arXiv subjects

Andrea Montanari

Publications and source records attributed to Andrea Montanari.

At least 109 records · Page 6Linked to original sources

arXiv2017

The Landscape of Empirical Risk for Non-convex Losses

Most high-dimensional estimation and prediction methods propose to minimize a cost function (empirical risk) that is written as a sum of losses associated to each data point. In this paper we focus on the case of non-convex losses, which is practically important but still poorly understood. Classical empirical process theory implies uniform convergence of the empirical risk to the population risk. While uniform convergence implies consistency of the resulting M-estimator, it does not ensure that the latter can be computed efficiently. In order to capture the complexity of computing M-estimators, we propose to study the landscape of the empirical risk, namely its stationary points and their properties. We establish uniform convergence of the gradient and Hessian of the empirical risk to their population counterparts, as soon as the number of samples becomes larger than the number of unknown parameters (modulo logarithmic factors). Consequently, good properties of the population risk can be carried to the empirical risk, and we can establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as non-convex binary classification, robust regression, and Gaussian mixture model, this result implies a complete characterization of the landscape of the empirical risk, and of the convergence properties of descent algorithms. We extend our analysis to the very high-dimensional setting in which the number of parameters exceeds the number of samples, and provide a characterization of the empirical risk landscape under a nearly information-theoretically minimal condition. Namely, if the number of samples exceeds the sparsity of the unknown parameters vector (modulo logarithmic factors), then a suitable uniform convergence result takes place. We apply this result to non-convex binary classification and robust regression in very high-dimension.

stat.ML↗

arXiv2016

Spectral algorithms for tensor completion

In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees have been proved, for third-order tensors, using the sixth level of the sum-of-squares (SOS) semidefinite programming hierarchy (Barak and Moitra, 2014). However, the SOS approach does not scale well to large problem instances. By contrast, spectral methods --- based on unfolding or matricizing the tensor --- are attractive for their low complexity, but have been believed to require a much larger sample size. This paper presents two main contributions. First, we propose a new unfolding-based method, which outperforms naive ones for symmetric $k$-th order tensors of rank $r$. For this result we make a study of singular space estimation for partially revealed matrices of large aspect ratio, which may be of independent interest. For third-order tensors, our algorithm matches the SOS method in terms of sample size (requiring about $rd^{3/2}$ revealed entries), subject to a worse rank condition ($r\ll d^{3/4}$ rather than $r\ll d^{3/2}$). We complement this result with a different spectral algorithm for third-order tensors in the overcomplete ($r\ge d$) regime. Under a random model, this second approach succeeds in estimating tensors of rank $d\le r \ll d^{3/2}$ from about $rd^{3/2}$ revealed entries.

cs.DS↗

arXiv2016

How Well Do Local Algorithms Solve Semidefinite Programs?

Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erdős-Renyi random graphs with bounded average degree $d>1$, and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor $2d^2/(2d^2+d-1)$ of the upper bound. In particular, the local algorithm is at most $8/9$ suboptimal, and $1+O(1/d)$ suboptimal for large degree. We then analyze a more sophisticated local algorithm, which aggregates information according to the harmonic measure on the limiting Galton-Watson (GW) tree. The resulting lower bound is expressed in terms of the conductance of the GW tree and matches surprisingly well the empirically determined SDP values on large-scale Erdős-Renyi graphs. We finally consider the planted partition model. In this case, purely local algorithms are known to fail, but they do succeed if a small amount of side information is available. Our results imply quantitative bounds on the threshold for partial recovery using SDP in this model.

