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Archive for August, 2013

The writings (and even papers/technical books) of Gian-Carlo Rota are perhaps amongst the most insightful that I have encountered in the past 3-4 years (maybe even more). Rota wrote to provoke, never resisting to finish a piece of writing with a rhetorical flourish even at the cost of injecting seeming inconsistency in his stance. I guess this is what you get when you have a first rate mathematician and philosopher endowed with an elegant; at times even devastating turn of phrase, with a huge axe to grind*.

The wisdom of G. C. Rota is best distilled in his book of essays, reviews and other thoughts: Indiscrete Thoughts and to some extent Discrete Thoughts. Perhaps I should review Indiscrete Thoughts in the next post, just to revisit some of those writings and my notes from them myself.
Rota in 1962

Rota in 1962

This post however is not about his writing in general as the title indicates. I recently discovered this excellent dialogue between Rota and David Sharp (1985). I found this on a lead from this László Lovász interview. Here he mentions that Rota’s combinatorics papers were an inspiration for him in his work to find more structure in combinatorics. From the David Sharp interview, here are two relevant excerpts (here too the above mentioned flourish is evident):
“Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
[…]
Much combinatorics of our day came out of an extraordinary coincidence. Disparate problems in combinatorics, ranging from problems in statistical mechanics to the problem of coloring a map, seem to bear no common features. However, they do have at least one common feature: their solution can be reduced to the problem of finding the roots of some polynomial or analytic function. The minimum number of colors required to properly color a map is given by the roots of a polynomial, called the chromatic polynomial; its value at N tells you in how many ways you can color the map with N colors. Similarly, the singularities of some complicated analytic function tell you the temperature at which a phase transition occurs in matter. The great insight, which is a long way from being understood, was to realize that the roots of the polynomials and analytic functions arising in a lot of combinatorial problems are the Betti numbers of certain surfaces related to the problem. Roughly speaking, the Betti numbers of a surface describe the number of different ways you can go around it. We are now trying to understand how this extraordinary coincidence comes about. If we do, we will have found a notable unification in mathematics.”
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*While having an axe to grind is a fairly common phrase. I got the idea of using it from an amazon review of Indiscrete thoughts (check link above). Because I really do think that that is the best description for a lot of his writings!

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Darwinian Evolution is a form of Computational Learning

The punchline of this book is perhaps: “Changing or increasing functionality of circuits in biological evolution is a form of computational learning“; although it also speaks of topics other than evolution, the underlying framework is of the Probably Approximately Correct model [1] from the theory of Machine Learning, from which the book gets its name.

"Probably Approximately Correct" by Leslie Valiant

“Probably Approximately Correct” by Leslie Valiant

[Clicking on the image above will direct you to the amazon page for the book]

I had first heard of this explicit connection between Machine Learning and Evolution in 2010 and have been quite fascinated by it since. It must be noted, however, that similar ideas have appeared in the past. It won’t be incorrect to say that usually they have been in the form of metaphor. It is another matter that this metaphor has generally been avoided for reasons I underline towards the end of this review. When I first heard about the idea it immediately made sense and like all great ideas, in hindsight looks completely obvious. Ergo, I was quite excited to see this book and preordered it months in advance.

This book is basically a popular science version and expansion of the ideas on the matter that appeared in a J-ACM article titled “Evolvability” in 2007 [2]. I have to say that I was expecting a bit more from the book than what it already had, in this sense I was somewhat disappointed. But at the same time, it can perhaps be a useful ‘starting’ book for those interested in the broad idea but without much background. We all have examples of books like this. Here’s one from me: about a couple of years ago I read a book on Knots by Alexei Sossinsky, the book got bad reviews from many serious mathematicians for not really being about mathematics, besides having many mistakes. But for a complete beginner, I thought it was an absolutely wonderful book, introducing the reader to the basic ideas very informally besides sharing his infectious enthusiasm for problems in Knot Theory. Coming back, in short if you have enough background of the basic notions of PAC Learning then the book might feel quite redundant in some chapters, but depending on how you read, it might still turn out to be a good read any way. Judging from some other (rude) reviews: If you are a professional nitpicker then this book is probably not worth your time since this book deals with ideas and not with a formal treatment of them. Given what it is aiming at, I think it is a good book.

