Lecture October 23. Scribes: Ruixin Qiang and Alana Shine
|
|
- Robert Cobb
- 5 years ago
- Views:
Transcription
1 CSCI699: Topics in Learning and Gae Theory Lecture October 23 Lecturer: Ilias Scribes: Ruixin Qiang and Alana Shine Today s topic is auction with saples. 1 Introduction to auctions Definition 1. In a single ite auction, there are k bidders bid for one ite, auction design is the proble of designing a truthful echnis to axiize certain objective. For different objectives, there are different results. Social Welfare (benefit the society): VCG echanis will be optial. In single ite case, it will be the second price auction. It is prior-free, i.e., does not require any assuption about the distribution of bidders value (not the cretiria of this lecture). Revenue (benefit the auctioneer): Myerson s algorith (1981) is optial for average case. In this case, there is guarantee only for Bayesian setting, i.e., we assue the bidders values v i are drawn fro certain distribution, (v 1, v 2,..., v k ) F. In this lecture, we will discuss revenue axiization. Also we assue the bidders values are independenlty drawn fro F i, i.e. F = F 1 F 2... F k. The Myerson s auction can be phrased as VCG echanis that axiizes virtual 1 F (v) f(v) values. The virtual value of a bidder with value v is defined to be φ(v) = v when the distribution with c.d.f. F, p.d.f. f is regular (φ(v) is strictly-increasing). The ain goal will be understanding the epirical version of Myerson s auction, naely, instead of knowing the distribution F, we can only draw saples fro it. This proble is hard in general without assuptions. For exaple, on a point with a large value on which the distribution assigns a very tiny probability, with certain nuber of saples we cannot observe the large value and thus cannot provide any guarantees. Therefore, we need assuptions about the class of distributions. 1
2 2 AUCTION WITH SAMPLES, FIRST NATURAL APPROACH 2 2 Auction with Saples, First natural approach There are two criteria for auctions with saples: Saple coplexity, i.e., the nuber of saples required. Coputational coplexity, i.e., the running tie of the algorith. Inforation theory provides a lower bound of saple coplexity S(n, ɛ, D). Here D is the class which we assue F is in. The algorith we designed should have saple coplexity in poly(s) and coputational coplexity in poly(s). Open proble: efficient algorith that has saple coplexity Õ(S). Here we have our first approach: 1. Take saples fro each F i ; 2. Get the epirical distribution F i ; 3. Run Myerson s auction with F i. This approach does not work but a variant of it will. 3 Learning distributions The proble of learning distributions is defined as: Definition 2. Given a faily of distributions D, and i.i.d. saples fro an distribution p D. Output a distribution h, s.t. with high probability d(h, p) ɛ. Here d(h, p) easures the distance between two distributions. Since the definition is generic, there are three probles we need to address. The distance functions we will discuss: total variation distance, Kologorov distance. Nuber of saples required (ɛ, D, d). For the with high probability, the randoness is fro the saples and the algorith.
3 3 LEARNING DISTRIBUTIONS Learning discrete distribution subject to total variation distance We assue D to be all discrete distributions over doain [n]. For two discrete distributions with p..f. p, q, the total variation distance between the is defined as d tv = ax A [n] p(a) q(a) = 1 2 p q 1 With saples S 1,..., S, the optial algorith is just using the epirical distribution, let N i = {j [] S j = i} : ˆp (i) = N i. When is large enough, we will have d tv (p, ˆp ) ɛ. To find a lower bound of, since d tv (p, ˆp ) is a rando variable, we can first find the that akes E [d tv (p, ˆp )] ɛ/c, then use Markov inequality to bound the tail probability: Pr[d tv (p, ˆp ) > ɛ] 1 C. ɛ. In the following proof, we will ignore the constant C since it is just a scale factor of Theore 3. When Ω( n ), the expected total variation distance between p and ˆp is upper bounded by ɛ. Proof. Since N i Bino(, p(i)), we have E [N i ] = p(i), Var[N i ] = p(i)(1 p(i)). E [d tv (p, ˆp )] 1 n E [ N i p(i) ] 1 n E [(N i p(i)) 2 ] = 1 n p(i)(1 p(i)) 1 n p(i) n
