arXiv subjects

David Gamarnik

Publications and source records attributed to David Gamarnik.

At least 55 records · Page 3Linked to original sources

arXiv2021

Algorithmic Obstructions in the Random Number Partitioning Problem

We consider the algorithmic problem of finding a near-optimal solution for the number partitioning problem (NPP). The NPP appears in many applications, including the design of randomized controlled trials, multiprocessor scheduling, and cryptography; and is also of theoretical significance. It possesses a so-called statistical-to-computational gap: when its input $X$ has distribution $\mathcal{N}(0,I_n)$, its optimal value is $Θ(\sqrt{n}2^{-n})$ w.h.p.; whereas the best polynomial-time algorithm achieves an objective value of only $2^{-Θ(\log^2 n)}$, w.h.p. In this paper, we initiate the study of the nature of this gap. Inspired by insights from statistical physics, we study the landscape of NPP and establish the presence of the Overlap Gap Property (OGP), an intricate geometric property which is known to be a rigorous evidence of an algorithmic hardness for large classes of algorithms. By leveraging the OGP, we establish that (a) any sufficiently stable algorithm, appropriately defined, fails to find a near-optimal solution with energy below $2^{-ω(n \log^{-1/5} n)}$; and (b) a very natural MCMC dynamics fails to find near-optimal solutions. Our simulations suggest that the state of the art algorithm achieving $2^{-Θ(\log^2 n)}$ is indeed stable, but formally verifying this is left as an open problem. OGP regards the overlap structure of $m-$tuples of solutions achieving a certain objective value. When $m$ is constant we prove the presence of OGP in the regime $2^{-Θ(n)}$, and the absence of it in the regime $2^{-o(n)}$. Interestingly, though, by considering overlaps with growing values of $m$ we prove the presence of the OGP up to the level $2^{-ω(\sqrt{n\log n})}$. Our proof of the failure of stable algorithms at values $2^{-ω(n \log^{-1/5} n)}$ employs methods from Ramsey Theory from the extremal combinatorics, and is of independent interest.

math.ST↗

arXiv2021

Self-Regularity of Non-Negative Output Weights for Overparameterized Two-Layer Neural Networks

We consider the problem of finding a two-layer neural network with sigmoid, rectified linear unit (ReLU), or binary step activation functions that "fits" a training data set as accurately as possible as quantified by the training error; and study the following question: \emph{does a low training error guarantee that the norm of the output layer (outer norm) itself is small?} We answer affirmatively this question for the case of non-negative output weights. Using a simple covering number argument, we establish that under quite mild distributional assumptions on the input/label pairs; any such network achieving a small training error on polynomially many data necessarily has a well-controlled outer norm. Notably, our results (a) have a polynomial (in $d$) sample complexity, (b) are independent of the number of hidden units (which can potentially be very high), (c) are oblivious to the training algorithm; and (d) require quite mild assumptions on the data (in particular the input vector $X\in\mathbb{R}^d$ need not have independent coordinates). We then leverage our bounds to establish generalization guarantees for such networks through \emph{fat-shattering dimension}, a scale-sensitive measure of the complexity class that the network architectures we investigate belong to. Notably, our generalization bounds also have good sample complexity (polynomials in $d$ with a low degree), and are in fact near-linear for some important cases of interest.

