# 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[torch.Tensor, torch.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.

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).

Return type

Optional[List[Callable[[torch.Tensor], torch.Tensor]]]

Example

>>> # constrain f(x)[0] <= 0
>>> A = torch.tensor([[1., 0.]])
>>> b = torch.tensor([[0.]])
>>> outcome_constraints = get_outcome_constraint_transforms((A, b))


## Containers¶

Containers to standardize inputs into models and acquisition functions.

class botorch.utils.containers.TrainingData(Xs, Ys, Yvars=None)[source]

Bases: object

Standardized container of model training data for models.

Properties:
Xs: A list of tensors, each of shape batch_shape x n_i x d,

where n_i is the number of training inputs for the i-th model.

Ys: A list of tensors, each of shape batch_shape x n_i x 1,

where n_i is the number of training observations for the i-th (single-output) model.

Yvars: A list of tensors, each of shape batch_shape x n_i x 1,

where n_i is the number of training observations of the observation noise for the i-th (single-output) model. If None, the observation noise level is unobserved.

Parameters
• Xs (List[torch.Tensor]) –

• Ys (List[torch.Tensor]) –

• Yvars (Optional[List[torch.Tensor]]) –

Return type

None

Xs: List[torch.Tensor]
Ys: List[torch.Tensor]
Yvars: Optional[List[torch.Tensor]] = None
classmethod from_block_design(X, Y, Yvar=None)[source]

Construct a TrainingData object from a block design description.

Parameters
• X (torch.Tensor) – A batch_shape x n x d tensor of training points (shared across all outcomes).

• Y (torch.Tensor) – A batch_shape x n x m tensor of training observations.

• Yvar (Optional[torch.Tensor]) – A batch_shape x n x m tensor of training noise variance observations, or None.

Returns

The TrainingData object (with is_block_design=True).

property is_block_design: bool

Indicates whether training data is a “block design”.

Block designs are designs in which all outcomes are observed at the same training inputs.

property X: torch.Tensor

The training inputs (block-design only).

This raises an UnsupportedError in the non-block-design case.

property Y: torch.Tensor

The training observations (block-design only).

This raises an UnsupportedError in the non-block-design case.

property Yvar: Optional[List[torch.Tensor]]

The training observations’s noise variance (block-design only).

This raises an UnsupportedError in the non-block-design case.

## Dispatcher¶

class botorch.utils.dispatcher.Dispatcher(name, doc=None, encoder=<class 'type'>)[source]

Bases: multipledispatch.dispatcher.Dispatcher

Clearing house for multiple dispatch functionality. This class extends <multipledispatch.Dispatcher> by: (i) generalizing the argument encoding convention during method lookup, (ii) implementing __getitem__ as a dedicated method lookup function.

Parameters
• name (str) – A string identifier for the Dispatcher instance.

• doc (Optional[str]) – A docstring for the multiply dispatched method(s).

• encoder (Callable[Any, Type]) – A callable that individually transforms the arguments passed at runtime in order to construct the key used for method lookup as tuple(map(encoder, args)). Defaults to type.

Return type

None

dispatch(*types)[source]

Method lookup strategy. Checks for an exact match before traversing the set of registered methods according to the current ordering.

Parameters

types (Type) – A tuple of types that gets compared with the signatures of registered methods to determine compatibility.

Returns

The first method encountered with a matching signature.

Return type

Callable

encode_args(args)[source]

Converts arguments into a tuple of types used during method lookup.

Parameters

args (Any) –

Return type

Tuple[Type]

help(*args, **kwargs)[source]

Prints the retrieved method’s docstring.

Parameters
• args (Any) –

• kwargs (Any) –

Return type

None

source(*args, **kwargs)[source]

Prints the retrieved method’s source types.

Return type

None

property encoder: Callable[Any, Type]
name
funcs
doc

## Low-Rank Cholesky Update Utils¶

botorch.utils.low_rank.extract_batch_covar(mt_mvn)[source]

Extract a batched independent covariance matrix from an MTMVN.

Parameters

mt_mvn (gpytorch.distributions.multitask_multivariate_normal.MultitaskMultivariateNormal) – A multi-task multivariate normal with a block diagonal covariance matrix.

Returns

A lazy covariance matrix consisting of a batch of the blocks of

the diagonal of the MultitaskMultivariateNormal.

Return type

gpytorch.lazy.lazy_tensor.LazyTensor

botorch.utils.low_rank.sample_cached_cholesky(posterior, baseline_L, q, base_samples, sample_shape, max_tries=6)[source]

Get posterior samples at the q new points from the joint multi-output posterior.

Parameters
• posterior (botorch.posteriors.gpytorch.GPyTorchPosterior) – The joint posterior is over (X_baseline, X).

• baseline_L (torch.Tensor) – The baseline lower triangular cholesky factor.

• q (int) – The number of new points in X.

• base_samples (torch.Tensor) – The base samples.

• sample_shape (torch.Size) – The sample shape.

• max_tries (int) – The number of tries for computing the Cholesky decomposition with increasing jitter.

Returns

A sample_shape x batch_shape x q x m-dim tensor of posterior

samples at the new points.

Return type

torch.Tensor

## 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 and an (optional) Tensor of size b x q x d 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[torch.Tensor]) – a 1-dimensional Tensor containing a weight for each task. If not provided, the identity mapping is used.

Returns

Transform function using the objective weights.

Return type

Callable[[torch.Tensor, Optional[torch.Tensor]], torch.Tensor]

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 non-negative objective.

This function uses a sigmoid approximation to an indicator function for each constraint.

Parameters
• obj (torch.Tensor) – A n_samples x b x q (x m’)-dim Tensor of objective values.

• constraints (List[Callable[[torch.Tensor], torch.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 (torch.Tensor) – A n_samples x b x q x m Tensor of samples drawn from the posterior.

• eta (float) – The temperature parameter for the sigmoid function.

Returns

A n_samples x b x q (x m’)-dim tensor of feasibility-weighted objectives.

Return type

torch.Tensor

botorch.utils.objective.soft_eval_constraint(lhs, eta=0.001)[source]

Element-wise evaluation of a constraint in a ‘soft’ fashion

value(x) = 1 / (1 + exp(x / eta))

Parameters
• lhs (torch.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.

Returns

Element-wise ‘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.

Return type

torch.Tensor

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 feasibility-weighting an objective for the case where the objective can be negative by using the following strategy: (1) Add M to make obj non-negative; (2) Apply constraints using the sigmoid approximation; (3) Shift by -M.

Parameters
• obj (torch.Tensor) – A n_samples x b x q (x m’)-dim Tensor of objective values.

