botorch.utils¶
Constraints¶
Helpers for handling outcome constraints.

botorch.utils.constraints.
get_outcome_constraint_transforms
(outcome_constraints)[source]¶ Create outcome constraint callables from outcome constraint tensors.
 Parameters
outcome_constraints (
Optional
[Tuple
[Tensor
,Tensor
]]) – A tuple of (A, b). For k outcome constraints and m outputs at f(x)`, A is k x m and b is k x 1 such that A f(x) <= b. Return type
Optional
[List
[Callable
[[Tensor
],Tensor
]]] Returns
A list of callables, each mapping a Tensor of size b x q x m to a tensor of size b x q, where m is the number of outputs of the model. Negative values imply feasibility. The callables support broadcasting (e.g. for calling on a tensor of shape mc_samples x b x q x m).
Example
>>> # constrain `f(x)[0] <= 0` >>> A = torch.tensor([[1., 0.]]) >>> b = torch.tensor([[0.]]) >>> outcome_constraints = get_outcome_constraint_transforms((A, b))
Objective¶
Helpers for handling objectives.

botorch.utils.objective.
get_objective_weights_transform
(weights)[source]¶ Create a linear objective callable from a set of weights.
Create a callable mapping a Tensor of size b x q x m to a Tensor of size b x q, where m is the number of outputs of the model using scalarization via the objective weights. This callable supports broadcasting (e.g. for calling on a tensor of shape mc_samples x b x q x m). For m = 1, the objective weight is used to determine the optimization direction.
 Parameters
weights (
Optional
[Tensor
]) – a 1dimensional Tensor containing a weight for each task. If not provided, the identity mapping is used. Return type
Callable
[[Tensor
],Tensor
] Returns
Transform function using the objective weights.
Example
>>> weights = torch.tensor([0.75, 0.25]) >>> transform = get_objective_weights_transform(weights)

botorch.utils.objective.
apply_constraints_nonnegative_soft
(obj, constraints, samples, eta)[source]¶ Applies constraints to a nonnegative objective.
This function uses a sigmoid approximation to an indicator function for each constraint.
 Parameters
obj (
Tensor
) – A n_samples x b x q Tensor of objective values.constraints (
List
[Callable
[[Tensor
],Tensor
]]) – A list of callables, each mapping a Tensor of size b x q x m to a Tensor of size b x q, where negative values imply feasibility. This callable must support broadcasting. Only relevant for multi output models (m > 1).samples (
Tensor
) – A b x q x m Tensor of samples drawn from the posterior.eta (
float
) – The temperature parameter for the sigmoid function.
 Return type
Tensor
 Returns
A n_samples x b x qdim tensor of feasibilityweighted objectives.

botorch.utils.objective.
soft_eval_constraint
(lhs, eta=0.001)[source]¶ Elementwise evaluation of a constraint in a ‘soft’ fashion
value(x) = 1 / (1 + exp(x / eta))
 Parameters
lhs (
Tensor
) – The left hand side of the constraint lhs <= 0.eta (
float
) – The temperature parameter of the softmax function. As eta grows larger, this approximates the Heaviside step function.
 Return type
Tensor
 Returns
Elementwise ‘soft’ feasibility indicator of the same shape as lhs. For each element x, value(x) > 0 as x becomes positive, and value(x) > 1 as x becomes negative.

botorch.utils.objective.
apply_constraints
(obj, constraints, samples, infeasible_cost, eta=0.001)[source]¶ Apply constraints using an infeasible_cost M for negative objectives.
This allows feasibilityweighting an objective for the case where the objective can be negative by usingthe following strategy: (1) add M to make obj nonnegative (2) apply constraints using the sigmoid approximation (3) shift by M
 Parameters
obj (
Tensor
) – A n_samples x b x q Tensor of objective values.constraints (
List
[Callable
[[Tensor
],Tensor
]]) – A list of callables, each mapping a Tensor of size b x q x m to a Tensor of size b x q, where negative values imply feasibility. This callable must support broadcasting. Only relevant for multi output models (m > 1).samples (
Tensor
) – A b x q x m Tensor of samples drawn from the posterior.infeasible_cost (
float
) – The infeasible value.eta (
float
) – The temperature parameter of the sigmoid function.
 Return type
Tensor
 Returns
A n_samples x b x qdim tensor of feasibilityweighted objectives.
Sampling¶
Utilities for MC and qMC sampling.

