arXiv subjects

Ankur Moitra

Publications and source records attributed to Ankur Moitra.

At least 55 records · Page 3Linked to original sources

arXiv2022

Minimax Rates for Robust Community Detection

In this work, we study the problem of community detection in the stochastic block model with adversarial node corruptions. Our main result is an efficient algorithm that can tolerate an $ε$-fraction of corruptions and achieves error $O(ε) + e^{-\frac{C}{2} (1 \pm o(1))}$ where $C = (\sqrt{a} - \sqrt{b})^2$ is the signal-to-noise ratio and $a/n$ and $b/n$ are the inter-community and intra-community connection probabilities respectively. These bounds essentially match the minimax rates for the SBM without corruptions. We also give robust algorithms for $\mathbb{Z}_2$-synchronization. At the heart of our algorithm is a new semidefinite program that uses global information to robustly boost the accuracy of a rough clustering. Moreover, we show that our algorithms are doubly-robust in the sense that they work in an even more challenging noise model that mixes adversarial corruptions with unbounded monotone changes, from the semi-random model.

cs.DS↗

arXiv2022

Robust Voting Rules from Algorithmic Robust Statistics

Maximum likelihood estimation furnishes powerful insights into voting theory, and the design of voting rules. However the MLE can usually be badly corrupted by a single outlying sample. This means that a single voter or a group of colluding voters can vote strategically and drastically affect the outcome. Motivated by recent progress in algorithmic robust statistics, we revisit the fundamental problem of estimating the central ranking in a Mallows model, but ask for an estimator that is provably robust, unlike the MLE. Our main result is an efficiently computable estimator that achieves nearly optimal robustness guarantees. In particular the robustness guarantees are dimension-independent in the sense that our overall accuracy does not depend on the number of alternatives being ranked. As an immediate consequence, we show that while the landmark Gibbard-Satterthwaite theorem tells us a strong impossiblity result about designing strategy-proof voting rules, there are quantitatively strong ways to protect against large coalitions if we assume that the remaining voters voters are honest and their preferences are sampled from a Mallows model. Our work also makes technical contributions to algorithmic robust statistics by designing new spectral filtering techniques that can exploit the intricate combinatorial dependencies in the Mallows model.

cs.DS↗

arXiv2022

From algorithms to connectivity and back: finding a giant component in random k-SAT

We take an algorithmic approach to studying the solution space geometry of relatively sparse random and bounded degree $k$-CNFs for large $k$. In the course of doing so, we establish that with high probability, a random $k$-CNF $Φ$ with $n$ variables and clause density $α= m/n \lesssim 2^{k/6}$ has a giant component of solutions that are connected in a graph where solutions are adjacent if they have Hamming distance $O_k(\log n)$ and that a similar result holds for bounded degree $k$-CNFs at similar densities. We are also able to deduce looseness results for random and bounded degree $k$-CNFs in a similar regime. Although our main motivation was understanding the geometry of the solution space, our methods have algorithmic implications. Towards that end, we construct an idealized block dynamics that samples solutions from a random $k$-CNF $Φ$ with density $α= m/n \lesssim 2^{k/52}$. We show this Markov chain can with high probability be implemented in polynomial time and by leveraging spectral independence, we also observe that it mixes relatively fast, giving a polynomial time algorithm to with high probability sample a uniformly random solution to a random $k$-CNF. Our work suggests that the natural route to pinning down when a giant component exists is to develop sharper algorithms for sampling solutions in random $k$-CNFs.

cs.DS↗

arXiv2022

Learning in Observable POMDPs, without Computationally Intractable Oracles

Much of reinforcement learning theory is built on top of oracles that are computationally hard to implement. Specifically for learning near-optimal policies in Partially Observable Markov Decision Processes (POMDPs), existing algorithms either need to make strong assumptions about the model dynamics (e.g. deterministic transitions) or assume access to an oracle for solving a hard optimistic planning or estimation problem as a subroutine. In this work we develop the first oracle-free learning algorithm for POMDPs under reasonable assumptions. Specifically, we give a quasipolynomial-time end-to-end algorithm for learning in "observable" POMDPs, where observability is the assumption that well-separated distributions over states induce well-separated distributions over observations. Our techniques circumvent the more traditional approach of using the principle of optimism under uncertainty to promote exploration, and instead give a novel application of barycentric spanners to constructing policy covers.

