0:00:21.746,0:00:28.678 On behalf of the Department of Computer Science and Automation at IISc 0:00:28.697,0:00:34.565 I would like to wish you all a very warm welcome to this first public lecture 0:00:34.715,0:00:38.976 on Foundations of Data Science by Professor Ravi Kannan 0:00:39.609,0:00:45.054 as all of you know, big data or data science 0:00:45.586,0:00:49.403 represent the latest wave in the computing revolution 0:00:50.554,0:00:55.559 now the volume, the complexity, and the variety of the data 0:00:55.559,0:00:59.804 that is now becoming available day after day 0:01:00.221,0:01:03.443 is now at an unprecedented scale 0:01:04.199,0:01:09.626 so if you are able to extract useful and actionable knowledge out of this data 0:01:09.890,0:01:13.490 that's going to be of a huge benefit 0:01:14.218,0:01:17.730 so this initiative, which is called the big data initiative 0:01:18.265,0:01:21.313 has just been launched in order to realize 0:01:21.791,0:01:28.092 the potential of this very exciting, and very fascinating, and very challenging area 0:01:29.423,0:01:33.624 now the primary objective of the big data initiative 0:01:34.478,0:01:39.379 is to build a very strong academic and research ecosystem 0:01:40.733,0:01:46.375 that would enable India's excellence in this fast emerging area 0:01:46.375,0:01:50.417 and take India to a leadership position in this area 0:01:51.482,0:01:54.844 now the faculty of the Department of Computer Science and Automation 0:01:54.844,0:01:59.562 and also the Indian Institute of Science have come together to launch this initiative 0:02:00.185,0:02:05.547 and they have lined up a series of ambitious programs as a part of this initiative 0:02:06.809,0:02:11.414 so the first of these initiatives is going to be the public lectures program 0:02:11.787,0:02:15.856 and the primary objective of the public lectures program 0:02:16.781,0:02:26.639 is to give an introduction, an exposure, to all the fascinating research works that are going on in this emerging area 0:02:27.137,0:02:31.868 and also to tell you more about the current state of the art in this area 0:02:32.698,0:02:37.924 now to give you the first public lecture we have today with us Professor Ravi Kannan 0:02:38.678,0:02:45.085 who is a legendary researcher in the area of algorithms and theoretical computer science for a long time 0:02:46.628,0:02:51.811 Professor Ravi Kannan is currently an adjunct faculty member of the Department of Computer Science and Automation 0:02:53.207,0:02:56.827 and as a principal researcher of the Microsoft Research India 0:02:57.199,0:03:02.403 he is leading a very successful group on algorithms at MSR India 0:03:03.717,0:03:11.783 and as all of you might know, he delivered the 2012 Professor I G Sarma memorial lecture 0:03:11.783,0:03:14.784 in the same faculty hall 0:03:14.784,0:03:21.723 and then in 2011 he was the recipient of the very prestigious 0:03:21.723,0:03:26.331 Donald Knuth Prize for his outstanding contributions in designing 0:03:26.331,0:03:30.653 creative algorithms for many computationally challenging problems 0:03:30.653,0:03:36.987 and lets give Professor Ravi Kannan a wide applause for the Donald Knuth Prize[br][audience applause] 0:03:38.762,0:03:44.588 and 20 years before that, way back in 1991 0:03:44.588,0:03:48.351 he was the recipient of the prestigious Fulkerson Prize 0:03:48.351,0:03:53.274 for his minor work on estimating the volume of convex sets 0:03:54.186,0:03:58.735 so here is a researcher who has made ground breaking contributions 0:03:58.735,0:04:01.837 on a variety of topics and algorithms 0:04:02.146,0:04:06.438 now he has designed approximation algorithms for India programming 0:04:07.085,0:04:10.132 and he has designed randomized algorithms for linereal algebra 0:04:10.698,0:04:14.742 and modus [sounds like "unclear" but that's not right] learning algorithms for convex sets 0:04:15.267,0:04:22.611 so he's truly, in my view, he is the {not english: sunil munaher gwaskil?} of the big data team 0:04:23.460,0:04:27.202 and with a wide applause, let's welcome Professor Ravi Kannan[br][audience applause] 0:04:27.202,0:04:33.271 and invite him to deliver this first public lecture, Professor Ravi Kannan[br][audience applause] 0:04:51.966,0:04:55.717 so thanks Professor Narahari for that gracious introduction 0:04:55.717,0:05:01.238 we await [Sulah Kandalkar?] but I'll start with [Rick Elsore?] 0:05:01.238,0:05:03.623 I want to thank also CSA, this is a very nice initiative 0:05:03.623,0:05:06.413 CSA has always been very dynamic 0:05:06.413,0:05:11.242 and sort of looking to latest advances in various fields 0:05:11.242,0:05:14.355 in addition to adding all the new exciting faculty 0:05:14.355,0:05:16.953 I think it's a very good initiative to start on big data 0:05:16.953,0:05:19.779 I should also recognize and thank Professor Shevade from CSA 0:05:19.779,0:05:24.480 for having this idea and getting us all together, thank you Shevade[br][applause] 0:05:29.734,0:05:34.725 so since this is the first lecture I should give you actually 0:05:34.725,0:05:37.886 a sort of bird's eye view of the whole field in some sense 0:05:37.886,0:05:39.644 I'm not very good at bird's eye view 0:05:39.644,0:05:43.476 so I will give you a couple of slides which are very general 0:05:43.476,0:05:46.672 and then zoom in on a piece of the action 0:05:46.672,0:05:51.090 once slice of the field if you will, and sort of focus on that 0:05:51.090,0:05:54.751 so the title of the talk is not big data, it's Foundations of Data Science 0:05:54.751,0:05:57.094 which is sort of related because it is actually the title of a book 0:05:57.094,0:05:58.970 I'm writing with John Hopcroft 0:05:58.970,0:06:02.903 in the process we had to think about first the following 0:06:02.903,0:06:08.447 what are really, what would you call the foundations of data science? 0:06:09.514,0:06:12.434 see the word "big data" connotes something very jazzy 0:06:12.434,0:06:15.592 and so that can have two interpretations 0:06:15.592,0:06:18.896 one is something cool or hot or whatever you call it 0:06:18.896,0:06:21.736 with perhaps not so much substance 0:06:21.736,0:06:24.032 one is something with a lot of substance 0:06:24.032,0:06:26.765 I will hope to convince you that it's both 0:06:26.765,0:06:28.964 it's hot but also has a lot of substance 0:06:28.964,0:06:31.532 and so the word "foundations of data science" this phrase 0:06:31.532,0:06:34.468 connotes perhaps the substance part of it 0:06:34.468,0:06:37.446 but the question is what are, what should you consider 0:06:37.446,0:06:40.020 to be part of the foundations of data science 0:06:40.020,0:06:43.425 or what should you teach if you're going to teach a course on data science 0:06:43.425,0:06:46.039 or foundations of data science, what should you teach 0:06:46.039,0:06:50.723 so here's something that if you haven't been to a talk with this sort of title 0:06:50.723,0:06:54.389 you won't have seen this but if you've been to a talk 0:06:54.389,0:06:58.659 almost certainly you've seen a curve like this which I then draw 0:06:59.097,0:07:03.899 this is exponential cx for some constant 0:07:05.555,0:07:07.960 so various forms of this curve 0:07:07.960,0:07:11.191 this is the rate of growth of data if you will 0:07:11.191,0:07:13.455 or the rate of growth of storage and speed and so on 0:07:13.455,0:07:15.157 there are different constants 0:07:15.157,0:07:19.821 now there is really a point, I'm not making sort of caricature out of this 0:07:19.821,0:07:25.972 but it is really a point in that it's not as if human knowledge has grown exponentially 0:07:25.972,0:07:30.514 but it is true that our ability to gather data, to store it 0:07:30.514,0:07:33.576 to annotate it has grown perhaps