botorch.utils.sampling.
manual_seed
(seed=None)[source]¶ Contextmanager for manual setting the torch.random seed.
 Parameters
seed (
Optional
[int
]) – The seed to set the random number generator to. Return type
Generator
[None
,None
,None
] Returns
Generator
Example
>>> with manual_seed(1234): >>> X = torch.rand(3)

botorch.utils.sampling.
construct_base_samples
(batch_shape, output_shape, sample_shape, qmc=True, seed=None, device=None, dtype=None)[source]¶ Construct base samples from a multivariate standard normal N(0, I_qo).
 Parameters
batch_shape (
Size
) – The batch shape of the base samples to generate. Typically, this is used with each dimension of size 1, so as to eliminate sampling variance across batches.output_shape (
Size
) – The output shape (q x m) of the base samples to generate.sample_shape (
Size
) – The sample shape of the samples to draw.qmc (
bool
) – If True, use quasiMC sampling (instead of iid draws).seed (
Optional
[int
]) – If provided, use as a seed for the RNG.
 Return type
Tensor
 Returns
A sample_shape x batch_shape x mutput_shape dimensional tensor of base samples, drawn from a N(0, I_qm) distribution (using QMC if qmc=True). Here output_shape = q x m.
Example
>>> batch_shape = torch.Size([2]) >>> output_shape = torch.Size([3]) >>> sample_shape = torch.Size([10]) >>> samples = construct_base_samples(batch_shape, output_shape, sample_shape)

botorch.utils.sampling.
construct_base_samples_from_posterior
(posterior, sample_shape, qmc=True, collapse_batch_dims=True, seed=None)[source]¶ Construct a tensor of normally distributed base samples.
 Parameters
posterior (
Posterior
) – A Posterior object.sample_shape (
Size
) – The sample shape of the samples to draw.qmc (
bool
) – If True, use quasiMC sampling (instead of iid draws).seed (
Optional
[int
]) – If provided, use as a seed for the RNG.
 Return type
Tensor
 Returns
A num_samples x 1 x q x m dimensional Tensor of base samples, drawn from a N(0, I_qm) distribution (using QMC if qmc=True). Here q and m are the same as in the posterior’s event_shape b x q x m. Importantly, this only obtain a single tbatch of samples, so as to not introduce any sampling variance across tbatches.
Example
>>> sample_shape = torch.Size([10]) >>> samples = construct_base_samples_from_posterior(posterior, sample_shape)

botorch.utils.sampling.
draw_sobol_samples
(bounds, n, q, seed=None)[source]¶ Draw qMC samples from the box defined by bounds.
 Parameters
bounds (
Tensor
) – A 2 x d dimensional tensor specifying box constraints on a ddimensional space, where bounds[0, :] and bounds[1, :] correspond to lower and upper bounds, respectively.n (
int
) – The number of (qbatch) samples.q (
int
) – The size of each qbatch.seed (
Optional
[int
]) – The seed used for initializing Owen scrambling. If None (default), use a random seed.
 Return type
Tensor
 Returns
A n x q x ddim tensor of qMC samples from the box defined by bounds.
Example
>>> bounds = torch.stack([torch.zeros(3), torch.ones(3)]) >>> samples = draw_sobol_samples(bounds, 10, 2)

botorch.utils.sampling.
draw_sobol_normal_samples
(d, n, device=None, dtype=None, seed=None)[source]¶ Draw qMC samples from a multivariate standard normal N(0, I_d)
A primary usecase for this functionality is to compute an QMC average of f(X) over X where each element of X is drawn N(0, 1).
 Parameters
d (
int
) – The dimension of the normal distribution.n (
int
) – The number of samples to return.device (
Optional
[device
]) – The torch device.dtype (
Optional
[dtype
]) – The torch dtype.seed (
Optional
[int
]) – The seed used for initializing Owen scrambling. If None (default), use a random seed.
 Return type
Tensor
 Returns
A tensor of qMC standard normal samples with dimension n x d with device and dtype specified by the input.
Example
>>> samples = draw_sobol_normal_samples(2, 10)

