Automating Mathematics

Working journal

Blending Pruning and Backprop

The problem

  • One way to evolve a system is the genetic algorithm style, where we consider the fitness of the final elements and prune based on this. This however has trivial credit assignment.
  • At the other extreme is a pure back-propogation. This has nice credit assignment, but done directly the support never changes.


  • Firstly, there must be an identity term in the evolution, so credit can go to objects just persisting.
  • Note that while the result of a big sum does not depend on terms that are zero, specifically $\mu(v)f$ with $\mu(v) = 0$, the gradient still depends on such terms. Specifically we can get a flow back to add weight to $v$
  • We decouple the big sum component, so that when we apply to an argument which is a distribution, we do not just take a set of functions depending on that distribution. We can instead take a specified support.
  • We take a big sum over all generated elements, to allow some of them to acquire weight, while computing the gradient.
  • We should periodically prune by weight, but only after flowing for long enough to allow for picking up weight.


  • We consider differentiable functions that also depend on a subset of V, by taking linear combinations including bigsums that are the union of the support of a distribution with the given subset.
  • For propagation, we just take an empty subset of V.
  • Once we have the image, we take its (essential) support as the given subset of V. We recompute the differentiable function. Note that the value is unchanged.
  • We use the gradient of the new function to back-propagate.
  • This gives a single loop, which we repeat to evolve the system.
  • Purge after looping for enough time.

Using the identity

  • The gradient of the identity does not depend on any form of a priori support.
  • Hence the gradient of a persistence term, i.e., identity multiplied by a weigh, attributes weight to everything in the final distribution.
  • Note: This lets us avoid the above complications.