• constraints (List[Callable[[torch.Tensor], torch.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 (torch.Tensor) – A n_samples x 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.

Returns

A n_samples x b x q (x m’)-dim tensor of feasibility-weighted objectives.

Return type

torch.Tensor

## Rounding¶

botorch.utils.rounding.approximate_round(X, tau=0.001)[source]

Diffentiable approximate rounding function.

This method is a piecewise approximation of a rounding function where each piece is a hyperbolic tangent function.

Parameters
• X (torch.Tensor) – The tensor to round to the nearest integer (element-wise).

• tau (float) – A temperature hyperparameter.

Returns

The approximately rounded input tensor.

Return type

torch.Tensor

## Sampling¶

Utilities for MC and qMC sampling.

References

Trikalinos2014polytope

T. A. Trikalinos and G. van Valkenhoef. Efficient sampling from uniform density n-polytopes. Technical report, Brown University, 2014.

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.

Returns

Generator

Return type

Generator[None, None, None]

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 multi-variate standard normal N(0, I_qo).

Parameters
• batch_shape (torch.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 (torch.Size) – The output shape (q x m) of the base samples to generate.

• sample_shape (torch.Size) – The sample shape of the samples to draw.

• qmc (bool) – If True, use quasi-MC sampling (instead of iid draws).

• seed (Optional[int]) – If provided, use as a seed for the RNG.

• device (Optional[torch.device]) –

• dtype (Optional[torch.dtype]) –

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.

Return type

torch.Tensor

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 (botorch.posteriors.posterior.Posterior) – A Posterior object.

• sample_shape (torch.Size) – The sample shape of the samples to draw.

• qmc (bool) – If True, use quasi-MC sampling (instead of iid draws).

• seed (Optional[int]) – If provided, use as a seed for the RNG.

• collapse_batch_dims (bool) –

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 t-batch of samples, so as to not introduce any sampling variance across t-batches.

Return type

torch.Tensor

Example

>>> sample_shape = torch.Size([10])
>>> samples = construct_base_samples_from_posterior(posterior, sample_shape)

botorch.utils.sampling.draw_sobol_samples(bounds, n, q, batch_shape=None, 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 d-dimensional space, where bounds[0, :] and bounds[1, :] correspond to lower and upper bounds, respectively.

• n (int) – The number of (q-batch) samples. As a best practice, use powers of 2.

• q (int) – The size of each q-batch.

• batch_shape (Optional[Iterable[int], torch.Size]) – The batch shape of the samples. If given, returns samples of shape n x batch_shape x q x d, where each batch is an n x q x d-dim tensor of qMC samples.

• seed (Optional[int]) – The seed used for initializing Owen scrambling. If None (default), use a random seed.

Returns

A n x batch_shape x q x d-dim tensor of qMC samples from the box defined by bounds.

Return type

Tensor

Example

>>> bounds = torch.stack([torch.zeros(3), torch.ones(3)])
>>> samples = draw_sobol_samples(bounds, 16, 2)

botorch.utils.sampling.draw_sobol_normal_samples(d, n, device=None, dtype=None, seed=None)[source]

Draw qMC samples from a multi-variate standard normal N(0, I_d)

A primary use-case 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. As a best practice, use powers of 2.

• device (Optional[torch.device]) – The torch device.

• dtype (Optional[torch.dtype]) – The torch dtype.

• seed (Optional[int]) – The seed used for initializing Owen scrambling. If None (default), use a random seed.

Returns

A tensor of qMC standard normal samples with dimension n x d with device and dtype specified by the input.

Return type

torch.Tensor

Example

>>> samples = draw_sobol_normal_samples(2, 16)

botorch.utils.sampling.sample_hypersphere(d, n=1, qmc=False, seed=None, device=None, dtype=None)[source]

Sample uniformly from a unit d-sphere.

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[torch.device]) – The torch device.

• dtype (Optional[torch.dtype]) – The torch dtype.

Returns

An n x d tensor of uniform samples from from the d-hypersphere.

Return type

torch.Tensor

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 d-simplex.

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[torch.device]) – The torch device.

• dtype (Optional[torch.dtype]) – The torch dtype.

Returns

An n x d tensor of uniform samples from from the d-simplex.

Return type

torch.Tensor

Example

>>> sample_simplex(d=3, n=10)

botorch.utils.sampling.sample_polytope(A, b, x0, n=10000, n0=100, seed=None)[source]

Hit and run sampler from uniform sampling points from a polytope, described via inequality constraints A*x<=b.

Parameters
• A (torch.Tensor) – A Tensor describing inequality constraints so that all samples satisfy Ax<=b.

• b (torch.Tensor) – A Tensor describing the inequality constraints so that all samples satisfy Ax<=b.

• x0 (torch.Tensor) – A d-dim Tensor representing a starting point of the chain satisfying the constraints.

• n (int) – The number of resulting samples kept in the output.

• n0 (int) – The number of burn-in samples. The chain will produce n+n0 samples but the first n0 samples are not saved.

• seed (Optional[int]) – The seed for the sampler. If omitted, use a random seed.

Returns

(n, d) dim Tensor containing the resulting samples.

Return type

torch.Tensor

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 (torch.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 non-negative, finite and have a non-zero 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[torch._C.Generator]) – A a pseudorandom number generator for sampling.

• out (Optional[torch.Tensor]) – The output tensor (optional). If provided, must be of size batch_shape x num_samples.

Returns

A batch_shape x num_samples tensor of samples.

Return type

torch.LongTensor

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

botorch.utils.sampling.find_interior_point(A, b, A_eq=None, b_eq=None)[source]

Find an interior point of a polytope via linear programming.

Parameters
• A (numpy.ndarray) – A n_ineq x d-dim numpy array containing the coefficients of the constraint inequalities.

• b (numpy.ndarray) – A n_ineq x 1-dim numpy array containing the right hand sides of the constraint inequalities.

• A_eq (Optional[numpy.ndarray]) – A n_eq x d-dim numpy array containing the coefficients of the constraint equalities.

• b_eq (Optional[numpy.ndarray]) – A n_eq x 1-dim numpy array containing the right hand sides of the constraint equalities.

Returns

A d-dim numpy array containing an interior point of the polytope. This function will raise a ValueError if there is no such point.

Return type

numpy.ndarray

This method solves the following Linear Program:

min -s subject to A @ x <= b - 2 * s, s >= 0, A_eq @ x = b_eq

In case the polytope is unbounded, then it will also constrain the slack variable s to s<=1.

class botorch.utils.sampling.PolytopeSampler(inequality_constraints=None, equality_constraints=None, bounds=None, interior_point=None)[source]

Bases: abc.ABC

Base class for samplers that sample points from a polytope.

