This weekend I ran across a blog post The Rodeo Clown Theory of Personal Development. The title comes from a hypothetical example of a goal you don’t know how to achieve: becoming a rodeo clown.

Let’s say you decide you want to be a rodeo clown. And let’s say you’re me and you have no idea how to be a rodeo clown. … Can you look at the possibilities currently available to you and imagine which of them might lead to an overall increase of rodeo clowndom in your life, even infinitesimally?

Each day you ask yourself what you can do that might lead you closer to your goal. This is a greedy algorithm: at each step you make as much progress toward your goal as you can. Greedy algorithms are not always optimal, but the premise above is that you have no idea what to do. In that case, doing whatever you can imagine that might might bring you a little closer to your goal is a smart thing to do.

Crossing the continent

If you’re in Los Angeles and your goal is to get to New York, you could start by walking east. You might run into a brick wall and decide that you should not take your approach too literally. Maybe you decide to buy a ticket for a bus that’s traveling east, even if you have to walk west to the bus station. Maybe the bus takes you to Houston. This isn’t the most direct route, but it’s progress.

When you don’t know what else to do, a greedy approach, especially one tempered with common sense, could be a very sensible way to start. Maybe it’ll put you in a position to learn of a more efficient approach. In the scenario above, maybe some stranger sees you walk into a wall, asks you what you’re trying to do, and tells you where the bus station is.

Sorting algorithms

The way most people sort a hand of playing cards is a sort of greedy algorithm. They scan for the highest card and put it first. Then they scan the remainder of the hand for the next highest card and so on. This algorithm is known as insertion sort. It’s not the most efficient algorithm, but it may be the most natural, and it works. And for a hand of five cards it’s about as good as any.

Anyone who has taken an algorithms class will know you can beat insertion sort with a more sophisticated algorithm such as heap sort. But sorting is a well-understood, one-dimensional problem. That’s why people long ago came up with solutions that are better than a greedy approach. But when a problem is not well understood and is high-dimensional, greedy algorithms are more likely to succeed, even if they’re not optimal.

Training neural nets

Training a neural network is a very high dimensional optimization problem, maybe on the order of a billion dimensions. And yet it is solved using stochastic gradient descent, which is essentially a greedy algorithm. At each iteration, you move in the direction that gives you the most improvement toward your goal. There’s a little more nuance to it than that, but that’s the basic idea.

How can gradient descent work so well in practice when in theory it could get stuck in a local minimum or a saddle point? John Carmack posted a heuristic explanation on Twitter a few weeks ago. A local critical point requires the “cooperation” of every variable in the function you’re optimizing. The more variables, the less likely this is to happen.

Carmack didn’t give a full explanation—he posted a tweet, not a journal article—but he does give an intuitive idea of what’s going on. At the global minimum, the variables cooperate for a good reason. He assumes the variables are unlikely to cooperate anywhere else, which is apparently often true when training neural nets.

Usually things get harder in higher dimensions, hence the “curse of dimensionality.” But there’s a sort of blessing of dimensionality here: you’re less likely to get stuck in a local minimum in higher dimensions.

Back to the rodeo

Training a neural network can be a huge undertaking, using millions of dollars worth of electricity, but it’s a mathematically defined problem. Life goals such as becoming a rodeo clown are not cleanly quantified problems. They are very high dimensional in the sense that there are a vast number of things you could do at any given time. You might argue by analogy with neural nets that the fact that there are so many dimensions means that you’re unlikely to get stuck in a local minimum. This is a loose analogy, one that easily breaks down, but still one worth thinking about.

Seth Godin once said “Remarkable work often comes from making choices when everyone else feels as though there is no choice.” What looks like a local minimum may be revealed to not be once you consider more options.

Image by Clarence Alford from Pixabay

There’s a little program called strings that searches for what appear to be strings inside binary file. I’ll refer to it as strings(1) to distinguish the program name from the common English word strings. [1]

What does strings(1) consider to be a string? By default it is a sequence of four or more bytes that correspond to printable ASCII strings. There are command options to change the sequence length and the character encoding.

There are 98 printable ASCII characters [2] and 256 possible values for an 8-bit byte, so the probability of a byte being a printable character is

p = 98/256 = 0.3828125.

This implies that the probability of strings(1) flagging a sequence of four random bytes as a string is p⁴ or about 2%.

How long a string might you find inside a binary file?

I ran strings(1) on a photograph and found a 46-character string:

   .IEC 61966-2.1 Default RGB colour space - sRGB

Of course this isn’t random. This string is part of the metadata stored inside JPEG images.

Next I encrypted the file so that no strings would be metadata and all strings would be false positives. The longest line was 12 characters:

    Z<Bq{7fH}~G9

How does this compare to what we might expect from a random file? I wrote about the probability of long runs a dozen years ago. In a file of n bytes, the expected length of the longest run of printable characters is approximately

In my case, the file had n = 203,308 bytes. The expected length of the longest run of printable characters would then be 12.2 characters, and so the actual length of the longest run is in line with what theory would have predicted.

[1] Unix documentation is separated into sections, and parentheses after a name specify the documentation section. Section 1 is for programs, Section 2 is for system calls, etc. So, for example, chmod(1) is the command line utility named chmod, and chmod(2) is the system call by the same name. Since command line utilities often have names that are common words, tacking (1) on the end helps distinguish program names from English words.

