[00:00:05]
>> Right. We're. Back. It. Fast. And it's now faculty that's what. Research he works on topics include. And I got to know him as a friend. At Cal Tech where he. Thanks Rachel. Ok so. Since it's a smaller audience I mean I'm hoping that I can take all of you along so you know if there are questions please stop me if there is I mean if you if you don't know what the right question is you know just point out something on the slate and say look I don't I don't get what that is piece tell me what that is again I'm happy to do that I.
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I don't have to like there's more sides and I can get through I'm happy to you know stop in between. Ok so so so this is well done with with a student of mine going John. A lot of this work is done by going John you know I'm just having the privilege of giving the stock but but but this work actually started while I was at Caltech in fact this was this was something that we started studying with Leonard and then you know.
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I think the problem the basic problem over here is very natural So let me just jump into what that is so what is partial function extension. Here's the problem so the problem is that I'm given that c. endpoints from some domain 130 n. and then given a value this is real value at each of those points if one through a friend so at the pointy I would like to have the value after by right.
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I'm also given some family script therefore functions think of these as let's say convex functions linear functions what have you. Some model monotone whatever so this is some implicitly given family script for functions and what I want to do is I want to figure out if there's a function in the family.
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Defined on the domain so there's a total function which basically extends this partial function which means that at the at at each of the points it should have value. Right so. It's a simple enough problem politically look at some quick examples so for a 1st example let's say that the domain is just a set of real numbers and my family f. I'm looking for convex functions right and there were the passion function says it says it had 0 The function would have value one at 2 which would have value $24.00 should have value to say that this is you know maybe 5 seconds and it's clear that no convex function can satisfy this.
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Value or 2 is too large or you know one of one of those are the values have to be changed so this is not extended. If I change the value add to this becomes extendable and what I want to point out is that there's no I mean at least in this case there's no unique extension you could think of any the lot of convex function which extend this conversion function Here's one where it may not be the one you thought of but this is a convex function with extensive special function.
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Another domain I'm going to be looking at is let's say subsets of a given set so in this case a.b.c. is my set and looking at subsets of this and the family f. I'm interested in is monotone functions. This one a partial function says. Sure if you're one of the late you think or this is this Ok.
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I don't know how I mean there's some colors and so. This looks. Yeah you have some late let's have some late Ok it's good. So yes I hope you can see the. So c. is a special function says that the empty set is value 0 b. has value 5 it has value 6 and a.b.c. has value 5 again this cannot possibly be extended to a monotone function because the value is larger than the value at a.b.c. right so this is again not extend the bill I change the value of one and I can now extend for example this is an extension.
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Ok so so so that So that's the basic problem I'm given a finite set of points with a value at each point and some domain I'm going to stick to 2 different domains either the m. dimensional reals or subsets of one through m. and given some family f. I want to find out if there exists a function in the family which extends my given partial function Ok.
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So that's my basic problem of partial function extension. In fact what I'm going to be interesting deficient algorithms for this I'm going to be looking at algorithms which run in time. And when. Just to just to keep in mind m. is my dimension of their domain so it's the number of it's the size of my set or it's the dimension of the end length space and is the number of observations in my partial function right so m. and then.
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I'm going to ignore bit complexity of the here. Just because like I said the there may not be a unique function where there were not be unique extensions so I am just right now I'm just interested in yes no Ok. Ok so so rate and I'm interested in politics Melton for this can you go on for a mile with them can you not.
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Questions about this right now Ok. I can also study an approximate version of this rates again I'm going to partially functional domain a family have and now I'm interested in something that approximates my partial function so I'm allowed some some some flexibility I want to minimize the alpha so that for some function f. you know that's that's the middle left and this inequality over here the value or the always lies between the given value and of times a given value right so so maybe if I'm allowed to flick a factor of 2 then I can find some function within my family this is generalization of that problem so this is approximate extension again I'm interested in the context of the problem is n.p. hard can approximate it and so on Ok So Ok so these are 2.
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Basic questions for this question a partial function extension. And a pops up all over the place I'm going to talk about 2 so the title of the talk was by someone content with applications someone talk about to applications I'm also interested in hearing your other places where you think this might be this might pop out this made this might be useful so but let me talk about 2 partly applications.