exponentially 0:07:33.576,0:07:36.794 and in fact so we do have exponential rate of growth of data 0:07:36.794,0:07:38.420 but I'm not going to draw you this curve 0:07:38.420,0:07:42.677 perhaps my suggestion is, what I'm going to put in the list here 0:07:42.677,0:07:46.065 the first few things, maybe the answer is no 0:07:46.065,0:07:49.291 they are not what I would consider the foundations of data science 0:07:49.291,0:07:51.415 but they are important never the less 0:07:51.415,0:07:54.179 you could also ask how do you collect and store 0:07:54.179,0:07:58.007 annotate and store data in various domains 0:07:58.007,0:08:02.743 in fact some of these responses will depend what domain will it dwell 0:08:02.743,0:08:07.231 be it astronomy, commerce. I put up three areas here in which we do have in fact 0:08:07.231,0:08:12.408 really a big growth in data that's been collected and stored 0:08:12.408,0:08:18.812 but I would like to suggest that this is not, for us, foundations of data science at the moment 0:08:18.812,0:08:22.311 partly because it's very important but it depends on the domain 0:08:22.311,0:08:25.633 so it's very specific to each domain perhaps 0:08:25.633,0:08:28.675 now you could also say basic statistics is important 0:08:28.675,0:08:34.153 because if you have a lot of data, more than you can handle simply by traditional algorithms 0:08:34.153,0:08:35.914 surely you must sample 0:08:35.914,0:08:39.018 that is you must draw a random sample of the data and deal with that 0:08:39.018,0:08:43.808 so sampling is important, so a study of sampling and basic statistics is perhaps important 0:08:43.808,0:08:48.021 now basic statistics is always important, but we have to assume that and move on 0:08:48.021,0:08:52.971 now the later parts of this are becoming progressively more "yes" answers 0:08:52.971,0:08:57.969 but perhaps here is a sort of mea culpa though I am a computer scientist 0:08:57.969,0:09:01.417 and in particular specialize in algorithms 0:09:01.417,0:09:09.017 for the last three decades perhaps, the area of algorithms focused on discrete structures 0:09:09.017,0:09:13.718 graphs and various other discrete structures, data structures are a big thing 0:09:13.718,0:09:17.048 if you've done computer science courses you've done [lifts?] arrays 0:09:17.048,0:09:20.008 and stacks and so on so forth, that was a big thing 0:09:20.008,0:09:26.078 now also we spent a couple of decades improving the algorithms for various discrete problems 0:09:26.078,0:09:28.521 like graph problems, shorter spots and flows 0:09:28.521,0:09:30.846 now these are very important going forward 0:09:30.846,0:09:36.170 but we have focused enough attention on these matters for the last thirty years 0:09:36.170,0:09:41.351 and learnt enough and progressed enough in some sense, which is very good 0:09:41.351,0:09:46.336 but yet ignored some other parts, I would like to say, somewhat controversially 0:09:46.336,0:09:50.861 we ignored some other parts in computer science, not overall in human knowledge, in computer science 0:09:50.861,0:09:55.870 linear algebra, numerical methods which have come out to be very important 0:09:55.870,0:10:02.376 so if you asked me twenty years ago whether computing eigenvalues or eigenvectors of a matrix 0:10:02.376,0:10:06.765 with a million non-zero entries is practical and important in the future 0:10:06.765,0:10:09.583 I would have said no, so would most computer scientists 0:10:09.583,0:10:12.878 but in fact this is what is used in basic search 0:10:12.878,0:10:18.935 so if you look at how Google does page rank or ranks the pages on the web it's based on eigenvalues 0:10:18.935,0:10:21.821 I won't tell you how but it's based on eigenvalues 0:10:21.821,0:10:25.921 so in fact it's been a surprise over the last decade or two 0:10:25.921,0:10:29.780 that what we might have thought wouldn't scale up, would not be important 0:10:29.780,0:10:31.367 has become important 0:10:31.367,0:10:38.610 one of them is this, sampling, not basic sampling but in fact most of this talk will be about sampling 0:10:38.610,0:10:45.189 but sampling in the context of big data is going to be different from what you might think of as statistical sampling 0:10:45.189,0:10:48.356 and that's important, you'll see that 0:10:48.356,0:10:53.974 and also very important it turns out is high dimensional geometry, you'll see some of this 0:10:53.974,0:11:00.453 so in fact even when you least expect, even for problems which don't look like that 0:11:00.453,0:11:04.814 it turns out those problems can be formulated, we'll see some of that 0:11:04.814,0:11:10.857 those problems can be formulated as problems to do with let's say ten thousand dimensional vectors 0:11:10.857,0:11:17.411 so what does ten thousand dimensional space look like, that turns out to be an important thing to worry about 0:11:17.411,0:11:23.940 and probability limit laws, central limit theorem, the very classical probability limit laws, they turn out to be very important 0:11:23.940,0:11:29.651 now you can sort of intuitively guess why, if you have lots of data you sort of learn to limit 0:11:29.651,0:11:34.256 right? and therefore such things should be more important than let's say basic statistics 0:11:34.256,0:11:38.486 and indeed in a lot of these areas we'll have to develop new tools 0:11:38.486,0:11:44.508 and we have developed some which handles this big data problem, that's what I'm hoping to convince you 0:11:46.894,0:11:53.957 so sampling, so it's clear that if you have a lot of data you have to sample 0:11:54.349,0:12:00.556 so that you get a subsample which is smaller and can be handled 0:12:00.556,0:12:05.054 but is it just picking uniformly at random a part of the data 0:12:05.054,0:12:08.343 is that going to be good enough, that's the question we want to ask right 0:12:08.343,0:12:10.666 so we have ten million pieces of data 0:12:10.666,0:12:13.956 just uniformly at random pick ten thousand of them and analyze them 0:12:13.956,0:12:15.672 is that what we are going to do 0:12:15.672,0:12:19.461 that's what we think of as sampling traditionally, perhaps the most elementary method 0:12:19.461,0:12:22.732 but here is a problem typically that we want to solve 0:12:22.732,0:12:27.033 it's an optimization problem, there are many of those this is one 0:12:27.033,0:12:30.312 you have a very large matrix A, m by n matrix 0:12:30.312,0:12:34.554 which we want to do what's called principal component analysis on A 0:12:34.554,0:12:37.308 if you're not familiar with this bear with me for a moment 0:12:37.308,0:12:40.730 you don't have to be at the moment, familiar with this, we'll see that a little more 0:12:40.730,0:12:44.104 but a lot of you are familiar with this I'm sure 0:12:44.104,0:12:51.438 so that just means we want to find a vector x, so x is an [n, or maybe m] vector of unit length 0:12:51.438,0:12:55.820 so that when I multiply A with x and take the length, that's as high as possible 0:12:55.820,0:12:59.477 so this turns out to be a basic linear algorithm problem, this is very important right? 0:12:59.477,0:13:04.943 so we want to maximize over all unit length vectors x the length of A 0:13:04.943,0:13:08.831 now A is very big so we can't do the whole problem perhaps 0:13:08.831,0:13:11.980 we can't even necessarily put A into random access memory 0:13:11.980,0:13:14.826 so it's going to be difficult to run traditional algorithms, what do we do 0:13:14.826,0:13:21.371 somehow we like to say we sample A to get a sample out of it: B 0:13:22.035,0:13:25.850 now B might have a smaller number of rows or columns maybe 0:13:25.850,0:13:28.636 more zeros or something makes it easier 0:13:29.089,0:13:31.901 we won't see particularly what at the moment 0:13:31.901,0:13:34.746 we want that B to have this property 0:13:35.197,0:13:38.434 see I don't know what x is going to be what I'm going to find 0:13:38.434,0:13:42.879 so I better have this property, that for every... red is important... 