importtorchfromtypingimportTuple,Listfromtorch._vmap_internalsimport_vmap# Utility functionsdef_as_tuple(inp,arg_name,fn_name):# Ensures that inp is a tuple of Tensors# Returns whether or not the original inp was a tuple and the tupled version of the inputis_inp_tuple=Trueifnotisinstance(inp,tuple):inp=(inp,)is_inp_tuple=Falsefori,elinenumerate(inp):ifnotisinstance(el,torch.Tensor):ifis_inp_tuple:raiseTypeError("The {} given to {} must be either a Tensor or a tuple of Tensors but the"" value at index {} has type {}.".format(arg_name,fn_name,i,type(el)))else:raiseTypeError("The {} given to {} must be either a Tensor or a tuple of Tensors but the"" given {} has type {}.".format(arg_name,fn_name,arg_name,type(el)))returnis_inp_tuple,inpdef_tuple_postprocess(res,to_unpack):# Unpacks a potentially nested tuple of Tensors# to_unpack should be a single boolean or a tuple of two booleans.# It is used to:# - invert _as_tuple when res should match the inp given to _as_tuple# - optionally remove nesting of two tuples created by multiple calls to _as_tupleifisinstance(to_unpack,tuple):assertlen(to_unpack)==2ifnotto_unpack[1]:res=tuple(el[0]forelinres)ifnotto_unpack[0]:res=res[0]else:ifnotto_unpack:res=res[0]returnresdef_grad_preprocess(inputs,create_graph,need_graph):# Preprocess the inputs to make sure they require gradient# inputs is a tuple of Tensors to preprocess# create_graph specifies if the user wants gradients to flow back to the Tensors in inputs# need_graph specifies if we internally want gradients to flow back to the Tensors in res# Note that we *always* create a new Tensor object to be able to see the difference between# inputs given as arguments and the same Tensors automatically captured by the user function.# Check this issue for more details on how that can happen: https://github.com/pytorch/pytorch/issues/32576res=[]forinpininputs:ifcreate_graphandinp.requires_grad:# Create at least a new Tensor object in a differentiable wayifnotinp.is_sparse:# Use .view_as() to get a shallow copyres.append(inp.view_as(inp))else:# We cannot use view for sparse Tensors so we cloneres.append(inp.clone())else:res.append(inp.detach().requires_grad_(need_graph))returntuple(res)def_grad_postprocess(inputs,create_graph):# Postprocess the generated Tensors to avoid returning Tensors with history when the user did not# request it.ifisinstance(inputs[0],torch.Tensor):ifnotcreate_graph:returntuple(inp.detach()forinpininputs)else:returninputselse:returntuple(_grad_postprocess(inp,create_graph)forinpininputs)def_validate_v(v,other,is_other_tuple):# This assumes that other is the correct shape, and v should match# Both are assumed to be tuples of Tensorsiflen(other)!=len(v):ifis_other_tuple:raiseRuntimeError("v is a tuple of invalid length: should be {} but got {}.".format(len(other),len(v)))else:raiseRuntimeError("The given v should contain a single Tensor.")foridx,(el_v,el_other)inenumerate(zip(v,other)):ifel_v.size()!=el_other.size():prepend=""ifis_other_tuple:prepend="Entry {} in ".format(idx)raiseRuntimeError("{}v has invalid size: should be {} but got {}.".format(prepend,el_other.size(),el_v.size()))def_check_requires_grad(inputs,input_type,strict):# Used to make all the necessary checks to raise nice errors in strict mode.ifnotstrict:returnifinput_typenotin["outputs","grad_inputs","jacobian","hessian"]:raiseRuntimeError("Invalid input_type to _check_requires_grad")fori,inpinenumerate(inputs):ifinpisNone:# This can only be reached for grad_inputs.raiseRuntimeError("The output of the user-provided function is independent of input {}."" This is not allowed in strict mode.".format(i))ifnotinp.requires_grad:ifinput_type=="hessian":raiseRuntimeError("The hessian of the user-provided function with respect to input {}"" is independent of the input. This is not allowed in strict mode."" You should ensure that your function is thrice differentiable and that"" the hessian depends on the inputs.".format(i))elifinput_type=="jacobian":raiseRuntimeError("While computing the hessian, found that the jacobian of the user-provided"" function with respect to input {} is independent of the input. This is not"" allowed in strict mode. You should ensure that your function is twice"" differentiable and that the jacobian depends on the inputs (this would be"" violated by a linear function for example).".format(i))elifinput_type=="grad_inputs":raiseRuntimeError("The gradient with respect to input {} is independent of the inputs of the"" user-provided function. This is not allowed in strict mode.".format(i))else:raiseRuntimeError("Output {} of the user-provided function does not require gradients."" The outputs must be computed in a differentiable manner from the input"" when running in strict mode.".format(i))def_autograd_grad(outputs,inputs,grad_outputs=None,create_graph=False,retain_graph=None):# Version of autograd.grad that accepts `None` in outputs and do not compute gradients for them.