#!/usr/bin/env python
import numpy as np
from fvgp import fvGP
from .gp_optimizer_base import GPOptimizerBase
# TODO (for gpCAM)
# see fvgp "gp.py" for TODOs
[docs]
class GPOptimizer(GPOptimizerBase):
"""
This class is an optimization extension of the :doc:`fvgp <fvgp:index>` package
for single-task (scalar-valued) Gaussian Processes.
Gaussian Processes can be initialized, trained, and conditioned; also
the posterior can be evaluated and used via an acquisition function,
and plugged into optimizers to find maxima.
V ... number of input points
D ... input space dimensionality
N ... arbitrary integers (N1, N2,...)
Parameters
----------
x_data : np.ndarray or list, optional
The input point positions. Shape (V x D), where D is the :py:attr:`fvgp.GP.index_set_dim`.
For single-task GPs, the index set dimension = input space dimension.
For multi-task GPs, the index set dimension = input space dimension + 1.
If dealing with non-Euclidean inputs
x_data should be a list, not a numpy array.
In this case, both the index set and the input space dim are set to 1.
If x_data is not provided here the GP will be initiated after `tell()`.
y_data : np.ndarray, optional
The values of the data points. Shape (V) or (V, N). If shape (V,N) the algorithm will run N independent GPs.
This is not to be confused with multi-task learning. In this case, all GPs have to have the same prior
mean function.
If not provided here the GP will be initiated after `tell()`.
init_hyperparameters : np.ndarray, optional
Vector of hyperparameters used to initiate the GP.
The default is an array of ones with the right length for the anisotropic Matern
kernel with automatic relevance determination (ARD). If `gp2Scale` is
enabled, the default kernel changes to the anisotropic Wendland kernel.
noise_variances : np.ndarray, optional
An numpy array defining the uncertainties/noise in the
`y_data` in form of a point-wise variance. Shape (V).
Note: if no noise_variances are provided here, the noise_function
callable will be used; if the callable is not provided, the noise variances
will be set to `abs(np.mean(y_data)) / 100.0`. If
noise covariances are required (correlated noise), make use of the `noise_function`.
Only provide a noise function OR `noise_variances`, not both.
If the shape of `y_data` is (V,N) the noise is still of shape (V), e.g., the outputs
must have the same noise in this scenario.
compute_device : str, optional
One of `cpu` or `gpu`, determines how linear algebra computations are executed. The default is `cpu`.
For `gpu`, pytorch or cupy has to be installed manually. For advanced options see `args`.
If `gp2Scale` is enabled but no kernel is provided, the choice of the `compute_device`
will be particularly important. In that case, the default Wendland kernel will be computed on
the cpu or the gpu which will significantly change the compute time depending on the compute
architecture.
kernel_function : Callable, optional
A symmetric positive definite covariance function (a kernel)
that calculates the covariance between
data points. It is a function of the form k(x1,x2,hyperparameters, [args]).
`args` is optional and is used to make `GPOptimizer.args` available.
The input `x1` is a N1 x D array of positions, `x2` is a N2 x D
array of positions, the hyperparameters argument
is a 1d array of length D+1 for the default kernel and of a different
length for user-defined kernels.
The default is a stationary anisotropic kernel
(`fvgp.GP.default_kernel`) which performs automatic relevance determination (ARD).
The output is a matrix, an N1 x N2 numpy array.
kernel_function_grad : Callable, optional
A function that calculates the derivative of the `kernel_function` with respect to the hyperparameters.
If provided, it will be used for local training (optimization) and can speed up the calculations.
It accepts as input `x1` (a N1 x D array of positions),
`x2` (a N2 x D array of positions) and
`hyperparameters` (a 1d array of length D+1 for the default kernel).
The default is an analytical gradient for the default kernel or a finite difference calculation otherwise.
If `ram_economy` is True, the function's input is x1, x2, hyperparameters (numpy array), and a direction (int).
The output is a numpy array of shape (len(hps) x N).
If `ram_economy` is `False`, the function's input is x1, x2, and hyperparameters.