Initialize PolytopeSampler.

Parameters
• inequality_constraints (Optional[Tuple[Tensor, Tensor]]) – Tensors (A, b) describing inequality constraints A @ x <= b, where A is a n_ineq_con x d-dim Tensor and b is a n_ineq_con x 1-dim Tensor, with n_ineq_con the number of inequalities and d the dimension of the sample space.

• equality_constraints (Optional[Tuple[Tensor, Tensor]]) – Tensors (C, d) describing the equality constraints C @ x = d, where C is a n_eq_con x d-dim Tensor and d is a n_eq_con x 1-dim Tensor with n_eq_con the number of equalities.

• bounds (Optional[Tensor]) – A 2 x d-dim tensor of box bounds, where inf (-inf) means that the respective dimension is unbounded above (below).

• interior_point (Optional[Tensor]) – A d x 1-dim Tensor presenting a point in the (relative) interior of the polytope. If omitted, determined automatically by solving a Linear Program.

Return type

None

feasible(x)[source]

Check whether a point is contained in the polytope.

Parameters

x (torch.Tensor) – A d x 1-dim Tensor.

Returns

True if x is contained inside the polytope (incl. its boundary), False otherwise.

Return type

bool

find_interior_point()[source]

Find an interior point of the polytope.

Returns

A d x 1-dim Tensor representing a point contained in the polytope. This function will raise a ValueError if there is no such point.

Return type

torch.Tensor

abstract draw(n=1, seed=None)[source]

Draw samples from the polytope.

Parameters
• n (int) – The number of samples.

• seed (Optional[int]) – The random seed.

Returns

A n x d Tensor of samples from the polytope.

Return type

torch.Tensor

class botorch.utils.sampling.HitAndRunPolytopeSampler(inequality_constraints=None, equality_constraints=None, bounds=None, interior_point=None, n_burnin=0)[source]

A sampler for sampling from a polyope using a hit-and-run algorithm.

A sampler for sampling from a polyope using a hit-and-run algorithm.

Parameters
• inequality_constraints (Optional[Tuple[Tensor, Tensor]]) – Tensors (A, b) describing inequality constraints A @ x <= b, where A is a n_ineq_con x d-dim Tensor and b is a n_ineq_con x 1-dim Tensor, with n_ineq_con the number of inequalities and d the dimension of the sample space.

• equality_constraints (Optional[Tuple[Tensor, Tensor]]) – Tensors (C, d) describing the equality constraints C @ x = d, where C is a n_eq_con x d-dim Tensor and d is a n_eq_con x 1-dim Tensor with n_eq_con the number of equalities.

• bounds (Optional[Tensor]) – A 2 x d-dim tensor of box bounds, where inf (-inf) means that the respective dimension is unbounded from above (below).

• interior_point (Optional[Tensor]) – A d x 1-dim Tensor representing a point in the (relative) interior of the polytope. If omitted, determined automatically by solving a Linear Program.

• n_burnin (int) – The number of burn in samples.

Return type

None

draw(n=1, seed=None)[source]

Draw samples from the polytope.

Parameters
• n (int) – The number of samples.

• seed (Optional[int]) – The random seed.

Returns

A n x d Tensor of samples from the polytope.

Return type

torch.Tensor

class botorch.utils.sampling.DelaunayPolytopeSampler(inequality_constraints=None, equality_constraints=None, bounds=None, interior_point=None)[source]

A polytope sampler using Delaunay triangulation.

This sampler first enumerates the vertices of the constraint polytope and then uses a Delaunay triangulation to tesselate its convex hull.

The sampling happens in two stages: 1. First, we sample from the set of hypertriangles generated by the Delaunay triangulation (i.e. which hyper-triangle to draw the sample from) with probabilities proportional to the triangle volumes. 2. Then, we sample uniformly from the chosen hypertriangle by sampling uniformly from the unit simplex of the appropriate dimension, and then computing the convex combination of the vertices of the hypertriangle according to that draw from the simplex.

The best reference (not exactly the same, but functionally equivalent) is [Trikalinos2014polytope]. A simple R implementation is available at https://github.com/gertvv/tesselample.

Initialize DelaunayPolytopeSampler.

Parameters
• inequality_constraints (Optional[Tuple[Tensor, Tensor]]) – Tensors (A, b) describing inequality constraints A @ x <= b, where A is a n_ineq_con x d-dim Tensor and b is a n_ineq_con x 1-dim Tensor, with n_ineq_con the number of inequalities and d the dimension of the sample space.

• equality_constraints (Optional[Tuple[Tensor, Tensor]]) – Tensors (C, d) describing the equality constraints C @ x = d, where C is a n_eq_con x d-dim Tensor and d is a n_eq_con x 1-dim Tensor with n_eq_con the number of equalities.

• bounds (Optional[Tensor]) – A 2 x d-dim tensor of box bounds, where inf (-inf) means that the respective dimension is unbounded from above (below).

• interior_point (Optional[Tensor]) – A d x 1-dim Tensor representing a point in the (relative) interior of the polytope. If omitted, determined automatically by solving a Linear Program.

Return type

None

Warning: The vertex enumeration performed in this algorithm can become extremely costly if there are a large number of inequalities. Similarly, the triangulation can get very expensive in high dimensions. Only use this algorithm for moderate dimensions / moderately complex constraint sets. An alternative is the HitAndRunPolytopeSampler.

draw(n=1, seed=None)[source]

Draw samples from the polytope.

Parameters
• n (int) – The number of samples.

• seed (Optional[int]) – The random seed.

Returns

A n x d Tensor of samples from the polytope.

Return type

torch.Tensor

botorch.utils.sampling.get_polytope_samples(n, bounds, inequality_constraints=None, equality_constraints=None, seed=None, thinning=32, n_burnin=10000)[source]

Sample from polytope defined by box bounds and (in)equality constraints.

This uses a hit-and-run Markov chain sampler.

TODO: make this method return the sampler object, to avoid doing burn-in every time we draw samples.

Parameters
• n (int) – The number of samples.

• bounds (torch.Tensor) – A 2 x d-dim tensor containing the box bounds.

• constraints (equality) – 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.

• constraints – 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.

• seed (Optional[int]) – The random seed.

• thinning (int) – The amount of thinning.

• n_burnin (int) – The number of burn-in samples for the Markov chain sampler.

• inequality_constraints (Optional[List[Tuple[torch.Tensor, torch.Tensor, float]]]) –

• equality_constraints (Optional[List[Tuple[torch.Tensor, torch.Tensor, float]]]) –

Returns

A n x d-dim tensor of samples.

