arXiv subjects

Ilias Zadik

Publications and source records attributed to Ilias Zadik.

50 records · Page 3Linked to original sources

arXiv2019

The Landscape of the Planted Clique Problem: Dense subgraphs and the Overlap Gap Property

In this paper we study the computational-statistical gap of the planted clique problem, where a clique of size $k$ is planted in an Erdos Renyi graph $G(n,\frac{1}{2})$ resulting in a graph $G\left(n,\frac{1}{2},k\right)$. The goal is to recover the planted clique vertices by observing $G\left(n,\frac{1}{2},k\right)$ . It is known that the clique can be recovered as long as $k \geq \left(2+ε\right)\log n $ for any $ε>0$, but no polynomial-time algorithm is known for this task unless $k=Ω\left(\sqrt{n} \right)$. Following a statistical-physics inspired point of view as an attempt to understand this computational-statistical gap, we study the landscape of the "sufficiently dense" subgraphs of $G$ and their overlap with the planted clique. Using the first moment method, we study the densest subgraph problems for subgraphs with fixed, but arbitrary, overlap size with the planted clique, and provide evidence of a phase transition for the presence of Overlap Gap Property (OGP) at $k=Θ\left(\sqrt{n}\right)$. OGP is a concept introduced originally in spin glass theory and known to suggest algorithmic hardness when it appears. We establish the presence of OGP when $k$ is a small positive power of $n$ by using a conditional second moment method. As our main technical tool, we establish the first, to the best of our knowledge, concentration results for the $K$-densest subgraph problem for the Erdos-Renyi model $G\left(n,\frac{1}{2}\right)$ when $K=n^{0.5-ε}$ for arbitrary $ε>0$. Finally, to study the OGP we employ a certain form of overparametrization, which is conceptually aligned with a large body of recent work in learning theory and optimization.

math.ST↗

arXiv2019

All-or-Nothing Phenomena: From Single-Letter to High Dimensions

We consider the linear regression problem of estimating a $p$-dimensional vector $β$ from $n$ observations $Y = X β+ W$, where $β_j \stackrel{\text{i.i.d.}}{\sim} π$ for a real-valued distribution $π$ with zero mean and unit variance, $X_{ij} \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,1)$, and $W_i\stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0, σ^2)$. In the asymptotic regime where $n/p \to δ$ and $ p/ σ^2 \to \mathsf{snr}$ for two fixed constants $δ, \mathsf{snr}\in (0, \infty)$ as $p \to \infty$, the limiting (normalized) minimum mean-squared error (MMSE) has been characterized by the MMSE of an associated single-letter (additive Gaussian scalar) channel. In this paper, we show that if the MMSE function of the single-letter channel converges to a step function, then the limiting MMSE of estimating $β$ in the linear regression problem converges to a step function which jumps from $1$ to $0$ at a critical threshold. Moreover, we establish that the limiting mean-squared error of the (MSE-optimal) approximate message passing algorithm also converges to a step function with a larger threshold, providing evidence for the presence of a computational-statistical gap between the two thresholds.

math.ST↗

arXiv2019

Inference in High-Dimensional Linear Regression via Lattice Basis Reduction and Integer Relation Detection

We focus on the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $β^*\in\mathbb{R}^p$ from its linear measurements, using a small number $n$ of samples. Unlike most of the literature, we make no sparsity assumption on $β^*$, but instead adopt a different regularization: In the noiseless setting, we assume $β^*$ consists of entries, which are either rational numbers with a common denominator $Q\in\mathbb{Z}^+$ (referred to as $Q$-rationality); or irrational numbers supported on a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-support assumption. Using a novel combination of the PSLQ integer relation detection, and LLL lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a $β^*\in\mathbb{R}^p$ enjoying the mixed-support assumption, from its linear measurements $Y=Xβ^*\in\mathbb{R}^n$ for a large class of distributions for the random entries of $X$, even with one measurement $(n=1)$. In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a $β^*\in\mathbb{R}^p$ enjoying $Q$-rationality, from its noisy measurements $Y=Xβ^*+W\in\mathbb{R}^n$, even with a single sample $(n=1)$. We further establish for large $Q$, and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample $(n=1)$ regime, where the sparsity-based methods such as LASSO and Basis Pursuit are known to fail. Furthermore, our results also reveal an algorithmic connection between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems.

