Optimizers

Optimizers implement the update rule used to modify the optimized parameters at each step of the optimization process. Dr.TVAM uses the Optimizer sub-classes from Mitsuba 3, therefore any optimizer available in Mitsuba 3 can be used in Dr.TVAM. We additionally provide two variants of the L-BFGS optimizer, which we will detail shortly.

Optimizers in mitsuba are instatiated as regular objects:

opt = mi.ad.Adam(lr=1e-2)

They maintain a dictionary-like interface to set and access optimization parameters. Marking a parameter as optimizable is done like so:

opt['x'] = mi.TensorXf(1.0)

From that point on, opt['x'] will be considered as a differentiable quantity, and will get a gradient after backpropagation. Please see the corresponding tutorial in the Mitsuba 3 documentation for more details.

Mitsuba 3 provides the following optimizers:

• SGD

• Adam

Additionally, we provide two variants of the L-BFGS optimizer:

L-BFGS (LBFGS)

This optimizer implements the classic limited-memory version of the BFGS algorithm. It is a quasi-Newton method that approximates the Hessian matrix of the loss function using past gradients.

The following additional parameters should be provided:

Key	Type	Description
m	int	The number of past gradients to store. Typical values aare between 5 and 10. A higher value will require more memory, as more gradients are stored, which can be prohibitive for large problems. Defaults to 5.
line_search_fn	function	The L-BFGS update determines the step size by performing a backtracking line search, which evaluates the loss for varying step sizes until one satisfying the Wolfe conditions is found. The line search function should take a dictionary of updated parameters as input, and return the loss value for those parameters.
wolfe	bool	Whether to use the Wolfe conditions for the line search or only the simpler Armijo rule. Defaults to False (i.e. Armijo rule).
search_it	int	The maximum number of iterations for the line search. Defaults to 20.

Linear L-BFGS (Linear LBFGS)

The main use case of Dr.TVAM is to optimize patterns for printing. In that case, the forward model is linear with respects to the patterns, which enables a nice performance optimization: the line search in L-BFGS requires evaluating the forward model at each step, which can be expensive. If the operation is linear, we only need to compute it once for the search direction \(d\):

\[\mathcal{L}(f(x + \alpha d)) = \mathcal{L}(f(x) + \alpha f(d))\]

Then, only the loss function \(\mathcal{L}\) needs to be evaluated at each line search step, which is much faster than evaluating the full forward model.

This optimizer requires the following additional parameters:

Key	Type	Description
m	int	The number of past gradients to store. Typical values aare between 5 and 10. A higher value will require more memory, as more gradients are stored, which can be prohibitive for large problems. Defaults to 5.
render_fn	function	The line search function from L-BFGS is now split in two parts. The render_fn evaluates the forward model, given a dictionary of parameters. It should return the recorded dose in the medium, i.e. the output of the rendering operation.
loss_fn	function	A function that takes as argument the recorded dose in the medium, and returns the loss value. This is the function that will be evaluated in the backtracking line search.
search_it	int	The maximum number of iterations for the line search. Defaults to 20.