Objective Function

I was listening to the audiobook of “Children of Dune” by Frank Herbert, and there is a section with an analytical quote that is meant to be an examination of Paul Maud’dib’s multidimensional view of time, and there is a particular part where it talks about the fact that any multidimensional space can also be viewed as an aggregate of independent dimensions and that therefore any point in a multidimensional space can be viewed as an aggregate of one-dimensional vectors or attributes.

This is true in some senses but not others. It got me thinking, “what is the difference between an n-space and a bag of n linear attributes?” I think for most people, the intuitive sense is that the difference is in the distance metric. A space is something you can (metaphorically or otherwise) move through. In physical space, there is a common notion of “distance” that applies regardless of whether you are going up-and-down, east-and-west, or north-and-south. You can ask the question: “Which is closer to Chicago, Nashville or New York?” Even though one is primarily south and the other is primarily east, you can calculate what “distance” means in both directions and compare them.

A person can be described by a bag of n linear attributes. Let’s pick two simple ones out of that bag: height and weight. Does it make sense to take a description of you in terms of your height and weight and call it a “point in a two-dimensional space?”

Mathematicians say yes! But I think most casual day-to-day observers might feel weird about that. Height and weight are very different, and it’s not intuitive how you’d relate these two measurements. I’m 6 feet tall and 180 pounds. Who is more similar to me: a person who is 5’11” and 180 pounds or a person who is 6′ and 175 pounds? The question doesn’t fully make sense. That’s where I think most intuitive people, most non-mathematician people, might feel more comfortable calling this kind of description a “bag of traits” rather than a “point in space”.

Mathematicians call it a “space”, but they call it a “non-metric” space. The word “metric” here just has to do with whether you can talk about “distance” in a meaningful and coherent way. You can take a list of vehicle makes and models and call it “vehicle space”, and mathematicians are fine with that as long as you acknowledge that it’s a non-metric space: there is no way I can come up with a formula I can use to calculate an exact number that represents how close a 2020 Tesla is to a 2019 Aston Martin, and whether that distance is closer or farther away than the distance to a 2010 Chevy Volt. Or at least, I can’t do it unless I impose assumptions about how a difference in manufacturing year is related to differences in make and model.

When you’re making decisions between things, sometimes you really want your descriptions of those things to be a space rather than a bag of attributes. Right? Buying a car is a good example of that. You know you care about safety, cost and performance. You’re looking at two cars, and one of them is a little cheaper and a little less safe. So now you think to yourself, “Fuck, I need a distance metric.”

You’re out looking at cars to buy. You turn to your friend or loved-one you have with you, and you say: “I need a distance metric.” That’s normal, right?

Because if each car is a bag of three linear vectors (safety, cost, performance) you can’t evaluate a “greater than” or “less than” comparison between cars. There may be a trade-off between cost and performance, but if you don’t know how to convert 550 foot-pounds per second to dollars, you’re hosed. You can’t actually compare two cars in a meaningful way.

So you treat it like a space. You come up with some way — usually intuitive, not mathematical — of assessing how many dollars each foot-pound per second is worth to you personally (or whatever).

It’s not objective. It’s personal. That doesn’t mean it’s “personal” in the sense of being emotional or irrational or arbitrary. There are plenty of tricks mathematicians can use to convert bags of attributes into metrics spaces.

There are two broad categories I can think of for doing this off the top of my head. I’ll call them “what’s normal?” and “what’s useful?”

The “what’s normal?” strategy goes like this. Take a group of people, whatever group you think is relevant for the conversation you’re having. Could be all people, could be people of a particular gender or age group or whatever. In that group, there’s a distribution of heights, and you can measure the center of that distribution, and the spread of that distribution. What’s average? How much variation is there? You can do the same thing with weights. Now, instead of talking about “weight” as pounds or kilograms, you can talk about a person’s weight relative to the whole group. This person’s weight is average. Or, this person weights more than 70% of that population. You can also talk abut height in the same relative kind of way. Instead of talking about feet, inches, or centimeters, it becomes: this person is shorter than 45% of the group we’re talking about.

The technical term, if you care, is “z-scoring”. You can z-score any measurement, and that effectively removes the “units” of measurement… because now the number you are using to describe the attribute is just a relative measurement, compared to the population as a whole. Data scientists love this, because it means you can now calculate a mathematical comparison. You have a distance function that puts height and weight on equal footing. You can use this to turn a bag of attributes into a space.

The “what’s useful?” strategy goes like this. If there is some outcome you want, then you can try to come up with a way of assessing the relationship between any particular attribute and that outcome. It can be as concrete as “long term economic return on investment” or as mushy as “how good I feel about the decision a year later.” Regardless, you’re converting units of measurement for things like safety, price and performance into a common unit of measurement: outcome.

The technical term, if you care, is “objective function”. Your objective function can be personal and complicated. If you don’t drive much so you feel like your risk of getting into an accident is low, and you’re worried about your bills but just need something, anything, to get you to work each day, then you’re objective function will evaluate the trade-off between cost and performance differently than someone who is not struggling with bills but instead is juggling how to get three kids to their various extra-curricular activities every week. Two different objective functions, that will calculate comparisons between cars in very different ways. But it also converts a bag of attributes into a space, because allows you to calculate a “distance function” uniformly for each of your attributes.

So, that’s what that passage in the “Dune” book made me think about. And that got me thinking about the election going on right now. We have Democratic primaries. Sanders, Warren and Biden. How do you decide which one you’re closest to? How do you decide the distance between them? They are all bags of attributes. People spend a lot of time talking about which attributes matter to them. That’s fine. But deciding which attributes “matter” is just the first step. I wish people also spent time thinking about their objective function. What’s your objective? How does this bag of attributes relate to your objective? How can you take that bag of attributes and turn it into a space that you can actually make decisions about?

I would love for more conversations about politics and voting to be framed in this way.

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