Yes, there is a scientifically proven formula for improving and learning quickly. For guitar (and all musical instruments) the formula is the same. Of course, it's not as simple as A + B = C, but it's not calculus either. Let's say you have a phrase of music you are learning or want to improve; here's how you should go about it:
1) PLAY PHRASE AS PRECISELY & RELAXED AS POSSIBLE
This means NO MISTAKES! And you need to be using as little effot as possible. Think of yourself as a brain surgeon in surgery. If you're too tense and not careful, you'll mess up. One tiny mistake and...very bad consequences. As a musician you need to have brain surgeon precision and hardly any tension.
2) REST & RELAX
A 2005 scientific study in "Experimental Brain Research" showed that waiting a few seconds between repetitions (e.g., between 2-5 seconds would be ideal for guitar) significantly increases learning speed and long-term rention. 1 second or less was shown to not have any benefit. So after you play your phrase, drop your hands/arms and relax for 2-5 seconds. DO NOT MINDLESSLY REPEAT!!
3) REPEAT YOUR PHRASE & PUSH YOURSELF
Always start your phrase from the last few notes of the previous phrase (assuming there is a previous phrase). You'll tactle your transitions right away and reduce practice time. When you repeat your phrase, do it as you did in STEP 1 above, but try to push yourself just a little. If you push too hard, you'll tense up and make mistakes, so push yourself just enough so that you maintain precision and relaxation. (You will make mistakes here. We'll talk about mistakes at the end of this post).
4) REST & RELAX AGAIN (same as step 2)
5) REPEAT STEPS 1-4 BETWEEN 5-10 TIMES ONLY
Scientific studies have found that most people need an AVERAGE of 5-7 repetitions for material to "sink in." In the end, it's not that cut and dry. The amount of repetitions really depends on both the complexity of the musical phrase and individual differences (e.g., how relaxed you are). Assuming you are correctly following STEPS 1-4 above, your repetition amount may be anywhere between 5-10 times. Your body (and fingers) will let you know. That said, DO NOT REPEAT MORE THAN 10 TIMES! Block practicing (or repeating the same thing over and over countless times) feels productive in the moment, but plenty of scienfitic studies have shown that this kind of repetition is actually either counterproductive or simply unproductive. In other words, you either get worse or don't get any better. Interleaved and spaced practice (STEP 6) are best!
6) STUDY SOMETHING ELSE USING THE SAME STEPS 1-5 (Interleaved & Spaced Practice)
Scientific studies have shown that interleaving or mixing up your practice material helps you retain information faster. After you practice your musical phrase from the steps above, stop, and do the same thing with a different musical phrase -- same or different piece of music. For my own practice, I've found that practicing 4-6 different musical phrases (depending on phrase lnegth) works well before going back to the beginning. My practice looks something like this:
- Musical phrase 1 (steps 1-5)
- Musical Phrase 2 (steps 1-5)
- Musical phrase 3 (steps 1-5)
- Musical phrase 4 (steps 1-5)
- Repeat X 2
I'd recommend trying 3 different musical phrases to start and adding in more if you have extra time.
Read about Interleaved Practice here:
We will make mistakes no matter what we do to avoid them -- especially in step 3 -- and THAT'S GREAT! Mistakes actually help us improve if we deal with them "deliberately" and use tem as learning opportunities (just like real life!). Remember, you're a brain-surgeon musician so you need to deal with mistakes like a brain surgeon in surgery. Find the mistake AS SOON AS IT HAPPENS and FIX IT ASAP! You'll first need to do some trial and error to find your own solution. When you have the solution, you need to play your phrase corectly 2-3 times (using your solution) to "erase" the mistake you made and make progress. For every 1 mistake you make, you need to play your phrase correctly 2-3 times so that your brain recognizes your soution as the preferred option.
Here's a great article from Bulletproof Musician on how experts get even better: