Torch differentiable

Contains methods for differentiating a pyTorch model. Most of the methods focus on ways to calculate matrix vector products. Moreover, it contains several methods to invert the Hessian vector product. These are used to calculate the influence of a training point on the model.

References

---

1. Koh, P.W., Liang, P., 2017. Understanding Black-box Predictions via Influence Functions. In: Proceedings of the 34th International Conference on Machine Learning, pp. 1885–1894. PMLR. ↩

2. Agarwal, N., Bullins, B., Hazan, E., 2017. Second-Order Stochastic Optimization for Machine Learning in Linear Time. In: Journal of Machine Learning Research, Vol. 18, pp. 1–40. JMLR. ↩

TorchTwiceDifferentiable(model, loss)

Bases: TwiceDifferentiable[torch.Tensor]

Wraps a torch.nn.Module and a loss function and provides methods to compute gradients and second derivative of the loss wrt. the model parameters

PARAMETER	DESCRIPTION
model	A (differentiable) function. TYPE: Module
loss	A differentiable scalar loss \( L(\hat{y}, y) \), mapping a prediction and a target to a real value. TYPE: Callable[[Tensor, Tensor], Tensor]

Source code in src/pydvl/influence/torch/torch_differentiable.py

def __init__(
    self,
    model: nn.Module,
    loss: Callable[[torch.Tensor, torch.Tensor], torch.Tensor],
):

    if model.training:
        logger.warning(
            "Passed model not in evaluation mode. This can create several issues in influence "
            "computation, e.g. due to batch normalization. Please call model.eval() before "
            "computing influences."
        )
    self.loss = loss
    self.model = model
    first_param = next(model.parameters())
    self.device = first_param.device
    self.dtype = first_param.dtype

parameters: List[torch.Tensor] property

RETURNS	DESCRIPTION
List[torch.Tensor]	All model parameters that require differentiating.

num_params: int property

Get the number of parameters of model f.

RETURNS	DESCRIPTION
int	Number of parameters. TYPE: int

grad(x, y, create_graph=False)

Calculates gradient of model parameters with respect to the model parameters.

PARAMETER	DESCRIPTION
x	A matrix [NxD] representing the features \( x_i \). TYPE: Tensor
y	A matrix [NxK] representing the target values \( y_i \). TYPE: Tensor
create_graph	If True, the resulting gradient tensor can be used for further differentiation. TYPE: bool DEFAULT: False

RETURNS	DESCRIPTION
Tensor	An array [P] with the gradients of the model.

Source code in src/pydvl/influence/torch/torch_differentiable.py

def grad(
    self, x: torch.Tensor, y: torch.Tensor, create_graph: bool = False
) -> torch.Tensor:
    r"""
    Calculates gradient of model parameters with respect to the model parameters.

    Args:
        x: A matrix [NxD] representing the features \( x_i \).
        y: A matrix [NxK] representing the target values \( y_i \).
        create_graph (bool): If True, the resulting gradient tensor can be used for further differentiation.

    Returns:
        An array [P] with the gradients of the model.
    """

    x = x.to(self.device)
    y = y.to(self.device)

    if create_graph and not x.requires_grad:
        x = x.requires_grad_(True)

    loss_value = self.loss(torch.squeeze(self.model(x)), torch.squeeze(y))
    grad_f = torch.autograd.grad(
        loss_value, self.parameters, create_graph=create_graph
    )
    return flatten_tensors_to_vector(grad_f)

hessian(x, y)

Calculates the explicit hessian of model parameters given data \(x\) and \(y\).

PARAMETER	DESCRIPTION
x	A matrix [NxD] representing the features \(x_i\). TYPE: Tensor
y	A matrix [NxK] representing the target values \(y_i\). TYPE: Tensor

RETURNS	DESCRIPTION
Tensor	A tensor representing the hessian of the loss with respect to the model parameters.

Source code in src/pydvl/influence/torch/torch_differentiable.py

def hessian(self, x: torch.Tensor, y: torch.Tensor) -> torch.Tensor:
    r"""
    Calculates the explicit hessian of model parameters given data \(x\) and \(y\).

    Args:
        x: A matrix [NxD] representing the features \(x_i\).
        y: A matrix [NxK] representing the target values \(y_i\).

    Returns:
        A tensor representing the hessian of the loss with respect to the model parameters.
    """

    def model_func(param):
        outputs = torch.func.functional_call(
            self.model,
            align_structure(
                {k: p for k, p in self.model.named_parameters() if p.requires_grad},
                param,
            ),
            (x.to(self.device),),
            strict=True,
        )
        return self.loss(outputs, y.to(self.device))

    params = flatten_tensors_to_vector(
        p.detach() for p in self.model.parameters() if p.requires_grad
    )
    return torch.func.hessian(model_func)(params)

mvp(grad_xy, v, backprop_on, *, progress=False) staticmethod

Calculates the second-order derivative of the model along directions v. This second-order derivative can be selected through the backprop_on argument.

PARAMETER	DESCRIPTION
grad_xy	An array [P] holding the gradients of the model parameters with respect to input \(x\) and labels \(y\), where P is the number of parameters of the model. It is typically obtained through self.grad. TYPE: Tensor
v	An array ([DxP] or even one-dimensional [D]) which multiplies the matrix, where D is the number of directions. TYPE: Tensor
progress	If True, progress will be printed. TYPE: bool DEFAULT: False
backprop_on	Tensor used in the second backpropagation (the first one is defined via grad_xy). TYPE: Tensor

RETURNS	DESCRIPTION
Tensor	A matrix representing the implicit matrix-vector product of the model along the given directions. The output shape is [DxM], with M being the number of elements of backprop_on.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@staticmethod
def mvp(
    grad_xy: torch.Tensor,
    v: torch.Tensor,
    backprop_on: torch.Tensor,
    *,
    progress: bool = False,
) -> torch.Tensor:
    r"""
    Calculates the second-order derivative of the model along directions v.
    This second-order derivative can be selected through the `backprop_on` argument.

