arXiv subjects

Christos Tzamos

Publications and source records attributed to Christos Tzamos.

At least 73 records · Page 4Linked to original sources

arXiv2018

Fast Modular Subset Sum using Linear Sketching

Given n positive integers, the Modular Subset Sum problem asks if a subset adds up to a given target t modulo a given integer m. This is a natural generalization of the Subset Sum problem (where m=+\infty) with ties to additive combinatorics and cryptography. Recently, in [Bringmann, SODA'17] and [Koiliaris and Xu, SODA'17], efficient algorithms have been developed for the non-modular case, running in near-linear pseudo-polynomial time. For the modular case, however, the best known algorithm by Koiliaris and Xu [Koiliaris and Xu, SODA'17] runs in time O~(m^{5/4}). In this paper, we present an algorithm running in time O~(m), which matches a recent conditional lower bound of [Abboud et al.'17] based on the Strong Exponential Time Hypothesis. Interestingly, in contrast to most previous results on Subset Sum, our algorithm does not use the Fast Fourier Transform. Instead, it is able to simulate the "textbook" Dynamic Programming algorithm much faster, using ideas from linear sketching. This is one of the first applications of sketching-based techniques to obtain fast algorithms for combinatorial problems in an offline setting.

cs.DS↗

arXiv2018

Certified Computation from Unreliable Datasets

A wide range of learning tasks require human input in labeling massive data. The collected data though are usually low quality and contain inaccuracies and errors. As a result, modern science and business face the problem of learning from unreliable data sets. In this work, we provide a generic approach that is based on \textit{verification} of only few records of the data set to guarantee high quality learning outcomes for various optimization objectives. Our method, identifies small sets of critical records and verifies their validity. We show that many problems only need $\text{poly}(1/\varepsilon)$ verifications, to ensure that the output of the computation is at most a factor of $(1 \pm \varepsilon)$ away from the truth. For any given instance, we provide an \textit{instance optimal} solution that verifies the minimum possible number of records to approximately certify correctness. Then using this instance optimal formulation of the problem we prove our main result: "every function that satisfies some Lipschitz continuity condition can be certified with a small number of verifications". We show that the required Lipschitz continuity condition is satisfied even by some NP-complete problems, which illustrates the generality and importance of this theorem. In case this certification step fails, an invalid record will be identified. Removing these records and repeating until success, guarantees that the result will be accurate and will depend only on the verified records. Surprisingly, as we show, for several computation tasks more efficient methods are possible. These methods always guarantee that the produced result is not affected by the invalid records, since any invalid record that affects the output will be detected and verified.

cs.GT↗

arXiv2018

Actively Avoiding Nonsense in Generative Models

A generative model may generate utter nonsense when it is fit to maximize the likelihood of observed data. This happens due to "model error," i.e., when the true data generating distribution does not fit within the class of generative models being learned. To address this, we propose a model of active distribution learning using a binary invalidity oracle that identifies some examples as clearly invalid, together with random positive examples sampled from the true distribution. The goal is to maximize the likelihood of the positive examples subject to the constraint of (almost) never generating examples labeled invalid by the oracle. Guarantees are agnostic compared to a class of probability distributions. We show that, while proper learning often requires exponentially many queries to the invalidity oracle, improper distribution learning can be done using polynomially many queries.

cs.LG↗

arXiv2018

A Converse to Banach's Fixed Point Theorem and its CLS Completeness

Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments. We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLS-complete. We thus provide the first natural complete problem for the class CLS, which was defined in [Daskalakis, Papadimitriou 2011] to capture the complexity of problems such as P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games.

cs.CC↗

arXiv2017

Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a "well-priced" condition, where each product's price is sufficiently high. We also provide an exact algorithm for $k$-additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers' preferences from their purchasing behavior.

cs.GT↗

arXiv2017

Strong Duality for a Multiple-Good Monopolist

We characterize optimal mechanisms for the multiple-good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure $μ$ derived from the buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand-bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for $n$ independent uniform items each supported on $[c,c+1]$ is a grand-bundling mechanism, as long as $c$ is sufficiently large, extending Pavlov's result for $2$ items [Pavlov'11]. At the same time, our characterization also implies that, for all $c$ and for all sufficiently large $n$, the optimal mechanism for $n$ independent uniform items supported on $[c,c+1]$ is not a grand bundling mechanism.

