The rapid rise of Generative AI (GenAI) tools has sparked debate over their role in complementing or replacing human workers across job contexts. We present a mathematical framework that models jobs, workers, and worker-job fit, introducing a novel decomposition of skills into decision-level and action-level subskills to reflect the complementary strengths of humans and GenAI. We analyze how changes in subskill abilities affect job success, identifying conditions for sharp transitions in success probability. We also establish sufficient conditions under which combining workers with complementary subskills significantly outperforms relying on a single worker. This explains phenomena such as productivity compression, where GenAI assistance yields larger gains for lower-skilled workers. We demonstrate the framework’s practicality using data from O*NET and Big-Bench Lite, aligning real-world data with our model via subskill-division methods. Our results highlight when and how GenAI complements human skills, rather than replacing them.

We introduce a framework for designing efficient diffusion models for $d$-dimensional symmetric-space Riemannian manifolds, including the torus, sphere, special orthogonal group and unitary group. Existing manifold diffusion models often depend on heat kernels, which lack closed-form expressions and require either $d$ gradient evaluations or exponential-in-$d$ arithmetic operations per training step. We introduce a new diffusion model for symmetric manifolds with a spatially-varying covariance, allowing us to leverage a projection of Euclidean Brownian motion to bypass heat kernel computations. Our training algorithm minimizes a novel efficient objective derived via Ito’s Lemma, allowing each step to run in $O(1)$ gradient evaluations and nearly-linear-in-$d$ ($O(d^{1.19})$) arithmetic operations, reducing the gap between diffusions on symmetric manifolds and Euclidean space. Manifold symmetries ensure the diffusion satisfies an "average-case" Lipschitz condition, enabling accurate and efficient sample generation. Empirically, our model outperforms prior methods in training speed and improves sample quality on synthetic datasets on the torus, special orthogonal group, and unitary group.

Meritocratic systems, from admissions to hiring, aim to impartially reward skill and effort. Yet persistent disparities across race, gender, and class challenge this ideal. Some attribute these gaps to structural inequality; others to individual choice. We develop a game-theoretic model in which candidates from different socioeconomic groups differ in their perceived post-selection value—shaped by social context and, increasingly, by AI-powered tools offering personalized career or salary guidance. Each candidate strategically chooses effort, balancing its cost against expected reward; effort translates into observable merit, and selection is based solely on merit. We characterize the unique Nash equilibrium in the large-agent limit and derive explicit formulas showing how valuation disparities and institutional selectivity jointly determine effort, representation, social welfare, and utility. We further propose a cost-sensitive optimization framework that quantifies how modifying selectivity or perceived value can reduce disparities without compromising institutional goals. Our analysis reveals a perception-driven bias: when perceptions of post-selection value differ across groups, these differences translate into rational differences in effort, propagating disparities backward through otherwise "fair" selection processes. While the model is static, it captures one stage of a broader feedback cycle linking perceptions, incentives, and outcomes—bridging rational-choice and structural explanations of inequality by showing how techno-social environments shape individual incentives in meritocratic systems.

We study the problem of constructing coresets for $(k, z)$-clustering when the input dataset is corrupted by stochastic noise drawn from a known distribution. In this setting, evaluating the quality of a coreset is inherently challenging, as the true underlying dataset is unobserved. To address this, we investigate coreset construction using surrogate error metrics that are tractable and provably related to the true clustering cost. We analyze a traditional metric from prior work and introduce a new error metric that more closely aligns with the true cost. Although our metric is defined independent of the noise distribution, it enables approximation guarantees that scale with the noise level. We design a coreset construction algorithm based on this metric and show that, under mild assumptions on the data and noise, enforcing an $\varepsilon$-bound under our metric yields smaller coresets and tighter guarantees on the true clustering cost than those obtained via classical metrics. In particular, we prove that the coreset size can improve by a factor of up to $\mathrm{poly}(k)$, where $n$ is the dataset size. Experiments on real-world datasets support our theoretical findings and demonstrate the practical advantages of our approach.

We study a two-institution stable matching model in which candidates from two distinct groups are evaluated using partially correlated signals that are group-biased. This extends prior work (which assumes institutions evaluate candidates in an identical manner) to a more realistic setting in which institutions rely on overlapping, but independently processed, criteria. These evaluations could consist of a variety of informative tools such as standardized tests, shared recommendation systems, or AI-based assessments with local noise. Two key parameters govern evaluations: the bias parameter $\beta \in (0,1]$, which models systematic disadvantage faced by one group, and the correlation parameter $\gamma \in [0,1]$, which captures the alignment between institutional rankings. We study the representation ratio $\mathcal{R}(\beta, \gamma)$, i.e., the ratio of disadvantaged to advantaged candidates selected by the matching process in this setting. Focusing on a regime in which all candidates prefer the same institution, we characterize the large-market equilibrium and derive a closed-form expression for the resulting representation ratio. Prior work shows that when $\gamma = 1$, this ratio scales linearly with $\beta$. In contrast, we show that $\mathcal{R}(\beta, \gamma)$ increases nonlinearly with $\gamma$ and even modest losses in correlation can cause sharp drops in the representation ratio. Our analysis identifies critical $\gamma$-thresholds where institutional selection behavior undergoes discrete transitions, and reveals structural conditions under which evaluator alignment or bias mitigation are most effective. Finally, we show how this framework and results enable interventions for fairness-aware design in decentralized selection systems.

Low-rank pseudoinverses are widely used to approximate matrix inverses in scalable machine learning, optimization, and scientific computing. However, real-world matrices are often observed with noise, arising from sampling, sketching, and quantization. The spectral-norm robustness of low-rank inverse approximations remains poorly understood. We systematically study the spectral-norm error $\| \tilde{A}_p^{-1} - A_p^{-1} \|$ for an $n\times n$ symmetric matrix $A$, where $A_p^{-1}$ denotes the best rank-\(p\) approximation of $A^{-1}$, and $\tilde{A} = A + E$ is a noisy observation. Under mild assumptions on the noise, we derive sharp non-asymptotic perturbation bounds that reveal how the error scales with the eigengap, spectral decay, and noise alignment with low-curvature directions of $A$. Our analysis introduces a novel application of contour integral techniques to the \emph{non-entire} function $f(z) = 1/z$, yielding bounds that improve over naive adaptations of classical full-inverse bounds by up to a factor of $\sqrt{n}$. Empirically, our bounds closely track the true perturbation error across a variety of real-world and synthetic matrices, while estimates based on classical results tend to significantly overpredict. These findings offer practical, spectrum-aware guarantees for low-rank inverse approximations in noisy computational environments.

A central challenge in machine learning is to understand how noise or measurement errors affect low-rank approximations, particularly in the spectral norm. This question is especially important in differentially private low-rank approximation, where one aims to preserve the top-$p$ structure of a data-derived matrix while ensuring privacy. Prior work often analyzes Frobenius norm error or changes in reconstruction quality, but these metrics can over- or under-estimate true subspace distortion. The spectral norm, by contrast, captures worst-case directional error and provides the strongest utility guarantees. We establish new high-probability spectral-norm perturbation bounds for symmetric matrices that refine the classical Eckart--Young--Mirsky theorem and explicitly capture interactions between a matrix $A \in \mathbb{R}^{n \times n}$ and an arbitrary symmetric perturbation $E$. Under mild eigengap and norm conditions, our bounds yield sharp estimates for $\| (A + E)_p - A_p \|$, where $A_p$ is the best rank-$p$ approximation of $A$, with improvements of up to a factor of $\sqrt{n}$. As an application, we derive improved utility guarantees for differentially private PCA, resolving an open problem in the literature. Our analysis relies on a novel contour bootstrapping method from complex analysis and extends it to a broad class of spectral functionals, including polynomials and matrix exponentials. Empirical results on real-world datasets confirm that our bounds closely track the actual spectral error under diverse perturbation regimes.

We consider the problem of approximating a \(d \times d\) covariance matrix M with a rank- k matrix under \((\varepsilon ,\delta)\) -differential privacy. We present and analyze a complex variant of the Gaussian mechanism and obtain upper bounds on the Frobenius norm of the difference between the matrix output by this mechanism and the best rank- k approximation to M . Our analysis provides improvements over previous bounds, particularly when the spectrum of M satisfies natural structural assumptions. The novel insight is to view the addition of Gaussian noise to a matrix as a continuous-time matrix Brownian motion. This viewpoint allows us to track the evolution of eigenvalues and eigenvectors of the matrix, which are governed by stochastic differential equations discovered by Dyson. These equations enable us to upper bound the Frobenius distance between the best rank- k approximation of M and that of a Gaussian perturbation of M as an integral that involves inverse eigenvalue gaps of the stochastically evolving matrix, as opposed to a sum of perturbation bounds obtained via Davis-Kahan-type theorems. Subsequently, again using the Dyson Brownian motion viewpoint, we show that the eigenvalues of the matrix M perturbed by Gaussian noise have large gaps with high probability. These results also contribute to the analysis of low-rank approximations under average-case perturbations, and to an understanding of eigenvalue gaps for random matrices, both of which may be of independent interest.