4 4 LEARNING DISTRIBUTION SUBJECT TO KOLMOGOROV DISTANCE 4 The second and last inequality is due to Cauchy-Schwarz inequality. When n, we have E [d tv (p, ˆp )] ɛ. The bound of n is tight, here we provide a constructive proof. Proof. We will construct a distribution faily D by perturbing the unifor distribution over [n]. For each pair (2i 1, 2i), set their probability to be either ( 1 ɛ, 1+ɛ) or n n ( 1+ɛ, 1 ɛ). Thus there will be n n 2n/2 different distributions in the faily. To learn a distribution h fro saples of p D, s.t. d tv (p, h) ɛ/4, we need to find the correct perturbing direction for at least n pairs. Finding the correct perturbation 4 direction for each pair requires O( 1 ) saples (by inforation theory). Thus in total, at least O( n ) saples are required. When n goes to infinity, there is no chance to use finite nuber of saples to learn a distribution and bound the total variation distance. However, for certain faily of distributions, we can learn distributions subject to the total variation distance less than ɛ, with saple coplexity poly(1/ɛ, log n) Distribution class D is all log-concave distributions over [n]. Then we can learn with Θ(1/ɛ 5/2 ). The hazard rate of a distribution is f(x)/(1 F (X)) where f is the PDF of the distribution and F is the CDF. If we have onotone none-decreasing hazard rate, we can learn it with O(log(n)/ɛ 3 ) saples. If I ake no assuptions about the distribution, to get error probability δ I need 1 n δ saples using our previous error analysis and arkov s inequality. With ore careful analysis, we can show that we only need ( ) n + log 1 δ Θ 4 Learning distribution subject to Kologorov Distance Instead of aking stronger assuptions on the distribution to achieve saple coplexity independent of saple support, we turn towards changing the etric we are using to learn fro total variation distance to Kologorov Distance. We have a rando variable X supported over R. Let F be the CDF of X. F (u) = P r[x u]
5 5 AUCTION WITH SAMPLES 5 Learning using Kologorov Distance eans d K (X, Y ) = ax u R F X(u) F Y (u) Theore 4. DKW Inequality For any X over R, we can learn with O( 1 ) up to d K (X, Y ) ɛ. Proof. Consider the epirical. ˆF (u) = 1 n 1{x i u} ax u R ˆF (u) F (u) 1 with probability at least Using chernoff bounds, 1 log( 10 ɛ ). We can also prove using artingales or chaining. 5 Auction with Saples Suppose we have a single bidder, single ite auction. Also, assue that the distribution of the bidder value is over [0, 1]. We will show that with O( 1 ) saples fro F I can achieve revenue at least OP T ɛ. By DKW using O( 1 ) saples for F, I can find ˆF such that sup u R F (u) ˆF (u) ɛ. We will use this ˆF to copute price ˆp that axiizes ˆp argax u [0,1] u(1 ˆF (u)). Recall that if we know F, optial p is exactly argax u [0,1] u(1 F (u)) and this p achieves optial revenue OP T. OP T = p (1 F (p )) Rev(ˆp) = ˆp(1 F (ˆp) ˆp(1 ˆF (ˆp) ɛ) = ˆp(1 ˆF (ˆp)) ɛˆp p (1 ˆF (p )) ɛˆp p (1 F (p ) ɛ) ɛˆp = p (1 F (p )) ɛ(ˆp + p ) = OP T 2ɛ With the last step using the fact that prices are bounded within the interval [0, 1]. But what if the prices are not bounded? Consider the following distribution over prices.
6 5 AUCTION WITH SAMPLES 6 The prices p is set to M 2 with probability 1 and 0 otherwise. In order to avoid saple M coplexity dependent on M, I need an assuption on the distribution. We now generalize to the single ite, k bidder odel. Assue that F is a product distribution so F = F 1 F 2 F k with v (1)...v (k) F. Theore 5. In the single ite auction with k bidders and independent regular distributions, if = Ω ( ) k 10 ɛ then there is a -saple auction coputable in polynoial-tie 7 in k and 1/ɛ with expected revenue at least OP T ɛ. Theore 6. Any such auction requires soe polynoial in k/ɛ saples. 1 F (x) A regular distribution is one where the virtual value x is non-decreasing. Let f(x) f i be the pdf of the distribution over prices for the ith bidder and F i the corresponding CDF. Then the virtual value v i of bidder i is 1 F i(v) f i. (v) Fact: If F i s are regular then Myerson coputes the VCG rule on virtual values. If all virtual values are less than zero, no one gets it. Otherwise the ite goes to the bidder with the highest virtual value.