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So the 2 applications are going to be learning and testing. I'm guessing a lot of you in department in this department in Georgia Tech have seen these 2 before let me go through the formal definitions. So packed learning So what is packed learning back spending is basically I would like to learn some family of functions how do I do that so I'm going to say this family f. is back notable if recently given some function f. start within the family and any distribution mewe over this to mean right so what I get to see are samples from this to me and I'm going to see pop all your family samples within this domain and I would like to learn this fact at this function if Start right what.
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What this means is so I'm given a start as an oracle. So it is not an article I'm given I'm given as samples so so each of these essays is drawn i.a.d. from my own lines to mission you and your employer from samples this learning algorithm takes all of these samples.
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And the value for this for the samples and it has to spit out some function f. where this function f. satisfies the following criteria that if I pick an us this from the distribution you then the function has to kind of be at most as much as the value of f. start and time should be at least as much as if the writer has to bound the correct value Ok so back to learning science the problem.
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Probably approximately correct. This is not actually packed learning this is something called the math learning but for the purpose of the start the distinction is not really important so. Let me just stick to the easier. Ok rate so it is clear so basic from f. start and getting following number of samples given the samples I want an algorithm to.
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Come up with a function so that for any new sample with high probability the value I get lies between I mean the value of the function later what I spit out and 5 times where it's better Ok. So that's back to learning now how does how does partial function extension come to this and particularly how does it help in getting your bones for this so.
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Ok So supposing that I can show the following Supposing I showed that. That basically given f. and given my domain there exist 2 things there exists some value are strictly larger than one and some subset of my domain which satisfies these 2 criteria firstly that my subset is large so my subset is super polynomial in the dimension.
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And what the 2nd point says is that for this subset of the domain I can choose any values f. I between one another. So so for the for the subset I'm going to pick any values between one another and let's say that no matter what values I pick for for the subset.
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This is extended will to a function in my script if I can choose any values you want to know are there still exists a function in that family. Which extends is given partial function if I show you this then. It's not hard to see that this family as cannot be packed learned within a factor smaller than ours.
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And this is like a 2 line thing let me just show you what this is. So Ok So given these 2 things humans value on a given the Super Bowl I'm with a subset of the domain let me be the uniform distribution of this right I mean basically the point is that since any value between one an hour is good and gives me a function in script if the learning algorithm really cannot do much since the value would produce it has to be at most as much as a star you know the only thing you can do is that it must guess one at the next set right whereas the actual left out is could be as large as ours so I get a factor of 5 Ok so we show these 2 things if I show that they exist as large subset and these are so that any real use between one and are expendable to the human family that gives me a lower bound on learning.
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This is used implicitly in previous papers as well I'm just making it explicit and then used to slip it around. It's Ok is this any questions about this. Ok good so this is one application lower bound for packed learning I'm going to talk about property testing now right.
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So this is the 2nd application property testing and again let me just say we're probably testing is here so so so give us a family of a function the game or domain for the applications the domain I'm going to stick to is the discrete domain subsets of one through him.
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This time I'm also given some epsilon and in learning I knew that my function I was trying to learn was in the family it was in the family in testing I want to figure out whether it's in the family or not and so given this parameter and I'm going to say that a function is epsilon far from the family if basically epsilon fraction of the points have to be changed for it to be in the family.
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If for this function f. start I need to change epsilon times to the m. values for it to be part of the family of functions. Ok so. Now what I need to do the problem I have is that now I'm given Oracle access to after so there's no distribution and while there's no implicit distribution.
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I need. I can choose my queries and based on these queries as one through escape I get the values back I want to determine if a star is in the family or if it's you know epsilon far away from the family if it's just far away from the family but if it's somewhere in between these 2 I'm Ok I can do anything I want Ok.
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So this is taken up a hill probably testing and here it's easier to see where why partially function extension plays a role because after you know the testing algorithm passes as one through ascii it gets back values what it has at the end of all these queries is a partial function it has as once risquÃ© the values of these points and basically.
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If you know if the values of the points and the values I have can be extended to a function in the family then I had better say that yes this function in the family because I have no evidence other ways so yes so basically I need to figure out whether this partial function can be extended to something within the family or not.