We consider the problem of sampling from a logconcave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a polytope $K:=${$\theta \in \mathbb{R}^d: A\theta \leq b$}, where $A\in \mathbb{R}^{m\times d}$ and $b \in \mathbb{R}^m$. The fastest-known algorithm for the setting when $f$ is $O(1)$-Lipschitz or $O(1)$-smooth runs in roughly $O(md \times md^{\omega -1})$ arithmetic operations, where the $md^{\omega -1}$ term arises because each Markov chain step requires computing a matrix inversion and determinant ($\omega \approx 2.37$ is the matrix multiplication constant). We present a nearly-optimal implementation of this Markov chain with per-step complexity that is roughly the number of non-zero entries of $A$ while the number of Markov chain steps remains the same. The key technical ingredients are 1) to show that the matrices that arise in this Dikin walk change slowly, 2) to deploy efficient linear solvers which can leverage this slow change to speed up matrix inversion by using information computed in previous steps, and 3) to speed up the computation of the determinantal term in the Metropolis filter step via a randomized Taylor series-based estimator. This result directly improves the runtime for applications that involve sampling from Gibbs distributions constrained to polytopes that arise in Bayesian statistics and private optimization.

This paper considers the scenario in which there are multiple institutions, each with a limited capacity for candidates, and candidates, each with preferences over the institutions. A central entity evaluates the utility of each candidate to the institutions, and the goal is to select candidates for each institution in a way that maximizes utility while also considering the candidates’ preferences. The paper focuses on the setting in which candidates are divided into multiple groups and the observed utilities of candidates in some groups are biased–systematically lower than their true utilities. The first result is that, in these biased settings, prior algorithms can lead to selections with sub-optimal true utility and significant discrepancies in the fraction of candidates from each group that get their preferred choices. Subsequently, an algorithm is presented along with proof that it produces selections that achieve near-optimal group fairness with respect to preferences while also nearly maximizing the true utility under distributional assumptions. Further, extensive empirical validation of these results in real-world and synthetic settings, in which the distributional assumptions may not hold, are presented.

We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$’th and the $k+1$’th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson’s stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.

We consider the problem of subset selection where one is given multiple rankings of items and the goal is to select the highest "quality" subset. Score functions from the multiwinner voting literature have been used to aggregate rankings into quality scores for subsets. We study this setting of subset selection problems when, in addition, rankings may contain systemic or unconscious biases toward a group of items. For a general model of input rankings and biases, we show that requiring the selected subset to satisfy group fairness constraints can improve the quality of the selection with respect to unbiased rankings. Importantly, we show that for fairness constraints to be effective, different multiwinner score functions may require a drastically different number of rankings: While for some functions, fairness constraints need an exponential number of rankings to recover a close-to-optimal solution, for others, this dependency is only polynomial. This result relies on a novel notion of "smoothness" of submodular functions in this setting that quantifies how well a function can "correctly" assess the quality of items in the presence of bias. The results in this paper can be used to guide the choice of multiwinner score functions for the subset selection setting considered here; we additionally provide a tool to empirically enable this.

Biases with respect to socially-salient attributes of individuals have been well documented in evaluation processes used in settings such as admissions and hiring. We view such an evaluation process as a transformation of a distribution of the true utility of an individual for a task to an observed distribution and model it as a solution to a loss minimization problem subject to an information constraint. Our model has two parameters that have been identified as factors leading to biases: the resource-information trade-off parameter in the information constraint and the risk-averseness parameter in the loss function. We characterize the distributions that arise from our model and study the effect of the parameters on the observed distribution. The outputs of our model enrich the class of distributions that can be used to capture variation across groups in the observed evaluations. We empirically validate our model by fitting real-world datasets and use it to study the effect of interventions in a downstream selection task. These results contribute to an understanding of the emergence of bias in evaluation processes and provide tools to guide the deployment of interventions to mitigate biases.

Given a Lipschitz or smooth convex function $f:K \to \mathbb{R}^d$ for a bounded polytope $K:=${ $\theta \in \mathbb{R}^d: A\theta \leq b$}, where $A\in \mathbb{R}^{m\times d}$ and $b \in \mathbb{R}^m$, we consider the problem of sampling from the log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to $K$. Interest in this problem derives from its applications to Bayesian inference and differential privacy. We present a generalization of the Dikin walk to this setting that requires at most $O((md + d L^2 R^2) \times md^{\omega-1} \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start. Here $L$ is the Lipschitz constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega \approx 2.37$ is the matrix-multiplication constant. This improves on the running time of prior works for a range of structured settings important for the aforementioned inference and privacy applications. Technically, we depart from previous Dikin walks by adding a soft-threshold regularizer derived from the Lipschitz or smoothness properties of $f$ to a barrier function for $K$ that allows our version of the Dikin walk to propose updates that have a high Metropolis acceptance ratio for $f$, while at the same time remaining inside the polytope $K$.

Subset selection tasks, arise in recommendation systems and search engines and ask to select a subset of items that maximize the value for the user. The values of subsets often display diminishing returns, and hence, submodular functions have been used to model them. If the inputs defining the submodular function are known, then existing algorithms can be used. In many applications, however, inputs have been observed to have social biases that reduce the utility of the output subset. Hence, interventions to improve the utility are desired. Prior works focus on maximizing linear functions—a special case of submodular functions—and show that fairness constraint-based interventions can not only ensure proportional representation but also achieve near-optimal utility in the presence of biases. We study the maximization of a family of submodular functions that capture functions arising in the aforementioned applications. Our first result is that, unlike linear functions, constraint-based interventions cannot guarantee any constant fraction of the optimal utility for this family of submodular functions. Our second result is an algorithm for submodular maximization. The algorithm provably outputs subsets that have near-optimal utility for this family under mild assumptions and that proportionally represent items from each group. In empirical evaluation, with both synthetic and real-world data, we observe that this algorithm improves the utility of the output subset for this family of submodular functions over baselines.

Consider the following optimization problem: Given $n \times n$ matrices $A$ and $\Lambda$, maximize $⟨A, U\Lambda U^*⟩$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix whose spectrum is the same as $\Lambda$ and, by setting $\Lambda$ to be appropriate diagonal matrices, one can recover matrix approximation problems such as PCA and rank-$k$ approximation. We study the problem of designing differentially private algorithms for this optimization problem in settings where the matrix $A$ is constructed using users’ private data. We give efficient and private algorithms that come with upper and lower bounds on the approximation error. Our results unify and improve upon several prior works on private matrix approximation problems. They rely on extensions of packing/covering number bounds for Grassmannians to unitary orbits which should be of independent interest.

We study a variant of a recently introduced min-max optimization framework where the max-player is constrained to update its parameters in a greedy manner until it reaches a first-order stationary point. Our equilibrium definition for this framework depends on a proposal distribution which the min-player uses to choose directions in which to update its parameters. We show that, given a smooth and bounded nonconvex-nonconcave objective function, access to any proposal distribution for the min-player’s updates, and stochastic gradient oracle for the max-player, our algorithm converges to the aforementioned approximate local equilibrium in a number of iterations that does not depend on the dimension. The equilibrium point found by our algorithm depends on the proposal distribution, and when applying our algorithm to train GANs we choose the proposal distribution to be a distribution of stochastic gradients. We empirically evaluate our algorithm on challenging nonconvex-nonconcave test-functions and loss functions arising in GAN training. Our algorithm converges on these test functions and, when used to train GANs, trains stably on synthetic and real-world datasets and avoids mode collapse.

For a $d$-dimensional log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a convex body $K$, the problem of outputting samples from a distribution $\nu$ which is $\varepsilon$-close in infinity-distance $\sup_{\theta \in K} |\log \frac{\nu(\theta)}{\pi(\theta)}|$ to $\pi$ arises in differentially private optimization. While sampling within total-variation distance $\varepsilon$ of $\pi$ can be done by algorithms whose runtime depends polylogarithmically on $\frac{1}{\varepsilon}$, prior algorithms for sampling in $\varepsilon$ infinity distance have runtime bounds that depend polynomially on $\frac{1}{\varepsilon}$. We bridge this gap by presenting an algorithm that outputs a point $\varepsilon$-close to $\pi$ in infinity distance that requires at most $\mathrm{poly}(\log \frac{1}{\varepsilon}, d)$ calls to a membership oracle for $K$ and evaluation oracle for $f$, when $f$ is Lipschitz. Our approach departs from prior works that construct Markov chains on a $\frac{1}{\varepsilon^2}$-discretization of $K$ to achieve a sample with $\varepsilon$ infinity-distance error, and present a method to directly convert continuous samples from $K$ with total-variation bounds to samples with infinity bounds. This approach also allows us to obtain an improvement on the dimension $d$ in the running time for the problem of sampling from a log-concave distribution on polytopes $K$ with infinity distance $\varepsilon$, by plugging in TV-distance running time bounds for the Dikin Walk Markov chain.