Computational and Statistical Learning Theory
Coputational and Statistical Learning Theory TTIC 31120 Prof. Nati Srebro Lecture 2: PAC Learning and VC Theory I Fro Adversarial Online to Statistical Three reasons to ove fro worst-case deterinistic
More informationComputational and Statistical Learning Theory
Coputational and Statistical Learning Theory Proble sets 5 and 6 Due: Noveber th Please send your solutions to learning-subissions@ttic.edu Notations/Definitions Recall the definition of saple based Radeacher
More information1 Proof of learning bounds
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #4 Scribe: Akshay Mittal February 13, 2013 1 Proof of learning bounds For intuition of the following theore, suppose there exists a
More information1 Generalization bounds based on Rademacher complexity
COS 5: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #0 Scribe: Suqi Liu March 07, 08 Last tie we started proving this very general result about how quickly the epirical average converges
More information1 Rademacher Complexity Bounds
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #10 Scribe: Max Goer March 07, 2013 1 Radeacher Coplexity Bounds Recall the following theore fro last lecture: Theore 1. With probability
More informationCS Lecture 13. More Maximum Likelihood
CS 6347 Lecture 13 More Maxiu Likelihood Recap Last tie: Introduction to axiu likelihood estiation MLE for Bayesian networks Optial CPTs correspond to epirical counts Today: MLE for CRFs 2 Maxiu Likelihood
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the bias-variance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
More informationE0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
More informationTight Information-Theoretic Lower Bounds for Welfare Maximization in Combinatorial Auctions
Tight Inforation-Theoretic Lower Bounds for Welfare Maxiization in Cobinatorial Auctions Vahab Mirrokni Jan Vondrák Theory Group, Microsoft Dept of Matheatics Research Princeton University Redond, WA 9805
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optiization and Approxiation Instructor: Moritz Hardt Eail: hardt+ee227c@berkeley.edu Graduate Instructor: Max Sichowitz Eail: sichow+ee227c@berkeley.edu October
More informationUnderstanding Machine Learning Solution Manual
Understanding Machine Learning Solution Manual Written by Alon Gonen Edited by Dana Rubinstein Noveber 17, 2014 2 Gentle Start 1. Given S = ((x i, y i )), define the ultivariate polynoial p S (x) = i []:y
More informationSymmetrization and Rademacher Averages
Stat 928: Statistical Learning Theory Lecture: Syetrization and Radeacher Averages Instructor: Sha Kakade Radeacher Averages Recall that we are interested in bounding the difference between epirical and
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE7C (Spring 018: Convex Optiization and Approxiation Instructor: Moritz Hardt Eail: hardt+ee7c@berkeley.edu Graduate Instructor: Max Sichowitz Eail: sichow+ee7c@berkeley.edu October 15,
More information1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
More informationMachine Learning Basics: Estimators, Bias and Variance
Machine Learning Basics: Estiators, Bias and Variance Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics in Basics
More informationE0 370 Statistical Learning Theory Lecture 5 (Aug 25, 2011)
E0 370 Statistical Learning Theory Lecture 5 Aug 5, 0 Covering Nubers, Pseudo-Diension, and Fat-Shattering Diension Lecturer: Shivani Agarwal Scribe: Shivani Agarwal Introduction So far we have seen how
More informationFixed-to-Variable Length Distribution Matching
Fixed-to-Variable Length Distribution Matching Rana Ali Ajad and Georg Böcherer Institute for Counications Engineering Technische Universität München, Gerany Eail: raa2463@gail.co,georg.boecherer@tu.de
More information1 Proving the Fundamental Theorem of Statistical Learning
THEORETICAL MACHINE LEARNING COS 5 LECTURE #7 APRIL 5, 6 LECTURER: ELAD HAZAN NAME: FERMI MA ANDDANIEL SUO oving te Fundaental Teore of Statistical Learning In tis section, we prove te following: Teore.