stat.ML↗

arXiv2020

The Overlap Gap Property in Principal Submatrix Recovery

We study support recovery for a $k \times k$ principal submatrix with elevated mean $λ/N$, hidden in an $N\times N$ symmetric mean zero Gaussian matrix. Here $λ>0$ is a universal constant, and we assume $k = N ρ$ for some constant $ρ\in (0,1)$. We establish that {there exists a constant $C>0$ such that} the MLE recovers a constant proportion of the hidden submatrix if $λ{\geq C} \sqrt{\frac{1}ρ \log \frac{1}ρ}$, {while such recovery is information theoretically impossible if $λ= o( \sqrt{\frac{1}ρ \log \frac{1}ρ} )$}. The MLE is computationally intractable in general, and in fact, for $ρ>0$ sufficiently small, this problem is conjectured to exhibit a \emph{statistical-computational gap}. To provide rigorous evidence for this, we study the likelihood landscape for this problem, and establish that for some $\varepsilon>0$ and $\sqrt{\frac{1}ρ \log \frac{1}ρ } \ll λ\ll \frac{1}{ρ^{1/2 + \varepsilon}}$, the problem exhibits a variant of the \emph{Overlap-Gap-Property (OGP)}. As a direct consequence, we establish that a family of local MCMC based algorithms do not achieve optimal recovery. Finally, we establish that for $λ> 1/ρ$, a simple spectral method recovers a constant proportion of the hidden submatrix.

math.PR↗

arXiv2020

Correlation Decay and the Absence of Zeros Property of Partition Functions

Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok \cite{barvinok2017combinatorics} for designing deterministic approximation algorithms for various graph counting and computing partition functions problems. Earlier methods for solving the same problem include the one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply sometimes coincide or nearly coincide. In this paper we show that this is more than just a coincidence. We establish that if the interpolation method is valid for a family of graphs satisfying the self-reducibility property, then this family exhibits a form of correlation decay property which is asymptotic Strong Spatial Mixing (SSM) at distances $ω(\log n)$, where $n$ is the number of nodes of the graph. This applies in particular to amenable graphs, such as graphs which are finite subsets of lattices. Our proof is based on a certain graph polynomial representation of the associated partition function. This representation is at the heart of the design of the polynomial time algorithms underlying the interpolation method itself. We conjecture that our result holds for all, and not just amenable graphs.

math.PR↗

arXiv2020

Stability, memory, and messaging tradeoffs in heterogeneous service systems

We consider a heterogeneous distributed service system, consisting of $n$ servers with unknown and possibly different processing rates. Jobs with unit mean and independent processing times arrive as a renewal process of rate $λn$, with $0<λ<1$, to the system. Incoming jobs are immediately dispatched to one of several queues associated with the $n$ servers. We assume that the dispatching decisions are made by a central dispatcher endowed with a finite memory, and with the ability to exchange messages with the servers. We study the fundamental resource requirements (memory bits and message exchange rate) in order for a dispatching policy to be {\bf maximally stable}, i.e., stable whenever the processing rates are such that the arrival rate is less than the total available processing rate. First, for the case of Poisson arrivals and exponential service times, we present a policy that is maximally stable while using a positive (but arbitrarily small) message rate, and $\log_2(n)$ bits of memory. Second, we show that within a certain broad class of policies, a dispatching policy that exchanges $o\big(n^2\big)$ messages per unit of time, and with $o(\log(n))$ bits of memory, cannot be maximally stable. Thus, as long as the message rate is not too excessive, a logarithmic memory is necessary and sufficient for maximal stability.

cs.PF↗

arXiv2020

Stationary Points of Shallow Neural Networks with Quadratic Activation Function

We consider the teacher-student setting of learning shallow neural networks with quadratic activations and planted weight matrix $W^*\in\mathbb{R}^{m\times d}$, where $m$ is the width of the hidden layer and $d\le m$ is the data dimension. We study the optimization landscape associated with the empirical and the population squared risk of the problem. Under the assumption the planted weights are full-rank we obtain the following results. First, we establish that the landscape of the empirical risk admits an "energy barrier" separating rank-deficient $W$ from $W^*$: if $W$ is rank deficient, then its risk is bounded away from zero by an amount we quantify. We then couple this result by showing that, assuming number $N$ of samples grows at least like a polynomial function of $d$, all full-rank approximate stationary points of the empirical risk are nearly global optimum. These two results allow us to prove that gradient descent, when initialized below the energy barrier, approximately minimizes the empirical risk and recovers the planted weights in polynomial-time. Next, we show that initializing below this barrier is in fact easily achieved when the weights are randomly generated under relatively weak assumptions. We show that provided the network is sufficiently overparametrized, initializing with an appropriate multiple of the identity suffices to obtain a risk below the energy barrier. At a technical level, the last result is a consequence of the semicircle law for the Wishart ensemble and could be of independent interest. Finally, we study the minimizers of the empirical risk and identify a simple necessary and sufficient geometric condition on the training data under which any minimizer has necessarily zero generalization error. We show that as soon as $N\ge N^*=d(d+1)/2$, randomly generated data enjoys this geometric condition almost surely, while that ceases to be true if $N<N^*$.

