arXiv subjects

Aravindan Vijayaraghavan

Publications and source records attributed to Aravindan Vijayaraghavan.

At least 37 records · Page 2Linked to original sources

arXiv2021

Efficient Algorithms for Learning Depth-2 Neural Networks with General ReLU Activations

We present polynomial time and sample efficient algorithms for learning an unknown depth-2 feedforward neural network with general ReLU activations, under mild non-degeneracy assumptions. In particular, we consider learning an unknown network of the form $f(x) = {a}^{\mathsf{T}}σ({W}^\mathsf{T}x+b)$, where $x$ is drawn from the Gaussian distribution, and $σ(t) := \max(t,0)$ is the ReLU activation. Prior works for learning networks with ReLU activations assume that the bias $b$ is zero. In order to deal with the presence of the bias terms, our proposed algorithm consists of robustly decomposing multiple higher order tensors arising from the Hermite expansion of the function $f(x)$. Using these ideas we also establish identifiability of the network parameters under minimal assumptions.

cs.LG↗

arXiv2021

Graph cuts always find a global optimum for Potts models (with a catch)

We prove that the $α$-expansion algorithm for MAP inference always returns a globally optimal assignment for Markov Random Fields with Potts pairwise potentials, with a catch: the returned assignment is only guaranteed to be optimal for an instance within a small perturbation of the original problem instance. In other words, all local minima with respect to expansion moves are global minima to slightly perturbed versions of the problem. On "real-world" instances, MAP assignments of small perturbations of the problem should be very similar to the MAP assignment(s) of the original problem instance. We design an algorithm that can certify whether this is the case in practice. On several MAP inference problem instances from computer vision, this algorithm certifies that MAP solutions to all of these perturbations are very close to solutions of the original instance. These results taken together give a cohesive explanation for the good performance of "graph cuts" algorithms in practice. Every local expansion minimum is a global minimum in a small perturbation of the problem, and all of these global minima are close to the original solution.

stat.ML↗

arXiv2021

Beyond Perturbation Stability: LP Recovery Guarantees for MAP Inference on Noisy Stable Instances

Several works have shown that perturbation stable instances of the MAP inference problem in Potts models can be solved exactly using a natural linear programming (LP) relaxation. However, most of these works give few (or no) guarantees for the LP solutions on instances that do not satisfy the relatively strict perturbation stability definitions. In this work, we go beyond these stability results by showing that the LP approximately recovers the MAP solution of a stable instance even after the instance is corrupted by noise. This "noisy stable" model realistically fits with practical MAP inference problems: we design an algorithm for finding "close" stable instances, and show that several real-world instances from computer vision have nearby instances that are perturbation stable. These results suggest a new theoretical explanation for the excellent performance of this LP relaxation in practice.

stat.ML↗

arXiv2021

Learning a mixture of two subspaces over finite fields

We study the problem of learning a mixture of two subspaces over $\mathbb{F}_2^n$. The goal is to recover the individual subspaces, given samples from a (weighted) mixture of samples drawn uniformly from the two subspaces $A_0$ and $A_1$. This problem is computationally challenging, as it captures the notorious problem of "learning parities with noise" in the degenerate setting when $A_1 \subseteq A_0$. This is in contrast to the analogous problem over the reals that can be solved in polynomial time (Vidal'03). This leads to the following natural question: is Learning Parities with Noise the only computational barrier in obtaining efficient algorithms for learning mixtures of subspaces over $\mathbb{F}_2^n$? The main result of this paper is an affirmative answer to the above question. Namely, we show the following results: 1. When the subspaces $A_0$ and $A_1$ are incomparable, i.e., $A_0$ and $A_1$ are not contained inside each other, then there is a polynomial time algorithm to recover the subspaces $A_0$ and $A_1$. 2. In the case when $A_1$ is a subspace of $A_0$ with a significant gap in the dimension i.e., $dim(A_1) \le αdim(A_0)$ for $α<1$, there is a $n^{O(1/(1-α))}$ time algorithm to recover the subspaces $A_0$ and $A_1$. Thus, our algorithms imply computational tractability of the problem of learning mixtures of two subspaces, except in the degenerate setting captured by learning parities with noise.

cs.DS↗

arXiv2020

Block Stability for MAP Inference

To understand the empirical success of approximate MAP inference, recent work (Lang et al., 2018) has shown that some popular approximation algorithms perform very well when the input instance is stable. The simplest stability condition assumes that the MAP solution does not change at all when some of the pairwise potentials are (adversarially) perturbed. Unfortunately, this strong condition does not seem to be satisfied in practice. In this paper, we introduce a significantly more relaxed condition that only requires blocks (portions) of an input instance to be stable. Under this block stability condition, we prove that the pairwise LP relaxation is persistent on the stable blocks. We complement our theoretical results with an empirical evaluation of real-world MAP inference instances from computer vision. We design an algorithm to find stable blocks, and find that these real instances have large stable regions. Our work gives a theoretical explanation for the widespread empirical phenomenon of persistency for this LP relaxation.

