def project_onto_l1_ball(x: ep.Tensor, eps: ep.Tensor) -> ep.Tensor:
"""Computes Euclidean projection onto the L1 ball for a batch. [#Duchi08]_
Adapted from the pytorch version by Tony Duan:
https://gist.github.com/tonyduan/1329998205d88c566588e57e3e2c0c55
Args:
x: Batch of arbitrary-size tensors to project, possibly on GPU
eps: radius of l-1 ball to project onto
References:
..[#Duchi08] Efficient Projections onto the l1-Ball for Learning in High Dimensions
John Duchi, Shai Shalev-Shwartz, Yoram Singer, and Tushar Chandra.
International Conference on Machine Learning (ICML 2008)
"""
original_shape = x.shape
x = flatten(x)
mask = (ep.norms.l1(x, axis=1) <= eps).astype(x.dtype).expand_dims(1)
mu = ep.flip(ep.sort(ep.abs(x)), axis=-1).astype(x.dtype)
cumsum = ep.cumsum(mu, axis=-1)
arange = ep.arange(x, 1, x.shape[1] + 1).astype(x.dtype)
rho = (ep.max(
((mu * arange >
(cumsum - eps.expand_dims(1)))).astype(x.dtype) * arange,
axis=-1,
) - 1)
# samples already under norm will have to select
rho = ep.maximum(rho, 0)
theta = (cumsum[ep.arange(x, x.shape[0]),
rho.astype(ep.arange(x, 1).dtype)] - eps) / (rho + 1.0)
proj = (ep.abs(x) - theta.expand_dims(1)).clip(min_=0, max_=ep.inf)
x = mask * x + (1 - mask) * proj * ep.sign(x)
return x.reshape(original_shape)
def mid_points(self, x0: ep.Tensor, x1: ep.Tensor, epsilons: ep.Tensor,
bounds) -> ep.Tensor:
# returns a point between x0 and x1 where
# epsilon = 0 returns x0 and epsilon = 1
# returns x1
# get epsilons in right shape for broadcasting
epsilons = epsilons.reshape(epsilons.shape + (1, ) * (x0.ndim - 1))
return epsilons * x1 + (1 - epsilons) * x0
def project_onto_l1_ball(x: ep.Tensor, eps: ep.Tensor):
"""
Compute Euclidean projection onto the L1 ball for a batch.
min ||x - u||_2 s.t. ||u||_1 <= eps
Inspired by the corresponding numpy version by Adrien Gaidon.
Adapted from the pytorch version by Tony Duan: https://gist.github.com/tonyduan/1329998205d88c566588e57e3e2c0c55
Parameters
----------
x: (batch_size, *) torch array
batch of arbitrary-size tensors to project, possibly on GPU
eps: float
radius of l-1 ball to project onto
Returns
-------
u: (batch_size, *) torch array
batch of projected tensors, reshaped to match the original
Notes
-----
The complexity of this algorithm is in O(dlogd) as it involves sorting x.
References
----------
[1] Efficient Projections onto the l1-Ball for Learning in High Dimensions
John Duchi, Shai Shalev-Shwartz, Yoram Singer, and Tushar Chandra.
International Conference on Machine Learning (ICML 2008)
"""
original_shape = x.shape
x = flatten(x)
mask = (ep.norms.l1(x, axis=1) < eps).astype(x.dtype).expand_dims(1)
mu = ep.flip(ep.sort(ep.abs(x)), axis=-1)
cumsum = ep.cumsum(mu, axis=-1)
arange = ep.arange(x, 1, x.shape[1] + 1)
rho = ep.max(
(mu * arange > (cumsum - eps.expand_dims(1))) * arange, axis=-1) - 1
theta = (cumsum[ep.arange(x, x.shape[0]), rho] - eps) / (rho + 1.0)
proj = (ep.abs(x) - theta.expand_dims(1)).clip(min_=0, max_=ep.inf)
x = mask * x + (1 - mask) * proj * ep.sign(x)
return x.reshape(original_shape)
def mid_points(
self,
x0: ep.Tensor,
x1: ep.Tensor,
epsilons: ep.Tensor,
bounds: Tuple[float, float],
):
# returns a point between x0 and x1 where
# epsilon = 0 returns x0 and epsilon = 1
delta = x1 - x0
min_, max_ = bounds
s = max_ - min_
# get epsilons in right shape for broadcasting
epsilons = epsilons.reshape(epsilons.shape + (1, ) * (x0.ndim - 1))
clipped_delta = ep.where(delta < -epsilons * s, -epsilons * s, delta)
clipped_delta = ep.where(clipped_delta > epsilons * s, epsilons * s,
clipped_delta)
return x0 + clipped_delta
def mid_points(
self,
x0: ep.Tensor,
x1: ep.Tensor,
epsilons: ep.Tensor,
bounds: Tuple[float, float],
) -> ep.Tensor:
# returns a point between x0 and x1 where
# epsilon = 0 returns x0 and epsilon = 1
# returns x1
# get epsilons in right shape for broadcasting
epsilons = epsilons.reshape(epsilons.shape + (1, ) * (x0.ndim - 1))
