Acquisition Functions

On this page, some information and references to the various acquisition functions are provided.

Info

If the acquisition function requires the best_f parameter, simply set it to 0.0, and Odyssey will find the current best observation by itself.

Info

The model, posterior_transform and maximize parameters can be ignored.

ExpectedImprovement

Taken from BoTorch ExpectedImprovement.

ExpectedImprovement

Bases: AnalyticAcquisitionFunction

Single-outcome Expected Improvement (analytic).

Computes classic Expected Improvement over the current best observed value, using the analytic formula for a Normal posterior distribution. Unlike the MC-based acquisition functions, this relies on the posterior at single test point being Gaussian (and require the posterior to implement mean and variance properties). Only supports the case of q=1. The model must be single-outcome.

EI(x) = E(max(f(x) - best_f, 0)),

where the expectation is taken over the value of stochastic function f at x.

Example

model = SingleTaskGP(train_X, train_Y) EI = ExpectedImprovement(model, best_f=0.2) ei = EI(test_X)

NOTE: It is strongly recommended to use LogExpectedImprovement instead of regular EI, because it solves the vanishing gradient problem by taking special care of numerical computations and can lead to substantially improved BO performance.

__init__

__init__(model: Model, best_f: Union[float, Tensor], posterior_transform: Optional[PosteriorTransform] = None, maximize: bool = True)

Single-outcome Expected Improvement (analytic).

Parameters:

• model (Model) –

  A fitted single-outcome model.

• best_f (Union[float, Tensor]) –

  Either a scalar or a b-dim Tensor (batch mode) representing the best function value observed so far (assumed noiseless).

• posterior_transform (Optional[PosteriorTransform], default: None ) –

  A PosteriorTransform. If using a multi-output model, a PosteriorTransform that transforms the multi-output posterior into a single-output posterior is required.

• maximize (bool, default: True ) –

  If True, consider the problem a maximization problem.

LogExpectedImprovement

Refer to BoTorch LogExpectedImprovement.

LogExpectedImprovement

Bases: AnalyticAcquisitionFunction

Logarithm of single-outcome Expected Improvement (analytic).

Computes the logarithm of the classic Expected Improvement acquisition function, in a numerically robust manner. In particular, the implementation takes special care to avoid numerical issues in the computation of the acquisition value and its gradient in regions where improvement is predicted to be virtually impossible.

See [Ament2023logei]_ for details. Formally,

LogEI(x) = log(E(max(f(x) - best_f, 0))),

where the expectation is taken over the value of stochastic function f at x.

Example

model = SingleTaskGP(train_X, train_Y) LogEI = LogExpectedImprovement(model, best_f=0.2) ei = LogEI(test_X)

__init__

__init__(model: Model, best_f: Union[float, Tensor], posterior_transform: Optional[PosteriorTransform] = None, maximize: bool = True)

Logarithm of single-outcome Expected Improvement (analytic).

Parameters:

• model (Model) –

  A fitted single-outcome model.

• best_f (Union[float, Tensor]) –

  Either a scalar or a b-dim Tensor (batch mode) representing the best function value observed so far (assumed noiseless).

• posterior_transform (Optional[PosteriorTransform], default: None ) –

  A PosteriorTransform. If using a multi-output model, a PosteriorTransform that transforms the multi-output posterior into a single-output posterior is required.

• maximize (bool, default: True ) –

  If True, consider the problem a maximization problem.

ProbabilityOfImprovement

Refer to BoTorch ProbabilityOfImprovement.

ProbabilityOfImprovement

Bases: AnalyticAcquisitionFunction

Single-outcome Probability of Improvement.

Probability of improvement over the current best observed value, computed using the analytic formula under a Normal posterior distribution. Only supports the case of q=1. Requires the posterior to be Gaussian. The model must be single-outcome.