Given a symmetric matrix $M$ and a vector $\lambda$, we present new bounds on the Frobenius-distance utility of the Gaussian mechanism for approximating $M$ by a matrix whose spectrum is $\lambda$, under $(\varepsilon,\delta)$-differential privacy. Our bounds depend on both $\lambda$ and the gaps in the eigenvalues of $M$, and hold whenever the top $k+1$ eigenvalues of $M$ have sufficiently large gaps. When applied to the problems of private rank-$k$ covariance matrix approximation and subspace recovery, our bounds yield improvements over previous bounds. Our bounds are obtained by viewing the addition of Gaussian noise as a continuous-time matrix Brownian motion. This viewpoint allows us to track the evolution of eigenvalues and eigenvectors of the matrix, which are governed by stochastic differential equations discovered by Dyson. These equations allow us to bound the utility as the square-root of a sum-of-squares of perturbations to the eigenvectors, as opposed to a sum of perturbation bounds obtained via Davis-Kahan-type theorems.

The fair-ranking problem, which asks to rank a given set of items to maximize utility subject to group fairness constraints, has received attention in the fairness, information retrieval, and machine learning literature. Recent works, however, observe that errors in socially-salient (including protected) attributes of items can significantly undermine fairness guarantees of existing fair-ranking algorithms and raise the problem of mitigating the effect of such errors. We study the fair-ranking problem under a model where socially-salient attributes of items are randomly and independently perturbed. We present a fair-ranking framework that incorporates group fairness requirements along with probabilistic information about perturbations in socially-salient attributes. We provide provable guarantees on the fairness and utility attainable by our framework and show that it is information-theoretically impossible to significantly beat these guarantees. Our framework works for multiple non-disjoint attributes and a general class of fairness constraints that includes proportional and equal representation. Empirically, we observe that, compared to baselines, our algorithm outputs rankings with higher fairness, and has a similar or better fairness-utility trade-off compared to baselines.

Abstract. We study the following problem: Given a continuous domain [Formula: see text] along with its convex hull [Formula: see text], a point [Formula: see text], and a measure [Formula: see text] on [Formula: see text], find the probability density over [Formula: see text] whose marginal is [Formula: see text] and that minimizes the KL divergence to the uniform density with respect to [Formula: see text]. Several distributions in mathematics, physics, statistics, and theoretical computer science arise by different settings of the parameters of this problem. We give a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a “balance” property of the measure [Formula: see text] on [Formula: see text], and exact algorithms for evaluating the dual and its gradient for several interesting settings of [Formula: see text] and [Formula: see text]. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL divergence minimizing distributions in several cases. Applications of our results include (1) an optimization characterization of the Goemans–Williamson measure [M. X. Goemans and D. P. Williamson, J ACM, 42 (1995), pp. 1115–1145] that is used to round a positive semidefinite matrix to a vector; (2) the computability of the entropic barrier for convex bodies, given a strong integration oracle, studied by [S. Bubeck and R. Eldan, Proc. Mach. Learn. Res. (PMLR), 40 (2015), p. 279], and (3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy [W. Band and J. L. Park, Found. Phys., 6 (1976), pp. 249–262; J. L. Park and W. Band, Found. Phys., 7 (1977), pp. 233–244; P. B. Slater, Phys. Lett. A, 159 (1991), pp. 411–414]; this corresponds to the case when [Formula: see text] is the set of rank-one projection matrices and [Formula: see text] is derived from the Haar measure on the unit sphere.

We present an optimization framework for learning a fair classifier in the presence of noisy perturbations in the protected attributes. Compared to prior work, our framework can be employed with a very general class of linear and linear-fractional fairness constraints, can handle multiple, non-binary protected attributes, and outputs a classifier that comes with provable guarantees on both accuracy and fairness. Empirically, we show that our framework can be used to attain either statistical rate or false positive rate fairness guarantees with a minimal loss in accuracy, even when the noise is large, in two real-world datasets.

We study fair classification in the presence of an omniscient adversary that, given an $\eta$, is allowed to choose an arbitrary $\eta$-fraction of the training samples and arbitrarily perturb their protected attributes. The motivation comes from settings in which protected attributes can be incorrect due to strategic misreporting, malicious actors, or errors in imputation; and prior approaches that make stochastic or independence assumptions on errors may not satisfy their guarantees in this adversarial setting. Our main contribution is an optimization framework to learn fair classifiers in this adversarial setting that comes with provable guarantees on accuracy and fairness. Our framework works with multiple and non-binary protected attributes, is designed for the large class of linear-fractional fairness metrics, and can also handle perturbations besides protected attributes. We prove near-tightness of our framework's guarantees for natural hypothesis classes: no algorithm can have significantly better accuracy and any algorithm with better fairness must have lower accuracy. Empirically, we evaluate the classifiers produced by our framework for statistical rate on real-world and synthetic datasets for a family of adversaries.

We study the problem of constructing coresets for clustering problems with time series data. This problem has gained importance across many fields including biology, medicine, and economics due to the proliferation of sensors facilitating real-time measurement and rapid drop in storage costs. In particular, we consider the setting where the time series data on $N$ entities is generated from a Gaussian mixture model with autocorrelations over $k$ clusters in $\mathbb{R}^d$. Our main contribution is an algorithm to construct coresets for the maximum likelihood objective for this mixture model. Our algorithm is efficient, and under a mild boundedness assumption on the covariance matrices of the underlying Gaussians, the size of the coreset is independent of the number of entities $N$ and the number of observations for each entity, and depends only polynomially on $k$, $d$ and $1/\varepsilon$, where $\varepsilon$ is the error parameter. We empirically assess the performance of our coreset with synthetic data.

Given two Hermitian matrices Y and Λ, the Harish-Chandra–Itzykson–Zuber (HCIZ) distribution is given by the density eTr(U Λ U*Y) with respect to the Haar measure on the unitary group. Random unitary matrices distributed according to the HCIZ distribution are important in various settings in physics and random matrix theory, but the problem of sampling efficiently from this distribution has remained open. We present two algorithms to sample matrices from distributions that are close to the HCIZ distribution. The first produces samples that are ξ-close in total variation distance, and the number of arithmetic operations required depends on poly(log1/ξ). The second produces samples that are ξ-close in infinity divergence, but with a poly(1/ξ) dependence. Our results have the following applications: 1) an efficient algorithm to sample from complex versions of matrix Langevin distributions studied in statistics, 2) an efficient algorithm to sample from continuous maximum entropy distributions over unitary orbits, which in turn implies an efficient algorithm to sample a pure quantum state from the entropy-maximizing ensemble representing a given density matrix, and 3) an efficient algorithm for differentially private rank-k approximation that comes with improved utility bounds for k>1.

Min-max optimization of a function f from [Formula: see text] to [Formula: see text] is an important framework for modeling robustness in adversarial settings with applications to optimization, economics, and deep learning. Oftentimes, f is nonconvex–nonconcave, and finding a global min-max point is computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max optimization formulation. However, many of these alternative solution concepts guarantee the existence of solution points only under strong assumptions on f, such as convexity or monotonicity of its gradient. We propose a new solution concept, the [Formula: see text]-greedy adversarial equilibrium, and show that it can serve as a computationally tractable alternative to min-max optimization. We prove the existence of such a point for any smooth bounded function with Lipschitz Hessian and give an algorithm that converges to an [Formula: see text]-greedy adversarial equilibrium in a number of evaluations of f, [Formula: see text], and [Formula: see text] that is polynomial in d, [Formula: see text], and the bounds of f and its Lipschitz constant. Funding: This work was supported by the National Science Foundation [Grant CCF-1908347]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0262 .

In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop, and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or antiferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, and change of functions). The algorithms for both these models are efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new “equilibrium" property that is shown to be satisfied by our algorithm at each step and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

We present a geometric approach toward derandomizing the isolation lemma of Mulmuley, Vazirani, and Vazirani. We construct a quasi-polynomial family of weights that isolate a vertex in any 0/1-polytope for which each face spans an affine space defined by a totally unimodular matrix. These polytopes are also called box-totally dual integral or principally box-integer. This includes the polytopes given by totally unimodular constraints and generalizes the recent derandomization of the isolation lemma for bipartite perfect matching and matroid intersection. We prove our result by associating a lattice to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of vectors of length within 3/2 of the shortest vector in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of circuits of size within 3/2 of the shortest circuit in a regular matroid. This is the technical core of the paper and relies on a variant of Seymour's decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids.

We show that for any regular matroid on m elements and $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a circuit in the matroid, is bounded by $m^{O(\alpha^2)}.$ This generalizes a result of Karger for the number of $\alpha$-minimum cuts in a graph. As a consequence, we obtain similar bounds on the number of $\alpha$-shortest vectors in totally unimodular lattices and on the number of $\alpha$-minimum weight code words in regular codes.

Data containing human or social attributes may over- or under-represent groups with respect to salient social attributes such as gender or race, which can lead to biases in downstream applications. This paper presents an algorithmic framework that can be used as a data preprocessing method towards mitigating such bias. Unlike prior work, it can efficiently learn distributions over large domains, controllably adjust the representation rates of protected groups and achieve target fairness metrics such as statistical parity, yet remains close to the empirical distribution induced by the given dataset. Our approach leverages the principle of maximum entropy {–} amongst all distributions satisfying a given set of constraints, we should choose the one closest in KL-divergence to a given prior. While maximum entropy distributions can succinctly encode distributions over large domains, they can be difficult to compute. Our main contribution is an instantiation of this framework for our set of constraints and priors, which encode our bias mitigation goals, and that runs in time polynomial in the dimension of the data. Empirically, we observe that samples from the learned distribution have desired representation rates and statistical rates, and when used for training a classifier incurs only a slight loss in accuracy while maintaining fairness properties.

