(This is the sixth in a series of posts on online optimization techniques and their “applications” to complexity theory, combinatorics and pseudorandomness. The plan for this series of posts is to alternate one post explaining a result from the theory of online convex optimization and one post explaining an “application.” The first two posts were about the technique of multiplicative weight updates and its application to “derandomizing” probabilistic arguments based on combining a Chernoff bound and a union bound. The third and fourth post were about the Follow-the-Regularized-Leader framework, and how it unifies multiplicative weights and gradient descent, and a “gradient descent view” of the Frieze-Kannan Weak Regularity Lemma. The fifth post was about the constrained version of the Follow-the-Regularized-Leader framework, and today we shall see how to apply that to a proof of the Impagliazzo Hard-Core Lemma.)
1. The Impagliazzo Hard-Core Lemma
The Impagliazzo Hard-Core Lemma is a striking result in the theory of average-case complexity. Roughly speaking, it says that if is a function that is “weakly” hard on average for a class of “efficiently computable” functions , that is, if, for some , we have that
then there is a subset of cardinality such that is “strongly” hard-on-average on , meaning that
for a small . Thus, the reason why functions from make a mistake in predicting at least a fraction of the times is that there is a “hard-core” set of inputs such that every function from makes a mistake about 1/2 of the times for the fraction of inputs coming from .
The result is actually not literally true as stated above, and it is useful to understand a counterexample, in order to motivate the correct statement. Suppose that contains just functions, and that each function differs from in exactly a fraction of inputs from , and that the set of mistakes are disjoint. Thus, for every set , no matter its size, there is a function that agrees with on at least a fraction of inputs from . The reason is that the sets of inputs on which the functions of differ from form a partition of , and so their intersections with form a partition of . By an averaging argument, one of those intersections must then contain at most elements of .
In the above example, however, if we choose any three distinct functions from , we have
So, although is weakly hard on average with respect to , we have that is not even worst-case hard for a slight extension of in which we allow functions obtained by simple compositions of a small number of functions of .
Theorem 1 (Impagliazzo Hard-Core Lemma) Let be a collection of functions , let a function, and let and be positive reals. Then at least one of the following conditions is true:
- ( is not weakly hard-on-average over with respect to a slight extension of ) There is a , an integer , and functions , such that
- ( is strongly hard-on-average over a set of density ) There is a set such that and
Where is equal to or depending on whether the boolean expression is true or false (the letter “” stands for “indicator” function of the truth of the expression).
2. Proving the Lemma
Impagliazzo’s proof had polynomial in both and , and an alternative proof discovered by Nisan has a stronger bound on of the order of . The proofs of Impagliazzo and Nisan did not immediately give a set of size (the set had size ), although this could be achieved by iterating their argument. An idea of Holenstein allows to prove the above statement in a more direct way.
Today we will see how to obtain the Impagliazzo Hard-Core Lemma from online optimization, as done by Barak, Hardt and Kale. Their proof achieves all the parameters claimed above, once combined with Holenstein’s ideas.
We say that a distribution (here “” stands for probability measure; we use this letter since we have already used last time to denote the Bregman divergence) has min-entropy at least if, for every , . In other words, the min-entropy of a distribution over a sample space is defined as
The uniform distribution over a set has min-entropy , and all distributions of min-entropy can be realized as a convex combination of distributions that are each uniform over a set of size , thus uniform distributions over large sets and large-min-entropy distributions are closely-related concepts. We will prove the following version of the hard-core lemma:
Theorem 2 (Impagliazzo Hard-Core Lemma — Min-Entropy Version) Let be a finite set, be a collection of functions , let a function, and let and be positive reals. Then at least one of the following conditions is true:
- ( is not weakly hard-on-average over with respect to ) There is a , an integer , and functions , such that
- ( is strongly hard-on-average on a distribution of min-entropy ) There is a distribution of min-entropy such that
Under minimal assumptions on (that it contains functions), the min-entropy version implies the set version, and the min-entropy version can be used as-is to derive most of the interesting consequences of the set version.
Let us restate it one more time.
