arXiv subjects

Nir Ailon

Publications and source records attributed to Nir Ailon.

34 records · Page 2Linked to original sources

arXiv2014

Reducing Dueling Bandits to Cardinal Bandits

We present algorithms for reducing the Dueling Bandits problem to the conventional (stochastic) Multi-Armed Bandits problem. The Dueling Bandits problem is an online model of learning with ordinal feedback of the form "A is preferred to B" (as opposed to cardinal feedback like "A has value 2.5"), giving it wide applicability in learning from implicit user feedback and revealed and stated preferences. In contrast to existing algorithms for the Dueling Bandits problem, our reductions -- named $\Doubler$, $\MultiSbm$ and $\DoubleSbm$ -- provide a generic schema for translating the extensive body of known results about conventional Multi-Armed Bandit algorithms to the Dueling Bandits setting. For $\Doubler$ and $\MultiSbm$ we prove regret upper bounds in both finite and infinite settings, and conjecture about the performance of $\DoubleSbm$ which empirically outperforms the other two as well as previous algorithms in our experiments. In addition, we provide the first almost optimal regret bound in terms of second order terms, such as the differences between the values of the arms.

cs.LG↗

arXiv2014

A tight lower bound instance for k-means++ in constant dimension

The k-means++ seeding algorithm is one of the most popular algorithms that is used for finding the initial $k$ centers when using the k-means heuristic. The algorithm is a simple sampling procedure and can be described as follows: Pick the first center randomly from the given points. For $i > 1$, pick a point to be the $i^{th}$ center with probability proportional to the square of the Euclidean distance of this point to the closest previously $(i-1)$ chosen centers. The k-means++ seeding algorithm is not only simple and fast but also gives an $O(\log{k})$ approximation in expectation as shown by Arthur and Vassilvitskii. There are datasets on which this seeding algorithm gives an approximation factor of $Ω(\log{k})$ in expectation. However, it is not clear from these results if the algorithm achieves good approximation factor with reasonably high probability (say $1/poly(k)$). Brunsch and Röglin gave a dataset where the k-means++ seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability that is exponentially small in $k$. However, this and all other known lower-bound examples are high dimensional. So, an open problem was to understand the behavior of the algorithm on low dimensional datasets. In this work, we give a simple two dimensional dataset on which the seeding algorithm achieves an $O(\log{k})$ approximation ratio with probability exponentially small in $k$. This solves open problems posed by Mahajan et al. and by Brunsch and Röglin.

cs.DS↗

arXiv2013

Online Ranking: Discrete Choice, Spearman Correlation and Other Feedback

Given a set $V$ of $n$ objects, an online ranking system outputs at each time step a full ranking of the set, observes a feedback of some form and suffers a loss. We study the setting in which the (adversarial) feedback is an element in $V$, and the loss is the position (0th, 1st, 2nd...) of the item in the outputted ranking. More generally, we study a setting in which the feedback is a subset $U$ of at most $k$ elements in $V$, and the loss is the sum of the positions of those elements. We present an algorithm of expected regret $O(n^{3/2}\sqrt{Tk})$ over a time horizon of $T$ steps with respect to the best single ranking in hindsight. This improves previous algorithms and analyses either by a factor of either $Ω(\sqrt{k})$, a factor of $Ω(\sqrt{\log n})$ or by improving running time from quadratic to $O(n\log n)$ per round. We also prove a matching lower bound. Our techniques also imply an improved regret bound for online rank aggregation over the Spearman correlation measure, and to other more complex ranking loss functions.

cs.LG↗

arXiv2013

A Lower Bound for Fourier Transform Computation in a Linear Model Over 2x2 Unitary Gates Using Matrix Entropy

Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem. All lower bounds so far have made strong restrictions on the computational model. One of the most well known results, by Morgenstern from 1973, provides an $Ω(n \log n)$ lower bound for the \emph{unnormalized} FFT when the constants used in the computation are bounded. The proof uses a potential function related to a determinant. The determinant of the unnormalized Fourier transform is $n^{n/2}$, and thus by showing that it can grow by at most a constant factor after each step yields the result. This classic result, however, does not explain why the \emph{normalized} Fourier transform, which has a unit determinant, should take $Ω(n\log n)$ steps to compute. In this work we show that in a layered linear circuit model restricted to unitary $2\times 2$ gates, one obtains an $Ω(n\log n)$ lower bound. The well known FFT works in this model. The main argument concluded from this work is that a potential function that might eventually help proving the $Ω(n\log n)$ conjectured lower bound for computation of Fourier transform is not related to matrix determinant, but rather to a notion of matrix entropy.

