Q&A with Andreas Griewank, the mind behind Automatic Differentiation
“Mathematics is the real science”, Andreas´ high school teacher told him, not knowing that he would be the person responsible for his choice of scientific field. Some years have gone by and now Andreas Griewank is widely viewed as being the creator of the field of Automatic Differentiation (AD), a very significant mathematics advancement that currently includes a community of some 80 researchers and some 8000 users world-wide.
Recently, Andreas sat down a few minutes to talk about the background and development of his field, how he sees education in Latin America and Ecuador and his most important goals as the Dean of Mathematical Sciences and Information Technology at Yachay Tech.
Yachay Tech (YT): What is Automatic Differentiation?
Andreas Griewank (AG): In school people spend considerable time calculating derivatives of functions. This represents a rather tedious effort in manipulating formulas. Of course, the idea behind is that you compute a rate of change that allows you to decide, say, when you are going uphill or downhill. Generally, even if you have a complex function given by a large formula that you can’t interpret visually, especially if it has several variables, you can still easily compute its value at a given point, i.e. setting of the variables. But the next question is: how does the value change with respect of the individual variables? That sensitivity is defined by one or more derivatives. Another way to understand the importance of derivatives is to note that in dealing with complicated functions one can approximate them locally by straight lines, whose slope is exactly given by the derivative. This is called “linearizing the function”, which yields a much easier way to analyze the function at least within some range of the variable values. People typically compute derivatives by hand or computer algebra systems, which mimic what you do by hand. However, frequently one finds that the formula for the derivative grows very rapidly in an combinatorial way, whereas by AD you can avoid this exponential growth. Most models people are interested in physics, economics and such, are computational simulation models for which AD works perfectly to calculate slopes with high accuracy.
YT: So, to sum up, the difficulty that AD resolves has to do with avoiding the manipulation of complex function formulas with exponential growth…
AG: Yes, AD achieves that but also helps to identify the most critical variables and for example the direction one should take to maximize a function.
YT: And what has been AD’s impact on the academic community?
AG: Well for a long time, a lot of people have been developing numerical methods for optimization, for equation solving, for the integration of ordinary differential equations, or a variant of that differential algebraic equations, always under the assumption that derivatives must be avoided because they are very costly or not available at all. They would say “oh, we are trying to develop models for realistic problems, and for this problems there is no chance in the world to get derivatives”. That was until about 20 years ago, and, of course, the people who were trying to avoid the use derivatives this way were reluctant when we came around with AD, and said “well, there evaluation is really not such a problem”. Nevertheless, in the following years, many people especially from engineering and computer science insisted that it was neither necessary or nor even beneficial to use derivatives at all, whereas a minority of computational scientists, especially mathematicians just used them naturally within software packages based on AD. Right now there is a relatively small community of around 80 academics who further develop theory and tools. They meet twice a year at regional workshops and who gather once every four years at the main AD conference, which took place in Oxford in early September. Our software tools are used worldwide by thousands of people and included in the standard software distributions like Coin-OR and Debian for Linux.
YT: All those pre grad students who have missed their careers because of the complicated differentiation rules would love your tools…
AG: My impression is actually, that the Latin American tradition puts too much emphasis on mechanical processes for manipulating formulas; this is a repetitive rule application naturally for computers to do, not for people. So maybe these failures are due to teaching that emphasizes too much these manipulations and calculations. I have seen that happen here; students fail courses, fail tests because they can’t solve several mechanical problems in two hours. They make too many little mistakes in the end get no or a wrong answer, even though they might understand the concepts and principles. This is why I’m a big fan of oral exams, which most people don’t agree because it is not traditional here. However I’ve seen quite a few students who in an oral exam could explain the mathematics perfectly well, but had repeatedly failed written exams, because they were too nervous or made one stupid mistake too many.
YT: Do you think this model drifts students away from the abstract conception of mathematics?
AG: Yes, but off course there are also people who are much happier if you train them, you give them a recipe, you put those questions on the exam and have them do homework with very similar or even identical problems. This suits some students because they are happy to repeat recipes; and they might even pass those courses but still not really understand what’s going on.
YT: But how to promote this abstract conception on students?
AG: What happens a few times too, is that, where education in universities is more abstract, like in Germany, some people are super-duper mathematic students in high school, but then in the university receive an almost purely theoretical class, they fail very badly. For example, they might be asked to prove under a certain set of axioms that (-1)*(-1) = 1, which everybody thinks they know. This axiomatic approach is a rigorous intellectual exercise, but historically mathematics evolved in a more heuristic fashion in response to the needs of society from measuring plots of land to the simulation of transmission lines. The latter task was performed in the late 1800s by an autodidactic electrical engineer called Oliver Heaviside, using methods that were shunned as unsound by the mathematicians and physicists of his days. Eventually they came around and mathematical theory was extended to accommodate his methods in a rigorous fashion. Of course this is an enlightening story that can be used to motivate students for the study of complex numbers and what is now called functional analysis. On the other hand some mathematical techniques like for example L’Hospital’s Rule are overstated in our calculus teaching, just because it lends itself to hundreds of exercises that can be solved by formula manipulation in class, homework and exams. Again, a public portal like WolframAlpha yields these results much more reliably and efficiently than the fastest math wizzard. Of course the students should understand the principle, but I don’t think in my mathematical research I’ve ever applied it. Yet it lends itself to manipulation of formulas and therefore to high school, maybe, tronco común* mathematics, he laughs.