    Args:
        grad_xy: An array [P] holding the gradients of the model parameters with respect to input
            \(x\) and labels \(y\), where P is the number of parameters of the model.
            It is typically obtained through `self.grad`.
        v: An array ([DxP] or even one-dimensional [D]) which multiplies the matrix,
            where D is the number of directions.
        progress: If True, progress will be printed.
        backprop_on: Tensor used in the second backpropagation
            (the first one is defined via grad_xy).

    Returns:
        A matrix representing the implicit matrix-vector product of the model along the given directions.
            The output shape is [DxM], with M being the number of elements of `backprop_on`.
    """

    device = grad_xy.device
    v = as_tensor(v, warn=False).to(device)
    if v.ndim == 1:
        v = v.unsqueeze(0)

    z = (grad_xy * Variable(v)).sum(dim=1)

    mvp = []
    for i in maybe_progress(range(len(z)), progress, desc="MVP"):
        mvp.append(
            flatten_tensors_to_vector(
                autograd.grad(z[i], backprop_on, retain_graph=True)
            )
        )
    return torch.stack([grad.contiguous().view(-1) for grad in mvp]).detach()

LowRankProductRepresentation dataclass

Representation of a low rank product of the form \(H = V D V^T\), where D is a diagonal matrix and V is orthogonal.

PARAMETER	DESCRIPTION
eigen_vals	Diagonal of D. TYPE: Tensor
projections	The matrix V. TYPE: Tensor

to(device)

Move the representing tensors to a device

Source code in src/pydvl/influence/torch/torch_differentiable.py

def to(self, device: torch.device):
    """
    Move the representing tensors to a device
    """
    return LowRankProductRepresentation(
        self.eigen_vals.to(device), self.projections.to(device)
    )

TorchTensorUtilities

Bases: TensorUtilities[Tensor, TorchTwiceDifferentiable]

einsum(equation, *operands) staticmethod

Sums the product of the elements of the input :attr:operands along dimensions specified using a notation based on the Einstein summation convention.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@staticmethod
def einsum(equation: str, *operands) -> torch.Tensor:
    """Sums the product of the elements of the input :attr:`operands` along dimensions specified using a notation
    based on the Einstein summation convention.
    """
    return torch.einsum(equation, *operands)

cat(a, **kwargs) staticmethod

Concatenates a sequence of tensors into a single torch tensor

Source code in src/pydvl/influence/torch/torch_differentiable.py

@staticmethod
def cat(a: Sequence[torch.Tensor], **kwargs) -> torch.Tensor:
    """Concatenates a sequence of tensors into a single torch tensor"""
    return torch.cat(a, **kwargs)

stack(a, **kwargs) staticmethod

Stacks a sequence of tensors into a single torch tensor

Source code in src/pydvl/influence/torch/torch_differentiable.py

@staticmethod
def stack(a: Sequence[torch.Tensor], **kwargs) -> torch.Tensor:
    """Stacks a sequence of tensors into a single torch tensor"""
    return torch.stack(a, **kwargs)

unsqueeze(x, dim) staticmethod

Add a singleton dimension at a specified position in a tensor.

PARAMETER	DESCRIPTION
x	A PyTorch tensor. TYPE: Tensor
dim	The position at which to add the singleton dimension. Zero-based indexing. TYPE: int

RETURNS	DESCRIPTION
Tensor	A new tensor with an additional singleton dimension.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@staticmethod
def unsqueeze(x: torch.Tensor, dim: int) -> torch.Tensor:
    """
    Add a singleton dimension at a specified position in a tensor.

    Args:
        x: A PyTorch tensor.
        dim: The position at which to add the singleton dimension. Zero-based indexing.

    Returns:
        A new tensor with an additional singleton dimension.
    """

    return x.unsqueeze(dim)

lanzcos_low_rank_hessian_approx(hessian_vp, matrix_shape, hessian_perturbation=0.0, rank_estimate=10, krylov_dimension=None, tol=1e-06, max_iter=None, device=None, eigen_computation_on_gpu=False, torch_dtype=None)

Calculates a low-rank approximation of the Hessian matrix of a scalar-valued function using the implicitly restarted Lanczos algorithm, i.e.:

\[ H_{\text{approx}} = V D V^T\]

where \(D\) is a diagonal matrix with the top (in absolute value) rank_estimate eigenvalues of the Hessian and \(V\) contains the corresponding eigenvectors.

PARAMETER	DESCRIPTION
hessian_vp	A function that takes a vector and returns the product of the Hessian of the loss function. TYPE: Callable[[Tensor], Tensor]
matrix_shape	The shape of the matrix, represented by the hessian vector product. TYPE: Tuple[int, int]
hessian_perturbation	Regularization parameter added to the Hessian-vector product for numerical stability. TYPE: float DEFAULT: 0.0
rank_estimate	The number of eigenvalues and corresponding eigenvectors to compute. Represents the desired rank of the Hessian approximation. TYPE: int DEFAULT: 10
krylov_dimension	The number of Krylov vectors to use for the Lanczos method. If not provided, it defaults to \( \min(\text{model.num_parameters}, \max(2 \times \text{rank_estimate} + 1, 20)) \). TYPE: Optional[int] DEFAULT: None
tol	The stopping criteria for the Lanczos algorithm, which stops when the difference in the approximated eigenvalue is less than tol. Defaults to 1e-6. TYPE: float DEFAULT: 1e-06
max_iter	The maximum number of iterations for the Lanczos method. If not provided, it defaults to \( 10 \cdot \text{model.num_parameters}\). TYPE: Optional[int] DEFAULT: None
device	The device to use for executing the hessian vector product. TYPE: Optional[torch.device] DEFAULT: None
eigen_computation_on_gpu	If True, tries to execute the eigen pair approximation on the provided device via cupy implementation. Ensure that either your model is small enough, or you use a small rank_estimate to fit your device's memory. If False, the eigen pair approximation is executed on the CPU with scipy's wrapper to ARPACK. TYPE: bool DEFAULT: False
torch_dtype	If not provided, the current torch default dtype is used for conversion to torch. TYPE: dtype DEFAULT: None

