Efficiently Evolving Juntas With Membership Queries

Recently, I showed that a simple genetic algorithm with uniform crossover can solve a noisy learning 7-parities problem efficiently—in O(n \textrm{ polylog}(n, \frac{1}{\epsilon})) time and O(\textrm{polylog}(n, \frac{1}{\epsilon})) membership queries. Learning parities is a subclass of a more general problem—learning juntas—that is of considerable interest to computational learning theorists because it neatly abstracts the general problem of learning in the presence of large amounts of irrelevant information. This is a very common problem in practice. A good example comes, “coincidentally”, from the field of genetic epidemiology—learning the genetic loci participating in the penetrance of a particular disease. (More about this “coincidence” in a bit)

The implicit concurrency hypothesis posits that the efficient learning of relevant variables in the presence of large numbers of irrelevant variables is precisely what gives evolution its power; so, understandably, when I read about the learning juntas problem a few weeks ago (on this blog), I sensed the possibility of a connection. Formally, a k-junta is a function f over n boolean variables, only j\,\,(\leq k\leq n) of which matter (i.e. are relevant) in the determination of the output of f. The j relevant variables are said to be the juntas of f, and the function f is completely characterized by its juntas and by a hidden boolean function h over j inputs. The output of f is just the output of h on the values of the j relevant variables of f (the values of the irrelevant variables are simply ignored). The problem of identifying the relevant variables is called the learning juntas problem.

Observe how this problem abstracts the genetic epidemiology problem mentioned above. The junta are the genetic loci participating in the presence/absence of the disease. The hidden function specifies which configurations of alleles at these loci cause the disease, and which ones do not. The problem of learning the relevant loci is the problem of learning juntas. Also note that the learning juntas problem generalizes the learning parities problem straightforwardly. Whereas the hidden function in the case of learning parities must be the parity function, the hidden function in the case of learning juntas can be any boolean function over the juntas.

Here is an evolution based algorithm, written in Python, that learns juntas given a membership query oracle. And here is a tutorial on its usage. For any value of k, the theory of implicit concurrency suggests that the k-junta for which it is hardest for a genetic algorithm to learn one or more relevant attributes (Mossel et al observed in Learning Juntas that to crack the problem, asymptotically speaking, one only needs to find a single relevant attribute) is the k-junta with a hidden parity function over k variables. This insight and my previous work on learning parities suggests that the evolutionary algorithm presented can learn the relevant attributes of k-juntas with small k in O(n\textrm{ polylog}(n, \frac{1}{\epsilon})) time and O(\textrm{polylog}(n, \frac{1}{\epsilon})) queries.

Implications for the Neo-Darwinian Synthesis

It seems like more than a coincidence that

  1. The Learning Juntas problem abstracts the genetic epidemiology problem mentioned earlier
  2. Evolutionary computation can efficiently solve the Learning Juntas problem for small numbers of juntas.

One is hard pressed not to hypothesize that evolution takes the form it takes precisely because this form allows it to solve a problem that is similar to the genetic epidemiology problem in all significant ways, save two :

  1. The problem is to learn the genetic loci participating in fitness/hotness not disease penetrance (often the same loci, but by no means always)
  2. Unlike computational geneticists, evolution has access to an active oracle.

Given the straightforwardness of this idea it may seem odd that it has not been forthcoming from Evolutionary Biology itself; that is until one realizes that a deep-rooted practice in Population Genetics actively steers theorists away. A vestige from the days before computers, when pen, paper, and mathematics was all one had, the conflation of the unit of inheritance with the unit of selection casts a long shadow. More about this in Sections 2.4.1 and 5.1 of my dissertation.

What now?

Despite the routine use of evolutionary algorithms in science and engineering, computer scientists have been unable to give a straight answer to a simple question: “Is there anything computationally efficient about evolution?” This question can now be answered in the affirmative. Recombinative evolutionary algorithms are capable of efficiently solving a broad, non-trivial problem: Learning k-Juntas with membership queries when k is small.

While fascinating from a biological perspective, this answer is not entirely satisfying to the engineer in me because there are conventional algorithms that solve the problem under consideration as efficiently; for example, the binary search algorithm due to Blum and Langley described in Section 5 of the paper by Mossel et al. A more involved algorithm due to Feldman solves the problem almost as efficiently, and does so non-adaptively and in the presence of random persistent classification error.

