Wednesday, December 03, 2008

CS theory in 60 minutes for highschool students

I might have the opportunity to give a talk at a high school, as part of a "what is science" style seminar. The audience would presumably tilt towards the more nerdy (since attendance is voluntary), but still probably wouldn't have more than a standard high school background in math or computer science (which means, likely, that they have never heard of computer science).  So I'd like to give a talk that introduces them to computer science as the study of computation, and I'd like to make it sound cool. So there should be perhaps some proofs of cool theorems, but all of the proofs should be simple enough to do by picture on a powerpoint slide. I'm looking for suggestions for what to present.

My first thought is a talk titled something like "What we can't do, infinity, and even bigger". It would cover two proofs that I think have a particularly high coolness to complicatedness ratio: Cantor's proof of the uncountability of reals, and Turing's proof of the undecidability of the halting problem. The outline would perhaps be something like:

1) Infinities are not all equal. There are lots of integers, but there are WAY more real numbers. Spend 20 minutes explaining diagonalization, and what it means.

2) Computer programs can do a lot, but not everything. 
Spend 20 minutes explaining the halting problem, and what it means to be undecidable. I'd probably avoid defining Turing Machines, since that seems like a mess, and just talk about it in terms of computer programs, which are hopefully a little more familiar

3) The punch line: Computer programs are countable, like the integers, but the problems that you might want to solve (languages) are uncountable, like the reals. So although computer programs can solve lots of things, actually, most things can't be solved. Hopefully this can be done in 10 minutes given that I've already talked about uncountability and undecidability.

I like these proofs since they don't require any machinery, and seem like they can be done visually with some animations. Thoughts?

Friday, November 28, 2008

Incentives, computation, and privacy

What do computer scientists bring to the study of game theory? One thing is simply a computer science style of analysis: worst case input, competitive ratios, approximation factors, etc. Perhaps the more interesting thing, however, is the study of how incentive constraints and computation interact: since we want to actually implement any mechanism we might dream up (in principle), it is subject not only to traditional game theoretic constraints such as truthfulness, but also standard computational limitations.

One fundamental question is: What can be computed by algorithms that satisfy constraint X, and it is natural to substitute "incentive compatibility" for "constraint X". An interesting study along these lines appeared in the last FOCS: "On The Hardness of Being Truthful"by Papadimitriou, Schapira, and Singer. This paper gives an example of a natural NP hard problem that can be well approximated efficiently, can be implemented truthfully, but cannot be efficiently well approximated by a truthful implementation: that is, they show that for this problem, truthfulness and computational limitation restrict what is algorithmically possible, although neither limitation is insurmountable on its own. 

They call the problem the "Combinatorial Public Projects Problem" (CPPP), although it is really just submodular maximization. In the CPPP, there are n agents, and m potential public goods. a mechanism for the CPP problem will select a subset of k of these goods. The agents have an opinion over which subsets are best: each agent i has a submodular valuation function f_i over sets of goods. The mechanism wishes to maximize the social welfare: it wants to pick the set of k goods that maximizes the sum of the agents valuation functions. We'll call this sum F.    Notice that since each f_i is submodular, F is submodular as well, and so the problem the mechanism faces is simply submodular maximization, and it is easy to show that the greedy algorithm (which selects at each stage the good that produces the largest marginal utility, until k goods are selected) achieves a (1-1/e) approximation. Not bad! But also not truthful -- agents may be incentivized to misreport their valuations. Whats more, the classic VCG mechanism achieves perfect social welfare, and enforces truthfulness -- but of course cannot be implemented efficiently, since it involves solving an NP hard problem exactly. What Papadimitriou, Schapira, and Singer show is that no truthful efficient algorithm can guarantee social welfare better by any polynomial factor than Sqrt(m) -- a huge gap from what is possible with either constraint alone.

Now whenever you see a striking impossibility result like this, its tempting to ask how resilient it is. Does it still hold if no player by himself constitutes a large fraction of the total social welfare? What if we relax truthfulness to approximate truthfulness, so that perhaps players can benefit by lying, but can't gain more than some small epsilon?

It turns out this particular problem is no longer so hard when we relax some of our constraints. Anupam Gupta, Katrina Ligett, Frank McSherry, Kunal Talwar, and I have a mostly unrelated paper, called "Differentially Private Approximation Algorithms". In this paper we also ask "What can be computed by algorithms that satisfy constraint X?", but in this paper, constraint X is differential privacy. We get mostly positive results: many NP hard problems, including the CPPP problem can be well approximated efficiently by privacy preserving algorithms. So what does this have to do with truthfulness?