RETURNS	DESCRIPTION
LowRankProductRepresentation	A LowRankProductRepresentation instance that contains the top (up until rank_estimate) eigenvalues and corresponding eigenvectors of the Hessian.

Source code in src/pydvl/influence/torch/torch_differentiable.py

def lanzcos_low_rank_hessian_approx(
    hessian_vp: Callable[[torch.Tensor], torch.Tensor],
    matrix_shape: Tuple[int, int],
    hessian_perturbation: float = 0.0,
    rank_estimate: int = 10,
    krylov_dimension: Optional[int] = None,
    tol: float = 1e-6,
    max_iter: Optional[int] = None,
    device: Optional[torch.device] = None,
    eigen_computation_on_gpu: bool = False,
    torch_dtype: torch.dtype = None,
) -> LowRankProductRepresentation:
    r"""
    Calculates a low-rank approximation of the Hessian matrix of a scalar-valued
    function using the implicitly restarted Lanczos algorithm, i.e.:

    \[ H_{\text{approx}} = V D V^T\]

    where \(D\) is a diagonal matrix with the top (in absolute value) `rank_estimate` eigenvalues of the Hessian
    and \(V\) contains the corresponding eigenvectors.

    Args:
        hessian_vp: A function that takes a vector and returns the product of
            the Hessian of the loss function.
        matrix_shape: The shape of the matrix, represented by the hessian vector
            product.
        hessian_perturbation: Regularization parameter added to the
            Hessian-vector product for numerical stability.
        rank_estimate: The number of eigenvalues and corresponding eigenvectors
            to compute. Represents the desired rank of the Hessian approximation.
        krylov_dimension: The number of Krylov vectors to use for the Lanczos
            method. If not provided, it defaults to
            \( \min(\text{model.num_parameters}, \max(2 \times \text{rank_estimate} + 1, 20)) \).
        tol: The stopping criteria for the Lanczos algorithm, which stops when
            the difference in the approximated eigenvalue is less than `tol`.
            Defaults to 1e-6.
        max_iter: The maximum number of iterations for the Lanczos method. If
            not provided, it defaults to \( 10 \cdot \text{model.num_parameters}\).
        device: The device to use for executing the hessian vector product.
        eigen_computation_on_gpu: If True, tries to execute the eigen pair
            approximation on the provided device via [cupy](https://cupy.dev/)
            implementation. Ensure that either your model is small enough, or you
            use a small rank_estimate to fit your device's memory. If False, the
            eigen pair approximation is executed on the CPU with scipy's wrapper to
            ARPACK.
        torch_dtype: If not provided, the current torch default dtype is used for
            conversion to torch.

    Returns:
        A [LowRankProductRepresentation][pydvl.influence.torch.torch_differentiable.LowRankProductRepresentation]
            instance that contains the top (up until rank_estimate) eigenvalues
            and corresponding eigenvectors of the Hessian.
    """

    torch_dtype = torch.get_default_dtype() if torch_dtype is None else torch_dtype

    if eigen_computation_on_gpu:
        try:
            import cupy as cp
            from cupyx.scipy.sparse.linalg import LinearOperator, eigsh
            from torch.utils.dlpack import from_dlpack, to_dlpack
        except ImportError as e:
            raise ImportError(
                f"Try to install missing dependencies or set eigen_computation_on_gpu to False: {e}"
            )

        if device is None:
            raise ValueError(
                "Without setting an explicit device, cupy is not supported"
            )

        def to_torch_conversion_function(x):
            return from_dlpack(x.toDlpack()).to(torch_dtype)

        def mv(x):
            x = to_torch_conversion_function(x)
            y = hessian_vp(x) + hessian_perturbation * x
            return cp.from_dlpack(to_dlpack(y))

    else:
        from scipy.sparse.linalg import LinearOperator, eigsh

        def mv(x):
            x_torch = torch.as_tensor(x, device=device, dtype=torch_dtype)
            y: NDArray = (
                (hessian_vp(x_torch) + hessian_perturbation * x_torch)
                .detach()
                .cpu()
                .numpy()
            )
            return y

        to_torch_conversion_function = partial(torch.as_tensor, dtype=torch_dtype)

    try:
        eigen_vals, eigen_vecs = eigsh(
            LinearOperator(matrix_shape, matvec=mv),
            k=rank_estimate,
            maxiter=max_iter,
            tol=tol,
            ncv=krylov_dimension,
            return_eigenvectors=True,
        )

    except ArpackNoConvergence as e:
        logger.warning(
            f"ARPACK did not converge for parameters {max_iter=}, {tol=}, {krylov_dimension=}, "
            f"{rank_estimate=}. \n Returning the best approximation found so far. Use those with care or "
            f"modify parameters.\n Original error: {e}"
        )

        eigen_vals, eigen_vecs = e.eigenvalues, e.eigenvectors

    eigen_vals = to_torch_conversion_function(eigen_vals)
    eigen_vecs = to_torch_conversion_function(eigen_vecs)

    return LowRankProductRepresentation(eigen_vals, eigen_vecs)

model_hessian_low_rank(model, training_data, hessian_perturbation=0.0, rank_estimate=10, krylov_dimension=None, tol=1e-06, max_iter=None, eigen_computation_on_gpu=False)

Calculates a low-rank approximation of the Hessian matrix of the model's loss function using the implicitly restarted Lanczos algorithm, i.e.

\[ H_{\text{approx}} = V D V^T\]

where \(D\) is a diagonal matrix with the top (in absolute value) rank_estimate eigenvalues of the Hessian and \(V\) contains the corresponding eigenvectors.