botorch.utils.sampling.
sample_hypersphere
(d, n=1, qmc=False, seed=None, device=None, dtype=None)[source]¶ Sample uniformly from a unit dsphere.
 Parameters
d (
int
) – The dimension of the hypersphere.n (
int
) – The number of samples to return.qmc (
bool
) – If True, use QMC Sobol sampling (instead of i.i.d. uniform).seed (
Optional
[int
]) – If provided, use as a seed for the RNG.device (
Optional
[device
]) – The torch device.dtype (
Optional
[dtype
]) – The torch dtype.
 Return type
Tensor
 Returns
An n x d tensor of uniform samples from from the dhypersphere.
Example
>>> sample_hypersphere(d=5, n=10)

botorch.utils.sampling.
sample_simplex
(d, n=1, qmc=False, seed=None, device=None, dtype=None)[source]¶ Sample uniformly from a dsimplex.
 Parameters
d (
int
) – The dimension of the simplex.n (
int
) – The number of samples to return.qmc (
bool
) – If True, use QMC Sobol sampling (instead of i.i.d. uniform).seed (
Optional
[int
]) – If provided, use as a seed for the RNG.device (
Optional
[device
]) – The torch device.dtype (
Optional
[dtype
]) – The torch dtype.
 Return type
Tensor
 Returns
An n x d tensor of uniform samples from from the dsimplex.
Example
>>> sample_simplex(d=3, n=10)

botorch.utils.sampling.
batched_multinomial
(weights, num_samples, replacement=False, generator=None, out=None)[source]¶ Sample from multinomial with an arbitrary number of batch dimensions.
 Parameters
weights (
Tensor
) – A batch_shape x num_categories tensor of weights. For each batch index i, j, …, this functions samples from a multinomial with input weights[i, j, …, :]. Note that the weights need not sum to one, but must be nonnegative, finite and have a nonzero sum.num_samples (
int
) – The number of samples to draw for each batch index. Must be smaller than num_categories if replacement=False.replacement (
bool
) – If True, samples are drawn with replacement.generator (
Optional
[Generator
]) – A a pseudorandom number generator for sampling.out (
Optional
[Tensor
]) – The output tensor (optional). If provided, must be of size batch_shape x num_samples.
 Return type
LongTensor
 Returns
A batch_shape x num_samples tensor of samples.
This is a thin wrapper around torch.multinomial that allows weight (input) tensors with an arbitrary number of batch dimensions (torch.multinomial only allows a single batch dimension). The calling signature is the same as for torch.multinomial.
Example
>>> weights = torch.rand(2, 3, 10) >>> samples = batched_multinomial(weights, 4) # shape is 2 x 3 x 4
Testing¶

class
botorch.utils.testing.
BotorchTestCase
(methodName='runTest')[source]¶ Bases:
unittest.case.TestCase
Basic test case for Botorch.
 This
sets the default device to be torch.device(“cpu”)
ensures that no warnings are suppressed by default.
Create an instance of the class that will use the named test method when executed. Raises a ValueError if the instance does not have a method with the specified name.

device
= device(type='cpu')¶

class
botorch.utils.testing.
BaseTestProblemBaseTestCase
[source]¶ Bases:
object

functions
: List[BaseTestProblem] = None¶


class
botorch.utils.testing.
SyntheticTestFunctionBaseTestCase
[source]¶ Bases:
botorch.utils.testing.BaseTestProblemBaseTestCase

functions
= None¶


class
botorch.utils.testing.
MockPosterior
(mean=None, variance=None, samples=None)[source]¶ Bases:
botorch.posteriors.posterior.Posterior
Mock object that implements dummy methods and feeds through specified outputs

property
device
¶ The torch device of the posterior.
 Return type
device

property
dtype
¶ The torch dtype of the posterior.
 Return type
dtype

property
event_shape
¶ The event shape (i.e. the shape of a single sample).
 Return type
Size

property
mean
¶ The mean of the posterior as a (b) x n x mdim Tensor.

property
variance
¶ The variance of the posterior as a (b) x n x mdim Tensor.

