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On behalf of the Department of Computer Science and Automation at IISc

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I would like to wish you all a very warm welcome to this first public lecture

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on Foundations of Data Science by Professor Ravi Kannan

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as all of you know, big data or data science

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represent the latest wave in the computing revolution

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now the volume, the complexity, and the variety of the data

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that is now becoming available day after day

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is now at an unprecedented scale

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so if you are able to extract useful and actionable knowledge out of this data

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that's going to be of a huge benefit

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so this initiative, which is called the big data initiative

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has just been launched in order to realize

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the potential of this very exciting, and very fascinating, and very challenging area

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now the primary objective of the big data initiative

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is to build a very strong academic and research ecosystem

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that would enable India's excellence in this fast emerging area

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and take India to a leadership position in this area

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now the faculty of the Department of Computer Science and Automation

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and also the Indian Institute of Science have come together to launch this initiative

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and they have lined up a series of ambitious programs as a part of this initiative

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so the first of these initiatives is going to be the public lectures program

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and the primary objective of the public lectures program

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is to give an introduction, an exposure, to all the fascinating research works that are going on in this emerging area

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and also to tell you more about the current state of the art in this area

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now to give you the first public lecture we have today with us Professor Ravi Kannan

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who is a legendary researcher in the area of algorithms and theoretical computer science for a long time

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Professor Ravi Kannan is currently an adjunct faculty member of the Department of Computer Science and Automation

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and as a principal researcher of the Microsoft Research India

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he is leading a very successful group on algorithms at MSR India

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and as all of you might know, he delivered the 2012 Professor I G Sarma memorial lecture

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in the same faculty hall

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and then in 2011 he was the recipient of the very prestigious

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Donald Knuth Prize for his outstanding contributions in designing

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creative algorithms for many computationally challenging problems

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and lets give Professor Ravi Kannan a wide applause for the Donald Knuth Prize
[audience applause]

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and 20 years before that, way back in 1991

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he was the recipient of the prestigious Fulkerson Prize

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for his minor work on estimating the volume of convex sets

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so here is a researcher who has made ground breaking contributions

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on a variety of topics and algorithms

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now he has designed approximation algorithms for India programming

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and he has designed randomized algorithms for linereal algebra

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and modus [sounds like "unclear" but that's not right] learning algorithms for convex sets

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so he's truly, in my view, he is the {not english: sunil munaher gwaskil?} of the big data team

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and with a wide applause, let's welcome Professor Ravi Kannan
[audience applause]

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and invite him to deliver this first public lecture, Professor Ravi Kannan
[audience applause]

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so thanks Professor Narahari for that gracious introduction

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we await [Sulah Kandalkar?] but I'll start with [Rick Elsore?]

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I want to thank also CSA, this is a very nice initiative

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CSA has always been very dynamic

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and sort of looking to latest advances in various fields

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in addition to adding all the new exciting faculty

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I think it's a very good initiative to start on big data

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I should also recognize and thank Professor Shevade from CSA

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for having this idea and getting us all together, thank you Shevade
[applause]

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so since this is the first lecture I should give you actually

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a sort of bird's eye view of the whole field in some sense

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I'm not very good at bird's eye view

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so I will give you a couple of slides which are very general

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and then zoom in on a piece of the action

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once slice of the field if you will, and sort of focus on that

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so the title of the talk is not big data, it's Foundations of Data Science

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which is sort of related because it is actually the title of a book

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I'm writing with John Hopcroft

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in the process we had to think about first the following

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what are really, what would you call the foundations of data science?

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see the word "big data" connotes something very jazzy

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and so that can have two interpretations

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one is something cool or hot or whatever you call it

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with perhaps not so much substance

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one is something with a lot of substance

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I will hope to convince you that it's both

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it's hot but also has a lot of substance

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and so the word "foundations of data science" this phrase

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connotes perhaps the substance part of it

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but the question is what are, what should you consider

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to be part of the foundations of data science

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or what should you teach if you're going to teach a course on data science

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or foundations of data science, what should you teach

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so here's something that if you haven't been to a talk with this sort of title

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you won't have seen this but if you've been to a talk

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almost certainly you've seen a curve like this which I then draw

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this is exponential cx for some constant

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so various forms of this curve

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this is the rate of growth of data if you will

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or the rate of growth of storage and speed and so on

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there are different constants

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now there is really a point, I'm not making sort of caricature out of this

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but it is really a point in that it's not as if human knowledge has grown exponentially

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but it is true that our ability to gather data, to store it

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to annotate it has grown perhaps exponentially

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and in fact so we do have exponential rate of growth of data

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but I'm not going to draw you this curve

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perhaps my suggestion is, what I'm going to put in the list here

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the first few things, maybe the answer is no

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they are not what I would consider the foundations of data science

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but they are important never the less

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you could also ask how do you collect and store

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annotate and store data in various domains

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in fact some of these responses will depend what domain will it dwell

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be it astronomy, commerce. I put up three areas here in which we do have in fact

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really a big growth in data that's been collected and stored

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but I would like to suggest that this is not, for us, foundations of data science at the moment

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partly because it's very important but it depends on the domain

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so it's very specific to each domain perhaps

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now you could also say basic statistics is important

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because if you have a lot of data, more than you can handle simply by traditional algorithms

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surely you must sample

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that is you must draw a random sample of the data and deal with that

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so sampling is important, so a study of sampling and basic statistics is perhaps important

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now basic statistics is always important, but we have to assume that and move on

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now the later parts of this are becoming progressively more "yes" answers

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but perhaps here is a sort of mea culpa though I am a computer scientist

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and in particular specialize in algorithms

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for the last three decades perhaps, the area of algorithms focused on discrete structures

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graphs and various other discrete structures, data structures are a big thing

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if you've done computer science courses you've done [lifts?] arrays

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and stacks and so on so forth, that was a big thing

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now also we spent a couple of decades improving the algorithms for various discrete problems

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like graph problems, shorter spots and flows

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now these are very important going forward

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but we have focused enough attention on these matters for the last thirty years

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and learnt enough and progressed enough in some sense, which is very good

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but yet ignored some other parts, I would like to say, somewhat controversially

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we ignored some other parts in computer science, not overall in human knowledge, in computer science

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linear algebra, numerical methods which have come out to be very important

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so if you asked me twenty years ago whether computing eigenvalues or eigenvectors of a matrix

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with a million non-zero entries is practical and important in the future

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I would have said no, so would most computer scientists

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but in fact this is what is used in basic search

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so if you look at how Google does page rank or ranks the pages on the web it's based on eigenvalues

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I won't tell you how but it's based on eigenvalues

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so in fact it's been a surprise over the last decade or two

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that what we might have thought wouldn't scale up, would not be important

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has become important

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one of them is this, sampling, not basic sampling but in fact most of this talk will be about sampling

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but sampling in the context of big data is going to be different from what you might think of as statistical sampling

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and that's important, you'll see that

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and also very important it turns out is high dimensional geometry, you'll see some of this

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so in fact even when you least expect, even for problems which don't look like that

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it turns out those problems can be formulated, we'll see some of that

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those problems can be formulated as problems to do with let's say ten thousand dimensional vectors

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so what does ten thousand dimensional space look like, that turns out to be an important thing to worry about

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and probability limit laws, central limit theorem, the very classical probability limit laws, they turn out to be very important

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now you can sort of intuitively guess why, if you have lots of data you sort of learn to limit

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right? and therefore such things should be more important than let's say basic statistics

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and indeed in a lot of these areas we'll have to develop new tools

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and we have developed some which handles this big data problem, that's what I'm hoping to convince you

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so sampling, so it's clear that if you have a lot of data you have to sample

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so that you get a subsample which is smaller and can be handled

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but is it just picking uniformly at random a part of the data

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is that going to be good enough, that's the question we want to ask right

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so we have ten million pieces of data

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just uniformly at random pick ten thousand of them and analyze them

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is that what we are going to do

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that's what we think of as sampling traditionally, perhaps the most elementary method

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but here is a problem typically that we want to solve

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it's an optimization problem, there are many of those this is one

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you have a very large matrix A, m by n matrix

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which we want to do what's called principal component analysis on A

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if you're not familiar with this bear with me for a moment

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you don't have to be at the moment, familiar with this, we'll see that a little more

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but a lot of you are familiar with this I'm sure

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so that just means we want to find a vector x, so x is an [n, or maybe m] vector of unit length

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so that when I multiply A with x and take the length, that's as high as possible

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so this turns out to be a basic linear algorithm problem, this is very important right?