# This has the extra constraint that inputs has to be a tupleassertisinstance(outputs,tuple)ifgrad_outputsisNone:grad_outputs=(None,)*len(outputs)assertisinstance(grad_outputs,tuple)assertlen(outputs)==len(grad_outputs)new_outputs:Tuple[torch.Tensor,...]=tuple()new_grad_outputs:Tuple[torch.Tensor,...]=tuple()forout,grad_outinzip(outputs,grad_outputs):ifoutisnotNoneandout.requires_grad:new_outputs+=(out,)new_grad_outputs+=(grad_out,)iflen(new_outputs)==0:# No differentiable output, we don't need to call the autograd enginereturn(None,)*len(inputs)else:returntorch.autograd.grad(new_outputs,inputs,new_grad_outputs,allow_unused=True,create_graph=create_graph,retain_graph=retain_graph)def_fill_in_zeros(grads,refs,strict,create_graph,stage):# Used to detect None in the grads and depending on the flags, either replace them# with Tensors full of 0s of the appropriate size based on the refs or raise an error.# strict and create graph allow us to detect when it is appropriate to raise an error# stage gives us information of which backward call we consider to give good error messageifstagenotin["back","back_trick","double_back","double_back_trick"]:raiseRuntimeError("Invalid stage argument '{}' to _fill_in_zeros".format(stage))res:Tuple[torch.Tensor,...]=tuple()fori,grads_iinenumerate(grads):ifgrads_iisNone:ifstrict:ifstage=="back":raiseRuntimeError("The output of the user-provided function is independent of ""input {}. This is not allowed in strict mode.".format(i))elifstage=="back_trick":raiseRuntimeError("The gradient with respect to the input is independent of entry {}"" in the grad_outputs when using the double backward trick to compute"" forward mode gradients. This is not allowed in strict mode.".format(i))elifstage=="double_back":raiseRuntimeError("The jacobian of the user-provided function is independent of ""input {}. This is not allowed in strict mode.".format(i))else:raiseRuntimeError("The hessian of the user-provided function is independent of ""entry {} in the grad_jacobian. This is not allowed in strict ""mode as it prevents from using the double backward trick to ""replace forward mode AD.".format(i))grads_i=torch.zeros_like(refs[i])else:ifstrictandcreate_graphandnotgrads_i.requires_grad:if"double"notinstage:raiseRuntimeError("The jacobian of the user-provided function is independent of ""input {}. This is not allowed in strict mode when create_graph=True.".format(i))else:raiseRuntimeError("The hessian of the user-provided function is independent of ""input {}. This is not allowed in strict mode when create_graph=True.".format(i))res+=(grads_i,)returnres# Public API
[docs]defvjp(func,inputs,v=None,create_graph=False,strict=False):r"""Function that computes the dot product between a vector ``v`` and the Jacobian of the given function at the point given by the inputs. Args: func (function): a Python function that takes Tensor inputs and returns a tuple of Tensors or a Tensor. inputs (tuple of Tensors or Tensor): inputs to the function ``func``. v (tuple of Tensors or Tensor): The vector for which the vector Jacobian product is computed. Must be the same size as the output of ``func``. This argument is optional when the output of ``func`` contains a single element and (if it is not provided) will be set as a Tensor containing a single ``1``. create_graph (bool, optional): If ``True``, both the output and result will be computed in a differentiable way. Note that when ``strict`` is ``False``, the result can not require gradients or be disconnected from the inputs. Defaults to ``False``. strict (bool, optional): If ``True``, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If ``False``, we return a Tensor of zeros as the vjp for said inputs, which is the expected mathematical value. Defaults to ``False``. Returns: output (tuple): tuple