property

class
botorch.utils.testing.
MockModel
(posterior)[source]¶ Bases:
botorch.models.model.Model
Mock object that implements dummy methods and feeds through specified outputs
Initializes internal Module state, shared by both nn.Module and ScriptModule.

posterior
(X, output_indices=None, observation_noise=False)[source]¶ Computes the posterior over model outputs at the provided points.
 Parameters
X (
Tensor
) – A b x q x ddim Tensor, where d is the dimension of the feature space, q is the number of points considered jointly, and b is the batch dimension.output_indices (
Optional
[List
[int
]]) – A list of indices, corresponding to the outputs over which to compute the posterior (if the model is multioutput). Can be used to speed up computation if only a subset of the model’s outputs are required for optimization. If omitted, computes the posterior over all model outputs.observation_noise (
bool
) – If True, add observation noise to the posterior.
 Return type
 Returns
A Posterior object, representing a batch of b joint distributions over q points and m outputs each.

property
num_outputs
¶ The number of outputs of the model.
 Return type
int

state_dict
()[source]¶ Returns a dictionary containing a whole state of the module.
Both parameters and persistent buffers (e.g. running averages) are included. Keys are corresponding parameter and buffer names.
 Returns
a dictionary containing a whole state of the module
 Return type
dict
Example:
>>> module.state_dict().keys() ['bias', 'weight']

load_state_dict
(state_dict=None, strict=False)[source]¶ Copies parameters and buffers from
state_dict
into this module and its descendants. Ifstrict
isTrue
, then the keys ofstate_dict
must exactly match the keys returned by this module’sstate_dict()
function. Parameters
state_dict (dict) – a dict containing parameters and persistent buffers.
strict (bool, optional) – whether to strictly enforce that the keys in
state_dict
match the keys returned by this module’sstate_dict()
function. Default:True
 Returns
missing_keys is a list of str containing the missing keys
unexpected_keys is a list of str containing the unexpected keys
 Return type
NamedTuple
withmissing_keys
andunexpected_keys
fields


class
botorch.utils.testing.
MockAcquisitionFunction
[source]¶ Bases:
object
Mock acquisition function object that implements dummy methods.

class
botorch.utils.testing.
MultiObjectiveTestProblemBaseTestCase
[source]¶ Bases:
botorch.utils.testing.BaseTestProblemBaseTestCase

functions
= None¶

Torch¶

class
botorch.utils.torch.
BufferDict
(buffers=None)[source]¶ Bases:
torch.nn.modules.module.Module
Holds buffers in a dictionary.
BufferDict can be indexed like a regular Python dictionary, but buffers it contains are properly registered, and will be visible by all Module methods.
BufferDict
is an ordered dictionary that respectsthe order of insertion, and
in
update()
, the order of the mergedOrderedDict
or anotherBufferDict
(the argument toupdate()
).
Note that
update()
with other unordered mapping types (e.g., Python’s plaindict
) does not preserve the order of the merged mapping. Parameters
buffers (iterable, optional) – a mapping (dictionary) of (string :
Tensor
) or an iterable of keyvalue pairs of type (string,Tensor
)
Example:
class MyModule(nn.Module): def __init__(self): super(MyModule, self).__init__() self.buffers = nn.BufferDict({ 'left': torch.randn(5, 10), 'right': torch.randn(5, 10) }) def forward(self, x, choice): x = self.buffers[choice].mm(x) return x
Initializes internal Module state, shared by both nn.Module and ScriptModule.

pop
(key)[source]¶ Remove key from the BufferDict and return its buffer.
 Parameters
key (string) – key to pop from the BufferDict

update
(buffers)[source]¶ Update the
BufferDict
with the keyvalue pairs from a mapping or an iterable, overwriting existing keys.Note
If
buffers
is anOrderedDict
, aBufferDict
, or an iterable of keyvalue pairs, the order of new elements in it is preserved. Parameters
buffers (iterable) – a mapping (dictionary) from string to
Tensor
, or an iterable of keyvalue pairs of type (string,Tensor
)
Transformations¶
Some basic data transformation helpers.

