Lifting Up the Edges of the Distribution

Since I started lifting weights, I’ve found curious the lack of any heuristics to calculate a one rep max based on a particular number of sets and reps. A number of calculations exist for a single set (see below chart from Wikipedia), but as pretty much all programming involves both sets and reps (e.g., bench press 5 sets of 5 reps at 225 pounds), being able to estimate the difficulty of a workout by translating the effort into a one rep max seems like it would be useful.

The main barrier to a formula including both sets and reps is the complexity of the relationship and the corresponding variation observed across different individuals. Even the one rep max approaches exhibit a significant amount of variation (especially at higher reps), and that’s without the added considerations of rest time and recharge rate.

That being said, it’s not too difficult to come up with a directionally accurate formula, as the relationship between additional sets and overall strength is clear cut. For example, we could modify the Epley formula with a multiplier for the number of sets.

Epley Formula (Original): 1RM = weight * (1 + 0.033 * reps)
Epley Formula with Sets: 1RM = weight * (1 + 0.033 * reps) * (1 + 0.01 * sets)

I’m far from certain about the exact value of the multiplier; 0.01 is a rough estimate based on the few data points I have available. However, I’m very confident that the multiplier is closer to 0.01 than to 0.1 (which would imply a 1RM nearly double that of a 5×5 max) or 0.001 (which would discount additional sets to essentially 0). Even more so, I’m confident in the direction of the relationship – doing more sets of the same lift unequivocally represents increased strength.

One interesting facet of this problem is that it’s essentially unexamined. A few different times I’ve tried searching online for formulas, and each time all I’ve been able to find are the formulas described above for a single set. This isn’t too surprising, as the difficulties mentioned mean it’s unlikely any single formula would have broad applicability, but it also means that the problem makes for a good “blind” test of large language model capabilities. While recent metrics such as 90th percentile performance on the LSAT, SAT, and GRE Verbal are extremely impressive, they’re lessened somewhat by the fact that these types of questions are commonplace within the training data for these models. Unfortunately, the scope of training means pretty much every common problem can be found within the training data, which is why my attention was drawn to the sets and reps formulation. 

Here’s a prompt to GPT-4 on the question and its response (bolding my own): 

Prompt

I’m familiar with many formulas to calculate a one rep powerlifting max based on reps of a particular weight, but I can’t find any which look at both reps and sets. Understanding that the relationship there will be more complex (and will also have a rest time component), could you lay out a potential approach and use it to calculate a one rep max for someone who can do 5x5x225 and someone who can do 1x5x225?

GPT-4 Response

Certainly. Determining a one rep max (1RM) considering both sets and reps is indeed more complex because it involves taking into account variables such as rest times, muscular fatigue, and an individual’s conditioning. Traditional 1RM calculation formulas primarily focus on the maximum weight that can be lifted for a specific number of repetitions (often to failure), not considering the number of sets performed.

However, we can modify the existing formulas a bit to account for the sets. We could potentially model this using a fatigue factor that decreases the effective weight lifted in subsequent sets. However, it’s essential to note that this is a very rough approximation, and there are many factors that it won’t take into account.

Here’s a possible approach:

1. Determine an initial 1RM using a standard formula, based on the first set. We’ll use the Epley formula for simplicity: 1RM = weight + (weight * reps * 0.0333). For example, for a 5×225 set, that would be: 225 + (225 * 5 * 0.0333) = 261.2 lbs.

2. Determine a fatigue factor. This is the tricky part and will largely be a guess. As an example, let’s suppose the fatigue factor is 2% per set (i.e., you can lift 2% less weight with each subsequent set). This is very rough and would vary widely from person to person and even from workout to workout.

3. Apply the fatigue factor to each subsequent set to calculate the effective weight, and then calculate an effective 1RM from each set.

4. Average the 1RMs from each set to get the final 1RM.

Let’s use this approach for the two examples:

Example 1 (5x5x225)

1. Calculate initial 1RM for first set: 225 + (225 * 5 * 0.0333) = 261.2 lbs

2. Let’s assume a fatigue factor of 2% per set

3. Apply fatigue factor to calculate effective 1RMs:
   1. Set 2: 225 * 0.98 = 220.5 lbs. Effective 1RM = 220.5 + (220.5 * 5 * 0.0333) = 256.3 lbs
   2. Set 3: 220.5 * 0.98 = 216.1 lbs. Effective 1RM = 216.1 + (216.1 * 5 * 0.0333) = 251.6 lbs
   3. Set 4: 216.1 * 0.98 = 211.8 lbs. Effective 1RM = 211.8 + (211.8 * 5 * 0.0333) = 247.0 lbs
   4. Set 5: 211.8 * 0.98 = 207.6 lbs. Effective 1RM = 207.6 + (207.6 * 5 * 0.0333) = 242.6 lbs
4. Average the 1RMs: (261.2 + 256.3 + 251.6 + 247.0 + 242.6) / 5 = 251.7 lbs

So, the estimated 1RM for a 5x5x225 workout would be 251.7 lbs using this model.

Example 2 (1x5x225)

In this case, there is only one set, so there’s no need to calculate the effective weight for subsequent sets or average multiple 1RMs. You can just calculate the 1RM from the single set using the Epley formula: 225 + (225 * 5 * 0.0333) = 261.2 lbs.

So, according to this model, someone who can do 5 sets of 5 reps with 225 lbs has an estimated 1RM of 251.7 lbs, while someone who can do just 1 set of 5 reps with the same weight has an estimated 1RM of 261.2 lbs.

Again, it’s important to note that this is a very rough approximation. There are many other factors at play in a real-world setting, such as individual physiology, training experience, rest between sets, nutrition, and more.

Well, that’s certainly a different way of looking at the problem. To summarize GPT-4’s approach, it’s saying that, due to fatigue, one can lift ~2% less each set (which actually seems pretty reasonable), and it’s then adjusting down the calculated 1RM for each set due to fatigue (which makes no sense). Someone who can lift 225 pounds 5 times after doing other activities has a 1RM of at least the amount implied by just lifting 225 pounds 5 times, not less, as suggested by this approach.

To make sure this wasn’t just a fluke response, I tried several slightly different variations of the question, and each time received a response similar to the above, with more reps resulting in a lower calculated 1RM (though the exact formula used differed). 

This issue reminds me of DALL-E 2’s inability to draw a chair with two legs. With their massive number of parameters and terabytes of training data, these models learn powerful behavior within a broad training distribution, but can only extrapolate so far beyond it. Two-legged chairs don’t make it into image training sets, and so even with DALL-E 2’s fairly robust concept of “two”, its much stronger concept of “chair = 3+ legs” prevailed. For GPT-4, the problem of incorporating sets and reps forced it out of its “comfort zone” (training distribution), and so it threw related concepts (like fatigue) at the problem fairly indiscriminately, resulting in its nonsensical answers. 

Perhaps this problem of extrapolation will be solved by further scale. While GPT-3 often seemed like it was just mimicking human text, GPT-4 often displays something like understanding, though it’s difficult to assess what portion of the difference is due to its larger training corpus. In principle, it seems possible for incremental data and training to drive a step change in the distillation of concepts, although that may just be because we haven’t yet figured out the principles governing this space. For now, we’ll have to rely on human ingenuity to figure out whether doing once or five times takes more effort.