A panel dataset contains features or observations for multiple individuals over multiple time periods and regression problems with panel data are common in statistics and applied ML. When dealing with massive datasets, coresets have emerged as a valuable tool from a computational, storage and privacy perspective, as one needs to work with and share much smaller datasets. However, results on coresets for regression problems thus far have only been available for cross-sectional data ($N$ individuals each observed for a single time unit) or longitudinal data (a single individual observed for $T>1$ time units), but there are no results for panel data ($N>1$, $T>1$). This paper introduces the problem of coresets to panel data settings; we first define coresets for several variants of regression problems with panel data and then present efficient algorithms to construct coresets of size that are independent of $N$ and $T$, and only polynomially depend on $1/\varepsilon$ (where $\varepsilon$ is the error parameter) and the number of regression parameters. Our approach is based on the Feldman-Langberg framework in which a key step is to upper bound the “total sensitivity” that is roughly the sum of maximum influences of all individual-time pairs taken over all possible choices of regression parameters. Empirically, we assess our approach with a synthetic and a real-world datasets; the coreset sizes constructed using our approach are much smaller than the full dataset and coresets indeed accelerate the running time of computing the regression objective.

Given a collection of n points in ℝ d , the goal of the (k,z)-clustering problem is to find a subset of k “centers” that minimizes the sum of the z-th powers of the Euclidean distance of each point to the closest center. Special cases of the (k,z)-clustering problem include the k-median and k-means problems. Our main result is a unified two-stage importance sampling framework that constructs an ε-coreset for the (k,z)-clustering problem. Compared to the results for (k,z)-clustering in [Feldman and Langberg, STOC 2011], our framework saves a ε2 d factor in the coreset size. Compared to the results for (k,z)-clustering in [Sohler and Woodruff, FOCS 2018], our framework saves a poly(k) factor in the coreset size and avoids the exp(k/ε) term in the construction time. Specifically, our coreset for k-median (z=1) has size Õ(ε−4 k) which, when compared to the result in [Sohler and Woodruff, STOC 2018], saves a k factor in the coreset size. Our algorithmic results rely on a new dimensionality reduction technique that connects two well-known shape fitting problems: subspace approximation and clustering, and may be of independent interest. We also provide a size lower bound of Ω(k· min{2 z/20,d }) for a 0.01-coreset for (k,z)-clustering, which has a linear dependence of size on k and an exponential dependence on z that matches our algorithmic results.

We initiate a study of the following problem: Given a continuous domain Ω along with its convex hull K, a point A ∈ K and a prior measure µ on Ω, find the probability density over Ω whose marginal is A and that minimizes the KL-divergence to µ. This framework gives rise to several extremal distributions that arise in mathematics, quantum mechanics, statistics, and theoretical computer science. Our technical contributions include a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a “balance” property of the measure µ on Ω, and exact algorithms for evaluating the dual and its gradient for several interesting settings of Ω and µ. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL-divergence minimizing distributions in several cases. Applications of our results include: 1) an optimization characterization of the Goemans-Williamson measure that is used to round a positive semidefinite matrix to a vector, 2) the computability of the entropic barrier for polytopes studied by Bubeck and Eldan, and 3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy by Band and Park in the 1970s: this corresponds to the case when Ω is the set of rank one projection matrices and µ corresponds to the Haar measure on the unit sphere. Our techniques generalize to the setting of rank k projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicable even beyond, to adjoint orbits of compact Lie groups.

The Langevin Markov chain algorithms are widely deployed methods to sample from distributions in challenging high-dimensional and non-convex statistics and machine learning applications. Despite this, current bounds for the Langevin algorithms are worse than those of competing algorithms in many important situations, for instance when sampling from weakly log-concave distributions, or when sampling or optimizing non-convex log-densities. We obtain improved bounds in many of these situations, showing that the Metropolis-adjusted Langevin algorithm (MALA) is faster than the best bounds for its competitor algorithms when the target distribution satisfies weak third- and fourth- order regularity properties associated with the input data. In many settings, our regularity conditions are weaker than the usual Euclidean operator norm regularity properties, allowing us to show faster bounds for a much larger class of distributions than would be possible with the usual Euclidean operator norm approach, including in statistics and machine learning applications where the data satisfy a certain incoherence condition. In particular, we show that using our regularity conditions one can obtain faster bounds for applications which include sampling problems in Bayesian logistic regression with weakly convex priors, and the nonconvex optimization problem of learning linear classifiers with zero-one loss functions. Our main technical contribution is an analysis of the Metropolis acceptance probability of MALA in terms of its “energy-conservation error," and a bound for this error in terms of third- and fourth- order regularity conditions. The combination of this higher-order analysis of the energy conservation error with the conductance method is key to obtaining bounds which have a sub-linear dependence on the dimension $d$ in the non-strongly logconcave setting.

Maximum entropy distributions with discrete support in $m$ dimensions arise in machine learning, statistics, information theory, and theoretical computer science. While structural and computational properties of max-entropy distributions have been extensively studied, basic questions such as: Do max-entropy distributions over a large support (e.g., $2^m$) with a specified marginal vector have succinct descriptions (polynomial-size in the input description)? and: Are entropy maximizing distributions “stable” under the perturbation of the marginal vector? have resisted a rigorous resolution. Here we show that these questions are related and resolve both of them. Our main result shows a ${\rm poly}(m, \log 1/\varepsilon)$ bound on the bit complexity of $\varepsilon$-optimal dual solutions to the maximum entropy convex program – for very general support sets and with no restriction on the marginal vector. Applications of this result include polynomial time algorithms to compute max-entropy distributions over several new and old polytopes for any marginal vector in a unified manner, a polynomial time algorithm to compute the Brascamp-Lieb constant in the rank-1 case. The proof of this result allows us to show that changing the marginal vector by $\delta$ changes the max-entropy distribution in the total variation distance roughly by a factor of ${\rm poly}(m, \log 1/\delta)\sqrt{\delta}$ – even when the size of the support set is exponential. Together, our results put max-entropy distributions on a mathematically sound footing – these distributions are robust and computationally feasible models for data.

This paper studies the problem of "isotropically rounding" a polytope K ⊆ R^n, that is, computing a linear transformation which makes the uniform distribution on the polytope have roughly identity covariance matrix. It is assumed that K ⊆ R^n is defined by m linear inequalities. We introduce a new variant of the ball walk Markov chain and show that, roughly, the expected number of arithmetic operations per-step of this Markov chain is O(m) that is sub-linear in the input size mn -- the per-step time of all prior Markov chains. Subsequently, we apply this new variant of the ball walk to obtain a rounding algorithm that gives a factor of √n improvement on the number of arithmetic operations over the previous bound which uses the hit-and-run algorithm. Since the cost of the rounding pre-processing step is in many cases the bottleneck in improving sampling or volume computation running time bounds, our results imply improved bounds for these tasks. Our algorithm achieves this improvement by a novel method of computing polytope membership, where one avoids checking inequalities which are estimated to have a very low probability of being violated. We believe that this method is likely to be of independent interest for constrained sampling and optimization problems.

In a recent study, Friedler et al. pointed out that several fair classification algorithms are not stable with respect to variations in the training set – a crucial consideration in several applications. Motivated by their work, we study the problem of designing classification algorithms that are both fair and stable. We propose an extended framework based on fair classification algorithms that are formulated as optimization problems, by introducing a stability-focused regularization term. Theoretically, we prove an additional stability guarantee, that was lacking in fair classification algorithms, and also provide an accuracy guarantee for our extended framework. Our accuracy guarantee can be used to inform the selection of the regularization parameter in our framework. We assess the benefits of our approach empirically by extending several fair classification algorithms that are shown to achieve the best balance between fairness and accuracy over the \textbf{Adult} dataset. Our empirical results show that our extended framework indeed improves the stability at only a slight sacrifice in accuracy.

Online advertising platforms are thriving due to the customizable audiences they offer advertisers. However, recent studies show that advertisements can be discriminatory with respect to the gender or race of the audience that sees the ad, and may inadvertently cross ethical and/or legal boundaries. To prevent this, we propose a constrained ad auction framework that maximizes the platform’s revenue conditioned on ensuring that the audience seeing an advertiser’s ad is distributed appropriately across sensitive types such as gender or race. Building upon Myerson’s classic work, we first present an optimal auction mechanism for a large class of fairness constraints. Finding the parameters of this optimal auction, however, turns out to be a non-convex problem. We show that this non-convex problem can be reformulated as a more structured non-convex problem with no saddle points or local-maxima; this allows us to develop a gradient-descent-based algorithm to solve it. Our empirical results on the A1 Yahoo! dataset demonstrate that our algorithm can obtain uniform coverage across different user types for each advertiser at a minor loss to the revenue of the platform, and a small change to the size of the audience each advertiser reaches.

In a recent work, \cite{chierichetti2017fair} studied the following ``fair'' variants of classical clustering problems such as k-means and k-median: given a set of n data points in R^d and a binary type associated to each data point, the goal is to cluster the points while ensuring that the proportion of each type in each cluster is roughly the same as its underlying proportion. Subsequent work has focused on either extending this setting to when each data point has multiple, non-disjoint sensitive types such as race and gender \cite{bera2019fair}, or to address the problem that the clustering algorithms in the above work do not scale well. The main contribution of this paper is an approach to clustering with fairness constraints that involve {\em multiple, non-disjoint} attributes, that is {\em also scalable}. Our approach is based on novel constructions of coresets: for the k-median objective, we construct an \eps-coreset of size O(\Gamma k^2 \eps^{-d}) where \Gamma is the number of distinct collections of groups that a point may belong to, and for the k-means objective, we show how to construct an \eps-coreset of size O(\Gamma k^3\eps^{-d-1}). The former result is the first known coreset construction for the fair clustering problem with the k-median objective, and the latter result removes the dependence on the size of the full dataset as in~\cite{schmidt2018fair} and generalizes it to multiple, non-disjoint attributes. Importantly, plugging our coresets into existing algorithms for fair clustering such as \cite{backurs2019scalable} results in the fastest algorithms for several cases. Empirically, we assess our approach over the \textbf{Adult} and \textbf{Bank} dataset, and show that the coreset sizes are much smaller than the full dataset; applying coresets indeed accelerates the running time of computing the fair clustering objective while ensuring that the resulting objective difference is small.

Given a sequence of convex functions $f_0, f_1, \ldots, f_T$, we study the problem of sampling from the Gibbs distribution $\pi_t \propto e^{-\sum_{k=0}^t f_k}$ for each epoch $t$ in an {\em online} manner. Interest in this problem derives from applications in machine learning, Bayesian statistics, and optimization where, rather than obtaining all the observations at once, one constantly acquires new data, and must continuously update the distribution. Our main result is an algorithm that generates roughly independent samples from $\pi_t$ for every epoch $t$ and, under mild assumptions, makes $\mathrm{polylog}(T)$ gradient evaluations per epoch. All previous results imply a bound on the number of gradient or function evaluations which is at least linear in $T$. Motivated by real-world applications, we assume that functions are smooth, their associated distributions have a bounded second moment, and their minimizer drifts in a bounded manner, but do not assume they are strongly convex. In particular, our assumptions hold for online Bayesian logistic regression, when the data satisfy natural regularity properties, giving a sampling algorithm with updates that are poly-logarithmic in $T$. In simulations, our algorithm achieves accuracy comparable to an algorithm specialized to logistic regression. Key to our algorithm is a novel stochastic gradient Langevin dynamics Markov chain with a carefully designed variance reduction step and constant batch size. Technically, lack of strong convexity is a significant barrier to analysis and, here, our main contribution is a martingale exit time argument that shows our Markov chain remains in a ball of radius roughly poly-logarithmic in $T$ for enough time to reach within $\epsilon$ of $\pi_t$.

We show that for any regular matroid on m elements and any α ≥ 1, the number of α-minimum circuits, or circuits whose size is at most an α-multiple of the minimum size of a circuit in the matroid is bounded by mO(α2). This generalizes a result of Karger for the number of α-minimum cuts in a graph. As a consequence, we obtain similar bounds on the number of α-shortest vectors in “totally unimodular” lattices and on the number of α-minimum weight codewords in “regular” codes.

In this paper, we study the problem of sampling from a graphical model when the model itself is changing dynamically with time. This problem derives its interest from a variety of inference, learning, and sampling settings in machine learning, computer vision, statistical physics, and theoretical computer science. While the problem of sampling from a static graphical model has received considerable attention, theoretical works for its dynamic variants have been largely lacking. The main contribution of this paper is an algorithm that can sample dynamically from a broad class of graphical models over discrete random variables. Our algorithm is parallel and Las Vegas: it knows when to stop and it outputs samples from the exact distribution. We also provide sufficient conditions under which this algorithm runs in time proportional to the size of the update, on general graphical models as well as well-studied specific spin systems. In particular we obtain, for the Ising model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the first dynamic sampling algorithms that can handle both edge and vertex updates (addition, deletion, change of functions), both efficient within regimes that are close to the respective uniqueness regimes, beyond which, even for the static and approximate sampling, no local algorithms were known or the problem itself is intractable. Our dynamic sampling algorithm relies on a local resampling algorithm and a new ``equilibrium'' property that is shown to be satisfied by our algorithm at each step, and enables us to prove its correctness. This equilibrium property is robust enough to guarantee the correctness of our algorithm, helps us improve bounds on fast convergence on specific models, and should be of independent interest.

We consider two non-convex formulations for computing the optimal constant in the Brascamp-Lieb inequality corresponding to a given datum and show that they are geodesically log-concave on the manifold of positive definite matrices endowed with the Riemannian metric corresponding to the Hessian of the log-determinant function. The first formulation is present in the work of Lieb [Lieb, 1990] and the second is new and inspired by the work of Bennett et al. [Bennett et al., 2008]. Recent work of Garg et al. [Ankit Garg et al., 2017] also implies a geodesically log-concave formulation of the Brascamp-Lieb constant through a reduction to the operator scaling problem. However, the dimension of the arising optimization problem in their reduction depends exponentially on the number of bits needed to describe the Brascamp-Lieb datum. The formulations presented here have dimensions that are polynomial in the bit complexity of the input datum.

We consider the problem of minimizing a convex objective function $F$ when one can only evaluate its noisy approximation $\hat{F}$. Unless one assumes some structure on the noise, $\hat{F}$ may be an arbitrary nonconvex function, making the task of minimizing $F$ intractable. To overcome this, prior work has often focused on the case when $F(x)-\hat{F}(x)$ is uniformly-bounded. In this paper we study the more general case when the noise has magnitude $\alpha F(x) + \beta$ for some $\alpha, \beta > 0$, and present a polynomial time algorithm that finds an approximate minimizer of $F$ for this noise model. Previously, Markov chains, such as the stochastic gradient Langevin dynamics, have been used to arrive at approximate solutions to these optimization problems. However, for the noise model considered in this paper, no single temperature allows such a Markov chain to both mix quickly and concentrate near the global minimizer. We bypass this by combining “simulated annealing" with the stochastic gradient Langevin dynamics, and gradually decreasing the temperature of the chain in order to approach the global minimizer. As a corollary one can approximately minimize a nonconvex function that is close to a convex function; however, the closeness can deteriorate as one moves away from the optimum.

Ranking algorithms are deployed widely to order a set of items in applications such as search engines, news feeds, and recommendation systems. Recent studies, however, have shown that, left unchecked, the output of ranking algorithms can result in decreased diversity in the type of content presented, promote stereotypes, and polarize opinions. In order to address such issues, we study the following variant of the traditional ranking problem when, in addition, there are fairness or diversity constraints. Given a collection of items along with 1) the value of placing an item in a particular position in the ranking, 2) the collection of sensitive attributes (such as gender, race, political opinion) of each item and 3) a collection of constraints that, for each k, bound the number of items with each attribute that are allowed to appear in the top k positions of the ranking, the goal is to output a ranking that maximizes the value with respect to the original rank quality metric while respecting the constraints. This problem encapsulates various well-studied problems related to bipartite and hypergraph matching as special cases and turns out to be hard to approximate even with simple constraints. Our main technical contributions are fast exact and approximation algorithms along with complementary hardness results that, together, come close to settling the approximability of this constrained ranking maximization problem. Unlike prior work on the constrained matching problems, our algorithm runs in linear time, even when the number of constraints is large, its approximation ratio does not depend on the number of constraints, and it produces solutions with small constraint violations. Our results rely on insights about the constrained matching problem when the objective satisfies properties that appear in common ranking metrics such as Discounted Cumulative Gain, Spearman's rho or Bradley-Terry.

Sampling methods that choose a subset of the data proportional to its diversity in the feature space are popular for data summarization. However, recent studies have noted the occurrence of bias {–} e.g., under or over representation of a particular gender or ethnicity {–} in such data summarization methods. In this paper we initiate a study of the problem of outputting a diverse and fair summary of a given dataset. We work with a well-studied determinantal measure of diversity and corresponding distributions (DPPs) and present a framework that allows us to incorporate a general class of fairness constraints into such distributions. Designing efficient algorithms to sample from these constrained determinantal distributions, however, suffers from a complexity barrier; we present a fast sampler that is provably good when the input vectors satisfy a natural property. Our empirical results on both real-world and synthetic datasets show that the diversity of the samples produced by adding fairness constraints is not too far from the unconstrained case.