cs.DM↗

arXiv2016

De-biasing the Lasso: Optimal Sample Size for Gaussian Designs

Performing statistical inference in high-dimension is an outstanding challenge. A major source of difficulty is the absence of precise information on the distribution of high-dimensional estimators. Here, we consider linear regression in the high-dimensional regime $p\gg n$. In this context, we would like to perform inference on a high-dimensional parameters vector $θ^*\in{\mathbb R}^p$. Important progress has been achieved in computing confidence intervals for single coordinates $θ^*_i$. A key role in these new methods is played by a certain debiased estimator $\hatθ^{\rm d}$ that is constructed from the Lasso. Earlier work establishes that, under suitable assumptions on the design matrix, the coordinates of $\hatθ^{\rm d}$ are asymptotically Gaussian provided $θ^*$ is $s_0$-sparse with $s_0 = o(\sqrt{n}/\log p )$. The condition $s_0 = o(\sqrt{n}/ \log p )$ is stronger than the one for consistent estimation, namely $s_0 = o(n/ \log p)$. We study Gaussian designs with known or unknown population covariance. When the covariance is known, we prove that the debiased estimator is asymptotically Gaussian under the nearly optimal condition $s_0 = o(n/ (\log p)^2)$. Note that earlier work was limited to $s_0 = o(\sqrt{n}/\log p)$ even for perfectly known covariance. The same conclusion holds if the population covariance is unknown but can be estimated sufficiently well, e.g. under the same sparsity conditions on the inverse covariance as assumed by earlier work. For intermediate regimes, we describe the trade-off between sparsity in the coefficients and in the inverse covariance of the design. We further discuss several applications of our results to high-dimensional inference. In particular, we propose a new estimator that is minimax optimal up to a factor $1+o_n(1)$ for i.i.d. Gaussian designs.

math.ST↗

arXiv2016

Sparse PCA via Covariance Thresholding

In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here each of the principal components $\mathbf{v}_1,\dots,\mathbf{v}_r$ has at most $s_0$ non-zero entries. We are particularly interested in the high dimensional regime wherein $p$ is comparable to, or even much larger than $n$. In an influential paper, \cite{johnstone2004sparse} introduced a simple algorithm that estimates the support of the principal vectors $\mathbf{v}_1,\dots,\mathbf{v}_r$ by the largest entries in the diagonal of the empirical covariance. This method can be shown to identify the correct support with high probability if $s_0\le K_1\sqrt{n/\log p}$, and to fail with high probability if $s_0\ge K_2 \sqrt{n/\log p}$ for two constants $0<K_1,K_2<\infty$. Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by \cite{KrauthgamerSPCA}. On the basis of numerical simulations (for the rank-one case), these authors conjectured that covariance thresholding correctly recover the support with high probability for $s_0\le K\sqrt{n}$ (assuming $n$ of the same order as $p$). We prove this conjecture, and in fact establish a more general guarantee including higher-rank as well as $n$ much smaller than $p$. Recent lower bounds \cite{berthet2013computational, ma2015sum} suggest that no polynomial time algorithm can do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before.

math.ST↗

arXiv2016

Performance of a community detection algorithm based on semidefinite programming

The problem of detecting communities in a graph is maybe one the most studied inference problems, given its simplicity and widespread diffusion among several disciplines. A very common benchmark for this problem is the stochastic block model or planted partition problem, where a phase transition takes place in the detection of the planted partition by changing the signal-to-noise ratio. Optimal algorithms for the detection exist which are based on spectral methods, but we show these are extremely sensible to slight modification in the generative model. Recently Javanmard, Montanari and Ricci-Tersenghi (arXiv:1511.08769) have used statistical physics arguments, and numerical simulations to show that finding communities in the stochastic block model via semidefinite programming is quasi optimal. Further, the resulting semidefinite relaxation can be solved efficiently, and is very robust with respect to changes in the generative model. In this paper we study in detail several practical aspects of this new algorithm based on semidefinite programming for the detection of the planted partition. The algorithm turns out to be very fast, allowing the solution of problems with $O(10^5)$ variables in few second on a laptop computer.

stat.ML↗

arXiv2016

A Grothendieck-type inequality for local maxima

A large number of problems in optimization, machine learning, signal processing can be effectively addressed by suitable semidefinite programming (SDP) relaxations. Unfortunately, generic SDP solvers hardly scale beyond instances with a few hundreds variables (in the underlying combinatorial problem). On the other hand, it has been observed empirically that an effective strategy amounts to introducing a (non-convex) rank constraint, and solving the resulting smooth optimization problem by ascent methods. This non-convex problem has --generically-- a large number of local maxima, and the reason for this success is therefore unclear. This paper provides rigorous support for this approach. For the problem of maximizing a linear functional over the elliptope, we prove that all local maxima are within a small gap from the SDP optimum. In several problems of interest, arbitrarily small relative error can be achieved by taking the rank constraint $k$ to be of order one, independently of the problem size.