The output is a numpy array of shape (len(hyperparameters) x N1 x N2). See `ram_economy`.
prior_mean_function : Callable, optional
A function f(x, hyperparameters, [args]) that evaluates the prior mean at a set of input position.
It accepts as input
an array of positions (of shape N1 x D) and hyperparameters (a 1d array of length D+1 for the default kernel).
Optionally, the third argument `args` can be defined.
The return value is a 1d array of length N1.
If prior_mean_function is provided, `fvgp.GP._default_mean_function` is used,
which is the average of the `y_data`.
prior_mean_function_grad : Callable, optional
A function that evaluates the gradient of the `prior_mean_function` at
a set of input positions with respect to the hyperparameters.
It accepts as input an array of positions (of size N1 x D) and hyperparameters
(a 1d array of length D+1 for the default kernel).
The return value is a 2d array of
shape (len(hyperparameters) x N1). If None is provided, either
zeros are returned since the default mean function does not depend on hyperparameters,
or a finite-difference approximation
is used if `prior_mean_function` is provided.
noise_function : Callable, optional
The noise function is a callable f(x,hyperparameters, [args]) that returns a
vector (1d np.ndarray) of len(x), a matrix of shape (length(x),length(x)) or a sparse matrix
of the same shape.
The third argument `args` is optional.
The input `x` is a numpy array of shape (N x D). The hyperparameter array is the same
that is communicated to mean and kernel functions.
Only provide a noise function OR a noise variance vector, not both.
noise_function_grad : Callable, optional
A function that evaluates the gradient of the `noise_function`
at an input position with respect to the hyperparameters.
It accepts as input an array of positions (of size N x D) and
hyperparameters (a 1d array of length D+1 for the default kernel).
The return value is a 2d np.ndarray of
shape (len(hyperparameters) x N) or a 3d np.ndarray of shape (len(hyperparameters) x N x N).
If None is provided, either
zeros are returned since the default noise function does not depend on
hyperparameters, or, if `noise_function` is provided but no noise function,
a finite-difference approximation will be used.
The same rules regarding `ram_economy` as for the kernel definition apply here.
That means the function will have an additional `direction` parameter.
gp2Scale: bool, optional
Turns on gp2Scale. This will distribute the covariance computations across multiple workers.
This is an advanced feature for HPC GPs up to 10
million data points. If gp2Scale is used, the default kernel is an anisotropic
Wendland kernel which is compactly supported. There are a few
things to consider (read on); this is an advanced option.
If no kernel is provided, the `compute_device` option should be revisited.
The default kernel will use the specified device to compute covariances.
The default is False.
gp2Scale_dask_client : dask.distributed.Client, optional
A dask client for gp2Scale.
On HPC architecture, this client is provided by the job script. Please have a look at the examples.
A local client is used as the default.
gp2Scale_batch_size : int, optional
Matrix batch size for distributed computing in gp2Scale. The default is 10000.
gp2Scale_linalg_mode : str, optional
One of `Chol`, `sparseLU`, `sparseCG`, `sparseMINRES`, `sparseSolve`, `sparseCGpre`
(incomplete LU preconditioner), or `sparseMINRESpre`. The default is None which amounts to
an automatic determination of the mode. For advanced customization options
this can also be an iterable with three callables: the first f(K), where K is the covariance matrix
to compute a factorization object
which is available in the second and third callable. The second being the linear solve f(obj, vec),
and the third being the logdet=f(obj). If a factorization object is not required, the first callable
should return the matrix itself (K).
calc_inv : bool, optional
If True, the algorithm calculates and stores the inverse of the covariance
matrix after each training or update of the dataset or hyperparameters,
which makes computing the posterior covariance faster (3-10 times).
For larger problems (>5000 data points), the use of inversion should be avoided due
to computational instability and costs. The default is
False. Note, the training will not use the
inverse for stability reasons. Storing the inverse is
a good option when the dataset is not too large and the posterior covariance is heavily used.
ram_economy : bool, optional
Only of interest if the gradient and/or Hessian of the log marginal likelihood is/are used for the training.
