Available Extensions
First order extensions
First-order extensions make it easier to extract information from the gradients being already backpropagated through the computational graph. They do not backpropagate additional information, and have small overhead. The implemented extensions are
BatchGrad
The individual gradients, rather than the sum over the samplesSumGradSquared
The second moment of the individual gradientVariance
The variance of the individual gradientsBatchL2Grad
The L2 norm of the individual gradients
- backpack.extensions.BatchGrad(subsampling: List[int] = None)
Individual gradients for each sample in a minibatch.
Stores the output in
grad_batch
as a[N x ...]
tensor, whereN
batch size and...
is the shape of the gradient.If
subsampling
is specified,N
is replaced by the number of active samples.Note
Beware of scaling issue
The individual gradients depend on the scaling of the overall function. Let
fᵢ
be the loss of thei
th sample, with gradientgᵢ
.BatchGrad
will return[g₁, …, gₙ]
if the loss is a sum,∑ᵢ₌₁ⁿ fᵢ
,[¹/ₙ g₁, …, ¹/ₙ gₙ]
if the loss is a mean,¹/ₙ ∑ᵢ₌₁ⁿ fᵢ
.
The concept of individual gradients is only meaningful if the objective is a sum of independent functions (no batchnorm).
- backpack.extensions.BatchL2Grad()
The squared L2 norm of individual gradients in the minibatch.
Stores the output in
batch_l2
as a tensor of size[N]
, whereN
is the batch size.Note
Beware of scaling issue
The individual L2 norm depends on the scaling of the overall function. Let
fᵢ
be the loss of thei
th sample, with gradientgᵢ
.BatchL2Grad
will return the L2 norm of[g₁, …, gₙ]
if the loss is a sum,∑ᵢ₌₁ⁿ fᵢ
,[¹/ₙ g₁, …, ¹/ₙ gₙ]
if the loss is a mean,¹/ₙ ∑ᵢ₌₁ⁿ fᵢ
.
- backpack.extensions.SumGradSquared()
The sum of individual-gradients-squared, or second moment of the gradient.
Stores the output in
sum_grad_squared
. Same dimension as the gradient.Note
Beware of scaling issue
The second moment depends on the scaling of the overall function. Let
fᵢ
be the loss of thei
th sample, with gradientgᵢ
.SumGradSquared
will return the sum of the squared[g₁, …, gₙ]
if the loss is a sum,∑ᵢ₌₁ⁿ fᵢ
,[¹/ₙ g₁, …, ¹/ₙ gₙ]
if the loss is a mean,¹/ₙ ∑ᵢ₌₁ⁿ fᵢ
.
- backpack.extensions.Variance()
Estimates the variance of the gradient using the samples in the minibatch.
Stores the output in
variance
. Same dimension as the gradient.Note
Beware of scaling issue
The variance depends on the scaling of the overall function. Let
fᵢ
be the loss of thei
th sample, with gradientgᵢ
.Variance
will return the variance of the vectors[g₁, …, gₙ]
if the loss is a sum,∑ᵢ₌₁ⁿ fᵢ
,[¹/ₙ g₁, …, ¹/ₙ gₙ]
if the loss is a mean,¹/ₙ ∑ᵢ₌₁ⁿ fᵢ
.
Second order extensions
Second-order extensions propagate additional information through the graph to extract structural or local approximations to second-order information. They are more expensive to run than a standard gradient backpropagation. The implemented extensions are
The diagonal of the Generalized Gauss-Newton (GGN)/Fisher information, using exact computation (
DiagGGNExact
) or Monte-Carlo approximation (DiagGGNMC
).Kronecker Block-Diagonal approximations of the GGN/Fisher
KFAC
,KFRA
,KFLR
.The diagonal of the Hessian
DiagHessian
The symmetric (square root) factorization of the GGN/Fisher information, using exact computation (
SqrtGGNExact
) or a Monte-Carlo (MC) approximation (SqrtGGNMC
)
- backpack.extensions.DiagGGNMC(mc_samples: int = 1)
Diagonal of the Generalized Gauss-Newton/Fisher.
Uses a Monte-Carlo approximation of the Hessian of the loss w.r.t. the model output.
Stores the output in
diag_ggn_mc
, has the same dimensions as the gradient.For a more precise but slower alternative, see
backpack.extensions.DiagGGNExact()
.