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so we want to maximize over all unit length vectors x the length of A

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now A is very big so we can't do the whole problem perhaps

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we can't even necessarily put A into random access memory

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so it's going to be difficult to run traditional algorithms, what do we do

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somehow we like to say we sample A to get a sample out of it: B

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now B might have a smaller number of rows or columns maybe

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more zeros or something makes it easier

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we won't see particularly what at the moment

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we want that B to have this property

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see I don't know what x is going to be what I'm going to find

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so I better have this property, that for every... red is important...

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for every x, Ax and Bx are nearly the same length

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if that were true I could deal with B instead of A right? if that were true

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if that were not true for every x I cannot do that

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because for every x I may find maybe a bad x for which B gives me the wrong answer right?

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so how do I do that, that's the kind of sampling we want to do

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we want the sample of A that's good for every x

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and x is an n dimensional vector, there are infinitely many of them

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but a rule of thumb it turns out, in n dimensions you can pretend there are only e to the n vectors

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it's not true, there are infinitely many, but you can pretend that there are e to the n vectors

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that's still a lot of vectors so this sampling better work for all the vectors, not just for one right?

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that's one issue, here's another issue that can come up in sampling

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big data means that the data is perhaps too big to be stored in one machine or one unit of memory, some memory

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so it may be distributed among many servers

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now important sampling, uniform sampling means everybody gets drawn with probability which is the same

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importance sampling means some items have a higher probability based on their importance

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so importance is sampling each item with probability proportional to its importance

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but then all servers must share the vectors of importance right?

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they must be sampling with the same vectors to be consistent

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now maybe the number of items are so large that this vector is costly to translate

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this is the kind of problem we are going to worry about a bit more, both of these problems

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by the way I should tell you the talk is not exhaustive

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I'm not going to tell you all the problems about big data

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that's why we have a whole series of lectures you can see here

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we'll have many more interesting lectures in the series in the coming months, one a month roughly

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so I'm going to cut to a slice of it and these are some of the problems we'll talk about in detail

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so consistency among various servers is one issue

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here is a very simple example that you might want to think about mentally, it's a mental exercise, a very simple one

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so I want an average of a set of real numbers, a large set of real numbers, a big set right?

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now you can just take a sample and take the average, that's an approximation

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but the variance may be very high, obviously right? variance is how to spread entries so I might be very high

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so how do we achieve low relative variance

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now one thing to focus your mind on is the cancellation problem

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suppose there are lots of big positive entries and lots of big negative entries and they're canceled out

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and the total was zero, the average was zero

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so if the answer is zero, if you have to make small relative error, you must get it exactly right

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right? if the answer is zero there is no error allowed

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allowing one percent error if the answer is zero I have to get exactly the answer

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so how do you do that with samples, ok that's our question

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now for average, plain simple average it's easy you can just keep a running sum if you read the whole thing

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but we'll have to worry about other statistics for which this is not easy right?

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so the problem is that the final answer will be very small and therefore the error allowed will be very small

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how do you do it with samples

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so these are some of the problems I deal with, I'll tell you some of the problems as we go along

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sampling is our first major issue, the second major issue we'll deal with in this talk is communication

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so again first one definition, what is "big data"? what's a "big data" problem?

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let's simply define it as data does not fit into random access memory

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so every computer has random access memory, if the data is big enough that doesn't fit in RAM then we call it big data

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that's the sort of working definition of it

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00:17:26,301 --> 00:17:30,915
so let's say we run a matrix algorithm, you all know traditional matrix algorithms

227
00:17:30,915 --> 00:17:38,214
as the algorithm is running along, a step of the algorithm may say "bring me the i j entry of the matrix"

228
00:17:38,214 --> 00:17:42,390
and in analyzing or implementing that we assume that can be done in unit time

229
00:17:42,390 --> 00:17:47,734
because if you had random access memory you can quickly access, that's what random access means

230
00:17:47,734 --> 00:17:52,455
you can quickly access let's say each entry of the matrix by putting down its index

231
00:17:52,455 --> 00:17:55,788
now this we cannot do if it doesn't fit into RAM

232
00:17:55,788 --> 00:18:01,754
so in fact the traditional paradigm of algorithm design assumes you can do it in unit time

233
00:18:01,754 --> 00:18:05,913
and that already fails that's the primitive that fails

234
00:18:05,913 --> 00:18:13,757
so people devised models that avoid this assumption, one model is the streaming model

235
00:18:13,757 --> 00:18:20,885
the streaming model says data is written on some external memory and it's steaming, it just goes by

236
00:18:20,885 --> 00:18:23,771
you can only read once from right to left

237
00:18:23,771 --> 00:18:30,168
I'll draw a picture here, I've been lazy, I should have drawn pictures on the slide but that takes a long time

238
00:18:30,168 --> 00:18:39,906
this is a stream if you will, the close part you are sitting here reading it you can only read once and it goes away

239
00:18:39,906 --> 00:18:46,855
well that per say is not good enough, there is a random access memory which is much smaller than this stream

240
00:18:46,855 --> 00:18:49,264
so I'll write this I've a very small

241
00:18:49,264 --> 00:18:55,201
so you can remember some of this stream, let's say by sampling you've sampled some of this stream

242
00:18:55,201 --> 00:18:57,521
uniform sampling is not going to do much

243
00:18:57,521 --> 00:19:03,776
good but there are sophisticated methods that do actually work and solve a lot of problems in this model

244
00:19:03,776 --> 00:19:12,328
it's I should say the streaming model was the object of study even before the phrase big data was coined

245
00:19:12,328 --> 00:19:18,727
it's studied for the last two decades and there's quite a body of very nice results

246
00:19:18,727 --> 00:19:26,313
I won't tell you the entire body the way I'm structuring this talk cuz I'm going to give you some more or less self-contained vignettes

247
00:19:26,313 --> 00:19:34,131
so we'll see one vignette from streaming data, streaming model, but I won't be able to show you a lot of stuff right?

248
00:19:35,009 --> 00:19:38,740
so now where does communication come in

249
00:19:38,740 --> 00:19:50,662
so the random access memory you can think of as communicating info information pertaining to the first part of the input to the second part

250
00:19:51,020 --> 00:19:54,338
so that's how the communication takes place

251
00:19:56,126 --> 00:20:00,042
so when I read up to this point this is all gone

252
00:20:02,835 --> 00:20:07,934
whatever information I need of that is stored in my random access memory

253
00:20:07,934 --> 00:20:16,615
so in some sense this bit is passing on is communicating some information to the second part by writing something on the random access memory

254
00:20:16,615 --> 00:20:23,973
that's one way to think of it so this analogy actually relates space and communication

255
00:20:23,973 --> 00:20:29,721
and this has been studied widely and it's is used to derive lower bounds in space

256
00:20:29,721 --> 00:20:37,398
ok a subject that I won't touch, very much into a [dista? distant?] high level concept somehow space is related to communication

257
00:20:37,398 --> 00:20:42,294
cuz you've got to write down what you want to communicate to the rest of the input [unintelligible word]

258
00:20:42,791 --> 00:20:50,946
so so far I will say it's been fairly up in the sky but one more slide of that and then we'll come to a very particular piece

259
00:20:50,946 --> 00:20:55,036
so streaming is one model that's been studied a lot

260
00:20:55,036 --> 00:21:00,205
a little less restrictive model of big data is that data is split among many servers

261
00:21:00,205 --> 00:21:03,448
but number of servers is much smaller

262
00:21:03,448 --> 00:21:14,767
now you might have, you could think of streaming as if each bit is sitting on a different computer, different server

263
00:21:15,866 --> 00:21:22,310
that's one way to do this, the only way they communicate is through RAM right? otherwise they cannot communicate

264
00:21:22,310 --> 00:21:30,463
now that's however if the length of this stream is n this is thinking of it as n servers

265
00:21:32,370 --> 00:21:36,700
now you can argue that a lot of real problems have data that's not so big

266
00:21:36,700 --> 00:21:41,412
in fact you can accommodate it on log n servers

267
00:21:41,412 --> 00:21:44,463
many smaller many fewer servers than the amount of data

268
00:21:44,463 --> 00:21:50,934
so that's a better model perhaps for a lot of data and that's one other model we'll study, we'll study both of these

269
00:21:52,236 --> 00:21:56,358
now again up to this point the talk has been about generalities

270
00:21:56,358 --> 00:22:01,277
now I want to zoom in on some examples which are self-contained examples for what is going on

271
00:22:05,043 --> 00:22:06,964
ok so let's start with an example

272
00:22:06,964 --> 00:22:11,273
this actually was a [starting/study?] example and motivated a lot of research on streaming algorithms