0:13:42.958,0:13:46.521 for every x, Ax and Bx are nearly the same length 0:13:46.578,0:13:52.114 if that were true I could deal with B instead of A right? if that were true 0:13:52.114,0:13:55.406 if that were not true for every x I cannot do that 0:13:55.406,0:14:00.322 because for every x I may find maybe a bad x for which B gives me the wrong answer right? 0:14:00.322,0:14:03.708 so how do I do that, that's the kind of sampling we want to do 0:14:03.708,0:14:07.430 we want the sample of A that's good for every x 0:14:07.430,0:14:11.297 and x is an n dimensional vector, there are infinitely many of them 0:14:11.297,0:14:18.539 but a rule of thumb it turns out, in n dimensions you can pretend there are only e to the n vectors 0:14:18.732,0:14:22.674 it's not true, there are infinitely many, but you can pretend that there are e to the n vectors 0:14:22.674,0:14:27.790 that's still a lot of vectors so this sampling better work for all the vectors, not just for one right? 0:14:27.790,0:14:32.650 that's one issue, here's another issue that can come up in sampling 0:14:32.650,0:14:39.895 big data means that the data is perhaps too big to be stored in one machine or one unit of memory, some memory 0:14:39.895,0:14:44.187 so it may be distributed among many servers 0:14:44.187,0:14:50.237 now important sampling, uniform sampling means everybody gets drawn with probability which is the same 0:14:50.237,0:14:55.540 importance sampling means some items have a higher probability based on their importance 0:14:55.540,0:15:00.116 so importance is sampling each item with probability proportional to its importance 0:15:00.116,0:15:04.443 but then all servers must share the vectors of importance right? 0:15:04.443,0:15:08.312 they must be sampling with the same vectors to be consistent 0:15:08.312,0:15:13.076 now maybe the number of items are so large that this vector is costly to translate 0:15:13.076,0:15:17.804 this is the kind of problem we are going to worry about a bit more, both of these problems 0:15:17.804,0:15:20.973 by the way I should tell you the talk is not exhaustive 0:15:20.973,0:15:24.201 I'm not going to tell you all the problems about big data 0:15:24.201,0:15:26.993 that's why we have a whole series of lectures you can see here 0:15:26.993,0:15:32.318 we'll have many more interesting lectures in the series in the coming months, one a month roughly 0:15:32.318,0:15:37.069 so I'm going to cut to a slice of it and these are some of the problems we'll talk about in detail 0:15:37.069,0:15:40.714 so consistency among various servers is one issue 0:15:40.714,0:15:46.385 here is a very simple example that you might want to think about mentally, it's a mental exercise, a very simple one 0:15:46.385,0:15:50.716 so I want an average of a set of real numbers, a large set of real numbers, a big set right? 0:15:50.716,0:15:55.426 now you can just take a sample and take the average, that's an approximation 0:15:55.426,0:16:01.571 but the variance may be very high, obviously right? variance is how to spread entries so I might be very high 0:16:01.571,0:16:05.149 so how do we achieve low relative variance 0:16:05.149,0:16:08.256 now one thing to focus your mind on is the cancellation problem 0:16:08.256,0:16:14.807 suppose there are lots of big positive entries and lots of big negative entries and they're canceled out 0:16:14.807,0:16:18.287 and the total was zero, the average was zero 0:16:18.287,0:16:23.234 so if the answer is zero, if you have to make small relative error, you must get it exactly right 0:16:23.234,0:16:25.642 right? if the answer is zero there is no error allowed 0:16:25.642,0:16:29.405 allowing one percent error if the answer is zero I have to get exactly the answer 0:16:29.405,0:16:34.468 so how do you do that with samples, ok that's our question 0:16:34.468,0:16:39.969 now for average, plain simple average it's easy you can just keep a running sum if you read the whole thing 0:16:39.969,0:16:44.466 but we'll have to worry about other statistics for which this is not easy right? 0:16:44.466,0:16:50.054 so the problem is that the final answer will be very small and therefore the error allowed will be very small 0:16:50.054,0:16:52.752 how do you do it with samples 0:16:52.980,0:16:57.872 so these are some of the problems I deal with, I'll tell you some of the problems as we go along 0:16:57.872,0:17:04.816 sampling is our first major issue, the second major issue we'll deal with in this talk is communication 0:17:04.816,0:17:12.486 so again first one definition, what is "big data"? what's a "big data" problem? 0:17:12.486,0:17:16.977 let's simply define it as data does not fit into random access memory 0:17:16.977,0:17:23.801 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 0:17:23.801,0:17:26.301 that's the sort of working definition of it 0:17:26.301,0:17:30.915 so let's say we run a matrix algorithm, you all know traditional matrix algorithms 0:17:30.915,0:17:38.214 as the algorithm is running along, a step of the algorithm may say "bring me the i j entry of the matrix" 0:17:38.214,0:17:42.390 and in analyzing or implementing that we assume that can be done in unit time 0:17:42.390,0:17:47.734 because if you had random access memory you can quickly access, that's what random access means 0:17:47.734,0:17:52.455 you can quickly access let's say each entry of the matrix by putting down its index 0:17:52.455,0:17:55.788 now this we cannot do if it doesn't fit into RAM 0:17:55.788,0:18:01.754 so in fact the traditional paradigm of algorithm design assumes you can do it in unit time 0:18:01.754,0:18:05.913 and that already fails that's the primitive that fails 0:18:05.913,0:18:13.757 so people devised models that avoid this assumption, one model is the streaming model 0:18:13.757,0:18:20.885 the streaming model says data is written on some external memory and it's steaming, it just goes by 0:18:20.885,0:18:23.771 you can only read once from right to left 0:18:23.771,0:18:30.168 I'll draw a picture here, I've been lazy, I should have drawn pictures on the slide but that takes a long time 0:18:30.168,0:18:39.906 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 0:18:39.906,0:18:46.855 well that per say is not good enough, there is a random access memory which is much smaller than this stream 0:18:46.855,0:18:49.264 so I'll write this I've a very small 0:18:49.264,0:18:55.201 so you can remember some of this stream, let's say by sampling you've sampled some of this stream 0:18:55.201,0:18:57.521 uniform sampling is not going to do much 0:18:57.521,0:19:03.776 good but there are sophisticated methods that do actually work and solve a lot of problems in this model 0:19:03.776,0:19:12.328 it's I should say the streaming model was the object of study even before the phrase big data was coined 0:19:12.328,0:19:18.727 it's studied for the last two decades and there's quite a body of very nice results 0:19:18.727,0:19:26.313 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 0:19:26.313,0:19:34.131 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? 