stat.ML↗

arXiv2020

Estimation of Monotone Multi-Index Models

In a multi-index model with $k$ index vectors, the input variables are transformed by taking inner products with the index vectors. A transfer function $f: \mathbb{R}^k \to \mathbb{R}$ is applied to these inner products to generate the output. Thus, multi-index models are a generalization of linear models. In this paper, we consider monotone multi-index models. Namely, the transfer function is assumed to be coordinate-wise monotone. The monotone multi-index model therefore generalizes both linear regression and isotonic regression, which is the estimation of a coordinate-wise monotone function. We consider the case of nonnegative index vectors. We provide an algorithm based on integer programming for the estimation of monotone multi-index models, and provide guarantees on the $L_2$ loss of the estimated function relative to the ground truth.

math.ST↗

arXiv2020

The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: Worst Case Examples

The Quantum Approximate Optimization Algorithm can be applied to search problems on graphs with a cost function that is a sum of terms corresponding to the edges. When conjugating an edge term, the QAOA unitary at depth p produces an operator that depends only on the subgraph consisting of edges that are at most p away from the edge in question. On random d-regular graphs, with d fixed and with p a small constant time log n, these neighborhoods are almost all trees and so the performance of the QAOA is determined only by how it acts on an edge in the middle of tree. Both bipartite random d-regular graphs and general random d-regular graphs locally are trees so the QAOA's performance is the same on these two ensembles. Using this we can show that the QAOA with $(d-1)^{2p} < n^A$ for any $A<1$, can only achieve an approximation ratio of 1/2 for Max-Cut on bipartite random d-regular graphs for d large. For Maximum Independent Set, in the same setting, the best approximation ratio is a d-dependent constant that goes to 0 as d gets big.

quant-ph↗

arXiv2020

The Quantum Approximate Optimization Algorithm Needs to See the Whole Graph: A Typical Case

The Quantum Approximate Optimization Algorithm can naturally be applied to combinatorial search problems on graphs. The quantum circuit has p applications of a unitary operator that respects the locality of the graph. On a graph with bounded degree, with p small enough, measurements of distant qubits in the state output by the QAOA give uncorrelated results. We focus on finding big independent sets in random graphs with dn/2 edges keeping d fixed and n large. Using the Overlap Gap Property of almost optimal independent sets in random graphs, and the locality of the QAOA, we are able to show that if p is less than a d-dependent constant times log n, the QAOA cannot do better than finding an independent set of size .854 times the optimal for d large. Because the logarithm is slowly growing, even at one million qubits we can only show that the algorithm is blocked if p is in single digits. At higher p the algorithm "sees" the whole graph and we have no indication that performance is limited.

quant-ph↗

arXiv2020

Neural Networks and Polynomial Regression. Demystifying the Overparametrization Phenomena