botorch.utils.transforms.
squeeze_last_dim
(Y)[source]¶ Squeeze the last dimension of a Tensor.
 Parameters
Y (
Tensor
) – A … x ddim Tensor. Return type
Tensor
 Returns
The input tensor with last dimension squeezed.
Example
>>> Y = torch.rand(4, 3) >>> Y_squeezed = squeeze_last_dim(Y)

botorch.utils.transforms.
standardize
(Y)[source]¶ Standardizes (zero mean, unit variance) a tensor by dim=2.
If the tensor is singledimensional, simply standardizes the tensor. If for some batch index all elements are equal (of if there is only a single data point), this function will return 0 for that batch index.
 Parameters
Y (
Tensor
) – A batch_shape x n x mdim tensor. Return type
Tensor
 Returns
The standardized Y.
Example
>>> Y = torch.rand(4, 3) >>> Y_standardized = standardize(Y)

botorch.utils.transforms.
normalize
(X, bounds)[source]¶ Minmax normalize X w.r.t. the provided bounds.
 Parameters
X (
Tensor
) – … x d tensor of databounds (
Tensor
) – 2 x d tensor of lower and upper bounds for each of the X’s d columns.
 Return type
Tensor
 Returns
 A … x ddim tensor of normalized data, given by
(X  bounds[0]) / (bounds[1]  bounds[0]). If all elements of X are contained within bounds, the normalized values will be contained within [0, 1]^d.
Example
>>> X = torch.rand(4, 3) >>> bounds = torch.stack([torch.zeros(3), 0.5 * torch.ones(3)]) >>> X_normalized = normalize(X, bounds)

botorch.utils.transforms.
unnormalize
(X, bounds)[source]¶ Unnormalizes X w.r.t. the provided bounds.
 Parameters
X (
Tensor
) – … x d tensor of databounds (
Tensor
) – 2 x d tensor of lower and upper bounds for each of the X’s d columns.
 Return type
Tensor
 Returns
 A … x ddim tensor of unnormalized data, given by
X * (bounds[1]  bounds[0]) + bounds[0]. If all elements of X are contained in [0, 1]^d, the unnormalized values will be contained within bounds.
Example
>>> X_normalized = torch.rand(4, 3) >>> bounds = torch.stack([torch.zeros(3), 0.5 * torch.ones(3)]) >>> X = unnormalize(X_normalized, bounds)

botorch.utils.transforms.
normalize_indices
(indices, d)[source]¶ Normalize a list of indices to ensure that they are positive.
 Parameters
indices (
Optional
[List
[int
]]) – A list of indices (may contain negative indices for indexing “from the back”).d (
int
) – The dimension of the tensor to index.
 Return type
Optional
[List
[int
]] Returns
A normalized list of indices such that each index is between 0 and d1, or None if indices is None.

botorch.utils.transforms.
t_batch_mode_transform
(expected_q=None)[source]¶ Factory for decorators taking a tbatched X tensor.
This method creates decorators for instance methods to transform an input tensor X to tbatch mode (i.e. with at least 3 dimensions). This assumes the tensor has a qbatch dimension. The decorator also checks the qbatch size if expected_q is provided.
 Parameters
expected_q (
Optional
[int
]) – The expected qbatch size of X. If specified, this will raise an AssertitionError if X’s qbatch size does not equal expected_q. Return type
Callable
[[Callable
[[Any
,Tensor
],Any
]],Callable
[[Any
,Tensor
],Any
]] Returns
The decorated instance method.
Example
>>> class ExampleClass: >>> @t_batch_mode_transform(expected_q=1) >>> def single_q_method(self, X): >>> ... >>> >>> @t_batch_mode_transform() >>> def arbitrary_q_method(self, X): >>> ...