More informationMULTIAGENT Resource Allocation (MARA) is the
EDIC RESEARCH PROPOSAL 1 Designing Negotiation Protocols for Utility Maxiization in Multiagent Resource Allocation Tri Kurniawan Wijaya LSIR, I&C, EPFL Abstract Resource allocation is one of the ain concerns
More informationBayesian Learning. Chapter 6: Bayesian Learning. Bayes Theorem. Roles for Bayesian Methods. CS 536: Machine Learning Littman (Wu, TA)
Bayesian Learning Chapter 6: Bayesian Learning CS 536: Machine Learning Littan (Wu, TA) [Read Ch. 6, except 6.3] [Suggested exercises: 6.1, 6.2, 6.6] Bayes Theore MAP, ML hypotheses MAP learners Miniu
More informationLecture 21. Interior Point Methods Setup and Algorithm
Lecture 21 Interior Point Methods In 1984, Kararkar introduced a new weakly polynoial tie algorith for solving LPs [Kar84a], [Kar84b]. His algorith was theoretically faster than the ellipsoid ethod and
More informationQuantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search
Quantu algoriths (CO 781, Winter 2008) Prof Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search ow we begin to discuss applications of quantu walks to search algoriths
More informationSupport Vector Machines. Machine Learning Series Jerry Jeychandra Blohm Lab
Support Vector Machines Machine Learning Series Jerry Jeychandra Bloh Lab Outline Main goal: To understand how support vector achines (SVMs) perfor optial classification for labelled data sets, also a
More informationCS599: Algorithm Design in Strategic Settings Fall 2012 Lecture 11: Ironing and Approximate Mechanism Design in Single-Parameter Bayesian Settings
CS599: Algorithm Design in Strategic Settings Fall 2012 Lecture 11: Ironing and Approximate Mechanism Design in Single-Parameter Bayesian Settings Instructor: Shaddin Dughmi Outline 1 Recap 2 Non-regular
More informationDetection and Estimation Theory
ESE 54 Detection and Estiation Theory Joseph A. O Sullivan Sauel C. Sachs Professor Electronic Systes and Signals Research Laboratory Electrical and Systes Engineering Washington University 11 Urbauer
More informationHandout 7. and Pr [M(x) = χ L (x) M(x) =? ] = 1.
Notes on Coplexity Theory Last updated: October, 2005 Jonathan Katz Handout 7 1 More on Randoized Coplexity Classes Reinder: so far we have seen RP,coRP, and BPP. We introduce two ore tie-bounded randoized
More informationPattern Recognition and Machine Learning. Learning and Evaluation for Pattern Recognition
Pattern Recognition and Machine Learning Jaes L. Crowley ENSIMAG 3 - MMIS Fall Seester 2017 Lesson 1 4 October 2017 Outline Learning and Evaluation for Pattern Recognition Notation...2 1. The Pattern Recognition
More information13.2 Fully Polynomial Randomized Approximation Scheme for Permanent of Random 0-1 Matrices
CS71 Randoness & Coputation Spring 018 Instructor: Alistair Sinclair Lecture 13: February 7 Disclaier: These notes have not been subjected to the usual scrutiny accorded to foral publications. They ay
More informationStochastic Subgradient Methods
Stochastic Subgradient Methods Lingjie Weng Yutian Chen Bren School of Inforation and Coputer Science University of California, Irvine {wengl, yutianc}@ics.uci.edu Abstract Stochastic subgradient ethods
More informationSoft Computing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis
Soft Coputing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis Beverly Rivera 1,2, Irbis Gallegos 1, and Vladik Kreinovich 2 1 Regional Cyber and Energy Security Center RCES
More informationSupplementary to Learning Discriminative Bayesian Networks from High-dimensional Continuous Neuroimaging Data
Suppleentary to Learning Discriinative Bayesian Networks fro High-diensional Continuous Neuroiaging Data Luping Zhou, Lei Wang, Lingqiao Liu, Philip Ogunbona, and Dinggang Shen Proposition. Given a sparse
More informationBayes Decision Rule and Naïve Bayes Classifier
Bayes Decision Rule and Naïve Bayes Classifier Le Song Machine Learning I CSE 6740, Fall 2013 Gaussian Mixture odel A density odel p(x) ay be ulti-odal: odel it as a ixture of uni-odal distributions (e.g.
More informationTesting Properties of Collections of Distributions
Testing Properties of Collections of Distributions Reut Levi Dana Ron Ronitt Rubinfeld April 9, 0 Abstract We propose a fraework for studying property testing of collections of distributions, where the
More informationLecture 4. 1 Examples of Mechanism Design Problems
CSCI699: Topics in Learning and Game Theory Lecture 4 Lecturer: Shaddin Dughmi Scribes: Haifeng Xu,Reem Alfayez 1 Examples of Mechanism Design Problems Example 1: Single Item Auctions. There is a single
More informationLecture Learning infinite hypothesis class via VC-dimension and Rademacher complexity;
CSCI699: Topics in Learning and Game Theory Lecture 2 Lecturer: Ilias Diakonikolas Scribes: Li Han Today we will cover the following 2 topics: 1. Learning infinite hypothesis class via VC-dimension and
More informationConsistent Multiclass Algorithms for Complex Performance Measures. Supplementary Material
Consistent Multiclass Algoriths for Coplex Perforance Measures Suppleentary Material Notations. Let λ be the base easure over n given by the unifor rando variable (say U over n. Hence, for all easurable
More informationBootstrapping Dependent Data
Bootstrapping Dependent Data One of the key issues confronting bootstrap resapling approxiations is how to deal with dependent data. Consider a sequence fx t g n t= of dependent rando variables. Clearly
More informationSupport recovery in compressed sensing: An estimation theoretic approach
Support recovery in copressed sensing: An estiation theoretic approach Ain Karbasi, Ali Horati, Soheil Mohajer, Martin Vetterli School of Coputer and Counication Sciences École Polytechnique Fédérale de
More informationThis model assumes that the probability of a gap has size i is proportional to 1/i. i.e., i log m e. j=1. E[gap size] = i P r(i) = N f t.