threshold = (bounds[1] - bounds[0]) * (1 - epsilons)
mask = (x1 - x0).abs() > threshold
new_x = ep.where(mask,
x0 + (x1 - x0).sign() * ((x1 - x0).abs() - threshold),
x0)
return new_x
def _binary_search(
self,
x_adv_flat: ep.Tensor,
mask: Union[ep.Tensor, List[bool]],
mask_indices: ep.Tensor,
indices: Union[ep.Tensor, List[int]],
adv_values: ep.Tensor,
non_adv_values: ep.Tensor,
original_shape: Tuple,
is_adversarial: Callable,
) -> ep.Tensor:
for i in range(10):
next_values = (adv_values + non_adv_values) / 2
x_adv_flat = ep.index_update(
x_adv_flat, (mask_indices, indices), next_values
)
is_adv = is_adversarial(x_adv_flat.reshape(original_shape))[mask]
adv_values = ep.where(is_adv, next_values, adv_values)
non_adv_values = ep.where(is_adv, non_adv_values, next_values)
return adv_values
def draw_proposals(bounds: Bounds, originals: ep.Tensor, perturbed: ep.Tensor,
unnormalized_source_directions: ep.Tensor,
source_directions: ep.Tensor, source_norms: ep.Tensor,
spherical_steps: ep.Tensor, source_steps: ep.Tensor,
surrogate_model: Model) -> Tuple[ep.Tensor, ep.Tensor]:
# remember the actual shape
shape = originals.shape
assert perturbed.shape == shape
assert unnormalized_source_directions.shape == shape
assert source_directions.shape == shape
# flatten everything to (batch, size)
originals = flatten(originals)
perturbed = flatten(perturbed)
unnormalized_source_directions = flatten(unnormalized_source_directions)
source_directions = flatten(source_directions)
N, D = originals.shape
assert source_norms.shape == (N, )
assert spherical_steps.shape == (N, )
assert source_steps.shape == (N, )
# draw from an iid Gaussian (we can share this across the whole batch)
eta = ep.normal(perturbed, (D, 1))
# make orthogonal (source_directions are normalized)
eta = eta.T - ep.matmul(source_directions, eta) * source_directions
assert eta.shape == (N, D)
pg_factor = 0.5
if not surrogate_model is None:
device = surrogate_model.device
projected_gradient = get_projected_gradients(perturbed.reshape(shape),
originals.reshape(shape),
0, surrogate_model)
projected_gradient = projected_gradient.reshape((N, D))
projected_gradient = torch.tensor(projected_gradient, device=device)
projected_gradient, restore_type = ep.astensor_(projected_gradient)
eta = (1. - pg_factor) * eta + pg_factor * projected_gradient
# rescale
norms = ep.norms.l2(eta, axis=-1)
assert norms.shape == (N, )
eta = eta * atleast_kd(spherical_steps * source_norms / norms, eta.ndim)
# project on the sphere using Pythagoras
distances = atleast_kd((spherical_steps.square() + 1).sqrt(), eta.ndim)
directions = eta - unnormalized_source_directions
spherical_candidates = originals + directions / distances
# clip
min_, max_ = bounds
spherical_candidates = spherical_candidates.clip(min_, max_)
# step towards the original inputs
new_source_directions = originals - spherical_candidates
assert new_source_directions.ndim == 2
new_source_directions_norms = ep.norms.l2(flatten(new_source_directions),
axis=-1)
# length if spherical_candidates would be exactly on the sphere
lengths = source_steps * source_norms
# length including correction for numerical deviation from sphere
lengths = lengths + new_source_directions_norms - source_norms
# make sure the step size is positive
lengths = ep.maximum(lengths, 0)
# normalize the length
lengths = lengths / new_source_directions_norms
lengths = atleast_kd(lengths, new_source_directions.ndim)
candidates = spherical_candidates + lengths * new_source_directions
# clip
candidates = candidates.clip(min_, max_)
# restore shape
candidates = candidates.reshape(shape)
spherical_candidates = spherical_candidates.reshape(shape)
return candidates, spherical_candidates
def atleast_kd(x: ep.Tensor, k) -> ep.Tensor:
shape = x.shape + (1, ) * (k - x.ndim)
return x.reshape(shape)
def flatten(x: ep.Tensor, keep=1) -> ep.Tensor:
shape = x.shape[:keep] + (-1, )
return x.reshape(shape)