Have modern ML techniques eclipsed evolution at the very thing it is good at?

I have a hunch that the answer is “no”. It would be fantastic to find generalizations of the learning juntas with membership queries problem that are more difficult for conventional algorithms, but remain efficiently solvable by evolution.

Relevant Computational Learning Theory papers
R.J. Lipton
[web] The Learning Juntas Problem
(An excellent introduction to the Learning Juntas problem)

E. Mossel, R. O’donnell, and R. Servedio
[pdf] Learning Juntas,
(Section 3.1 is especially relevant if you want to reverse engineering the posted Python algorithm)

A. Blum, L. Hellerstein, and N. Littlestone
[pdf] Learning in the Presence of Finitely or Infitely Many Irrelevant Attributes

V. Feldman
[pdf] Attribute-Efficient and Non-adaptive Learning of Parities and DNF Expressions

Relevant papers on Implicit Concurrency
K.M. Burjorjee
[pdf] The Fundamental Learning Problem that Genetic Algorithms with Uniform Crossover Solve Efficiently and Repeatedly as Evolution Proceeds

K.M. Burjorjee
[pdf] Explaining Optimization in Genetic Algorithms with Uniform Crossover

Relevant papers on genetic epidemiology
J.H. Moore, L.W. Hahn, M.D. Ritchie, T.A. Thornton, and B.C. White
[web] Routine Discovery of Complex Genetic Models using Genetic Algorithms

J.H. Moore
[web] The ubiquitous nature of epistasis in determining susceptibility to common human diseases

Last Updated: September 18, 2013

Theoretical Bonafides For Implicit Concurrency

A commonly asked question since the early days of evolutionary computation is “What are Genetic Algorithms good for?” What, in other words, is the thing that genetic algorithms (GAs) do well? The prevailing answer for almost two decades was “Hierarchical Building Block Assembly”, also known as the building block hypothesis. This hypothesis generated tremendous excitement in its day and helped jumpstart the field of genetic algorithmics. However it fell into disrepute during the ’90s because of a lack of theoretical support and (more damningly) anomalies in the empirical record.

Since then, The Question (“What are GAs good for?”) has gone unanswered—or, to be precise, remains poorly answered—The Building Block Hypothesis continues on as a placeholder in textbooks. More recently, The Question is increasingly going unasked by evolutionary computation theorists, in large part because EC theorists are now wary of entertaining claims about genetic algorithms that cannot be formally deduced, and because Genetic algorithms used in practice (ones with finite but non-unitary recombinative populations) have proven frustratingly resistant to conventional analytic approaches.

The following paper squares off with Das Question, and for genetic algorithms with uniform crossover (UGAs), ventures an answer: Implicit Concurrent Multivariate Effect Evaluation—implicit concurrency for short—is what UGAs are good for. Implicit concurrency is compared with implicit parallelism (the “engine of optimization” according to the building block hypothesis), and the former is revealed to be more powerful.

Empirical evidence for implicit concurrency can be found here and here. And additional empirical evidence is easily obtained for one-off problem instances—just run a genetic algorithm on the appropriately chosen instance. The paper below takes a different tack. It establishes theoretical support (across an infinite set of problem instances) for the claim that implicit concurrency is a form of efficient computational learning. It achieves this end by demonstrating that implicit concurrency can be used to obtain low bounds on the time and query complexity of a UGA based algorithm that solves a constrained version of a well recognized problem from the computational learning literature: Learning parities with a noisy membership query oracle.

The difficulty of formally analyzing genetic algorithms is circumvented with an empirico-symmetry-analytic proof comprised of

  1. A formal part (no knowledge of genetic algorithms required)
  2. An accessible symmetry argument
  3. A simple statistical hypothesis testing based rejection of a null hypothesis with very high significance p<10^{-100}

The paper is here:
The Fundamental Learning Problem that GAs with UX Solve Efficiently and Repeatedly as Evolution Proceeds

The code used to conduct the experiment is here:
https://github.com/kburjorj/speedyGApy/blob/master/soda14.py

Implicit Concurrency in Genetic Algorithms

Genetic Algorithmics remains a niche area within Artificial Intelligence largely because the field has not identified a computation of some kind that genetic algorithms perform efficiently. The skepticism with which GAs are regarded is understandable. If, after decades of research and thousands of published articles, we cannot say what genetic algorithms do efficiently, then it is hardly surprising that the field would be held at arm’s length by the wider AI community.