PARAMETER	DESCRIPTION
model	A PyTorch model instance that is twice differentiable, wrapped into TorchTwiceDifferential. The Hessian will be calculated with respect to this model's parameters. TYPE: TorchTwiceDifferentiable
training_data	A DataLoader instance that provides the model's training data. Used in calculating the Hessian-vector products. TYPE: DataLoader
hessian_perturbation	Optional regularization parameter added to the Hessian-vector product for numerical stability. TYPE: float DEFAULT: 0.0
rank_estimate	The number of eigenvalues and corresponding eigenvectors to compute. Represents the desired rank of the Hessian approximation. TYPE: int DEFAULT: 10
krylov_dimension	The number of Krylov vectors to use for the Lanczos method. If not provided, it defaults to min(model.num_parameters, max(2*rank_estimate + 1, 20)). TYPE: Optional[int] DEFAULT: None
tol	The stopping criteria for the Lanczos algorithm, which stops when the difference in the approximated eigenvalue is less than tol. Defaults to 1e-6. TYPE: float DEFAULT: 1e-06
max_iter	The maximum number of iterations for the Lanczos method. If not provided, it defaults to 10*model.num_parameters. TYPE: Optional[int] DEFAULT: None
eigen_computation_on_gpu	If True, tries to execute the eigen pair approximation on the provided device via cupy implementation. Make sure, that either your model is small enough or you use a small rank_estimate to fit your device's memory. If False, the eigen pair approximation is executed on the CPU by scipy wrapper to ARPACK. TYPE: bool DEFAULT: False

RETURNS	DESCRIPTION
LowRankProductRepresentation	A LowRankProductRepresentation instance that contains the top (up until rank_estimate) eigenvalues and corresponding eigenvectors of the Hessian.

Source code in src/pydvl/influence/torch/torch_differentiable.py

def model_hessian_low_rank(
    model: TorchTwiceDifferentiable,
    training_data: DataLoader,
    hessian_perturbation: float = 0.0,
    rank_estimate: int = 10,
    krylov_dimension: Optional[int] = None,
    tol: float = 1e-6,
    max_iter: Optional[int] = None,
    eigen_computation_on_gpu: bool = False,
) -> LowRankProductRepresentation:
    r"""
    Calculates a low-rank approximation of the Hessian matrix of the model's loss function using the implicitly
    restarted Lanczos algorithm, i.e.

    \[ H_{\text{approx}} = V D V^T\]

    where \(D\) is a diagonal matrix with the top (in absolute value) `rank_estimate` eigenvalues of the Hessian
    and \(V\) contains the corresponding eigenvectors.


    Args:
        model: A PyTorch model instance that is twice differentiable, wrapped into `TorchTwiceDifferential`.
            The Hessian will be calculated with respect to this model's parameters.
        training_data: A DataLoader instance that provides the model's training data.
            Used in calculating the Hessian-vector products.
        hessian_perturbation: Optional regularization parameter added to the Hessian-vector product
            for numerical stability.
        rank_estimate: The number of eigenvalues and corresponding eigenvectors to compute.
            Represents the desired rank of the Hessian approximation.
        krylov_dimension: The number of Krylov vectors to use for the Lanczos method.
            If not provided, it defaults to min(model.num_parameters, max(2*rank_estimate + 1, 20)).
        tol: The stopping criteria for the Lanczos algorithm, which stops when the difference
            in the approximated eigenvalue is less than `tol`. Defaults to 1e-6.
        max_iter: The maximum number of iterations for the Lanczos method. If not provided, it defaults to
            10*model.num_parameters.
        eigen_computation_on_gpu: If True, tries to execute the eigen pair approximation on the provided
            device via cupy implementation.
            Make sure, that either your model is small enough or you use a
            small rank_estimate to fit your device's memory.
            If False, the eigen pair approximation is executed on the CPU by scipy wrapper to
            ARPACK.

    Returns:
        A [LowRankProductRepresentation][pydvl.influence.torch.torch_differentiable.LowRankProductRepresentation]
            instance that contains the top (up until rank_estimate) eigenvalues
            and corresponding eigenvectors of the Hessian.
    """
    raw_hvp = get_hvp_function(
        model.model, model.loss, training_data, use_hessian_avg=True
    )

    return lanzcos_low_rank_hessian_approx(
        hessian_vp=raw_hvp,
        matrix_shape=(model.num_params, model.num_params),
        hessian_perturbation=hessian_perturbation,
        rank_estimate=rank_estimate,
        krylov_dimension=krylov_dimension,
        tol=tol,
        max_iter=max_iter,
        device=model.device if hasattr(model, "device") else None,
        eigen_computation_on_gpu=eigen_computation_on_gpu,
    )

solve_linear(model, training_data, b, hessian_perturbation=0.0)

Given a model and training data, it finds x such that \(Hx = b\), with \(H\) being the model hessian.