botorch.utils.transforms.
concatenate_pending_points
(method)[source]¶ Decorator concatenating X_pending into an acquisition function’s argument.
This decorator works on the forward method of acquisition functions taking a tensor X as the argument. If the acquisition function has an X_pending attribute (that is not None), this is concatenated into the input X, appropriately expanding the pending points to match the batch shape of X.
Example
>>> class ExampleAcquisitionFunction: >>> @concatenate_pending_points >>> @t_batch_mode_transform() >>> def forward(self, X): >>> ...
 Return type
Callable
[[Any
,Tensor
],Any
]

botorch.utils.transforms.
match_batch_shape
(X, Y)[source]¶ Matches the batch dimension of a tensor to that of another tensor.
 Parameters
X (
Tensor
) – A batch_shape_X x q x d tensor, whose batch dimensions that correspond to batch dimensions of Y are to be matched to those (if compatible).Y (
Tensor
) – A batch_shape_Y x q’ x d tensor.
 Return type
Tensor
 Returns
A batch_shape_Y x q x d tensor containing the data of X expanded to the batch dimensions of Y (if compatible). For instance, if X is b’’ x b’ x q x d and Y is b x q x d, then the returned tensor is b’’ x b x q x d.
Example
>>> X = torch.rand(2, 1, 5, 3) >>> Y = torch.rand(2, 6, 4, 3) >>> X_matched = match_batch_shape(X, Y) >>> X_matched.shape torch.Size([2, 6, 5, 3])
Feasible Volume¶

botorch.utils.feasible_volume.
get_feasible_samples
(samples, inequality_constraints=None)[source]¶ Checks which of the samples satisfy all of the inequality constraints.
 Parameters
samples (
Tensor
) – A sample size x d size tensor of feature samples, where d is a feature dimension.constraints (inequality) – A list of tuples (indices, coefficients, rhs), with each tuple encoding an inequality constraint of the form sum_i (X[indices[i]] * coefficients[i]) >= rhs.
 Return type
Tuple
[Tensor
,float
] Returns
2element tuple containing
Samples satisfying the linear constraints.
Estimated proportion of samples satisfying the linear constraints.

botorch.utils.feasible_volume.
get_outcome_feasibility_probability
(model, X, outcome_constraints, threshold=0.1, nsample_outcome=1000, seed=None)[source]¶ Monte Carlo estimate of the feasible volume with respect to the outcome constraints.
 Parameters
model (
Model
) – The model used for sampling the posterior.X (
Tensor
) – A tensor of dimension batchshape x 1 x d, where d is feature dimension.outcome_constraints (
List
[Callable
[[Tensor
],Tensor
]]) – A list of callables, each mapping a Tensor of dimension sample_shape x batchshape x q x m to a Tensor of dimension sample_shape x batchshape x q, where negative values imply feasibility.threshold (
float
) – A lower limit for the probability of posterior samples feasibility.nsample_outcome (
int
) – The number of samples from the model posterior.seed (
Optional
[int
]) – The seed for the posterior sampler. If omitted, use a random seed.
 Return type
float
 Returns
Estimated proportion of features for which posterior samples satisfy given outcome constraints with probability above or equal to the given threshold.

botorch.utils.feasible_volume.
estimate_feasible_volume
(bounds, model, outcome_constraints, inequality_constraints=None, nsample_feature=1000, nsample_outcome=1000, threshold=0.1, verbose=False, seed=None, device=None, dtype=None)[source]¶ Monte Carlo estimate of the feasible volume with respect to feature constraints and outcome constraints.
 Parameters
bounds (
Tensor
) – A 2 x d tensor of lower and upper bounds for each column of X.model (
Model
) – The model used for sampling the outcomes.outcome_constraints (
List
[Callable
[[Tensor
],Tensor
]]) – A list of callables, each mapping a Tensor of dimension sample_shape x batchshape x q x m to a Tensor of dimension sample_shape x batchshape x q, where negative values imply feasibility.constraints (inequality) – A list of tuples (indices, coefficients, rhs), with each tuple encoding an inequality constraint of the form sum_i (X[indices[i]] * coefficients[i]) >= rhs.
nsample_feature (
int
) – The number of feature samples satisfying the bounds.nsample_outcome (
int
) – The number of outcome samples from the model posterior.threshold (
float
) – A lower limit for the probability of outcome feasibilityseed (
Optional
[int
]) – The seed for both feature and outcome samplers. If omitted, use a random seed.verbose (
bool
) – An indicator for whether to log the results.
 Returns
 Estimated proportion of volume in feature space that is
feasible wrt the bounds and the inequality constraints (linear).
 Estimated proportion of feasible features for which
posterior samples (outcome) satisfies the outcome constraints with probability above the given threshold.
 Return type
2element tuple containing
MultiObjective Utilities¶
Box Decompositions¶
Algorithms for partitioning the nondominated space into rectangles.
References
 Couckuyt2012(1,2)
I. Couckuyt, D. Deschrijver and T. Dhaene, “Towards Efficient Multiobjective Optimization: Multiobjective statistical criterions,” 2012 IEEE Congress on Evolutionary Computation, Brisbane, QLD, 2012, pp. 18.