273
00:22:11,273 --> 00:22:17,557
we have a large network with lots of computers sending messages to each other

274
00:22:18,438 --> 00:22:21,029
there are routers which route the message

275
00:22:21,029 --> 00:22:27,395
so there are n computers, n is very large, perhaps in a million

276
00:22:28,243 --> 00:22:32,939
but there are routers which are square boxes, maybe there are only ten of them or a hundred of them

277
00:22:32,939 --> 00:22:40,835
and so a message from here to here might go through this router and they might send it to this and they might send it to that

278
00:22:40,835 --> 00:22:46,628
and let's say the message finally wants to go there, it'll go there so it goes through three routers and then goes to the end destination

279
00:22:47,392 --> 00:22:54,700
so we won't worry about the mechanics of routing let's just study what happens here for the logs of the messages

280
00:22:54,700 --> 00:22:58,037
so the last router that sends it to the destination

281
00:22:58,037 --> 00:23:01,637
the last one keeps a log of the message

282
00:23:01,857 --> 00:23:09,499
the log just has for us the sending computers name i, and the length of the message, that's the log it keeps

283
00:23:10,076 --> 00:23:18,271
and you want to analyze these logs, you want to do statistics on these logs to better improve the efficiency of the network that's there that's all A

284
00:23:19,315 --> 00:23:25,901
so let's say Ai is the total length of all the messages sent by computer i in one day let's say

285
00:23:25,901 --> 00:23:31,498
and you want statistics of Ai, the statistics are important to again optimize the network

286
00:23:31,498 --> 00:23:33,833
that's the problem we want to solve

287
00:23:33,833 --> 00:23:39,718
now statistics one thing you want to do is perhaps the average of Ai or the total of Ai

288
00:23:39,718 --> 00:23:42,732
but you may want the variance, you may want the second moment

289
00:23:42,732 --> 00:23:46,498
so that's just 1 over n, there are n computers

290
00:23:46,498 --> 00:23:53,958
and n Ai, you sum the Ai squared and then take the average right? let's say we want to do that

291
00:23:54,410 --> 00:23:56,506
now it's that simple let's see what happens

292
00:23:56,506 --> 00:23:59,574
the trouble is no router knows Ai in full

293
00:23:59,574 --> 00:24:06,789
so i was this computer, it might have sent messages which have gone through many routers

294
00:24:06,789 --> 00:24:15,458
and some of them went there, some of them went there, some of them went there, so nobody knows the full value of Ai right?

295
00:24:15,928 --> 00:24:22,765
so if you want to find this you seem to have to communicate all the partial Ai and that requires O(n) space

296
00:24:22,765 --> 00:24:27,868
because there are n computers, you have to communicate for each one the total from each router

297
00:24:27,868 --> 00:24:31,854
so think of the problem actually just for two routers, already interesting

298
00:24:31,854 --> 00:24:35,102
now let's get to the routers formulated abstractly

299
00:24:35,102 --> 00:24:38,521
each router has just an n vector so this is where vectors come in

300
00:24:38,521 --> 00:24:41,335
there are no vectors to start with right? this not a geometry problem

301
00:24:41,335 --> 00:24:43,911
but here I am going to make up vectors

302
00:24:43,911 --> 00:24:48,077
so each router you can think of as having an n vector

303
00:24:48,077 --> 00:24:54,412
and the components tell you the total length of all messages from each computer that's been logged by this router

304
00:24:54,412 --> 00:25:02,558
so for computer i all the messages that were logged with this router, that total is kept there let's say that's a vector

305
00:25:02,558 --> 00:25:08,557
so again vectors have come in surreptitiously they have nothing to do with the problem but we've formulated the vectors

306
00:25:08,557 --> 00:25:17,712
we'll see the advantage of that, it's not a bookkeeping device that I put down vectors it's for keeping track of this it's more than a bookkeeping device

307
00:25:17,712 --> 00:25:23,283
so what we want to find I say alas inner sum n over t

308
00:25:23,283 --> 00:25:30,693
so we want to take the sum over all routers of the total length of message logged by that router coming from i

309
00:25:30,693 --> 00:25:35,167
I want to sum inside first, then square and then add

310
00:25:35,167 --> 00:25:37,981
it would have been nice if the sum was outside

311
00:25:37,981 --> 00:25:46,830
the trouble is the sum is inside so you have to first total over all routers for each i, square, and then sum over all i right?

312
00:25:46,830 --> 00:25:51,075
and that's a problem because these route this information is not contained in any body

313
00:25:51,075 --> 00:25:53,787
now if you don't follow the exact mechanics that's fine

314
00:25:53,787 --> 00:25:57,259
I'm going to abstract it even more that makes life a lot simpler

315
00:25:57,259 --> 00:26:00,047
so here's a vector problem this is the abstract problem we want to solve

316
00:26:00,047 --> 00:26:04,196
there's an n component vector, bold face is vectors

317
00:26:04,206 --> 00:26:06,646
a sub t residing in server t

318
00:26:06,695 --> 00:26:13,084
and you want to take the sum and you want to estimate the length squared of the sum right? sum of squares of length

319
00:26:13,239 --> 00:26:16,821
we want to sum the vectors component wise and then take the sum of squares

320
00:26:17,072 --> 00:26:20,359
now if we had the whole vector and we could write it down this is a trivial problem

321
00:26:20,359 --> 00:26:28,606
but we don't want to write down the whole vector so we want to use very little communication compared to the number of components, how do we do that

322
00:26:28,606 --> 00:26:35,460
ok so maybe I can sample k of the n components of the vector v right?

323
00:26:35,460 --> 00:26:38,622
v has too many components I don't want to write them all down

324
00:26:38,622 --> 00:26:41,734
maybe I can sample k and just collect information on that

325
00:26:41,734 --> 00:26:45,647
is that any good and the answer is a gentle no the variance can be very high

326
00:26:45,647 --> 00:26:51,138
some components can be much higher than other components so this will be a loss if you do that

327
00:26:51,138 --> 00:26:57,690
well here is a beautiful theorem which is a geometry theorem, now you see that geometry is necessary and useful

328
00:26:57,690 --> 00:27:00,505
called a Johnson Lindenstrauss Theorem

329
00:27:00,505 --> 00:27:05,396
if you perhaps don't remember anything from this talk hopefully you can take this home if you didn't see it before

330
00:27:05,396 --> 00:27:07,754
so this theorem says the following

331
00:27:07,754 --> 00:27:15,877
if I pick a matrix R which has a small number of rows, k is supposed to be small

332
00:27:15,877 --> 00:27:22,811
matrix with independent random random unit vectors, so they're independent random vectors with unit length as rows

333
00:27:22,859 --> 00:27:31,108
then I take the length of R times v and that tells me the length of v but there is a little scale factor here don't worry about that

334
00:27:31,108 --> 00:27:34,106
which is a known factor, we know this so we don't care

335
00:27:34,456 --> 00:27:39,502
it tells me the length of v, so a little schematic picture here

336
00:27:42,538 --> 00:27:46,609
so we had a vector x with a lot of components, n components

337
00:27:47,019 --> 00:27:52,468
and we multiply it by an R which is a very small number of rows

338
00:27:52,468 --> 00:27:57,127
it becomes now a k vector, this is R x

339
00:27:58,346 --> 00:28:01,691
and it's enough to find the length of that right?

340
00:28:01,691 --> 00:28:09,574
so what we did was again we had to find the length of some n things squared, instead this says a sample of k things will do

341
00:28:09,574 --> 00:28:12,331
not a plain sample that won't do right?

342
00:28:12,331 --> 00:28:17,125
but this kind of random vector then multiplied it will do

343
00:28:18,187 --> 00:28:22,361
the proof is not difficult I mean obviously I won't do the proof here but the proof is not difficult

344
00:28:22,361 --> 00:28:27,528
but what it says is that a sample of k components, plain components won't do

345
00:28:27,528 --> 00:28:30,764
but a sample of k combinations of components will do

346
00:28:37,187 --> 00:28:47,909
so now the trouble is this if all servers happen to have the same random matrix R they can do their own R times a t and send these vectors

347
00:28:47,909 --> 00:28:51,587
now these are only k vectors they are very small ok?

348
00:28:51,587 --> 00:29:00,691
you can send these vectors over to a central processor which can sum the vectors and since things are linear it can find R times v and compute the length

349
00:29:00,691 --> 00:29:06,513
but you need to know the same R, everybody needs the same R ok?

