PYTHON · NUMPY

The integral you cannot do

Show why fitting an energy-based model needs sampling, by watching the partition function become intractable in dimension.

01Challenge

Normalise an energy by quadrature in one dimension, then in five, and read the cost.

Predict firstAt fifty-one points per axis, how many energy evaluations does a seven-joint arm need to normalise by quadrature?

02Model

An energy defines a probability only after you divide by the sum of the exponential over every possible answer. That sum is the partition function, and it is the reason this whole story happens. In one dimension you can compute it by quadrature at negligible cost. In the dimension of a robot action it is a sum over a continuum in every joint at once, and the cost of quadrature is the grid resolution raised to the number of dimensions. There is no cleverness in the loop that removes this; it is the volume of the space. So the field does the only other thing available, which is to avoid computing the normaliser and instead estimate the gradient using samples drawn from the model. That decision, forced by an integral, is what put a Markov chain in the inner loop.

03Guided practice

Compute the partition function by quadrature at increasing dimension and record the evaluation count.

04Feedback

PASS WHEN You watched the evaluation count grow as resolution raised to the dimension.

If your run fails, check:

Fifty-one points per axis is a coarse grid, and seven joints still puts the count past a trillion. The problem is the volume, not the arithmetic.
Nothing in this is a bad algorithm. It is the definition of a normalising constant over a continuous space.
Because the integral is out of reach, the gradient is estimated from samples instead. That substitution is the hinge of the whole course.

05Retrieve & space

What exactly is the partition function summing over?
Why does a faster processor not rescue quadrature here?
What does the field do instead, and what does that put in the inner loop?

06Mastery & project

Explain why an energy-based model needs sampling, in terms of the dimension of the answer space.

Estimate the action dimension of a system you work on and compute the quadrature cost at fifty points per axis. Report the exponent, not the number.

GO FURTHER · THE LIVE PLAYGROUND

Open the lottery bench ↗ · One energy landscape, two substrates, and the crossover between them.

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PYTHON · NUMPY

The integral you cannot do

One energy, a grid, and dimensions from one to six.

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