stat.ML↗

arXiv2020

Adversarial robustness via robust low rank representations

Adversarial robustness measures the susceptibility of a classifier to imperceptible perturbations made to the inputs at test time. In this work we highlight the benefits of natural low rank representations that often exist for real data such as images, for training neural networks with certified robustness guarantees. Our first contribution is for certified robustness to perturbations measured in $\ell_2$ norm. We exploit low rank data representations to provide improved guarantees over state-of-the-art randomized smoothing-based approaches on standard benchmark datasets such as CIFAR-10 and CIFAR-100. Our second contribution is for the more challenging setting of certified robustness to perturbations measured in $\ell_\infty$ norm. We demonstrate empirically that natural low rank representations have inherent robustness properties, that can be leveraged to provide significantly better guarantees for certified robustness to $\ell_\infty$ perturbations in those representations. Our certificate of $\ell_\infty$ robustness relies on a natural quantity involving the $\infty \to 2$ matrix operator norm associated with the representation, to translate robustness guarantees from $\ell_2$ to $\ell_\infty$ perturbations. A key technical ingredient for our certification guarantees is a fast algorithm with provable guarantees based on the multiplicative weights update method to provide upper bounds on the above matrix norm. Our algorithmic guarantees improve upon the state of the art for this problem, and may be of independent interest.

cs.LG↗

arXiv2020

Efficient Tensor Decomposition

This chapter studies the problem of decomposing a tensor into a sum of constituent rank one tensors. While tensor decompositions are very useful in designing learning algorithms and data analysis, they are NP-hard in the worst-case. We will see how to design efficient algorithms with provable guarantees under mild assumptions, and using beyond worst-case frameworks like smoothed analysis.

cs.DS↗

arXiv2020

Estimating Principal Components under Adversarial Perturbations

Robustness is a key requirement for widespread deployment of machine learning algorithms, and has received much attention in both statistics and computer science. We study a natural model of robustness for high-dimensional statistical estimation problems that we call the adversarial perturbation model. An adversary can perturb every sample arbitrarily up to a specified magnitude $δ$ measured in some $\ell_q$ norm, say $\ell_\infty$. Our model is motivated by emerging paradigms such as low precision machine learning and adversarial training. We study the classical problem of estimating the top-$r$ principal subspace of the Gaussian covariance matrix in high dimensions, under the adversarial perturbation model. We design a computationally efficient algorithm that given corrupted data, recovers an estimate of the top-$r$ principal subspace with error that depends on a robustness parameter $κ$ that we identify. This parameter corresponds to the $q \to 2$ operator norm of the projector onto the principal subspace, and generalizes well-studied analytic notions of sparsity. Additionally, in the absence of corruptions, our algorithmic guarantees recover existing bounds for problems such as sparse PCA and its higher rank analogs. We also prove that the above dependence on the parameter $κ$ is almost optimal asymptotically, not just in a minimax sense, but remarkably for every instance of the problem. This instance-optimal guarantee shows that the $q \to 2$ operator norm of the subspace essentially characterizes the estimation error under adversarial perturbations.

cs.LG↗

arXiv2020

Scheduling Precedence-Constrained Jobs on Related Machines with Communication Delay

We consider the problem of scheduling $n$ precedence-constrained jobs on $m$ uniformly-related machines in the presence of an arbitrary, fixed communication delay $ρ$. We consider a model that allows job duplication, i.e. processing of the same job on multiple machines, which, as we show, can reduce the length of a schedule (i.e., its makespan) by a logarithmic factor. Our main result is an $O(\log m \log ρ/ \log \log ρ)$-approximation algorithm for minimizing makespan, assuming the minimum makespan is at least $ρ$. Our algorithm is based on rounding a linear programming relaxation for the problem, which includes carefully designed constraints capturing the interaction among communication delay, precedence requirements, varying speeds, and job duplication. Our result builds on two previous lines of work, one with communication delay but identical machines (Lepere, Rapine 2002) and the other with uniformly-related machines but no communication delay (Chudak, Shmoys 1999). We next show that the integrality gap of our mathematical program is $Ω(\sqrt{\log ρ})$. Our gap construction employs expander graphs and exploits a property of robust expansion and its generalization to paths of longer length. Finally, we quantify the advantage of duplication in scheduling with communication delay. We show that the best schedule without duplication can have makespan $Ω(ρ/\log ρ)$ or $Ω(\log m/\log\log m)$ or $Ω(\log n/\log \log n)$ times that of an optimal schedule allowing duplication. Nevertheless, we present a polynomial time algorithm to transform any schedule to a schedule without duplication at the cost of a $O(\log^2 n \log m)$ factor increase in makespan. Together with our makespan approximation algorithm for schedules allowing duplication, this also yields a polylogarithmic-approximation algorithm for the setting where duplication is not allowed.