Return type

torch.Tensor

botorch.utils.sampling.sparse_to_dense_constraints(d, constraints)[source]

Convert parameter constraints from a sparse format into a dense format.

This method converts sparse triples of the form (indices, coefficients, rhs) to constraints of the form Ax >= b or Ax = b.

Parameters
• d (int) – The input dimension.

• constraints (inequality) – A list of tuples (indices, coefficients, rhs), with each tuple encoding an (in)equality constraint of the form sum_i (X[indices[i]] * coefficients[i]) >= rhs or sum_i (X[indices[i]] * coefficients[i]) = rhs.

Returns

• A: A n_constraints x d-dim tensor of coefficients.

• b: A n_constraints x 1-dim tensor of right hand sides.

Return type

A two-element tuple containing

## Sampling from GP priors¶

class botorch.utils.gp_sampling.GPDraw(model, seed=None)[source]

Bases: torch.nn.modules.module.Module

Convenience wrapper for sampling a function from a GP prior.

This wrapper implicitly defines the GP sample as a self-updating function by keeping track of the evaluated points and respective base samples used during the evaluation.

This does not yet support multi-output models.

Construct a GP function sampler.

Parameters
• model (Model) – The Model defining the GP prior.

• seed (Optional[int]) –

Return type

None

property Xs: torch.Tensor

A (batch_shape) x n_eval x d-dim tensor of locations at which the GP was evaluated (or None if the sample has never been evaluated).

property Ys: torch.Tensor

A (batch_shape) x n_eval x d-dim tensor of associated function values (or None if the sample has never been evaluated).

forward(X)[source]

Evaluate the GP sample function at a set of points X.

Parameters

X (torch.Tensor) – A batch_shape x n x d-dim tensor of points

Returns

The value of the GP sample at the n points.

Return type

torch.Tensor

training: bool
class botorch.utils.gp_sampling.RandomFourierFeatures(kernel, input_dim, num_rff_features, sample_shape=None)[source]

Bases: torch.nn.modules.module.Module

A class that represents Random Fourier Features.

Initialize RandomFourierFeatures.

Parameters
• kernel (Kernel) – The GP kernel.

• input_dim (int) – The input dimension to the GP kernel.

• num_rff_features (int) – The number of fourier features.

• sample_shape (Optional[torch.Size]) – The shape of a single sample. For a single-element torch.Size object, this is simply the number of RFF draws.

Return type

None

forward(X)[source]

Get fourier basis features for the provided inputs. Note that if sample_shape has been passed, then the rightmost subset of the batch shape of the input should be sample_shape.

Parameters

X (torch.Tensor) – input tensor of shape (batch_shape) x n x input_dim

Returns

A Tensor of shape (batch_shape) x n x rff

Return type

torch.Tensor

training: bool
botorch.utils.gp_sampling.get_deterministic_model_multi_samples(weights, bases)[source]

Get a batched deterministic model that batch evaluates n_samples function samples. This supports multi-output models as well.

Parameters
• weights (List[torch.Tensor]) – a list of weights with num_outputs elements. Each weight is of shape (batch_shape_input) x n_samples x num_rff_features, where (batch_shape_input) is the batch shape of the inputs used to obtain the posterior weights.

• bases (List[botorch.utils.gp_sampling.RandomFourierFeatures]) – a list of RandomFourierFeatures with num_outputs elements. Each basis has a sample shape of n_samples.

• n_samples – the number of function samples.

Returns

A batched GenericDeterministicModels that batch evaluates n_samples function samples.

Return type

botorch.models.deterministic.GenericDeterministicModel

botorch.utils.gp_sampling.get_deterministic_model(weights, bases)[source]

Get a deterministic model using the provided weights and bases for each output.

Parameters
Returns

A deterministic model.

Return type

botorch.models.deterministic.GenericDeterministicModel

botorch.utils.gp_sampling.get_weights_posterior(X, y, sigma_sq)[source]

Sample bayesian linear regression weights.

Parameters
• X (torch.Tensor) – a (batch_shape) x n x num_rff_features-dim tensor of inputs

• y (torch.Tensor) – a (batch_shape) x n-dim tensor of outputs

• sigma_sq (float) – the noise variance

Returns

The posterior distribution over the weights.

Return type

torch.distributions.multivariate_normal.MultivariateNormal

botorch.utils.gp_sampling.get_gp_samples(model, num_outputs, n_samples, num_rff_features=500)[source]

Sample functions from GP posterior using RFFs. The returned GenericDeterministicModel effectively wraps num_outputs models, each of which has a batch shape of n_samples. Refer get_deterministic_model_multi_samples for more details.

Parameters
• model (botorch.models.model.Model) – The model.

• num_outputs (int) – The number of outputs.

• n_samples (int) – The number of functions to be sampled IID.

• num_rff_features (int) – The number of random Fourier features.

Returns

A batched GenericDeterministicModel that batch evaluates n_samples sampled functions.

Return type

botorch.models.deterministic.GenericDeterministicModel

## Testing¶

class botorch.utils.testing.BotorchTestCase(methodName='runTest')[source]

Bases: unittest.case.TestCase

Basic test case for Botorch.

This
1. sets the default device to be torch.device(“cpu”)

2. 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')
setUp()[source]

Hook method for setting up the test fixture before exercising it.

class botorch.utils.testing.BaseTestProblemBaseTestCase[source]

Bases: object

functions: List[botorch.test_functions.base.BaseTestProblem]
test_forward()[source]
class botorch.utils.testing.SyntheticTestFunctionBaseTestCase[source]
test_optimal_value()[source]
test_optimizer()[source]
functions: List[botorch.test_functions.base.BaseTestProblem]
class botorch.utils.testing.MockPosterior(mean=None, variance=None, samples=None)[source]

Mock object that implements dummy methods and feeds through specified outputs

property device: torch.device

The torch device of the posterior.

property dtype: torch.dtype

The torch dtype of the posterior.

property event_shape: torch.Size

The event shape (i.e. the shape of a single sample).

property mean

The mean of the posterior as a (b) x n x m-dim Tensor.

property variance

The variance of the posterior as a (b) x n x m-dim Tensor.

rsample(sample_shape=None, base_samples=None)[source]

Mock sample by repeating self._samples. If base_samples is provided, do a shape check but return the same mock samples.

Parameters
• sample_shape (Optional[torch.Size]) –

• base_samples (Optional[torch.Tensor]) –

Return type

torch.Tensor

class botorch.utils.testing.MockModel(posterior)[source]

Mock object that implements dummy methods and feeds through specified outputs

Initializes internal Module state, shared by both nn.Module and ScriptModule.

