in order to change the average statistics of the system in polynomial time? And how many nodes are needed to obstruct approximate convergence of the dynamics? Our results can be viewed as studying the robustness of classical sampling methods and are thus related to robust inference. The proofs connect to classical theory of Glauber dynamics from statistical physics.

math.PR↗

arXiv2023

Robust Model Selection and Nearly-Proper Learning for GMMs

In learning theory, a standard assumption is that the data is generated from a finite mixture model. But what happens when the number of components is not known in advance? The problem of estimating the number of components, also called model selection, is important in its own right but there are essentially no known efficient algorithms with provable guarantees let alone ones that can tolerate adversarial corruptions. In this work, we study the problem of robust model selection for univariate Gaussian mixture models (GMMs). Given $\textsf{poly}(k/ε)$ samples from a distribution that is $ε$-close in TV distance to a GMM with $k$ components, we can construct a GMM with $\widetilde{O}(k)$ components that approximates the distribution to within $\widetilde{O}(ε)$ in $\textsf{poly}(k/ε)$ time. Thus we are able to approximately determine the minimum number of components needed to fit the distribution within a logarithmic factor. Prior to our work, the only known algorithms for learning arbitrary univariate GMMs either output significantly more than $k$ components (e.g. $k/ε^2$ components for kernel density estimates) or run in time exponential in $k$. Moreover, by adapting our techniques we obtain similar results for reconstructing Fourier-sparse signals.

cs.DS↗

arXiv2023

A New Approach to Learning Linear Dynamical Systems

Linear dynamical systems are the foundational statistical model upon which control theory is built. Both the celebrated Kalman filter and the linear quadratic regulator require knowledge of the system dynamics to provide analytic guarantees. Naturally, learning the dynamics of a linear dynamical system from linear measurements has been intensively studied since Rudolph Kalman's pioneering work in the 1960's. Towards these ends, we provide the first polynomial time algorithm for learning a linear dynamical system from a polynomial length trajectory up to polynomial error in the system parameters under essentially minimal assumptions: observability, controllability, and marginal stability. Our algorithm is built on a method of moments estimator to directly estimate Markov parameters from which the dynamics can be extracted. Furthermore, we provide statistical lower bounds when our observability and controllability assumptions are violated.

math.OC↗

arXiv2022

Distilling Model Failures as Directions in Latent Space

Existing methods for isolating hard subpopulations and spurious correlations in datasets often require human intervention. This can make these methods labor-intensive and dataset-specific. To address these shortcomings, we present a scalable method for automatically distilling a model's failure modes. Specifically, we harness linear classifiers to identify consistent error patterns, and, in turn, induce a natural representation of these failure modes as directions within the feature space. We demonstrate that this framework allows us to discover and automatically caption challenging subpopulations within the training dataset. Moreover, by combining our framework with off-the-shelf diffusion models, we can generate images that are especially challenging for the analyzed model, and thus can be used to perform synthetic data augmentation that helps remedy the model's failure modes. Code available at https://github.com/MadryLab/failure-directions

cs.LG↗