cs.DS↗

arXiv2019

On Robustness to Adversarial Examples and Polynomial Optimization

We study the design of computationally efficient algorithms with provable guarantees, that are robust to adversarial (test time) perturbations. While there has been an proliferation of recent work on this topic due to its connections to test time robustness of deep networks, there is limited theoretical understanding of several basic questions like (i) when and how can one design provably robust learning algorithms? (ii) what is the price of achieving robustness to adversarial examples in a computationally efficient manner? The main contribution of this work is to exhibit a strong connection between achieving robustness to adversarial examples, and a rich class of polynomial optimization problems, thereby making progress on the above questions. In particular, we leverage this connection to (a) design computationally efficient robust algorithms with provable guarantees for a large class of hypothesis, namely linear classifiers and degree-2 polynomial threshold functions (PTFs), (b) give a precise characterization of the price of achieving robustness in a computationally efficient manner for these classes, (c) design efficient algorithms to certify robustness and generate adversarial attacks in a principled manner for 2-layer neural networks. We empirically demonstrate the effectiveness of these attacks on real data.

cs.LG↗

arXiv2019

Smoothed Analysis in Unsupervised Learning via Decoupling

Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in unsupervised learning and high-dimensional data analysis. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decompositions and learning mixtures of Gaussians, such guarantees have been hard to obtain for several other important problems in unsupervised learning. A core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries, that are given by low-degree polynomials of a few base underlying random variables. In this work, we address this challenge by obtaining high-confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind. We then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in unsupervised learning: 1. Robust subspace recovery, when the fraction $α$ of inliers in the d-dimensional subspace $T \subset \mathbb{R}^n$ is at least $α> (d/n)^\ell$ for any constant integer $\ell>0$. This contrasts with the known worst-case intractability when $α< d/n$, and the previous smoothed analysis result which needed $α> d/n$ (Hardt and Moitra, 2013). 2. Learning overcomplete hidden markov models, where the size of the state space is any polynomial in the dimension of the observations. This gives the first polynomial time guarantees for learning overcomplete HMMs in a smoothed analysis model. 3. Higher order tensor decompositions, where we generalize the so-called FOOBI algorithm of Cardoso to find order-$\ell$ rank-one tensors in a subspace. This allows us to obtain polynomially robust decomposition algorithms for $2\ell$'th order tensors with rank $O(n^{\ell})$.

cs.DS↗

arXiv2018

Towards Learning Sparsely Used Dictionaries with Arbitrary Supports

Dictionary learning is a popular approach for inferring a hidden basis or dictionary in which data has a sparse representation. Data generated from the dictionary A (an n by m matrix, with m > n in the over-complete setting) is given by Y = AX where X is a matrix whose columns have supports chosen from a distribution over k-sparse vectors, and the non-zero values chosen from a symmetric distribution. Given Y, the goal is to recover A and X in polynomial time. Existing algorithms give polytime guarantees for recovering incoherent dictionaries, under strong distributional assumptions both on the supports of the columns of X, and on the values of the non-zero entries. In this work, we study the following question: Can we design efficient algorithms for recovering dictionaries when the supports of the columns of X are arbitrary? To address this question while circumventing the issue of non-identifiability, we study a natural semirandom model for dictionary learning where there are a large number of samples $y=Ax$ with arbitrary k-sparse supports for x, along with a few samples where the sparse supports are chosen uniformly at random. While the few samples with random supports ensures identifiability, the support distribution can look almost arbitrary in aggregate. Hence existing algorithmic techniques seem to break down as they make strong assumptions on the supports. Our main contribution is a new polynomial time algorithm for learning incoherent over-complete dictionaries that works under the semirandom model. Additionally the same algorithm provides polynomial time guarantees in new parameter regimes when the supports are fully random. Finally using these techniques, we also identify a minimal set of conditions on the supports under which the dictionary can be (information theoretically) recovered from polynomial samples for almost linear sparsity, i.e., $k=\tilde{O}(n)$.