math.ST↗

arXiv2019

High-Dimensional Regression with Binary Coefficients. Estimating Squared Error and a Phase Transition

We consider a sparse linear regression model Y=Xβ^{*}+W where X has a Gaussian entries, W is the noise vector with mean zero Gaussian entries, and β^{*} is a binary vector with support size (sparsity) k. Using a novel conditional second moment method we obtain a tight up to a multiplicative constant approximation of the optimal squared error \min_β\|Y-Xβ\|_{2}, where the minimization is over all k-sparse binary vectors β. The approximation reveals interesting structural properties of the underlying regression problem. In particular, a) We establish that n^*=2k\log p/\log (2k/σ^{2}+1) is a phase transition point with the following "all-or-nothing" property. When n exceeds n^{*}, (2k)^{-1}\|β_{2}-β^*\|_0\approx 0, and when n is below n^{*}, (2k)^{-1}\|β_{2}-β^*\|_0\approx 1, where β_2 is the optimal solution achieving the smallest squared error. With this we prove that n^{*} is the asymptotic threshold for recovering β^* information theoretically. b) We compute the squared error for an intermediate problem \min_β\|Y-Xβ\|_{2} where minimization is restricted to vectors βwith \|β-β^{*}\|_0=2k ζ, for ζ\in [0,1]. We show that a lower bound part Γ(ζ) of the estimate, which corresponds to the estimate based on the first moment method, undergoes a phase transition at three different thresholds, namely n_{\text{inf,1}}=σ^2\log p, which is information theoretic bound for recovering β^* when k=1 and σis large, then at n^{*} and finally at n_{\text{LASSO/CS}}. c) We establish a certain Overlap Gap Property (OGP) on the space of all binary vectors βwhen n\le ck\log p for sufficiently small constant c. We conjecture that OGP is the source of algorithmic hardness of solving the minimization problem \min_β\|Y-Xβ\|_{2} in the regime n<n_{\text{LASSO/CS}}.

stat.ML↗

arXiv2019

Sparse High-Dimensional Linear Regression. Algorithmic Barriers and a Local Search Algorithm

We consider a sparse high dimensional regression model where the goal is to recover a $k$-sparse unknown vector $β^*$ from $n$ noisy linear observations of the form $Y=Xβ^*+W \in \mathbb{R}^n$ where $X \in \mathbb{R}^{n \times p}$ has iid $N(0,1)$ entries and $W \in \mathbb{R}^n$ has iid $N(0,σ^2)$ entries. Under certain assumptions on the parameters, an intriguing assymptotic gap appears between the minimum value of $n$, call it $n^*$, for which the recovery is information theoretically possible, and the minimum value of $n$, call it $n_{\mathrm{alg}}$, for which an efficient algorithm is known to provably recover $β^*$. In \cite{gamarnikzadik} it was conjectured that the gap is not artificial, in the sense that for sample sizes $n \in [n^*,n_{\mathrm{alg}}]$ the problem is algorithmically hard. We support this conjecture in two ways. Firstly, we show that the optimal solution of the LASSO provably fails to $\ell_2$-stably recover the unknown vector $β^*$ when $n \in [n^*,c n_{\mathrm{alg}}]$, for some sufficiently small constant $c>0$. Secondly, we establish that $n_{\mathrm{alg}}$, up to a multiplicative constant factor, is a phase transition point for the appearance of a certain Overlap Gap Property (OGP) over the space of $k$-sparse vectors. The presence of such an Overlap Gap Property phase transition, which originates in statistical physics, is known to provide evidence of an algorithmic hardness. Finally we show that if $n>C n_{\mathrm{alg}}$ for some large enough constant $C>0$, a very simple algorithm based on a local search improvement rule is able both to $\ell_2$-stably recover the unknown vector $β^*$ and to infer correctly its support, adding it to the list of provably successful algorithms for the high dimensional linear regression problem.