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Could Ok so. Hopefully this is Ok. Let me skip to. Some of talk about well 3 different classes of functions. I would like or some other to some modern convex. So. Let me just put down the definitions I mean some are just says that if you put 22 sets together nothing magical happens as in t. do not mislead they don't complement each other so the value of the Union is that most the sum of the values of the 2 sets right.
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Some oddly has this property of diminishing marginal returns I mean this is a definition that I like i'm basically says if I add I to s. the game is larger than if I add it to a larger set in this case the said s. Union j.. And we all know what convex is someone else with this.
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Ok so so now I'm going to look at it. Like a passion function extension and probably testing learning with respect to these 3 classes of functions. Let me to suck a little bit of our previous work. So so the so the problem of partially function extensions. What I'm interested is the complexity of the problem and protest in figuring out for the time algorithms for figuring out if it's expendable or not that's not something which is studied earlier this is something that we're studying you know.
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Roughly for the 1st time. What clearly what has been studied earlier is learning and testing rate so so most of the previous results of learning and testing let me just say what these are so for some other 2 functions it's known that that that you can learn it to roughly a factor of or a square with them so can at all in previous work showed an upper bound of square root and log and so you can learn it within the factors created him not them and the short of the word want to screw them by logging so there's about a log square and difference and one of our contribution is going to be to tighten this gap between the 2 of.
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Was a dimension yes and is the number of House visions m. is the dimensions. Yeah yeah I mean it's. So so and so and will only appear only looking at partial function expansion of the ways is no way. Ok So Ok so for somebody to have the answer is approximately square to them the upper bone adds log.
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Divides by law again for somebody the functions for learning doesn't Ok so what what what what bulk and how we should as long as someone function is nice by nice I mean it has to be non-negative in monotone in this case you can show an upper bound or square with them so you can learn it within a square root and factor again and load one of to the one thirds.
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And. Then one that looked at the problem of testing some of the functions you know like given a function with some order or not and they gave a tester which makes these many queries Yeah. Yeah yeah sorry. Right that's a good question Ok so yeah because because because this upper bones were bones I mean it's good to ramble epsilon is less than one anyway Ok So rates so it's Epsilon to the power of minus square to 10 log.
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And I'll compare this down to something else. I want to say that this is a party later which I think is largely open testing as somewhat of a function in fact a lot of things to do with make you think that people have studied similar functions to death rate but I but I think there's a lot of work to be done.
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For complex functions people have this problem parsing function extension has been looked at in convex analysis I'm sure there are people over here who can tell me about this but anyway so basically in connection Alice is the largest in the complexity so usually they'll be my question function will be some infinite set and their interest in characterizing when it can be extended to a convex function.
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Which is different from our case because I'm working with a finite set and I'm looking at complexity so not really going to compare. To what's going on it now says but it has been studied and there are some you know balance between the 2. Ok so let me now talk about our results Ok for somebody to functions so the 1st thing we show is that it's going to complete the past function extension problem is going to complete that means 2 things firstly the kohen t. part of it means that there is a certificate of known extendibility rights if you give me a partial function.
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I can give you a poly time where if I will certify it of known membership insularity of non-extant ability to survive and then of course there can be complete mean that it's also in the heart you can't really enjoy it and you can't really verifiable model of the usual assumptions.
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For approximate extension we should tie downs of feet of luggage. For learning we saw all of this is tied together going to liberty to learn this for learning like we said so so the earlier Bond of the world was created and by law again we improve the stroller born of just square root them which unfortunately still leaves a gap of law game with the upper bound but but this is an improvement in what was known earlier.
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For testing we give a tester that makes these many queries just for comparison sake if this log one would have sort of outside the square root then this would be the same as the bound for some model of functions that I showed over here. This is the 1st non-trivial test affords about the functions which make these many queries and like I said it's it's stately better than the.
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Pound tester for some modeler functions that yes approximate extension reference my attention means that that that basically I'm allowed destructibility So I want to find the tightest alpha so that the function in the family you know is at most the given value and times is at least the given value.
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Is that the director for Ok Ok. Right so right so so so like I said so so we have we. For Saturday functions we make contributions to all 3. This is clearly not the last word for this I think there are better testers available I don't know what those are.
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For some model of functions so yeah so like I said someone who functions for somebody to function real to give this this characterization you know it's going to be complete I commuted on the proximity so one of the functions I we barely know anything about partial function extension.