In their 2007 FOCS paper "Mechanism Design via Differential Privacy", Mcsherry and and Talwar observed that mechanisms that satisfy differential privacy are also approximately truthful: privacy guarantees that any single player who changes his input does not change the probability of any output by more than a (1+epsilon) factor; therefore, he cannot increase his utility by more than a (1+epsilon) factor by deviating from a truthful input. In general, lets say that a mechanism satisfies gamma-truthfulness if no agent can benefit by more than an additive gamma factor by deviating from truthful behavior. So suppose in the CPP problem we scale every agent's valuation function so that her maximum valuation is 1. As a corollary of one of our approximation results, we show that for any gamma, there exists an efficient, gamma-truthful mechanism that achieves an additive O(k log m * log(1/gamma)/gamma) approximation factor. (Contrasting with the impossibility result of a multiplicative Sqrt(m) factor for 0-truthful mechanisms). In particular, for instances in which we have OPT = Ω(k log m * log(1/gamma)/gamma), this is an approximately truthful constant factor approximation algorithm.

I think that computer scientists are just starting to scratch the surface of the interplay between game theory and computation. It seems that there is a rich theory of incentive compatible computation waiting to be discovered, and that algorithmic game theory will prove to fit in nicely with the rest of theoretical computer science as a discipline that studies the fundamental abilities and limits of computation.


Friday, November 21, 2008

Computer science in the news

The New York Times magazine has a feature this week on the NetFlix challenge:

Its actually puts computer science in a pretty nice light -- it makes it seem very exciting. In particular, it seems to take the position that complex machine learning systems have something approaching real intelligence -- or at least that they are complex enough that they aren't fully understood by their own creators:

There’s a sort of unsettling, alien quality to their computers’ results. When the teams examine the ways that singular value decomposition is slotting movies into categories, sometimes it makes sense to them — as when the computer highlights what appears to be some essence of nerdiness in a bunch of sci-fi movies. But many categorizations are now so obscure that they cannot see the reasoning behind them. Possibly the algorithms are finding connections so deep and subconscious that customers themselves wouldn’t even recognize them. At one point, Chabbert showed me a list of movies that his algorithm had discovered share some ineffable similarity; it includes a historical movie, “Joan of Arc,” a wrestling video, “W.W.E.: SummerSlam 2004,” the comedy “It Had to Be You” and a version of Charles Dickens’s “Bleak House.”  For the life of me, I can’t figure out what possible connection they have, but Chabbert assures me that this singular value decomposition scored 4 percent higher than Cinematch — so it must be doing something right.

The author is definitely giving the computer too much credit for these anomylous clusters (and holds up "singular value decomposition" throughout the article as some sort of magical technique), but the point made is the right one. The difference between real intelligence and "just an algorithm" does in large part seem to be whether or not you can anticipate what its going to do. When you create a program that can come up with output that surprises you, thats pretty cool.

Saturday, November 08, 2008

Auctions with Unknown Supply

Its been a long time since my last post; I spent the summer at Microsoft Research in Silicon Valley, which was tons of fun, and since then I've been writing up some of the work I did there. Oh, and I've also been wasting an inordinate amount of time reading political blogs. But all of these things are now over, and its time for a post. Today, I'll write about one of the cool projects I worked on at Microsoft, with Moshe Babaioff and Liad Blumrosen.

Lots of people have heard talks about truthful auctions by now: A seller has some object he wants to sell, and bidders each have some private value for the item. The seller wants to sell the item to the highest bidder. How should he do it? If he just asks for bids and sells to the highest bidder, charging the bidder his own bid, buyers will lie about how much they value the item, and the item may not actually go to the person who values it the most. A classic result of Vickrey says that the seller should solicit bids, and give the item to the highest bidder, but charge him the second highest price. Then, the bidders can't do any better for themselves by lying about their valuations -- they should all just tell the truth! Auctions that have this property are called truthful auctions. Its easy to generalize this to the case in which the seller has not just a single item, but instead is selling k identical items.

This is all well and good -- the problem of selling items seems solved. And now that auctions are used to sell billions of advertisements on the internet, the theory of truthful auctions can be put to good use.

Unfortunately, there are some characteristics of these auctions that haven't been modelled. Even if we abstract away the details of particular ad systems (different slots worth different amounts, etc.), the items we are selling are page views, and page views are peculiar items indeed: First and foremost, we don't know how many items we will have available to sell! The items arrive online as users make clicks, and since we can't delay showing the user a page, we must decide on the allocation and price of each item as soon as it arrives, before we know how many future items will arrive. So the auctions we really want to study are online. And unlike previous work on online auctions, the auctioneer knows who the bidders are -- its the items that arrive online.