0:19:35.009,0:19:38.740 so now where does communication come in 0:19:38.740,0:19:50.662 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 0:19:51.020,0:19:54.338 so that's how the communication takes place 0:19:56.126,0:20:00.042 so when I read up to this point this is all gone 0:20:02.835,0:20:07.934 whatever information I need of that is stored in my random access memory 0:20:07.934,0:20:16.615 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 0:20:16.615,0:20:23.973 that's one way to think of it so this analogy actually relates space and communication 0:20:23.973,0:20:29.721 and this has been studied widely and it's is used to derive lower bounds in space 0:20:29.721,0:20:37.398 ok a subject that I won't touch, very much into a [dista? distant?] high level concept somehow space is related to communication 0:20:37.398,0:20:42.294 cuz you've got to write down what you want to communicate to the rest of the input [unintelligible word] 0:20:42.791,0:20:50.946 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 0:20:50.946,0:20:55.036 so streaming is one model that's been studied a lot 0:20:55.036,0:21:00.205 a little less restrictive model of big data is that data is split among many servers 0:21:00.205,0:21:03.448 but number of servers is much smaller 0:21:03.448,0:21:14.767 now you might have, you could think of streaming as if each bit is sitting on a different computer, different server 0:21:15.866,0:21:22.310 that's one way to do this, the only way they communicate is through RAM right? otherwise they cannot communicate 0:21:22.310,0:21:30.463 now that's however if the length of this stream is n this is thinking of it as n servers 0:21:32.370,0:21:36.700 now you can argue that a lot of real problems have data that's not so big 0:21:36.700,0:21:41.412 in fact you can accommodate it on log n servers 0:21:41.412,0:21:44.463 many smaller many fewer servers than the amount of data 0:21:44.463,0:21:50.934 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 0:21:52.236,0:21:56.358 now again up to this point the talk has been about generalities 0:21:56.358,0:22:01.277 now I want to zoom in on some examples which are self-contained examples for what is going on 0:22:05.043,0:22:06.964 ok so let's start with an example 0:22:06.964,0:22:11.273 this actually was a [starting/study?] example and motivated a lot of research on streaming algorithms 0:22:11.273,0:22:17.557 we have a large network with lots of computers sending messages to each other 0:22:18.438,0:22:21.029 there are routers which route the message 0:22:21.029,0:22:27.395 so there are n computers, n is very large, perhaps in a million 0:22:28.243,0:22:32.939 but there are routers which are square boxes, maybe there are only ten of them or a hundred of them 0:22:32.939,0:22:40.835 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 0:22:40.835,0:22:46.628 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 0:22:47.392,0:22:54.700 so we won't worry about the mechanics of routing let's just study what happens here for the logs of the messages 0:22:54.700,0:22:58.037 so the last router that sends it to the destination 0:22:58.037,0:23:01.637 the last one keeps a log of the message 0:23:01.857,0:23:09.499 the log just has for us the sending computers name i, and the length of the message, that's the log it keeps 0:23:10.076,0:23:18.271 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 0:23:19.315,0:23:25.901 so let's say Ai is the total length of all the messages sent by computer i in one day let's say 0:23:25.901,0:23:31.498 and you want statistics of Ai, the statistics are important to again optimize the network 0:23:31.498,0:23:33.833 that's the problem we want to solve 0:23:33.833,0:23:39.718 now statistics one thing you want to do is perhaps the average of Ai or the total of Ai 0:23:39.718,0:23:42.732 but you may want the variance, you may want the second moment 0:23:42.732,0:23:46.498 so that's just 1 over n, there are n computers 0:23:46.498,0:23:53.958 and n Ai, you sum the Ai squared and then take the average right? let's say we want to do that 0:23:54.410,0:23:56.506 now it's that simple let's see what happens 0:23:56.506,0:23:59.574 the trouble is no router knows Ai in full 0:23:59.574,0:24:06.789 so i was this computer, it might have sent messages which have gone through many routers 0:24:06.789,0:24:15.458 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? 0:24:15.928,0:24:22.765 so if you want to find this you seem to have to communicate all the partial Ai and that requires O(n) space 0:24:22.765,0:24:27.868 because there are n computers, you have to communicate for each one the total from each router 0:24:27.868,0:24:31.854 so think of the problem actually just for two routers, already interesting 0:24:31.854,0:24:35.102 now let's get to the routers formulated abstractly 0:24:35.102,0:24:38.521 each router has just an n vector so this is where vectors come in 0:24:38.521,0:24:41.335 there are no vectors to start with right? this not a geometry problem 0:24:41.335,0:24:43.911 but here I am going to make up vectors 0:24:43.911,0:24:48.077 so each router you can think of as having an n vector 0:24:48.077,0:24:54.412 and the components tell you the total length of all messages from each computer that's been logged by this router 0:24:54.412,0:25:02.558 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 0:25:02.558,0:25:08.557 so again vectors have come in surreptitiously they have nothing to do with the problem but we've formulated the vectors 0:25:08.557,0:25:17.712 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 0:25:17.712,0:25:23.283 so what we want to find I say alas inner sum n over t 0:25:23.283,0:25:30.693 so we want to take the sum over all routers of the total length of message logged by that router coming from i 0:25:30.693,0:25:35.167 I want to sum inside first, then square and then add 0:25:35.167,0:25:37.981 it would have been nice if the sum was outside 0:25:37.981,0:25:46.830 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? 0:25:46.830,0:25:51.075 and that's a problem because these route this information is not contained in any body 0:25:51.075,0:25:53.787 now if you don't follow the exact mechanics that's fine 0:25:53.787,0:25:57.259 I'm going to abstract it even more that makes life a lot simpler 0:25:57.259,0:26:00.047 so here's a vector problem this is the abstract problem we want to solve 0:26:00.047,0:26:04.196 there's an n component vector, bold face is vectors 0:26:04.206,0:26:06.646 a sub t residing in server t 0:26:06.695,0:26:13.084 and you want to take the sum and you want to estimate the length squared of the sum right? sum of squares of length 0:26:13.239,0:26:16.821 we want to sum the vectors component wise and then take the sum of squares 0:26:17.072,0:26:20.359 now if we had the whole vector and we could write it down this is a trivial problem 0:26:20.359,0:26:28.606 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 0:26:28.606,0:26:35.460 ok so maybe I can sample k of the n components of the vector v right? 0:26:35.460,0:26:38.622 v has too many components I don't want to write them all down 0:26:38.622,0:26:41.734 maybe I can sample k and just collect information on that 0:26:41.734,0:26:45.647 is that any good and the answer is a gentle no the variance can be very high 0:26:45.647,0:26:51.138 some components can be much higher than other components so this will be a loss if you do that 0:26:51.138,0:26:57.690 well here is a beautiful theorem which is a geometry theorem, now you see that geometry is necessary and useful 0:26:57.690,0:27:00.505 called a Johnson Lindenstrauss Theorem 0:27:00.505,0:27:05.396 if you perhaps don't remember anything from this talk hopefully you can take this home if you didn't see it before 0:27:05.396,0:27:07.754 so this theorem says the following 0:27:07.754,0:27:15.877 if I pick a matrix R which has a small number of rows, k is supposed to be small 0:27:15.877,0:27:22.811 matrix with independent random random unit vectors, so they're independent random vectors with unit length as rows 0:27:22.859,0:27:31.108 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 0:27:31.108,0:27:34.106 which is a known factor, we know this so we don't care 0:27:34.456,0:27:39.502 it tells me the length of v, so a little schematic picture here 0:27:42.538,0:27:46.609 so we had a vector x with a lot of components, n components 0:27:47.019,0:27:52.468 and we multiply it by an R which is a very small number of rows 0:27:52.468,0:27:57.127 it becomes now a k vector, this is R x 0:27:58.346,0:28:01.691 and it's enough to find the length of that right? 