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I I don't know whether it lies in here I don't know whether it whether it's n.p. hard whether it's in current b. I have absolutely no idea. A few things we can show is that supposing my partially function forms a 90 chain right so no set is contained within the other then.
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It doesn't matter what values you give me it is always expandable I can always give you some one function you only think of that that the sum of us will not be nice like we defined it will not be monotone nonnegative So if you just if you just added phases of some model t. definition.
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It can always be extended. Because of this and because of what we said earlier about large sets and you know values between one through our it follows that these general some of the functions cannot be learned within any factor. I have something to say about testing as well I'm just going to say that until later so you know forgive me for that right now.
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And Ok And lastly for convex function so convex functions. This is Ok so there's interest in the economy so again so is approximate extension is integrated even if you give me something which is not which cannot be extended to correct function acknowledge tell you what is the closest convex function to that partial function so this can be done in politics that's that's that's good Here's what but here's the interesting.
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Footnote to that maybe. So like I said there are many possible extensions rates or so here is a partial function extension there are many possible extensions there is a particular extension which has been studied in context analysis as well and is the one you think of when you see these points like this is the one that I think of at least.
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And what and the property that this partly extension satisfy is that you know within the context of these given points this extension is maximal. Any other any other extension you know will live below the these lines. Ok so if I if I so Ok so if I give you a partial function which is extended so this you know the existence canonical extension and I ask you well Ok so it's extended now here's this point now tell me the value of this canonical extension at this point it turns out that this question is empty heart so I mean I can tell you that there exists an extension that is a con canonical extension great but I can't tell you the value of these things you know and I'll define it only relate on a basically it's extension which is which is maximal within the convex hull and given that it's maximal within it's minimal outside.
[00:24:02]
If you basically have a lower envelop of the points and that's what. Ok this I thought was kind of curious. Ok anyway so so. A bunch of results I hope to drill down into some of these. Ok so I'm actually not going to talk about the convex functions part but I maybe I just need to some additive and some more letting those tools to maybe the more interesting parts.
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So Ok so so the what's about are 2 functions out. And this is this is the partial function extension theorem that it's going to be hard there's a tight bond of your game and approximate extension I'm not going to prove this I'm going to focus more on the learning interesting parts of it.
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But all of these results they're based on like I said this characterization of non extendibility So what what does this certificate of non extendibility say. What it says is that if and only if the special function cannot be extended then the following has to happen there exists a set we just contained in a bunch of other sets and the value of this set is strictly larger than the sum of the values of you know these are the sets that contain the set in the union.
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Let me take a quick example for this for example this partial function cannot be extended to us about it a function because Ab is contained in the union of a c. in b.c. but the value at a.b. strickly larger than one percent right. Ok so this is yes this is going to stick this in your mind this is going to be we were going to use this basically a set has to be contained in the universe of other sets and the value of that set has to be strictly larger than the sum of values.
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Of those other sets right so these 2 conditions are to be satisfied. Ok so this characterizes non extendibility 2 separate functions Ok but Ok so now let's see how this gets used in learning and testing. So this is not extended Will I'm going to skip the proof of this theorem.
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If you don't approximation algorithms a bunch this is the state of log and comes from set covered items of Set Cover. Yeah Ok. Great so right so so about the learning go around and again recall what you said earlier that for learning your bones apart from the stations we show that there's a large family this is one through our lab about a list of the schools for that.
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So what this uses is is this is. I this comment on real things called our cover free families so let me define what these are so. Ok So looking at subsets of one through them I will say that some family of sets subsets is an arc of a free if we simply no set is contained in the union of our other sets.
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So if I pick any artist one sets then one of them is contained in the union not the others rights this is called an I mean no are sets covers any other set Ok so this is called an arc of the family these were I guess as is true of most in common to these were studied berdache as well who presented some bones.
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Rate Ok so so and so slightly definition of out of the family is so not booked I pick a place once that's And I'm saying that if this is contained Then fact they must be strictly larger than are Ok this is a state a different definition and same thing.
[00:27:48]
And here's the basic lemma So basically if I start off with an arc of a free family then this partial function then I can choose any values so this r. is this parameter over here as long as it use any values within one are for the sets in this family this cannot I mean this is always expendable to us America function.
