In this post we will construct a “spectral sparsifier” of a given hypergraph in a way that is similar to how Spielman and Srivastava construct spectral graph sparsifiers. We will assign a probability to each hyperedge, we will sample each hyperedge with probability , and we will weigh it by if selected. We will then bound the “spectral error” of this construction in terms of the supremum of a Gaussian process using Talagrand’s comparison inequality and finally bound the supremum of the Gaussian process (which will involve matrices) using matrix Chernoff bounds. This is joint work with Nikhil Bansal and Ola Svensson.
This is the second in a series of posts explaining a result on hypergraph sparsification that uses Talagrand’s work on Gaussian processes.
In the previous post we talked about Gaussian and sub-Gaussian processes and generic chaining.
In this post we talk about the Spielman-Srivastava probabilistic construction of graph sparsifiers. Their analysis requires a bound on the largest eigenvalue of a certain random matrix, that can be derived from matrix Chernoff bounds.
We will then make our life harder and we will also derive an analysis of the Spielman-Srivastava construction by casting the largest eigenvalue of that random matrix as the sup of a sub-Gaussian process, and then we will apply the machinery from the previous post.
This will be more complicated than it needs to be, but the payoff will be that, as will be shown in the next post, this more complicated proof will also apply, with some changes, to the setting of hypergraphs.
Welcome to phase two of in theory, in which we again talk about math. I spent last Fall teaching two courses and getting settled, I mostly traveled in January and February, and I have spent the last two months on my sofa catching up on TV series. Hence I will reach back to last Spring, when I learned about Talagrand’s machinery of generic chaining and majorizing measures from Nikhil Bansal, in the context of our work with Ola Svensson on graph and hypergraph sparsification. Here I would like to record what I understood about the machinery, and in a follow-up post I plan to explain the application to hypergraph sparsification.
I am recruiting for two postdoctoral positions, each for one year renewable to a second, to work with me at Bocconi University on topics related to average-case analysis of algorithms, approximation algorithms, and combinatorial constructions.
The positions have a very competitive salary and relocation benefits. Funding for travel is available.
Application information is at this link. The deadline is December 15. If you apply, please also send me an email (L.Trevisan at unibocconi.it) to let me know.
ITC replaces the International Conference on Information Theoretic Security (ICITS), which was dedicated to the same topic and ran 2005-2017. ITC can be seen as a reboot of ICITS with a new name, a new steering committee and a renewed excitement. (beware: there is a fake website for ICITS 2019 created by a known fraudulent organization)
The first ITC conference will take place in Boston, MA on June 17-19, 2020 (just before STOC). The submission deadline for ITC 2020 is Dec 16, 2019 and the call for papers (including a nomination procedure for the greatest hits track) is available here: https://itcrypto.github.io/2020.html
Please submit your best work to ITC 2020! We hope to see many of you there!
In this post we return to the generic form of the FTRL online optimization algorithm. If the cost functions are linear, as they will be in all the applications that I plan to talk about, the algorithm is:
where is the convex set of feasible solutions that the algorithm is allowed to produce, is the linear loss function at time , and is the strictly convex regularizer.
If we have an unconstrained problem, that is, if , then the optimization problem (1) has a unique solution: the such that
and we can usually both compute efficiently in an algorithm and reason about effectively in an analysis.
Unfortunately, we are almost always interested in constrained settings, and then it becomes difficult both to compute and to reason about it.
A very nice special case happens when the regularizer acts as a barrier function for , that is, the (norm of the) gradient of goes to infinity when one approaches the boundary of . In such a case, it is impossible for the minimum of (1) to occur at the boundary and the solution will be again the unique in such that
We swept this point under the rug when we studied FTRL with negative-entropy regularizer in the settings of experts, in which is the set of probability distributions. When we proceeded to solve (1) using Lagrange multipliers, we ignored the non-negativity constraints. The reason why it was ok to do so was that the negative-entropy is a barrier function for the non-negative orthant .
Another important special case occurs when the regularizer is a multiple of length-squared. In this case, we saw that we could “decouple” the optimization problem by first solving the unconstrained optimization problem, and then projecting the solution of the unconstrained problem to :
Then we have the closed-form solution and, depending on the set , the projection might also have a nice closed-form, as in the case that comes up in results related to regularity lemmas.
As we will see today, this approach of solving the unconstrained problem and then projecting on works for every regularizer, for an appropriate notion of projection called the Bregman projection (the projection will depend on the regularizer).
To define the Bregman projection, we will first define the Bregman divergence with respect to the regularizer , which is a non-negative “distance” defined on (or possibly a subset of for which the regularizer is a barrier function). Then, the Bregman projection of on is defined as .