The Vickery auction maximizes social welfare -- the sum of the valuations of the bidders who recieve items. A basic question is how well we can approximate the optimal social welfare when items arrive online. At first blush, it seems that perhaps we can achieve optimal social welfare: After all, if we knew bidders' valuations, the greedy algorithm which simply allocated items as they came in to the unsatisfied bidder with the highest bid would achieve this. The problem is designing a payment rule that incentivizes bidders to actually bid their true values.

But lets step back -- what do we mean, anyway, when we say that the supply is unknown to the mechanism? There are several things we might mean. Perhaps, as is standard in the analysis of online algorithms, we mean that the supply is adversarial -- items arrive one at a time, but may stop at any time, and we want good guarantees on social welfare regardless of how many items arrive. However, this might not be totally realistic. After all, if we are Microsoft (or, lets see... Google?), we're running lots of these auctions every day, and we have good distributional information about how many items arrive each day. So another thing we might mean is that the total number of items that will arrive in any given day is drawn from a distribution that is known to the auctioneer. In this case, we only want good welfare guarantees in expectation over this distribution.  

And this gets to the heart of our result: distributional information is -really- important. What we show is that in the adversarial setting, its impossible to get a constant approximation to social welfare. In fact, no deterministic truthful mechanism gets better than the trivial n-approximation in the worst case. Even randomized truthful mechanisms can't achieve better than an Ω(loglog(n)) approximation.  On the other hand, in the stochastic setting, it is possible to get a constant approximation to social welfare. For any distribution that satisfies a technical condition known as monotone hazard rate (and many distributions do), there is a deterministic truthful mechanism which achieves a ~ 16 approximation. 

So what is the take home message? Pay attention to distributional information, at least when it comes to your supply. Economists often make distributional assumptions about the values of the bidders, and our result shows that this is less important in our setting -- we don't need this to get a good approximation to welfare. This is good, because a search engine has direct access to the distribution over clicks and page views, but must do some guessing to figure out the true values of its bidders (as distinct from their bids). And this is how it should be -- if we have a model in which what we want to accomplish is impossible, we should fix it with the information that we have right under our noses.

Saturday, September 13, 2008

Grim Trigger

It seems that the late army scientist turned anthrax suspect Bruce Ivins was not just a microbiologist, he was a game theorist. In an article today in the New York Times, we learn that before the FBI focused its anthrax investigation on him, he wrote a will, requesting that in the event of his death, his remains be cremated and scattered. But talk is cheap, and he wasn't sure that his wife and children would honor his request. 

It turns out that Ms. Ivins, his wife, was an anti-abortion activist and a former president of Frederick County Right to Life, a pro-life organization. So Ivins wrote the following into his will:
“If my remains are not cremated and my ashes are not scattered or spread on the ground, I give to Planned Parenthood of Maryland $50,000"
or about 1/3 of his estate. 

His concern seems to have stemmed from the fact that cremation is discouraged by Catholic doctrine, and in the case of cremation, ashes must be buried in sacred ground rather than scattered. But the threat-from-beyond-the-grave has worked, and it seems that Ivin will get his wish. So consider this a victory of game theory over theology.

Sunday, May 04, 2008

Presidential politics

Since my last post, there was a presidential primary in Pennsylvania, and despite running into Bill Clinton hanging out in front of my polling station, I voted for Barack Obama. One of the things that has particularly bothered me about the Republican administration over the last 8 years (excluding an ill-conceived war, the loss of a major US city, etc.) has been the prevailing atmosphere of anti-intellectualism. Our administration has cut funding for basic research, and has dismissed scientific evidence when making policy decisions. It has come to the point where distinguished educational credentials are liabilities when running for public office. Part of why I like Obama is that he speaks in complete sentences rather than in soundbites, and seems in general to hold education and reason in high regard (this has caused him to be attacked as elitist). Although Hillary has proposed significant increases in scientific funding (including raising the stipend of the NSF fellowship to $40,000!), and although she is also a product of an elite educational system, she has recently recast herself as a populist. Unfortunately, that seems to mean distancing herself from expert knowledge. I was disappointed to find this article: Clinton Spurns "Elite" Economists on Gas Tax. When did Elite become a bad word? A quote from the article:

Clinton used her appearance on ABC's "This Week" to raise questions about Obama's ability to connect with working-class Americans while dismissing economists who have said her plan to suspend gas taxes over the summer would do little good.

"I'm not going to put my lot in with economists," the New York senator said when asked to name a credible economist who supported her proposal.

"We've got to get out of this mind-set where somehow elite opinion is always on the side of doing things that really disadvantage the vast majority of Americans," said Clinton, a former first lady who would be the first woman president.

In other news, my Price of Malice paper was rejected from EC. Well, no worries. I'm not about to put my lot in with computer scientists.