cs.GT↗

arXiv2017

Ten Steps of EM Suffice for Mixtures of Two Gaussians

The Expectation-Maximization (EM) algorithm is a widely used method for maximum likelihood estimation in models with latent variables. For estimating mixtures of Gaussians, its iteration can be viewed as a soft version of the k-means clustering algorithm. Despite its wide use and applications, there are essentially no known convergence guarantees for this method. We provide global convergence guarantees for mixtures of two Gaussians with known covariance matrices. We show that the population version of EM, where the algorithm is given access to infinitely many samples from the mixture, converges geometrically to the correct mean vectors, and provide simple, closed-form expressions for the convergence rate. As a simple illustration, we show that, in one dimension, ten steps of the EM algorithm initialized at infinity result in less than 1\% error estimation of the means. In the finite sample regime, we show that, under a random initialization, $\tilde{O}(d/ε^2)$ samples suffice to compute the unknown vectors to within $ε$ in Mahalanobis distance, where $d$ is the dimension. In particular, the error rate of the EM based estimator is $\tilde{O}\left(\sqrt{d \over n}\right)$ where $n$ is the number of samples, which is optimal up to logarithmic factors.

stat.ML↗

arXiv2017

Truthful Facility Location with Additive Errors

We address the problem of locating facilities on the $[0,1]$ interval based on reports from strategic agents. The cost of each agent is her distance to the closest facility, and the global objective is to minimize either the maximum cost of an agent or the social cost. As opposed to the extensive literature on facility location which considers the multiplicative error, we focus on minimizing the worst-case additive error. Minimizing the additive error incentivizes mechanisms to adapt to the size of the instance. I.e., mechanisms can sacrifice little efficiency in small instances (location profiles in which all agents are relatively close to one another), in order to gain more [absolute] efficiency in large instances. We argue that this measure is better suited for many manifestations of the facility location problem in various domains. We present tight bounds for mechanisms locating a single facility in both deterministic and randomized cases. We further provide several extensions for locating multiple facilities.

cs.GT↗

arXiv2016

Improving Viterbi is Hard: Better Runtimes Imply Faster Clique Algorithms

The classic algorithm of Viterbi computes the most likely path in a Hidden Markov Model (HMM) that results in a given sequence of observations. It runs in time $O(Tn^2)$ given a sequence of $T$ observations from a HMM with $n$ states. Despite significant interest in the problem and prolonged effort by different communities, no known algorithm achieves more than a polylogarithmic speedup. In this paper, we explain this difficulty by providing matching conditional lower bounds. We show that the Viterbi algorithm runtime is optimal up to subpolynomial factors even when the number of distinct observations is small. Our lower bounds are based on assumptions that the best known algorithms for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight $k$-Clique problem in edge-weighted graphs are essentially tight. Finally, using a recent algorithm by Green Larsen and Williams for online Boolean matrix-vector multiplication, we get a $2^{Ω(\sqrt {\log n})}$ speedup for the Viterbi algorithm when there are few distinct transition probabilities in the HMM.

cs.CC↗

arXiv2016

Faster Sublinear Algorithms using Conditional Sampling

A conditional sampling oracle for a probability distribution D returns samples from the conditional distribution of D restricted to a specified subset of the domain. A recent line of work (Chakraborty et al. 2013 and Cannone et al. 2014) has shown that having access to such a conditional sampling oracle requires only polylogarithmic or even constant number of samples to solve distribution testing problems like identity and uniformity. This significantly improves over the standard sampling model where polynomially many samples are necessary. Inspired by these results, we introduce a computational model based on conditional sampling to develop sublinear algorithms with exponentially faster runtimes compared to standard sublinear algorithms. We focus on geometric optimization problems over points in high dimensional Euclidean space. Access to these points is provided via a conditional sampling oracle that takes as input a succinct representation of a subset of the domain and outputs a uniformly random point in that subset. We study two well studied problems: k-means clustering and estimating the weight of the minimum spanning tree. In contrast to prior algorithms for the classic model, our algorithms have time, space and sample complexity that is polynomial in the dimension and polylogarithmic in the number of points. Finally, we comment on the applicability of the model and compare with existing ones like streaming, parallel and distributed computational models.