PARAMETER	DESCRIPTION
model	A model wrapped in the TwiceDifferentiable interface. TYPE: TorchTwiceDifferentiable
training_data	A DataLoader containing the training data. TYPE: DataLoader
b	A vector or matrix, the right hand side of the equation \(Hx = b\). TYPE: Tensor
hessian_perturbation	Regularization of the hessian. TYPE: float DEFAULT: 0.0

RETURNS	DESCRIPTION
InverseHvpResult	Instance of InverseHvpResult, having an array that solves the inverse problem, i.e. it returns \(x\) such that \(Hx = b\), and a dictionary containing information about the solution.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@InversionRegistry.register(TorchTwiceDifferentiable, InversionMethod.Direct)
def solve_linear(
    model: TorchTwiceDifferentiable,
    training_data: DataLoader,
    b: torch.Tensor,
    hessian_perturbation: float = 0.0,
) -> InverseHvpResult:
    r"""
    Given a model and training data, it finds x such that \(Hx = b\), with \(H\) being the model hessian.

    Args:
        model: A model wrapped in the TwiceDifferentiable interface.
        training_data: A DataLoader containing the training data.
        b: A vector or matrix, the right hand side of the equation \(Hx = b\).
        hessian_perturbation: Regularization of the hessian.

    Returns:
        Instance of [InverseHvpResult][pydvl.influence.twice_differentiable.InverseHvpResult],
            having an array that solves the inverse problem, i.e. it returns \(x\) such that \(Hx = b\),
            and a dictionary containing information about the solution.
    """

    all_x, all_y = [], []
    for x, y in training_data:
        all_x.append(x)
        all_y.append(y)
    hessian = model.hessian(torch.cat(all_x), torch.cat(all_y))
    matrix = hessian + hessian_perturbation * torch.eye(
        model.num_params, device=model.device
    )
    info = {"hessian": hessian}
    return InverseHvpResult(x=torch.linalg.solve(matrix, b.T).T, info=info)

solve_batch_cg(model, training_data, b, hessian_perturbation=0.0, *, x0=None, rtol=1e-07, atol=1e-07, maxiter=None, progress=False)

Given a model and training data, it uses conjugate gradient to calculate the inverse of the Hessian Vector Product. More precisely, it finds x such that \(Hx = b\), with \(H\) being the model hessian. For more info, see Wikipedia.

PARAMETER	DESCRIPTION
model	A model wrapped in the TwiceDifferentiable interface. TYPE: TorchTwiceDifferentiable
training_data	A DataLoader containing the training data. TYPE: DataLoader
b	A vector or matrix, the right hand side of the equation \(Hx = b\). TYPE: Tensor
hessian_perturbation	Regularization of the hessian. TYPE: float DEFAULT: 0.0
x0	Initial guess for hvp. If None, defaults to b. TYPE: Optional[torch.Tensor] DEFAULT: None
rtol	Maximum relative tolerance of result. TYPE: float DEFAULT: 1e-07
atol	Absolute tolerance of result. TYPE: float DEFAULT: 1e-07
maxiter	Maximum number of iterations. If None, defaults to 10*len(b). TYPE: Optional[int] DEFAULT: None
progress	If True, display progress bars. TYPE: bool DEFAULT: False

RETURNS	DESCRIPTION
InverseHvpResult	Instance of InverseHvpResult, having a matrix of shape [NxP] with each line being a solution of \(Ax=b\), and a dictionary containing information about the convergence of CG, one entry for each line of the matrix.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@InversionRegistry.register(TorchTwiceDifferentiable, InversionMethod.Cg)
def solve_batch_cg(
    model: TorchTwiceDifferentiable,
    training_data: DataLoader,
    b: torch.Tensor,
    hessian_perturbation: float = 0.0,
    *,
    x0: Optional[torch.Tensor] = None,
    rtol: float = 1e-7,
    atol: float = 1e-7,
    maxiter: Optional[int] = None,
    progress: bool = False,
) -> InverseHvpResult:
    r"""
    Given a model and training data, it uses conjugate gradient to calculate the
    inverse of the Hessian Vector Product. More precisely, it finds x such that \(Hx =
    b\), with \(H\) being the model hessian. For more info, see
    [Wikipedia](https://en.wikipedia.org/wiki/Conjugate_gradient_method).

    Args:
        model: A model wrapped in the TwiceDifferentiable interface.
        training_data: A DataLoader containing the training data.
        b: A vector or matrix, the right hand side of the equation \(Hx = b\).
        hessian_perturbation: Regularization of the hessian.
        x0: Initial guess for hvp. If None, defaults to b.
        rtol: Maximum relative tolerance of result.
        atol: Absolute tolerance of result.
        maxiter: Maximum number of iterations. If None, defaults to 10*len(b).
        progress: If True, display progress bars.

    Returns:
        Instance of [InverseHvpResult][pydvl.influence.twice_differentiable.InverseHvpResult],
            having a matrix of shape [NxP] with each line being a solution of \(Ax=b\),
            and a dictionary containing information about the convergence of CG,
            one entry for each line of the matrix.
    """

    total_grad_xy = 0
    total_points = 0
    for x, y in maybe_progress(training_data, progress, desc="Batch Train Gradients"):
        grad_xy = model.grad(x, y, create_graph=True)
        total_grad_xy += grad_xy * len(x)
        total_points += len(x)
    backprop_on = model.parameters
    reg_hvp = lambda v: model.mvp(
        total_grad_xy / total_points, v, backprop_on
    ) + hessian_perturbation * v.type(torch.float64)
    batch_cg = torch.zeros_like(b)
    info = {}
    for idx, bi in enumerate(maybe_progress(b, progress, desc="Conjugate gradient")):
        batch_result, batch_info = solve_cg(
            reg_hvp, bi, x0=x0, rtol=rtol, atol=atol, maxiter=maxiter
        )
        batch_cg[idx] = batch_result
        info[f"batch_{idx}"] = batch_info
    return InverseHvpResult(x=batch_cg, info=info)

solve_cg(hvp, b, *, x0=None, rtol=1e-07, atol=1e-07, maxiter=None)

Conjugate gradient solver for the Hessian vector product.