350
00:29:06,513 --> 00:29:11,444
how do we do that, we need a lot of communication it seems to agree on one R

351
00:29:11,444 --> 00:29:17,870
and this is going to be this has been a simple problem and I want to isolate this problem now, forget the vector problem

352
00:29:17,870 --> 00:29:22,580
how to share randomness this is a very important problem it turns out in many areas

353
00:29:22,580 --> 00:29:31,694
how can many servers share the same they have to agree on the same n bit random string without transmitting random R right?

354
00:29:31,694 --> 00:29:35,279
they all have to have exactly the same random string how do they do that

355
00:29:38,122 --> 00:29:42,302
the string has n bits of information so it seems to need n bits of communication

356
00:29:42,598 --> 00:29:51,670
we'll see how to get around that but for many applications of this problem as well as other problems in distributed computing, complexity and cryptography

357
00:29:51,670 --> 00:29:57,924
we don't need all the components to be independent we need only what's called 4-way independence

358
00:29:57,924 --> 00:30:03,119
now 4-way independence is every set of 4 or fewer bits are independent

359
00:30:03,369 --> 00:30:07,440
so full independence is the entire thing should be independent mutually

360
00:30:07,440 --> 00:30:15,229
but we only need every collection of 4 bits or 3 bits or 2 bits or 1 bit to be independent, that's enough it turns out

361
00:30:15,229 --> 00:30:26,383
now 4-way independent things are easier to get coding theory, and coding theory gives us this

362
00:30:26,596 --> 00:30:28,573
it's actually a very classic code

363
00:30:31,876 --> 00:30:36,768
so here's one way to get the 4-way independent stream very long stream but a 4-way independent

364
00:30:36,768 --> 00:30:50,419
I pick at random coefficients, 4 coefficients and form a degree 3 polynomial, a not plus a 1 x plus a 2 x squared plus a 3 x cubed it's a real polynomial right? just the polynomial of degree 3

365
00:30:50,419 --> 00:30:56,815
then I evaluate this polynomial at n points, p at 1, p at 2, and p at n, and so on

366
00:30:56,815 --> 00:31:02,619
what happens is that this string which is much longer than 4, right? same length, is 4-way independent

367
00:31:02,619 --> 00:31:06,011
and you have to do it over a finite field but this is the idea

368
00:31:06,011 --> 00:31:14,414
there are four degrees of freedom right? four degrees of independence, a 1, a not, a 1, a 3, a 4, and that suffices to give 4-way independence

369
00:31:14,414 --> 00:31:20,723
now these are called pseudo-random sequences and many of you will notice now the answer for the communication problem is simple

370
00:31:20,723 --> 00:31:26,558
you just generate the seed, some central location, and transmit the seed

371
00:31:26,558 --> 00:31:30,436
a not, a 1, a 2, a 3, centrally and transmit to all servers

372
00:31:30,436 --> 00:31:36,884
they can individually find the string, the strings will all be the same right? it's the same string so they'll all be the same

373
00:31:36,884 --> 00:31:43,085
so they've agreed on an n length random string with just very little transmission

374
00:31:43,518 --> 00:31:48,313
and that the communicating randomness like this we've said is an important derivative

375
00:31:50,540 --> 00:31:56,556
so finishing the vector problem as I've said there was an important paper due to Alon Szegedy about fifteen years back

376
00:31:56,556 --> 00:32:02,403
which started the field of streaming algorithms so I want to go through what we have more or less summarizing what we have

377
00:32:02,403 --> 00:32:10,060
there's a central processor which generates a seed for the random matrix, it transmits the seeds to all the servers

378
00:32:10,060 --> 00:32:16,924
so let's just find, apply R to their individual so they build up the whole matrix from the seed

379
00:32:16,924 --> 00:32:25,430
apply this and then send it to the central processor which sums up all these things and then finds the length

380
00:32:26,422 --> 00:32:35,615
if server t did not have it explicitly, so it's possible that server t did not have it explicitly but we had only the streaming model

381
00:32:36,612 --> 00:32:46,835
it only has a log of every entry but the logs are in arbitrary order, they're not sorted, so they're not sorted altogether for one i

382
00:32:46,835 --> 00:33:01,038
but whenever I see that the i source has sent a message of length a then a little thought will tell you that all I have to do to R times a t is update it by adding a times column i of R to R times a t

383
00:33:01,038 --> 00:33:12,906
again it's not important to remember the details but the point is whatever order I give these logs I can update my matrix vector product and you can do this in the streaming model once I have the seeds

384
00:33:15,476 --> 00:33:37,136
now that [someone asks a question, I can't hear] so v is not available locally [audience speaks again] we have only these a t, v is the sume of what's available locally so we use the vector sum of vectors available at each server

385
00:33:37,136 --> 00:33:49,668
and we want to be able to estimate the length of the sum rather than each individual one, ask me other questions that you have we can go slow

386
00:33:49,668 --> 00:33:52,976
[someone may be asking a question, I can't hear anything just a long pause]

387
00:33:56,665 --> 00:34:10,019
I'm talking about scenarios where each server has a certain amount of information but we want the statistics of the total and we are not allowed to just transmit the whole vector that's too costly

388
00:34:13,744 --> 00:34:17,295
briefly how to [say one/save on?] communication

389
00:34:21,586 --> 00:34:32,802
so I want to study something called higher order correlations. let's say you have time-series data of n events and over T times because n and T are supposed to be large this is again large data

390
00:34:32,992 --> 00:34:44,330
so Aij is 1 over 0 depending on whether event Ei happened at time t j, so whether the [IT?] event happened at time j if so then Aij is 1 over 0

391
00:34:44,498 --> 00:34:52,849
so this is time series right? time series just means for each event I have a bit, I have a long string of zeros and ones which tells you when the event happened

392
00:34:52,849 --> 00:35:05,417
now pairwise correlations wants to know how many pairs of events co-occured, so you want triples Ei, E1, E2, t, so that E1 and E2 both occur at time t

393
00:35:05,417 --> 00:35:14,890
so if E1 and E2 both co-occur at time t, how many times do they co-occur? add them up, this is clearly of interest. I won't motivate it but I'll give you some examples in a minute

394
00:35:14,890 --> 00:35:29,852
but it turns out higher order correlations are also of interest, we have four events E1 through E4, and t times three time steps, and we want to know how many such things where all four occurred at all these three times right?

395
00:35:29,852 --> 00:35:45,174
it turns out there is something actually [national or natural?] motivations which I won't describe in detail, these higher order correlations are important in a lot of contexts one is neuron styling but uh we'll see a couple of other examples on the slide

396
00:35:45,174 --> 00:35:59,992
now big data again means that events are split up on servers, so no server knows all the events, each server knows some of the events. so each server knows some of these time series, nobody knows all of them.

397
00:36:01,618 --> 00:36:18,069
another example of this is from commercial data right? customers buying products, each customer is on one server and you want to know for instance how many triples of customers and products are there so that all three bought quite a bit of these two

398
00:36:18,069 --> 00:36:30,545
so these kind so of analysis are necessary for instance they optimize what products should be put where and so on and so forth. again the details of what the exact problem is not important but it's this correlation that we want to know

399
00:36:33,835 --> 00:36:45,951
so documents and terms in an information retrieval are co-occurrence of terms trigrams or triples of terms, whether they co-occur and how many times they co-occur are important, that's a very exact similar problem to this

400
00:36:48,274 --> 00:36:54,386
here's the abstract formulation, you have a bipartite graph so i can draw it on the board

401
00:37:04,708 --> 00:37:23,055
so one side we have events, the other side we have time steps, and you put an arrow if it occurred. so some things occurred some things didn't.

402
00:37:26,700 --> 00:37:35,860
this whole graph is not presented on any server. servers only know for one particular event what are all the time steps that it happened, perhaps these two.

403
00:37:37,268 --> 00:37:47,305
and some other server, maybe the elliptic server might know some of the other stuff, the square server might know some of this, that's the situation

404
00:37:51,780 --> 00:38:05,223
so uh and you can think of this ok here is a concrete version of this problem, estimate the number of (2, 4) sub-graphs with each of the two left vertices connected to at least three of the four right vertices

405
00:38:05,223 --> 00:38:17,086
so it's again not very important exactly what happens but here's a graph with two vertices on the left and four on the left, four on the right, and you want to study the connections

406
00:38:17,086 --> 00:38:26,339
you want to do the averages of this over all sets of two and four right? that's total you want the statistics of it, and the point is the data is split

407
00:38:27,122 --> 00:38:38,349
so you can formulate this as a bipartite graph with left and right vertices, it turns out this problem, and i'm going to go over this quickly if you don't follow it don't worry about it

408
00:38:38,349 --> 00:38:48,006
this problem is more or less exactly the vectors problem for reasons that should become clear, so it's a vector problem in a very high number of dimensions

409
00:38:48,006 --> 00:38:54,876
so if there are 10 right vertices you form a 10 to the 4 component vector v and it's that vector's length we want.

