DROP: a Workload-Aware Optimizer for Dimensionality Reduction
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Are All Distributions Easy?
Are all distributions easy? Emanuele Viola∗ November 5, 2009 Abstract Complexity theory typically studies the complexity of computing a function h(x): f0; 1gn ! f0; 1gm of a given input x. We advocate the study of the complexity of generating the distribution h(x) for uniform x, given random bits. Our main results are: • There are explicit AC0 circuits of size poly(n) and depth O(1) whose output n P distribution has statistical distance 1=2 from the distribution (X; i Xi) 2 f0; 1gn × f0; 1; : : : ; ng for uniform X 2 f0; 1gn, despite the inability of these P circuits to compute i xi given x. Previous examples of this phenomenon apply to different distributions such as P n+1 (X; i Xi mod 2) 2 f0; 1g . We also prove a lower bound independent from n on the statistical distance be- tween the output distribution of NC0 circuits and the distribution (X; majority(X)). We show that 1 − o(1) lower bounds for related distributions yield lower bounds for succinct data structures. • Uniform randomized AC0 circuits of poly(n) size and depth d = O(1) with error can be simulated by uniform randomized circuits of poly(n) size and depth d + 1 with error + o(1) using ≤ (log n)O(log log n) random bits. Previous derandomizations [Ajtai and Wigderson '85; Nisan '91] increase the depth by a constant factor, or else have poor seed length. Given the right tools, the above results have technically simple proofs. ∗Supported by NSF grant CCF-0845003. Email: [email protected] 1 Introduction Complexity theory, with some notable exceptions, typically studies the complexity of com- puting a function h(x): f0; 1gn ! f0; 1gm of a given input x. -
Typical Stability
Typical Stability Raef Bassily∗ Yoav Freundy Abstract In this paper, we introduce a notion of algorithmic stability called typical stability. When our goal is to release real-valued queries (statistics) computed over a dataset, this notion does not require the queries to be of bounded sensitivity – a condition that is generally assumed under differential privacy [DMNS06, Dwo06] when used as a notion of algorithmic stability [DFH+15b, DFH+15c, BNS+16] – nor does it require the samples in the dataset to be independent – a condition that is usually assumed when generalization-error guarantees are sought. Instead, typical stability requires the output of the query, when computed on a dataset drawn from the underlying distribution, to be concentrated around its expected value with respect to that distribution. Typical stability can also be motivated as an alternative definition for database privacy. Like differential privacy, this notion enjoys several important properties including robustness to post-processing and adaptive composition. However, privacy is guaranteed only for a given family of distributions over the dataset. We also discuss the implications of typical stability on the generalization error (i.e., the difference between the value of the query computed on the dataset and the expected value of the query with respect to the true data distribution). We show that typical stability can control generalization error in adaptive data analysis even when the samples in the dataset are not necessarily independent and when queries to be computed are not necessarily of bounded- sensitivity as long as the results of the queries over the dataset (i.e., the computed statistics) follow a distribution with a “light” tail. -
An Efficient Boosting Algorithm for Combining Preferences Raj