math.OC↗

arXiv2016

Phase Transitions in Semidefinite Relaxations

Statistical inference problems arising within signal processing, data mining, and machine learning naturally give rise to hard combinatorial optimization problems. These problems become intractable when the dimensionality of the data is large, as is often the case for modern datasets. A popular idea is to construct convex relaxations of these combinatorial problems, which can be solved efficiently for large scale datasets. Semidefinite programming (SDP) relaxations are among the most powerful methods in this family, and are surprisingly well-suited for a broad range of problems where data take the form of matrices or graphs. It has been observed several times that, when the `statistical noise' is small enough, SDP relaxations correctly detect the underlying combinatorial structures. In this paper we develop asymptotic predictions for several `detection thresholds,' as well as for the estimation error above these thresholds. We study some classical SDP relaxations for statistical problems motivated by graph synchronization and community detection in networks. We map these optimization problems to statistical mechanics models with vector spins, and use non-rigorous techniques from statistical mechanics to characterize the corresponding phase transitions. Our results clarify the effectiveness of SDP relaxations in solving high-dimensional statistical problems.

cond-mat.stat-mech↗

arXiv2015

Semidefinite Programs on Sparse Random Graphs and their Application to Community Detection

Denote by $A$ the adjacency matrix of an Erdos-Renyi graph with bounded average degree. We consider the problem of maximizing $\langle A-E\{A\},X\rangle$ over the set of positive semidefinite matrices $X$ with diagonal entries $X_{ii}=1$. We prove that for large (bounded) average degree $d$, the value of this semidefinite program (SDP) is --with high probability-- $2n\sqrt{d} + n\, o(\sqrt{d})+o(n)$. For a random regular graph of degree $d$, we prove that the SDP value is $2n\sqrt{d-1}+o(n)$, matching a spectral upper bound. Informally, Erdos-Renyi graphs appear to behave similarly to random regular graphs for semidefinite programming. We next consider the sparse, two-groups, symmetric community detection problem (also known as planted partition). We establish that SDP achieves the information-theoretically optimal detection threshold for large (bounded) degree. Namely, under this model, the vertex set is partitioned into subsets of size $n/2$, with edge probability $a/n$ (within group) and $b/n$ (across). We prove that SDP detects the partition with high probability provided $(a-b)^2/(4d)> 1+o_{d}(1)$, with $d= (a+b)/2$. By comparison, the information theoretic threshold for detecting the hidden partition is $(a-b)^2/(4d)> 1$: SDP is nearly optimal for large bounded average degree. Our proof is based on tools from different research areas: $(i)$ A new `higher-rank' Grothendieck inequality for symmetric matrices; $(ii)$ An interpolation method inspired from statistical physics; $(iii)$ An analysis of the eigenvectors of deformed Gaussian random matrices.

cs.DM↗

arXiv2015

The LASSO risk for gaussian matrices

We consider the problem of learning a coefficient vector x_0\in R^N from noisy linear observation y=Ax_0+w \in R^n. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator x'. In this case, a popular approach consists in solving an L1-penalized least squares problem known as the LASSO or Basis Pursuit DeNoising (BPDN). For sequences of matrices A of increasing dimensions, with independent gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications.

math.ST↗

arXiv2015

Convergence rates of sub-sampled Newton methods

We consider the problem of minimizing a sum of $n$ functions over a convex parameter set $\mathcal{C} \subset \mathbb{R}^p$ where $n\gg p\gg 1$. In this regime, algorithms which utilize sub-sampling techniques are known to be effective. In this paper, we use sub-sampling techniques together with low-rank approximation to design a new randomized batch algorithm which possesses comparable convergence rate to Newton's method, yet has much smaller per-iteration cost. The proposed algorithm is robust in terms of starting point and step size, and enjoys a composite convergence rate, namely, quadratic convergence at start and linear convergence when the iterate is close to the minimizer. We develop its theoretical analysis which also allows us to select near-optimal algorithm parameters. Our theoretical results can be used to obtain convergence rates of previously proposed sub-sampling based algorithms as well. We demonstrate how our results apply to well-known machine learning problems. Lastly, we evaluate the performance of our algorithm on several datasets under various scenarios.

stat.ML↗

arXiv2015

The set of solutions of random XORSAT formulae

The XOR-satisfiability (XORSAT) problem requires finding an assignment of $n$ Boolean variables that satisfy $m$ exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing $n$ variables and $m$ clauses of size $k$. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as $k$-satisfiability ($k$-SAT), hypergraph bicoloring and graph coloring. For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random $k$-XORSAT. In particular, we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is intimately tied to the construction of specific subgraphs of the hypergraph associated with an instance of $k$-XORSAT. In order to study such subgraphs, we establish novel local weak convergence results for them.