0:28:01.691,0:28:09.574 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 0:28:09.574,0:28:12.331 not a plain sample that won't do right? 0:28:12.331,0:28:17.125 but this kind of random vector then multiplied it will do 0:28:18.187,0:28:22.361 the proof is not difficult I mean obviously I won't do the proof here but the proof is not difficult 0:28:22.361,0:28:27.528 but what it says is that a sample of k components, plain components won't do 0:28:27.528,0:28:30.764 but a sample of k combinations of components will do 0:28:37.187,0:28:47.909 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 0:28:47.909,0:28:51.587 now these are only k vectors they are very small ok? 0:28:51.587,0:29:00.691 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 0:29:00.691,0:29:06.513 but you need to know the same R, everybody needs the same R ok? 0:29:06.513,0:29:11.444 how do we do that, we need a lot of communication it seems to agree on one R 0:29:11.444,0:29:17.870 and this is going to be this has been a simple problem and I want to isolate this problem now, forget the vector problem 0:29:17.870,0:29:22.580 how to share randomness this is a very important problem it turns out in many areas 0:29:22.580,0:29:31.694 how can many servers share the same they have to agree on the same n bit random string without transmitting random R right? 0:29:31.694,0:29:35.279 they all have to have exactly the same random string how do they do that 0:29:38.122,0:29:42.302 the string has n bits of information so it seems to need n bits of communication 0:29:42.598,0:29:51.670 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 0:29:51.670,0:29:57.924 we don't need all the components to be independent we need only what's called 4-way independence 0:29:57.924,0:30:03.119 now 4-way independence is every set of 4 or fewer bits are independent 0:30:03.369,0:30:07.440 so full independence is the entire thing should be independent mutually 0:30:07.440,0:30:15.229 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 0:30:15.229,0:30:26.383 now 4-way independent things are easier to get coding theory, and coding theory gives us this 0:30:26.596,0:30:28.573 it's actually a very classic code 0:30:31.876,0:30:36.768 so here's one way to get the 4-way independent stream very long stream but a 4-way independent 0:30:36.768,0:30:50.419 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 0:30:50.419,0:30:56.815 then I evaluate this polynomial at n points, p at 1, p at 2, and p at n, and so on 0:30:56.815,0:31:02.619 what happens is that this string which is much longer than 4, right? same length, is 4-way independent 0:31:02.619,0:31:06.011 and you have to do it over a finite field but this is the idea 0:31:06.011,0:31:14.414 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 0:31:14.414,0:31:20.723 now these are called pseudo-random sequences and many of you will notice now the answer for the communication problem is simple 0:31:20.723,0:31:26.558 you just generate the seed, some central location, and transmit the seed 0:31:26.558,0:31:30.436 a not, a 1, a 2, a 3, centrally and transmit to all servers 0:31:30.436,0:31:36.884 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 0:31:36.884,0:31:43.085 so they've agreed on an n length random string with just very little transmission 0:31:43.518,0:31:48.313 and that the communicating randomness like this we've said is an important derivative 0:31:50.540,0:31:56.556 so finishing the vector problem as I've said there was an important paper due to Alon Szegedy about fifteen years back 0:31:56.556,0:32:02.403 which started the field of streaming algorithms so I want to go through what we have more or less summarizing what we have 0:32:02.403,0:32:10.060 there's a central processor which generates a seed for the random matrix, it transmits the seeds to all the servers 0:32:10.060,0:32:16.924 so let's just find, apply R to their individual so they build up the whole matrix from the seed 0:32:16.924,0:32:25.430 apply this and then send it to the central processor which sums up all these things and then finds the length 0:32:26.422,0:32:35.615 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 0:32:36.612,0:32:46.835 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 0:32:46.835,0:33:01.038 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 0:33:01.038,0:33:12.906 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 0:33:15.476,0:33:37.136 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 0:33:37.136,0:33:49.668 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 0:33:49.668,0:33:52.976 [someone may be asking a question, I can't hear anything just a long pause] 0:33:56.665,0:34:10.019 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 0:34:13.744,0:34:17.295 briefly how to [say one/save on?] communication 0:34:21.586,0:34:32.802 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 0:34:32.992,0:34:44.330 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 0:34:44.498,0:34:52.849 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 0:34:52.849,0:35:05.417 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 0:35:05.417,0:35:14.890 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 0:35:14.890,0:35:29.852 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? 0:35:29.852,0:35:45.174 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 0:35:45.174,0:35:59.992 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. 0:36:01.618,0:36:18.069 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 0:36:18.069,0:36:30.545 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 0:36:33.835,0:36:45.951 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 0:36:48.274,0:36:54.386 here's the abstract formulation, you have a bipartite graph so i can draw it on the board 0:37:04.708,0:37:23.055 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. 0:37:26.700,0:37:35.860 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. 0:37:37.268,0:37:47.305 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 0:37:51.780,0:38:05.223 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 0:38:05.223,0:38:17.086 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 0:38:17.086,0:38:26.339 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 0:38:27.122,0:38:38.349 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 0:38:38.349,0:38:48.006 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 0:38:48.006,0:38:54.876 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. 0:38:54.876,0:39:03.825 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 0:39:03.825,0:39:18.452 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 0:39:18.452,0:39:23.508 so this is an example where the number of dimensions is even bigger than what you started with 0:39:24.628,0:39:35.512 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 0:39:35.512,0:39:45.271 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 0:39:45.271,0:39:54.978 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 0:39:54.978,0:40:04.698 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? 0:40:04.698,0:40:15.672 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 0:40:15.672,0:40:32.991 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 0:40:32.991,0:40:44.371 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, 0:40:44.371,0:40:54.471 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 0:40:54.471,0:41:10.043 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? 