His Highness is Naked. Time for some Real Clothes

Interestingly, the thing genetic algorithms do efficiently—concurrent multivariate effect evaluation (implicit concurrency for short)—is not difficult to describe or demonstrate. It’s been under our noses ever since schemata and schema partitions were defined as concepts, but for various reasons, which I won’t go into here, it’s escaped identification and dissemination.

Implicit concurrency (not to be confused with implicit parallelism) is a marvelous phenomenon that should be more widely appreciated than it currently is. This blog post provides a quick introduction. For a more in depth treatment check out  my FOGA 2013 paper Explaining Optimization in Genetic Algorithms with Uniform Crossover (slides here). And if you really want to have at it, my dissertation is here.

Schema Partitions and Effects

Let’s begin with a quick primer on schemata and schema partitions. Let S=\{0,1\}^\ell be a search space consisting of binary strings of length \ell. Let \mathcal I be some set of indices between 1 and \ell, i.e.  \mathcal I\subseteq\{1,\ldots,\ell\}. Then  \mathcal I represents a partition of S into 2^{|\mathcal I|} subsets called schemata (singular schema) as in the following example: suppose \ell=4, and \mathcal I=\{1,3\}, then \mathcal I partitions S into four schemata:

0*0* = \{0000, 0001, 0100, 0101\}

0*1* = \{0010, 0011, 0110, 0111\}

1*0* = \{1000, 1001, 1100, 1101\}

1*1* = \{1010, 1011, 1110, 1111\}

where the symbol * stands for ‘wildcard’. Partitions of this type are called schema partitions. As we’ve already seen, schemata can be expressed using templates, for example,  0*1*. The same goes for schema partitions. For example \#*\#* denotes the schema partition represented by the index set \mathcal I. Here the symbol \# stands for ‘defined bit’. The order of a schema partition is simply the cardinality of the index set that defines the partition (in our running example, it is |\mathcal I| = 2). Clearly, schema partitions of lower order are coarser than schema partitions of higher order.

Let us define the effect of a schema partition to be the variance of the average fitness values of the constituent schemata under sampling from the uniform distribution over each schema. So for example, the effect of the schema partition \#*\#*=\{0*0*\,,\, 0*1*\,,\, 1*0*\,,\, 1*1*\} is

\frac{1}{4}\sum\limits_{i=0}^1\sum\limits_{j=0}^1(F(i*j*)-F(****))^2

where the operator F gives the average fitness of a schema under sampling from the uniform distribution.

You’re now well poised to understand implicit concurrency. Before we get to a description, a brief detour to provide some motivation: We’re going to do a thought experiment in which we examine how effects change with the coarseness of schema partitions. Let [\mathcal I] denote the schema partition represented by some index set \mathcal I. Consider a search space S=\{0,1\}^\ell with \ell=10^6, and let \mathcal I =\{1,\ldots,10^6\}. Then [\mathcal I] is the finest possible partition of S; one where each schema in the partition has just one point. Consider what happens to the effect of [\mathcal I] as we start removing elements from \mathcal I. It should be relatively easy to see that the effect of [\mathcal I] decreases monotonically. Why? Because we’re averaging over points that used to be in separate partitions. Don’t proceed further until you convince yourself that coarsening a partition tends to decrease its effect.

Finally, observe that the number of schema partitions of order o is {\ell \choose o}. So for \ell = 10^6,  the number of schema partitions of order 2,3,4 and 5 are on the order of 10^{11}, 10^{17}, 10^{23}, and 10^{28} respectively. The take away from our thought experiment is this: while a search space may have vast numbers of coarse schema partitions, most of them will have negligible effects (due to averaging). In other words, while coarse schema partitions are numerous, ones with non-negligible effects are rare.

So what exactly does a genetic algorithm do efficiently? Using experiments and symmetry arguments I’ve demonstrated that a genetic algorithm with uniform crossover can concurrently sift through vast numbers of coarse schema partitions and identify partitions with non-negligible  effects. In other words, a genetic algorithm with uniform crossover can implicitly perform multitudes of effect/no-effect multifactor analyses and can efficiently identify interacting loci with non-negligible effects.

