gp_opt: Gaussian process-based optimisation

In adaptr: Adaptive Trial Simulator

Gaussian process-based optimisation

Description

Used internally. Simple Gaussian process-based Bayesian optimisation function, used to find the next value to evaluate (as x) in the calibrate_trial() function. Uses only a single input dimension, which may be rescaled to the ⁠[0, 1]⁠ range by the function, and a covariance structure based on absolute distances between values, raised to a power (pow) and subsequently divided by lengthscale before the inverse exponentiation of the resulting matrix is used. The pow and lengthscale hyperparameters consequently control the smoothness by controlling the rate of decay between correlations with distance.
The optimisation algorithm uses bi-directional uncertainty bounds in an acquisition function that suggests the next target to evaluate, with wider uncertainty bounds (higher kappa) leading to increased 'exploration' (i.e., the function is more prone to suggest new target values where the uncertainty is high and often further from the best evaluation so far) and narrower uncertainty bounds leading to increased 'exploitation' (i.e., the function is more prone to suggest new target values relatively close to the mean predictions from the model).
The dir argument controls whether the suggested value (based on both uncertainty bounds) should be the value closest to target in either direction (dir = 0), at or above target (dir > 0), or at or below target (dir < 0), if any, are preferred.
When the function being evaluated is noise-free and monotonically increasing or decreasing, the optimisation function can narrow the range of predictions based on the input evaluations (narrow = TRUE), leading to a finer grid of potential new targets to suggest compared to when predictions are spaced over the full range.
If the new value at which to evaluate the function suggested has already been evaluated, random noise will be added to ensure evaluation at a new value (if narrow is FALSE, noise will be based on a random draw from a normal distribution with the current suggested value as mean and the standard deviation of the x values as SD, truncated to the range of x-values; if narrow is TRUE, a new value drawn from a uniform distribution within the current narrowed range will be suggested. For both strategies, the process will be repeated until the suggested value is 'new').
The Gaussian process model used is partially based on code from Gramacy 2020 (with permission), see References.

Usage

gp_opt(
  x,
  y,
  target,
  dir = 0,
  resolution = 5000,
  kappa = 1.96,
  pow = 1.95,
  lengthscale = 1,
  scale_x = TRUE,
  noisy = FALSE,
  narrow = FALSE
)

Arguments

x	numeric vector, the previous values where the function being calibrated was evaluated.
y	numeric vector, the corresponding results of the previous evaluations at the x values (must be of the same length as x).
target	single numeric value, the desired target value for the calibration process.
dir	single numeric value (default 0), used when selecting the next value to evaluate at. See which_nearest() for further description.
resolution	single integer (default 5000), size of the grid at which the predictions used to select the next value to evaluate at are made. Note: memory use and time will substantially increase with higher values.
kappa	single numeric value ⁠> 0⁠ (default 1.96), used for the width of uncertainty bounds (based on the Gaussian process posterior predictive distribution), which are used to select the next value to evaluate at.
pow	single numerical value, passed to cov_mat() and controls the smoothness of the Gaussian process. Should be between 1 (no smoothness, piecewise straight lines between each subsequent x/y-coordinate if lengthscale described below is 1) and 2; defaults to 1.95, which leads to slightly faster decay of correlations when x values are internally scaled to the ⁠[0, 1]⁠-range compared to 2.
lengthscale	single numerical value (default 1) or numerical vector of length 2; all values must be finite and non-negative. If a single value is provided, this will be used as the lengthscale hyperparameter and passed directly to cov_mat(). If a numerical vector of length 2 is provided, the second value must be higher than the first and the optimal lengthscale in this range will be found using an optimisation algorithm. If any value is 0, a minimum amount of noise will be added as lengthscales must be ⁠> 0⁠. Controls smoothness/decay in combination with pow.
scale_x	single logical value; if TRUE (the default) the x-values will be scaled to the ⁠[0, 1]⁠ range according to the minimum/maximum values provided. If FALSE, the model will use the original scale. If distances on the original scale are small, scaling may be preferred. The returned values will always be on the original scale.
noisy	single logical value. If FALSE (the default), a noiseless process is assumed, and interpolation between values is performed (i.e., with no uncertainty at the evaluated x-values); if TRUE, the y-values are assumed to come from a noisy process, and regression is performed (i.e., some uncertainty at the evaluated x-values will be included in the predictions, with the amount estimated using an optimisation algorithm).
narrow	single logical value. If FALSE (the default), predictions are evenly spread over the full x-range. If TRUE, the prediction grid will be spread evenly over an interval consisting of the two x-values with corresponding y-values closest to the target in opposite directions. This setting should only be used if noisy is FALSE and only if the function can safely be assumed to be only monotonically increasing or decreasing, in which case this will lead to a faster search and a smoother prediction grid in the relevant region without increasing memory use.

Value

List containing two elements, next_x, a single numerical value, the suggested next x value at which to evaluate the function, and predictions, a data.frame with resolution rows and the four columns: x, the x grid values where predictions are made; y_hat, the predicted means, and lub and uub, the lower and upper uncertainty bounds of the predictions according to kappa.

References

Gramacy RB (2020). Chapter 5: Gaussian Process Regression. In: Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences. Chapman Hall/CRC, Boca Raton, Florida, USA. Available online.

Greenhill S, Rana S, Gupta S, Vellanki P, Venkatesh S (2020). Bayesian Optimization for Adaptive Experimental Design: A Review. IEEE Access, 8, 13937-13948. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/ACCESS.2020.2966228")}

adaptr documentation built on May 29, 2024, 7:48 a.m.