cs.CC↗

arXiv2013

Breaking the Small Cluster Barrier of Graph Clustering

This paper investigates graph clustering in the planted cluster model in the presence of {\em small clusters}. Traditional results dictate that for an algorithm to provably correctly recover the clusters, {\em all} clusters must be sufficiently large (in particular, $\tildeΩ(\sqrt{n})$ where $n$ is the number of nodes of the graph). We show that this is not really a restriction: by a more refined analysis of the trace-norm based recovery approach proposed in Jalali et al. (2011) and Chen et al. (2012), we prove that small clusters, under certain mild assumptions, do not hinder recovery of large ones. Based on this result, we further devise an iterative algorithm to recover {\em almost all clusters} via a "peeling strategy", i.e., recover large clusters first, leading to a reduced problem, and repeat this procedure. These results are extended to the {\em partial observation} setting, in which only a (chosen) part of the graph is observed.The peeling strategy gives rise to an active learning algorithm, in which edges adjacent to smaller clusters are queried more often as large clusters are learned (and removed). From a high level, this paper sheds novel insights on high-dimensional statistics and learning structured data, by presenting a structured matrix learning problem for which a one shot convex relaxation approach necessarily fails, but a carefully constructed sequence of convex relaxationsdoes the job.

cs.LG↗

arXiv2013

Fast and RIP-optimal transforms

We study constructions of $k \times n$ matrices $A$ that both (1) satisfy the restricted isometry property (RIP) at sparsity $s$ with optimal parameters, and (2) are efficient in the sense that only $O(n\log n)$ operations are required to compute $Ax$ given a vector $x$. Our construction is based on repeated application of independent transformations of the form $DH$, where $H$ is a Hadamard or Fourier transform and $D$ is a diagonal matrix with random $\{+1,-1\}$ elements on the diagonal, followed by any $k \times n$ matrix of orthonormal rows (e.g.\ selection of $k$ coordinates). We provide guarantees (1) and (2) for a larger regime of parameters for which such constructions were previously unknown. Additionally, our construction does not suffer from the extra poly-logarithmic factor multiplying the number of observations $k$ as a function of the sparsity $s$, as present in the currently best known RIP estimates for partial random Fourier matrices and other classes of structured random matrices.

math.NA↗

arXiv2012

Active Learning Using Smooth Relative Regret Approximations with Applications

The disagreement coefficient of Hanneke has become a central data independent invariant in proving active learning rates. It has been shown in various ways that a concept class with low complexity together with a bound on the disagreement coefficient at an optimal solution allows active learning rates that are superior to passive learning ones. We present a different tool for pool based active learning which follows from the existence of a certain uniform version of low disagreement coefficient, but is not equivalent to it. In fact, we present two fundamental active learning problems of significant interest for which our approach allows nontrivial active learning bounds. However, any general purpose method relying on the disagreement coefficient bounds only fails to guarantee any useful bounds for these problems. The tool we use is based on the learner's ability to compute an estimator of the difference between the loss of any hypotheses and some fixed "pivotal" hypothesis to within an absolute error of at most $\eps$ times the

cs.LG↗

arXiv2012

A note on: No need to choose: How to get both a PTAS and Sublinear Query Complexity

We revisit various PTAS's (Polynomial Time Approximation Schemes) for minimization versions of dense problems, and show that they can be performed with sublinear query complexity. This means that not only do we obtain a (1+eps)-approximation to the NP-Hard problems in polynomial time, but also avoid reading the entire input. This setting is particularly advantageous when the price of reading parts of the input is high, as is the case, for examples, where humans provide the input. Trading off query complexity with approximation is the raison d'etre of the field of learning theory, and of the ERM (Empirical Risk Minimization) setting in particular. A typical ERM result, however, does not deal with computational complexity. We discuss two particular problems for which (a) it has already been shown that sublinear querying is sufficient for obtaining a (1 + eps)-approximation using unlimited computational power (an ERM result), and (b) with full access to input, we could get a (1+eps)-approximation in polynomial time (a PTAS). Here we show that neither benefit need be sacrificed. We get a PTAS with efficient query complexity.

cs.CC↗

arXiv2012

Active Learning of Custering with Side Information Using $\eps$-Smooth Relative Regret Approximations

