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At the conference I was at last week, I was asked by a mathematician about the exact statement of the Impagliazzo HardCore Lemma, and whether I could state it without saying “algorithm.”
So I gave the following statement:
Theorem 1 (Impagliazzo HardCore Lemma — Basic Version) Let be a class of boolean functions , let be an arbitrary function, and let be arbitrary parameters.
Then at least one of the following conditions is true:
 is globally well approximated by a simple composition of a few functions from .
That is, there are functions , and a “simple” function such that
 is locally almost orthogonal to .
That is there is a subset of density at least such that for every ,
The theorem has been strengthened in a couple of ways in work of Klivans and Servedio and Holenstein. The “complexity” parameter needs to be only which is conjectured (or known?) to be tight, and the density of the set can be made , which is tight. In all proofs is really simple, being more or less a threshold function applied to the sum of the .
Stating the result in this way (which is, of course, standard) raises a few questions that I find rather interesting.
In the last post, I stated the following generalization of the weak regularity lemma:
Theorem (Low Complexity Approximation, TTV Version) Let be a probability space, a bounded function, a collection of bounded functions , and an approximation parameter.
Then there is a function such that
 has low complexity relative to : there are functions and coefficients such that
 is indistinguishable from by , that is,
(Last time, I mentioned that our proof handled only boolean functions ; now we can handle arbitrary bounded functions, and with an “energydecrease” style proof, this will appear in the next online revision of the paper.)
This seems to be a useful tool, limited only by one’s creativity in choosing the functions and then making use of the properties of .
As already discussed,
 if one takes to be the edges of a complete graph, and the set of indicator variables of cuts, then the existence of gives the weak regularity lemma of Frieze and Kannan; and
 if one takes to be the set of circuits of size at most , and normalizes and to be probability distributions, one gets that for every probability distribution of high entropy there is a (nonuniformly) efficiently samplable and computable distribution that is indistinguishable from by circuits of size .
In this post I’ll show how to use it to give short proofs of the Hardcore Lemma of Impagliazzo and the Dense Model Theorem of Green, Tao and Ziegler. Both proofs also have, at least in hindsight, a sense of “inevitability,” meaning that given the LowComplexity Approximation Theorem, and given what we want to prove, both proofs get to the point in a most economical and natural way.
 The Impagliazzo Hardcore Lemma. We have already mentioned that if is “hardonaverage” for , then cannot be an approximation in the sense of being close to on most inputs. What, then, about the points on which and differ? They form an Impagliazzo Hardcore Set for , as described next.
Let be a function that is weakly hard on average for a class of algorithms . Suppose, specifically, that for every algorithm of complexity relative to we have
and, more generally, for fractional , we have
Then, construct an approximating function of complexity
relative to and such that and are indistinguishable by . Note that, even though is “indistinguishable” from , it is also “far” from , as in (1).Define a probability distribution that assigns to point a probability proportional to . (If were boolean, this would be the uniform distribution over points on which and differ.) Then this distribution is dense in the uniform distribution, meaning that every point has probability at most . Observe also that we have
for every , because and have the same sign and , so we have
and so is a hardcore distribution, because the above expression is equivalent to
 The Dense Model Theorem. Suppose that is a pseudorandom set with respect to functions that have bounded complexity relative to , and let be a dense subset of , .
To find a dense model of , we take to be the characteristic function of , and we let be the lowcomplexity approximation, but using the uniform distribution on as .. Now suppose for simplicity that is boolean, and that is the set of inputs of on which is 1. We want to argue that is a dense model of . By assuming without loss of generality that contains the allone function, we get from the indistinguishability of and that
and from the pseudorandomness of we have
and so and is indeed dense.
For the indistinguishability of and , take any function , and observe that
where we use both the indistinguishability of and under distribution , and the fact that the distributions and are indistinguishable by functions of bounded complexity.
