PYTHON · NUMPY

What a faster simulator is worth

A simulator gets 19.91x faster. Which algorithm's training run benefits most: a model-free method that treats the simulator as a black box, or an analytic-gradient method that differentiates through it?

Decompose published training runtimes into the time spent simulating and the time spent learning, and show that a simulator speedup is worth most to the method that needs it least.

01Challenge

A new differentiable simulator is 19.91 times faster than the one it replaced, and the method built on it wins every task. Work out how much of each algorithm's training run the simulator actually is, what the speedup is worth to each, and what a perfect simulator would still leave on the table.

Predict firstA simulator gets 19.91x faster. Which algorithm's training run benefits most: a model-free method that treats the simulator as a black box, or an analytic-gradient method that differentiates through it?

02Model

You have met the simulator as something to trust or distrust. Here it is as something to pay for, and the question is which part of a training run you are actually paying for. A 2025 paper introduces a parallel differentiable multiphysics simulator that runs 19.91 times faster than the non-parallel implementation it replaced, and a new algorithm on top of it that takes gradients straight through the physics. The algorithm wins every task. The paper also does something papers rarely do: it times the simulator on its own, with random actions and no policy attached, and separately times each algorithm's whole training run on the same task and machine. Those two tables together let you split every run into two parts, the seconds spent simulating and the seconds spent on everything else, which for a gradient method means recording the tape and running the backward pass through contact. What you find is Amdahl's law with a twist. All five algorithms spend the same hour inside the simulator, because they share the same number of environments and the same horizon. For the model-free baseline that hour is most of the run. For the method that differentiates through the physics it is a small fraction, so the speedup the paper advertises mostly accelerates its competitors. The method is still better. The table simply cannot show you where its time goes, and you can.

03Guided practice

1. The simulator's seconds are shared. All five algorithms use 32 environments and a horizon of 32 on this task, so all five spend FWD_S seconds simulating. What differs is everything after. 2. One line. The share is FWD_S / (runtime_hours * 3600). It has to land between 0 and 1. If SAPO comes out near 6.5 you divided the wrong way round. 3. The ceiling is 1/(1 - share). That is how much faster a run can get if the simulator becomes free, which is Amdahl's law. It is the same shape as the ceiling on relocating a decision in PAI-360. 4. Then re-score per hour. The paper matches algorithms by environment steps, which is fair for comparing sample efficiency. Dividing by wall clock instead does not change the winner, but it changes the size of the win by exactly the ratio of the two runtimes.

04Feedback

PASS WHEN You split five published training runs into simulation and learning, and found that a 19.91x simulator is worth 16.49x to the method that treats it as a black box and 3.91x to the method that differentiates through it.

If your run fails, check:

The share must be between 0 and 1. If it is larger than 1 for SAPO, the division is inverted.
The runtime column is in hours and FWD_S is in seconds. One of them needs a factor of 3600.
Define sim_share(runtime_hours) returning one number, the fraction of the run spent in the forward simulation.

05Retrieve & space

The simulator was timed with random actions and no policy. What does that let you attribute to the learning, and what would you lose if it had been timed inside a training loop?
SAPO's ceiling is 1.18x. Name the change the paper itself says would raise it, and explain why that change targets the other 84.61% of the run.
The paper compares algorithms at an equal number of environment steps. When is that the right comparison, and when would a person paying for the workstation want the per-hour one?

06Mastery & project

You can take any published speedup to one component of a pipeline, find the fraction of the whole that component represents for each user of it, and state both the speedup each user actually receives and the ceiling no improvement to that component can pass.

Profile one training run of your own, or read the timings of one you trust, and split it into simulation and everything else. State the ceiling a perfect simulator would give you. Then name the one component outside the simulator you would optimise first, and say what its own ceiling is.

GO FURTHER · THE LIVE PLAYGROUND

The paper ↗ · Xing, Luk and Oh, ICLR 2025: the Rewarped simulator and SAPO, with the simulator timed on its own in Table 16 and every algorithm's run timed in Table 18.
What a simulator is worth ↗ · The companion question: not how fast the simulator is, but how many hardware trials its answers can replace.
PAI-360 · What a Decision Costs ↗ · The same 1/(1 - f) ceiling, applied to relocating a decision instead of to speeding up a simulator.

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PYTHON · NUMPY

What a faster simulator is worth

One task, one machine, two published tables, and the part of a training run a faster simulator cannot reach.

You read

the arrays and values already in scope

You change

the code you write in each cell

Fixed

the dataset and the checks that grade you