Clustering is considered a non-supervised learning setting, in which the goal is to partition a collection of data points into disjoint clusters. Often a bound $k$ on the number of clusters is given or assumed by the practitioner. Many versions of this problem have been defined, most notably $k$-means and $k$-median. An underlying problem with the unsupervised nature of clustering it that of determining a similarity function. One approach for alleviating this difficulty is known as clustering with side information, alternatively, semi-supervised clustering. Here, the practitioner incorporates side information in the form of "must be clustered" or "must be separated" labels for data point pairs. Each such piece of information comes at a "query cost" (often involving human response solicitation). The collection of labels is then incorporated in the usual clustering algorithm as either strict or as soft constraints, possibly adding a pairwise constraint penalty function to the chosen clustering objective. Our work is mostly related to clustering with side information. We ask how to choose the pairs of data points. Our analysis gives rise to a method provably better than simply choosing them uniformly at random. Roughly speaking, we show that the distribution must be biased so as more weight is placed on pairs incident to elements in smaller clusters in some optimal solution. Of course we do not know the optimal solution, hence we don't know the bias. Using the recently introduced method of $\eps$-smooth relative regret approximations of Ailon, Begleiter and Ezra, we can show an iterative process that improves both the clustering and the bias in tandem. The process provably converges to the optimal solution faster (in terms of query cost) than an algorithm selecting pairs uniformly.

cs.LG↗

arXiv2011

An Active Learning Algorithm for Ranking from Pairwise Preferences with an Almost Optimal Query Complexity

We study the problem of learning to rank from pairwise preferences, and solve a long-standing open problem that has led to development of many heuristics but no provable results for our particular problem. Given a set $V$ of $n$ elements, we wish to linearly order them given pairwise preference labels. A pairwise preference label is obtained as a response, typically from a human, to the question "which if preferred, u or v?$ for two elements $u,v\in V$. We assume possible non-transitivity paradoxes which may arise naturally due to human mistakes or irrationality. The goal is to linearly order the elements from the most preferred to the least preferred, while disagreeing with as few pairwise preference labels as possible. Our performance is measured by two parameters: The loss and the query complexity (number of pairwise preference labels we obtain). This is a typical learning problem, with the exception that the space from which the pairwise preferences is drawn is finite, consisting of ${n\choose 2}$ possibilities only. We present an active learning algorithm for this problem, with query bounds significantly beating general (non active) bounds for the same error guarantee, while almost achieving the information theoretical lower bound. Our main construct is a decomposition of the input s.t. (i) each block incurs high loss at optimum, and (ii) the optimal solution respecting the decomposition is not much worse than the true opt. The decomposition is done by adapting a recent result by Kenyon and Schudy for a related combinatorial optimization problem to the query efficient setting. We thus settle an open problem posed by learning-to-rank theoreticians and practitioners: What is a provably correct way to sample preference labels? To further show the power and practicality of our solution, we show how to use it in concert with an SVM relaxation.

cs.DS↗

arXiv2010

An Improved Algorithm for Bipartite Correlation Clustering

Bipartite Correlation clustering is the problem of generating a set of disjoint bi-cliques on a set of nodes while minimizing the symmetric difference to a bipartite input graph. The number or size of the output clusters is not constrained in any way. The best known approximation algorithm for this problem gives a factor of 11. This result and all previous ones involve solving large linear or semi-definite programs which become prohibitive even for modestly sized tasks. In this paper we present an improved factor 4 approximation algorithm to this problem using a simple combinatorial algorithm which does not require solving large convex programs. The analysis extends a method developed by Ailon, Charikar and Alantha in 2008, where a randomized pivoting algorithm was analyzed for obtaining a 3-approximation algorithm for Correlation Clustering, which is the same problem on graphs (not bipartite). The analysis for Correlation Clustering there required defining events for structures containing 3 vertices and using the probability of these events to produce a feasible solution to a dual of a certain natural LP bounding the optimal cost. It is tempting here to use sets of 4 vertices, which are the smallest structures for which contradictions arise for Bipartite Correlation Clustering. This simple idea, however, appears to be evasive. We show that, by modifying the LP, we can analyze algorithms which take into consideration subgraph structures of unbounded size. We believe our techniques are interesting in their own right, and may be used for other problems as well.