PARAMETER	DESCRIPTION
hvp	A callable Hvp, operating with tensors of size N. TYPE: Callable[[Tensor], Tensor]
b	A vector or matrix, the right hand side of the equation \(Hx = b\). TYPE: Tensor
x0	Initial guess for hvp. TYPE: Optional[torch.Tensor] DEFAULT: None
rtol	Maximum relative tolerance of result. TYPE: float DEFAULT: 1e-07
atol	Absolute tolerance of result. TYPE: float DEFAULT: 1e-07
maxiter	Maximum number of iterations. If None, defaults to 10*len(b). TYPE: Optional[int] DEFAULT: None

RETURNS	DESCRIPTION
InverseHvpResult	Instance of InverseHvpResult, with a vector x, solution of \(Ax=b\), and a dictionary containing information about the convergence of CG.

Source code in src/pydvl/influence/torch/torch_differentiable.py

def solve_cg(
    hvp: Callable[[torch.Tensor], torch.Tensor],
    b: torch.Tensor,
    *,
    x0: Optional[torch.Tensor] = None,
    rtol: float = 1e-7,
    atol: float = 1e-7,
    maxiter: Optional[int] = None,
) -> InverseHvpResult:
    r"""
    Conjugate gradient solver for the Hessian vector product.

    Args:
        hvp: A callable Hvp, operating with tensors of size N.
        b: A vector or matrix, the right hand side of the equation \(Hx = b\).
        x0: Initial guess for hvp.
        rtol: Maximum relative tolerance of result.
        atol: Absolute tolerance of result.
        maxiter: Maximum number of iterations. If None, defaults to 10*len(b).

    Returns:
        Instance of [InverseHvpResult][pydvl.influence.twice_differentiable.InverseHvpResult],
            with a vector x, solution of \(Ax=b\), and a dictionary containing
            information about the convergence of CG.
    """

    if x0 is None:
        x0 = torch.clone(b)
    if maxiter is None:
        maxiter = len(b) * 10

    y_norm = torch.sum(torch.matmul(b, b)).item()
    stopping_val = max([rtol**2 * y_norm, atol**2])

    x = x0
    p = r = (b - hvp(x)).squeeze().type(torch.float64)
    gamma = torch.sum(torch.matmul(r, r)).item()
    optimal = False

    for k in range(maxiter):
        if gamma < stopping_val:
            optimal = True
            break
        Ap = hvp(p).squeeze()
        alpha = gamma / torch.sum(torch.matmul(p, Ap)).item()
        x += alpha * p
        r -= alpha * Ap
        gamma_ = torch.sum(torch.matmul(r, r)).item()
        beta = gamma_ / gamma
        gamma = gamma_
        p = r + beta * p

    info = {"niter": k, "optimal": optimal, "gamma": gamma}
    return InverseHvpResult(x=x, info=info)

solve_lissa(model, training_data, b, hessian_perturbation=0.0, *, maxiter=1000, dampen=0.0, scale=10.0, h0=None, rtol=0.0001, progress=False)

Uses LISSA, Linear time Stochastic Second-Order Algorithm, to iteratively approximate the inverse Hessian. More precisely, it finds x s.t. \(Hx = b\), with \(H\) being the model's second derivative wrt. the parameters. This is done with the update

\[H^{-1}_{j+1} b = b + (I - d) \ H - \frac{H^{-1}_j b}{s},\]

where \(I\) is the identity matrix, \(d\) is a dampening term and \(s\) a scaling factor that are applied to help convergence. For details, see (Koh and Liang, 2017)¹ and the original paper (Agarwal et. al.)².

PARAMETER	DESCRIPTION
model	A model wrapped in the TwiceDifferentiable interface. TYPE: TorchTwiceDifferentiable
training_data	A DataLoader containing the training data. TYPE: DataLoader
b	A vector or matrix, the right hand side of the equation \(Hx = b\). TYPE: Tensor
hessian_perturbation	Regularization of the hessian. TYPE: float DEFAULT: 0.0
maxiter	Maximum number of iterations. TYPE: int DEFAULT: 1000
dampen	Dampening factor, defaults to 0 for no dampening. TYPE: float DEFAULT: 0.0
scale	Scaling factor, defaults to 10. TYPE: float DEFAULT: 10.0
h0	Initial guess for hvp. TYPE: Optional[torch.Tensor] DEFAULT: None
rtol	tolerance to use for early stopping TYPE: float DEFAULT: 0.0001
progress	If True, display progress bars. TYPE: bool DEFAULT: False

RETURNS	DESCRIPTION
InverseHvpResult	Instance of InverseHvpResult, with a matrix of shape [NxP] with each line being a solution of \(Ax=b\), and a dictionary containing information about the accuracy of the solution.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@InversionRegistry.register(TorchTwiceDifferentiable, InversionMethod.Lissa)
def solve_lissa(
    model: TorchTwiceDifferentiable,
    training_data: DataLoader,
    b: torch.Tensor,
    hessian_perturbation: float = 0.0,
    *,
    maxiter: int = 1000,
    dampen: float = 0.0,
    scale: float = 10.0,
    h0: Optional[torch.Tensor] = None,
    rtol: float = 1e-4,
    progress: bool = False,
) -> InverseHvpResult:
    r"""
    Uses LISSA, Linear time Stochastic Second-Order Algorithm, to iteratively
    approximate the inverse Hessian. More precisely, it finds x s.t. \(Hx = b\),
    with \(H\) being the model's second derivative wrt. the parameters.
    This is done with the update

    \[H^{-1}_{j+1} b = b + (I - d) \ H - \frac{H^{-1}_j b}{s},\]

    where \(I\) is the identity matrix, \(d\) is a dampening term and \(s\) a scaling
    factor that are applied to help convergence. For details, see
    (Koh and Liang, 2017)<sup><a href="#koh_liang_2017">1</a></sup> and the original paper
    (Agarwal et. al.)<sup><a href="#agarwal_secondorder_2017">2</a></sup>.