cs.DS↗

arXiv2016

A Size-Free CLT for Poisson Multinomials and its Applications

An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We show that any $(n,k)$-PMD is ${\rm poly}\left({k\over σ}\right)$-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on $n$ from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structural characterization of PMDs shown in recent work by Daskalakis, Kamath, and Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS for approximate Nash equilibria in anonymous games, significantly improving the state of the art, and matching qualitatively the running time dependence on $n$ and $1/\varepsilon$ of the best known algorithm for two-strategy anonymous games. Our new CLT also enables the construction of covers for the set of $(n,k)$-PMDs, which are proper and whose size is shown to be essentially optimal. Our cover construction combines our CLT with the Shapley-Folkman theorem and recent sparsification results for Laplacian matrices by Batson, Spielman, and Srivastava. Our cover size lower bound is based on an algebraic geometric construction. Finally, leveraging the structural properties of the Fourier spectrum of PMDs we show that these distributions can be learned from $O_k(1/\varepsilon^2)$ samples in ${\rm poly}_k(1/\varepsilon)$-time, removing the quasi-polynomial dependence of the running time on $1/\varepsilon$ from the algorithm of Daskalakis, Kamath, and Tzamos.

cs.DS↗

arXiv2016

Tight Hardness Results for Maximum Weight Rectangles

Given $n$ weighted points (positive or negative) in $d$ dimensions, what is the axis-aligned box which maximizes the total weight of the points it contains? The best known algorithm for this problem is based on a reduction to a related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in time $O(n^d)$. It was conjectured [Barbay et al., CCCG'13] that this runtime is tight up to subpolynomial factors. We answer this conjecture affirmatively by providing a matching conditional lower bound. We also provide conditional lower bounds for the special case when points are arranged in a grid (a well studied problem known as Maximum Subarray problem) as well as for other related problems. All our lower bounds are based on assumptions that the best known algorithms for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique problem in edge-weighted graphs are essentially optimal.

cs.DS↗

arXiv2015

On the Structure, Covering, and Learning of Poisson Multinomial Distributions

An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We prove a structural characterization of these distributions, showing that, for all $\varepsilon >0$, any $(n, k)$-Poisson multinomial random vector is $\varepsilon$-close, in total variation distance, to the sum of a discretized multidimensional Gaussian and an independent $(\text{poly}(k/\varepsilon), k)$-Poisson multinomial random vector. Our structural characterization extends the multi-dimensional CLT of Valiant and Valiant, by simultaneously applying to all approximation requirements $\varepsilon$. In particular, it overcomes factors depending on $\log n$ and, importantly, the minimum eigenvalue of the PMD's covariance matrix from the distance to a multidimensional Gaussian random variable. We use our structural characterization to obtain an $\varepsilon$-cover, in total variation distance, of the set of all $(n, k)$-PMDs, significantly improving the cover size of Daskalakis and Papadimitriou, and obtaining the same qualitative dependence of the cover size on $n$ and $\varepsilon$ as the $k=2$ cover of Daskalakis and Papadimitriou. We further exploit this structure to show that $(n,k)$-PMDs can be learned to within $\varepsilon$ in total variation distance from $\tilde{O}_k(1/\varepsilon^2)$ samples, which is near-optimal in terms of dependence on $\varepsilon$ and independent of $n$. In particular, our result generalizes the single-dimensional result of Daskalakis, Diakonikolas, and Servedio for Poisson Binomials to arbitrary dimension.

cs.DS↗

arXiv2015

Who to Trust for Truthfully Maximizing Welfare?

We introduce a general approach based on \emph{selective verification} and obtain approximate mechanisms without money for maximizing the social welfare in the general domain of utilitarian voting. Having a good allocation in mind, a mechanism with verification selects few critical agents and detects, using a verification oracle, whether they have reported truthfully. If yes, the mechanism produces the desired allocation. Otherwise, the mechanism ignores any misreports and proceeds with the remaining agents. We obtain randomized truthful (or almost truthful) mechanisms without money that verify only $O(\ln m / ε)$ agents, where $m$ is the number of outcomes, independently of the total number of agents, and are $(1-ε)$-approximate for the social welfare. We also show that any truthful mechanism with a constant approximation ratio needs to verify $Ω(\log m)$ agents. A remarkable property of our mechanisms is \emph{robustness}, namely that their outcome depends only on the reports of the truthful agents.