- backpack.extensions.DiagGGNExact()
Diagonal of the Generalized Gauss-Newton/Fisher.
Uses the exact Hessian of the loss w.r.t. the model output.
Stores the output in
diag_ggn_exact
, has the same dimensions as the gradient.For a faster but less precise alternative, see
backpack.extensions.DiagGGNMC()
.
- backpack.extensions.BatchDiagGGNMC(mc_samples: int = 1)
Individual diagonal of the Generalized Gauss-Newton/Fisher.
Uses a Monte-Carlo approximation of the Hessian of the loss w.r.t. the model output.
Stores the output in
diag_ggn_mc_batch
as a[N x ...]
tensor, whereN
is the batch size and...
is the shape of the gradient.For a more precise but slower alternative, see
backpack.extensions.BatchDiagGGNExact()
.
- backpack.extensions.BatchDiagGGNExact()
Individual diagonal of the Generalized Gauss-Newton/Fisher.
Uses the exact Hessian of the loss w.r.t. the model output.
Stores the output in
diag_ggn_exact_batch
as a[N x ...]
tensor, whereN
is the batch size and...
is the shape of the gradient.
- backpack.extensions.KFAC(mc_samples=1)
Approximate Kronecker factorization of the Generalized Gauss-Newton/Fisher using Monte-Carlo sampling.
Stores the output in
kfac
as a list of Kronecker factors.If there is only one element, the item represents the GGN/Fisher approximation itself.
If there are multiple elements, they are arranged in the order such that their Kronecker product represents the Generalized Gauss-Newton/Fisher approximation.
The dimension of the factors depends on the layer, but the product of all row dimensions (or column dimensions) yields the dimension of the layer parameter.
Note
The literature uses column-stacking as vectorization convention, but
torch
defaults to a row-major storing scheme of tensors. The order of factors might differs from the presentation in the literature.Implements the procedures described by
Optimizing Neural Networks with Kronecker-factored Approximate Curvature by James Martens and Roger Grosse, 2015.
A Kronecker-factored approximate Fisher matrix for convolution layers by Roger Grosse and James Martens, 2016
- backpack.extensions.KFLR()
Approximate Kronecker factorization of the Generalized Gauss-Newton/Fisher using the full Hessian of the loss function w.r.t. the model output.
Stores the output in
kflr
as a list of Kronecker factors.If there is only one element, the item represents the GGN/Fisher approximation itself.
If there are multiple elements, they are arranged in the order such that their Kronecker product represents the Generalized Gauss-Newton/Fisher approximation.
The dimension of the factors depends on the layer, but the product of all row dimensions (or column dimensions) yields the dimension of the layer parameter.
Note
The literature uses column-stacking as vectorization convention. This is in contrast to the default row-major storing scheme of tensors in
torch
. Therefore, the order of factors differs from the presentation in the literature.Implements the procedures described by
Practical Gauss-Newton Optimisation for Deep Learning by Aleksandar Botev, Hippolyt Ritter and David Barber, 2017.
Extended for convolutions following
A Kronecker-factored approximate Fisher matrix for convolution layers by Roger Grosse and James Martens, 2016
- backpack.extensions.KFRA()
Approximate Kronecker factorization of the Generalized Gauss-Newton/Fisher using the full Hessian of the loss function w.r.t. the model output and averaging after every backpropagation step.
Stores the output in
kfra
as a list of Kronecker factors.If there is only one element, the item represents the GGN/Fisher approximation itself.
If there are multiple elements, they are arranged in the order such that their Kronecker product represents the Generalized Gauss-Newton/Fisher approximation.
The dimension of the factors depends on the layer, but the product of all row dimensions (or column dimensions) yields the dimension of the layer parameter.
Note
The literature uses column-stacking as vectorization convention. This is in contrast to the default row-major storing scheme of tensors in
torch
. Therefore, the order of factors differs from the presentation in the literature.Practical Gauss-Newton Optimisation for Deep Learning by Aleksandar Botev, Hippolyt Ritter and David Barber, 2017.
Extended for convolutions following
A Kronecker-factored approximate Fisher matrix for convolution layers by Roger Grosse and James Martens, 2016
- backpack.extensions.DiagHessian()
BackPACK extension that computes the Hessian diagonal.
Stores the output in
diag_h
, has the same dimensions as the gradient.Warning
Very expensive on networks with non-piecewise linear activations.
- backpack.extensions.BatchDiagHessian()
BackPACK extensions that computes the per-sample (individual) Hessian diagonal.