Unfortunately, it is not so easy to reason about Bregman projections either, but the notion of Bregman divergence offers a way to reinterpret the FTRL algorithm from another point of view, called mirror descent. Via this reinterpretation, we will prove the regret bound
which carries the intuition that the regret comes from a combination of the “distance” of our initial solution from the offline optimum and of the “stability” of the algorithm, that is, the “distance” between consecutive soltuions. Nicely, the above bound measures both quantities using the same “distance” function.
We now discuss how to view proofs of certain regularity lemmas as applications of the FTRL methodology.
The Szemeredi Regularity Lemma states (in modern language) that every dense graph is well approximate by a graph with a very simple structure, made of the (edge-disjoint) union of a constant number of weighted complete bipartite subgraphs. The notion of approximation is a bit complicated to describe, but it enables the proof of counting lemmas, which show that, for example, the number of triangles in the original graph is well approximated by the (appropriately weighted) number of triangles in the approximating graph.
Analogous regularity lemmas, in which an arbitrary object is approximated by a low-complexity object, have been proved for hypergraphs, for subsets of abelian groups (for applications to additive combinatorics), in an analytic setting (for applications to graph limits) and so on.
The weak regularity lemma of Frieze and Kannan provides, as the name suggests, a weaker kind of approximation than the one promised by Szemeredi’s lemma, but one that is achievable with a graph that has a much smaller number of pieces. If is the “approximation error” that one is willing to tolerate, Szemeredi’s lemma constructs a graph that is the union of a weighted complete bipartite subgraphs where the height of the tower of exponentials is polynomial in . In the Frieze-Kannan construction, that number is cut down to a single exponential . This result too can be generalized to graph limits, subsets of groups, and so on.
With Tulsiani and Vadhan, we proved an abstract version of the Frieze-Kannan lemma (which can be applied to graphs, functions, distributions, etc.) in which the “complexity” of the approximation is . In the graph case, the approximating graph is still the union of complete bipartite subgraphs, but it has a more compact representation. One consequence of this result is that for every high-min-entropy distribution , there is an efficiently samplable distribution with the same min-entropy as , that is indistinguishable from . Such a result could be taken to be a proof that what GANs attempt to achieve is possible in principle, except that our result requires an unrealistically high entropy (and we achieve “efficient samplability” and “indistinguishability” only in a weak sense).
All these results are proved with a similar strategy: one starts from a trivial approximator, for example the empty graph, and then repeats the following iteration: if the current approximator achieves the required approximation, then we are done; otherwise take a counterexample, and modify the approximator using the counterexample. Then one shows that:
- The number of iterations is bounded, by keeping track of an appropriate potential function;
- The “complexity” of the approximator does not increase too much from iteration to iteration.
Typically, the number of iterations is , and the difference between the various results is given by whether at each iteration the “complexity” increases exponentially, or by a multiplicative factor, or by an additive term.
Like in the post on pseudorandom constructions, one can view such constructions as an online game between a “builder” and an “inspector,” except that now the online optimization algorithm will play the role of the builder, and the inspector is the one acting as an adversary. The bound on the number of rounds comes from the fact that the online optimization algorithms that we have seen so far achieve amortized error per round after rounds, so it takes rounds for the error bound to go below .
We will see that the abstract weak regularity lemma of my paper with Tulsiani and Vadhan (and hence the graph weak regularity lemma of Frieze and Kannan) can be immediately deduced from the theory developed in the previous post.
When I was preparing these notes, I was asked by several people if the same can be done for Szemeredi’s lemma. I don’t see a natural way of doing that. For such results, one should maybe use the online optimization techniques as a guide rather than as a black box. In general, iterative arguments (in which one constructs an object through a series of improvements) require the choice of a potential function, and an argument about how much the potential function changes at every step. The power of the FTRL method is that it creates the potential function and a big part of the analysis automatically and, even where it does not work directly, it can serve as an inspiration.
One could imagine a counterfactual history in which people first proved the weak regularity lemma using online optimization out of the box, as we do in this post, and then decided to try and use an L2 potential function and an iterative method to get the Szemeredi lemma, subsequently trying to see what happens if the potential function is entropy, thus discovering Jacob Fox’s major improvement on the “triangle removal lemma,” which involves the construction of an approximator that just approximates the number of triangles.