Multiwinner voting rules are used to select a small representative subset of candidates or items from a larger set given the preferences of voters. However, if candidates have sensitive attributes such as gender or ethnicity (when selecting a committee), or specified types such as political leaning (when selecting a subset of news items), an algorithm that chooses a subset by optimizing a multiwinner voting rule may be unbalanced in its selection -- it may under or over represent a particular gender or political orientation in the examples above. We introduce an algorithmic framework for multiwinner voting problems when there is an additional requirement that the selected subset should be ``fair'' with respect to a given set of attributes. Our framework provides the flexibility to (1) specify fairness with respect to multiple, non-disjoint attributes (e.g., ethnicity and gender) and (2) specify a score function. We study the computational complexity of this constrained multiwinner voting problem for monotone and submodular score functions and present several approximation algorithms and matching hardness of approximation results for various attribute group structure and types of score functions. We also present simulations that suggest that adding fairness constraints may not affect the scores significantly when compared to the unconstrained case.

We present a prototype for a news search engine that presents balanced viewpoints across liberal and conservative articles with the goal of depolarizing content and allowing users to escape their filter bubble. The balancing is done according to flexible user-defined constraints, and leverages recent advances in constrained bandit optimization. We showcase our balanced news feed by displaying it side-by-side with the news feed produced by a traditional (polarized) feed.

Hamiltonian Monte Carlo (HMC) is a widely deployed method to sample from high-dimensional distributions in Statistics and Machine learning. HMC is known to run very efficiently in practice and its popular second-order ``leapfrog" implementation has long been conjectured to run in $d^{1/4}$ gradient evaluations. Here we show that this conjecture is true when sampling from strongly log-concave target distributions that satisfy a weak third-order regularity property associated with the input data. Our regularity condition is weaker than the Lipschitz Hessian property and allows us to show faster convergence bounds for a much larger class of distributions than would be possible with the usual Lipschitz Hessian constant alone. Important distributions that satisfy our regularity condition include posterior distributions used in Bayesian logistic regression for which the data satisfies an ``incoherence" property. Our result compares favorably with the best available bounds for the class of strongly log-concave distributions, which grow like $d^{{1}/{2}}$ gradient evaluations with the dimension. Moreover, our simulations on synthetic data suggest that, when our regularity condition is satisfied, leapfrog HMC performs better than its competitors -- both in terms of accuracy and in terms of the number of gradient evaluations it requires.

Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in theoretical computer science and machine learning. DPPs define probability distributions over subsets of a given ground set, they exhibit interesting properties such as negative correlation, and, unlike other models of negative correlation such as Markov random fields, have efficient algorithms for sampling. When applied to kernel methods in machine learning, DPPs favor subsets of the given data with more diverse features. However, many real-world applications require efficient algorithms to sample from DPPs with additional constraints on the sampled subset, e.g., partition or matroid constraints that are important from the viewpoint of ensuring priors, resource or fairness constraints on the sampled subset. Whether one can efficiently sample from DPPs in such constrained settings is an important problem that was first raised in a survey of DPPs for machine learning by Kulesza and Taskar and studied in some recent works. The main contribution of this paper is the first resolution of the complexity of sampling from DPPs with constraints. On the one hand, we give exact efficient algorithms for sampling from constrained DPPs when the description of the constraints is in unary; this includes special cases of practical importance such as a small number of partition, knapsack or budget constraints. On the other hand, we prove that when the constraints are specified in binary, this problem is #P-hard via a reduction from the problem of computing mixed discriminants; implying that it may be unlikely that there is an FPRAS. Technically, our algorithmic result benefits from viewing the constrained sampling problem via the lens of polynomials and we obtain our complexity results by providing an equivalence between computing mixed discriminants and sampling from partition constrained DPPs. As a consequence, we obtain a few corollaries of independent interest: 1) An algorithm to count, sample (and, hence, optimize) over the base polytope of regular matroids when there are additional (succinct) budget constraints and, 2) An algorithm to evaluate and compute mixed characteristic polynomials, that played a central role in the resolution of the Kadison-Singer problem, for certain special cases.

Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors v1, ..., vm∈ ℝdand a constraint family B ⊆ 2[m], find a set S ∈ B that maximizes the squared volume of the simplex spanned by the vectors in S. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is given a collection of feature vectors that represent data such as documents or images. The volume of a collection of vectors is used as a measure of their diversity, and partition or matroid constraints over [m] are imposed in order to ensure resource or fairness constraints. Even with a simple cardinality constraint (B = (r[m])), the r problem becomes NP-hard and has received much attention starting with a result by Khachiyan [1] who gave an rO(r)approximation algorithm for this problem. Recently, Nikolov and Singh [2] presented a convex program and showed how it can be used to estimate the value of the most diverse set when there are multiple cardinality constraints (i.e., when B corresponds to a partition matroid). Their proof of the integrality gap of the convex program relied on an inequality by Gurvits [3], and was recently extended to regular matroids [4], [5]. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms - that also output a set - remained open. The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that arise from these functions which allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity where anti-concentration phenomena has recently been deployed.

We present a Gaussian random walk in a polytope that starts at a point inside and continues until it gets absorbed at a vertex. Our main result is that the probability distribution induced on the vertices by this random walk has strong negative dependence properties for matroid polytopes. Such distributions are highly sought after in randomized algorithms as they imply concentration properties. Our random walk is simple to implement, computationally efficient and can be viewed as an algorithm to round the starting point in an unbiased manner. The proof relies on a simple inductive argument that synthesizes the combinatorial structure of matroid polytopes with the geometric structure of multivariate Gaussian distributions. Our result not only implies a long line of past results in a unified and transparent manner, but also implies new results about constructing negatively associated distributions for all matroids.

We study a distributed learning process observed in human groups and other social animals. This learning process appears in settings in which each individual in a group is trying to decide over time, in a distributed manner, which option to select among a shared set of options. Specifically, we consider a stochastic dynamics in a group in which every individual selects an option in the following two-step process: (1) select a random individual and observe the option that individual chose in the previous time step, and (2) adopt that option if its stochastic quality was good at that time step. Various instantiations of such distributed learning appear in nature, and have also been studied in the social science literature. From the perspective of an individual, an attractive feature of this learning process is that it is a simple heuristic that requires extremely limited computational capacities. But what does it mean for the group -- could such a simple, distributed and essentially memoryless process lead the group as a whole to perform optimally? We show that the answer to this question is yes -- this distributed learning is highly effective at identifying the best option and is close to optimal for the group overall. Our analysis also gives quantitative bounds that show fast convergence of these stochastic dynamics. We prove our result by first defining a (stochastic) infinite population version of these distributed learning dynamics and then combining its strong convergence properties along with its relation to the finite population dynamics. Prior to our work the only theoretical work related to such learning dynamics has been either in deterministic special cases or in the asymptotic setting. Finally, we observe that our infinite population dynamics is a stochastic variant of the classic multiplicative weights update (MWU) method. Consequently, we arrive at the following interesting converse: the learning dynamics on a finite population considered here can be viewed as a novel distributed and low-memory implementation of the classic MWU method.

Several fundamental optimization and counting problems arising in computer science, mathematics and physics can be reduced to one of the following computational tasks involving polynomials and set systems: given an oracle access to an m-variate real polynomial g and to a family of (multi-)subsets ℬ of [m], (1) compute the sum of coefficients of monomials in g corresponding to all the sets that appear in B(1), or find S ε ℬ such that the monomial in g corresponding to S has the largest coefficient in g. Special cases of these problems, such as computing permanents and mixed discriminants, sampling from determinantal point processes, and maximizing sub-determinants with combinatorial constraints have been topics of much recent interest in theoretical computer science.

Over the last two centuries there have been tremendous scientific and mathematical advances in our understanding of evolution, life and its mysteries. Recently, the relatively new and powerful tool of computation has joined forces to develop this understanding further: the underlying tenet is that several natural processes, including evolution itself, can be viewed as computing or optimizing something - evolution is computation. Furthermore, as in computation, efficiency is an important consideration in evolution. As many of these evolutionary processes are described using the language of dynamical systems, this entails understanding how quickly such systems can attain their equilibria. This endeavor not only has the potential to give us fundamental insights into life, it holds the promise that we will unveil new computational models and techniques. In this talk we will see some vignettes of this interplay between evolution and computation.

In this paper we study the mixing time of evolutionary Markov chains over populations of a fixed size (N) in which each individual can be one of m types. These Markov chains have the property that they are guided by a dynamical system from the m-dimensional probability simplex to itself. Roughly, given the current state of the Markov chain, which can be viewed as a probability distribution over the m types, the next state is generated by applying this dynamical system to this distribution, and then sampling from it N times. Many processes in nature, from biology to sociology, are evolutionary and such chains can be used to model them. In this study, the mixing time is of particular interest as it determines the speed of evolution and whether the statistics of the steady state can be efficiently computed. In a recent result [Panageas, Srivastava, Vishnoi, Soda, 2016], it was suggested that the mixing time of such Markov chains is connected to the geometry of this guiding dynamical system. In particular, when the dynamical system has a fixed point which is a global attractor, then the mixing is fast. The limit sets of dynamical systems, however, can exhibit more complex behavior: they could have multiple fixed points that are not necessarily stable, periodic orbits, or even chaos. Such behavior arises in important evolutionary settings such as the dynamics of sexual evolution and that of grammar acquisition. In this paper we prove that the geometry of the dynamical system can also give tight mixing time bounds when the dynamical system has multiple fixed points and periodic orbits. We show that the mixing time continues to remain small in the presence of several unstable fixed points and is exponential in N when there are two or more stable fixed points. As a consequence of our results, we obtain a phase transition result for the mixing time of the sexual/grammar model mentioned above. We arrive at the conclusion that in the interesting parameter regime for these models, i.e., when there are multiple stable fixed points, the mixing is slow. Our techniques strengthen the connections between Markov chains and dynamical systems and we expect that the tools developed in this paper should have a wider applicability.