Let’s Play a Game

It’s actually quite easy (not to mention, cool and fun) to visualize a genetic algorithm as it identifies such loci. Let’s play a game. Consider a stochastic function that takes bitstrings of length 200 as input and returns an output that depends on the values of the bits of at just four indices. These four indices are fixed; they can be any one of the {\ell \choose 4} combinations of four indices between between 1 and 200. Given some bitstring, if the parity of the bits at these indices is 1 (i.e. if the sum of the four bits is odd) then the stochastic function returns a value drawn from the magenta distribution (see below). Otherwise, it returns a value drawn from the black distribution. The four indices are said to be pivotal. All other indices are said to be non-pivotal.

parityDistribs

As per the discussion in the first part of this post, the set of pivotal indices is the dual of a schema partition of order 4. Of all the schema partitions of order 4 or less, only this partition has a non-zero effect. All other schema partitions of order 4 or less have no effect. (Verify for yourself that this is true) In other words, in the world of effect evaluation, parity is a Needle in a Haystack (NIAH) problem. The kind of problem that seems closed to all approaches save brute force.

Now for the rules of the game: Say I give you query access to the stochastic function just described, but I do not tell you what four indices are pivotal. You are free to query the function with any bitstring 200 bits long as many times as you want. Your job is to recover the pivotal indices I picked, i.e. to identify the only schema partition of order 4 or less with a non-negligible effect.

Take a moment to think about how you would do it? What is the time and query complexity of your method?

What Not Breaking a Sweat Looks Like

The animation below shows what happens when a genetic algorithm with uniform crossover is applied to the stochastic function just described. Each dot displays the proportion of 1’s in the population at a locus. Note that it’s trivial to just “read off” the proportion of 0s at each locus. The four pivotal loci are marked by red dots. Of course, the genetic algorithm does not “know” that these loci are special. It only has “query access” to the stochastic function.

As the animation shows, after 500 generations you can simply “read off” the four loci I picked by examining the proportion of 1s to 0s in the population at each locus. Yay! You’ve just seen implicit concurrency in action. The chromosome size in this case is 200. So there are {200 \choose 4} possible combinations of four loci. From all of these possibilities, the genetic algorithm managed to identify the correct one within five hundred generations. Let’s put implicit concurrency through it’s paces. I’m going to tack on an additional 800 non-pivotal loci while leaving the indices of the four pivotal loci unchanged. Check out what happens:

[Note: more dots in the animation below does not mean a bigger population or more queries. More dots just means more loci under consideration. The population size and total number of queries remain the same]

So despite a quintupling in the number of bits, entailing an increase in the number of coarse schema partitions of order 4 to {1000 \choose 4}, the genetic algorithm solves the problem with no increase in the number of queries. Not bad. (Of course we’re talking about a single run of a stochastic process. And yes, it’s a representative run. See chapter 3 of my dissertation to get a sense for the shape of the underlying process)

Let’s take it up another notch, and increase the length of the bitstrings to 10,000. So now we’re looking at  {10000 \choose 4} \sim 10^{18} combinations of four loci. That’s on the order of a million trillion combinations. This time round, let’s also change the locations of the 4 pivotal loci. Will the genetic algorithm find them in 500 generations or less?

How’s that for not breaking a sweat? Don’t be deceived by the ease with which the genetic algorithm finds the answer. This is not an easy problem.

Intrigued? [If you've read this far, you should be] To run the experiments yourself download speedyGApy, and run it with

python speedyGA.py --fitnessFunction seap --bitstringLength <go-wild!>

noting that the increase in “wall clock” time between generations as you increase bitstringLength is due to an increase in the running time of everything (including query evaluation). The number of queries (i.e. number of fitness evaluations), however, stays the same. To learn how a genetic algorithm parlays implicit concurrency into a general-purpose global search heuristic called hyperclimbing, read my FOGA 2013 paper Explaining Optimization in Genetic Algorithms with Uniform Crossover (slides here).