This proof is appealingly intuitive, in the sense that if we expect to be indistinguishable from a large set, then when we try to approximate the characteristic function of we will end up with a low complexity function that is spread around much of , thus defining a dense model. It also shows that “relative” versions of the Regularity Lemma, such as the Regularity Lemma for subgraphs of an expander, may be derived from the regular Lemma by the above argument. A disadvantage of the argument is that it does not establish the stronger form of the Dense Model Theorem suggested by Impagliazzo, in which there is no set , but we require to have the “pseudodensity” requirement that for every lowcomplexity bounded function ,
which follows immediately if has density in a pseudorandom set , but that is a seemingly weaker property. (The relative Regularity Lemma in graphs had long be known to hold under such a pseudodensity assumption.)
In a previous post, I described abstract forms of the weak regularity lemma, in which we start from an arbitrary bounded function and an arbitrary set of “structured” bounded functions , and we construct an “approximatng” function that has “low complexity” relative to and is “indistinguishable” from by functions from . We had two proofs, both generalizations of arguments due to Frieze and Kannan: in one proof is not bounded, but its complexity is , where is the approximation parameter; in the second proof, is bounded but it has exponential complexity in the approximation parameter.
In a new paper by Madhur Tulsiani, Salil Vadhan, and I, we give a new proof that gives an approximating function that, at the same time, is bounded and has low complexity. This has a number of applications, which I will describe in a future post.
(Note that, in the statement below, is required to contain Boolean functions, rather than bounded ones. It is possible, however, to “reduce” the bounded case to the boolean case via the following observation: if is a family of bounded functions, is the family of boolean functions , and and are indistinguishable according to , then they are also indistinguishable according to .)
Theorem (Low Complexity Approximation, TTV Version) Let be a probability space, a bounded function, a collection of boolean functions , and an approximation parameter.
Then there is a function such that
 has low complexity relative to : there are functions and coefficients such that
 is indistinguishable from by , that is,
That is, is simply a linear combination of functions from , whose value is truncated to be between and .
Recall that, when we do not require to be bounded, can be constructed via the following algorithm:
Algorithm FK
 0;
 while such that
 return
And the analysis shows that, if we call , then . Setting shows that the algorithm stops within steps.
Our proof proceeds by doing exactly the same, but making sure at every step that is bounded.
Algorithm TTV
 0;
 while such that
 return
The problem with the old analysis is that now we could conceivably have a step in which for all , and so , and thus , and we have no energy decrease. To get to such a state, however, there must have been about steps in which changed in the same way as in the FriezeKannan case. I think there should be a sort of “amortized analysis” that would “charge” the lack of change of at certain steps to other steps in which it indeed changes, and establish that, for every time step ,
Unfortunately I don’t know how to construct such a proof. Our proof, instead, follows step by step Impagliazzo’s proof of the Impagliazzo Hardcore Set Lemma, which employs a very similar algorithm (in a quite different context). As shown by Klivans and Servedio, the algorithm in Impagliazzo’s proof can be seen as a boosting algorithm in learning theory (and every other known boosting algorithm can be used to prove Impagliazzo’s Hardcore Lemma); this is why we think of our proof as a “boosting” proof.
I must confess, I have never really understood Impagliazzo’s proof, and so I can’t say I really understand ours, other than being able to reproduce the steps.
The idea is to consider the quantity , which depends on and , and to see how it behaves summed over , for a fixed , and how it behaves for a fixed , summed over .
Suppose the algorithm is still running after steps. Then, because of the way we define the termination condition, we have, for every
and the crux of the proof is to show that for every we have
So if we sum (1) over and average (2) over , we get
and setting gives .
Inequality (2), the main step of the proof, is the part for which I have little intuition. One breaks the summation into groups of time steps, depending on the value of ; there are groups (because the value of changes by discrete increments of , and is between and ) and each one is shown to contribute , where is the number of time steps in the group.
It is perhaps instructive to translate the FriezeKannan proof to this setup. In the FriezeKannan algorithm, we have
and so
which is analogous to our inequality (2).