Theorem 3 (Impagliazzo Hard-Core Lemma — Min-Entropy Version) Let be a finite set, be a collection of functions , let a function, and let and be positive reals. Suppose that for every distribution of min-entropy we have
Then there is a , an integer , and functions , such that
As in previous posts, we are going to think about a game between a “builder” that works toward the construction of and an “inspector” that looks for defects in the construction. More specifically, at every round , the inspector is going to pick a distribution of min-entropy and the builder is going to pick a function . The loss function, that the inspector wants to minimize, is
The inspector runs the agile online mirror descent algorithm with the constraint of picking distributions of the required min-entropy, and using the entropy regularizer; the builder always chooses a function such that that
which is always a possible choice given the assumptions of our theorem.
Just by plugging the above setting into the analysis from the previous post, we get that if we play this online game for steps, the builder picks functions such that, for every distribution of min-entropy , we have
We will prove that (1) holds in the next section, but we emphasize again that it is just a matter of mechanically using the analysis from the previous post. Impagliazzo’s proof relies, basically, on playing the game using lazy mirror descent with regularization, and he obtains a guarantee like the one above after steps.
What do we do with (1)? Impagliazzo’s original reasoning was to define
and to consider the set of “bad” inputs such that . We have
The min-entropy of the uniform distribution over is , and this needs to be less than , so we conclude that happens for at most a fraction of elements of .
This is qualitatively what we promised, but it is off by a factor of 2 from what we stated above. The factor of 2 comes from a subsequent idea of Holenstein. In Holenstein’s analysis, we sort elements of according to
and he lets be the set of elements of for which the above quantity is smallest, and he shows that if we properly pick an integer and define
then will be equal to for all and also for at least half the , meaning that for at least a fraction of the input. Since this is a bit outside the scope of this series of posts, we will not give an exposition of Holenstein’s argument.
3. Analysis of the Online Game
It remains to show that we can achieve (1) with of the order of . As we said, we play a game in which, at every step
- The “inspector” player picks a distribution of min-entropy at least , that is, it picks a number for each such that .
- The “builder” player picks a function , whose existence is guaranteed by the assumption of the theorem, such that
and defines the loss function
- The “inspector” is charged the loss .
We analyze what happens if the inspector plays the strategy defined by agile mirror descent with negative entropy regularizer. Namely, we define the regularizer
for a choice of that we will fix later. The corresponding Bregman divergence is
and we work over the space of distributions of min-entropy .
The agile online mirror descent algorithm is
so that is the uniform distribution, and for
Solving the first step of agile online mirror descent, we have
Using the analysis from the previous post, for every distribution in , and every number of steps, we have the regret bound
and we can bound
where, in the last step, we used the fact the quantity in parenthesis is either 0 or which is , and that because is a distribution.
Overall, the regret is bounded by
where the last inequality comes from an optimized choice of .
Recall that we choose the functions so that for every , so for every
and by choosing of the order of we get
It remains to observe that
so we have that for every distribution of min-entropy at least it holds that
which is the statement that we promised and from which the Impagliazzo Hard-Core Lemma follows.
4. Some Final Remarks
After Impagliazzo circulated a preliminary version of his paper, Nisan had the following idea: consider the game that we define above, in which a builder picks an , an inspector picks a distribution of the prescribed min-entropy, and the loss for the inspector is given by . We can think of it as a zero-sum game if we also assign a gain to the builder.
If the builder plays second, there is a strategy that guarantees a gain that is at least , and so there must be a mixed strategy, that is, a distribution over functions in , that guarantees such a gain even if the builder plays first. In other words, for all distributions of the prescribed min-entropy we have
Nisan then observes that we can sample functions and have, with high probability
and the sampling bound on can be improved to order of with the same conclusion.
Basically, what we have been doing today is to come up with an algorithm that finds an approximate solution for the LP that defines the optimal mixed strategy for the game, and to design the algorithm is such a way that the solution is very sparse.
This is a common feature of other applications of online optimization techniques to find “sparse approximations”: one sets up an optimization problem whose objective function measures the “approximation error” of a given solution. The object we want to approximate is the optimum of the optimization problem, and we use variants of mirror descent to prove the existence of a sparse solution that is a good approximation.