[A request: Please help me get the word out about Implicit Concurrency. Tweet/retweet, blog/reblog, like, email, and mention on mailing lists. As always, if you have questions/comments, holler]

FOGA 2013 Slides

Optimization by Genetic Algorithms with uniform crossover is one of the deep mysteries of Evolutionary Computation. At first glance, the efficacy of uniform crossover, an extremely disruptive form of variation, makes no sense. Yet, genetic algorithms with uniform crossover often outperform genetic algorithms with variation operators that induce tight linkage between genetic loci.

At the Foundations of Genetic Algorithms Conference XII (January 16-20, Adelaide), I proposed an explanation for the efficacy of uniform crossover. The hypothesis presented posits that genetic algorithms with uniform crossover implement an intuitive, general purpose, global optimization heuristic called Hyperclimbing extraordinarily efficiently. The final version of the paper is available here. A generalization of the Hyperclimbing Hypothesis that explains optimization by genetic algorithms with less disruptive forms of crossover appears in my dissertation.

The presentation contains several animations, such as the one below, that serve as proof of concept for Hyperclimbing Hypothesis. To view the animations, click on the “Watch On YouTube” links in the slides above or just download the presentation (93 Mb) and view it in Adobe Acrobat Reader. Note: the pdf reader in Chrome will not display the animations.

Download SpeedyGApy to run the experiments yourself (easy, and highly recommended).

SpeedyGA Ported to Python

Today, I ported SpeedyGA from Matlab to Python. SpeedyGApy is a configurable, single-file, barebones, vectorized, numpy + matplotlib based genetic algorithm that rips.

SpeedyGApy Github Repo

The following video shows the kind of animation that gets displayed when speedyGA is run on a staircase function. Animations like this one serve as proof-of-concept for the Hyperclimbing Hypothesis, a new explanation for optimization in genetic algorithms with uniform crossover. Once you’ve downloaded speedyGApy, you can run this, and other experiments for yourself with the following command:

$ python speedyGA.py --fitnessFunction staircase

Generative Fixation

One of the mainstays of human engineering is the idea of hierarchical assembly—the assembly of useful low level modules (a.k.a. building blocks) into useful modules at higher levels. Artifacts ranging from nuclear submarines to enterprise software are all constructed in this fashion. The building block hypothesis holds that genetic algorithms also construct solutions using hierarchical assembly, and that the basic building blocks used by these algorithms are short chromosomal snippets that confer above average fitness. Tantalizing as this hypothesis may be, it is based on strong assumptions about the distribution of fitness over a search space. I’ve criticized these assumptions in my dissertation and have proposed a different hypothesis based on weaker assumptions. This alternate hypothesis is grounded in a different metaphor—generative fixation.

Though not as ubiquitously recognizable as hierarchical assembly is, generative fixation underlies progress in many areas. Like in the video industry for instance, which only really took off once the VHS/Betamax war had run its course. The “fixation” of VHS within this industry in essence generated new opportunities for advancement. For example, it permitted the development of the video rental business, which brought films from the back rooms of studio houses into living rooms everywhere. Had the tussle between VHS and Betamax continued, many of these opportunities might not have presented themselves. The economics of supporting two formats simultaneously would have been economically crippling for the fledgling industry.

VHS v. Betamax was a case where two contestants competed for dominance along a single dimension—the format accepted by video players. Consider a scenario consisting of multiple dimensions and multiple contestants in each dimension. Suppose, moreover, that no contestant in any dimension is outright superior. That is, the superioriority/non-superiority of the contestants in each dimension is dependent on the state of the contests in the other dimensions. This scenario describes the most commonly arising situation in natural evolution. Here, the “dimensions” are genetic loci, and the “contestants” are alleles. (Statistically, the scenario just described is more likely than one where one or more locus has an allele that is superior regardless of the frequencies of the alleles at other loci.)

The danger in such cases is that progress will stall because no contestant will come to dominate its dimension. The generative fixation hypothesis holds that in a system undergoing recombinative evolutionary dynamics, progress will continue. Although no locus has an allele that is outright superior, a small number of  alleles belonging to different dimensions that play nice together (i.e. confer above average fitness as a group) will come to dominate their respective dimensions. In doing so, they will set the stage for the next multi-dimensional contest over the remaining dimensions. And so on.

I’ve demonstrated the genetic algorithm’s ability to scaleably identify and fix synergistic sets of unlinked non-competing alleles in a recent manuscript and in my dissertation.