0:41:12.705,0:41:24.286 [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 0:41:24.788,0:41:36.238 [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 0:41:37.012,0:41:43.296 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 0:41:43.848,0:42:01.796 [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 0:42:01.796,0:42:11.594 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. 0:42:14.022,0:42:26.837 anything else? [unintelligible question] yeah so the answer to that is I'm a theoretician. [audience laughs] 0:42:27.843,0:42:33.392 well you know this is the model in which you can. k typically wants to be small 0:42:33.392,0:42:44.607 [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 0:42:45.235,0:42:53.090 [unintelligible comment] no I thought you meant this scale [laughter] this scale is also [unintelligible] 0:42:53.090,0:43:00.032 anything else? ok good. so let me quickly do principal component analysis. 0:43:00.032,0:43:10.816 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 0:43:10.816,0:43:20.429 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? 0:43:20.429,0:43:32.634 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 0:43:32.634,0:43:39.387 Unit Vector y perpendicular to the first principal component maximizing this called the second principal component and so on 0:43:39.405,0:43:43.347 again a lot of you are familiar with this, if not just think of it as an optimization point 0:43:44.959,0:43:48.198 very Nice Linear Algebra Theory, for many problems you want to find these 0:43:48.819,0:43:53.475 now big data the matrix x of A may not be stored all on one server 0:43:54.162,0:44:01.905 here is a simple model we can think of, there are many servers, each stores a similar dimension matrix, 0:44:01.905,0:44:08.988 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 0:44:08.998,0:44:14.782 and you want to deal with the sum of all the matrices but you don't want to communicate all of that right? 0:44:14.782,0:44:22.234 so you want to find principal components of the sum but communicate only a tiny fraction of the whole data amongst them 0:44:25.059,0:44:32.953 so PCA for distributed data, server t has matrix A t you want to sum and take the principal component 0:44:32.953,0:44:40.849 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 0:44:40.849,0:44:54.138 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 0:44:54.138,0:45:05.676 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 0:45:05.676,0:45:10.266 I really want the length of A times x but it's enough to do this 0:45:10.266,0:45:15.463 R makes it much smaller, we erased that, R makes it much smaller so we can communicate better 0:45:15.463,0:45:25.381 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 0:45:25.381,0:45:37.196 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 0:45:37.196,0:45:46.356 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 0:45:46.356,0:45:54.466 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 0:45:54.466,0:46:03.746 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 0:46:03.746,0:46:13.374 so there's one random matrix R so again the picture was, oh I think I have a picture on the next slide 0:46:13.672,0:46:24.345 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 0:46:24.345,0:46:36.583 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 0:46:36.583,0:46:44.334 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 0:46:44.334,0:46:53.203 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 0:46:53.203,0:47:05.920 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 0:47:05.920,0:47:13.261 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 0:47:13.261,0:47:17.532 machine learning in a distributed setting is an active area 0:47:17.532,0:47:23.726 there are dynamic versions of these questions you can ask where the data is subject to updates 0:47:23.726,0:47:28.931 there are some papers here, I point out only one reference but there are many references here 0:47:29.620,0:47:44.503 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 0:47:44.503,0:47:51.785 so this is called sparsification which is another twist on sampling 0:47:51.785,0:48:02.618 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? 0:48:02.618,0:48:09.121 now you can think about RA as combinations of the rows of A right? RA is the combination of the rows of A 0:48:10.045,0:48:28.615 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? 0:48:31.232,0:48:34.627 how about graphs? because graphs are a special case of the matrices 0:48:34.627,0:48:38.616 so here is a picture, a description of what a graph looks like 0:48:38.616,0:48:41.495 now I've gone to columns and to rows excuse me 0:48:41.495,0:48:46.168 so here's a graph, it's represented by this edge [nor?] adjacency matrix 0:48:46.168,0:48:53.874 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 0:48:53.874,0:49:02.967 similarly edge 2 goes from B to C, and that's that edge, B to C and so on 0:49:02.967,0:49:10.646 I put down this matrix and uh here's a cut which cuts the graph into two pieces 0:49:10.882,0:49:16.118 I look at all the edges going across the cut, these three edges now going across that cut 0:49:16.118,0:49:20.786 it turns out, and this is a calculation that you don't have to do at this point 0:49:20.786,0:49:33.520 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 0:49:33.520,0:49:41.707 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 0:49:41.707,0:49:47.705 so from this we can formulate the following problem which is exactly the same as what I had before 0:49:47.705,0:49:55.624 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 0:49:55.624,0:49:58.631 I don't want the combination, I want a subset 0:49:58.865,0:50:06.080 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 0:50:06.412,0:50:11.991 if n vertices or have n choose two edges, that's many many edges 0:50:11.991,0:50:19.432 can I sample a subset of edges, so in this case I have I think fifteen edges instead of [tensials?] two 0:50:19.432,0:50:29.909 and perhaps I weight them, I make them thicker, so that I want to guarantee that any cut I make 0:50:29.909,0:50:34.244 so there are n vertices I can cut them into two to the n possible ways 0:50:34.244,0:50:39.228 any cut I make here, and I make here, has the same value 0:50:39.228,0:50:47.593 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 0:50:47.593,0:50:58.906 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 0:50:58.906,0:51:07.872 that's a problem we might want to solve and um here's a problem with this setup 0:51:07.872,0:51:10.636 here is a pathological graph 0:51:10.636,0:51:19.256 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 0:51:19.256,0:51:23.967 here is a cut just cutting it into two pieces, there is one edge crossing it 0:51:24.998,0:51:32.390 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? 