Global and Local Maxima

A function reaches a global maximum where its value is at least as high as at every other point in its domain. It reaches a local maximum where its value is at least as high as at nearby points. A global maximum can also be a local maximum, and several points can share the same maximum value. Some functions have no maximum at all.

Picture a landscape with several hills. Walking uphill from your starting point may take you to the top of a nearby hill without taking you to the highest point in the landscape. In computer science, one such search method is called “hill climbing,” because it repeatedly moves to a better nearby solution. It searches the landscape; it does not create it. As Brian Christian and Tom Griffiths explain, “the search through a space of solutions, some better and some worse, is commonly thought of in terms of a landscape with hills and valleys, where your goal is to reach the highest peak.”

If every small change makes a solution worse, an approach that accepts only immediate improvements can get stuck. A broader search might require a different starting point, a larger change, or a temporary loss:

We may need to temporarily worsen our solution if we want to continue searching for improvements.

—Brian Christian and Tom Griffiths

In “Escaping Local Maxima,” Dave Rael describes the “chasms between where we are and where we could be [are] opportunities to improve by finding ways of moving from a place where progress is flat to find a new slope to climb.” The practical question is whether the changes you are considering merely refine the current approach or open up a different set of possibilities.

The Jogbra story

In 1977, Lisa Lindahl developed a working sports-bra prototype. Turning it into a business brought unfamiliar problems: production, sales, distribution, and marketing. Lindahl and her partners tried approaches that failed, including a mittens line called Jogmit, while finding a market for Jogbra in sporting goods stores. Later, selling the company to Playtex brought another transition.

The story illustrates the limits of improving only what you already know how to do. A good prototype did not settle the manufacturing or sales questions. Lindahl had to learn skills for which success with the original design provided little preparation. She describes the experience this way:

“You make your plans looking upward toward your goal, only to reach it and find that what you thought was the pinnacle, the ceiling of your endeavor, was in fact only the floor of your next level.”

The analogy has limits. Business success is not a fixed landscape that can be fully mapped, and a failed experiment is not necessarily a step toward a better outcome. The Jogbra account shows learning through changes in approach; it does not prove that every setback is necessary or beneficial.

Deciding whether to search further

Start by defining what you want to improve. In engineering it might be efficiency within constraints on cost and reliability. In a career decision it might be a combination of pay, time, and work you value. Without a clear objective, calling a different option a higher peak says little.

Then ask what you have actually compared. A design can be the best of several small variations while a different architecture remains unexplored. A job can be attractive among the roles you know without being the best job you could possibly have. Usually you cannot establish a global maximum in such open-ended choices.

Search has costs as well as potential benefits. A small prototype, trial project, or limited period of retraining can help you investigate an alternative without committing everything to it. Decide what evidence would justify a larger move and how much temporary loss you can afford. A plateau is a reason to consider other approaches, not a promise that a better outcome lies beyond it.