CS 493: Algoriths for Massive Data Sets Feb 2, 2002 Local Models, Bloo Filter Scribe: Qin Lv Local Models In global odels, every inverted file entry is copressed with the sae odel. This work wells when
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationLecture 12: Ensemble Methods. Introduction. Weighted Majority. Mixture of Experts/Committee. Σ k α k =1. Isabelle Guyon
Lecture 2: Enseble Methods Isabelle Guyon guyoni@inf.ethz.ch Introduction Book Chapter 7 Weighted Majority Mixture of Experts/Coittee Assue K experts f, f 2, f K (base learners) x f (x) Each expert akes
More informationarxiv: v1 [cs.ds] 3 Feb 2014
arxiv:40.043v [cs.ds] 3 Feb 04 A Bound on the Expected Optiality of Rando Feasible Solutions to Cobinatorial Optiization Probles Evan A. Sultani The Johns Hopins University APL evan@sultani.co http://www.sultani.co/
More informationLearnability and Stability in the General Learning Setting
Learnability and Stability in the General Learning Setting Shai Shalev-Shwartz TTI-Chicago shai@tti-c.org Ohad Shair The Hebrew University ohadsh@cs.huji.ac.il Nathan Srebro TTI-Chicago nati@uchicago.edu
More informationESE 523 Information Theory
ESE 53 Inforation Theory Joseph A. O Sullivan Sauel C. Sachs Professor Electrical and Systes Engineering Washington University 11 Urbauer Hall 10E Green Hall 314-935-4173 (Lynda Marha Answers) jao@wustl.edu
More informationProbability Distributions
Probability Distributions In Chapter, we ephasized the central role played by probability theory in the solution of pattern recognition probles. We turn now to an exploration of soe particular exaples
More informationBipartite subgraphs and the smallest eigenvalue
Bipartite subgraphs and the sallest eigenvalue Noga Alon Benny Sudaov Abstract Two results dealing with the relation between the sallest eigenvalue of a graph and its bipartite subgraphs are obtained.
More informationSolutions 1. Introduction to Coding Theory - Spring 2010 Solutions 1. Exercise 1.1. See Examples 1.2 and 1.11 in the course notes.
Solutions 1 Exercise 1.1. See Exaples 1.2 and 1.11 in the course notes. Exercise 1.2. Observe that the Haing distance of two vectors is the iniu nuber of bit flips required to transfor one into the other.
More informationComputable Shell Decomposition Bounds
Coputable Shell Decoposition Bounds John Langford TTI-Chicago jcl@cs.cu.edu David McAllester TTI-Chicago dac@autoreason.co Editor: Leslie Pack Kaelbling and David Cohn Abstract Haussler, Kearns, Seung
More informationIntroduction to Discrete Optimization
Prof. Friedrich Eisenbrand Martin Nieeier Due Date: March 9 9 Discussions: March 9 Introduction to Discrete Optiization Spring 9 s Exercise Consider a school district with I neighborhoods J schools and
More informationIntroduction to Machine Learning. Recitation 11
Introduction to Machine Learning Lecturer: Regev Schweiger Recitation Fall Seester Scribe: Regev Schweiger. Kernel Ridge Regression We now take on the task of kernel-izing ridge regression. Let x,...,
More information3.3 Variational Characterization of Singular Values
3.3. Variational Characterization of Singular Values 61 3.3 Variational Characterization of Singular Values Since the singular values are square roots of the eigenvalues of the Heritian atrices A A and
More informationOutperforming the Competition in Multi-Unit Sealed Bid Auctions
Outperforing the Copetition in Multi-Unit Sealed Bid Auctions ABSTRACT Ioannis A. Vetsikas School of Electronics and Coputer Science University of Southapton Southapton SO17 1BJ, UK iv@ecs.soton.ac.uk
More informationFairness via priority scheduling
Fairness via priority scheduling Veeraruna Kavitha, N Heachandra and Debayan Das IEOR, IIT Bobay, Mubai, 400076, India vavitha,nh,debayan}@iitbacin Abstract In the context of ulti-agent resource allocation
More informationRisk & Safety in Engineering. Dr. Jochen Köhler
Risk & afety in Engineering Dr. Jochen Köhler Contents of Today's Lecture Introduction to Classical Reliability Theory tructural Reliability The fundaental case afety argin Introduction to Classical Reliability
