# Miscellaneous Ops¶

The pyro.ops module implements high-level utilities that are mostly independent of the rest of Pyro.

class DualAveraging(prox_center=0, t0=10, kappa=0.75, gamma=0.05)[source]

Bases: object

Dual Averaging is a scheme to solve convex optimization problems. It belongs to a class of subgradient methods which uses subgradients to update parameters (in primal space) of a model. Under some conditions, the averages of generated parameters during the scheme are guaranteed to converge to an optimal value. However, a counter-intuitive aspect of traditional subgradient methods is “new subgradients enter the model with decreasing weights” (see $$[1]$$). Dual Averaging scheme solves that phenomenon by updating parameters using weights equally for subgradients (which lie in a dual space), hence we have the name “dual averaging”.

This class implements a dual averaging scheme which is adapted for Markov chain Monte Carlo (MCMC) algorithms. To be more precise, we will replace subgradients by some statistics calculated during an MCMC trajectory. In addition, introducing some free parameters such as t0 and kappa is helpful and still guarantees the convergence of the scheme.

References

[1] Primal-dual subgradient methods for convex problems, Yurii Nesterov

[2] The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, Matthew D. Hoffman, Andrew Gelman

Parameters: prox_center (float) – A “prox-center” parameter introduced in $$[1]$$ which pulls the primal sequence towards it. t0 (float) – A free parameter introduced in $$[2]$$ that stabilizes the initial steps of the scheme. kappa (float) – A free parameter introduced in $$[2]$$ that controls the weights of steps of the scheme. For a small kappa, the scheme will quickly forget states from early steps. This should be a number in $$(0.5, 1]$$. gamma (float) – A free parameter which controls the speed of the convergence of the scheme.
step(g)[source]

Updates states of the scheme given a new statistic/subgradient g.

Parameters: g (float) – A statistic calculated during an MCMC trajectory or subgradient.
get_state()[source]

Returns the latest $$x_t$$ and average of $$\left\{x_i\right\}_{i=1}^t$$ in primal space.

velocity_verlet(z, r, potential_fn, step_size, num_steps=1)[source]

Second order symplectic integrator that uses the velocity verlet algorithm.

Parameters: Return tuple (z_next, r_next): z (dict) – dictionary of sample site names and their current values (type Tensor). r (dict) – dictionary of sample site names and corresponding momenta (type Tensor). potential_fn (callable) – function that returns potential energy given z for each sample site. The negative gradient of the function with respect to z determines the rate of change of the corresponding sites’ momenta r. step_size (float) – step size for each time step iteration. num_steps (int) – number of discrete time steps over which to integrate. final position and momenta, having same types as (z, r).
single_step_velocity_verlet(z, r, potential_fn, step_size, z_grads=None)[source]

A special case of velocity_verlet integrator where num_steps=1. It is particular helpful for NUTS kernel.

Parameters: Return tuple (z_next, r_next, z_grads, potential_energy): z_grads (torch.Tensor) – optional gradients of potential energy at current z. next position and momenta, together with the potential energy and its gradient w.r.t. z_next.
newton_step_2d(loss, x, trust_radius=None)[source]

Performs a Newton update step to minimize loss on a batch of 2-dimensional variables, optionally regularizing to constrain to a trust region.

loss must be twice-differentiable as a function of x. If loss is 2+d-times differentiable, then the return value of this function is d-times differentiable.

When loss is interpreted as a negative log probability density, then the return value of this function can be used to construct a Laplace approximation MultivariateNormal(mode,cov).

Warning

Take care to detach the result of this function when used in an optimization loop. If you forget to detach the result of this function during optimization, then backprop will propagate through the entire iteration process, and worse will compute two extra derivatives for each step.

Example use inside a loop:

x = torch.zeros(1000, 2)  # arbitrary initial value
for step in range(100):
x = x.detach()          # block gradients through previous steps
x.requires_grad = True  # ensure loss is differentiable wrt x
loss = my_loss_function(x)
x = newton_step_2d(loss, x, trust_radius=1.0)
# the final x is still differentiable

Parameters: loss (torch.Tensor) – A scalar function of x to be minimized. x (torch.Tensor) – A dependent variable with rightmost size of 2. trust_radius (float) – An optional trust region trust_radius. The updated value mode of this function will be within trust_radius of the input x. A pair (mode, cov) where mode is an updated tensor of the same shape as the original value x, and cov is an esitmate of the covariance 2x2 matrix with cov.shape == x.shape[:-1] + (2,2). tuple