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So why is that. Let's look at activation supposing it's not expendable then 2 things have to exist and they have to exist some k. plus one set so that firstly there are from from from a certificate give us 1st is contained in the union of these cases and the value over here what we strictly larger than the sum of these values right so this is our certificate of knowledge and ability this we said has to exist.
[00:28:37]
But there's a problem firstly I know that there's an archive in family right because this arc of a free family because this is contained in the union of k. sets they for k. must be strictly to the not right. But the 2nd thing is I know that all of these values are at least one right so then the value of this must be strictly greater than our close one but that's a problem because I wanted to sing values between one another it's so the certificate or non explain really cannot exist so basically if I start off with an arc of a free family and I assign values between one and not I always get something which extends to us as America function.
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Create. Now I'm just going to plug in some recent work on our families so what was shown is that as long as our is little or square root time there exists an arc of a large article family was with the size super polynomial and. Yeah I mean basically you plug all the same and you will get the square root of m. lower bound on learning some other to functions right so basically this shows that there exists a super polynomial family that for any value between one and squared him is extended with and therefore you get a line about.
[00:29:58]
Ok so. This is kind of nice because we could I mean you know like you kind of looked around for bounds in our families and it turns out these guys had you know just obtained this nice result so we kind of plugged it in for a lot you can get results.
[00:30:19]
Ok so let me talk a little about testing I'm not going to get into technical details but I'm just going to show what a test what the tester does it I want you to do proofs of this so this is a characterization it's not that important right now and we'll use this diagram a little bit later on as well so you know I mean my empty set is at the bottom of my set of my you know the one through Am is at the top and these levels correspond to subsets of a fixed size.
[00:30:53]
So here's what the tester does it's going to pick out this bind which is you know embedded 2 plus square root them logo and what absolute on a number 2 minus and then just as the following so it chooses a point in this band it looks at all subsets of this you know we're which lie again within within the span.
[00:31:14]
It carries all of the sets and this gives me a partial function it gives me a bunch of points with values values for each of the points I'm just going to check my characterization of severity functions for you know this partial function. If it's I was fired then I have to reject.
[00:31:36]
And I do this one of the absolute times if I haven't rejected at the end and I just accept it so again I mean basically the core of this is you get a partial function you check the characterisation if it's if it's satisfying then you detect. I want to point out that this is a horribly inefficient tester.
[00:31:58]
Because I'm picking something like these many sets at a time and then I'm running this. The sky tradition which basically looks at a whole bunch of subsets see if it's contained and so on. So even with these many Querrey is. If there's a less computational expensive test that would be nice as well.
[00:32:20]
But in terms of credit complexity this at least matches what we have as a model function Ok. I think I'm going to skip the convex function part of it. So I mean I guess the only point that I want to make over here is that. So let me put all of this over here I mean so for comics functions what happens is that you can read on an LP You can read on the dual of the LP and it turns out so this canonical extension that I spoke about basically it says that.
[00:33:00]
There's no notation on the board let's not get lost in that basically for this dual LP There's a dual polyhedron my dual polyhedron has some vertices right and I am and this trying to find the vertex which maximizes some linear function Ok so basically given a point x. if I want the value of this canonical extension I want to find the vertex of this jewel polyhedron which maximizes sum function this is this is a linear function but anyway.
[00:33:35]
For the empty hardness here's the problem. I don't know how to optimize over vertices of a polyhedron I can automate over the poly Hadrian that's fine right but if you just listed below vertices I don't know how to do that which which basically will give me the n.p. hardness over here.
[00:33:55]
Because I just have to optimize over to Caesar in there for him to. Sorry that was too vague I can say a few words after and your question were. So so so this function surfaces so I mean this is a linear function in x. So this. Yes yes yes yes so this is the kid vertex this is so these these values are act you know this is at the kids' vertex both of these are at the airport x..
[00:34:35]
O.. How is the 1st new k. term defined. I mean so this is the value at the vertex rate I mean. I'm not sure I understand the question fully so so so so so did muse on the why is defined the vertices and m. and m. and m. maximizing new plus x. transpose.