Before attempting to sketch a skiagram of the main content of the book: One of the main subthemes of the book, constantly emphasized is to look at computer science as a kind of an enabling tool to study natural science. This is oft ignored, perhaps because of the reason that CS curricula are rarely designed with any natural science component in them and hence there is no impetus for aspiring computer scientists to view them from the computational lens. On the other hand,  the relation of computer science with mathematics has become quite well established. As a technology the impact of Computer Science has been tremendous. All this is quite remarkable given the fact that just about a century ago the notion of a computation was not even well defined. Unrelated to the book: More recently people have started taking the idea of digital physics (i.e. physics from a solely computable/digital perspective) seriously. But in the other natural sciences its usage is still woeful. Valiant draws upon the example of Alan Turing as a natural scientist and not just as a computer scientist to make this point. Alan Turing was more interested in natural phenomenon (intelligence, limits of mechanical calculation, pattern formation etc) and used tools from Computer Science to study them, a fact that is evident from his publications. That Turing was trying to understand natural phenomenon was remarked in his obituary by Max Neumann by summarizing the body of his work as: “the extent and the limitations of mechanistic explanations of nature”.

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The book begins with a delightful and quite famous quote by John von Neumann (through this paragraph I also discovered the context to the quote). This paragraph also adequately summarizes the motivation for the book very well:

“In 1947 John von Neumann, the famously gifted mathematician, was keynote speaker at the first annual meeting of the Association for Computng Machinery. In his address he said that future computers would get along with just a dozen instruction types, a number known to be adequate for expressing all of mathematics. He went on to say that one need not be surprised at this small number, since 1000 words were known to be adequate for most situations in real life, and mathematics was only a small part of life, and a very simple part at that. The audience responded with hilarity. This provoked von Neumann to respond: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is”

Though counterintuitive, von Neumann’s quip contains an obvious truth. Einstein’s theory of general relativity is simple in the sense that one can write the essential content on one line as a single equation. Understanding its meaning, derivation, and consequences requires more extensive study and effort. However, this formal simplicity is striking and powerful. The power comes from the implied generality, that knowledge of one equation alone will allow one to make accurate predictions about a host of situations not even connected when the equation was first written down.

Most aspects of life are not so simple. If you want to succeed in a job interview, or in making an investment, or in choosing a life partner, you can be quite sure that there is no equation that will guarantee you success. In these endeavors it will not be possible to limit the pieces of knowledge that might be relevant to any one definable source. And even if you had all the relevant knowledge, there may be no surefire way of combining it to yield the best decision.

This book is predicated on taking this distinction seriously […]”

In a way, aspects of life as mentioned above appear theoryless, in the sense that there seems to be no mathematical or scientific theory like relativity for them. Something which is quantitative, definitive and short. Note that these are generally not “theoryless” in the sense that there exists no theory at all since obviously people can do all the tasks mentioned in a complex world quite effectively. A specific example is of how organisms adapt and species evolve without having a theory of the environment as such. How can such coping mechanisms come into being in the first place is the main question asked in the book.

Let’s stick to the specific example of biological evolution. Clearly, it is one of the central ideas in biology and perhaps one of the most important theories in science that changed the way we look at the world. But inspite of its importance, Valiant claims (and correctly in my opinion) that evolution is not understood well in a quantitative sense. Evidence that convinces us of its correctness is of the following sort: Biologists usually show a sequence of evolving objects; stages, where the one coming later is more complex than the previous. Since this is studied mostly via the fossil record there is always a lookout for missing links between successive stages (As a side note: Animal eyes is an extremely fascinating and very readable book that delves with this question but specifically dealing with the evolution of the eye. This is particularly interesting given that due to the very nature of the eye, there can be no fossil records for them). Darwin had remarked that numerous successive paths are necessary – that is, if it was not possible to trace a path from an earlier form to a more complicated form then it was hard to explain how it came about. But again, as Valiant stresses, this is not really an explanation of evolution. It is more of an “existence proof” and not a quantitative explanation. That is, even if there is evidence for the existence of a path, one can’t really say that a certain path is likely to be followed just because it exists. As another side note: Watch this video on evolution by Carl Sagan from the beautiful COSMOS