An Efficient Boosting Algorithm for Combining Preferences by Raj Dharmarajan Iyer Jr. Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1999 © Massachusetts Institute of Technology 1999. All rights reserved. A uth or .............. ..... ..... .................................. Department of E'ectrical ngineering and Computer Science August 24, 1999 C ertified by .. ................. ...... .. .............. David R. Karger Associate Professor Thesis Supervisor Accepted by............... ........ Arthur C. Smith Chairman, Departmental Committe Graduate Students MACHU OF TEC Lo NOV LIBRARIES An Efficient Boosting Algorithm for Combining Preferences by Raj Dharmarajan Iyer Jr. Submitted to the Department of Electrical Engineering and Computer Science on August 24, 1999, in partial fulfillment of the requirements for the degree of Master of Science Abstract The problem of combining preferences arises in several applications, such as combining the results of different search engines. This work describes an efficient algorithm for combining multiple preferences. We first give a formal framework for the problem. We then describe and analyze a new boosting algorithm for combining preferences called RankBoost. We also describe an efficient implementation of the algorithm for certain natural cases. We discuss two experiments we carried out to assess the performance of RankBoost. In the first experi- ment, we used the algorithm to combine different WWW search strategies, each of which is a query expansion for a given domain. For this task, we compare the performance of Rank- Boost to the individual search strategies. The second experiment is a collaborative-filtering task for making movie recommendations. -
An Introduction to Johnson–Lindenstrauss Transforms
An Introduction to Johnson–Lindenstrauss Transforms Casper Benjamin Freksen∗ 2nd March 2021 Abstract Johnson–Lindenstrauss Transforms are powerful tools for reducing the dimensionality of data while preserving key characteristics of that data, and they have found use in many fields from machine learning to differential privacy and more. This note explains whatthey are; it gives an overview of their use and their development since they were introduced in the 1980s; and it provides many references should the reader wish to explore these topics more deeply. The text was previously a main part of the introduction of my PhD thesis [Fre20], but it has been adapted to be self contained and serve as a (hopefully good) starting point for readers interested in the topic. 1 The Why, What, and How 1.1 The Problem Consider the following scenario: We have some data that we wish to process but the data is too large, e.g. processing the data takes too much time, or storing the data takes too much space. A solution would be to compress the data such that the valuable parts of the data are kept and the other parts discarded. Of course, what is considered valuable is defined by the data processing we wish to apply to our data. To make our scenario more concrete let us say that our data consists of vectors in a high dimensional Euclidean space, R3, and we wish to find a transform to embed these vectors into a lower dimensional space, R<, where < 3, so that we arXiv:2103.00564v1 [cs.DS] 28 Feb 2021 ≪ can apply our data processing in this lower dimensional space and still get meaningful results. -
Adversarial Robustness of Streaming Algorithms Through Importance Sampling
Adversarial Robustness of Streaming Algorithms through Importance Sampling Vladimir Braverman Avinatan Hassidim Yossi Matias Mariano Schain Google∗ Googley Googlez Googlex Sandeep Silwal Samson Zhou MITO CMU{ June 30, 2021 Abstract Robustness against adversarial attacks has recently been at the forefront of algorithmic design for machine learning tasks. In the adversarial streaming model, an adversary gives an algorithm a sequence of adaptively chosen updates u1; : : : ; un as a data stream. The goal of the algorithm is to compute or approximate some predetermined function for every prefix of the adversarial stream, but the adversary may generate future updates based on previous outputs of the algorithm. In particular, the adversary may gradually learn the random bits internally used by an algorithm to manipulate dependencies in the input. This is especially problematic as many important problems in the streaming model require randomized algorithms, as they are known to not admit any deterministic algorithms that use sublinear space. In this paper, we introduce adversarially robust streaming algorithms for central machine learning and algorithmic tasks, such as regression and clustering, as well as their more general counterparts, subspace embedding, low-rank approximation, and coreset construction. For regression and other numerical linear algebra related tasks, we consider the row arrival streaming model. Our results are based on a simple, but powerful, observation that many importance sampling-based algorithms give rise to adversarial robustness which is in contrast to sketching based algorithms, which are very prevalent in the streaming literature but suffer from adversarial attacks. In addition, we show that the well-known merge and reduce paradigm in streaming is adversarially robust. -