[00:35:13]
Whatever this value of the vertex is so. Yeah. Sure Ok. Yeah sure sure Ok. First No what makes this n.p. hard is I don't have a polytope is there were I have is a polyhedron. Yeah. Rate No So again I can maximize a linear function over polyhedron that's fine but I but what I want to do is find is not find the optimal value is for is to find the optimal vertex So for example it could be that the optimal value goes off into infinity which is which which which will actually be the case but I don't want to go off into infinity to.
[00:36:10]
Yes. Exactly exactly exactly exactly so I don't know which vertex you know because yeah I mean if I do some takes if I do whatever the what except a well just what I will often definity may not be the optimal vertex Ok. Could again you know please stop me from audience to run and we're going to.
[00:36:39]
Blah blah blah Ok So Ok so this is talk about some of the functions. So. These are some model of functions again to this is a definition I just want to. Like the way that I'm going to looking at some of the function is in terms of these squares triangles diamonds Mr not ultra diamonds and basically the the this probably says that the sum of these 2 values must be at least some of these 2 values it's on too many points as we have used some of the talk in the and.
[00:37:11]
And the bottom point. This is what I said about property testing and let me just refine this a little bit because what we lose we present a lot of bones for testing some model of functions but it's for a particular class of property testers So let me just again this is this should be fairly natural.
[00:37:32]
So so the class I'm interested to know what donors proximity of oblivious testers which just means that basically they don't depend upon epsilon where absolutely the proximity to my class let me just you know my my purity ignores Epsilon to begin with so it just looks at the function so it has a basic test which to main depends only on script f. and the domain b. for an example of that let's look at the somewhat sub sub model of the tester.
[00:38:02]
This is a square tester and it does the obvious thing you know like what is the easiest thing we do you have this condition about somewhere like this so you randomly pick a square you pick s 2 points which are not in this you get a square and then you just check if the square satisfies the somewhere Larry condition right if the sum of these 2 values is actually the sum of these 2 animals this does not depend upon epsilon at all right I mean this is just independent of epsilon.
[00:38:31]
And this is the square tester for some levity which I just you know talked about in terms of the parts so it. Yes that's exactly right thank you right so so basically what appealed it does is it has this test independent of epsilon but the detection probably the probably that I actually find you know a bad said given that there are these bad sets will depend upon epsilon clearly rate everything in the sum model that it's not going to find any.
[00:39:07]
So the what I call the action property there's a basic test addiction probably tells me how likely I am in terms of Epsilon to find something which smells wrong which which is incorrect right and if there's my objection probably will be well to get a constant problem of you know finding out what this function is is not some model I repeat the states when I would be one of our detection probably teams right so this is what is called a proximity oblivious Tester a few things I want to point out that each of these basic tests are independent of the previous one great so when I do a basic test when I pick the next square I don't look at the square I pick previously entrain compare the values over there.
[00:39:46]
That's going to be important. Is Ok Do we understand what the beauty is roughly. So for again so for example for the for the square test or what they showed was if you just randomly pick a square do the steps the detection probably is epsilon to the square root m. Le Guen so therefore you need to understand one of these many times and we get that found.
[00:40:13]
This is what is called a 4 query beauty because the basic test consists of creating these 4 points. Yes yes. Yes exactly exactly exactly that exactly right so I mean there is this there is this I mean basically we're mapping violations to Epsilon you know how. Things should be and this exactly gives us what what relations.
[00:40:50]
Ok good so yes and so this is what is known as the for Kerry beauty. Let me give another example of a puti So this is a modern mystery tester and again you know. What is the natural thing you do you want to test as something more tone or not I'm going to pick unless I'm going to be can I not nest compare the 2 values right and the value it as you know should be larger than the other one.
[00:41:15]
And for this gold record all showed that the direction probably would have said over and therefore to get a constant probability of detection you just run this in what Absolom times Ok right good Ok so. So this hour we talked about here's what we show. So here's what was known only it is not what we show so so what was.
[00:41:45]
The one that showed was this for a very purity you just talked about which has a direction from will be as shown. So they give an upper bound which is fine. They give a lower bound for this particular square tested do you load one of epsilon Square. And but that means that there is this big gap between the 2 right so so.
[00:42:09]
Yes So this showed that addiction probably has at least this much because you know that's what the base to test run and the show that it's at most this much but between the 2 clearly there's a whole yes so we don't know what the right answer is. We're not going to close this bone but we will say something about this pond.