410
00:38:54,876 --> 00:39:03,825
so component v i is the number of left vertices connected to at least three of the four right vertices, again you need not follow exactly all the details but it's very simple

411
00:39:03,825 --> 00:39:18,452
and the answer we want is more or less the squared length of v, it's the same problem that we had earlier but in a very high dimensional space, so if there are n vertices here it will be n [choose?] four dimensions so n to the four dimensional space

412
00:39:18,452 --> 00:39:23,508
so this is an example where the number of dimensions is even bigger than what you started with

413
00:39:24,628 --> 00:39:35,512
now it turns out that can be computed by the previous algorithm but you don't in the neuron firing case it turns out two is not enough, you want higher order correlations than two

414
00:39:35,512 --> 00:39:45,271
so then uh sorry so what we did is only for two, what do we do for higher order? now I'm going to go over this quickly but in the abstract setting now

415
00:39:45,271 --> 00:39:54,978
we have a vector v with n non-negative components, we want.. which is actually not in one place it's a some of vectors on different servers, again the proof is same as before

416
00:39:54,978 --> 00:40:04,698
but now we want to estimate the k [moment?] we want the sum of the k powers, k is larger than two perhaps, k is three, four, k is something else, how do we do that?

417
00:40:04,698 --> 00:40:15,672
the methods for two don't apply, this beautiful theorem [which aries?] doesn't work any more, it turns out that sort of theorem only works for two no more for anything higher than two

418
00:40:15,672 --> 00:40:32,991
but in a recent paper we showed that it can be done with not too much communication, it turns out to be enough for each server to do its own important sampling according to its own vectors' components raised to the k power

419
00:40:32,991 --> 00:40:44,371
so I would have liked to draw samples according to [sum or some?] vector to the k power, it suffices it's important to do it individually each server and then exchange a bit of communication,

420
00:40:44,371 --> 00:40:54,471
this is not trivial, but I won't tell you the details, but important sampling per server is enough, that's the point of the story

421
00:40:54,471 --> 00:41:10,043
I want to go to the next topic, perhaps if there are any questions briefly we can, I don't know whether there are any, if I've gone fast enough that maybe you're all lost which is fine then [audience laughs] any questions?

422
00:41:12,705 --> 00:41:24,286
[someone asks a question, unintelligible] so important sampling, uniform sampling is every item is uniformly likely to be picked, important sampling means the probabilities are not uniform that's all

423
00:41:24,788 --> 00:41:36,238
[another question, unintelligible] how do you pick the importance? in this case it's proportional in the previous slide it was proportional to this k power v i

424
00:41:37,012 --> 00:41:43,296
that is what we'd like to do but we don't have the vector v so instead it turns out each server its own

425
00:41:43,848 --> 00:42:01,796
[unintelligible question] what to have any server? [audience clarifies] ah it's uh maybe um I'd want to adjust because you can speed up competition, but here we are looking at data being so big that it is being put on many servers

426
00:42:01,796 --> 00:42:11,594
or in a network routing case there are different routers they just log messages right? each one. if you want them all to collect in one place you need a huge amount of memory so you can't do that.

427
00:42:14,022 --> 00:42:26,837
anything else? [unintelligible question] yeah so the answer to that is I'm a theoretician. [audience laughs]

428
00:42:27,843 --> 00:42:33,392
well you know this is the model in which you can. k typically wants to be small

429
00:42:33,392 --> 00:42:44,607
[unintelligible question] of the dimension of the number of rows it's logarithmic actually, it's only logarithmic, it's much smaller than [m or n?] so Johnson-Lindenstrauss theorem would logarithmic

430
00:42:45,235 --> 00:42:53,090
[unintelligible comment] no I thought you meant this scale [laughter] this scale is also [unintelligible]

431
00:42:53,090 --> 00:43:00,032
anything else? ok good. so let me quickly do principal component analysis.

432
00:43:00,032 --> 00:43:10,816
so we have a large matrix in this case and the matrix you can think of with each row is a data point, there are m points in n space, n dimensions

433
00:43:10,816 --> 00:43:20,429
the number of points is much larger than dimensions, so maybe they all live in ten dimensional space and there are a million points, or two hundred dimensional space there are a million points right?

434
00:43:20,429 --> 00:43:32,634
so basic concept of linear algebra again this is a recapsulating this you want to find the Unit Vector x which maximizes the length of Ax, that's called the first principal component

435
00:43:32,634 --> 00:43:39,387
Unit Vector y perpendicular to the first principal component maximizing this called the second principal component and so on

436
00:43:39,405 --> 00:43:43,347
again a lot of you are familiar with this, if not just think of it as an optimization point

437
00:43:44,959 --> 00:43:48,198
very Nice Linear Algebra Theory, for many problems you want to find these

438
00:43:48,819 --> 00:43:53,475
now big data the matrix x of A may not be stored all on one server

439
00:43:54,162 --> 00:44:01,905
here is a simple model we can think of, there are many servers, each stores a similar dimension matrix,

440
00:44:01,905 --> 00:44:08,988
so the whole dimension is there, but maybe there are a lot of zeros in each server so the data [in each of them?] is smaller

441
00:44:08,998 --> 00:44:14,782
and you want to deal with the sum of all the matrices but you don't want to communicate all of that right?

442
00:44:14,782 --> 00:44:22,234
so you want to find principal components of the sum but communicate only a tiny fraction of the whole data amongst them

443
00:44:25,059 --> 00:44:32,953
so PCA for distributed data, server t has matrix A t you want to sum and take the principal component

444
00:44:32,953 --> 00:44:40,849
how do we do that? again uh so I want to keep the theme to this random uh sampling of Johnson-Lindenstrauss in some sense

445
00:44:40,849 --> 00:44:54,138
we use that, that told us that there's a random, if you pick a random matrix R then the length of x for every vector x, R times A times x

446
00:44:54,138 --> 00:45:05,676
now in the old setup we multiply R by a vector, well A times x is a vector right? so R times A times x the length is estimate, is a good estimate of the length of A times x

447
00:45:05,676 --> 00:45:10,266
I really want the length of A times x but it's enough to do this

448
00:45:10,266 --> 00:45:15,463
R makes it much smaller, we erased that, R makes it much smaller so we can communicate better

449
00:45:15,463 --> 00:45:25,381
but now if this is only true for one x then it wouldn't help us, but if it were true for every x, this is the optimization problem that I first pointed out

450
00:45:25,381 --> 00:45:37,196
if it were true for every x we could just solve the problem on R times A instead of for A right? because x I find for R times A should also be good for A if this kind of relation is true for every A

451
00:45:37,196 --> 00:45:46,356
and in fact it turns out well the number of x's is exponential in n but Johnson-Lindenstrauss, now this is one more step

452
00:45:46,356 --> 00:45:54,466
gives us a low enough probability of failure for one x that we get this actually works for every x simultaneously, for all the x's simultaneously

453
00:45:54,466 --> 00:46:03,746
not only is this true for one x it's true for every x at the same time, ok that's asking for a lot because there are many many x's but that's given to us

454
00:46:03,746 --> 00:46:13,374
so there's one random matrix R so again the picture was, oh I think I have a picture on the next slide

455
00:46:13,672 --> 00:46:24,345
there's one random matrix R so that R times A times x and A times x are similar length for every x, where big n is not very big

456
00:46:24,345 --> 00:46:36,583
so a picture here would make it clearer, here is a big matrix, here is R times A which is far fewer rows for the same number of columns but has a very nice property

457
00:46:36,583 --> 00:46:44,334
that for any vector x that you can think of the length of A applied to x and the length of RA applied to x are similar

458
00:46:44,334 --> 00:46:53,203
that's quite a striking property, so anything you want to do with A you can do with R times A because all these lengths are preserved

459
00:46:53,203 --> 00:47:05,920
so that finishes PCA which is a very brief thing, now the area of distributed data problems, as far as big data goes a big part of big data is that data is distributed

460
00:47:05,920 --> 00:47:13,261
so there are many problems that people are currently studying, how to do linear non-linear problems, optimization problems is an important thing in this model

461
00:47:13,261 --> 00:47:17,532
machine learning in a distributed setting is an active area

462
00:47:17,532 --> 00:47:23,726
there are dynamic versions of these questions you can ask where the data is subject to updates

463
00:47:23,726 --> 00:47:28,931
there are some papers here, I point out only one reference but there are many references here

464
00:47:29,620 --> 00:47:44,503
ok good, I want to do one more topic uh I picked this topic partly because it relates to it, partly because it's very beautiful recent work and one of the co-authors is in Bangalore actually so I thought it'd be nice to do

465
00:47:44,503 --> 00:47:51,785
so this is called sparsification which is another twist on sampling

466
00:47:51,785 --> 00:48:02,618
now we already saw that if you have a matrix A which is big we can compress it by using a random matrix A in front, you get R times A right?