[2] p could be a little larger if you consider some of the control characters to be printable.

Arshad Khan left a comment on my post on the less and more utilities saying “on ubuntu if I do less on a pdf file, it shows me the text contents of the pdf.”

Apparently this is an undocumented feature of GNU less. It works, but I don’t see anything about it in the man page documentation [1].

Not all versions of less do this. On my Mac, less applied to a PDF gives a warning saying “… may be a binary file. See it anyway?” If you insist, it will dump gibberish to the command line.

A more portable way to extract text from a PDF would be to use something like the pypdf Python module:

    from pypdf import PdfReader

    reader = PdfReader("myfile.pdf")
    for page in reader.pages:
        print(page.extract_text()) 

The pypdf documentation gives several options for how to extract text. The documentation also gives a helpful discussion of why it’s not always clear what extracting text from a PDF should mean. Should captions and page numbers be extracted? What about tables? In what order should text elements be extracted?

PDF files are notoriously bad as a data exchange format. When you extract text from a PDF, you’re likely not using the file in a way its author intended, maybe even in a way the author tried to discourage.

Related post: Your PDF may reveal more than you intend

[1] Update: Several people have responded saying that that less isn’t extracting the text from a PDF, but lesspipe is. That would explain why it’s not a documented feature of less. But it’s not clear how lesspipe is implicitly inserting itself.

Further update: Thanks to Konrad Hinsen for pointing me to this explanation. less reads an environment variable LESSOPEN for a preprocessor to run on its arguments, and that variable is, on some systems, set to lesspipe.

I recently ran across a tweet from Allen Downey saying

So much of 20th century statistics was just a waste of time, computing precise answers to useless questions.

He’s right. I taught mathematical statistics at GSBS [1, 2] several times, and each time I taught it I became more aware of how pointless some of the material was.

I do believe mathematical statistics is useful, even some parts whose usefulness isn’t immediately obvious, but there were other parts of the curriculum I couldn’t justify spending time on [3].

Fun and profit

I’ll say this in partial defense of computing precise answers to useless questions: it can be fun and good for your career.

Mathematics problems coming out of statistics can be more interesting, and even more useful, than the statistical applications that motivate them. Several times in my former career a statistical problem of dubious utility lead to an interesting mathematical puzzle.

Solving practical research problems in statistics is hard, and can be hard to publish. If research addresses a practical problem that a reviewer isn’t aware of, a journal may reject it. The solution to a problem in mathematical statistics, regardless of its utility, is easier to publish.

Private sector

Outside of academia there is less demand for precise answers to useless questions. A consultant can be sure that a client finds a specific project useful because they’re willing to directly spend money on it. Maybe the client is mistaken, but they’re betting that they’re not.

Academics get grants for various lines of research, but this isn’t the same kind of feedback because the people who approve grants are not spending their own money. Imagine a grant reviewer saying “I think this research is so important, I’m not only going to approve this proposal, I’m going to write the applicant a $5,000 personal check.”

Consulting clients may be giving away someone else’s money too, but they have a closer connection to the source of the funds than a bureaucrat has when dispensing tax money.

Notes

[1] When I taught there, GSBS was The University of Texas Graduate School of Biomedical Sciences. I visited their website this morning, and apparently GSBS is now part of, or at least co-branded with, MD Anderson Cancer Center.

There was a connection between GSBS and MDACC at the time. Some of the GSBS instructors, like myself, were MDACC employees who volunteered to teach a class.

[2] Incidentally, there was a connection between GSBS and Allen Downey: one of my students was his former student, and he told me what a good teacher Allen was.

[3] I talk about utility a lot in this post, but I’m not a utilitarian. There are good reasons to learn things that are not obviously useful. But the appeal of statistics is precisely its utility, and so statistics that isn’t useful is particularly distasteful.

Pure math is beautiful (and occasionally useful) and applied math is useful (and occasionally beautiful). But there’s no reason to study fake applied math that is neither beautiful or useful.

Yesterday I stumbled on the fact that the size of Jupiter is roughly the geometric mean between the sizes of Earth and the Sun. That’s not too surprising: in some sense (i.e. on a logarithmic scale) Jupiter is the middle sized object in our solar system.

What I find more surprising is that a systematic search finds mean relationships that are far more accurate. The radius of Jupiter is within 5% of the geometric mean of the radii of the Earth and Sun. But all the mean relations below have an error less than 1%.

The radius of Mercury equals the geometric mean of the radii of the Moon and Mars, within 0.052%.

The radius of Mars equals the harmonic mean of the radii of the Moon and Jupiter, within 0.08%.

The radius of Uranus equals the arithmetic-geometric mean of the radii of Earth and Saturn, within 0.0018%.

See the links below for more on AM, GM, and AGM.

Now let’s look at masses.

The mass of Earth is the geometric mean of the masses of Mercury and Neptune, within 2.75%. This is the least accurate approximation in this post.

The mass of Pluto is the harmonic mean of the masses of the Moon and Mars, within 0.7%.

The mass of Uranus is the arithmetic-geometric mean of the masses of of the Moon and Saturn, within 0.54%.

Related posts

• Python code for means
• Arithmetic-geometric mean
• Springs, resistors, and harmonic means