[00:42:30]
Ok so let me talk about our results these 2 have only talked about so far for. For testing here the 2 bones we showed So 1st we showed that any constant Querrey purity proximal of obvious tested including for example the square tester of said in one that must have taken probably order of epsilon Square right so any set any constant way to test it has to be run at least $1.00 to $1.00 absolute squared times.
[00:43:01]
The 2nd thing we show is that yes. Yes. Yes yes yes yes exactly I mean there might be an 8 you know point as to which does better but the sort that it doesn't. The other thing we show which is I guess. Well Ok so the other thing we show is that even with the subjects financial any subbasements inquiry purity test or anything which has less than well to do a little off.
[00:43:36]
Still cannot have linear detection probably so selegiline as the holy grail you would like to run your purity you want to get epsilon times this shows that along at a subway or some expansion you have to you have to at least run it want to would have slowed to the one or $1.00 times so you can get lean years beauty's as long as you have some books mentioned.
[00:43:59]
The 2 things on c 1011 which I mentioned where the general is of the lower bound of the Shire the more that there is was for just the parts list greatest are we doing the same down but for this more general class of concentrate if you notice. The other thing the other thing is that this new alone separates someone like you from uni actually so so for linearity remember we had this we had this this this beauty which which had direction probably.
[00:44:27]
Well it had epsilon of where I am if you run it m. times then you get epsilon detection probability we show that was a multi You cannot have that right so this separates the 2 and it's going to mention this I won't get into details but basically this holds for a class of functions called me a traitor and functions as well so these are the 1st lower bounds for me trying functions.
[00:44:48]
The subclass of some order functions but let me not get into that. Ok so what I'm going to do now is talk a little bit about this this 2nd result again like the goal is going to be to show some pretty pictures and maybe give some idea of what the Lord wants for this little exercise should be an opportunity to show your tester and let me show you what you were born for just as you like as well Ok so here's the pyramid I mean let's with everyone what I see you know something is a 1.2 you know one of a question which I always had where they had one point to come from you know like 2 is nice 5 is nice war where did one point to come from so Ok so I'm going to show you and point to comes from.
[00:45:34]
So what we're going to do is like this is almost or lower bronze work is that I'm going to construct this large family of functions the it's going to put a parent raised by this art with our. Art So our sets of size and weight are they true for every set of size and by 2 I'm going to have a function in my family.
[00:45:53]
Which will satisfy the following properties that basically I'm going to draw this are uniformly at random right so my test is going to be presented with some effort. And basically this gesture in order to reject the function to say that this function is not I mean like all of these are going to be not submodel right but to reject this function the test is basically going to have to find 2 sets in b. for which the union gives me out.
[00:46:21]
Right. But the problem is our is one uniformly at random so basically it's not going to be really high chance of finding these things so so if I have a q. query puti. Well so I mean the number of unions that I get is going to be q. squared right so they're probably rejections going someplace you squared over the number of choices that I have for our which is 2 to the m. So there's going to be a probably reduction so if q. is mentioned then destroy your addiction is going to be 2 to mind Sam.
[00:46:52]
Ok so so far so good. What I will also show is that each of these functions is this is far from some mortality to the minus 5 m. by 6 so this is the fraction remember which means a number of points which which changes to the m I 6.
[00:47:12]
So as a function of epsilon this is what it turns out to be because 5 m. by 6 you rated you raised 1.2 you get to a minus. This is roughly what happens there are 3 things to show Firstly the construction of these functions secondly that basically I must find in b. in order to reject the you know the function and 2nd and thirdly this distance so let me start with the construction and the other 2 are going to follow very naturally so we construct the family would satisfy these these these properties.
[00:47:53]
So here is. My diagram again for subsets of one through m.. I'm going to choose a more stylized depiction So here's my out are as is am I do this is all subsets of r. and there is no empty set at the bottom. Is of a construction is going to look like I'm going to pick 3 sets you want to do with 3 all of which are subsets of odd they have size them by 3 and the property is that the union of any 2 will give me out.
[00:48:22]
So the union of if you want to need to gives me are so so so these areas are all sets which are which contain it and are contained within our rights and when you hear some you hear me. As an example let's see if r. is one through 6 these are this is one choice for a Wanted to a 3 you can check that the unifying gives me yet.