More informationWill Monroe August 9, with materials by Mehran Sahami and Chris Piech. image: Arito. Parameter learning
Will Monroe August 9, 07 with aterials by Mehran Sahai and Chris Piech iage: Arito Paraeter learning Announceent: Proble Set #6 Goes out tonight. Due the last day of class, Wednesday, August 6 (before
More informationStability Bounds for Non-i.i.d. Processes
tability Bounds for Non-i.i.d. Processes Mehryar Mohri Courant Institute of Matheatical ciences and Google Research 25 Mercer treet New York, NY 002 ohri@cis.nyu.edu Afshin Rostaiadeh Departent of Coputer
More informationOn the Inapproximability of Vertex Cover on k-partite k-uniform Hypergraphs
On the Inapproxiability of Vertex Cover on k-partite k-unifor Hypergraphs Venkatesan Guruswai and Rishi Saket Coputer Science Departent Carnegie Mellon University Pittsburgh, PA 1513. Abstract. Coputing
More informationA Low-Complexity Congestion Control and Scheduling Algorithm for Multihop Wireless Networks with Order-Optimal Per-Flow Delay
A Low-Coplexity Congestion Control and Scheduling Algorith for Multihop Wireless Networks with Order-Optial Per-Flow Delay Po-Kai Huang, Xiaojun Lin, and Chih-Chun Wang School of Electrical and Coputer
More informationList Scheduling and LPT Oliver Braun (09/05/2017)
List Scheduling and LPT Oliver Braun (09/05/207) We investigate the classical scheduling proble P ax where a set of n independent jobs has to be processed on 2 parallel and identical processors (achines)
More informationNew Bounds for Learning Intervals with Implications for Semi-Supervised Learning
JMLR: Workshop and Conference Proceedings vol (1) 1 15 New Bounds for Learning Intervals with Iplications for Sei-Supervised Learning David P. Helbold dph@soe.ucsc.edu Departent of Coputer Science, University
More informationResearch Article On the Isolated Vertices and Connectivity in Random Intersection Graphs
International Cobinatorics Volue 2011, Article ID 872703, 9 pages doi:10.1155/2011/872703 Research Article On the Isolated Vertices and Connectivity in Rando Intersection Graphs Yilun Shang Institute for
More informationNon-Parametric Non-Line-of-Sight Identification 1
Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,
More informationLearnability of Gaussians with flexible variances
Learnability of Gaussians with flexible variances Ding-Xuan Zhou City University of Hong Kong E-ail: azhou@cityu.edu.hk Supported in part by Research Grants Council of Hong Kong Start October 20, 2007
More informationMixed Robust/Average Submodular Partitioning
Mixed Robust/Average Subodular Partitioning Kai Wei 1 Rishabh Iyer 1 Shengjie Wang 2 Wenruo Bai 1 Jeff Biles 1 1 Departent of Electrical Engineering, University of Washington 2 Departent of Coputer Science,
More information1 Identical Parallel Machines
FB3: Matheatik/Inforatik Dr. Syaantak Das Winter 2017/18 Optiizing under Uncertainty Lecture Notes 3: Scheduling to Miniize Makespan In any standard scheduling proble, we are given a set of jobs J = {j
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationEquilibria on the Day-Ahead Electricity Market
Equilibria on the Day-Ahead Electricity Market Margarida Carvalho INESC Porto, Portugal Faculdade de Ciências, Universidade do Porto, Portugal argarida.carvalho@dcc.fc.up.pt João Pedro Pedroso INESC Porto,
More informationLecture 20 November 7, 2013
CS 229r: Algoriths for Big Data Fall 2013 Prof. Jelani Nelson Lecture 20 Noveber 7, 2013 Scribe: Yun Willia Yu 1 Introduction Today we re going to go through the analysis of atrix copletion. First though,
More informationA Note on Scheduling Tall/Small Multiprocessor Tasks with Unit Processing Time to Minimize Maximum Tardiness
A Note on Scheduling Tall/Sall Multiprocessor Tasks with Unit Processing Tie to Miniize Maxiu Tardiness Philippe Baptiste and Baruch Schieber IBM T.J. Watson Research Center P.O. Box 218, Yorktown Heights,
More informationSupplement to: Subsampling Methods for Persistent Homology