We study the Poincare-Bendixson theorem for two-dimensional continuous dynamical systems in compact domains from the point of view of computation, seeking algorithms for finding the limit cycle promised by this classical result. We start by considering a discrete analogue of this theorem and show that both finding a point on a limit cycle, and determining if a given point is on one, are PSPACE-complete. For the continuous version, we show that both problems are uncomputable in the real complexity sense; i.e., their complexity is arbitrarily high. Subsequently, we introduce a notion of an approximate cycle and prove an approximate Poincare-Bendixson theorem guaranteeing that some orbits come very close to forming a cycle in the absence of approximate fixpoints; surprisingly, it holds for all dimensions. The corresponding computational problem defined in terms of arithmetic circuits is PSPACE-complete.

In this paper we study dynamics inspired by Physarum polycephalum (a slime mold) for solving linear programs [NTY00, IJNT11, JZ12]. These dynamics are arrived at by a local and mechanistic interpretation of the inner workings of the slime mold and a global optimization perspective has been lacking even in the simplest of instances. Our first result is an interpretation of the dynamics as an optimization process. We show that Physarum dynamics can be seen as a steepest-descent type algorithm on a certain Riemannian manifold. Moreover, we prove that the trajectories of Physarum are in fact paths of optimizers to a parametrized family of convex programs, in which the objective is a linear cost function regularized by an entropy barrier. Subsequently, we rigorously establish several important properties of solution curves of Physarum. We prove global existence of such solutions and show that they have limits, being optimal solutions of the underlying LP. Finally, we show that the discretization of the Physarum dynamics is efficient for a class of linear programs, which include unimodular constraint matrices. Thus, together, our results shed some light on how nature might be solving instances of perhaps the most complex problem in P: linear programming.

In this paper we prove that the mixing time of a broad class of evolutionary dynamics in finite, unstructured populations is roughly logarithmic in the size of the state space. An important special case of such a stochastic process is the Wright-Fisher model from evolutionary biology (with selection and mutation) on a population of size N over m genotypes. Our main result implies that the mixing time of this process is O(log N) for all mutation rates and fitness landscapes, and solves the main open problem from [4]. In particular, it significantly extends the main result in [18] who proved this for m = 2. Biologically, such models have been used to study the evolution of viral populations with applications to drug design strategies countering them. Here the time it takes for the population to reach a steady state is important both for the estimation of the steady-state structure of the population as well in the modeling of the treatment strength and duration. Our result, that such populations exhibit rapid mixing, makes both of these approaches sound. Technically, we make a novel connection between Markov chains arising in evolutionary dynamics and dynamical systems on the probability simplex. This allows us to use the local and global stability properties of the fixed points of such dynamical systems to construct a contractive coupling in a fairly general setting. We expect that our mixing time result would be useful beyond the evolutionary biology setting, and the techniques used here would find applications in bounding the mixing times of Markov chains which have a natural underlying dynamical system.

In the last few years, there has been a significant interest in the computational abilities of Physarum polycephalum (a slime mold). This arose from a remarkable experiment which showed that this organism can compute shortest paths in a maze [10]. Subsequently, the workings of Physarum were mathematically modeled as a dynamical system and algorithms inspired by this model were proposed to solve several graph problems: shortest paths, flows, and linear programs to name a few. Indeed, computer scientists have initiated a rigorous study of these dynamics and a first step towards this was taken by [1,2] who proved that the Physarum dynamics for the shortest path problem are efficient (when edge-lengths are polynomially bounded). In this paper, we take this further: we prove that the discrete time Physarum-dynamics can also efficiently solve the uncapacitated mincost flow problems on undirected and directed graphs; problems that are non-trivial generalizations of shortest path. This raises the tantalizing possibility that nature, via evolution, developed algorithms that efficiently solve some of the most complex computational problems, about a billion years before we did.

In this paper we study the mixing times of a Wright-Fisher model for an asexual population. Typically, such models consist of a population of several genotypes where each genotype reproduces at a different rate, genotypes can mutate to each other, and the population is subject to the evolutionary pressure of selection. While such models have been used extensively to study different types of populations, recently, such stochastic finite population models have been used to model viral populations with the goal of understanding the effect of mutagenic drugs. Here, the time it takes for the population to reach a steady state is important both for carrying out simulations and to determine treatment strength and duration. Despite their importance, and having been widely studied, there has been a lack of such bounds for many relevant ranges of model parameters even for the case of two genotypes (single locus); primarily due to their difficulty, see [Eth11].

In this article, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into ℓ 1 with constant distortion. We show that for an arbitrarily small constant δ > 0, for all large enough n , there is an n -point negative type metric which requires distortion at least (log log n ) 1/6-δ to embed into ℓ 1 . Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC), establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (nonuniform) S PARSEST C UT problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an “integrality gap instance” for S PARSEST C UT . Towards this, we first construct an integrality gap instance for a natural SDP relaxation of U NIQUE G AMES . Then we “simulate” the PCP reduction and “translate” the integrality gap instance of U NIQUE G AMES to an integrality gap instance of S PARSEST C UT . This enables us to prove a (log log n ) 1/6-δ integrality gap for S PARSEST C UT , which is known to be equivalent to the metric embedding lower bound.

We study the problem of computing max-entropy distributions over a discrete set of objects subject to observed marginals. There has been a tremendous amount of interest in such distributions due to their applicability in areas such as statistical physics, economics, biology, information theory, machine learning, combinatorics and algorithms. However, a rigorous and systematic study of how to compute such distributions has been lacking. Since the underlying set of discrete objects can be exponential in the input size, the first question in such a study is if max-entropy distributions have polynomially-sized descriptions. We start by giving a structural result which shows that such succinct descriptions exist under very general conditions. Subsequently, we use techniques from convex programming to give a meta-algorithm that can efficiently (approximately) compute max-entropy distributions provided one can efficiently (approximately) count the underlying discrete set. Thus, we can translate a host of existing counting algorithms, developed in an unrelated context, into algorithms that compute max-entropy distributions. Conversely, we prove that counting oracles are necessary for computing max-entropy distributions: we show how algorithms that compute max-entropy distributions can be converted into counting algorithms.

We prove that for an arbitrarily small constant $\varepsilon>0,$ the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor $2^{\log ^{1-\varepsilon}n}$, under the assumption that NP $\not \subseteq$ SIZE$(2^{\log^{O(1/\varepsilon)} n})$.

No abstract available.

In this paper we consider the problem of computing market equilibria in the Fisher setting for utility models such as spending constraint and perfect, price-discrimination. These models were inspired from modern e-commerce settings and attempt to bridge the gap between the computationally hard but realistic separable, piecewise-linear and concave utility model and, the tractable but less relevant linear utility case. While there are polynomial time algorithms known for these problems, the question of whether there exist polynomial time Simplex-like algorithms has remained elusive, even for linear markets. Such algorithms are desirable due to their conceptual simplicity, ease of implementation and practicality. This paper takes a significant step towards this goal by presenting the first Simplex-like algorithms for these markets assuming a positive resolution of an algebraic problem of Cucker, Koiran and Smale. Unconditionally, our algorithms are FPTASs; they compute prices and allocations such that each buyer derives at least a -fraction of the utility at a true market equilibrium, and their running times are polynomial in the input length and 1/ε. We start with convex programs which capture market equilibria in each setting and, in a systematic way, convert them into linear complementarity problem (LCP) formulations. Then, departing from previous approaches which try to pivot on a single polyhedron associated to the LCP obtained, we carefully construct a polynomial-length sequence of polyhedra, one containing the other, such that starting from an optimal solution to one allows us to obtain an optimal solution to the next in the sequence in a polynomial number of complementary pivot steps. Our framework to convert a convex program into an LCP and then come up with a Simplex-like algorithm that moves on a sequence of connected polyhedra may be of independent interest.

A randomized polynomial time algorithm is presented which, for every simple, connected, k-regular graph on n vertices, finds a tour that visits every vertex and has length at most (1 + √(64/1n k)) n with high probability. The proof follows simply from results developed in the context of permanents; Egorychev's and Falikman's theorem which lower bounds the permanent of a doubly stochastic matrix and the polynomial time algorithm of Jerrum, Sinclair and Vigoda which samples a near-random, perfect matching from a bipartite graph. The techniques in this paper suggest new permanent-based approaches for TSP which could be useful in attacking other interesting cases of TSP.

We prove that for an arbitrarily small constant ε>0, assuming NP⊈ DTIME (2logO 1-ε n), the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor better than 2log1-ε n. This improves upon the previous hardness factor of (log n)δ for some δ>0 due to [AKKV05].