Parameters

posterior (MockPosterior) –

Return type

None

posterior(X, output_indices=None, posterior_transform=None, observation_noise=False)[source]

Computes the posterior over model outputs at the provided points.

Note: The input transforms should be applied here using

self.transform_inputs(X) after the self.eval() call and before any model.forward or model.likelihood calls.

Parameters
• X (torch.Tensor) – A b x q x d-dim 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 multi-output). 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.

• posterior_transform (Optional[botorch.acquisition.objective.PosteriorTransform]) – An optional PosteriorTransform.

Returns

A Posterior object, representing a batch of b joint distributions over q points and m outputs each.

Return type

botorch.utils.testing.MockPosterior

property num_outputs: int

The number of outputs of the model.

property batch_shape: torch.Size

The batch shape of the model.

This is a batch shape from an I/O perspective, independent of the internal representation of the model (as e.g. in BatchedMultiOutputGPyTorchModel). For a model with m outputs, a test_batch_shape x q x d-shaped input X to the posterior method returns a Posterior object over an output of shape broadcast(test_batch_shape, model.batch_shape) x q x m.

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. Parameters and buffers set to None are not included.

Warning

Currently state_dict() also accepts positional arguments for destination, prefix and keep_vars in order. However, this is being deprecated and keyword arguments will be enforced in future releases.

Warning

Please avoid the use of argument destination as it is not designed for end-users.

Parameters
• destination (dict, optional) – If provided, the state of module will be updated into the dict and the same object is returned. Otherwise, an OrderedDict will be created and returned. Default: None.

• prefix (str, optional) – a prefix added to parameter and buffer names to compose the keys in state_dict. Default: ''.

• keep_vars (bool, optional) – by default the Tensor s returned in the state dict are detached from autograd. If it’s set to True, detaching will not be performed. Default: False.

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. If strict is True, then the keys of state_dict must exactly match the keys returned by this module’s state_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’s state_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 with missing_keys and unexpected_keys fields

Note

If a parameter or buffer is registered as None and its corresponding key exists in state_dict, load_state_dict() will raise a RuntimeError.

class botorch.utils.testing.MockAcquisitionFunction[source]

Bases: object

Mock acquisition function object that implements dummy methods.

set_X_pending(X_pending=None)[source]
Parameters

X_pending (Optional[torch.Tensor]) –

class botorch.utils.testing.MultiObjectiveTestProblemBaseTestCase[source]
test_attributes()[source]
test_max_hv()[source]
test_ref_point()[source]
functions: List[botorch.test_functions.base.BaseTestProblem]
class botorch.utils.testing.ConstrainedMultiObjectiveTestProblemBaseTestCase[source]
test_num_constraints()[source]
test_evaluate_slack_true()[source]
functions: List[botorch.test_functions.base.BaseTestProblem]

## 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 respects

• the order of insertion, and

• in update(), the order of the merged OrderedDict or another BufferDict (the argument to update()).

Note that update() with other unordered mapping types (e.g., Python’s plain dict) does not preserve the order of the merged mapping.

Parameters

buffers (iterable, optional) – a mapping (dictionary) of (string : Tensor) or an iterable of key-value 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.

clear()[source]

Remove all items from the BufferDict.

pop(key)[source]

Remove key from the BufferDict and return its buffer.

Parameters

key (string) – key to pop from the BufferDict

keys()[source]

Return an iterable of the BufferDict keys.

items()[source]

Return an iterable of the BufferDict key/value pairs.

values()[source]

Return an iterable of the BufferDict values.

update(buffers)[source]

Update the BufferDict with the key-value pairs from a mapping or an iterable, overwriting existing keys.

Note

If buffers is an OrderedDict, a BufferDict, or an iterable of key-value pairs, the order of new elements in it is preserved.

Parameters

buffers (iterable) – a mapping (dictionary) from string to Tensor, or an iterable of key-value pairs of type (string, Tensor)

extra_repr()[source]

Set the extra representation of the module

To print customized extra information, you should re-implement this method in your own modules. Both single-line and multi-line strings are acceptable.

training: bool

## Transformations¶

Some basic data transformation helpers.

botorch.utils.transforms.squeeze_last_dim(Y)[source]

Squeeze the last dimension of a Tensor.

Parameters

Y (torch.Tensor) – A … x d-dim Tensor.

Returns

The input tensor with last dimension squeezed.

Return type

torch.Tensor

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 single-dimensional, simply standardizes the tensor. If for some batch index all elements are equal (or if there is only a single data point), this function will return 0 for that batch index.

Parameters

Y (torch.Tensor) – A batch_shape x n x m-dim tensor.

Returns

The standardized Y.

Return type

torch.Tensor

Example

>>> Y = torch.rand(4, 3)
>>> Y_standardized = standardize(Y)

botorch.utils.transforms.normalize(X, bounds)[source]

Min-max normalize X w.r.t. the provided bounds.

Parameters
• X (torch.Tensor) – … x d tensor of data

• bounds (torch.Tensor) – 2 x d tensor of lower and upper bounds for each of the X’s d columns.

Returns

A … x d-dim 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.

Return type

torch.Tensor

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]

Un-normalizes X w.r.t. the provided bounds.

Parameters
• X (torch.Tensor) – … x d tensor of data

• bounds (torch.Tensor) – 2 x d tensor of lower and upper bounds for each of the X’s d columns.

Returns

A … x d-dim 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 un-normalized values will be contained within bounds.

Return type

torch.Tensor

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.

Returns

A normalized list of indices such that each index is between 0 and d-1, or None if indices is None.

Return type

Optional[List[int]]

botorch.utils.transforms.is_fully_bayesian(model)[source]

Check if at least one model is a SaasFullyBayesianSingleTaskGP

Parameters
• model (Model) – A BoTorch model (may be a ModelList or ModelListGP)

• d – The dimension of the tensor to index.

Returns

True if at least one model is a SaasFullyBayesianSingleTaskGP

Return type

bool

botorch.utils.transforms.t_batch_mode_transform(expected_q=None, assert_output_shape=True)[source]

Factory for decorators enabling consistent t-batch behavior.

This method creates decorators for instance methods to transform an input tensor X to t-batch mode (i.e. with at least 3 dimensions). This assumes the tensor has a q-batch dimension. The decorator also checks the q-batch size if expected_q is provided, and the output shape if assert_output_shape is True.

Parameters
• expected_q (Optional[int]) – The expected q-batch size of X. If specified, this will raise an AssertionError if X’s q-batch size does not equal expected_q.

