The Score Function Estimator

import torch

# True target distribution probabilities
true_probs = torch.tensor([0.1, 0.6, 0.3])

# Optimisation parameters
n_batch = 1000
n_iter = 3000
lr = 0.01

def loss(x):
    """
    The target loss, a negative log-likelihood for a
    categorical distribution with the given probabilities.
    """
    return -torch.distributions.Multinomial(
        total_count=1, probs=true_probs).log_prob(x)


# Set the seed for reproducibility
torch.manual_seed(42)

# Initialize the parameter estimates
theta_hat = torch.nn.Parameter(torch.tensor([0., 0., 0.]))
optimizer = torch.optim.Adam([theta_hat], lr=lr)

for epoch in range(n_iter):
    optimizer.zero_grad()
    # Sample from the estimated distribution
    x_sample = torch.distributions.Multinomial(
        1, logits=theta_hat).sample((n_batch,))
    # evaluate log density at the sample points
    log_p_theta_x = torch.distributions.Multinomial(
        1, logits=theta_hat).log_prob(x_sample)
    # Evaluate the target function at the sample points
    f_hat = loss(x_sample)

    # Compute the gradient of the log density wrt parameters.
    # The `grad_outputs` multiply the `f_hat` by gradient directly.
    grad_log_p_theta_x = torch.autograd.grad(
        outputs=log_p_theta_x,
        inputs=theta_hat,
        grad_outputs=torch.ones_like(log_p_theta_x),
        create_graph=True)[0]

    # The final gradients are weighted over the sample points
    final_gradients = (
        f_hat.detach().unsqueeze(1)
        * grad_log_p_theta_x
      ).mean(dim=0)
    theta_hat.grad = final_gradients

    optimizer.step()

    if epoch % 100 == 0:
        print(f"Epoch {epoch}, Estimated Probs:"
          f"{torch.softmax(theta_hat, dim=0).detach().numpy()}")

# Display the final estimated probabilities
estimated_final_probs = torch.softmax(theta_hat, dim=0)
print("Final Estimated Probabilities: "
  f" {estimated_final_probs.detach().numpy()}"
  f" (True Probabilities: {true_probs.detach().numpy()}")