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objectives
Objectives

In BoTorch, an objective is a module that allows for convenient transformation of model outputs into a scalar function to be optimized. Typical use cases for this are the scalarization of outputs for a multi-output model (see e.g. 1), or optimization subject to outcome constraints, which can be achieved by weighting the objective by the probability of feasibility 2.

When using classical analytic formulations of acquisition functions, one needs to be careful that the transformation results in a posterior distribution of the transformed outputs that still satisfies the assumptions of the analytic formulation. For instance, to use standard Expected Improvement on a transformed output of a model, the transformation needs to be affine (because Gaussians are closed under affine transformations). When using MC-based acquisition functions, however, fewer assumptions are required, and one can apply general transformations to the model outputs with relative impunity so long gradients can be back-propagated through the transformation.

All BoTorch objectives are derived from MCAcquisitionObjective. BoTorch implements several MC-based objectives, including LinearMCObjective for linear combinations of model outputs, and ConstrainedMCObjective for constrained objectives (using a sigmoid approximation for the constraints).

Using custom objectives

Utilizing GenericMCObjective

The GenericMCObjective allows simply using a generic callable to implement an ad-hoc objective. The callable is expected to map a sample_shape x batch_shape x q x o-dimensional tensor of posterior samples and an (optional) batch_shape x q x d-dimensional tensor of inputs to a sample_shape x batch_shape x q-dimensional tensor of sampled objective values.

For instance, say you have a multi-output model with $o=2$ outputs, and you want to optimize a $obj(y) = 1 - \|y - y_0\|_2$, where $y_0 \in \mathbb{R}^2$. For this you would use the following custom objective (here we can ignore the ninputs $X$ as the objective does not depend on it):

obj = lambda xi, X: 1 - torch.norm(xi - y_0, dim=-1)
mc_objective = GenericMCObjective(obj)

Implementing a custom objective module

Instead of using GenericMCObjective, you can also implement your own MCAcquisitionObjective modules to make them easier to re-use, or support more complex logic. The only thing required to implement is a forward method that takes in a sample_shape x batch_shape x q x o-dimensional tensor of posterior samples and maps it to a sample_shape x batch_shape x q-dimensional tensor of sampled objective values.

A custom objective module of the above example would be

class MyCustomObjective(MCAcquisitionObjective):

    def forward(self, samples, X=None):
      return 1 - torch.norm(samples - y_0, dim=-1)

Footnotes

  1. B. Paria, K. Kandasamy, and B. Póczos. A Flexible Multi-Objective Bayesian Optimization Approach using Random Scalarizations. ArXiv, 2018. ↩

  2. B. Letham, B. Karrer, G. Ottoni and Bakshy, E. Constrained Bayesian Optimization with Noisy Experiments. Bayesian Analysis, 2018. ↩