Related to this, one of the first questions that one might ask, and indeed was asked by Darwin himself: Why has evolution taken as much time as it has? How much time would have sufficed for all the complexity that we see around us to evolve? This question infact bothered Darwin a lot in his day. In On the Origins of Species he originally gave an estimate of the Earth’s age to be at least 300 million years, implying indirectly, that there was enough time. This estimate was immediately thrashed by Lord Kelvin, one of the pre-eminent British physicists of the time, who estimated the age of the Earth to be only about 24 million years. This caused Darwin to withdraw the estimate from his book. However, this estimate greatly worried Darwin as he thought 24 million years just wasn’t enough time. To motivate on the same theme Valiant writes the following:

“[…] William Paley, in a highly influential book, Natural Theology (1802) , argued that life, as complex as it is, could not have come into being without the help of a designer. Numerous lines of evidence have become available in the two centuries since, through genetics and the fossil record, that persuade professional biologists that existing life forms on Earth are indeed related and have indeed evolved. This evidence contradicts Paley’s conclusion, but it does not directly address his argument. A convincing direct counterargument to Paley’s would need a specific evolution mechanism to be demonstrated capable of giving rise to the quantity and quality of the complexity now found in biology, within the time and resources believed to have been available. […]”

A specific, albeit more limited version of this question might be: Consider the human genome, which has about 3 billion base pairs. Now, if evolution is a search problem, as it naturally appears to be, then why did the evolution of this genome not take exponential time? If it would have taken exponential time then clearly such evolution could not have happened in any time scale since the origin of the earth. Thus, a more pointed question to ask would be: What circuits could evolve in sub-exponential time (and on a related note, what circuits are evolvable only in exponential time?). Given the context, the idea of thinking about this in circuits might seem a little foreign. But on some more thought it is quite clear that the idea of a circuit is natural when it comes to modeling such systems (at least in principle). For example: One might think of the genome as a circuit, just as how one might think of the complex protein interaction networks and networks of neurons in the brain as circuits that update themselves in response to the environment.

The last line is essentially the key idea of adaptation, that entities interact with the environment and update themselves (hopefully to cope better) in response. But the catch is that the world/environment is far too complex for simple entities (relatively speaking), with limited computational power, to have a theory for. Hence, somehow the entity will have to cope without really “understanding” the environment (it can only be partially modeled) and improve their functionality. The key thing to pay attention here is the interaction or the interface between the limited computational entity in question and the environment. The central idea in Valiant’s thesis is to think of and model this interaction as a form of computational learning. The entity will absorb information about the world and “learn” so that in the future it “can do better”. A lot of Biology can be thought of as the above: Complex behaviours in environments. Wherein by complex behaviours we mean that in different circumstances, we (or alternately our limited computational entity) might behave completely differently. Complicated in the sense that there can’t possibly be a look up table for modeling such complex interactions. Such interactions-responses can just be thought of as complicated functions e.g. how would a deer react could be seen as a function of some sensory inputs. Or for another example: Protein circuits. The human body has about 25000 proteins. How much of a certain protein is expressed is a function depending on the quantities of other proteins in the body.  This function is quite complicated, certainly not something like a simple linear regression. Thus there are two main questions to be asked: One, how did these circuits come into being without there being a designer to actually design them? Two, How do such circuits maintain themselves? That is, each “node” in protein circuits is a function and as circumstances change it might be best to change the way they work. How could have such a thing evolved?

Given the above, one might ask another question: At what rate can functionality increase or adapt under the Darwinian process? Valiant (like many others, such as Chaitin. See this older blog post for a short discussion) comes to the conclusion that the Darwinian evolution is a very elegant computational process. And since it is so, with the change in the environment there has to be a quantitative way of saying how much rate of change can be kept up with and what environments are unevolvable for the entity. It is not hard to see that this is essentially a question in computer science and no other discipline has the tools to deal with it.

In so far that (biological) interactions-responses might be thought of as complicated functions and that the limited computational entity that is trying to cope has to do better in the future, this is just machine learning! This idea, that changing or increasing functionality of circuits in biological evoution is a form of computational learning, is perhaps very obvious in hindsight. This (changing functionality) is done in Machine Learning in the following sense: We want to acquire complicated functions without explicitly programming for them, from examples and labels (or “correct” answers). This looks at exactly at the question at how complex mechanisms can evolve without someone designing it (consider a simple perceptron learning algorithm for a toy example to illustrate this). In short: We generate a hypothesis and if it doesn’t match our expectations (in performance) then we update the hypothesis by a computational procedure. Just based on a single example one might be able to change the hypothesis. One could draw an analogy to evolution where “examples” could be experiences, and the genome is the circuit that is modified over a period of time. But note that this is not how it looks like in evolution because the above (especially drawing to the perceptron example) sounds more Lamarckian. What the Darwinian process says is that we don’t change the genome directly based on experiences. What instead happens is that we make a lot of copies of the genome which are then tested in the environment with the better one having a higher fitness. Supervised Machine Learning as drawn above is very lamarckian and not exactly Darwinian.