467
00:48:02,618 --> 00:48:09,121
now you can think about RA as combinations of the rows of A right? RA is the combination of the rows of A

468
00:48:10,045 --> 00:48:28,615
here is a, so instead of RA, so combining the rows of A is it possible to just take a sample, a subset of rows, not combinations but a subset of rows maybe with weights and make that do this job?

469
00:48:31,232 --> 00:48:34,627
how about graphs? because graphs are a special case of the matrices

470
00:48:34,627 --> 00:48:38,616
so here is a picture, a description of what a graph looks like

471
00:48:38,616 --> 00:48:41,495
now I've gone to columns and to rows excuse me

472
00:48:41,495 --> 00:48:46,168
so here's a graph, it's represented by this edge [nor?] adjacency matrix

473
00:48:46,168 --> 00:48:53,874
so edge 1 goes from A to B so I put a plus one on A, minus one on B, there's vertix A and vertix B

474
00:48:53,874 --> 00:49:02,967
similarly edge 2 goes from B to C, and that's that edge, B to C and so on

475
00:49:02,967 --> 00:49:10,646
I put down this matrix and uh here's a cut which cuts the graph into two pieces

476
00:49:10,882 --> 00:49:16,118
I look at all the edges going across the cut, these three edges now going across that cut

477
00:49:16,118 --> 00:49:20,786
it turns out, and this is a calculation that you don't have to do at this point

478
00:49:20,786 --> 00:49:33,520
it turns out I can represent the cut by a vector which puts ones on the vertices on the left bank, these two a and d, and zero for the vertices on the right bank, b and c of the cut

479
00:49:33,520 --> 00:49:41,707
and if I take the length of the vector vA squared that's a [slice?] of the cut, it's always true but you have to prove this

480
00:49:41,707 --> 00:49:47,705
so from this we can formulate the following problem which is exactly the same as what I had before

481
00:49:47,705 --> 00:49:55,624
so I want now but a subset of edges, a subset of columns, so that the length of vB and the length of vA are close

482
00:49:55,624 --> 00:49:58,631
I don't want the combination, I want a subset

483
00:49:58,865 --> 00:50:06,080
so in pictures this is for graphs here is a graph on ten vertices which has all the [tensions or tensials?] to edges, so it's a lot of edges

484
00:50:06,412 --> 00:50:11,991
if n vertices or have n choose two edges, that's many many edges

485
00:50:11,991 --> 00:50:19,432
can I sample a subset of edges, so in this case I have I think fifteen edges instead of [tensials?] two

486
00:50:19,432 --> 00:50:29,909
and perhaps I weight them, I make them thicker, so that I want to guarantee that any cut I make

487
00:50:29,909 --> 00:50:34,244
so there are n vertices I can cut them into two to the n possible ways

488
00:50:34,244 --> 00:50:39,228
any cut I make here, and I make here, has the same value

489
00:50:39,228 --> 00:50:47,593
I take the total number of edges crossing this cut, I take the total weight of edges crossing this cut, they must be the same, roughly the same

490
00:50:47,593 --> 00:50:58,906
so again I want to sparsify the graph, choose a subset of edges, so that every cut here has roughly the same weight as every, the corresponding cut here

491
00:50:58,906 --> 00:51:07,872
that's a problem we might want to solve and um here's a problem with this setup

492
00:51:07,872 --> 00:51:10,636
here is a pathological graph

493
00:51:10,636 --> 00:51:19,256
if you have a graph which is very dense here and very dense here and only one edge connecting the two this is called a dumbell graph

494
00:51:19,256 --> 00:51:23,967
here is a cut just cutting it into two pieces, there is one edge crossing it

495
00:51:24,998 --> 00:51:32,390
I'd better get this edge in my sample otherwise I'll get a zero instead of one, that's not good, that's not good relative error right?

496
00:51:32,834 --> 00:51:36,117
zero is not within relative error one percent or one

497
00:51:36,308 --> 00:51:40,069
so if I want to sparsify this graph I'd better always pick the same

498
00:51:40,197 --> 00:51:43,679
so uniform sampling of the edges will not do

499
00:51:43,871 --> 00:51:49,220
I have to do something else and a beautiful theorem of Spielmann, Teng and Srivatsava

500
00:51:49,220 --> 00:51:51,620
Nikhil Srivatsava is at Microsoft in Bangalore

501
00:51:51,630 --> 00:51:57,341
uh says that uh if you have an n by m matrix where m is larger than n

502
00:51:57,341 --> 00:52:04,608
you could, there is a probability distribution it cannot be uniform I have to make sure I put a high probability on this edge

503
00:52:04,608 --> 00:52:15,353
there is a probability distribution we can put on the rows of A so that if we do IID sampling of a certain number of rows

504
00:52:15,353 --> 00:52:20,435
only on log n rows, a small number of rows think of it, according to the distributions

505
00:52:20,435 --> 00:52:24,218
then for every x length of Ax and length of Bx are the same

506
00:52:24,218 --> 00:52:29,554
and that turns out answers the cut problem for a reason I won't actually describe

507
00:52:29,554 --> 00:52:37,342
and if A actually came from a graph like the last one, sampling probabilities are proportional to the electrical resistances

508
00:52:37,342 --> 00:52:44,350
when you view the graph as a resister network and in fact sampling can be done in a nearly linear time

509
00:52:44,350 --> 00:52:56,098
so again I have a graph I must sample a certain set of edges, a small set of edges, so that every cut is correctly represented and they say we can do that provided you choose the right probabilities

510
00:52:56,098 --> 00:53:02,894
you cannot do that if you choose uniform probability right? and the probabilities are proportional to electrical resistences

511
00:53:02,894 --> 00:53:12,791
so this theorem they proved about five years back, there's been a lot of work on this area and one of the people who's worked on this area is Ramesh Hariharan

512
00:53:12,791 --> 00:53:23,500
who is our next speaker in this series he will talk next month, perhaps not about this but something else uh but he is also from Bangalore so he has done quite a bit of work on this also

513
00:53:23,500 --> 00:53:38,083
now this led to something quite unexpected uh but proved about two months ago and that's the uh which also Srivatsava was, Nikhil Srivatsava was one of the co-authors

514
00:53:38,083 --> 00:53:47,506
there was a beautiful result that settles a classic mathematics problem and the problem is actually very important in quantum theory as well as operator theory

515
00:53:47,506 --> 00:53:53,154
well it's one of those few things which can be stated very simply to do with vectors

516
00:53:53,154 --> 00:53:59,027
I won't be able to describe what connection it has to the previous theorem but it does, it's actually connected to the previous theorem

517
00:53:59,027 --> 00:54:05,839
so if big data you thought was only going to let you handle big data well here is something very fundamental that came out of

518
00:54:05,839 --> 00:54:11,911
not necessarily looking staring at big data but something that has to do with compression and sparsification

519
00:54:11,911 --> 00:54:21,624
so the theorem here says the following you have a finite set of vectors, this is going somewhat far afield but I want to tell you this theorem because it's a spectacular achievement right?

