navigation links

variance_training module

Optimization of invertible flows in the weighted dataset problem using the DKL loss

Reminder: we have a dataset (x,p(x),f(x)) such that - x ~ p(x) - we want to learn a model that draws points according to f(x), which is positive, and known up to normalization

We want to optimize a function q(x) such that doing importance sampling to compute f(x) with it minimizes the variance.

The variance of the importance sampling estimator is our proto-loss

pL(f,q) = ∫ dx q(x) (f(x)/q(x))^2 - (∫ dx q(x) f(x)/q(x))^2

= ∫ dx (f(x)^2/q(x)) - I(f)^2

where I(f) is the integral we want to compute and is independent of q, so our real loss is

L(f,q) = ∫ dx f(x)^2/q(x)

Which we further can compute using importance sampling from p(x):

L(f,q) = ∫ dx p(x) f(x)^2/q(x)/p(x)

Which we can compute from our dataset as the expectation value

L(f,q) = E(f(x)^2/(q(x) p(x)), x~p(x)

class BasicStatefulVarTrainer(*args, **kwargs)

Bases: zunis.training.weighted_dataset.weighted_dataset_trainer.BasicStatefulTrainer

Basic stateful trainer based on the variance loss

class BasicVarTrainer(*args, **kwargs)

Bases: zunis.training.weighted_dataset.weighted_dataset_trainer.BasicTrainer

Basic trainer based on the variance loss

weighted_variance_loss(fx, px, logqx)

Proxy variance loss for the integral of a function f using importance sampling from q, but where the variance is estimated with importance sampling from p.

We want to optimize a function q(x) such that doing importance sampling to compute f(x) with it minimizes the variance.

The variance of the importance sampling estimator is our proto-loss

pL(f,q) = ∫ dx q(x) (f(x)/q(x))^2 - (∫ dx q(x) f(x)/q(x))^2

= ∫ dx (f(x)^2/q(x)) - I(f)^2

where I(f) is the integral we want to compute and is independent of q, so our real loss is

L(f,q) = ∫ dx f(x)^2/q(x)

Which we further can compute using importance sampling from p(x):

L(f,q) = ∫ dx p(x) f(x)^2/q(x)/p(x)

Which we can compute from our dataset as the expectation value