Stores the output in
diag_h_batch
as a[N x ...]
tensor, whereN
is the batch size and...
is the parameter shape.Warning
Very expensive on networks with non-piecewise linear activations.
- backpack.extensions.SqrtGGNExact(subsampling: List[int] = None)
Exact matrix square root of the generalized Gauss-Newton/Fisher.
Uses the exact Hessian of the loss w.r.t. the model output.
Stores the output in
sqrt_ggn_exact
, has shape[C, N, param.shape]
, whereC
is the model output dimension (number of classes for classification problems) andN
is the batch size. If sub-sampling is enabled,N
is replaced by the number of active samples,len(subsampling)
.For a faster but less precise alternative, see
backpack.extensions.SqrtGGNMC()
.Note
(Relation to the GGN/Fisher) For each parameter,
param.sqrt_ggn_exact
can be viewed as a[C * N, param.numel()]
matrix. Concatenating this matrix over all parameters results in a matrixVᵀ
, which is the GGN/Fisher’s matrix square root, i.e.G = V Vᵀ
.
- backpack.extensions.SqrtGGNMC(mc_samples: int = 1, subsampling: List[int] = None)
Approximate matrix square root of the generalized Gauss-Newton/Fisher.
Uses a Monte-Carlo (MC) approximation of the Hessian of the loss w.r.t. the model output.
Stores the output in
sqrt_ggn_mc
, has shape[M, N, param.shape]
, whereM
is the number of Monte-Carlo samples andN
is the batch size. If sub-sampling is enabled,N
is replaced by the number of active samples,len(subsampling)
.For a more precise but slower alternative, see
backpack.extensions.SqrtGGNExact()
.Note
(Relation to the GGN/Fisher) For each parameter,
param.sqrt_ggn_mc
can be viewed as a[M * N, param.numel()]
matrix. Concatenating this matrix over all parameters results in a matrixVᵀ
, which is the approximate GGN/Fisher’s matrix square root, i.e.G ≈ V Vᵀ
.
Block-diagonal curvature products
These extensions do not compute information directly, but give access to functions to compute matrix-matrix products with block-diagonal approximations of the Hessian.
Extensions propagate functions through the computation graph. In contrast to standard gradient computation, the graph is retained during backpropagation (this results in higher memory consumption). The cost of one matrix-vector multiplication is on the order of one backward pass.
Implemented extensions are matrix-free curvature-matrix multiplication with the block-diagonal of the Hessian, generalized Gauss-Newton (GGN)/Fisher, and positive-curvature Hessian. They are formalized by the concept of Hessian backpropagation, described in:
Modular Block-diagonal Curvature Approximations for Feedforward Architectures by Felix Dangel, Stefan Harmeling, Philipp Hennig, 2020.
- backpack.extensions.HMP(savefield='hmp')
Matrix-free multiplication with the block-diagonal Hessian.
Stores the multiplication function in
hmp
.For a parameter of shape
[...]
the function receives and returns a tensor of shape[V, ...]
. Each vector slice across the leading dimension is multiplied with the block-diagonal Hessian.
- backpack.extensions.GGNMP(savefield='ggnmp')
Matrix-free Multiplication with the block-diagonal generalized Gauss-Newton/Fisher.
Stores the multiplication function in
ggnmp
.For a parameter of shape
[...]
the function receives and returns a tensor of shape[V, ...]
. Each vector slice across the leading dimension is multiplied with the block-diagonal GGN/Fisher.
- backpack.extensions.PCHMP(savefield='pchmp', modify='clip')
Matrix-free multiplication with the block-diagonal positive-curvature Hessian (PCH).
Stores the multiplication function in
pchmp
.For a parameter of shape
[...]
the function receives and returns a tensor of shape[V, ...]
. Each vector slice across the leading dimension is multiplied with the block-diagonal positive curvature Hessian.The PCH is proposed in
BDA-PCH: Block-Diagonal Approximation of Positive-Curvature Hessian for Training Neural Networks by Sheng-Wei Chen, Chun-Nan Chou and Edward Y. Chang, 2018.
There are different concavity-eliminating modifications which can be selected by the modify argument (“abs” or “clip”).
Note
The name positive-curvature Hessian may be misleading. While the PCH is always positive semi-definite (PSD), it does not refer to the projection of the exact Hessian on to the space of PSD matrices.