• assert_output_shape (bool) – If True, this will raise an AssertionError if the output shape does not match either the t-batch shape of X, or the acqf.model.batch_shape for acquisition functions using batched models.

Returns

The decorated instance method.

Return type

Callable[[Callable[[AcquisitionFunction, Any], Any]], Callable[[AcquisitionFunction, Any], Any]]

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):
>>>         ...

Parameters

method (Callable[[Any, torch.Tensor], Any]) –

Return type

Callable[[Any, torch.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 (torch.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 (torch.Tensor) – A batch_shape_Y x q’ x d 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.

Return type

torch.Tensor

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])

botorch.utils.transforms.convert_to_target_pre_hook(module, *args)[source]

Pre-hook for automatically calling .to(X) on module prior to forward

## 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 (torch.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.

• inequality_constraints (Optional[List[Tuple[torch.Tensor, torch.Tensor, float]]]) –

Returns

2-element tuple containing

• Samples satisfying the linear constraints.

• Estimated proportion of samples satisfying the linear constraints.

Return type

Tuple[torch.Tensor, float]

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 (botorch.models.model.Model) – The model used for sampling the posterior.

• X (torch.Tensor) – A tensor of dimension batch-shape x 1 x d, where d is feature dimension.

• outcome_constraints (List[Callable[[torch.Tensor], torch.Tensor]]) – A list of callables, each mapping a Tensor of dimension sample_shape x batch-shape x q x m to a Tensor of dimension sample_shape x batch-shape 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.

Returns

Estimated proportion of features for which posterior samples satisfy given outcome constraints with probability above or equal to the given threshold.

Return type

float

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 (torch.Tensor) – A 2 x d tensor of lower and upper bounds for each column of X.

• model (botorch.models.model.Model) – The model used for sampling the outcomes.

• outcome_constraints (List[Callable[[torch.Tensor], torch.Tensor]]) – A list of callables, each mapping a Tensor of dimension sample_shape x batch-shape x q x m to a Tensor of dimension sample_shape x batch-shape 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 feasibility

• seed (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.

• inequality_constraints (Optional[List[Tuple[torch.Tensor, torch.Tensor, float]]]) –

• device (Optional[torch.device]) –

• dtype (Optional[torch.dtype]) –

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

2-element tuple containing

## Abstract Box Decompositions¶

Box decomposition algorithms.

References

Lacour17(1,2,3,4,5,6)

R. Lacour, K. Klamroth, C. Fonseca. A box decomposition algorithm to compute the hypervolume indicator. Computers & Operations Research, Volume 79, 2017.

class botorch.utils.multi_objective.box_decompositions.box_decomposition.BoxDecomposition(ref_point, sort, Y=None)[source]

Bases: torch.nn.modules.module.Module, abc.ABC

An abstract class for box decompositions.

Note: Internally, we store the negative reference point (minimization).

Initialize BoxDecomposition.

Parameters
• ref_point (Tensor) – A m-dim tensor containing the reference point.

• sort (bool) – A boolean indicating whether to sort the Pareto frontier.

• Y (Optional[Tensor]) – A (batch_shape) x n x m-dim tensor of outcomes.

Return type

None

property pareto_Y: torch.Tensor

This returns the non-dominated set.

Returns

A n_pareto x m-dim tensor of outcomes.

property ref_point: torch.Tensor

Get the reference point.

Returns

A m-dim tensor of outcomes.

property Y: torch.Tensor

Get the raw outcomes.

Returns

A n x m-dim tensor of outcomes.

partition_space()[source]

Compute box decomposition.

Return type

None

abstract get_hypercell_bounds()[source]

Get the bounds of each hypercell in the decomposition.

Returns

A 2 x num_cells x num_outcomes-dim tensor containing the

lower and upper vertices bounding each hypercell.

Return type

torch.Tensor

update(Y)[source]

Update non-dominated front and decomposition.

By default, the partitioning is recomputed. Subclasses can override this functionality.

Parameters

Y (torch.Tensor) – A (batch_shape) x n x m-dim tensor of new, incremental outcomes.

Return type

None

reset()[source]

Reset non-dominated front and decomposition.

Return type

None

abstract compute_hypervolume()[source]

Compute hypervolume that is dominated by the Pareto Froniter.

Returns

A (batch_shape)-dim tensor containing the hypervolume dominated by

each Pareto frontier.

Return type

torch.Tensor

training: bool
class botorch.utils.multi_objective.box_decompositions.box_decomposition.FastPartitioning(ref_point, Y=None)[source]

Bases: botorch.utils.multi_objective.box_decompositions.box_decomposition.BoxDecomposition, abc.ABC

A class for partitioning the (non-)dominated space into hyper-cells.

Note: this assumes maximization. Internally, it multiplies outcomes by -1 and performs the decomposition under minimization.

This class is abstract to support to two applications of Alg 1 from [Lacour17]: 1) partitioning the space that is dominated by the Pareto frontier and 2) partitioning the space that is not dominated by the Pareto frontier.

Initialize FastPartitioning.

Parameters
• ref_point (Tensor) – A m-dim tensor containing the reference point.

• Y (Optional[Tensor]) – A (batch_shape) x n x m-dim tensor

Return type

None

update(Y)[source]

Update non-dominated front and decomposition.

Parameters

Y (torch.Tensor) – A (batch_shape) x n x m-dim tensor of new, incremental outcomes.

Return type

None

partition_space()[source]

Compute box decomposition.

Return type

None

get_hypercell_bounds()[source]

Get the bounds of each hypercell in the decomposition.

Returns

A 2 x (batch_shape) x num_cells x m-dim tensor containing the

lower and upper vertices bounding each hypercell.

Return type

torch.Tensor

training: bool

## Box Decomposition List¶

Box decomposition container.

class botorch.utils.multi_objective.box_decompositions.box_decomposition_list.BoxDecompositionList(*box_decompositions)[source]

Bases: torch.nn.modules.module.Module

A list of box decompositions.

Initialize the box decomposition list.

Parameters
• *box_decompositions – An variable number of box decompositions

• box_decompositions (BoxDecomposition) –

Return type

None

Example

>>> bd1 = FastNondominatedPartitioning(ref_point, Y=Y1)
>>> bd2 = FastNondominatedPartitioning(ref_point, Y=Y2)
>>> bd = BoxDecompositionList(bd1, bd2)

property pareto_Y: List[torch.Tensor]

This returns the non-dominated set.

Note: Internally, we store the negative pareto set (minimization).

Returns

A list where the ith element is the n_pareto_i x m-dim tensor

of pareto optimal outcomes for each box_decomposition i.

property ref_point: torch.Tensor

Get the reference point.