[00:48:45]
And let's just focus on one of those diamonds over there. So like I said this is all set that contains a one and is contained by are the size of this is they 4 to the m. by 6 right so I mean in my 3 elements are fixed I can choose to include or exclude in my 2 minus and my 3 which is in my 6.
[00:49:09]
Ok so here's what my function looks like so given the fixed odds this is what my function looks like basically everywhere it's linear except within these 2 diamonds it's the size of the set minus one Ok. And basically if you pick it it's not some model because if you pick any point inside you know.
[00:49:35]
Above a one in any point with about it too and you draw the corresponding square that square you know does not satisfy the condition we had earlier ill give you this one I'm not going to go through the matter here but that's what we're so so so so so so any point over here any point over here will give you a problem something which is not expendable so this is not some model of I guess the the interesting thing is that if you leave out any of these that if let's say you just did that everywhere else it was as a 1st minus one except for superset of a 2 then that would in fact be somewhat Hiller which means that the distance to somewhere like to how many points need to be changed is exactly you know one of these shaded areas which is to do the m. by 6.
[00:50:22]
I mean I've shown an upper bound on the distance you can also show lower bounds but let me just not do that for now. Ok so so show the construction I've showed this distance thing now we just need to show this. This is not so hard because I mean everywhere else the function is defined as the sailor said that's linear that skill is the mortal hate the only way that you can get a certificate of non somewhat Latapy is if you get a square like this right which means that for basically a test in order to get a certificate of non-celebrity has to pick a set over here it has to pick a set over here.
[00:51:02]
A necessary condition for that is that it must pick 2 sets for which the union gives me out which is exactly what I'm stating here and this gives me pounds and so on I'm skipping a lot of details but please give some idea on the door on a clear question about any of this Ok.
[00:51:34]
I'm going to stop over here a bit with the stuff I just want to mention a few things so these are a result I talked about just when the stated again more generally we talked about convex and some of the functions I want to say that this is a kind of theme that that can generate I have been working on there's a bunch of other results of examples we have results of coverage functions as well for some model of functions defined not just on your subsets of one through m. but on General Latta sees as well.
[00:52:07]
I'm happy to mail you the paper and I'm happy to talk about it as you like but let me go into what I think is the interesting part which is the huge number of open questions. Right so like I said it was somewhat late extension so we said something about testing we said something we're learning we don't know a lot about some of the extension rates are just you know given given a passion function is this going to be standard somewhat of a function or not I don't know whether it's n.p. I don't know whether it's going to be I don't know whether it's n.p. hard I don't know much about that and I would there's something that we did invest some time on so you know if anybody has any ideas I'm happy to hear those.
[00:52:44]
Like you said even for this issue the one that tested there's a large gap between upper and lower bounds but in general more generally lower bones better upper bounds anything is welcome Similarly for some other 2 functions and more generally you know. The special function extension I think if it comes up in other places well I'm happy to hear about other applications you may have.
[00:53:08]
But I'll stop here thanks. So. Yeah yeah. Yes. So no. All. This is just. You know yeah so but we are but but the harness doesn't translate so so so that question is hard it's hard yeah so that way that so so I mean it's a so they're basically asking a very structured So for example if you just look at a graph character.
[00:53:59]
That is hard that is hard to hear you have so many children functions for middle class the meter reading functions we can show that the problem is hard. You know exactly exactly yes so I. Wow. Yeah that's I mean so what. This. Yeah actually that's why we start looking at convexity.
[00:54:36]
We don't know how to use it like we weren't able to translate what we know what connects into somewhat later I mean the problem that for can we do life extension you need to know I have a specific bunch of points where the values are but. Just what.
[00:54:58]
I like. I like is a source of this problem I'm just given a set of points right suppose function and given the one through the end. I don't have freedom of choosing for their points. I think yes but I think that might be too general a statement to be so I mean Ok so what I did not save over here was that the underlying thing we use in all of these proofs is yeah I was lemma I was me Max thing and we would basically translates.
[00:55:55]
Like a randomized construction into lower bones but but that kind of progress is as you said that if you choose randomly over and over large enough said then that will give you some kind of lower bounds. I don't know if like I said I think that statement itself might be a little too general to be of more specific use I mean it's a good thing to keep in mind.
[00:56:18]
But you have to give you know give more details for it to be actually. I think here thanks.