    Args:
        model: A model wrapped in the TwiceDifferentiable interface.
        training_data: A DataLoader containing the training data.
        b: A vector or matrix, the right hand side of the equation \(Hx = b\).
        hessian_perturbation: Regularization of the hessian.
        maxiter: Maximum number of iterations.
        dampen: Dampening factor, defaults to 0 for no dampening.
        scale: Scaling factor, defaults to 10.
        h0: Initial guess for hvp.
        rtol: tolerance to use for early stopping
        progress: If True, display progress bars.

    Returns:
        Instance of [InverseHvpResult][pydvl.influence.twice_differentiable.InverseHvpResult], with a matrix of shape [NxP] with each line being a solution of \(Ax=b\),
            and a dictionary containing information about the accuracy of the solution.
    """

    if h0 is None:
        h_estimate = torch.clone(b)
    else:
        h_estimate = h0
    shuffled_training_data = DataLoader(
        training_data.dataset, training_data.batch_size, shuffle=True
    )

    def lissa_step(
        h: torch.Tensor, reg_hvp: Callable[[torch.Tensor], torch.Tensor]
    ) -> torch.Tensor:
        """Given an estimate of the hessian inverse and the regularised hessian
        vector product, it computes the next estimate.

        Args:
            h: An estimate of the hessian inverse.
            reg_hvp: Regularised hessian vector product.

        Returns:
            The next estimate of the hessian inverse.
        """
        return b + (1 - dampen) * h - reg_hvp(h) / scale

    for _ in maybe_progress(range(maxiter), progress, desc="Lissa"):
        x, y = next(iter(shuffled_training_data))
        grad_xy = model.grad(x, y, create_graph=True)
        reg_hvp = (
            lambda v: model.mvp(grad_xy, v, model.parameters) + hessian_perturbation * v
        )
        residual = lissa_step(h_estimate, reg_hvp) - h_estimate
        h_estimate += residual
        if torch.isnan(h_estimate).any():
            raise RuntimeError("NaNs in h_estimate. Increase scale or dampening.")
        max_residual = torch.max(torch.abs(residual / h_estimate))
        if max_residual < rtol:
            break
    mean_residual = torch.mean(torch.abs(residual / h_estimate))
    logger.info(
        f"Terminated Lissa with {max_residual*100:.2f} % max residual."
        f" Mean residual: {mean_residual*100:.5f} %"
    )
    info = {
        "max_perc_residual": max_residual * 100,
        "mean_perc_residual": mean_residual * 100,
    }
    return InverseHvpResult(x=h_estimate / scale, info=info)

solve_arnoldi(model, training_data, b, hessian_perturbation=0.0, *, rank_estimate=10, krylov_dimension=None, low_rank_representation=None, tol=1e-06, max_iter=None, eigen_computation_on_gpu=False)

Solves the linear system Hx = b, where H is the Hessian of the model's loss function and b is the given right-hand side vector. It employs the implicitly restarted Arnoldi method for computing a partial eigen decomposition, which is used fo the inversion i.e.

\[x = V D^{-1} V^T b\]

where \(D\) is a diagonal matrix with the top (in absolute value) rank_estimate eigenvalues of the Hessian and \(V\) contains the corresponding eigenvectors.

PARAMETER	DESCRIPTION
model	A PyTorch model instance that is twice differentiable, wrapped into TorchTwiceDifferential. The Hessian will be calculated with respect to this model's parameters. TYPE: TorchTwiceDifferentiable
training_data	A DataLoader instance that provides the model's training data. Used in calculating the Hessian-vector products. TYPE: DataLoader
b	The right-hand side vector in the system Hx = b. TYPE: Tensor
hessian_perturbation	Optional regularization parameter added to the Hessian-vector product for numerical stability. TYPE: float DEFAULT: 0.0
rank_estimate	The number of eigenvalues and corresponding eigenvectors to compute. Represents the desired rank of the Hessian approximation. TYPE: int DEFAULT: 10
krylov_dimension	The number of Krylov vectors to use for the Lanczos method. Defaults to min(model's number of parameters, max(2 times rank_estimate + 1, 20)). TYPE: Optional[int] DEFAULT: None
low_rank_representation	An instance of LowRankProductRepresentation containing a previously computed low-rank representation of the Hessian. If provided, all other parameters are ignored; otherwise, a new low-rank representation is computed using provided parameters. TYPE: Optional[LowRankProductRepresentation] DEFAULT: None
tol	The stopping criteria for the Lanczos algorithm. Ignored if low_rank_representation is provided. TYPE: float DEFAULT: 1e-06
max_iter	The maximum number of iterations for the Lanczos method. Ignored if low_rank_representation is provided. TYPE: Optional[int] DEFAULT: None
eigen_computation_on_gpu	If True, tries to execute the eigen pair approximation on the model's device via a cupy implementation. Ensure the model size or rank_estimate is appropriate for device memory. If False, the eigen pair approximation is executed on the CPU by the scipy wrapper to ARPACK. TYPE: bool DEFAULT: False

RETURNS	DESCRIPTION
InverseHvpResult	Instance of InverseHvpResult, having the solution vector x that satisfies the system \(Ax = b\), where \(A\) is a low-rank approximation of the Hessian \(H\) of the model's loss function, and an instance of LowRankProductRepresentation, which represents the approximation of H.