Note: Internally, we store the negative reference point (minimization).

Returns

A n_box_decompositions x m-dim tensor of outcomes.

get_hypercell_bounds()[source]

Get the bounds of each hypercell in the decomposition.

Returns

A 2 x n_box_decompositions x num_cells x num_outcomes-dim tensor

containing the lower and upper vertices bounding each hypercell.

Return type

torch.Tensor

update(Y)[source]

Update the partitioning.

Parameters

Y (Union[List[torch.Tensor], torch.Tensor]) – A n_box_decompositions x n x num_outcomes-dim tensor or a list where the ith element contains the new points for box_decomposition i.

Return type

None

compute_hypervolume()[source]

Compute hypervolume that is dominated by the Pareto Froniter.

Returns

A (batch_shape)-dim tensor containing the hypervolume dominated by

each Pareto frontier.

Return type

torch.Tensor

training: bool

## Box Decomposition Utilities¶

Utilities for box decomposition algorithms.

botorch.utils.multi_objective.box_decompositions.utils.compute_local_upper_bounds(U, Z, z)[source]

Compute local upper bounds.

Note: this assumes minimization.

This uses the incremental algorithm (Alg. 1) from [Lacour17].

Parameters
• U (torch.Tensor) – A n x m-dim tensor containing the local upper bounds.

• Z (torch.Tensor) – A n x m x m-dim tensor containing the defining points.

• z (torch.Tensor) – A m-dim tensor containing the new point.

Returns

• A new n’ x m-dim tensor local upper bounds.

• A n’ x m x m-dim tensor containing the defining points.

Return type

2-element tuple containing

botorch.utils.multi_objective.box_decompositions.utils.get_partition_bounds(Z, U, ref_point)[source]

Get the cell bounds given the local upper bounds and the defining points.

This implements Equation 2 in [Lacour17].

Parameters
• Z (torch.Tensor) – A n x m x m-dim tensor containing the defining points. The first dimension corresponds to u_idx, the second dimension corresponds to j, and Z[u_idx, j] is the set of definining points Z^j(u) where u = U[u_idx].

• U (torch.Tensor) – A n x m-dim tensor containing the local upper bounds.

• ref_point (torch.Tensor) – A m-dim tensor containing the reference point.

Returns

A 2 x num_cells x m-dim tensor containing the lower and upper vertices

bounding each hypercell.

Return type

torch.Tensor

botorch.utils.multi_objective.box_decompositions.utils.update_local_upper_bounds_incremental(new_pareto_Y, U, Z)[source]

Update the current local upper with the new pareto points.

This assumes minimization.

Parameters
• new_pareto_Y (torch.Tensor) – A n x m-dim tensor containing the new Pareto points.

• U (torch.Tensor) – A n’ x m-dim tensor containing the local upper bounds.

• Z (torch.Tensor) – A n x m x m-dim tensor containing the defining points.

Returns

• A new n’ x m-dim tensor local upper bounds.

• A n’ x m x m-dim tensor containing the defining points

Return type

2-element tuple containing

botorch.utils.multi_objective.box_decompositions.utils.compute_non_dominated_hypercell_bounds_2d(pareto_Y_sorted, ref_point)[source]

Compute an axis-aligned partitioning of the non-dominated space for 2 objectives.

Parameters
• pareto_Y_sorted (torch.Tensor) – A (batch_shape) x n_pareto x 2-dim tensor of pareto outcomes that are sorted by the 0th dimension in increasing order. All points must be better than the reference point.

• ref_point (torch.Tensor) – A (batch_shape) x 2-dim reference point.

Returns

A 2 x (batch_shape) x n_pareto + 1 x m-dim tensor of cell bounds.

Return type

torch.Tensor

botorch.utils.multi_objective.box_decompositions.utils.compute_dominated_hypercell_bounds_2d(pareto_Y_sorted, ref_point)[source]

Compute an axis-aligned partitioning of the dominated space for 2-objectives.

Parameters
• pareto_Y_sorted (torch.Tensor) – A (batch_shape) x n_pareto x 2-dim tensor of pareto outcomes that are sorted by the 0th dimension in increasing order.

• ref_point (torch.Tensor) – A 2-dim reference point.

Returns

A 2 x (batch_shape) x n_pareto x m-dim tensor of cell bounds.

Return type

torch.Tensor

## Dominated Partitionings¶

Algorithms for partitioning the dominated space into hyperrectangles.

class botorch.utils.multi_objective.box_decompositions.dominated.DominatedPartitioning(ref_point, Y=None)[source]

Partition dominated space into axis-aligned hyperrectangles.

This uses the Algorithm 1 from [Lacour17].

Example

>>> bd = DominatedPartitioning(ref_point, Y)


Initialize FastPartitioning.

Parameters
• ref_point (Tensor) – A m-dim tensor containing the reference point.

• Y (Optional[Tensor]) – A (batch_shape) x n x m-dim tensor

Return type

None

compute_hypervolume()[source]

Compute hypervolume that is dominated by the Pareto Frontier.

Returns

A (batch_shape)-dim tensor containing the hypervolume dominated by

each Pareto frontier.

Return type

torch.Tensor

training: bool

## Hypervolume¶

Hypervolume Utilities.

References

Fonseca2006(1,2)

C. M. Fonseca, L. Paquete, and M. Lopez-Ibanez. An improved dimension-sweep algorithm for the hypervolume indicator. In IEEE Congress on Evolutionary Computation, pages 1157-1163, Vancouver, Canada, July 2006.

Ishibuchi2011

H. Ishibuchi, N. Akedo, and Y. Nojima. A many-objective test problem for visually examining diversity maintenance behavior in a decision space. Proc. 13th Annual Conf. Genetic Evol. Comput., 2011.

botorch.utils.multi_objective.hypervolume.infer_reference_point(pareto_Y, max_ref_point=None, scale=0.1, scale_max_ref_point=False)[source]

Get reference point for hypervolume computations.

This sets the reference point to be ref_point = nadir - 0.1 * range when there is no pareto_Y that is better than the reference point.

[Ishibuchi2011] find 0.1 to be a robust multiplier for scaling the nadir point.

Note: this assumes maximization of all objectives.

Parameters
• pareto_Y (torch.Tensor) – A n x m-dim tensor of Pareto-optimal points.

• max_ref_point (Optional[torch.Tensor]) – A m dim tensor indicating the maximum reference point.

• scale (float) – A multiplier used to scale back the reference point based on the range of each objective.

• scale_max_ref_point (bool) – A boolean indicating whether to apply scaling to the max_ref_point based on the range of each objective.