class
botorch.utils.multi_objective.box_decomposition.
NondominatedPartitioning
(num_outcomes, Y=None, alpha=0.0, eps=None)[source]¶ Bases:
torch.nn.modules.module.Module
A class for partitioning the nondominated space into hypercells.
Note: this assumes maximization. Internally, it multiplies by 1 and performs the decomposition under minimization. TODO: use maximization internally as well.
Note: it is only feasible to use this algorithm to compute an exact decomposition of the nondominated space for m<5 objectives (alpha=0.0).
The alpha parameter can be increased to obtain an approximate partitioning faster. The alpha is a fraction of the total hypervolume encapsuling the entire pareto set. When a hypercell’s volume divided by the total hypervolume is less than alpha, we discard the hypercell. See Figure 2 in [Couckuyt2012] for a visual representation.
This PyTorch implementation of the binary partitioning algorithm ([Couckuyt2012]) is adapted from numpy/tensorflow implementation at: https://github.com/GPflow/GPflowOpt/blob/master/gpflowopt/pareto.py.
TODO: replace this with a more efficient decomposition. E.g. https://link.springer.com/content/pdf/10.1007/s10898019007987.pdf
Initialize NondominatedPartitioning.
 Parameters
num_outcomes (
int
) – The number of outcomesY (
Optional
[Tensor
]) – A n x mdim tensoralpha (
float
) – a thresold fraction of total volume used in an approximate decomposition.eps (
Optional
[float
]) – a small value for numerical stability

property
eps
¶  Return type
float

property
pareto_Y
¶ This returns the nondominated set.
Note: Internally, we store the negative pareto set (minimization).
 Return type
Tensor
 Returns
A n_pareto x mdim tensor of outcomes.

update
(Y)[source]¶ Update nondominated front and decomposition.
 Parameters
Y (
Tensor
) – A n x mdim tensor of outcomes. Return type
None

binary_partition_non_dominated_space
()[source]¶ Partition the nondominated space into disjoint hypercells.
This method works for an arbitrary number of outcomes, but is less efficient than partition_non_dominated_space_2d for the 2outcome case.

partition_non_dominated_space_2d
()[source]¶ Partition the nondominated space into disjoint hypercells.
This direct method works for m=2 outcomes.
 Return type
None

get_hypercell_bounds
(ref_point)[source]¶ Get the bounds of each hypercell in the decomposition.
 Parameters
ref_point (
Tensor
) – A mdim tensor containing the reference point. Return type
Tensor
 Returns
 A 2 x num_cells x num_outcomesdim tensor containing the
lower and upper vertices bounding each hypercell.

compute_hypervolume
(ref_point)[source]¶ Compute the hypervolume for the given reference point.
Note: This assumes minimization.
This method computes the hypervolume of the nondominated space and computes the difference between the hypervolume between the ideal point and hypervolume of the nondominated space.
Note there are much more efficient alternatives for computing hypervolume when m > 2 (which do not require partitioning the nondominated space). Given such a partitioning, this method is quite fast.
 Parameters
ref_point (
Tensor
) – A mdim tensor containing the reference point. Return type
float
 Returns
The dominated hypervolume.
Hypervolume¶
Hypervolume Utilities.
References
 Fonseca2006(1,2)
C. M. Fonseca, L. Paquete, and M. LopezIbanez. An improved dimensionsweep algorithm for the hypervolume indicator. In IEEE Congress on Evolutionary Computation, pages 11571163, Vancouver, Canada, July 2006.