If True, components of the derivative of the log marginal likelihood are
calculated sequentially, leading to a slow-down
but much less RAM usage. If the derivative of the kernel (and noise function) with
respect to the hyperparameters (kernel_function_grad) is
going to be provided, it has to be tailored: for `ram_economy=True` it should be
of the form f(x, hyperparameters, direction)
and return a 2d numpy array of shape len(x1) x len(x2).
If `ram_economy=False`, the function should be of the form f(x, hyperparameters)
and return a numpy array of shape
H x len(x1) x len(x2), where H is the number of hyperparameters.
CAUTION: This array will be stored and is very large.
cost_function : Callable, optional
A function encoding the cost of motion through the input
space and the cost of a measurement. Its inputs
are an `origin` (np.ndarray of size V x D), `x`
(np.ndarray of size V x D), and the value of `cost_func_params`;
`origin` is the starting position, and `x` is the
destination position. The return value is a 1d array of
length V describing the costs as floats. The 'score' from
acquisition_function is divided by this
returned cost to determine the next measurement point.
The default in no-op.
logging : bool
If true, logging is enabled. The default is False.
args: dict, optional
Advanced options. Recognized keys are:
- "random_logdet_lanczos_degree" : int; default = 20
- "random_logdet_error_rtol" : float; default = 0.01
- "random_logdet_verbose" : True/False; default = False
- "random_logdet_print_info" : True/False; default = False
- "sparse_minres_tol" : float
- "cg_minres_tol" : float
- "random_logdet_lanczos_compute_device" : str; default = "cpu"/"gpu"
- "Chol_factor_compute_device" : str; default = "cpu"/"gpu"
- "update_Chol_factor_compute_device": str; default = "cpu"/"gpu"
- "Chol_solve_compute_device" : str; default = "cpu"/"gpu"
- "Chol_logdet_compute_device" : str; default = "cpu"/"gpu"
- "GPU_engine" : str; default = "torch"/"cupy"
All other keys will be stored and are available as part of the object instance and
in kernel, mean, and noise functions.
Attributes
----------
x_data : np.ndarray
Datapoint positions.
y_data : np.ndarray
Datapoint values.
noise_variances : np.ndarray
Datapoint observation variances.
hyperparameters : np.ndarray
Current hyperparameters in use.
K : np.ndarray
Current prior covariance matrix of the GP.
m : np.ndarray
Current prior mean vector.
V : np.ndarray
the noise covariance matrix.
"""
def __init__(
self,
x_data=None,
y_data=None,
init_hyperparameters=None,
noise_variances=None,
compute_device="cpu",
kernel_function=None,
kernel_function_grad=None,
noise_function=None,
noise_function_grad=None,
prior_mean_function=None,
prior_mean_function_grad=None,
gp2Scale=False,
gp2Scale_dask_client=None,
gp2Scale_batch_size=10000,
gp2Scale_linalg_mode=None,
calc_inv=False,
ram_economy=False,
cost_function=None,
logging=False,
args=None
):
super().__init__(x_data=x_data,
y_data=y_data,
init_hyperparameters=init_hyperparameters,
noise_variances=noise_variances,
compute_device=compute_device,
kernel_function=kernel_function,
kernel_function_grad=kernel_function_grad,
noise_function=noise_function,
noise_function_grad=noise_function_grad,
prior_mean_function=prior_mean_function,
prior_mean_function_grad=prior_mean_function_grad,
gp2Scale=gp2Scale,
gp2Scale_dask_client=gp2Scale_dask_client,
gp2Scale_batch_size=gp2Scale_batch_size,
gp2Scale_linalg_mode=gp2Scale_linalg_mode,
calc_inv=calc_inv,
ram_economy=ram_economy,
cost_function=cost_function,
logging=logging,
multi_task=False,
args=args)
[docs]
class fvGPOptimizer(GPOptimizerBase, fvGP):
"""
This class is an optimization extension of the :doc:`fvgp <fvgp:index>`
package for multi-task (vector-valued) Gaussian Processes.
Gaussian Processes can be initialized, trained, and conditioned; also
the posterior can be evaluated and used via an acquisition function,
and plugged into optimizers to find maxima.