Private Center Points and Learning of Halfspaces
Proceedings of Machine Learning Research vol 99:1–14, 2019 32nd Annual Conference on Learning Theory Private Center Points and Learning of Halfspaces Amos Beimel [email protected] Ben-Gurion University Shay Moran [email protected] Princeton University Kobbi Nissim [email protected] Georgetown University Uri Stemmer [email protected] Ben-Gurion University Editors: Alina Beygelzimer and Daniel Hsu Abstract We present a private agnostic learner for halfspaces over an arbitrary finite domain X ⊂ d with ∗ R sample complexity poly(d; 2log jXj). The building block for this learner is a differentially pri- ∗ vate algorithm for locating an approximate center point of m > poly(d; 2log jXj) points – a high dimensional generalization of the median function. Our construction establishes a relationship between these two problems that is reminiscent of the relation between the median and learning one-dimensional thresholds [Bun et al. FOCS ’15]. This relationship suggests that the problem of privately locating a center point may have further applications in the design of differentially private algorithms. We also provide a lower bound on the sample complexity for privately finding a point in the convex hull. For approximate differential privacy, we show a lower bound of m = Ω(d+log∗ jXj), whereas for pure differential privacy m = Ω(d log jXj). Keywords: Differential privacy, Private PAC learning, Halfspaces, Quasi-concave functions. 1. Introduction Machine learning models are often trained on sensitive personal information, e.g., when analyzing healthcare records or social media data. There is hence an increasing awareness and demand for pri- vacy preserving machine learning technology. -
Adversarially Robust Property-Preserving Hash Functions
Adversarially robust property-preserving hash functions The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Boyle, Elette et al. “Adversarially robust property-preserving hash functions.” Paper in the Leibniz International Proceedings in Informatics, LIPIcs, 124, 16, 10th Innovations in Theoretical Computer Science (ITCS 2019), San Diego, California, January 10-12, 2019, Leibniz Center for Informatics: 1-20 © 2019 The Author(s) As Published 10.4230/LIPIcs.ITCS.2019.16 Publisher Leibniz Center for Informatics Version Final published version Citable link https://hdl.handle.net/1721.1/130258 Terms of Use Creative Commons Attribution 4.0 International license Detailed Terms https://creativecommons.org/licenses/by/4.0/ Adversarially Robust Property-Preserving Hash Functions Elette Boyle1 IDC Herzliya, Kanfei Nesharim Herzliya, Israel [email protected] Rio LaVigne2 MIT CSAIL, 32 Vassar Street, Cambridge MA, 02139 USA [email protected] Vinod Vaikuntanathan3 MIT CSAIL, 32 Vassar Street, Cambridge MA, 02139 USA [email protected] Abstract Property-preserving hashing is a method of compressing a large input x into a short hash h(x) in such a way that given h(x) and h(y), one can compute a property P (x, y) of the original inputs. The idea of property-preserving hash functions underlies sketching, compressed sensing and locality-sensitive hashing. Property-preserving hash functions are usually probabilistic: they use the random choice of a hash function from a family to achieve compression, and as a consequence, err on some inputs. Traditionally, the notion of correctness for these hash functions requires that for every two inputs x and y, the probability that h(x) and h(y) mislead us into a wrong prediction of P (x, y) is negligible. -
Adversarial Laws of Large Numbers and Optimal Regret in Online Classification