Thus, there is something unsatisfying in the analogy to supervised learning. There is a clear notion of a target in the same. Then one might ask, what is the target of evolution? Evolution is thought of as an undirected process. Without a goal. This is true in a very important sense however this is incorrect. Evolution does not have a goal in the sense that it wants to evolve humans or elephants etc. But it certainly does have a target. This target is “good behaviour” (where behaviour is used very loosely) that would aid in the survival of the entity in an environment. Thus, this target is the “ideal” function (which might be quite complicated) that tells us how to behave in what situation. This is already incorporated in the study of evolution by notions such as fitness that encode that various functions are more beneficial. Thus evolution can be framed as a kind of machine learning problem based on the notion of Darwinian feedback. That is, make many copies of the “current genome” and let the one with good performance in the real world win. More specifically, this is a limited kind of PAC Learning.  If you call your current hypothesis your genome, then your genome does not depend on your experiences. Variants to this genome are generated by a polynomial time randomized Turing Machine. To illuminate the difference with supervised learning, we come back to a point made earlier that PAC Learning is essentially Lamarckian i.e. we have a hidden function that we want to learn. One however has examples and labels corresponding to this hidden function, these could be considered “queries” to this function. We then process these example/label pairs and learn a “good estimate” of this hidden function in polynomial time. It is slightly different in Evolvability. Again, we have a hidden “ideal” function f(x). The examples are genomes. However, how we can find out more about the target is very limited, since one can empirically only observe the aggregate goodness of the genome (a performance function). The task then is to mutate the genome so that it’s functionality improves. So the main difference with the usual supervised learning is that one could query the hidden function in a very limited way: That is we can’t act on a single example and have to take aggregate statistics into account.

Then one might ask what can one prove using this model? Valiant demonstrates some examples. For example, Parity functions are not evolvable for uniform distribution while monotone disjunctions are actually evolvable. This function is ofcourse very non biological but it does illustrate the main idea: That the ideal function together with the real world distribution imposes a fitness landscape and that in some settings we can show that some functions can evolve and some can not. This in turn illustrates that evolution is a computational process independent of substrate.

In the same sense as above it can be shown that Boolean Conjunctions are not evolvable for all distributions. There has also been similar work on real valued functions off late which is not reported in detail in the book. Another recent work that is only mentioned in passing towards the end is the study of multidimensional space of actions that deal with sets of functions (not just one function) that might be evolvable together. This is an interesting line of work as it is pretty clear that biological evolution deals with the evolution of a set of functions together and not functions in isolation.

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Overall I think the book is quite good. Although I would rate it 3/5. Let me explain. Clearly this book is aimed at the non expert. But this might be disappointing to those who bought the book because of the fact that this recent area of work, of studying evolution through the lens of computational learning, is very exciting and intellectually interesting. The book is also aimed at biologists, and considering this, the learning sections of the book are quite dumbed down. But at the same time, I think the book might fail to impress most of them any way. I think this is because generally biologists (barring a small subset) have a very different way of thinking (say as compared to the mathematicians or computer scientists) especially through the computational lens. I have had some arguments about the main ideas in the book over the past couple of years with some biologist friends who take the usage of “learning” to mean that what is implied is that evolution is a directed process. It would have been great if the book would have spent more time on this particular aspect. Also, the book explicitly states that it is about quantitative aspects of evolution and has nothing to do with speciation, population dynamics and other rich areas of study. However, I have already seen some criticism of the book by biologists on this premise.

As far as I am concerned, as an admirer of Prof. Leslie Valiant’s range and quality of contributions, I would have preferred if the book went into more depth. Just to have a semi-formal monograph on the study of evolution using the tools of PAC Learning right from the person who initiated this area of study. However this is just a selfish consideration.

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References:

[1] A Theory of the Learnable, Leslie Valiant, Communications of the ACM, 1984 (PDF).

[2] Evolvability, Leslie Valiant, Journal of the ACM, 2007 (PDF).

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