with: func_output (tuple of Tensors or Tensor): output of ``func(inputs)`` vjp (tuple of Tensors or Tensor): result of the dot product with the same shape as the inputs. Example: >>> def exp_reducer(x): ... return x.exp().sum(dim=1) >>> inputs = torch.rand(4, 4) >>> v = torch.ones(4) >>> vjp(exp_reducer, inputs, v) (tensor([5.7817, 7.2458, 5.7830, 6.7782]), tensor([[1.4458, 1.3962, 1.3042, 1.6354], [2.1288, 1.0652, 1.5483, 2.5035], [2.2046, 1.1292, 1.1432, 1.3059], [1.3225, 1.6652, 1.7753, 2.0152]])) >>> vjp(exp_reducer, inputs, v, create_graph=True) (tensor([5.7817, 7.2458, 5.7830, 6.7782], grad_fn=<SumBackward1>), tensor([[1.4458, 1.3962, 1.3042, 1.6354], [2.1288, 1.0652, 1.5483, 2.5035], [2.2046, 1.1292, 1.1432, 1.3059], [1.3225, 1.6652, 1.7753, 2.0152]], grad_fn=<MulBackward0>)) >>> def adder(x, y): ... return 2 * x + 3 * y >>> inputs = (torch.rand(2), torch.rand(2)) >>> v = torch.ones(2) >>> vjp(adder, inputs, v) (tensor([2.4225, 2.3340]), (tensor([2., 2.]), tensor([3., 3.]))) """withtorch.enable_grad():is_inputs_tuple,inputs=_as_tuple(inputs,"inputs","vjp")inputs=_grad_preprocess(inputs,create_graph=create_graph,need_graph=True)outputs=func(*inputs)is_outputs_tuple,outputs=_as_tuple(outputs,"outputs of the user-provided function","vjp")_check_requires_grad(outputs,"outputs",strict=strict)ifvisnotNone:_,v=_as_tuple(v,"v","vjp")v=_grad_preprocess(v,create_graph=create_graph,need_graph=False)_validate_v(v,outputs,is_outputs_tuple)else:iflen(outputs)!=1oroutputs[0].nelement()!=1:raiseRuntimeError("The vector v can only be None if the ""user-provided function returns ""a single Tensor with a single element.")enable_grad=Trueifcreate_graphelsetorch.is_grad_enabled()withtorch.set_grad_enabled(enable_grad):grad_res=_autograd_grad(outputs,inputs,v,create_graph=create_graph)vjp=_fill_in_zeros(grad_res,inputs,strict,create_graph,"back")# Cleanup objects and return them to the useroutputs=_grad_postprocess(outputs,create_graph)vjp=_grad_postprocess(vjp,create_graph)return_tuple_postprocess(outputs,is_outputs_tuple),_tuple_postprocess(vjp,is_inputs_tuple)
[docs]defjvp(func,inputs,v=None,create_graph=False,strict=False):r"""Function that computes the dot product between the Jacobian of the given function at the point given by the inputs and a vector ``v``. Args: func (function): a Python function that takes Tensor inputs and returns a tuple of Tensors or a Tensor. inputs (tuple of Tensors or Tensor): inputs to the function ``func``. v (tuple of Tensors or Tensor): The vector for which the Jacobian vector product is computed. Must be the same size as the input of ``func``. This argument is optional when the input to ``func`` contains a single element and (if it is not provided) will be set as a Tensor containing a single ``1``. create_graph (bool, optional): If ``True``, both the output and result will be computed in a differentiable way. Note that when ``strict`` is ``False``, the result can not require gradients or be disconnected from the inputs. Defaults to ``False``. strict (bool, optional): If ``True``, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If ``False``, we return a Tensor of zeros as the jvp for said inputs, which is the expected mathematical value. Defaults to ``False``. Returns: output (tuple): tuple with: func_output (tuple of Tensors or Tensor): output of ``func(inputs)`` jvp (tuple of Tensors or Tensor): result of the dot product with the same shape as the output. Example: >>> def exp_reducer(x): ... return x.exp().sum(dim=1) >>> inputs = torch.rand(4, 4) >>> v = torch.ones(4, 4) >>> jvp(exp_reducer, inputs, v) (tensor([6.3090, 4.6742, 7.9114, 8.2106]), tensor([6.3090, 4.6742, 7.9114, 8.2106])) >>> jvp(exp_reducer, inputs, v, create_graph=True) (tensor([6.3090, 4.6742, 7.9114, 8.2106], grad_fn=<SumBackward1>), tensor([6.3090, 4.6742, 7.9114, 8.2106], grad_fn=<SqueezeBackward1>)) >>> def adder(x, y): ... return 2 * x + 3 * y >>> inputs = (torch.rand(2), torch.rand(2)) >>> v = (torch.ones(2), torch.ones(2)) >>> jvp(adder, inputs, v) (tensor([2.2399, 2.5005]), tensor([5., 5.])) Note: The jvp is currently computed by using the backward of the backward (sometimes called the double backwards trick) as we don't have support for forward mode AD in PyTorch at the moment. """withtorch.enable_grad():is_inputs_tuple,inputs=_as_tuple(inputs,"inputs","jvp")inputs=_grad_preprocess(inputs,create_graph=create_graph,need_graph=True)ifvisnotNone:_,v=_as_tuple(v,"v","jvp")v=_grad_preprocess(v,create_graph=create_graph,need_graph=False)_validate_v(v,inputs,is_inputs_tuple)else:iflen(inputs)!