cs.LG↗

arXiv2022

Provably Auditing Ordinary Least Squares in Low Dimensions

Measuring the stability of conclusions derived from Ordinary Least Squares linear regression is critically important, but most metrics either only measure local stability (i.e. against infinitesimal changes in the data), or are only interpretable under statistical assumptions. Recent work proposes a simple, global, finite-sample stability metric: the minimum number of samples that need to be removed so that rerunning the analysis overturns the conclusion, specifically meaning that the sign of a particular coefficient of the estimated regressor changes. However, besides the trivial exponential-time algorithm, the only approach for computing this metric is a greedy heuristic that lacks provable guarantees under reasonable, verifiable assumptions; the heuristic provides a loose upper bound on the stability and also cannot certify lower bounds on it. We show that in the low-dimensional regime where the number of covariates is a constant but the number of samples is large, there are efficient algorithms for provably estimating (a fractional version of) this metric. Applying our algorithms to the Boston Housing dataset, we exhibit regression analyses where we can estimate the stability up to a factor of $3$ better than the greedy heuristic, and analyses where we can certify stability to dropping even a majority of the samples.

stat.ML↗

arXiv2022

Algorithms from Invariants: Smoothed Analysis of Orbit Recovery over $SO(3)$

In this work we study orbit recovery over $SO(3)$, where the goal is to recover a function on the sphere from noisy, randomly rotated copies of it. We assume that the function is a linear combination of low-degree spherical harmonics. This is a natural abstraction for the problem of recovering the three-dimensional structure of a molecule through cryo-electron tomography. For provably learning the parameters of a generative model, the method of moments is the standard workhorse of theoretical machine learning. It turns out that there is a natural incarnation of the method of moments for orbit recovery based on invariant theory. Bandeira et al. [BBSK+18] used invariant theory to give tight bounds on the sample complexity in terms of the noise level. However many of the key challenges remain: Can we prove bounds on the sample complexity that are polynomial in $n$, the dimension of the signal? The bounds in [BBSK+18] hide constants that have an unspecified dependence on $n$ and only hold in the limit as $σ^2 \rightarrow \infty$ where $σ^2$ is the variance of the noise. Moreover can we give efficient algorithms? We revisit these challenges from the perspective of smoothed analysis, where we assume that the coefficients of the signal, in the basis of spherical harmonics, are subject to small random perturbations. Our main result is a quasi-polynomial time algorithm for orbit recovery over $SO(3)$ in this model. Our approach is based on frequency marching, which sequentially solves linear systems to find higher degree coefficients. Our main technical contribution is to show that these linear systems have unique solutions, are well-conditioned, and that the error can be made to compound over at most a logarithmic number of rounds. We believe that our work takes an important first step towards uncovering the algorithmic implications of invariant theory.

cs.DS↗

arXiv2022

Spoofing Generalization: When Can't You Trust Proprietary Models?

In this work, we study the computational complexity of determining whether a machine learning model that perfectly fits the training data will generalizes to unseen data. In particular, we study the power of a malicious agent whose goal is to construct a model g that fits its training data and nothing else, but is indistinguishable from an accurate model f. We say that g strongly spoofs f if no polynomial-time algorithm can tell them apart. If instead we restrict to algorithms that run in $n^c$ time for some fixed $c$, we say that g c-weakly spoofs f. Our main results are 1. Under cryptographic assumptions, strong spoofing is possible and 2. For any c> 0, c-weak spoofing is possible unconditionally While the assumption of a malicious agent is an extreme scenario (hopefully companies training large models are not malicious), we believe that it sheds light on the inherent difficulties of blindly trusting large proprietary models or data.

cs.LG↗

arXiv2022

Planning in Observable POMDPs in Quasipolynomial Time

Partially Observable Markov Decision Processes (POMDPs) are a natural and general model in reinforcement learning that take into account the agent's uncertainty about its current state. In the literature on POMDPs, it is customary to assume access to a planning oracle that computes an optimal policy when the parameters are known, even though the problem is known to be computationally hard. Almost all existing planning algorithms either run in exponential time, lack provable performance guarantees, or require placing strong assumptions on the transition dynamics under every possible policy. In this work, we revisit the planning problem and ask: are there natural and well-motivated assumptions that make planning easy? Our main result is a quasipolynomial-time algorithm for planning in (one-step) observable POMDPs. Specifically, we assume that well-separated distributions on states lead to well-separated distributions on observations, and thus the observations are at least somewhat informative in each step. Crucially, this assumption places no restrictions on the transition dynamics of the POMDP; nevertheless, it implies that near-optimal policies admit quasi-succinct descriptions, which is not true in general (under standard hardness assumptions). Our analysis is based on new quantitative bounds for filter stability -- i.e. the rate at which an optimal filter for the latent state forgets its initialization. Furthermore, we prove matching hardness for planning in observable POMDPs under the Exponential Time Hypothesis.