Source code in src/pydvl/influence/torch/torch_differentiable.py

@InversionRegistry.register(TorchTwiceDifferentiable, InversionMethod.Arnoldi)
def solve_arnoldi(
    model: TorchTwiceDifferentiable,
    training_data: DataLoader,
    b: torch.Tensor,
    hessian_perturbation: float = 0.0,
    *,
    rank_estimate: int = 10,
    krylov_dimension: Optional[int] = None,
    low_rank_representation: Optional[LowRankProductRepresentation] = None,
    tol: float = 1e-6,
    max_iter: Optional[int] = None,
    eigen_computation_on_gpu: bool = False,
) -> InverseHvpResult:
    r"""
    Solves the linear system Hx = b, where H is the Hessian of the model's loss function and b is the given
    right-hand side vector.
    It employs the [implicitly restarted Arnoldi method](https://en.wikipedia.org/wiki/Arnoldi_iteration) for
    computing a partial eigen decomposition, which is used fo the inversion i.e.

    \[x = V D^{-1} V^T b\]

    where \(D\) is a diagonal matrix with the top (in absolute value) `rank_estimate` eigenvalues of the Hessian
    and \(V\) contains the corresponding eigenvectors.

    Args:
        model: A PyTorch model instance that is twice differentiable, wrapped into
            [TorchTwiceDifferential][pydvl.influence.torch.torch_differentiable.TorchTwiceDifferentiable].
            The Hessian will be calculated with respect to this model's parameters.
        training_data: A DataLoader instance that provides the model's training data.
            Used in calculating the Hessian-vector products.
        b: The right-hand side vector in the system Hx = b.
        hessian_perturbation: Optional regularization parameter added to the Hessian-vector
            product for numerical stability.
        rank_estimate: The number of eigenvalues and corresponding eigenvectors to compute.
            Represents the desired rank of the Hessian approximation.
        krylov_dimension: The number of Krylov vectors to use for the Lanczos method.
            Defaults to min(model's number of parameters, max(2 times rank_estimate + 1, 20)).
        low_rank_representation: An instance of
            [LowRankProductRepresentation][pydvl.influence.torch.torch_differentiable.LowRankProductRepresentation]
            containing a previously computed low-rank representation of the Hessian. If provided, all other parameters
            are ignored; otherwise, a new low-rank representation is computed
            using provided parameters.
        tol: The stopping criteria for the Lanczos algorithm.
            Ignored if `low_rank_representation` is provided.
        max_iter: The maximum number of iterations for the Lanczos method.
            Ignored if `low_rank_representation` is provided.
        eigen_computation_on_gpu: If True, tries to execute the eigen pair approximation on the model's device
            via a cupy implementation. Ensure the model size or rank_estimate is appropriate for device memory.
            If False, the eigen pair approximation is executed on the CPU by the scipy wrapper to ARPACK.

    Returns:
        Instance of [InverseHvpResult][pydvl.influence.torch.torch_differentiable.InverseHvpResult],
            having the solution vector x that satisfies the system \(Ax = b\),
            where \(A\) is a low-rank approximation of the Hessian \(H\) of the model's loss function, and an instance
            of [LowRankProductRepresentation][pydvl.influence.torch.torch_differentiable.LowRankProductRepresentation],
            which represents the approximation of H.
    """

    b_device = b.device if hasattr(b, "device") else torch.device("cpu")

    if low_rank_representation is None:
        if b_device.type == "cuda" and not eigen_computation_on_gpu:
            raise ValueError(
                "Using 'eigen_computation_on_gpu=False' while 'b' is on a 'cuda' device is not supported. "
                "To address this, consider the following options:\n"
                " - Set eigen_computation_on_gpu=True if your model and data are small enough "
                "and if 'cupy' is available in your environment.\n"
                " - Move 'b' to the CPU with b.to('cpu').\n"
                " - Precompute a low rank representation and move it to the 'b' device using:\n"
                "     low_rank_representation = model_hessian_low_rank(model, training_data, ..., "
                "eigen_computation_on_gpu=False).to(b.device)"
            )

        low_rank_representation = model_hessian_low_rank(
            model,
            training_data,
            hessian_perturbation=hessian_perturbation,
            rank_estimate=rank_estimate,
            krylov_dimension=krylov_dimension,
            tol=tol,
            max_iter=max_iter,
            eigen_computation_on_gpu=eigen_computation_on_gpu,
        )
    else:
        if b_device.type != low_rank_representation.device.type:
            raise RuntimeError(
                f"The devices for 'b' and 'low_rank_representation' do not match.\n"
                f" - 'b' is on device: {b_device}\n"
                f" - 'low_rank_representation' is on device: {low_rank_representation.device}\n"
                f"\nTo resolve this, consider moving 'low_rank_representation' to '{b_device}' by using:\n"
                f"low_rank_representation = low_rank_representation.to(b.device)"
            )

        logger.info("Using provided low rank representation, ignoring other parameters")

    result = low_rank_representation.projections @ (
        torch.diag_embed(1.0 / low_rank_representation.eigen_vals)
        @ (low_rank_representation.projections.t() @ b.t())
    )
    return InverseHvpResult(
        x=result.t(),
        info={
            "eigenvalues": low_rank_representation.eigen_vals,
            "eigenvectors": low_rank_representation.projections,
        },
    )

---

Last update: 2023-09-02
Created: 2023-09-02