cs.LG↗

arXiv2018

Optimality of Approximate Inference Algorithms on Stable Instances

Approximate algorithms for structured prediction problems---such as LP relaxations and the popular alpha-expansion algorithm (Boykov et al. 2001)---typically far exceed their theoretical performance guarantees on real-world instances. These algorithms often find solutions that are very close to optimal. The goal of this paper is to partially explain the performance of alpha-expansion and an LP relaxation algorithm on MAP inference in Ferromagnetic Potts models (FPMs). Our main results give stability conditions under which these two algorithms provably recover the optimal MAP solution. These theoretical results complement numerous empirical observations of good performance.

stat.ML↗

arXiv2017

Clustering Stable Instances of Euclidean k-means

The Euclidean k-means problem is arguably the most widely-studied clustering problem in machine learning. While the k-means objective is NP-hard in the worst-case, practitioners have enjoyed remarkable success in applying heuristics like Lloyd's algorithm for this problem. To address this disconnect, we study the following question: what properties of real-world instances will enable us to design efficient algorithms and prove guarantees for finding the optimal clustering? We consider a natural notion called additive perturbation stability that we believe captures many practical instances. Stable instances have unique optimal k-means solutions that do not change even when each point is perturbed a little (in Euclidean distance). This captures the property that the k-means optimal solution should be tolerant to measurement errors and uncertainty in the points. We design efficient algorithms that provably recover the optimal clustering for instances that are additive perturbation stable. When the instance has some additional separation, we show an efficient algorithm with provable guarantees that is also robust to outliers. We complement these results by studying the amount of stability in real datasets and demonstrating that our algorithm performs well on these benchmark datasets.

cs.LG↗

arXiv2017

Clustering Semi-Random Mixtures of Gaussians

Gaussian mixture models (GMM) are the most widely used statistical model for the $k$-means clustering problem and form a popular framework for clustering in machine learning and data analysis. In this paper, we propose a natural semi-random model for $k$-means clustering that generalizes the Gaussian mixture model, and that we believe will be useful in identifying robust algorithms. In our model, a semi-random adversary is allowed to make arbitrary "monotone" or helpful changes to the data generated from the Gaussian mixture model. Our first contribution is a polynomial time algorithm that provably recovers the ground-truth up to small classification error w.h.p., assuming certain separation between the components. Perhaps surprisingly, the algorithm we analyze is the popular Lloyd's algorithm for $k$-means clustering that is the method-of-choice in practice. Our second result complements the upper bound by giving a nearly matching information-theoretic lower bound on the number of misclassified points incurred by any $k$-means clustering algorithm on the semi-random model.

cs.DS↗

arXiv2017

On Learning Mixtures of Well-Separated Gaussians

We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of $k$ standard spherical Gaussians, and the goal is to estimate the means up to accuracy $δ$ using $poly(k,d, 1/δ)$ samples. In this work, we study the following question: what is the minimum separation needed between the means for solving this task? The best known algorithm due to Vempala and Wang [JCSS 2004] requires a separation of roughly $\min\{k,d\}^{1/4}$. On the other hand, Moitra and Valiant [FOCS 2010] showed that with separation $o(1)$, exponentially many samples are required. We address the significant gap between these two bounds, by showing the following results. 1. We show that with separation $o(\sqrt{\log k})$, super-polynomially many samples are required. In fact, this holds even when the $k$ means of the Gaussians are picked at random in $d=O(\log k)$ dimensions. 2. We show that with separation $Ω(\sqrt{\log k})$, $poly(k,d,1/δ)$ samples suffice. Note that the bound on the separation is independent of $δ$. This result is based on a new and efficient "accuracy boosting" algorithm that takes as input coarse estimates of the true means and in time $poly(k,d, 1/δ)$ outputs estimates of the means up to arbitrary accuracy $δ$ assuming the separation between the means is $Ω(\min\{\sqrt{\log k},\sqrt{d}\})$ (independently of $δ$). We also present a computationally efficient algorithm in $d=O(1)$ dimensions with only $Ω(\sqrt{d})$ separation. These results together essentially characterize the optimal order of separation between components that is needed to learn a mixture of $k$ spherical Gaussians with polynomial samples.

cs.DS↗

arXiv2016

Learning Communities in the Presence of Errors

We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige---Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add $o(n)$ edges. Our work answers this question affirmatively even in the case of $k>2$ communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is $εm$ close to SBM in the Kullback---Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities.

cs.DS↗

arXiv2015

Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + Ω(1/\sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a \emph{quantum} algorithm to find an assignment satisfying a $\frac{1}{2} + Ω(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $μ+ Ω(1/\sqrt{D})$ fraction of constraints, where $μ$ is the fraction that would be satisfied by a uniformly random assignment.

cs.CC↗