Entropic Mirror Simplex Update
Update probability weights multiplicatively while paying an explicit KL movement cost.
Free theory articleUpdate probability weights multiplicatively while paying an explicit KL movement cost.
Concept & interpretation
Mirror descent offers a principled way to optimize functions over constrained spaces. When updating weights on a probability simplex, the standard Euclidean gradient descent can easily push weights out of bounds. The entropic mirror update uses the Kullback-Leibler (KL) divergence as its proximity measure, resulting in multiplicative updates that naturally remain within the simplex.
Experimenting with the Update
You can interactively explore the entropic mirror descent using the Entropic Mirror Simplex Update module:
- Set Prior to "Uniform" and Losses to "Increasing". Observe how the trace moves towards the coordinate with the lowest loss (Coordinate 1).
- Increase the Learning rate to "2". The trajectory becomes steeper, reaching a more concentrated distribution at the same number of steps.
- Change Prior to "Sparse". Notice that Coordinate 3 remains exactly at zero; the multiplicative update cannot revive a component once it has zero probability.