In the Impagliazzo hardcore set theorem we are a given a function such that every algorithm in a certain class makes errors at least a fraction of the times when given a random input. We think of as small, and so of as exhibiting a weak form of averagecase complexity. We want to find a large set such that is averagecase hard in a stronger sense when restricted to . This stronger form of averagecase complexity will be that no efficient algorithm can make noticeably fewer errors while computing on than a trivial algorithm that always outputs the same value regardless of the input. The formal statement of what we are trying to do (see also the discussion in this previous post) is:
Impagliazzo HardCore Set Theorem, “Constructive Version”
Let be a boolean function, be a size parameter, be given. Then there is a size parameter such that the following happens.Suppose that for every function computable by a circuit of size we have
Then there is a set such that: (i) is recognizable by circuits of size ; (ii) , and in fact the number of in such that is at least , and so is the number of in such that ; and (iii) for every computable by a circuit of size ,
Our approach will be to look for a “regular partition” of . We shall construct a partition of such that: (i) given , we can efficiently compute what is the block that belongs to; (ii) the number of blocks does not depend on ; (iii) restricted to most blocks behaves like a random function of the same density. (By “density” of a function we mean the fraction of inputs on which the function evaluates to one.)
In particular, we will use the following form of (iii): for almost all the blocks , no algorithm has advantage more than over a constant predictor in computing in .
Let be the union of all majority0 blocks (that is, of blocks such that takes the value 0 on a majority of elements of ) and let be the union of all majority1 blocks.
I want to claim that no algorithm can do noticeably better on than the constant algorithm that always outputs 0. Indeed, we know that within (almost) all of the blocks that compose no algorithm can do noticeably better than the always0 algorithm, so this must be true for a stronger reason for the union. The same is true for , with reference to the constant algorithm that always outputs 1. Also, if the partition is efficiently computable, then(in a nonuniform setting) and are efficiently recognizable. It remains to argue that either or is large and not completely unbalanced.
Recalling that we are in a nonuniform setting (where by “algorithms” we mean “circuits”) and that the partition is efficiently computable, the following is a well defined efficient algorithm for attempting to compute :
Algorithm. Local Majority
On input :
determine the block that belongs to;
output if ;
otherwise output 0
(The majority values of in the various blocks are just a set of bits that can be hardwired into the circuit.)
We assumed that every efficient algorithm must make at least a fraction of errors. The set of inputs where the Local Majority algorithm makes mistakes is the union, over all blocks , of the “minority inputs” of the block . (If is the majority value of in a block , then the “minority inputs” of are the set of inputs such that .)
Let be the set of minority inputs (those where our algorithm makes a mistake) in and be the set of minority inputs in . Then at least one of and must have size at least , because the size of their union is at least . If has size at least , then has all the properties of the set we are looking for.
It remains to construct the partition. We describe an iterative process to construct it. We begin with the trivial partition where . At a generic step of the construction, we have a partition , and we consider as above. Let be such that . If there is no algorithm that has noticeable advantage in computing over , we are done. Otherwise, if there is such an algorithm , we refine the partition by splitting each block according to the values that takes on the elements of the block.
After steps of this process, the partition has the following form: there are functions and each of the (at most) blocks of the partition corresponds to a bit string and it contains all inputs such that . In particular, the partition is efficiently computable.
We need to argue that this process terminates with . To this end, we define a potential function that measures the “imbalance” of inside the blocks the partition
and we can show that this potential function increases by at least at each step of the iteration. Since the potential function can be at most 1, the bound on the number of iterations follows.
A reader familiar with the proof of the Szemeredi Regularity Lemma will recognize the main ideas of iterative partitioning, of using a “counterexample” to the regularity property required of the final partition to do a refinement step, and of using a potential function argument to bound the number of refinement steps.
In which way can we see them as “finitary ergodic theoretic” techniques? As somebody who does not know anything about ergodic theory, I may not be in an ideal position to answer this question. But this kind of difficulty has not stopped me before, so I may attempt to answer this question in a future post.