class
botorch.utils.multi_objective.hypervolume.
Hypervolume
(ref_point)[source]¶ Bases:
object
Hypervolume computation dimension sweep algorithm from [Fonseca2006].
Adapted from Simon Wessing’s implementation of the algorithm (Variant 3, Version 1.2) in [Fonseca2006] in PyMOO: https://github.com/msucoinlab/pymoo/blob/master/pymoo/vendor/hv.py
Maximization is assumed.
TODO: write this in C++ for faster looping.
Initialize hypervolume object.
 Parameters
ref_point (
Tensor
) – mdim Tensor containing the reference point.

property
ref_point
¶ Get reference point (for maximization).
 Return type
Tensor
 Returns
A mdim tensor containing the reference point.

botorch.utils.multi_objective.hypervolume.
sort_by_dimension
(nodes, i)[source]¶ Sorts the list of nodes inplace by the specified objective.
 Parameters
nodes (
List
[Node
]) – A list of Nodesi (
int
) – The index of the objective to sort by
 Return type
None

class
botorch.utils.multi_objective.hypervolume.
Node
(m, dtype, device, data=None)[source]¶ Bases:
object
Node in the MultiList data structure.
Initialize MultiList.
 Parameters
m (
int
) – The number of objectivesdtype (
dtype
) – The dtypedevice (
device
) – The devicedata (
Optional
[Tensor
]) – The tensor data to be stored in this Node.

class
botorch.utils.multi_objective.hypervolume.
MultiList
(m, dtype, device)[source]¶ Bases:
object
A special data structure used in hypervolume computation.
It consists of several doubly linked lists that share common nodes. Every node has multiple predecessors and successors, one in every list.
Initialize m doubly linked lists.
 Parameters
m (
int
) – number of doubly linked listsdtype (
dtype
) – the dtypedevice (
device
) – the device

append
(node, index)[source]¶ Appends a node to the end of the list at the given index.
 Parameters
node (
Node
) – the new nodeindex (
int
) – the index where the node should be appended.
 Return type
None

extend
(nodes, index)[source]¶ Extends the list at the given index with the nodes.
 Parameters
nodes (
List
[Node
]) – list of nodes to append at the given index.index (
int
) – the index where the nodes should be appended.
 Return type
None

reinsert
(node, index, bounds)[source]¶ Reinserts the node at its original position.
Reinserts the node at its original position in all lists in [0, ‘index’] before it was removed. This method assumes that the next and previous nodes of the node that is reinserted are in the list.
 Parameters
node (
Node
) – The nodeindex (
int
) – The upper bound on the range of indicesbounds (
Tensor
) – A 2 x mdim tensor bounds on the objectives
 Return type
None
Pareto¶

botorch.utils.multi_objective.pareto.
is_non_dominated
(Y)[source]¶ Computes the nondominated front.
Note: this assumes maximization.
 Parameters
Y (
Tensor
) – a (batch_shape) x n x mdim tensor of outcomes. Return type
Tensor
 Returns
A (batch_shape) x ndim boolean tensor indicating whether each point is nondominated.
Scalarization¶
Helper utilities for constructing scalarizations.
References
 Knowles2005(1,2)
J. Knowles, “ParEGO: a hybrid algorithm with online landscape approximation for expensive multiobjective optimization problems,” in IEEE Transactions on Evolutionary Computation, vol. 10, no. 1, pp. 5066, Feb. 2006.

botorch.utils.multi_objective.scalarization.
get_chebyshev_scalarization
(weights, Y, alpha=0.05)[source]¶ Construct an augmented Chebyshev scalarization.
Outcomes are first normalized to [0,1] and then an augmented Chebyshev scalarization is applied.
 Augmented Chebyshev scalarization:
objective(y) = min(w * y) + alpha * sum(w * y)
Note: this assumes maximization.
See [Knowles2005] for details.
This scalarization can be used with qExpectedImprovement to implement qParEGO as proposed in [Daulton2020].
 Parameters
weights (
Tensor
) – A mdim tensor of weights.Y (
Tensor
) – A n x mdim tensor of observed outcomes, which are used for scaling the outcomes to [0,1].alpha (
float
) – Parameter governing the influence of the weighted sum term. The default value comes from [Knowles2005].
 Return type
Callable
[[Tensor
],Tensor
] Returns
Transform function using the objective weights.
Example
>>> weights = torch.tensor([0.75, 0.25]) >>> transform = get_aug_chebyshev_scalarization(weights, Y)