V ... number of input points
Di... input space dimensionality
No... number of outputs
N ... arbitrary integers (N1, N2,...)
The main logic of :doc:`fvgp <fvgp:index>` is that any multi-task GP is just a single-task GP
over a Cartesian product space of input and output space, as long as the kernel
is flexible enough, so prepare to work on your kernel. This is the best
way to give the user optimal control and power. At various instances, for example
prior-mean function, noise function, and kernel function definitions, you will
see that the input `x` is defined over this combined space.
For example, if your input space is a Euclidean 2d space and your output
is labelled [0,1], the input to the mean, kernel, and noise functions might be
x =
[[0.2, 0.3,0],[0.9,0.6,0],
[0.2, 0.3,1],[0.9,0.6,1]]
This has to be understood and taken into account when customizing gpCAM for multi-task
use. The examples will provide deeper insights.
Parameters
----------
x_data : np.ndarray | list, optional
The input point positions. Shape (V x Di), where Di is the :py:attr:`fvgp.fvGP.input_set_dim`.
For multi-task GPs, the index set dimension = input space dimension + 1.
If dealing with non-Euclidean inputs
x_data should be a list, not a numpy array.
In this case, both the index set and the input space dim are set to 1.
y_data : np.ndarray
The values of the data points. Shape (V,No).
It is possible that not every entry in `x_data`
has all corresponding tasks available. In that case `y_data` may have np.nan as the corresponding entries.
init_hyperparameters : np.ndarray, optional
Vector of hyperparameters used to initiate the GP.
The default is an array of ones with the right length for the anisotropic Matern
kernel with automatic relevance determination (ARD). The task direction is
simply considered a separate dimension. If gp2Scale is
enabled, the default kernel changes to the anisotropic Wendland kernel.
noise_variances : np.ndarray, optional
An numpy array defining the uncertainties/noise in the
`y_data` in form of a point-wise variance. Shape (V, No).
If `y_data` has np.nan entries, the corresponding
`noise_variances` have to be np.nan.
Note: if no noise_variances are provided here, the noise_function
callable will be used; if the callable is not provided, the noise variances
will be set to `abs(np.mean(y_data)) / 100.0`. If
noise covariances are required (correlated noise), make use of the `noise_function`.
Only provide a noise function OR `noise_variances`, not both.
compute_device : str, optional
One of `cpu` or `gpu`, determines how linear algebra computations are executed. The default is `cpu`.
For `gpu`, pytorch or cupy has to be installed manually. For advanced options see `args`.
If `gp2Scale` is enabled but no kernel is provided, the choice of the `compute_device`
will be particularly important. In that case, the default Wendland kernel will be computed on
the cpu or the gpu which will significantly change the compute time depending on the compute
architecture.
kernel_function : Callable, optional
A symmetric positive definite covariance function (a kernel)
that calculates the covariance between
data points. It is a function of the form k(x1,x2,hyperparameters, [args]).
`args` is optional and is used to make `fvgp.gp.args` available.
The input `x1` a N1 x Di+1 array of positions, `x2` is a N2 x Di+1
array of positions, the hyperparameters argument
is a 1d array of length N depending on how many hyperparameters are initialized.
The default is a stationary anisotropic kernel
(`fvgp.GP.default_kernel`) which performs automatic relevance determination (ARD). The task
direction is simply considered an additional dimension. This kernel should only be used for tests and in the
simplest of cases.
The output is a matrix, an N1 x N2 numpy array.
kernel_function_grad : Callable, optional
A function that calculates the derivative of the `gp_kernel_function` with respect to the hyperparameters.
If provided, it will be used for local training (optimization) and can speed up the calculations.
It accepts as input `x1` (a N1 x Di + 1 array of positions),
`x2` (a N2 x Di + 1 array of positions) and
`hyperparameters` (a 1d array of length Di+2 for the default kernel).
The default is a finite difference calculation.
If `ram_economy` is True, the function's input is x1, x2,hyperparameters (numpy array), and a direction (int).
The output is a numpy array of shape (len(hps) x N).
