The notion that failure is ultimately a good thing has launched about as many clichés in the 21st century as Elon Musk has launched rockets. From "failing up" to being "sharpened by fire," we love to make failure sound like a tasty appetizer before a big, satisfying meal.

I think I just came up with another failure cliché there. 

So by now we all get the idea that failure is inevitable and our fails are something we should learn from, but what is the ideal amount of failure? We can't just be failing all the time, right?

As it turns out, there may be a sweet spot for failing, according to new research out from a team led by the University of Arizona with help from researchers at Brown University, Princeton, and the University of California, Los Angeles. 

Their new study, out in the journal Nature Communications and titled "The Eighty Five Percent Rule for Optimal Learning," makes the case that getting it wrong 15 percent of the time is actually the "sweet spot" for learning.

"These ideas that were out there in the education field--that there is this 'zone of proximal difficulty,' in which you ought to be maximizing your learning--we've put that on a mathematical footing," said lead author and Arizona professor of psychology and cognitive science Robert Wilson, in a release.

The researchers conducted a series of machine-learning experiments in which computers were taught to identify handwritten numbers. The computers learned fastest when the difficulty was set to a level in which the system got it wrong 15 percent of the time. 

"If you have an error rate of 15 percent or accuracy of 85 percent, you are always maximizing your rate of learning in these two-choice tasks," Wilson said, adding that the 85 percent rule was also seen to hold in previous studies of animal learning.

He says the rule could also apply to how people learn through experience, using the example of a radiologist learning to tell the difference between images of tumors and non-tumors. 

"You get better at figuring out there's a tumor in an image over time, and you need experience and you need examples to get better," Wilson said. "If I give really easy examples, you get 100 percent right all the time and there's nothing left to learn. If I give really hard examples, you'll be 50 percent correct and still not learning anything new, whereas if I give you something in between, you can be at this sweet spot where you are getting the most information from each particular example."

So what does the 85 Percent Rule mean for those us who aren't dealing in tumors? Well, it actually comes back to one of those old failure clichés: If you aren't failing, you aren't trying. 

"If you are taking classes that are too easy and acing them all the time, then you probably aren't getting as much out of a class as someone who's struggling but managing to keep up," Wilson said.

Learning comes from challenges and challenges come with a risk of failure. What's new is that we now know that risk should be at about 15 percent.