Finding the Maximum Peak of Success when Iterating and Pivoting
If you’re a fan of the Lean approach to product development, you’ve probably heard of the local maximum problem. In his book Scaling Lean, Ash Maurya warns of the perils of getting stuck in a local maximum when doing small iterations while searching for improvements in business model or product.
The success of a product experiment can be represented by a mathematical surface with mountains and valleys were the height of the surface represents the amount of success. Unfortunately, this surface isn’t a priori visible, otherwise every business could jump directly to the highest peak and simply build the optimal, most profitable perfect solution from the get go.
At the core of the Lean approach is the Build -> Measure -> Learn cycle where each iteration or pivot is a jump to a new area to explore in the space of products and solutions.
The problem of finding the peak is often compared to the hill climbing problem from computer science . You can get a sense of it by imagining trying to find the highest peak by doing small jumps in the upwards sloping direction. Once you reach a small peak you stall there.
Entrepreneurial success is sought in a very irregular multidimensional space. There are so many things you can try that don’t work it’s scary. The true home-runs, are not predictably placed but can only be found by jumping around the space, measuring the results of each iteration and hopefully learning some insight as you go to help aim the next jumps.
Hill climbing is a very primitive algorithm. Computer science has much better ways to navigate hidden irregular curves. One of the more sophisticated method is called the Metropolis-Hastings (MH) algorithm. It offers a solution to the problem of getting stuck. This algorithm is part of the Monte Carlo family of algorithms, named after the casino because it’s based on the act of taking random samples, a bit like throwing dices in a casino. It is usually used for curves which can’t be easily calculated otherwise. Like we do in Lean, it finds the shape of a curve and its peaks by doing calculated random jumps.
Metropolis Hastings is one of the best method for highly dimensional problems where trying completely random points is too slow.
The essence of the MH algorithm comes down to defining the size of the jumps through the space as well as the origin of each jumps.
Basically for each iteration, two decisions need to be made:
1. How far you want to jump for your next attempt and how different and original an hypothesis do you want to try?
2. Do you pivot from your very last attempt or if that last attempt was a failure, do you reject it, back up to the better solution you had before and try a variation on that one instead?
The MH algorithm has an equation that ties 1. and 2.. I am going to try to give an intuitive sense of how it works.
A common method for 1. is for the size of the jumps to follow a bell curve influenced by a prior. That is a high number of small jumps around the current position to best guess positions, a medium number of medium length jumps and occasional long jumps to unlikely positions. Short jumps change only a few variables and only by a little. Long jumps can change multiple variables in multiple dimensions, features, technologies, marketing and distribution channels, ideal customers etc., and change them more. The longer jumps are often called pivots and the shorter jumps called iterations but the delineation is subjective.
How do you decide where its best to start your next jump from? Basically MH says that if your last jump had better results than the one before, always jump from that newest point. Intuitively, it means that you probably had some improvements that are worth keeping for the next attempt. This is the most common case.
What if the last attempt gave a worst outcome than the one before? Then it comes down to how much of a failure it was and how unlikely the jump was. If you did a long jump, tried something fairly different, original or unintuitive and ended up with a slightly worse result, it’s possible that you ended up in a valley next to a new very high peak and you should do a few (likely smaller) iterations around this new location to make sure you don’t miss the peak). If this isn’t fruitful, you will eventually want to long jump away from that area to a safer best guess.
If the jump was an obvious guess yet it made things worse, or if the jump made things a lot worse, then discard this attempt, revert to the previous position and pivot from there instead.
Those are, vaguely, the probabilistic rules of Metropolis–Hastings.
The nice thing about it is that it is guaranteed (under reasonable assumptions and at the limit of an infinite number of tries) to not get stuck on a local maximum and to cover the whole search space spending proportionately more time on peaks than valleys, allowing you to discover those peaks.
Theoretically if the space is too irregular, and has vast valleys between promising regions you can still get stuck in a particular region for a very long time but it is still a massive improvement over hill climbing which is likely to quickly get stuck in a mediocre spot forever.
For finding business models, crossing very long valleys would be akin to changing industry and technology completely and you probably don’t want to do that anyways. Practically you are stuck in your your region. MH fits this reality and is good for exploring a multi peak local-ish region.
Of course, I’m not the first one to advocate for an approach that spends time in mediocre outcome valleys. Seth Godin for example, makes a good case for spending unpleasant time there in order to get to higher peaks. To me, the MH algorithm makes a compelling mathematical case that this kind of intuition is correct.
The approach works best if it integrates human knowledge and intuition. Entrepreneurs are usually smart and can generally do much better than random jumps. This kind of intuition is what is known as the prior in probabilistic methods. Basically you can tune the length and direction of your jumps to go more towards regions that your knowledge and experience tell you are better. Occasionally though, you should try something a bit more crazy, something you didn’t necessarily think would work.
As a math lover, it makes me happy that there exists a probabilistic algorithm that I can use as a metaphor to guide product or business pivots and that it provides a provably correct common sense approach to tackle this type of complex and difficult problem.