Smoothed Complexity Theory

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

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 $\alpha$ of inliers in the d-dimensional subspace $T \subset \mathbb{R}^n$ is at least $\alpha > (d/n)^\ell$ for any constant integer $\ell>0$. This contrasts with the known worst-case intractability when $\alpha< d/n$, and the previous smoothed analysis result which needed $\alpha > d/n$ (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 $O(n^{\ell})$.Comment: 44 page

Smoothed Analysis of Dynamic Networks

We generalize the technique of smoothed analysis to distributed algorithms in dynamic network models. Whereas standard smoothed analysis studies the impact of small random perturbations of input values on algorithm performance metrics, dynamic graph smoothed analysis studies the impact of random perturbations of the underlying changing network graph topologies. Similar to the original application of smoothed analysis, our goal is to study whether known strong lower bounds in dynamic network models are robust or fragile: do they withstand small (random) perturbations, or do such deviations push the graphs far enough from a precise pathological instance to enable much better performance? Fragile lower bounds are likely not relevant for real-world deployment, while robust lower bounds represent a true difficulty caused by dynamic behavior. We apply this technique to three standard dynamic network problems with known strong worst-case lower bounds: random walks, flooding, and aggregation. We prove that these bounds provide a spectrum of robustness when subjected to smoothing---some are extremely fragile (random walks), some are moderately fragile / robust (flooding), and some are extremely robust (aggregation).Comment: 20 page

Modular smoothed analysis

Spielman’s smoothed complexity - a hybrid between worst and average case complexity measures - relies on perturbations of input instances to determine where average-case behavior turns to worst-case. The paper proposes a method supporting modular smoothed analysis. The method, involving a novel permutation model, is developed for the discrete case, focusing on randomness preserving algorithms. This approach simplifies the smoothed analysis and achieves greater precession in the expression of the smoothed complexity, where a recurrence equation is obtained as opposed to bounds. Moreover, the approach addresses, in this context, the formation of input instances–an open problem in smoothed complexity. To illustrate the method, we determine the modular smoothed complexity of Quicksort

Smoothed Efficient Algorithms and Reductions for Network Coordination Games

Worst-case hardness results for most equilibrium computation problems have raised the need for beyond-worst-case analysis. To this end, we study the smoothed complexity of finding pure Nash equilibria in Network Coordination Games, a PLS-complete problem in the worst case. This is a potential game where the sequential-better-response algorithm is known to converge to a pure NE, albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial) smoothed complexity when the underlying game graph is a complete (resp. arbitrary) graph, and every player has constantly many strategies. We note that the complete graph case is reminiscent of perturbing all parameters, a common assumption in most known smoothed analysis results. Second, we define a notion of smoothness-preserving reduction among search problems, and obtain reductions from $2$-strategy network coordination games to local-max-cut, and from $k$-strategy games (with arbitrary $k$) to local-max-cut up to two flips. The former together with the recent result of [BCC18] gives an alternate $O(n^8)$-time smoothed algorithm for the $2$-strategy case. This notion of reduction allows for the extension of smoothed efficient algorithms from one problem to another. For the first set of results, we develop techniques to bound the probability that an (adversarial) better-response sequence makes slow improvements on the potential. Our approach combines and generalizes the local-max-cut approaches of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful definition of the matrix which captures the increase in potential, a tighter union bound on adversarial sequences, and balancing it with good enough rank bounds. We believe that the approach and notions developed herein could be of interest in addressing the smoothed complexity of other potential and/or congestion games

On smoothed analysis of quicksort and Hoare's find

We provide a smoothed analysis of Hoare's find algorithm, and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect or one-sided quicksort - is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoare’s find needs is Theta(n^2), the average-case number is Theta(n). We analyze what happens between these two extremes by providing a smoothed analysis. In the first perturbation model, an adversary specifies a sequence of n numbers of [0,1], and then, to each number of the sequence, we add a random number drawn independently from the interval [0,d]. We prove that Hoare's find needs Theta(n/(d+1) sqrt(n/d) + n) comparisons in expectation if the adversary may also specify the target element (even after seeing the perturbed sequence) and slightly fewer comparisons for finding the median. In the second perturbation model, each element is marked with a probability of p, and then a random permutation is applied to the marked elements. We prove that the expected number of comparisons to find the median is Omega((1−p)n/p log n). Finally, we provide lower bounds for the smoothed number of comparisons of quicksort and Hoare’s find for the median-of-three pivot rule, which usually yields faster algorithms than always selecting the first element: The pivot is the median of the first, middle, and last element of the sequence. We show that median-of-three does not yield a significant improvement over the classic rule

Polynomial-time Tensor Decompositions with Sum-of-Squares

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete 3-tensors and learning overcomplete dictionaries with constant relative sparsity. We also give the first robust analysis for decomposing overcomplete 4-tensors in the smoothed analysis model. A key ingredient of our analysis is to establish small spectral gaps in moment matrices derived from solutions to sum-of-squares relaxations. To enable this analysis we augment sum-of-squares relaxations with spectral analogs of maximum entropy constraints.Comment: to appear in FOCS 201