PI(x) = P(y >= best_f), y ~ f(x)

Example

model = SingleTaskGP(train_X, train_Y) PI = ProbabilityOfImprovement(model, best_f=0.2) pi = PI(test_X)

__init__

__init__(model: Model, best_f: Union[float, Tensor], posterior_transform: Optional[PosteriorTransform] = None, maximize: bool = True)

Single-outcome Probability of Improvement.

Parameters:

• model (Model) –

  A fitted single-outcome model.

• best_f (Union[float, Tensor]) –

  Either a scalar or a b-dim Tensor (batch mode) representing the best function value observed so far (assumed noiseless).

• posterior_transform (Optional[PosteriorTransform], default: None ) –

  A PosteriorTransform. If using a multi-output model, a PosteriorTransform that transforms the multi-output posterior into a single-output posterior is required.

• maximize (bool, default: True ) –

  If True, consider the problem a maximization problem.

LogProbabilityOfImprovement

Refer to BoTorch LogProbabilityOfImprovement.

LogProbabilityOfImprovement

Bases: AnalyticAcquisitionFunction

Single-outcome Log Probability of Improvement.

Logarithm of the probability of improvement over the current best observed value, computed using the analytic formula under a Normal posterior distribution. Only supports the case of q=1. Requires the posterior to be Gaussian. The model must be single-outcome.

The logarithm of the probability of improvement is numerically better behaved than the original function, which can lead to significantly improved optimization of the acquisition function. This is analogous to the common practice of optimizing the log likelihood of a probabilistic model - rather the likelihood - for the sake of maximium likelihood estimation.

logPI(x) = log(P(y >= best_f)), y ~ f(x)

Example

model = SingleTaskGP(train_X, train_Y) LogPI = LogProbabilityOfImprovement(model, best_f=0.2) log_pi = LogPI(test_X)

__init__

__init__(model: Model, best_f: Union[float, Tensor], posterior_transform: Optional[PosteriorTransform] = None, maximize: bool = True)

Single-outcome Probability of Improvement.

Parameters:

• model (Model) –

  A fitted single-outcome model.

• best_f (Union[float, Tensor]) –

  Either a scalar or a b-dim Tensor (batch mode) representing the best function value observed so far (assumed noiseless).

• posterior_transform (Optional[PosteriorTransform], default: None ) –

  A PosteriorTransform. If using a multi-output model, a PosteriorTransform that transforms the multi-output posterior into a single-output posterior is required.

• maximize (bool, default: True ) –

  If True, consider the problem a maximization problem.

UpperConfidenceBound

Refer to BoTorch UpperConfidenceBound.

UpperConfidenceBound

Bases: AnalyticAcquisitionFunction

Single-outcome Upper Confidence Bound (UCB).

Analytic upper confidence bound that comprises of the posterior mean plus an additional term: the posterior standard deviation weighted by a trade-off parameter, beta. Only supports the case of q=1 (i.e. greedy, non-batch selection of design points). The model must be single-outcome.

UCB(x) = mu(x) + sqrt(beta) * sigma(x), where mu and sigma are the posterior mean and standard deviation, respectively.

Example

model = SingleTaskGP(train_X, train_Y) UCB = UpperConfidenceBound(model, beta=0.2) ucb = UCB(test_X)

__init__

__init__(model: Model, beta: Union[float, Tensor], posterior_transform: Optional[PosteriorTransform] = None, maximize: bool = True) -> None

Single-outcome Upper Confidence Bound.

Parameters:

• model (Model) –

  A fitted single-outcome GP model (must be in batch mode if candidate sets X will be)

• beta (Union[float, Tensor]) –

  Either a scalar or a one-dim tensor with b elements (batch mode) representing the trade-off parameter between mean and covariance

• posterior_transform (Optional[PosteriorTransform], default: None ) –

  A PosteriorTransform. If using a multi-output model, a PosteriorTransform that transforms the multi-output posterior into a single-output posterior is required.

• maximize (bool, default: True ) –

  If True, consider the problem a maximization problem.