Adversarial Laws of Large Numbers and Optimal Regret in Online Classification Noga Alon* Omri Ben-Eliezer† Yuval Dagan‡ Shay Moran§ Moni Naor¶ Eylon Yogev|| Abstract Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed sub-population is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by Ben-Eliezer and Yogev (2020), and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are online learnable. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of Littlestone’s dimension, thus resolving the main open question from Ben-David, Pal,´ and Shalev-Shwartz (2009), which was also posed by Rakhlin, Sridharan, and Tewari (2015). *Department of Mathematics, Princeton University, Princeton, New Jersey, USA and Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv, Israel. Research supported in part by NSF grant DMS-1855464, BSF grant 2018267 and the Simons Foundation. Email: [email protected]. †Center for Mathematical Sciences and Applications, Harvard University, Massachusetts, USA. Research partially conducted while the author was at Weizmann Institute of Science, supported in part by a grant from the Israel Science Foundation (no. -
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T G¨ P 2012 C N Deadline: December 31, 2011 The Gödel Prize for outstanding papers in the area of theoretical computer sci- ence is sponsored jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery, Special Inter- est Group on Algorithms and Computation Theory (ACM-SIGACT). The award is presented annually, with the presentation taking place alternately at the Inter- national Colloquium on Automata, Languages, and Programming (ICALP) and the ACM Symposium on Theory of Computing (STOC). The 20th prize will be awarded at the 39th International Colloquium on Automata, Languages, and Pro- gramming to be held at the University of Warwick, UK, in July 2012. The Prize is named in honor of Kurt Gödel in recognition of his major contribu- tions to mathematical logic and of his interest, discovered in a letter he wrote to John von Neumann shortly before von Neumann’s death, in what has become the famous P versus NP question. The Prize includes an award of USD 5000. AWARD COMMITTEE: The winner of the Prize is selected by a committee of six members. The EATCS President and the SIGACT Chair each appoint three members to the committee, to serve staggered three-year terms. The committee is chaired alternately by representatives of EATCS and SIGACT. The 2012 Award Committee consists of Sanjeev Arora (Princeton University), Josep Díaz (Uni- versitat Politècnica de Catalunya), Giuseppe Italiano (Università a˘ di Roma Tor Vergata), Mogens Nielsen (University of Aarhus), Daniel Spielman (Yale Univer- sity), and Eli Upfal (Brown University). ELIGIBILITY: The rule for the 2011 Prize is given below and supersedes any di fferent interpretation of the parametric rule to be found on websites on both SIGACT and EATCS. -
Theory and Algorithms for Modern Problems in Machine Learning and an Analysis of Markets
Theory and Algorithms for Modern Problems in Machine Learning and an Analysis of Markets by Ashish Rastogi A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Computer Science Courant Institute of Mathematical Sciences New York University May 2008 Richard Cole—Advisor Mehryar Mohri—Advisor °c Ashish Rastogi All Rights Reserved, 2008 To the most wonderful parents in the whole world, Mrs. Asha Rastogi and Mr. Shyam Lal Rastogi iv Acknowledgements First and foremost, I would like to thank my advisors, Professor Richard Cole and Professor Mehryar Mohri, for their unwavering support, guidance and constant encouragement. They have been inspiring mentors and much of what lies in the following pages can be credited to them. Working under their supervision has been one of the most enriching experiences of my life. I would also like to thank Professor Joel Spencer, Professor Arun Sun- dararajan, Professor Subhash Khot and Dr. Corinna Cortes for agreeing to serve as members on my thesis committee. Professor Spencer’s class on Random Graphs remains one of the most stim- ulating courses I undertook as a graduate student. Internships at Google through the summers of 2005, 2006 and 2007 were some of the most enjoy- able periods of my graduate school life. Many thanks are due to Dr. Corinna Cortes for providing me with the opportunity to work on several challenging problems at Google. Research initiated during these internships culminated in the development of ideas that form the bulk of this thesis. I would also like to thank my peers from the graduate school. -