The Impagliazzo hardcore set theorem is one of the bits of magic of complexity theory. Say you have a function such that every efficient algorithm makes errors at least of the times when computing on a random input. (We’ll think of as exhibiting a weak form of averagecase complexity.) Clearly, different algorithms will fail on a different of the inputs, and it seems that, intuitively, there should be functions for which no particular input is harder than any particular other input, per se. It’s just that whenever you try to come up with an algorithm, some set of mistakes, dependent on the algorithmic technique, will arise.
As a good example, think of the process of generating at random, by deciding for every input to set with probability and with probability . (Make the choices independently for different inputs.) With very high probability, every efficient algorithm fails with probability at least about , but, if we look at every efficiently recognizable large set , we see that takes the value 1 on approximately of the elements of , and so the trivial algorithm that always outputs 1 has a pretty good success probability.
Consider, however, the set of size that you get by taking the inputs such that plus a random sample of inputs such that . Then we can see that no efficient algorithm can compute on much better than of the inputs of . This is the highest form of averagecase complexity for a boolean function: on such a set no algorithm does much better in computing than an algorithm that makes a random guess.
The Impagliazzo hardcore theorem states that it is always possible to find such a set where the averagecase hardness is “concentrated.” Specifically, it states that if every efficient algorithm fails to compute on a fraction of inputs, then there is a set of size such that every efficient algorithm fails to compute on at least a fraction of the elements of . This is true for every , and if “efficient” is quantified as “circuits of size ” in the premise, then “efficient” is quantified as “circuits of size ” in the conclusion.
The example of the biased random function given above implies that, if one wants to prove the theorem for arbitrary , then the set cannot be efficiently computable itself. (The example does not forbid, however, that be efficiently computable given oracle access to , or that a random element of be samplable given a sampler for the distribution for uniform .)
A number of proofs of the hard core theorem are known, and connections have been found with the process of boosting in learning theory and with the construction and the decoding of certain errorcorrecting codes. Here is a precise statement.

Impagliazzo HardCore Set Theorem
Let be a boolean function, be a size parameter, be given. Then there is a such that the following happens.
Suppose that for every function computable by a circuit of size we have
Then there is a set of size such that for every function computable by a circuit of size we have
Using the “finitary ergodic theoretic” approach of iterative partitioning, we (Omer Reingold, Madhur Tulsiani, Salil Vadhan and I) are able to prove the following variant.

Impagliazzo HardCore Set Theorem, “Constructive Version”
Let be a boolean function, be a size parameter, be given. Then there is a such that the following happens.
Suppose that for every function computable by a circuit of size we have
Then there is a set such that: (i) is recognizable by circuits of size ; (ii) , and in fact the number of in such that is at least , and so is the number of in such that ; and (iii) for every computable by a circuit of size ,
The difference is that is now an efficiently recognizable set (which is good), but we are not able to derive the same strong averagecase complexity of in (which, as discussed as the beginning, is impossible in general). Instead of proving that a “random guess algorithm” is nearoptimal on , we prove that a “fixed answer algorithm” is nearoptimal on . That is, instead of saying that no algorithm can do better than a random guess, we say that no algorithm can do better than either always outputting 0 or always outputting 1. Note that this conclusion is meaningless if is, say, always equal to 1 on , but in our construction we have that is not exceedingly biased on , and if , say, then the conclusion is quite nontrivial.
One can also find a set with the same type of averagecase complexity as in the original Impagliazzo result by putting into a size sample of elements of such that and an equal size sample of elements of such that equals 1. (Alternatively, put in all the elements of on which achieves the minority value of in , then add a random sample of as many elements achieving the majority value.) Then we recover the original statement except that is exponential instead of polynomial. [Update: constructing is somewhat more complicated than we originally thought, the details are in the paper.]
Coming up next, the proof of the “constructive hard core set theorem” and my attempt at explaining what the techniques have to do with “finitary ergodic theory.”
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