Presentation at the University of Washington: Optimization by Hyperclimbing

Yesterday, I presented my research on genetic algorithms at the University of Washington.

Talk abstract

My slides

More Debate, Please!

“… there are many issues in computing that inspire differing
opinions. We would be better off highlighting the differences
rather than pretending they do not exist”
–Moshe Y. Vardi

In an article entitled “More Debate, Please!”, in the January, 2010 issue of Communications of the ACM, Moshe Y. Vardi, editor-in-chief of Communications, writes:

`Vigorous debate, I believe, exposes all sides of an issue—their strengths and weaknesses. It helps us reach more knowledgeable conclusions. To quote Benjamin Franklin: “When Truth and Error have fair play, the former is always an overmatch for the latter.”’[1]

Vardi goes on to say that as he solicited ideas for the 2008 relaunch of Communications, he was frequently told to keep controversial topics front and center. “Let blood spill over the pages of Communications,” a member of a focus group jokingly urged [1].

When attempting to publish my doctoral research in evolutionary computation journals, I found the sentiments expressed by Vardi to be in short supply. The reviewers seemed much more invested in not rocking the boat than in fostering a climate in which prevailing assumptions can be challenged, and alternate ideas expressed transparently. They seemed, in short, to be inured to the poverty of the field’s foundations, and, for the most part, had little tolerance for someone with a bone to pick with the status quo. “Fall in line, or have your work be rejected,” was the overarching message.

One way this unfortunate state of affairs may be addressed is through the institution of a forum like the Point/Counterpoint section introduced to Communications by Vardi in 2008—a forum where the various controversies that mark our field are periodically featured, and the different sides of each controversy given, as Benjamin Franklin put it, “fair play”. There are several contentious topics in EC. Tapped correctly, many of  these topics can be powerful vehicles for learning—not just about the workings of evolutionary algorithms, but, also, about the workings of a vibrant intellectual community. Right now, instead of vigorous, open, ongoing debates in the EC literature, uneasy truces prevail. The community, by and large, steps around the the really big points of contention. Researchers talk past each other to niche audiences. And, if my experience is anything to go by, new lines of criticism, and new modes of analysis are hastily dismissed.

In the absence of a written record of ongoing controversies, new entrants to the field will not have access to the various positions involved. Pressed for time, and confronting the reality of “publish or perish”, most will fall back on the opinions and practices of their advisors. It doesn’t take much to see that in an environment like this, opportunities for learning and advancement will frequently be missed.

A forum for open, ongoing, collegial debate would  bring awareness, and transparency to the controversies in our field. It would also (one hopes) inculcate a more welcoming attitude toward alternate approaches, conclusions, and critiques.

Two topics for debate:

EC Theory and First Hitting Time:  Is it problematic that so much contemporary theoretical  work in EC focuses on “first hitting time”, i.e., the number of fitness evaluations required to find a global optimum? Do we look at first hitting time only because there currently isn’t a well developed, and generally accepted theoretical framework for examining adaptation (the generation of fitter points over time)? If so, isn’t the study of first hitting time a lot  like the proverbial search for one’s house keys under the light of a street lamp just because it happens to be dark in one’s house?

The Building Block Hypothesis: Can the building block hypothesis be reconciled with the widely reported utility of uniform crossover? If yes, how? If no, can we—more to the point, should we—be comfortable with this knowledge given the considerable influence of the building block hypothesis on contemporary evolutionary computation research?

What other topics have been under-addressed in the evolutionary computation literature? Leave a comment with your opinion, or a link to your own blog post.

[1] Moshe Y. Vardi. More debate please!, In Communications of the ACM 53(1):5, 2010

J

Hyperclimbing and Decimation

In recent years, probabilistic inference algorithms such as survey propagation and belief propagation have been shown to be remarkably effective at tackling large, random instances of SAT, and other combinatorial optimization problems that lie beyond the reach of previous approaches. These inference algorithms belong to a class of techniques called decimation strategies. Decimation strategies monotonically reduce the size of a problem instance by iteratively fixing partial solutions (partial variable assignments in the case of SAT).

The generative fixation hypothesis essentially states that genetic algorithms work by efficiently implementing a decimation strategy called hyperclimbing.

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