cs.LG↗

arXiv2021

Tensor Completion Made Practical

Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees, based on solving large semidefinite programs which are impractical to run, or make strong assumptions such as requiring the factors to be nearly orthogonal. In this paper we introduce a new variant of alternating minimization, which in turn is inspired by understanding how the progress measures that guide convergence of alternating minimization in the matrix setting need to be adapted to the tensor setting. We show strong provable guarantees, including showing that our algorithm converges linearly to the true tensors even when the factors are highly correlated and can be implemented in nearly linear time. Moreover our algorithm is also highly practical and we show that we can complete third order tensors with a thousand dimensions from observing a tiny fraction of its entries. In contrast, and somewhat surprisingly, we show that the standard version of alternating minimization, without our new twist, can converge at a drastically slower rate in practice.

cs.DS↗

arXiv2021

Learning GMMs with Nearly Optimal Robustness Guarantees

In this work we solve the problem of robustly learning a high-dimensional Gaussian mixture model with $k$ components from $ε$-corrupted samples up to accuracy $\widetilde{O}(ε)$ in total variation distance for any constant $k$ and with mild assumptions on the mixture. This robustness guarantee is optimal up to polylogarithmic factors. The main challenge is that most earlier works rely on learning individual components in the mixture, but this is impossible in our setting, at least for the types of strong robustness guarantees we are aiming for. Instead we introduce a new framework which we call {\em strong observability} that gives us a route to circumvent this obstacle.

cs.LG↗

arXiv2021

Kalman Filtering with Adversarial Corruptions

Here we revisit the classic problem of linear quadratic estimation, i.e. estimating the trajectory of a linear dynamical system from noisy measurements. The celebrated Kalman filter gives an optimal estimator when the measurement noise is Gaussian, but is widely known to break down when one deviates from this assumption, e.g. when the noise is heavy-tailed. Many ad hoc heuristics have been employed in practice for dealing with outliers. In a pioneering work, Schick and Mitter gave provable guarantees when the measurement noise is a known infinitesimal perturbation of a Gaussian and raised the important question of whether one can get similar guarantees for large and unknown perturbations. In this work we give a truly robust filter: we give the first strong provable guarantees for linear quadratic estimation when even a constant fraction of measurements have been adversarially corrupted. This framework can model heavy-tailed and even non-stationary noise processes. Our algorithm robustifies the Kalman filter in the sense that it competes with the optimal algorithm that knows the locations of the corruptions. Our work is in a challenging Bayesian setting where the number of measurements scales with the complexity of what we need to estimate. Moreover, in linear dynamical systems past information decays over time. We develop a suite of new techniques to robustly extract information across different time steps and over varying time scales.

stat.ML↗

arXiv2021

Can Q-Learning be Improved with Advice?

Despite rapid progress in theoretical reinforcement learning (RL) over the last few years, most of the known guarantees are worst-case in nature, failing to take advantage of structure that may be known a priori about a given RL problem at hand. In this paper we address the question of whether worst-case lower bounds for regret in online learning of Markov decision processes (MDPs) can be circumvented when information about the MDP, in the form of predictions about its optimal $Q$-value function, is given to the algorithm. We show that when the predictions about the optimal $Q$-value function satisfy a reasonably weak condition we call distillation, then we can improve regret bounds by replacing the set of state-action pairs with the set of state-action pairs on which the predictions are grossly inaccurate. This improvement holds for both uniform regret bounds and gap-based ones. Further, we are able to achieve this property with an algorithm that achieves sublinear regret when given arbitrary predictions (i.e., even those which are not a distillation). Our work extends a recent line of work on algorithms with predictions, which has typically focused on simple online problems such as caching and scheduling, to the more complex and general problem of reinforcement learning.

