Growth by Learning: Does Progress Accelerate with Mastery?

Scott Young wrote a good post about two types of progress when learning something new: logarithmic and exponential.

Anything you try to improve will have a growth curve. Imagine you ran everyday and you tracked your speed to finish a 5-mile course. Smoothing out the noise, over enough time you’d probably get a graph like this:

Logarithmic Curve

Here, improvement works on a logarithmic scale. As you get better, it gets harder and harder to improve. Elite athletes expend enormous effort to shave seconds off their best times. Novice athletes can shave minutes with just a little practice.

Logarithmic growth is the first type of growth. This is where you see a lot of progress in the beginning, but continuing progress is more difficult.

Now imagine a different graph. This time you’ve build a new website you update regularly and you’re measuring subscribers. This graph would likely look very different:

This is exponential growth, the second type of growth. Website traffic is often exponential because as a blog attracts more readers, there are more opportunities for word about the blog to spread. A blog with zero traffic also has zero word of mouth.

I’ve noticed most things tend to be either logarithmic or exponential growth. Despite this, linear progress is what most people expect. We tend to expect things to move in the same direction or rate as they have in the past. This violation of our expectation leads to some mistakes in how we set goals and act on them.

 Scott later offers advice on how to tell whether a given activity is one or the other:

The easiest way to tell is to look at how other people have progressed in that field. Don’t pay attention to their rates, just pay attention to the shape of their growth trajectory. Is it the kind that slows down with mastery or speeds up?

I’ve written about related themes in the past. Here’s my post about efforts where you see continuous, ongoing improvement vs. quantum leaps. Here’s my post about how formal schooling is an information rich environment where you receive constant feedback on how you’re doing, whereas in the real world you sometimes have to go months without knowing if you’re on the right track.

6 Responses to Growth by Learning: Does Progress Accelerate with Mastery?

  1. Ted Sanders says:

    I really, really dislike the characterization of growth into log and exponential. Why not sublinear and superlinear? Those categories are much more general and don’t carry any of the implications that logarithmic and exponential curves do.

    I really can’t to authentically communicate how much this bothers me. It reminds me of people who loosely throw around terms like power laws. If your argument still works with more general mathematical terms, then why invoke specific ones? In general, I think you only obfuscate and weaken your argument by (wrongly) tying it to specific functions.

  2. Scott Young says:


    Thanks for the mention!


    It’s a fair criticism, but my point was to make an analogy. Exponential growth has a compounding feature, which people familiar with interest payments can readily relate. Quadratic growth and factorial growth are both superlinear but lack that characteristic. The point wasn’t to conflate exponential with all types of superlinear growth, but to, at least attempt, to claim that the exponential curve as a category of growth is an important one to understand and remember.


  3. Dave Carlson says:

    It’s also important to consider what you’re measuring, which you hinted at in your post. Many things follow a logarithmic curve, where empirical progress grows more slowly as you approach mastery. However, if you’re measuring performance relative to others, I think it’s a much more linear distribution. To continue your elite athlete example, as you approach the flat portion of the logarithmic curve, each second becomes more meaningful. That one second will move you ahead of a lot more people than a one-second improvement would have when you were at a very pedestrian level.

    For another example, look at hitting in baseball. A lot of major league pitchers were the best hitters on their high school (and perhaps even college) teams, but struggle in the majors because they simply don’t get as much time to practice. Position players receive extra practice, which may help them change their bat’s position by a fraction of a centimeter or speed their swing or reaction time by a hundredth of a second, but those slight differences are often the difference between hitting .200 and hitting .300. So though the empirical gains are small, the effects are large when you get to an elite level. So it’s meaningful to consider what your desired goals are – to be the best at something, or simply have proficiency, and that can shape the kind of curve you learn to anticipate.

    • Ben Casnocha says:

      Excellent point that the difference between a very small change in performance will have a greater impact in the pros. And so the importance of this overall idea goes back to your goals and likely situation, as you say.

  4. DaveJ says:

    A few points. First, exponential growth basically never continues that way, because of finite resources / customers / competitors / etc. They eventually start to converge toward a maximum.

    Indeed, if you look at the full time scale on both things like athletic performance and things like blogs, they both have a sigmoid (s-curve) shape, such a this: They start out slowly (learning is slow at first because everything is new; subscriber growth is slow because there is not a base that is communicating). Then, there is a steeper growth period that may look exponential or otherwise super-linear – or even steeply linear. Finally, some sort of saturation is approached – either because we get close to the human limits of performance, the entire target market has awareness, etc.

    The lesson is that it’s not a question of *which kind* of curve you’re on, but *where you are on the curve*.

    • Ben Casnocha says:

      Agree with the saturation point, but I’m not sure this means the *kind* of trajectory your own — and its characteristics — until you reach saturation isn’t also something to reflect upon as you learn a new skill.

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