Suppleent to: Subsapling Methods for Persistent Hoology A. Technical results In this section, we present soe technical results that will be used to prove the ain theores. First, we expand the notation
More informationCharacterizing the Sample Complexity of Private Learners
Characterizing the Saple Coplexity of ivate Learners Aos Beiel ept of Coputer Science Ben-Gurion University beiel@csbguacil Kobbi Nissi ept of Coputer Science Ben-Gurion University & Harvard University
More informationBest Arm Identification: A Unified Approach to Fixed Budget and Fixed Confidence
Best Ar Identification: A Unified Approach to Fixed Budget and Fixed Confidence Victor Gabillon Mohaad Ghavazadeh Alessandro Lazaric INRIA Lille - Nord Europe, Tea SequeL {victor.gabillon,ohaad.ghavazadeh,alessandro.lazaric}@inria.fr
More informationVC Dimension and Sauer s Lemma
CMSC 35900 (Spring 2008) Learning Theory Lecture: VC Diension and Sauer s Lea Instructors: Sha Kakade and Abuj Tewari Radeacher Averages and Growth Function Theore Let F be a class of ±-valued functions
More informationOn Process Complexity
On Process Coplexity Ada R. Day School of Matheatics, Statistics and Coputer Science, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand, Eail: ada.day@cs.vuw.ac.nz Abstract Process
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 11 10/15/2008 ABSTRACT INTEGRATION I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 11 10/15/2008 ABSTRACT INTEGRATION I Contents 1. Preliinaries 2. The ain result 3. The Rieann integral 4. The integral of a nonnegative
More informationEstimating Average-Case Learning Curves Using Bayesian, Statistical Physics and VC Dimension Methods
Estiating Average-Case Learning Curves Using Bayesian, Statistical Physics and VC Diension Methods David Haussler University of California Santa Cruz, California Manfred Opper Institut fur Theoretische
More informationComputable Shell Decomposition Bounds
Journal of Machine Learning Research 5 (2004) 529-547 Subitted 1/03; Revised 8/03; Published 5/04 Coputable Shell Decoposition Bounds John Langford David McAllester Toyota Technology Institute at Chicago
More informationShannon Sampling II. Connections to Learning Theory
Shannon Sapling II Connections to Learning heory Steve Sale oyota echnological Institute at Chicago 147 East 60th Street, Chicago, IL 60637, USA E-ail: sale@athberkeleyedu Ding-Xuan Zhou Departent of Matheatics,
More informationFeature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationGraphical Models in Local, Asymmetric Multi-Agent Markov Decision Processes
Graphical Models in Local, Asyetric Multi-Agent Markov Decision Processes Ditri Dolgov and Edund Durfee Departent of Electrical Engineering and Coputer Science University of Michigan Ann Arbor, MI 48109
More informationAnalyzing Simulation Results
Analyzing Siulation Results Dr. John Mellor-Cruey Departent of Coputer Science Rice University johnc@cs.rice.edu COMP 528 Lecture 20 31 March 2005 Topics for Today Model verification Model validation Transient
More informationCSE525: Randomized Algorithms and Probabilistic Analysis May 16, Lecture 13
CSE55: Randoied Algoriths and obabilistic Analysis May 6, Lecture Lecturer: Anna Karlin Scribe: Noah Siegel, Jonathan Shi Rando walks and Markov chains This lecture discusses Markov chains, which capture
More informatione-companion ONLY AVAILABLE IN ELECTRONIC FORM
OPERATIONS RESEARCH doi 10.1287/opre.1070.0427ec pp. ec1 ec5 e-copanion ONLY AVAILABLE IN ELECTRONIC FORM infors 07 INFORMS Electronic Copanion A Learning Approach for Interactive Marketing to a Custoer
More informationFoundations of Machine Learning Boosting. Mehryar Mohri Courant Institute and Google Research
Foundations of Machine Learning Boosting Mehryar Mohri Courant Institute and Google Research ohri@cis.nyu.edu Weak Learning Definition: concept class C is weakly PAC-learnable if there exists a (weak)
More informationRademacher Complexity Margin Bounds for Learning with a Large Number of Classes
Radeacher Coplexity Margin Bounds for Learning with a Large Nuber of Classes Vitaly Kuznetsov Courant Institute of Matheatical Sciences, 25 Mercer street, New York, NY, 002 Mehryar Mohri Courant Institute