math.ST↗

arXiv2019

The All-or-Nothing Phenomenon in Sparse Linear Regression

We study the problem of recovering a hidden binary $k$-sparse $p$-dimensional vector $β$ from $n$ noisy linear observations $Y=Xβ+W$ where $X_{ij}$ are i.i.d. $\mathcal{N}(0,1)$ and $W_i$ are i.i.d. $\mathcal{N}(0,σ^2)$. A closely related hypothesis testing problem is to distinguish the pair $(X,Y)$ generated from this structured model from a corresponding null model where $(X,Y)$ consist of purely independent Gaussian entries. In the low sparsity $k=o(p)$ and high signal to noise ratio $k/σ^2=Ω\left(1\right)$ regime, we establish an `All-or-Nothing' information-theoretic phase transition at a critical sample size $n^*=2 k\log \left(p/k\right) /\log \left(1+k/σ^2\right)$, resolving a conjecture of \cite{gamarnikzadik}. Specifically, we show that if $\liminf_{p\to \infty} n/n^*>1$, then the maximum likelihood estimator almost perfectly recovers the hidden vector with high probability and moreover the true hypothesis can be detected with a vanishing error probability. Conversely, if $\limsup_{p\to \infty} n/n^*<1$, then it becomes information-theoretically impossible even to recover an arbitrarily small but fixed fraction of the hidden vector support, or to test hypotheses strictly better than random guess. Our proof of the impossibility result builds upon two key techniques, which could be of independent interest. First, we use a conditional second moment method to upper bound the Kullback-Leibler (KL) divergence between the structured and the null model. Second, inspired by the celebrated area theorem, we establish a lower bound to the minimum mean squared estimation error of the hidden vector in terms of the KL divergence between the two models.

math.ST↗

arXiv2019

A simple bound on the BER of the MAP decoder for massive MIMO systems

The deployment of massive MIMO systems has revived much of the interest in the study of the large-system performance of multiuser detection systems. In this paper, we prove a non-trivial upper bound on the bit-error rate (BER) of the MAP detector for BPSK signal transmission and equal-power condition. In particular, our bound is approximately tight at high-SNR. The proof is simple and relies on Gordon's comparison inequality. Interestingly, we show that under the assumption that Gordon's inequality is tight, the resulting BER prediction matches that of the replica method under the replica symmetry (RS) ansatz. Also, we prove that, when the ratio of receive to transmit antennas exceeds $0.9251$, the replica prediction matches the matched filter lower bound (MFB) at high-SNR. We corroborate our results by numerical evidence.

cs.IT↗

arXiv2018

High Dimensional Linear Regression using Lattice Basis Reduction

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector $β^*$ from $n$ noisy linear observations $Y=Xβ^*+W \in \mathbb{R}^n$, for known $X \in \mathbb{R}^{n \times p}$ and unknown $W \in \mathbb{R}^n$. Unlike most of the literature on this model we make no sparsity assumption on $β^*$. Instead we adopt a regularization based on assuming that the underlying vectors $β^*$ have rational entries with the same denominator $Q \in \mathbb{Z}_{>0}$. We call this $Q$-rationality assumption. We propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm. We establish that under the $Q$-rationality assumption, our algorithm recovers exactly the vector $β^*$ for a large class of distributions for the iid entries of $X$ and non-zero noise $W$. We prove that it is successful under small noise, even when the learner has access to only one observation ($n=1$). Furthermore, we prove that in the case of the Gaussian white noise for $W$, $n=o\left(p/\log p\right)$ and $Q$ sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.

math.ST↗

arXiv2018

Private Algorithms Can Always Be Extended

We consider the following fundamental question on $ε$-differential privacy. Consider an arbitrary $ε$-differentially private algorithm defined on a subset of the input space. Is it possible to extend it to an $ε'$-differentially private algorithm on the whole input space for some $ε'$ comparable with $ε$? In this note we answer affirmatively this question for $ε'=2ε$. Our result applies to every input metric space and space of possible outputs. This result originally appeared in a recent paper by the authors [BCSZ18]. We present a self-contained version in this note, in the hopes that it will be broadly useful.