If `ram_economy` is `False`, the function's input is x1, x2, and hyperparameters.
The output is a numpy array of shape (len(hyperparameters) x N1 x N2). See `ram_economy`.
prior_mean_function : Callable, optional
A function f(x, hyperparameters, [args]) that evaluates the prior mean at a set of input position.
It accepts as input
an array of positions (of shape N1 x Di+1) and
hyperparameters (a 1d array of length Di+2 for the default kernel).
Optionally, the third argument `args` can be defined.
The return value is a 1d array of length N1. If None is provided,
`fvgp.GP._default_mean_function` is used, which is the average of the `y_data`.
prior_mean_function_grad : Callable, optional
A function that evaluates the gradient of the `prior_mean_function` at
a set of input positions with respect to the hyperparameters.
It accepts as input an array of positions (of size N1 x Di+1) and hyperparameters
(a 1d array of length Di+2 for the default kernel).
The return value is a 2d array of
shape (len(hyperparameters) x N1). If None is provided, either
zeros are returned since the default mean function does not depend on hyperparameters,
or a finite-difference approximation
is used if `prior_mean_function` is provided.
noise_function : Callable, optional
The noise function is a callable f(x,hyperparameters, [args]) that returns a
vector (1d np.ndarray) of len(x), a matrix of shape (length(x),length(x)) or a sparse matrix
of the same shape.
The third argument `args` is optional.
The input `x` is a numpy array of shape (N x Di+1). The hyperparameter array is the same
that is communicated to mean and kernel functions.
Only provide a noise function OR a noise variance vector, not both.
noise_function_grad : Callable, optional
A function that evaluates the gradient of the `noise_function`
at an input position with respect to the hyperparameters.
It accepts as input an array of positions (of size N x Di+1) and
hyperparameters (a 1d array of length D+1 for the default kernel).
The return value is a 2d np.ndarray of
shape (len(hyperparameters) x N) or a 3d np.ndarray of shape (len(hyperparameters) x N x N).
If None is provided, either
zeros are returned since the default noise function does not depend on
hyperparameters, or, if `noise_function` is provided but no noise function,
a finite-difference approximation will be used.
The same rules regarding `ram_economy` as for the kernel definition apply here.
That means the function will have an additional `direction` parameter.
gp2Scale: bool, optional
Turns on gp2Scale. This will distribute the covariance computations across multiple workers.
This is an advanced feature for HPC GPs up to 10
million data points. If gp2Scale is used, the default kernel is an anisotropic
Wendland kernel which is compactly supported. There are a few
things to consider (read on); this is an advanced option.
If no kernel is provided, the `compute_device` option should be revisited.
The default kernel will use the specified device to compute covariances.
The default is False.
gp2Scale_dask_client : dask.distributed.Client, optional
A dask client for gp2Scale.
On HPC architecture, this client is provided by the job script. Please have a look at the examples.
A local client is used as the default.
gp2Scale_batch_size : int, optional
Matrix batch size for distributed computing in gp2Scale. The default is 10000.
gp2Scale_linalg_mode : str, optional
One of `Chol`, `sparseLU`, `sparseCG`, `sparseMINRES`, `sparseSolve`, `sparseCGpre`
(incomplete LU preconditioner), or `sparseMINRESpre`. The default is None which amounts to
an automatic determination of the mode. For advanced customization options
this can also be an iterable with three callables: the first f(K), where K is the covariance matrix
to compute a factorization object
which is available in the second and third callable. The second being the linear solve f(obj, vec),
and the third being the logdet=f(obj). If a factorization object is not required, the first callable
should return the matrix itself (K).
calc_inv : bool, optional
If True, the algorithm calculates and stores the inverse of the covariance
matrix after each training or update of the dataset or hyperparameters,
which makes computing the posterior covariance faster (3-10 times).
For larger problems (>5000 data points), the use of inversion should be avoided due
to computational instability and costs. The default is
False. Note, the training will not use the
inverse for stability reasons. Storing the inverse is
a good option when the dataset is not too large and the posterior covariance is heavily used.
ram_economy : bool, optional
Only of interest if the gradient and/or Hessian of the log marginal likelihood is/are used for the training.