Sunday, February 10, 2008

Malice, Anarchy, and Regret

Its been a long time since my last post -- I've been busy doing a lot of things, including attending part of this very interesting workshop on data privacy, and laboriously adding my own random coins to the PhD admissions process at CMU. Now that the next class has been admitted (Welcome!), I will hopefully have some time to get back to research.

There are some interesting things going on in the world of privacy, but that is a topic for another post. Since the terminology used by the algorithmic game theory community is much more colorful, I'll write a little about something I have been thinking about recently: The price of malice!

Recall that the price of anarchy is a measure of the degradation of social welfare in the face of selfish behavior. Specifically, it is the worst case ratio of the optimal social welfare to the social welfare in a Nash equilibrium. One of the problems with this measure is that it is really quite brittle -- it assumes that everyone is perfectly self interested and rational. But in actual games (say, computer networks), although many people are selfish and wish to minimize their own delay, others exhibit different behaviors. Some are oblivious to network traffic, and so may seem to be irrational. Some are explicitly malicious, and seek to increase the delay of others (see denial of service attacks). Enter the price of malice: an attempt to define a quantity that measures the effect on social welfare of these malicious players. So how should we define it?

Well, its been studied in several papers, and they define it differently! Moscibroda, Schmid, and Wattenhofer define a price of malice that is parameterized by the number of malicious players, k. PoM(k) is the ratio of the cost in a game with k malicious players, and n-k rational (selfish) players, to the worst case social cost with n rational (selfish) players. Babaioff, Kleinberg, and Papadimitriou give a different, unparameterized definition (also, of course, called the price of malice!) Their price of malice is essentially the first derivative of PoM(k), evaluated at k=0: That is, it is the worst case marginal cost of converting an epsilon fraction of the players from selfish to malicious.

But what does it mean to have a game with rational and malicious players? Both papers define an equilibrium model by assigning the malicious players a utility function equal to the negation of the social welfare function. This is natural when we are talking about Nash Equilibrium, but leads to perhaps an awkward definition of a malicious player: because the malicious players are myopically seeking to minimize social welfare, sometimes malicious players can fall into equilibria with rational players that actually improve social welfare (as compared to equilibria with rational players alone)! Both papers observe this phenomenon, and Babaioff, Kleinberg, and Papadimitriou call it "the windfall of malice". This is a pretty cool and counterintuitive phenomenon, but somehow it doesn't seem like a satisfying definition of malicious play -- after all, if a truly malicious adversary was helping you by acting adversarialy, he could (if nothing else) act `rationally' and avoid giving you this boost. What we really want is an adversary who we allow to behave arbitrarily -- we want to take the worst case over all possible adversaries, who might engage in sophisticated counterspeculation.

So, to add to the clutter, in a recent paper, I have given yet another definition of the price of malice. I use the framework from the recent paper on the price of total anarchy to define the price of malice (an analogue of Moscibroda et al.'s definition), and the differential price of malice (an analogue of Babaioff et al.'s definition) with weaker assumptions. First, imagine that players are playing in a repeated game, rather than a one-shot game. This allows us to weaken the assumption that players are playing according to a one-shot Nash equilibrium and consider each player individually, assuming only that each rational player experiences no "regret". That is, no player "regrets" not playing some fixed action -- each player is doing at least as well as she would have been had she retroactively changed her play history to repeatedly play any fixed action. Note that by the definition of a Nash equilibrium, if all players were repeatedly playing according to a Nash, all players would be experiencing no regret (and so this is a weaker assumption than that players play according to a Nash equilibrium). This weakened assumption has two benefits. First, it makes the predicted behavior computationally plausable: while finding a Nash equilibrium is PPAD complete in general games, there are efficient algorithms players can use to unilaterally guarantee that they have regret quickly tending to 0. But that isn't the main point here: It also allows us to define rational play on a player by player basis, rather than defining rational play over an entire population (by saying that they all play according to a Nash equilibrium). This allows us to analyze games in which only a fraction of the players are `rational', and the rest are behaving arbitrarily. So we can now study the price of malice -- in which k players are Byzantine -- players about whom we make no assumptions, and who may engage in sophisticated malicious play. Now that we are taking the worst case over all possible adversaries, we can no longer observe a windfall of malice. This should be what we want -- after all, we want to bound the cost of malicious players who may be smart enough not to help us out.

Of course, depending on the situation, our adversaries might be myopic, and so the equilibria models of the price of malice also are interesting quantities to study. Fortunately, the no-regret definitions are backwards compatable: since they make strictly weaker assumptions, upper bounds on the price of malice and differential price of malice in the no-regret model also serve as upper bounds on Moscibroda et al.'s and Babaioff et al.'s price of malice (respectively).