0:51:32.834,0:51:36.117 zero is not within relative error one percent or one 0:51:36.308,0:51:40.069 so if I want to sparsify this graph I'd better always pick the same 0:51:40.197,0:51:43.679 so uniform sampling of the edges will not do 0:51:43.871,0:51:49.220 I have to do something else and a beautiful theorem of Spielmann, Teng and Srivatsava 0:51:49.220,0:51:51.620 Nikhil Srivatsava is at Microsoft in Bangalore 0:51:51.630,0:51:57.341 uh says that uh if you have an n by m matrix where m is larger than n 0:51:57.341,0:52:04.608 you could, there is a probability distribution it cannot be uniform I have to make sure I put a high probability on this edge 0:52:04.608,0:52:15.353 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 0:52:15.353,0:52:20.435 only on log n rows, a small number of rows think of it, according to the distributions 0:52:20.435,0:52:24.218 then for every x length of Ax and length of Bx are the same 0:52:24.218,0:52:29.554 and that turns out answers the cut problem for a reason I won't actually describe 0:52:29.554,0:52:37.342 and if A actually came from a graph like the last one, sampling probabilities are proportional to the electrical resistances 0:52:37.342,0:52:44.350 when you view the graph as a resister network and in fact sampling can be done in a nearly linear time 0:52:44.350,0:52:56.098 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 0:52:56.098,0:53:02.894 you cannot do that if you choose uniform probability right? and the probabilities are proportional to electrical resistences 0:53:02.894,0:53:12.791 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 0:53:12.791,0:53:23.500 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 0:53:23.500,0:53:38.083 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 0:53:38.083,0:53:47.506 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 0:53:47.506,0:53:53.154 well it's one of those few things which can be stated very simply to do with vectors 0:53:53.154,0:53:59.027 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 0:53:59.027,0:54:05.839 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 0:54:05.839,0:54:11.911 not necessarily looking staring at big data but something that has to do with compression and sparsification 0:54:11.911,0:54:21.624 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? 0:54:21.624,0:54:28.507 um by these people, so if you have a finite set of vectors which are in what's called inertial position 0:54:28.507,0:54:33.108 inertial position means the set of vectors is cheap 0:54:33.108,0:54:43.421 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 0:54:43.421,0:54:55.397 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 0:54:55.397,0:55:01.415 so that needs to be true and no vector should be big, so vectors are all small in length 0:55:01.415,0:55:08.564 then you can always partition this set of vectors into two sets that are about half inertial 0:55:08.564,0:55:19.237 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 0:55:19.237,0:55:28.927 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 0:55:28.927,0:55:34.674 so it turns out again this settles actually a very long standing problem in operative theory 0:55:34.674,0:55:40.936 now this has something to do with graph sparsification in fact that was their starting point 0:55:40.936,0:55:53.358 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 0:55:53.358,0:56:03.270 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 0:56:03.270,0:56:14.897 but I believe that's all so I'm done, we are now onto questions if you want [audience applauds] 0:56:36.759,0:56:40.242 {moderator} so we're open for questions now 0:56:40.961,0:56:53.113 {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 0:56:53.113,0:57:07.035 {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?] 0:57:07.583,0:57:20.094 {audience} [first sentence not at mike, unintelligible] ...true for all time that we have used computers right? why is that called big data? 0:57:20.498,0:57:29.974 {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 0:57:29.974,0:57:36.489 I store my graph for my matrix or my input for any problem in RAM and then I can quickly access various things 0:57:36.489,0:57:43.831 we never, well not never, we almost never worried about access problems if data was not stored [in area?] 0:57:43.831,0:57:48.443 we did sometimes, there is some literature, but we didn't sort of seriously worry about that 0:57:48.443,0:57:53.456 but big data means even in theory we better seriously worry about that somehow 0:57:53.456,0:57:56.221 {moderator} one other person 0:57:56.221,0:58:09.619 {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?] 0:58:09.619,0:58:15.822 {Kannan} yeah so you say our optimal sampling rate that we need, we don't often know optimal answer 0:58:15.822,0:58:21.787 [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 0:58:21.787,0:58:24.840 but in a few cases we do know the optimal answer 0:58:24.840,0:58:26.435 {audience} well what is that kind of [br] 0:58:26.435,0:58:34.279 {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 0:58:34.969,0:58:39.485 {audience} in your presentation you are dealing with [well in?] numerical data and maybe you are not talking of 0:58:39.699,0:58:48.767 {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} 0:58:48.767,0:58:56.169 but partly, partly is the following, uh machine learning in general or other subjects have feature vectors 0:58:56.169,0:59:03.648 feature vectors can have Boolean fields but generally is much nicer to deal with real data because there's a lot more structure 0:59:03.648,0:59:13.429 {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 0:59:13.429,0:59:22.073 {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 0:59:22.073,0:59:32.083 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 0:59:32.083,0:59:37.456 the dot product is the number of common so there are connections like this which can be spotted 0:59:37.456,0:59:43.857 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 0:59:43.857,0:59:50.504 so uh there's a one in the ideal position if the [ite?] were a page linked to [jates?] web page 0:59:50.504,0:59:57.999 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 0:59:57.999,1:00:04.406 the reason it works is distance is squared and dot products make sense as if they were real even though they're Booleans 1:00:15.693,1:00:23.457 {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 1:00:23.457,1:00:25.890 {Kannan} completely removed from reality I'm sorry {audience laughs} but yeah that's true 1:00:25.890,1:00:29.841 {audience} sir where does theory and practice meet and how do they meet? 1:00:29.841,1:00:40.357 {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 1:00:40.357,1:00:51.104 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?} 1:00:51.104,1:00:59.203 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 1:00:59.203,1:01:05.427 well in practice that's too big but however it guides practice in a sense that the whole method comes from theory 1:01:05.427,1:01:11.069 now in practice you can go and say I'll use only five log n over epsilon and that works beautifully 1:01:11.069,1:01:18.927 so, ok, we need two things, we need empirical studies to translate this theory into practice, because the bounds may be prohibitive 1:01:18.927,1:01:24.446 we also need the other way mechanism, in practice people use certain heuristics or algorithms 1:01:24.446,1:01:30.100 I mean people are happy using the k means algorithm for anything they want, it always works 1:01:30.100,1:01:35.115 theoreticians are puzzled that it always works and we are still trying to prove that there are cases where it works 1:01:35.115,1:01:41.254 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 1:01:41.254,1:01:45.416 but that's how all this vector stuff came from theory in some sense 1:01:46.075,1:01:52.252 {audience} so the problem on [insulate?] comparative analysis on distributed data so what are the objectives there? 1:01:52.252,1:01:55.988 what are the things that you want to compress it to within maybe a target rate? 1:01:55.988,1:02:00.862 and then of all possible ways, all those R matrices, all possible ways by which you can compress it to [dot or target?] size 1:02:00.862,1:02:07.317 so are there uh analytical lower bounds on the amount of communication that is needed and which is possible? 