520
00:54:21,624 --> 00:54:28,507
um by these people, so if you have a finite set of vectors which are in what's called inertial position

521
00:54:28,507 --> 00:54:33,108
inertial position means the set of vectors is cheap

522
00:54:33,108 --> 00:54:43,421
inertial position means you take any vector x and sum of the squares of x dot v, dot product over the set that's exactly the length squared

523
00:54:43,421 --> 00:54:55,397
so you may think of as this as the energy of x and the direction v, so this says the energy of x together along the directions, and t is exactly the length squared

524
00:54:55,397 --> 00:55:01,415
so that needs to be true and no vector should be big, so vectors are all small in length

525
00:55:01,415 --> 00:55:08,564
then you can always partition this set of vectors into two sets that are about half inertial

526
00:55:08,564 --> 00:55:19,237
so this was about x squared, each set is about half inertial so for every x I take the sum of vectors in the first part of the energy squared in the direction

527
00:55:19,237 --> 00:55:28,927
that's approximately x squared over two for the first set, so it must be true for the second set also right? because the total is x squared

528
00:55:28,927 --> 00:55:34,674
so it turns out again this settles actually a very long standing problem in operative theory

529
00:55:34,674 --> 00:55:40,936
now this has something to do with graph sparsification in fact that was their starting point

530
00:55:40,936 --> 00:55:53,358
this actually says that not only can you cut up a graph, can you sparsify a graph, you can actually split it up into very sparse pieces

531
00:55:53,358 --> 00:56:03,270
you can split it up into many pieces each of which is sparse, but while there's some conditionals that's what this theorem ends up saying in a way that I won't be able to completely describe

532
00:56:03,270 --> 00:56:14,897
but I believe that's all so I'm done, we are now onto questions if you want [audience applauds]

533
00:56:36,759 --> 00:56:40,242
{moderator} so we're open for questions now

534
00:56:40,961 --> 00:56:53,113
{audience} sir you are taking the [unintelligible, just one word] approximately the length of the x and the R x so [our theory? (two more words)] how the x alone with [several words unintelligible] what would be the order of that approximately

535
00:56:53,113 --> 00:57:07,035
{Kannan} oh in terms of epsilon? so in terms of the [vertiver?] epsilon required the number of rows will grow as one over epsilon squared, so not too bad, and in terms of n it's only [longer or logr or log R?]

536
00:57:07,583 --> 00:57:20,094
{audience} [first sentence not at mike, unintelligible] ...true for all time that we have used computers right? why is that called big data?

537
00:57:20,498 --> 00:57:29,974
{Kannan} no, ok so uh good point. so it's true, it has been true in practice for a long time, but in theory the paradigm we had for algorithms was

538
00:57:29,974 --> 00:57:36,489
I store my graph for my matrix or my input for any problem in RAM and then I can quickly access various things

539
00:57:36,489 --> 00:57:43,831
we never, well not never, we almost never worried about access problems if data was not stored [in area?]

540
00:57:43,831 --> 00:57:48,443
we did sometimes, there is some literature, but we didn't sort of seriously worry about that

541
00:57:48,443 --> 00:57:53,456
but big data means even in theory we better seriously worry about that somehow

542
00:57:53,456 --> 00:57:56,221
{moderator} one other person

543
00:57:56,221 --> 00:58:09,619
{audience} sampling, is there any connection theoretically to this [micro?] sampling about minimum frequency of sampling equal and in big data as to theoretical work on is there something [ecoland?]

544
00:58:09,619 --> 00:58:15,822
{Kannan} yeah so you say our optimal sampling rate that we need, we don't often know optimal answer

545
00:58:15,822 --> 00:58:21,787
[niquis?] was nice to help me know the optimal answer, we often don't know the optimal answer, we know lower and upper bounds which don't quite meet

546
00:58:21,787 --> 00:58:24,840
but in a few cases we do know the optimal answer

547
00:58:24,840 --> 00:58:26,435
{audience} well what is that kind of 


548
00:58:26,435 --> 00:58:34,279
{Kannan} is work in that kind of thing? the lower ones are very hard to come by that's the problem [niquis?] sampling is something else, lower bounds were also possible there

549
00:58:34,969 --> 00:58:39,485
{audience} in your presentation you are dealing with [well in?] numerical data and maybe you are not talking of

550
00:58:39,699 --> 00:58:48,767
{Kannan} so good point good point, throughout the talk I only focus on numerical data partly to spite my friends who focused on Boolean variables for too long {audience laughs}

551
00:58:48,767 --> 00:58:56,169
but partly, partly is the following, uh machine learning in general or other subjects have feature vectors

552
00:58:56,169 --> 00:59:03,648
feature vectors can have Boolean fields but generally is much nicer to deal with real data because there's a lot more structure

553
00:59:03,648 --> 00:59:13,429
{audience} but there's some of this uh uh you know text or even pictures really or audio or all that, can they be converted into numerical

554
00:59:13,429 --> 00:59:22,073
{Kannan} actually that's a very good question, so for instance if you have zero one, so for things occurring or not occurring zero or one

555
00:59:22,073 --> 00:59:32,083
now the numerical distance squared between those two is of course the number of flips or the dot product for instance even though they are Boolean

556
00:59:32,083 --> 00:59:37,456
the dot product is the number of common so there are connections like this which can be spotted

557
00:59:37,456 --> 00:59:43,857
now it turns out, actually it is a very good point you raised, it turns out for instance for the vet matrix you have the point of making some hypertext links

558
00:59:43,857 --> 00:59:50,504
so uh there's a one in the ideal position if the [ite?] were a page linked to [jates?] web page

559
00:59:50,504 --> 00:59:57,999
now this is purely a Boolean thing but you take that matrix and do eigenvalue analysis of it, that's how you get page rank

560
00:59:57,999 --> 01:00:04,406
the reason it works is distance is squared and dot products make sense as if they were real even though they're Booleans

561
01:00:15,693 --> 01:00:23,457
{audience} sir I've heard a lot of big data talks and oftentimes they're about the systems aspect of big data and it was fascinating to hear the foundational aspects of big data

562
01:00:23,457 --> 01:00:25,890
{Kannan} completely removed from reality I'm sorry {audience laughs} but yeah that's true

563
01:00:25,890 --> 01:00:29,841
{audience} sir where does theory and practice meet and how do they meet?

564
01:00:29,841 --> 01:00:40,357
{Kannan} actually that's a good question so for instance uh this idea of using Johnson-Lindenstrauss right? really in a way comes from theory

565
01:00:40,357 --> 01:00:51,104
ok, so, I mean in fact, I probably want to claim I'm not probably even sure, I would I would think theoreticians are the first to think of random [theoreticians?] because they're {sermos arah?}

566
01:00:51,104 --> 01:00:59,203
ok so we prove this theorem that says if you take a number of rows, hundred log n over epsilon squared then I can prove that this works

567
01:00:59,203 --> 01:01:05,427
well in practice that's too big but however it guides practice in a sense that the whole method comes from theory

568
01:01:05,427 --> 01:01:11,069
now in practice you can go and say I'll use only five log n over epsilon and that works beautifully

569
01:01:11,069 --> 01:01:18,927
so, ok, we need two things, we need empirical studies to translate this theory into practice, because the bounds may be prohibitive

570
01:01:18,927 --> 01:01:24,446
we also need the other way mechanism, in practice people use certain heuristics or algorithms

571
01:01:24,446 --> 01:01:30,100
I mean people are happy using the k means algorithm for anything they want, it always works

572
01:01:30,100 --> 01:01:35,115
theoreticians are puzzled that it always works and we are still trying to prove that there are cases where it works

573
01:01:35,115 --> 01:01:41,254
but in trying to prove that maybe we come up with twists that are interesting in practice so it's probably both ways that it would go

574
01:01:41,254 --> 01:01:45,416
but that's how all this vector stuff came from theory in some sense

575
01:01:46,075 --> 01:01:52,252
{audience} so the problem on [insulate?] comparative analysis on distributed data so what are the objectives there?

576
01:01:52,252 --> 01:01:55,988
what are the things that you want to compress it to within maybe a target rate?

577
01:01:55,988 --> 01:02:00,862
and then of all possible ways, all those R matrices, all possible ways by which you can compress it to [dot or target?] size

578
01:02:00,862 --> 01:02:07,317
so are there uh analytical lower bounds on the amount of communication that is needed and which is possible?

579
01:02:07,317 --> 01:02:15,132
{Kannan} yes there are lower bounds actually and it turns out this algorithm had meets the upper and lower

580
01:02:15,132 --> 01:02:21,029
this is one of these algorithms which actually is tight with the lower bounds, we're there for that problem already

581
01:02:21,300 --> 01:02:31,225
so I can give you references but roughly for, well I won't tell you the exact lower bound but yeah in this case the lower and upper bounds meet

582
01:02:32,340 --> 01:02:36,482
so also for the v i to the k there are lower bounds there are lower bounds that meet the upper bounds

583
01:02:42,777 --> 01:02:50,313
{audience} this [bipartite?] graph you draw graph you draw you say that each server has its own respective information but how do you combine them finally together?

584
01:02:51,064 --> 01:02:56,696
{Kannan} so that's still the important sampling, so each of the, what the model of what I said was each server has important sampling

585
01:02:56,696 --> 01:03:05,589
just on its own events, and somehow that's enough it turns out with some exchanges and rejection sampling which I didn't go into

586
01:03:05,589 --> 01:03:11,400
that's enough to do something which is as if importance sampling on the whole thing with the sum

587
01:03:12,015 --> 01:03:15,651
so there's some work to be shown there, I didn't show it

588
01:03:17,824 --> 01:03:21,957
I'll move to the other side, no I won't move to the other side {laughter}

589
01:03:22,717 --> 01:03:32,090
{audience} ok so on this geometric embedding of symbolic uh problems and so on uh you know of course in logic you can also embed many problems in logic

590
01:03:32,296 --> 01:03:37,129
is there a sort of a sampling theorem there that is striking in logic?