More informationEstimating Parameters for a Gaussian pdf
Pattern Recognition and achine Learning Jaes L. Crowley ENSIAG 3 IS First Seester 00/0 Lesson 5 7 Noveber 00 Contents Estiating Paraeters for a Gaussian pdf Notation... The Pattern Recognition Proble...3
More informationBounds on the Minimax Rate for Estimating a Prior over a VC Class from Independent Learning Tasks
Bounds on the Miniax Rate for Estiating a Prior over a VC Class fro Independent Learning Tasks Liu Yang Steve Hanneke Jaie Carbonell Deceber 01 CMU-ML-1-11 School of Coputer Science Carnegie Mellon University
More informationA Note on Online Scheduling for Jobs with Arbitrary Release Times
A Note on Online Scheduling for Jobs with Arbitrary Release Ties Jihuan Ding, and Guochuan Zhang College of Operations Research and Manageent Science, Qufu Noral University, Rizhao 7686, China dingjihuan@hotail.co
More informationPseudo-marginal Metropolis-Hastings: a simple explanation and (partial) review of theory
Pseudo-arginal Metropolis-Hastings: a siple explanation and (partial) review of theory Chris Sherlock Motivation Iagine a stochastic process V which arises fro soe distribution with density p(v θ ). Iagine
More informationUniversal algorithms for learning theory Part II : piecewise polynomial functions
Universal algoriths for learning theory Part II : piecewise polynoial functions Peter Binev, Albert Cohen, Wolfgang Dahen, and Ronald DeVore Deceber 6, 2005 Abstract This paper is concerned with estiating
More informationAN OPTIMAL SHRINKAGE FACTOR IN PREDICTION OF ORDERED RANDOM EFFECTS
Statistica Sinica 6 016, 1709-178 doi:http://dx.doi.org/10.5705/ss.0014.0034 AN OPTIMAL SHRINKAGE FACTOR IN PREDICTION OF ORDERED RANDOM EFFECTS Nilabja Guha 1, Anindya Roy, Yaakov Malinovsky and Gauri
More informationA note on the multiplication of sparse matrices
Cent. Eur. J. Cop. Sci. 41) 2014 1-11 DOI: 10.2478/s13537-014-0201-x Central European Journal of Coputer Science A note on the ultiplication of sparse atrices Research Article Keivan Borna 12, Sohrab Aboozarkhani
More informationarxiv: v3 [cs.lg] 7 Jan 2016
Efficient and Parsionious Agnostic Active Learning Tzu-Kuo Huang Alekh Agarwal Daniel J. Hsu tkhuang@icrosoft.co alekha@icrosoft.co djhsu@cs.colubia.edu John Langford Robert E. Schapire jcl@icrosoft.co
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationWorst-case efficiency ratio in false-name-proof combinatorial auction mechanisms
Worst-case efficiency ratio in false-nae-proof cobinatorial auction echaniss Atsushi Iwasaki 1, Vincent Conitzer, Yoshifusa Oori 1, Yuko Sakurai 1, Taiki Todo 1, Mingyu Guo, and Makoto Yokoo 1 1 Kyushu
More informationUsing EM To Estimate A Probablity Density With A Mixture Of Gaussians
Using EM To Estiate A Probablity Density With A Mixture Of Gaussians Aaron A. D Souza adsouza@usc.edu Introduction The proble we are trying to address in this note is siple. Given a set of data points
More informationCollision-based Testers are Optimal for Uniformity and Closeness
Electronic Colloquiu on Coputational Coplexity, Report No. 178 (016) Collision-based Testers are Optial for Unifority and Closeness Ilias Diakonikolas Theis Gouleakis John Peebles Eric Price USC MIT MIT
More informationEstimating Entropy and Entropy Norm on Data Streams
Estiating Entropy and Entropy Nor on Data Streas Ait Chakrabarti 1, Khanh Do Ba 1, and S. Muthukrishnan 2 1 Departent of Coputer Science, Dartouth College, Hanover, NH 03755, USA 2 Departent of Coputer
More informationAn Improved Particle Filter with Applications in Ballistic Target Tracking
Sensors & ransducers Vol. 72 Issue 6 June 204 pp. 96-20 Sensors & ransducers 204 by IFSA Publishing S. L. http://www.sensorsportal.co An Iproved Particle Filter with Applications in Ballistic arget racing
More informationEnsemble Based on Data Envelopment Analysis
Enseble Based on Data Envelopent Analysis So Young Sohn & Hong Choi Departent of Coputer Science & Industrial Systes Engineering, Yonsei University, Seoul, Korea Tel) 82-2-223-404, Fax) 82-2- 364-7807
More information