cs.DM↗

arXiv2015

Computational Implications of Reducing Data to Sufficient Statistics

Given a large dataset and an estimation task, it is common to pre-process the data by reducing them to a set of sufficient statistics. This step is often regarded as straightforward and advantageous (in that it simplifies statistical analysis). I show that -on the contrary- reducing data to sufficient statistics can change a computationally tractable estimation problem into an intractable one. I discuss connections with recent work in theoretical computer science, and implications for some techniques to estimate graphical models.

stat.CO↗

arXiv2015

Asymptotic Mutual Information for the Two-Groups Stochastic Block Model

We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph $G$ from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit `single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing.

cs.IT↗

arXiv2015

Finding One Community in a Sparse Graph

We consider a random sparse graph with bounded average degree, in which a subset of vertices has higher connectivity than the background. In particular, the average degree inside this subset of vertices is larger than outside (but still bounded). Given a realization of such graph, we aim at identifying the hidden subset of vertices. This can be regarded as a model for the problem of finding a tightly knitted community in a social network, or a cluster in a relational dataset. In this paper we present two sets of contributions: $(i)$ We use the cavity method from spin glass theory to derive an exact phase diagram for the reconstruction problem. In particular, as the difference in edge probability increases, the problem undergoes two phase transitions, a static phase transition and a dynamic one. $(ii)$ We establish rigorous bounds on the dynamic phase transition and prove that, above a certain threshold, a local algorithm (belief propagation) correctly identify most of the hidden set. Below the same threshold \emph{no local algorithm} can achieve this goal. However, in this regime the subset can be identified by exhaustive search. For small hidden sets and large average degree, the phase transition for local algorithms takes an intriguingly simple form. Local algorithms succeed with high probability for ${\rm deg}_{\rm in} - {\rm deg}_{\rm out} > \sqrt{{\rm deg}_{\rm out}/e}$ and fail for ${\rm deg}_{\rm in} - {\rm deg}_{\rm out} < \sqrt{{\rm deg}_{\rm out}/e}$ (with ${\rm deg}_{\rm in}$, ${\rm deg}_{\rm out}$ the average degrees inside and outside the community). We argue that spectral algorithms are also ineffective in the latter regime. It is an open problem whether any polynomial time algorithms might succeed for ${\rm deg}_{\rm in} - {\rm deg}_{\rm out} < \sqrt{{\rm deg}_{\rm out}/e}$.

stat.ML↗

arXiv2015

A Perspective on Future Research Directions in Information Theory

Information theory is rapidly approaching its 70th birthday. What are promising future directions for research in information theory? Where will information theory be having the most impact in 10-20 years? What new and emerging areas are ripe for the most impact, of the sort that information theory has had on the telecommunications industry over the last 60 years? How should the IEEE Information Theory Society promote high-risk new research directions and broaden the reach of information theory, while continuing to be true to its ideals and insisting on the intellectual rigor that makes its breakthroughs so powerful? These are some of the questions that an ad hoc committee (composed of the present authors) explored over the past two years. We have discussed and debated these questions, and solicited detailed inputs from experts in fields including genomics, biology, economics, and neuroscience. This report is the result of these discussions.

cs.IT↗

arXiv2015

Extremal Cuts of Sparse Random Graphs

For Erdős-Rényi random graphs with average degree $γ$, and uniformly random $γ$-regular graph on $n$ vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are $n\Big(\fracγ{4} + {\sf P}_* \sqrt{\fracγ{4}} + o(\sqrtγ)\Big) + o(n)$ while the size of the minimum bisection is $n\Big(\fracγ{4}-{\sf P}_*\sqrt{\fracγ{4}} + o(\sqrtγ)\Big) + o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with ${\sf P}_* \approx 0.7632$ standing for the ground state energy of the latter, expressed analytically via Parisi's formula.

math.PR↗

arXiv2015

Universality in polytope phase transitions and message passing algorithms

We consider a class of nonlinear mappings $\mathsf{F}_{A,N}$ in $\mathbb{R}^N$ indexed by symmetric random matrices $A\in\mathbb{R}^{N\times N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333-366]. Within information theory, they are known as "approximate message passing" algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $\mathsf{F}$ for polynomial functions $\mathsf{F}$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.

math.PR↗