If True, components of the derivative of the log marginal likelihood are
calculated sequentially, leading to a slow-down
but much less RAM usage. If the derivative of the kernel (and noise function) with
respect to the hyperparameters (kernel_function_grad) is
going to be provided, it has to be tailored: for `ram_economy=True` it should be
of the form f(x, hyperparameters, direction)
and return a 2d numpy array of shape len(x1) x len(x2).
If `ram_economy=False`, the function should be of the form f(x, hyperparameters)
and return a numpy array of shape
H x len(x1) x len(x2), where H is the number of hyperparameters.
CAUTION: This array will be stored and is very large.
cost_function : Callable, optional
A function encoding the cost of motion through the input
space and the cost of a measurement. Its inputs
are an `origin` (np.ndarray of size V x D), `x`
(np.ndarray of size V x D), and the value of `cost_func_params`;
`origin` is the starting position, and `x` is the
destination position. The return value is a 1d array of
length V describing the costs as floats. The 'score' from
acquisition_function is divided by this
returned cost to determine the next measurement point.
The default is no-op.
logging : bool
If true, logging is enabled. The default is False.
args: dict, optional
Advanced options. Recognized keys are:
- "random_logdet_lanczos_degree" : int; default = 20
- "random_logdet_error_rtol" : float; default = 0.01
- "random_logdet_verbose" : True/False; default = False
- "random_logdet_print_info" : True/False; default = False
- "sparse_minres_tol" : float
- "cg_minres_tol" : float
- "random_logdet_lanczos_compute_device" : str; default = "cpu"/"gpu"
- "Chol_factor_compute_device" : str; default = "cpu"/"gpu"
- "update_Chol_factor_compute_device": str; default = "cpu"/"gpu"
- "Chol_solve_compute_device" : str; default = "cpu"/"gpu"
- "Chol_logdet_compute_device" : str; default = "cpu"/"gpu"
- "GPU_engine" : str; default = "torch"/"cupy"
All other keys will be stored and are available as part of the object instance.
Attributes
----------
x_data : np.ndarray | list
Datapoint positions
y_data : np.ndarray
Datapoint values
noise_variances : np.ndarray
Datapoint observation variances.
fvgp_x_data : np.ndarray | list
Data points from the fvgp point of view.
fvgp_y_data : np.ndarray
The data values from the fvgp point of view.
fvgp_noise_variances : np.ndarray
Observation variances from the fvgp point of view.
hyperparameters : np.ndarray
Current hyperparameters in use.
K : np.ndarray
Current prior covariance matrix of the GP
m : np.ndarray
Current prior mean vector.
V : np.ndarray
the noise covariance matrix or a vector.
"""
def __init__(
self,
x_data=None,
y_data=None,
init_hyperparameters=None,
noise_variances=None,
compute_device="cpu",
kernel_function=None,
kernel_function_grad=None,
noise_function=None,
noise_function_grad=None,
prior_mean_function=None,
prior_mean_function_grad=None,
gp2Scale=False,
gp2Scale_dask_client=None,
gp2Scale_batch_size=10000,
gp2Scale_linalg_mode=None,
calc_inv=False,
ram_economy=False,
cost_function=None,
logging=False,
args=None
):
super().__init__(x_data=x_data,
y_data=y_data,
init_hyperparameters=init_hyperparameters,
noise_variances=noise_variances,
compute_device=compute_device,
kernel_function=kernel_function,
kernel_function_grad=kernel_function_grad,
noise_function=noise_function,
noise_function_grad=noise_function_grad,
prior_mean_function=prior_mean_function,
prior_mean_function_grad=prior_mean_function_grad,
gp2Scale=gp2Scale,
gp2Scale_dask_client=gp2Scale_dask_client,
gp2Scale_batch_size=gp2Scale_batch_size,
gp2Scale_linalg_mode=gp2Scale_linalg_mode,
calc_inv=calc_inv,
ram_economy=ram_economy,
cost_function=cost_function,
logging=logging,
multi_task=True,
args=args)