We give a novel spectral approximation algorithm for the balanced (edge-)separator problem that, given a graph G, a constant balance b ∈ (0,1/2], and a parameter γ, either finds an Ω(b)-balanced cut of conductance O(√γ) in G, or outputs a certificate that all b-balanced cuts in G have conductance at least γ, and runs in time ~O(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(-L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrix-exponential-vector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(-A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector v, in time roughly ~O(mA), independent of the norm of A, where mA is the number of non-zero entries of A. This uses, in a non-trivial way, the result of Spielman and Teng on inverting symmetric and diagonally-dominant matrices in ~O(mA) time. Finally, using old and new uniform approximations to e-x we show how to obtain, via the Lanczos method, a simple algorithm to compute exp(-A)v for symmetric PSD matrices that runs in time roughly O(tA⋅ √norm(A)), where tA is the time required for the computation of the vector Aw for given vector w. As an application, we obtain a simple and practical algorithm, with output conductance O(√γ), for balanced separator that runs in time O(m/√γ). This latter algorithm matches the running time, but improves on the approximation guarantee of the Evolving-Sets-based algorithm by Andersen and Peres for balanced separator.

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, μ c, however, is not known. Application of the quasispecies theory to determine μ c poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and μ c. We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated μ c to be 7 x 10(-5)-1 x 10(-4) substitutions/site/replication, ≈ 2-6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, μ c increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of μ c may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.

We present a modification of “Normalized Cuts” to incorporate priors which can be used for constrained image segmentation. Compared to previous generalizations of “Normalized Cuts” which incorporate constraints, our technique has two advantages. First, we seek solutions which are sufficiently “correlated” with priors which allows us to use noisy top-down information, for example from an object detector. Second, given the spectral solution of the unconstrained problem, the solution of the constrained one can be computed in small additional time, which allows us to run the algorithm in an interactive mode. We compare our algorithm to other graph cut based algorithms and highlight the advantages.

The subspace approximation problem Subspace(k, p) asks for a k dimensional linear subspace that fits a given set of m points in ℝn optimally. The error for fitting is a generalization of the least squares fit and uses the ℓp norm of the distances (ℓ2 distances) of the points from the subspace, e.g., p = ∞ means minimizing the ℓ2 distance of the farthest point from the subspace. Previous work on subspace approximation considers either the case of small or constant k and p [27, 11, 14] or the case of p = ∞ [16, 8, 17, 7, 24, 23, 29]. In this paper, we study the algorithms and hardness for Subspace(k, p) in the natural range 1 ≤ k ≤ n and 2 ≤ p ≤ ∞. Our results are as follows. Extending the convex relaxation and rounding techniques of Varadarajan, Venkatesh, Ye and Zhang [29], we give a polynomial time approximation algorithm for Subspace(k, p), for any k and any p ≥ 2, with an approximation guarantee of roughly , where is the pth moment of a standard normal variable. This improves to γp for k = n − 1. We exhibit a simple integrality gap (or “rank gap”) instance for our convex relaxation giving a gap of γp(1 − ε), for any constant ε > 0. We show that, assuming the Unique Games Conjecture, the subspace approximation problem is hard to approximate within a factor better than γp(1 − ε), for any constant ε > 0. Our hardness reduction involves a dictatorship test which is somewhat different from “long code” based tests used in reductions from Unique Games, and seems better suited for problems of a continuous nature.

In a beautiful result, Raghavendra established optimal Unique Games Conjecture (UGC)-based inapproximability for a large class of constraint satisfaction problems (CSPs). In the class of CSPs he considers, of which Maximum Cut is a prominent example, the goal is to find an assignment which maximizes a weighted fraction of constraints satisfied. He gave a generic semi-definite program (SDP) for this class of problems and showed how the approximability of each problem is determined by the corresponding SDP (upto an arbitrarily small additive error) assuming the UGC. He noted that his techniques do no apply to CSPs with strict constraints (all of which must be satisfied) such as Vertex Cover. In this paper we address the approximability of these strict-CSPs. In the class of CSPs we consider, one is given a set of constraints over a set of variables, and a cost function over the assignments, the goal is to find an assignment to the variables of minimum cost which satisfies all the constraints. We present a generic linear program (LP) for a large class of strict-CSPs and give a systematic way to convert integrality gaps for this LP into UGC-based inapproximability results. Some important problems whose approximability our framework captures are Vertex Cover, Hypergraph Vertex Cover, k-partite-Hypergraph Vertex Cover, Independent Set and other covering and packing problems over q-ary alphabets, and a scheduling problem. For the covering and packing problems, which occur quite commonly in practice as well, we provide a matching rounding algorithm, thus settling their approximability upto an arbitrarily small additive error.

In this paper, we consider the following graph partitioning problem: The input is an undirected graph G = (V, E), a balance parameter b ∊ (0, 1/2] and a target conductance value γ ∊ (0, 1). The output is a cut which, if non-empty, is of conductance at most O(f), for some function f(G, γ), and which is either balanced or well correlated with all cuts of conductance at most γ. In a seminal paper, Spielman and Teng [16] gave an -time algorithm for and used it to decompose graphs into a collection of near-expanders [18]. We present a new spectral algorithm for this problem which runs in time for f = √γ. Our result yields the first nearly-linear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primal-dual semidefinite-programming (SDP) approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than γ, somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale's [3] framework to bound the running time. Our result contains technical ingredients that may be of independent interest.

We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander.

In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cut-matching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a poly-logarithmic number of single-commodity max-flow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω(√log n) in the cut-matching game framework. These results suggest that the simple and concrete abstraction of the cut-matching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut.

Arora, Rao and Vazirani [2] showed that the standard semi-definite programming (SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an integrality gap of O(√log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture) by constructing an Ω(log log n) integrality gap instance. Khot and Vishnoi [16] had earlier disproved the non-uniform version of the ARV-Conjecture.A simple "stretching" of the integrality gap instance for the Sparsest Cut problem serves as an Ω(log log n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in [6, 11], where it was shown that its integrality gap is bounded from above by O(√log n log log n).

We study the following question: What is the smallest t such that every symmetric boolean function on k variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t? We exclude the constant functions for which there is no such t and the parity functions for which t has to be k. Let τ(k) be the smallest such t. The main contribution of this paper is a proof of the following self similar nature of this question: If τ(l) ≤ s, then for any ɛ> 0 and � for k ≥ k0(l, ɛ), τ(k) ≤ k s+1 l+1 + ɛ Coupling this result with a computer based search which establishes τ(30) = 2, one obtains that for large enough k, τ(k) ≤ 3k/31. The motivation for our work is to understand the complexity of learning symmetric juntas. A k-junta is a boolean function of n variables that depends only on an unknown subset of k variables. If f is symmetric in the variables it depends on, it is called a symmetric k-junta. Our results imply an algorithm to learn the class of symmetric k-juntas, in the uniform PAC learning model, in time approximately n 3k 31. This improves on a result of Mossel, O’Donnell and Servedio in [11], who show that symmetric k-juntas can be ∗ Research supported by NSF grants CCR-0002299 and CCF-0431023.

We show that, unless NP/spl sube/DTIME(2/sup poly log(n)/) the closest vector problem with pre-processing, for /spl lscr//sub p/ norm for any p /spl ges/ 1, is hard to approximate within a factor of (log n)/sup 1/p - /spl epsi//' /P for any /spl epsi/ > 0. This improves the previous best factor of 3/sup 1/p/ - /spl epsi/ due to Regev (2004). Our results also imply that under the same complexity assumption, the nearest codeword problem with pre-processing is hard to approximate within a factor of (log n)/sup 1 - /spl epsi//' for any /spl epsi/ > 0.

In this paper, we disprove the following conjecture due to Goemans (1997) and Linial (2002): "Every negative type metric embeds into with constant distortion." We show that for every /spl delta/ > 0, and for large enough n, there is an n-point negative type metric which requires distortion at-least (log log n) /sup 1/6-/spl delta// to embed into l/sub 1/. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot (2002), establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies super-constant hardness results for (non-uniform) sparsest cut and minimum uncut problems. It is already known that the UGC also implies an optimal hardness result for maximum cut (2004). Though these hardness results depend on the UGC, the integrality gap instances rely "only" on the PCP reductions for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of unique games. Then, we "simulate" the PCP reduction and "translate"the integrality gap instance of unique games to integrality gap instances for the respective cut problems! This enables us to prove a (log log n) /sup 1/6-/spl delta// integrality gap for (nonuniform) sparsest cut and minimum uncut, and an optimal integrality gap for maximum cut. All our SDP solutions satisfy the so-called "triangle inequality" constraints. This also shows, for the first time, that the triangle inequality constraints do not add any power to the Goemans-Williamson's SDP relaxation of maximum cut. The integrality gap for sparsest cut immediately implies a lower bound for embedding negative type metrics into l/sub i/. It also disproves the non-uniform version of Arora, Rao and Vazirani's Conjecture (2004), asserting that the integrality gap of the sparsest cut SDP, with the triangle inequality constraints, is bounded from above by a constant.