The multiplicative weights algorithm is simple to define and analyze, and it has several applications, but both its definition and its analysis seem to come out of nowhere. We mentioned that all the quantities arising in the algorithm and its analysis have statistical physics interpretations, but even this observation brings up more questions than it answers. The Gibbs distribution, for example, does put more weight on lower-energy states, and so it makes sense in an optimization setting, but to get good approximations one wants to use lower temperatures, while the distributions used by the multiplicative weights algorithms have temperature , where is the final “amortized” regret bound, so that one uses, quite counterintuitively, higher temperatures for better approximations.
Furthermore, it is not clear how we would generalize the ideas of multiplicative weights to the case in which the set of feasible solutions is anything other than the set of distributions.
Today we discuss the “Follow the Regularized Leader” method, which provides a framework to design and analyze online algorithms in a versatile and well-motivated way. We will then see how we can “discover” the definition and analysis of multiplicative weights, and how to “discover” another online algorithm which can be seen as a generalization of projected gradient descent (that is, one can derive the projected gradient descent algorithm and its analysis from this other online algorithm).
Today we will see how to use the analysis of the multiplicative weights algorithm in order to construct pseudorandom sets.
The method will yield constructions that are optimal in terms of the size of the pseudorandom set, but not very efficient, although there is at least one case (getting an “almost pairwise independent” pseudorandom generator) in which the method does something that I am not sure how to replicate with other techniques.
Mostly, the point of this post is to illustrate a concept that will reoccur in more interesting contexts: that we can use an online optimization algorithm in order to construct a combinatorial object satisfying certain desired properties. The idea is to run a game between a “builder” against an “inspector,” in which the inspector runs the online optimization algorithm with the goal of finding a violated property in what the builder is building, and the builder plays the role of the adversary selecting the cost functions, with the advantage that it gets to build a piece of the construction after seeing what property the “inspector” is looking for. By the regret analysis of the online optimization problem, if the builder did well at each round against the inspector, then it will do well also against the “offline optimum” that looks for a violated property after seeing the whole construction. For example, the construction of graph sparsifiers by Allen-Zhu, Liao and Orecchia can be cast in this framework.
(In some other applications, it will be the “builder” that runs the algorithm and the “inspector” who plays the role of the adversary. This will be the case of the Frieze-Kannan weak regularity lemma and of the Impagliazzo hard-core lemma. In those cases we capitalize on the fact that we know that there is a very good offline optimum, and we keep going for as long as the adversary is able to find violated properties in what the builder is constructing. After a sufficiently large number of rounds, the regret experienced by the algorithm would exceed the general regret bound, so the process must terminate in a small number of rounds. I have been told that this is just the “dual view” of what I described in the previous paragraph.)
But, back the pseudorandom sets: if is a collection of boolean functions , for example the functions computed by circuits of a certain type and a certain size, then a multiset is -pseudorandom for if, for every , we have
That is, sampling uniformly from , which we can do with random bits, is as good as sampling uniformly from , which requires bits, as far as the functions in are concerned.
It is easy to use Chernoff bounds and union bounds to argue that there is such a set of size , so that we can sample from it using only random bits.
We will prove this result (while also providing an “algorithm” for the construction) using multiplicative weights.
The multiplicative weights or hedge algorithm is the most well known and most frequently rediscovered algorithm in online optimization.
The problem it solves is usually described in the following language: we want to design an algorithm that makes the best possible use of the advice coming from self-described experts. At each time step , the algorithm has to decide with what probability to follow the advice of each of the experts, that is, the algorithm has to come up with a probability distribution where and . After the algorithm makes this choice, it is revealed that following the advice of expert at time leads to loss , so that the expected loss of the algorithm at time is . A loss can be negative, in which case its absolute value can be interpreted as a profit.
After steps, the algorithm “regrets” that it did not just always follow the advice of the expert that, with hindsight, was the best one, so that the regret of the algorithm after steps is
This corresponds to the instantiation of the framework we described in the previous post to the special case in which the set of feasible solutions is the set of probability distributions over the sample space and in which the loss functions are linear functions of the form . In order to bound the regret, we also have to bound the “magnitude” of the loss functions, so in the following we will assume that for all and all we have , and otherwise we can scale everything by a known upper bound on .
We now describe the algorithm.
The algorithm maintains at each step a vector of weights which is initialized as . The algorithm performs the following operations at time :
That is, the weight of expert at time is , and the probability of following the advice of expert at time is proportional to the weight. The parameter is hardwired into the algorithm and we will optimize it later. Note that the algorithm gives higher weight to experts that produced small losses (or negative losses of large absolute value) in the past, and thus puts higher probability on such experts.
We will prove the following bound.
Theorem 1 Assuming that for all and we have , for every , after steps the multiplicative weight algorithm experiences a regret that is always bounded as
In particular, if , by setting we achieve a regret bound