math.ST↗

arXiv2018

Revealing Network Structure, Confidentially: Improved Rates for Node-Private Graphon Estimation

Motivated by growing concerns over ensuring privacy on social networks, we develop new algorithms and impossibility results for fitting complex statistical models to network data subject to rigorous privacy guarantees. We consider the so-called node-differentially private algorithms, which compute information about a graph or network while provably revealing almost no information about the presence or absence of a particular node in the graph. We provide new algorithms for node-differentially private estimation for a popular and expressive family of network models: stochastic block models and their generalization, graphons. Our algorithms improve on prior work, reducing their error quadratically and matching, in many regimes, the optimal nonprivate algorithm. We also show that for the simplest random graph models ($G(n,p)$ and $G(n,m)$), node-private algorithms can be qualitatively more accurate than for more complex models---converging at a rate of $\frac{1}{ε^2 n^{3}}$ instead of $\frac{1}{ε^2 n^2}$. This result uses a new extension lemma for differentially private algorithms that we hope will be broadly useful.

math.ST↗

arXiv2018

Orthogonal Machine Learning: Power and Limitations

Double machine learning provides $\sqrt{n}$-consistent estimates of parameters of interest even when high-dimensional or nonparametric nuisance parameters are estimated at an $n^{-1/4}$ rate. The key is to employ Neyman-orthogonal moment equations which are first-order insensitive to perturbations in the nuisance parameters. We show that the $n^{-1/4}$ requirement can be improved to $n^{-1/(2k+2)}$ by employing a $k$-th order notion of orthogonality that grants robustness to more complex or higher-dimensional nuisance parameters. In the partially linear regression setting popular in causal inference, we show that we can construct second-order orthogonal moments if and only if the treatment residual is not normally distributed. Our proof relies on Stein's lemma and may be of independent interest. We conclude by demonstrating the robustness benefits of an explicit doubly-orthogonal estimation procedure for treatment effect.

cs.LG↗

arXiv2017

Mixed-integer convex representability

We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.

math.OC↗

arXiv2013

Universal Padé approximants and their behaviour on the boundary

There are several kinds of universal Taylor series. In one such kind the universal approximation is required at every boundary point of the domain of definition $\OO$ of the universal function $f$. In another kind the universal approximation is not required at any point of $\partial\OO$ but in this case the universal function $f$ can be taken smooth on $\oO$ and, moreover, it can be approximated by it's Taylor partial sums on every compact subset of $\oO$. Similar generic phenomena hold when the partial sums of the Taylor expansion of the universal function are replaced by some Padé approximants of it. In the present paper we show that in the case of Padé approximants, if $\OO$ is an open set and $S,T$ are two subsets of $\partial\OO$ that satisfy some conditions, then there exists a universal function $f\in H(\OO)$ which is smooth on $\OO\cup S$ and has some Padé approximants that approximate $f$ on each compact subset of $\OO\cup S$ and simultaneously obtain universal approximation on each compact subset of $(\C\sm\oO)\cup T$. A sufficient condition for the above to happen is $\oS\cap\oT=\emptyset$, while a necessary and sufficient condition is not known.

math.CV↗

arXiv2012

Padé Approximants, density of rational functions in $\bbb{A^\infty(\OO)}$ and smoothness of the integration operator

First we establish some generic universalities for Padé approximants in the closure $X^\infty(\OO)$ in $A^\infty(\OO)$ of all rational functions with poles off $\oO$, the closure taken in $\C$ of the domain $\OO\subset\C$.\ Next we give sufficient conditions on $\OO$ so that $X^\infty(\OO)=A^\infty(\OO)$.\ Some of these conditions imply that, even if the boundary $\partial\OO$ of a Jordan domain $\OO$ has infinite length, the integration operator on $\OO$ preserves $H^\infty(\OO)$ and $A(\OO)$ as well.\ We also give an example of a Jordan domain $\OO$ and a function $f\in A(\OO)$, such that its antiderivative is not bounded on $\OO$.\ Finally we restate these results for Volterra operators on the open unit disc $D$ and we complete them by some generic results.

math.CV↗