*The tronco común is the name given to the first two years of pre-graduate programs at Yachay Tech where all students take the same classes regardless of his or her major. These first two years have a strong emphasis of mathematics for all.
YT: As dean of the School of Mathematical Sciences and Information Technology at Yachay Tech, how do you plan on choosing the right teaching styles for success?
AG: By understanding the needs. Right now we have more students in the Software and Computer Engineering career than in the Mathematics career. Historically, computer science separated itself from mathematics around 30 years ago, most of the early computer scientists where mathematicians, physicists or chemists, but now they are computer scientists. So a fairly strong separation has been established in many places, for example, at the Humboldt University, where I taught until last year. There have been really fantastic developments in information theory and computational complexity. Which you can also view as mathematics in some sense but it really grew out of computer science. Now some topics like computational geometry, image processing, etc., are being studied in some groups by by computer scientists and in some places by mathematicians. So I think that a complete separation between mathematics and computer science should should be avoided. On the other hand there are classical fields that are separated in both cases. So for the school I would like to have, as long as we are small, an emphasis in computational aspects both from the Information Technology side and the Mathematical Sciences side. Off course that opens up the opportunity to involve everybody else: physics, chemists, geologists and so on. In conclusion, I would like to keep this things together and, rather than having a bipolar relation between both fields, having a large number of working areas where people from both traditions can work together. Also cooperating with other schools is fundamental. Unfortunately for some time mathematical modeling gained a bit of a bad reputation, when it was mistakenly viewed as being primarily dedicated to financial mathematics, whose models are very sophisticated but not very much connected to reality if only for a lack of data. We should try to avoid unfounded expectation and reliance in the power of mathematical modeling in the future. Of course in may other fields of science and engineering computer simulations have become almost eerily precise even taking into account stochastic effects.
YT: It is actually fundamental to connect science with society, so how does this field of science contribute to social change?
AG: In contrast to financial mathematics, theoretically more basic methods related to operations research, like for example the modeling of value chains, the optimization of traffic flows and so forth have been found to be very useful. Consequently, they became more and more important, finally getting used by all the big companies, making it hard for smaller companies and developing countries to keep up. Therefore, one of the ideals would be to generate software technology that can also help the smaller or medium size manufacturers.
YT: And also, social change has been part of who you are for quite a long time since you also have experience in political activism due to your time in Die Linke….
AG: Yes, Die Linke has a foundation called The Rosa Luxemburg Foundation. This institution does all kinds of things, and one of those activities is to give scholarships for people to study in pregraduate and postgraduate education. This party, especially the part of the foundation, has traditionally been in some sense a classical sociological, left wing, critic of capitalism organization; so I fought very hard to convince many of the alumni of this organization, often in social sciences departments, that the organization should also sponsor people in more fields of research that can contribute to society. In particular, medicine, for example, and engineering. Loosely speaking, we could still have this critical left wing vision of things but contribute to society more directly. Quite often, when the moment to choose which students were the ones to receive the scholarship, the traditional criteria was to analyze how involved this person was to the left wing movement. Naturally, if someone wanted to study a very complex field of science, the time to engage in this type of activism was reduced. Therefore, the percentage of scientists, even more so physicians or economical experts, remained very low. So I did a little statistical study of the selection method and found out that the academic qualification had very little to do in the process, but to have this study accepted by the organization was practically impossible…, too much quantitative information.
YT: Are you still trying to connect mathematical models to society change? How will you do that at Yachay Tech?
AG: Mathematics has traditionally being dealing with modeling. It has been used for economics and social modeling. So in designing our courses here at Yachay Tech, there is always calculus and there is always linear algebra. But, following the suggestion of the American Mass Society and the Industrial Applied Mathematics, we know the third basic course has to be related to data analysis. You can’t ignore it, everything is being optimized by data collection and data analysis. So you can introduce a few parameters and use this model to optimize a lot of processes. The use of Artificial Intelligence has changed a great number of things. The use of Artificial Neural Networks, for example, is something that Google is doing and that keeps on surprising me in many ways. But this comes with responsibilities related to conserving information to good use, and understanding the limitation of these things. I mean, a lot of models work very well for a while, for a certain specific problems but may really mislead you in a different contexts. So we know that we have a responsibility to society, to analyze this methods and also, of course, the question of privacy and ethical data use, because that has become a real problem for society now.