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Multi-Fidelity and Prior Optimizers#

This section concerns optimizers that use both Multi-Fidelity and Priors. They combine the advantages and disadvantages of both methods to exploit all available information. For a detailed explanation of Multi-Fidelity and Priors, please refer here.

Optimizers using Multi-Fidelity and Priors#

1 PriorBand#

PriorBand is an extension of HyperBand that utilizes expert Priors to choose the next configuration.

PriorBand's sampling module \(\mathcal{E}_\pi\) balances the influence of the Prior, the incumbent configurations and randomness to select configurations.

PriorBand's Sampler
The PriorBand sampling module balances the influence of the Prior, the \(1/\eta\) incumbent configurations and randomness to select configurations. (Image Source: PriorBand-paper, Jan 27, 2025)

The Prior sampling \(p_\pi\) is most meaningful at full fidelity and when not much data is available yet, while the incumbent sampling \(p_{\hat{\lambda}}\), coming from actual data, is most significant but sparse, and random sampling \(p_{\mathcal{U}}\) is needed for exploration, especially at lower fidelities. This results in these inital sampling probabilities when there is no incument yet:

\[ p_{\mathcal{U}}=1/(1+\eta^r)\\ p_\pi=1-p_{\mathcal{U}}\\ p_{\hat{\lambda}}=0 \]

where \(\eta\) is the promotion-hyperparameter from HyperBand and \(r\) is the current fidelity level (rung), showing the decay of the random sampling probability with increasing fidelity.

When there is an incumbent, the probabilities are adjusted to:

\[ p_{\mathcal{U}}=1/(1+\eta^r)\\ p_\pi=p_\pi\cdot\mathcal{S}_{\hat{\lambda}}/(\mathcal{S}_\pi+\mathcal{S}_{\hat{\lambda}})\\ p_{\hat{\lambda}}=p_{\hat{\lambda}}\cdot\mathcal{S}_{\pi}/(\mathcal{S}_\pi+\mathcal{S}_{\hat{\lambda}}) \]

where \(\mathcal{S}_\pi\) and \(\mathcal{S}_{\hat{\lambda}}\) are the summed probabilities of the top \(1/\eta\) configurations under Prior and incumbent sampling, respectively. This way, the balance is shifted towards the distribution that would have yielded the best configurations so far. Crucially, this compensates for potentially bad Priors, as the incumbent sampling will take over when it has proven to be better.

Practical Tips
  • PriorBand is a good choice when you have a Prior but are wary of its quality and you can utilize Multi-Fidelity.

Info

PriorBand is chosen as the default optimizer in NePS when there is both Prior and Multi-Fidelity information available.

Model-based PriorBand#

PriorBand can also be extended with a model, where after \(n\) evaluations, a BO model is trained to advise the sampling module.