cs.GT↗

arXiv2015

Optimal Pricing is Hard

We show that computing the revenue-optimal deterministic auction in unit-demand single-buyer Bayesian settings, i.e. the optimal item-pricing, is computationally hard even in single-item settings where the buyer's value distribution is a sum of independently distributed attributes, or multi-item settings where the buyer's values for the items are independent. We also show that it is intractable to optimally price the grand bundle of multiple items for an additive bidder whose values for the items are independent. These difficulties stem from implicit definitions of a value distribution. We provide three instances of how different properties of implicit distributions can lead to intractability: the first is a #P-hardness proof, while the remaining two are reductions from the SQRT-SUM problem of Garey, Graham, and Johnson. While simple pricing schemes can oftentimes approximate the best scheme in revenue, they can have drastically different underlying structure. We argue therefore that either the specification of the input distribution must be highly restricted in format, or it is necessary for the goal to be mere approximation to the optimal scheme's revenue instead of computing properties of the scheme itself.

cs.GT↗

arXiv2015

Mechanism Design via Optimal Transport

Optimal mechanisms have been provided in quite general multi-item settings, as long as each bidder's type distribution is given explicitly by listing every type in the support along with its associated probability. In the implicit setting, e.g. when the bidders have additive valuations with independent and/or continuous values for the items, these results do not apply, and it was recently shown that exact revenue optimization is intractable, even when there is only one bidder. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare and the problem is challenging even in the two-item case. In this paper, we provide a framework for designing optimal mechanisms using optimal transport theory and duality theory. We instantiate our framework to obtain conditions under which only pricing the grand bundle is optimal in multi-item settings (complementing the work of [Manelli and Vincent 2006], as well as to characterize optimal two-item mechanisms. We use our results to derive closed-form descriptions of the optimal mechanism in several two-item settings, exhibiting also a setting where a continuum of lotteries is necessary for revenue optimization but a closed-form representation of the mechanism can still be found efficiently using our framework.

cs.GT↗

arXiv2014

The Value of Knowing Your Enemy

Many auction settings implicitly or explicitly require that bidders are treated equally ex-ante. This may be because discrimination is philosophically or legally impermissible, or because it is practically difficult to implement or impossible to enforce. We study so-called {\em anonymous} auctions to understand the revenue tradeoffs and to develop simple anonymous auctions that are approximately optimal. We consider digital goods settings and show that the optimal anonymous, dominant strategy incentive compatible auction has an intuitive structure --- imagine that bidders are randomly permuted before the auction, then infer a posterior belief about bidder i's valuation from the values of other bidders and set a posted price that maximizes revenue given this posterior. We prove that no anonymous mechanism can guarantee an approximation better than O(n) to the optimal revenue in the worst case (or O(log n) for regular distributions) and that even posted price mechanisms match those guarantees. Understanding that the real power of anonymous mechanisms comes when the auctioneer can infer the bidder identities accurately, we show a tight O(k) approximation guarantee when each bidder can be confused with at most k "higher types". Moreover, we introduce a simple mechanism based on n target prices that is asymptotically optimal and build on this mechanism to extend our results to m-unit auctions and sponsored search.

cs.GT↗

arXiv2013

Strategy-Proof Facility Location for Concave Cost Functions

We consider k-Facility Location games, where n strategic agents report their locations on the real line, and a mechanism maps them to k facilities. Each agent seeks to minimize his connection cost, given by a nonnegative increasing function of his distance to the nearest facility. Departing from previous work, that mostly considers the identity cost function, we are interested in mechanisms without payments that are (group) strategyproof for any given cost function, and achieve a good approximation ratio for the social cost and/or the maximum cost of the agents. We present a randomized mechanism, called Equal Cost, which is group strategyproof and achieves a bounded approximation ratio for all k and n, for any given concave cost function. The approximation ratio is at most 2 for Max Cost and at most n for Social Cost. To the best of our knowledge, this is the first mechanism with a bounded approximation ratio for instances with k > 2 facilities and any number of agents. Our result implies an interesting separation between deterministic mechanisms, whose approximation ratio for Max Cost jumps from 2 to unbounded when k increases from 2 to 3, and randomized mechanisms, whose approximation ratio remains at most 2 for all k. On the negative side, we exclude the possibility of a mechanism with the properties of Equal Cost for strictly convex cost functions. We also present a randomized mechanism, called Pick the Loser, which applies to instances with k facilities and n = k+1 agents, and for any given concave cost function, is strongly group strategyproof and achieves an approximation ratio of 2 for Social Cost.

cs.GT↗