In the context of neural network models, overparametrization refers to the phenomena whereby these models appear to generalize well on the unseen data, even though the number of parameters significantly exceeds the sample sizes, and the model perfectly fits the in-training data. A conventional explanation of this phenomena is based on self-regularization properties of algorithms used to train the data. In this paper we prove a series of results which provide a somewhat diverging explanation. Adopting a teacher/student model where the teacher network is used to generate the predictions and student network is trained on the observed labeled data, and then tested on out-of-sample data, we show that any student network interpolating the data generated by a teacher network generalizes well, provided that the sample size is at least an explicit quantity controlled by data dimension and approximation guarantee alone, regardless of the number of internal nodes of either teacher or student network. Our claim is based on approximating both teacher and student networks by polynomial (tensor) regression models with degree depending on the desired accuracy and network depth only. Such a parametrization notably does not depend on the number of internal nodes. Thus a message implied by our results is that parametrizing wide neural networks by the number of hidden nodes is misleading, and a more fitting measure of parametrization complexity is the number of regression coefficients associated with tensorized data. In particular, this somewhat reconciles the generalization ability of neural networks with more classical statistical notions of data complexity and generalization bounds. Our empirical results on MNIST and Fashion-MNIST datasets indeed confirm that tensorized regression achieves a good out-of-sample performance, even when the degree of the tensor is at most two.

stat.ML↗

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

The Overlap Gap Property and Approximate Message Passing Algorithms for $p$-spin models

We consider the algorithmic problem of finding a near ground state (near optimal solution) of a $p$-spin model. We show that for a class of algorithms broadly defined as Approximate Message Passing (AMP), the presence of the Overlap Gap Property (OGP), appropriately defined, is a barrier. We conjecture that when $p\ge 4$ the model does indeed exhibits OGP (and prove it for the space of binary solutions). Assuming the validity of this conjecture, as an implication, the AMP fails to find near ground states in these models, per our result. We extend our result to the problem of finding pure states by means of Thouless, Anderson and Palmer (TAP) based iterations, which is yet another example of AMP type algorithms. We show that such iterations fail to find pure states approximately, subject to the conjecture that the space of pure states exhibits the OGP, appropriately stated, when $p\ge 4$.

math.PR↗

arXiv2019

Computing the partition function of the Sherrington-Kirkpatrick model is hard on average

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington-Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless $P= \#P$, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction ($1/n^{O(1)}$) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding $P=\#P$. The computational model that we adopt is {\em finite-precision arithmetic}, where the algorithmic inputs are truncated first to a certain level $N$ of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. \cite{cai1999hardness} for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan \cite{sudan1996maximum}, for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime $p$. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different {\em real-valued} computational model, e.g. using a Blum-Shub-Smale machine \cite{blum1988theory} operating over real-valued inputs.

math.PR↗

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

Sparse High-Dimensional Isotonic Regression

We consider the problem of estimating an unknown coordinate-wise monotone function given noisy measurements, known as the isotonic regression problem. Often, only a small subset of the features affects the output. This motivates the sparse isotonic regression setting, which we consider here. We provide an upper bound on the expected VC entropy of the space of sparse coordinate-wise monotone functions, and identify the regime of statistical consistency of our estimator. We also propose a linear program to recover the active coordinates, and provide theoretical recovery guarantees. We close with experiments on cancer classification, and show that our method significantly outperforms standard methods.

math.ST↗

arXiv2018

Explicit construction of RIP matrices is Ramsey-hard

Matrices $Φ\in\R^{n\times p}$ satisfying the Restricted Isometry Property (RIP) are an important ingredient of the compressive sensing methods. While it is known that random matrices satisfy the RIP with high probability even for $n=\log^{O(1)}p$, the explicit construction of such matrices defied the repeated efforts, and the most known approaches hit the so-called $\sqrt{n}$ sparsity bottleneck. The notable exception is the work by Bourgain et al \cite{bourgain2011explicit} constructing an $n\times p$ RIP matrix with sparsity $s=Θ(n^{{1\over 2}+ε})$, but in the regime $n=Ω(p^{1-δ})$. In this short note we resolve this open question in a sense by showing that an explicit construction of a matrix satisfying the RIP in the regime $n=O(\log^2 p)$ and $s=Θ(n^{1\over 2})$ implies an explicit construction of a three-colored Ramsey graph on $p$ nodes with clique sizes bounded by $O(\log^2 p)$ -- a question in the extremal combinatorics which has been open for decades.

math.PR↗