1:02:07.317,1:02:15.132 {Kannan} yes there are lower bounds actually and it turns out this algorithm had meets the upper and lower 1:02:15.132,1:02:21.029 this is one of these algorithms which actually is tight with the lower bounds, we're there for that problem already 1:02:21.300,1:02:31.225 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 1:02:32.340,1:02:36.482 so also for the v i to the k there are lower bounds there are lower bounds that meet the upper bounds 1:02:42.777,1:02:50.313 {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? 1:02:51.064,1:02:56.696 {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 1:02:56.696,1:03:05.589 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 1:03:05.589,1:03:11.400 that's enough to do something which is as if importance sampling on the whole thing with the sum 1:03:12.015,1:03:15.651 so there's some work to be shown there, I didn't show it 1:03:17.824,1:03:21.957 I'll move to the other side, no I won't move to the other side {laughter} 1:03:22.717,1:03:32.090 {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 1:03:32.296,1:03:37.129 is there a sort of a sampling theorem there that is striking in logic? 1:03:39.107,1:03:41.154 {Kannan} for logic uh?[br] 1:03:41.431,1:03:43.911 {audience} so for theorem proving or for [combine this/compactness?] 1:03:44.894,1:03:50.349 {Kannan} actually I don't know, uh um, the difficulty for theorem proving you want the exact thing right? 1:03:50.349,1:03:54.479 so the closest thing I can think of at the programming languages people, to verify a problem 1:03:54.479,1:03:58.348 I mean you would like a theorem that says the problem is always true which is often difficult to prove 1:03:58.348,1:04:05.659 they do have sampling methods of generating random inputs on which it's enough to check so you can assert with high probability 1:04:05.659,1:04:13.468 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 1:04:13.468,1:04:22.326 {audience} yeah but there could be other things like uh finding deep resolvants or steps in theorem proving [br]{Kannan} that's right 1:04:22.326,1:04:30.360 {audience} that could come from[br]{Kannan} yeah maybe some steps can be subject to sampling but I don't know much of anything along those lines actually 1:04:43.244,1:04:53.623 {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 1:04:53.623,1:04:58.920 so to what extent all these algorithms have been implemented? 1:05:00.140,1:05:07.013 um say I'm from a softer background and I'm trying to implement these things with respect to a [field?] problem 1:05:07.297,1:05:13.142 to what extent are these implemented so that they're useable? 1:05:13.359,1:05:19.996 {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 1:05:19.996,1:05:31.240 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 1:05:31.240,1:05:41.929 of how the machines interact which Google built and there have now been maybe two or three year old studies of the matrix algorithms 1:05:41.929,1:05:47.869 um streaming I would guess is implemented but I don't know precisely 1:05:47.869,1:05:52.558 uh the matrix algorithm you don't want to solve it's uh you have large matrices and you want to do this 1:05:55.558,1:06:06.089 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 1:06:08.388,1:06:14.490 {moderator} like just to add to that other question to that actually the one of the streams of activity here is to take 1:06:14.490,1:06:20.666 very well established algorithms like clustering, single value decomposition, that have been around for awhile 1:06:20.666,1:06:26.557 but put them on a sort of more firm foundational basis where you understand when they work, when they don't 1:06:26.557,1:06:31.492 how you can sample them to make them faster etcetera, that's hat's clearly one of the streams 1:06:31.492,1:06:39.756 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? 1:06:39.756,1:06:48.716 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? 1:06:48.716,1:06:55.667 {Kannan} yeah uh certainly, graphical models, relief propagation for examples, there there's some theory but not enough 1:06:55.667,1:07:03.701 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 1:07:03.701,1:07:09.791 uh it is supposed to be spectacular. no theory establishing that as such 1:07:09.791,1:07:16.309 topic modeling is another area that, one of our future speakers Chiranjib works on 1:07:16.309,1:07:24.333 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 1:07:24.333,1:07:30.150 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 1:07:30.150,1:07:36.881 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} 1:07:36.881,1:07:43.101 but we can't prove anything I mean it makes us unhappy but that might be one, another way to go also 1:07:49.184,1:07:55.739 {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 1:07:55.739,1:08:03.089 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? 1:08:03.089,1:08:06.227 so I can implement to the application side also? 1:08:06.227,1:08:13.428 {Kannan} ok so that's a a question, more generally you might have a, you might have big data that you have to annotate 1:08:13.428,1:08:18.201 I mean that's a huge problem, I mean in uh biology, in astronomy, and so on, there's a huge problem 1:08:18.201,1:08:22.145 I don't know whether there is a sort of simple solution for that 1:08:22.145,1:08:25.198 obviously not because every domain has to do its [unintelligible] 1:08:25.198,1:08:30.627 now as far as these specific things how to convert them into vectors, these more or less come as these logs right? 1:08:30.627,1:08:35.182 it was just messages, source and for instance, source and number of packets 1:08:35.182,1:08:39.459 it's very easy to run through the log and build your vectors, that's what we were saying 1:08:39.459,1:08:47.450 so in some of these instances it is easy, but more generally the problem of how to make data not only machine readable 1:08:47.450,1:08:55.261 but intelligible sort of is, is a big problem for which I don't think I have a solution 1:09:00.077,1:09:02.792 so this is the quiet, I can come to the quiet side and then 1:09:03.163,1:09:06.737 {moderator} all the questions, were almost on that side yes, the questions come from that side 1:09:06.911,1:09:08.924 {Kannan} unless we are done with time 1:09:09.143,1:09:12.937 {moderator} any other questions?[br]{silence} 1:09:17.494,1:09:27.355 {moderator} ok so if there are no more questions then we'll um thank Ravi again for uh {applause} very impressive talk 1:09:28.327,1:09:30.266 {Kannan} oh thank you 1:09:33.647,1:09:35.994 {presenter} this is very small gift for a very big lecture [br]{laughter}[br] 1:09:36.952,1:09:39.429 {Kannan} well the next one will say big data on it [br]{laughter} 1:09:39.668,1:09:44.264 {off camera} I'd just like to mention and those of you who don't know Ravi is writing, he mentioned at the beginning 1:09:44.264,1:09:47.721 a foundational book on this topic and uh some of it might be available on the web uh 1:09:47.721,1:09:49.480 {Kannan} oh it's available to download yeah 1:09:49.480,1:09:56.829 {off camera} at least even for me who's been in this community for awhile to have all of these different algorithms strung together 1:09:56.829,1:10:05.802 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 1:10:05.802,1:10:08.472 so I would encourage all of you to take a look at it and uh 1:10:08.472,1:10:11.145 {Kannan} yeah please feel free to download it and uh use it anyway 1:10:11.145,1:10:15.051 {off camera} thanks Ravi again and uh I'll turn it over to Professor 1:10:15.051,1:10:23.338 ok so we do have uh coffee and things right next door so please help yourself there 1:10:23.338,1:10:28.741 and then we'll meet again in a month's time for the next lecture and the details will be up on the web 1:10:28.741,1:10:32.300 {Kannan} so thanks for coming and thanks thanks for the questions and...[br]{applause}