cs.DS↗

arXiv2010

Self-Improving Algorithms

We investigate ways in which an algorithm can improve its expected performance by fine-tuning itself automatically with respect to an unknown input distribution D. We assume here that D is of product type. More precisely, suppose that we need to process a sequence I_1, I_2, ... of inputs I = (x_1, x_2, ..., x_n) of some fixed length n, where each x_i is drawn independently from some arbitrary, unknown distribution D_i. The goal is to design an algorithm for these inputs so that eventually the expected running time will be optimal for the input distribution D = D_1 * D_2 * ... * D_n. We give such self-improving algorithms for two problems: (i) sorting a sequence of numbers and (ii) computing the Delaunay triangulation of a planar point set. Both algorithms achieve optimal expected limiting complexity. The algorithms begin with a training phase during which they collect information about the input distribution, followed by a stationary regime in which the algorithms settle to their optimized incarnations.

cs.DS↗

arXiv2010

Almost Optimal Unrestricted Fast Johnson-Lindenstrauss Transform

The problems of random projections and sparse reconstruction have much in common and individually received much attention. Surprisingly, until now they progressed in parallel and remained mostly separate. Here, we employ new tools from probability in Banach spaces that were successfully used in the context of sparse reconstruction to advance on an open problem in random pojection. In particular, we generalize and use an intricate result by Rudelson and Vershynin for sparse reconstruction which uses Dudley's theorem for bounding Gaussian processes. Our main result states that any set of $N = \exp(\tilde{O}(n))$ real vectors in $n$ dimensional space can be linearly mapped to a space of dimension $k=O(\log N\polylog(n))$, while (1) preserving the pairwise distances among the vectors to within any constant distortion and (2) being able to apply the transformation in time $O(n\log n)$ on each vector. This improves on the best known $N = \exp(\tilde{O}(n^{1/2}))$ achieved by Ailon and Liberty and $N = \exp(\tilde{O}(n^{1/3}))$ by Ailon and Chazelle. The dependence in the distortion constant however is believed to be suboptimal and subject to further investigation. For constant distortion, this settles the open question posed by these authors up to a $\polylog(n)$ factor while considerably simplifying their constructions.

cs.DS↗

arXiv2008

A Simple Linear Ranking Algorithm Using Query Dependent Intercept Variables

The LETOR website contains three information retrieval datasets used as a benchmark for testing machine learning ideas for ranking. Algorithms participating in the challenge are required to assign score values to search results for a collection of queries, and are measured using standard IR ranking measures (NDCG, precision, MAP) that depend only the relative score-induced order of the results. Similarly to many of the ideas proposed in the participating algorithms, we train a linear classifier. In contrast with other participating algorithms, we define an additional free variable (intercept, or benchmark) for each query. This allows expressing the fact that results for different queries are incomparable for the purpose of determining relevance. The cost of this idea is the addition of relatively few nuisance parameters. Our approach is simple, and we used a standard logistic regression library to test it. The results beat the reported participating algorithms. Hence, it seems promising to combine our approach with other more complex ideas.

cs.IR↗

arXiv2007

An efficient reduction of ranking to classification

This paper describes an efficient reduction of the learning problem of ranking to binary classification. The reduction guarantees an average pairwise misranking regret of at most that of the binary classifier regret, improving a recent result of Balcan et al which only guarantees a factor of 2. Moreover, our reduction applies to a broader class of ranking loss functions, admits a simpler proof, and the expected running time complexity of our algorithm in terms of number of calls to a classifier or preference function is improved from $Ω(n^2)$ to $O(n \log n)$. In addition, when the top $k$ ranked elements only are required ($k \ll n$), as in many applications in information extraction or search engines, the time complexity of our algorithm can be further reduced to $O(k \log k + n)$. Our reduction and algorithm are thus practical for realistic applications where the number of points to rank exceeds several thousands. Much of our results also extend beyond the bipartite case previously studied. Our rediction is a randomized one. To complement our result, we also derive lower bounds on any deterministic reduction from binary (preference) classification to ranking, implying that our use of a randomized reduction is essentially necessary for the guarantees we provide.

cs.LG↗

arXiv2002

Torsion points on curves and common divisors of a^k-1 and b^k-1

We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1 and b-1 are coprime, then a^k-1 and b^k-1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do this we use a result of Lang's on the finiteness of torsion points on algebraic curves. We also give a matrix analogue of these results, where for a unimodular matrix A, we look at the greatest common divisor of the elements of the matrix A^k-I.

math.NT↗