591
01:03:39,107 --> 01:03:41,154
{Kannan} for logic uh?


592
01:03:41,431 --> 01:03:43,911
{audience} so for theorem proving or for [combine this/compactness?]

593
01:03:44,894 --> 01:03:50,349
{Kannan} actually I don't know, uh um, the difficulty for theorem proving you want the exact thing right?

594
01:03:50,349 --> 01:03:54,479
so the closest thing I can think of at the programming languages people, to verify a problem

595
01:03:54,479 --> 01:03:58,348
I mean you would like a theorem that says the problem is always true which is often difficult to prove

596
01:03:58,348 --> 01:04:05,659
they do have sampling methods of generating random inputs on which it's enough to check so you can assert with high probability

597
01:04:05,659 --> 01:04:13,468
even sometimes that's not enough there also there are ways of coming up but not full fledged frugal ways but they have heuristic ways of coming up with a test state

598
01:04:13,468 --> 01:04:22,326
{audience} yeah but there could be other things like uh finding deep resolvants or steps in theorem proving 
{Kannan} that's right

599
01:04:22,326 --> 01:04:30,360
{audience} that could come from
{Kannan} yeah maybe some steps can be subject to sampling but I don't know much of anything along those lines actually

600
01:04:43,244 --> 01:04:53,623
{audience (also first and only female voice to speak so far)} look this is really not uh um [deduhdinesys?] question as such uh um think of me as a layman asking question to uh you know professor

601
01:04:53,623 --> 01:04:58,920
so to what extent all these algorithms have been implemented?

602
01:05:00,140 --> 01:05:07,013
um say I'm from a softer background and I'm trying to implement these things with respect to a [field?] problem

603
01:05:07,297 --> 01:05:13,142
to what extent are these implemented so that they're useable?

604
01:05:13,359 --> 01:05:19,996
{Kannan} ok uh I don't know a lot about where the streaming algorithms have been implemented but perhaps they've been, they're around

605
01:05:19,996 --> 01:05:31,240
the matrix algorithm is actually, people have begun to study in these statistical architectures I know you're familiar with MapReduce that's one particular discrete uh distributed architecture

606
01:05:31,240 --> 01:05:41,929
of how the machines interact which Google built and there have now been maybe two or three year old studies of the matrix algorithms

607
01:05:41,929 --> 01:05:47,869
um streaming I would guess is implemented but I don't know precisely

608
01:05:47,869 --> 01:05:52,558
uh the matrix algorithm you don't want to solve it's uh you have large matrices and you want to do this

609
01:05:55,558 --> 01:06:06,089
oh and clustering is another example where which people have implemented and k means, I didn't talk about that, in in MapReduce and other settings there has been some study

610
01:06:08,388 --> 01:06:14,490
{moderator} like just to add to that other question to that actually the one of the streams of activity here is to take

611
01:06:14,490 --> 01:06:20,666
very well established algorithms like clustering, single value decomposition, that have been around for awhile

612
01:06:20,666 --> 01:06:26,557
but put them on a sort of more firm foundational basis where you understand when they work, when they don't

613
01:06:26,557 --> 01:06:31,492
how you can sample them to make them faster etcetera, that's hat's clearly one of the streams

614
01:06:31,492 --> 01:06:39,756
along that line I had a question which would be um are there, are there methods that are very well established like analog there is [really?] clustering etcetera?

615
01:06:39,756 --> 01:06:48,716
which clearly have an impact in practice but do not have a mathematical foundation yet which we all need to look at, examples of those?

616
01:06:48,716 --> 01:06:55,667
{Kannan} yeah uh certainly, graphical models, relief propagation for examples, there there's some theory but not enough

617
01:06:55,667 --> 01:07:03,701
uh deep learning is a big, so there's a subject called deep learning that um machine learning people have devised over the last maybe ten years

618
01:07:03,701 --> 01:07:09,791
uh it is supposed to be spectacular. no theory establishing that as such

619
01:07:09,791 --> 01:07:16,309
topic modeling is another area that, one of our future speakers Chiranjib works on

620
01:07:16,309 --> 01:07:24,333
is another area where there is a lot of heuristic methods, we perhaps have few proofs and uh things need to be proved along those lines so

621
01:07:24,333 --> 01:07:30,150
I mean I mean there are a lot, so, so I'd say, so both ways right, one question was how to take theory to practice

622
01:07:30,150 --> 01:07:36,881
but there's the other way that practical people, I mean they are generally happier people because whatever they use it seems to work {audience laughter}

623
01:07:36,881 --> 01:07:43,101
but we can't prove anything I mean it makes us unhappy but that might be one, another way to go also

624
01:07:49,184 --> 01:07:55,739
{audience} there is another basic question from my side like uh even if we go with all of these algorithm uh being a kind of uh

625
01:07:55,739 --> 01:08:03,089
if I look at from the softer [bound?] of a few [of our applications?] [bound?] a few, how are you going to get the data into this [theorem?] like with the matrices theorem?

626
01:08:03,089 --> 01:08:06,227
so I can implement to the application side also?

627
01:08:06,227 --> 01:08:13,428
{Kannan} ok so that's a a question, more generally you might have a, you might have big data that you have to annotate

628
01:08:13,428 --> 01:08:18,201
I mean that's a huge problem, I mean in uh biology, in astronomy, and so on, there's a huge problem

629
01:08:18,201 --> 01:08:22,145
I don't know whether there is a sort of simple solution for that

630
01:08:22,145 --> 01:08:25,198
obviously not because every domain has to do its [unintelligible]

631
01:08:25,198 --> 01:08:30,627
now as far as these specific things how to convert them into vectors, these more or less come as these logs right?

632
01:08:30,627 --> 01:08:35,182
it was just messages, source and for instance, source and number of packets

633
01:08:35,182 --> 01:08:39,459
it's very easy to run through the log and build your vectors, that's what we were saying

634
01:08:39,459 --> 01:08:47,450
so in some of these instances it is easy, but more generally the problem of how to make data not only machine readable

635
01:08:47,450 --> 01:08:55,261
but intelligible sort of is, is a big problem for which I don't think I have a solution

636
01:09:00,077 --> 01:09:02,792
so this is the quiet, I can come to the quiet side and then

637
01:09:03,163 --> 01:09:06,737
{moderator} all the questions, were almost on that side yes, the questions come from that side

638
01:09:06,911 --> 01:09:08,924
{Kannan} unless we are done with time

639
01:09:09,143 --> 01:09:12,937
{moderator} any other questions?
{silence}

640
01:09:17,494 --> 01:09:27,355
{moderator} ok so if there are no more questions then we'll um thank Ravi again for uh {applause} very impressive talk

641
01:09:28,327 --> 01:09:30,266
{Kannan} oh thank you

642
01:09:33,647 --> 01:09:35,994
{presenter} this is very small gift for a very big lecture 
{laughter}


643
01:09:36,952 --> 01:09:39,429
{Kannan} well the next one will say big data on it 
{laughter}

644
01:09:39,668 --> 01:09:44,264
{off camera} I'd just like to mention and those of you who don't know Ravi is writing, he mentioned at the beginning

645
01:09:44,264 --> 01:09:47,721
a foundational book on this topic and uh some of it might be available on the web uh

646
01:09:47,721 --> 01:09:49,480
{Kannan} oh it's available to download yeah

647
01:09:49,480 --> 01:09:56,829
{off camera} at least even for me who's been in this community for awhile to have all of these different algorithms strung together

648
01:09:56,829 --> 01:10:05,802
under a very nice umbrella that relates now to the, to the things in the world is, is certainly, certainly very interesting, very you know foundational

649
01:10:05,802 --> 01:10:08,472
so I would encourage all of you to take a look at it and uh

650
01:10:08,472 --> 01:10:11,145
{Kannan} yeah please feel free to download it and uh use it anyway

651
01:10:11,145 --> 01:10:15,051
{off camera} thanks Ravi again and uh I'll turn it over to Professor

652
01:10:15,051 --> 01:10:23,338
ok so we do have uh coffee and things right next door so please help yourself there

653
01:10:23,338 --> 01:10:28,741
and then we'll meet again in a month's time for the next lecture and the details will be up on the web

654
01:10:28,741 --> 01:10:32,300
{Kannan} so thanks for coming and thanks thanks for the questions and...
{applause}