Returns

A m-dim tensor containing the reference point.

Return type

torch.Tensor

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/msu-coinlab/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) – m-dim Tensor containing the reference point.

Return type

None

property ref_point: torch.Tensor

Get reference point (for maximization).

Returns

A m-dim tensor containing the reference point.

compute(pareto_Y)[source]

Compute hypervolume.

Parameters

pareto_Y (torch.Tensor) – A n x m-dim tensor of pareto optimal outcomes

Returns

The hypervolume.

Return type

float

botorch.utils.multi_objective.hypervolume.sort_by_dimension(nodes, i)[source]

Sorts the list of nodes in-place by the specified objective.

Parameters
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 objectives

• dtype (torch.dtype) – The dtype

• device (torch.device) – The device

• data (Optional[Tensor]) – The tensor data to be stored in this Node.

Return type

None

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 lists

• dtype (torch.dtype) – the dtype

• device (torch.device) – the device

Return type

None

append(node, index)[source]

Appends a node to the end of the list at the given index.

Parameters
Return type

None

extend(nodes, index)[source]

Extends the list at the given index with the nodes.

Parameters
Return type

None

remove(node, index, bounds)[source]

Removes and returns ‘node’ from all lists in [0, ‘index’].

Parameters
Return type

botorch.utils.multi_objective.hypervolume.Node

reinsert(node, index, bounds)[source]

Re-inserts the node at its original position.

Re-inserts 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
Return type

None

## Non-dominated Partitionings¶

Algorithms for partitioning the non-dominated 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. 1-8.

class botorch.utils.multi_objective.box_decompositions.non_dominated.NondominatedPartitioning(ref_point, Y=None, alpha=0.0)[source]

A class for partitioning the non-dominated space into hyper-cells.

Note: this assumes maximization. Internally, it multiplies outcomes 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 non-dominated 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/s10898-019-00798-7.pdf

Initialize NondominatedPartitioning.

Parameters
• ref_point (Tensor) – A m-dim tensor containing the reference point.

• Y (Optional[Tensor]) – A (batch_shape) x n x m-dim tensor.

• alpha (float) – A thresold fraction of total volume used in an approximate decomposition.

Return type

None

Example

>>> bd = NondominatedPartitioning(ref_point, Y=Y1)

get_hypercell_bounds()[source]

Get the bounds of each hypercell in the decomposition.

Parameters

ref_point – A (batch_shape) x m-dim tensor containing the reference point.

Returns

A 2 x num_cells x m-dim tensor containing the

lower and upper vertices bounding each hypercell.

Return type

torch.Tensor

compute_hypervolume()[source]

Compute the hypervolume for the given reference point.

This method computes the hypervolume of the non-dominated space and computes the difference between the hypervolume between the ideal point and hypervolume of the non-dominated space.

Returns

(batch_shape)-dim tensor containing the dominated hypervolume.

Return type

torch.Tensor

training: bool
class botorch.utils.multi_objective.box_decompositions.non_dominated.FastNondominatedPartitioning(ref_point, Y=None)[source]

A class for partitioning the non-dominated space into hyper-cells.

Note: this assumes maximization. Internally, it multiplies by -1 and performs the decomposition under minimization.

This class is far more efficient than NondominatedPartitioning for exact box partitionings

This class uses the two-step approach similar to that in [Yang2019], where:
1. first, Alg 1 from [Lacour17] is used to find the local lower bounds

for the maximization problem

2. second, the local lower bounds are used as the Pareto frontier for the

minimization problem, and [Lacour17] is applied again to partition the space dominated by that Pareto frontier.

Initialize FastNondominatedPartitioning.

Parameters
• ref_point (Tensor) – A m-dim tensor containing the reference point.

• Y (Optional[Tensor]) – A (batch_shape) x n x m-dim tensor.

Return type

None

Example

>>> bd = FastNondominatedPartitioning(ref_point, Y=Y1)

compute_hypervolume()[source]

Compute hypervolume that is dominated by the Pareto Froniter.

Returns

A (batch_shape)-dim tensor containing the hypervolume dominated by

each Pareto frontier.

training: bool

## Pareto¶

botorch.utils.multi_objective.pareto.is_non_dominated(Y, deduplicate=True)[source]

Computes the non-dominated front.

Note: this assumes maximization.

For small n, this method uses a highly parallel methodology that compares all pairs of points in Y. However, this is memory intensive and slow for large n. For large n (or if Y is larger than 5MB), this method will dispatch to a loop-based approach that is faster and has a lower memory footprint.

Parameters
• Y (torch.Tensor) – A (batch_shape) x n x m-dim tensor of outcomes.

• deduplicate (bool) – A boolean indicating whether to only return unique points on the pareto frontier.

Returns

A (batch_shape) x n-dim boolean tensor indicating whether each point is non-dominated.

Return type

torch.Tensor

## Scalarization¶

Helper utilities for constructing scalarizations.

References

Knowles2005(1,2)

J. Knowles, “ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems,” in IEEE Transactions on Evolutionary Computation, vol. 10, no. 1, pp. 50-66, Feb. 2006.

botorch.utils.multi_objective.scalarization.get_chebyshev_scalarization(weights, Y, alpha=0.05)[source]

Construct an augmented Chebyshev scalarization.

Augmented Chebyshev scalarization:

objective(y) = min(w * y) + alpha * sum(w * y)

Outcomes are first normalized to [0,1] for maximization (or [-1,0] for minimization) and then an augmented Chebyshev scalarization is applied.

Note: this assumes maximization of the augmented Chebyshev scalarization. Minimizing/Maximizing an objective is supported by passing a negative/positive weight for that objective. To make all w * y’s have positive sign such that they are comparable when computing min(w * y), outcomes of minimization objectives are shifted from [0,1] to [-1,0].

See [Knowles2005] for details.

This scalarization can be used with qExpectedImprovement to implement q-ParEGO as proposed in [Daulton2020qehvi].

Parameters
• weights (torch.Tensor) – A m-dim tensor of weights. Positive for maximization and negative for minimization.

• Y (torch.Tensor) – A n x m-dim tensor of observed outcomes, which are used for scaling the outcomes to [0,1] or [-1,0].

• alpha (float) – Parameter governing the influence of the weighted sum term. The default value comes from [Knowles2005].

Returns

Transform function using the objective weights.

Return type

Callable[[torch.Tensor, Optional[torch.Tensor]], torch.Tensor]

Example

>>> weights = torch.tensor([0.75, -0.25])
>>> transform = get_aug_chebyshev_scalarization(weights, Y)
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