40Th ACM Symposium on Theory of Computing (STOC 2008) Saturday
40th ACM Symposium on Theory of Computing (STOC 2008) Saturday, May 17, 2008 7:00pm – 10:00pm Registration (Conference Centre) 8:00pm – 10:00pm Reception (Palm Court) Sunday, May 18, 2008 8:00am – 5:00pm Registration (Conference Centre) 8:10am – 8:35am Breakfast (Conference Centre) Session 1A Session 1B (Theatre) (Saanich Room) Chair: Venkat Guruswami Chair: David Shmoys (Cornell (University of Washington and University) Institute for Advanced Study) 8:35am - 8:55am Parallel Repetition in Projection The Complexity of Temporal Games and a Concentration Bound Constraint Satisfaction Problems Anup Rao Manuel Bodirsky, Jan Kara 9:00am - 9:20am SDP Gaps and UGC Hardness for An Effective Ergodic Theorem Multiway Cut, 0-Extension and and Some Applications Metric Labeling Satyadev Nandakumar Rajeskar Manokaran, Joseph (Seffi) Naor, Prasad Raghavendra, Roy Schwartz 9:25am – 9:45am Unique Games on Expanding Algorithms for Subset Selection Constraint Graphs are Easy in Linear Regression Sanjeev Arora, Subhash A. Khot, Abhimanyu Das, David Kempe Alexandra Kolla, David Steurer, Madhur Tulsiani, Nisheeth Vishnoi 9:45am - 10:10am Break Session 2 (Theatre) Chair: Joan Feigenbaum (Yale University) 10:10am - 11:10am Rethinking Internet Routing Invited talk by Jennifer Rexford (Princeton University) 11:10am - 11:20am Break Session 3A Session 3B (Theatre) (Saanich Room) Chair: Xiaotie Deng (City Chair: Anupam Gupta (Carnegie University of Hong Kong) Mellon University) The Pattern Matrix Method for 11:20am – 11:40am Interdomain Routing and Games Lower Bounds on Quantum Hagay Levin, Michael Schapira, Communication Aviv Zohar Alexander A. Sherstov 11:45am – 12:05pm Optimal approximation for the Classical Interaction Cannot Submodular Welfare Problem in Replace a Quantum Message the value oracle model Dmitry Gavinsky Jan Vondrak 12:10pm – 12:30pm Optimal Mechanism Design and Span-program-based quantum Money Burning algorithm for evaluating formulas Jason Hartline, Tim Ben W. -
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SIGMOD/PODS 2003 Final Program San Diego, California June 8-13, 2003 Answer Queries, Rada Chirkova, Chen Li Streams, Abhinandan Das, Johannes Sunday, June 8th Gehrke, Mirek Riedewald 8:30-17:30 The Impact of the Constant Complement DEBS Workshop (Windsor) Approach Towards View Updating, Jens SIGMOD Research Session 3: OLAP PKC50 Workshop (Hampton) Lechtenboerger (Garden Salon 1) MPDS Workshop (Sheffield) Chair: Latha Colby View-Based Query Containment, Diego 19:30-21:30 Calvanese, Giuseppe De Giacomo, Maurizio Best Paper: Spreadsheets in RDBMS for SIGMOD/PODS Welcoming Reception Lenzerini, Moshe Y. Vardi OLAP, Abhinav Gupta, Andy Witkowski, (Garden Salon 1) Gregory Dorman, Srikanth Bellamkonda, The View Selection Problem for XML Tolga Bozkaya, Nathan Folkert, Lei Sheng, Content Based Routing, Ashish Kumar Sankar Subramanian Monday, June 9th Gupta, Alon Y. Halevy, Dan Suciu 8:00-8:45 QC-Trees: An Efficient Summary Structure Continental Breakfast (Atlas Foyer) 19:00-22:00 for Semantic OLAP, Laks Lakshmanan, Jian SIGMOD New DB Faculty Symposium Pei, Yan Zhao 8:45-10:00 (Windsor & Sheffield) PODS Session 1: Invited Talk SIGMOD Industrial Session 1: Web (Hampton & Sheffield) 21:00-23:00 Services (Sheffield) Chair: Tova Milo PODS Business Meeting Chair: Surajit Chaudhuri (Garden Salon 1) E-services: A Look Behind the Curtain, Rick The Future of Web Services - I, Adam Hull Bosworth (BEA Systems) Tuesday, June 10th 10:00-10:15 8:00-8:45 The Future of Web Services - II, Felipe Morning Coffee Break (Atlas Foyer) Continental Breakfast (Atlas Foyer) Cabrera (Microsoft) 10:15-11:15 8:45-9:45 SIGMOD Demos: Group A (Crescent) PODS Session 2: Award Papers SIGMOD Keynote (Regency Ballroom) (Hampton & Sheffield) Chair: Alon Halevy A System for Watermarking Relational Chair: Catriel Beeri Databases, Rakesh Agrawal, Jerry Kiernan The Database Course Qua Poster Child For Best Paper: An Information-Theoretic More Efficient Education, Jeffrey D.