cs.LG↗

arXiv2021

Rigorous Guarantees for Tyler's M-estimator via quantum expansion

Estimating the shape of an elliptical distribution is a fundamental problem in statistics. One estimator for the shape matrix, Tyler's M-estimator, has been shown to have many appealing asymptotic properties. It performs well in numerical experiments and can be quickly computed in practice by a simple iterative procedure. Despite the many years the estimator has been studied in the statistics community, there was neither a tight non-asymptotic bound on the rate of the estimator nor a proof that the iterative procedure converges in polynomially many steps. Here we observe a surprising connection between Tyler's M-estimator and operator scaling, which has been intensively studied in recent years in part because of its connections to the Brascamp-Lieb inequality in analysis. We use this connection, together with novel results on quantum expanders, to show that Tyler's M-estimator has the optimal rate up to factors logarithmic in the dimension, and that in the generative model the iterative procedure has a linear convergence rate even without regularization.

cs.DS↗

arXiv2021

Settling the Robust Learnability of Mixtures of Gaussians

This work represents a natural coalescence of two important lines of work: learning mixtures of Gaussians and algorithmic robust statistics. In particular we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights (bounded fractionality) and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving dimension-independent polynomial identifiability through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation.

cs.DS↗

arXiv2021

Online and Distribution-Free Robustness: Regression and Contextual Bandits with Huber Contamination

In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for both linear regression and contextual bandits. In fact our algorithms succeed where conventional methods fail. In particular we show strong lower bounds against Huber regression and more generally any convex M-estimator. Our approach is based on a novel alternating minimization scheme that interleaves ordinary least-squares with a simple convex program that finds the optimal reweighting of the distribution under a spectral constraint. Our results obtain essentially optimal dependence on the contamination level $η$, reach the optimal breakdown point, and naturally apply to infinite dimensional settings where the feature vectors are represented implicitly via a kernel map.

cs.LG↗

arXiv2021

Polynomial time guarantees for the Burer-Monteiro method

The Burer-Monteiro method is one of the most widely used techniques for solving large-scale semidefinite programs (SDP). The basic idea is to solve a nonconvex program in $Y$, where $Y$ is an $n \times p$ matrix such that $X = Y Y^T$. In this paper, we show that this method can solve SDPs in polynomial time in a smoothed analysis setting. More precisely, we consider an SDP whose domain satisfies some compactness and smoothness assumptions, and slightly perturb the cost matrix and the constraints. We show that if $p \gtrsim \sqrt{2(1+η)m}$, where $m$ is the number of constraints and $η>0$ is any fixed constant, then the Burer-Monteiro method can solve SDPs to any desired accuracy in polynomial time, in the setting of smooth analysis. Our bound on $p$ approaches the celebrated Barvinok-Pataki bound in the limit as $η$ goes to zero, beneath which it is known that the nonconvex program can be suboptimal. Previous analyses were unable to give polynomial time guarantees for the Burer-Monteiro method, since they either assumed that the criticality conditions are satisfied exactly, or ignored the nontrivial problem of computing an approximately feasible solution. We address the first problem through a novel connection with tubular neighborhoods of algebraic varieties. For the feasibility problem we consider a least squares formulation, and provide the first guarantees that do not rely on the restricted isometry property.

math.OC↗

arXiv2021

No-go Theorem for Acceleration in the Hyperbolic Plane

In recent years there has been significant effort to adapt the key tools and ideas in convex optimization to the Riemannian setting. One key challenge has remained: Is there a Nesterov-like accelerated gradient method for geodesically convex functions on a Riemannian manifold? Recent work has given partial answers and the hope was that this ought to be possible. Here we dash these hopes. We prove that in a noisy setting, there is no analogue of accelerated gradient descent for geodesically convex functions on the hyperbolic plane. Our results apply even when the noise is exponentially small. The key intuition behind our proof is short and simple: In negatively curved spaces, the volume of a ball grows so fast that information about the past gradients is not useful in the future.

math.OC↗

arXiv2021

Learning to Sample from Censored Markov Random Fields

We study learning Censor Markov Random Fields (abbreviated CMRFs). These are Markov Random Fields where some of the nodes are censored (not observed). We present an algorithm for learning high-temperature CMRFs within o(n) transportation distance. Crucially our algorithm makes no assumption about the structure of the graph or the number or location of the observed nodes. We obtain stronger results for high girth high-temperature CMRFs as well as computational lower bounds indicating that our results can not be qualitatively improved.

cs.LG↗