=1orinputs[0].nelement()!=1:raiseRuntimeError("The vector v can only be None if the input to ""the user-provided function is a single Tensor ""with a single element.")outputs=func(*inputs)is_outputs_tuple,outputs=_as_tuple(outputs,"outputs of the user-provided function","jvp")_check_requires_grad(outputs,"outputs",strict=strict)# The backward is linear so the value of grad_outputs is not important as# it won't appear in the double backward graph. We only need to ensure that# it does not contain inf or nan.grad_outputs=tuple(torch.zeros_like(out,requires_grad=True)foroutinoutputs)grad_inputs=_autograd_grad(outputs,inputs,grad_outputs,create_graph=True)_check_requires_grad(grad_inputs,"grad_inputs",strict=strict)ifcreate_graph:withtorch.enable_grad():grad_res=_autograd_grad(grad_inputs,grad_outputs,v,create_graph=create_graph)jvp=_fill_in_zeros(grad_res,outputs,strict,create_graph,"back_trick")else:grad_res=_autograd_grad(grad_inputs,grad_outputs,v,create_graph=create_graph)jvp=_fill_in_zeros(grad_res,outputs,strict,create_graph,"back_trick")# Cleanup objects and return them to the useroutputs=_grad_postprocess(outputs,create_graph)jvp=_grad_postprocess(jvp,create_graph)return_tuple_postprocess(outputs,is_outputs_tuple),_tuple_postprocess(jvp,is_outputs_tuple)
def_construct_standard_basis_for(tensors:Tuple[torch.Tensor,...],tensor_numels:Tuple[int,...])->Tuple[torch.Tensor,...]:# This function:# - constructs a N=sum(tensor_numels) standard basis. i.e. an NxN identity matrix.# - Splits the identity matrix into chunks with each chunk size determined by `tensor_numels`.# - Each chunk corresponds to one tensor. The chunk has the same dtype and# device as the tensor## For example, with tensor_numels = [1, 2, 1], this function returns:# ( tensor([[1], tensor([[0, 0], tensor([[0],# [0], [1, 0], [0],# [0], [0, 1], [0],# [0]]) , [0, 0]]) , [1]]) )## Precondition: tensor_numels == tuple(tensor.numel() for tensor in tensors)# Precondition: tensors always has at least one element.## See NOTE: [Computing jacobian with vmap and grad for multiple tensors]# for context behind this function. All the pre-conditions are guarded for# in torch.autograd.functional.jacobian.assertlen(tensors)==len(tensor_numels)assertlen(tensors)>0total_numel=sum(tensor_numels)diag_start_indices=(0,*torch.tensor(tensor_numels).cumsum(dim=0)[:-1].neg().unbind())chunks=tuple(tensor.new_zeros(total_numel,tensor_numel)fortensor,tensor_numelinzip(tensors,tensor_numels))forchunk,diag_start_idxinzip(chunks,diag_start_indices):chunk.diagonal(diag_start_idx).fill_(1)returnchunks
[docs]defjacobian(func,inputs,create_graph=False,strict=False,vectorize=False):r"""Function that computes the Jacobian of a given function. Args: func (function): a Python function that takes Tensor inputs and returns a tuple of Tensors or a Tensor. inputs (tuple of Tensors or Tensor): inputs to the function ``func``. create_graph (bool, optional): If ``True``, the Jacobian will be computed in a differentiable manner. Note that when ``strict`` is ``False``, the result can not require gradients or be disconnected from the inputs. Defaults to ``False``. strict (bool, optional): If ``True``, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If ``False``, we return a Tensor of zeros as the jacobian for said inputs, which is the expected mathematical value. Defaults to ``False``. vectorize (bool, optional): This feature is experimental, please use at your own risk. When computing the jacobian, usually we invoke ``autograd.grad`` once per row of the jacobian. If this flag is ``True``, we use the vmap prototype feature as the backend to vectorize calls to ``autograd.grad`` so we only invoke it once instead of once per row. This should lead to performance improvements in many use cases, however, due to this feature being incomplete, there may be performance cliffs. Please use `torch._C._debug_only_display_vmap_fallback_warnings(True)` to show any performance warnings and file us issues if warnings exist for your use case. Defaults to ``False``. Returns: Jacobian (Tensor or nested tuple of Tensors): if there is a single input and output, this will be a single Tensor containing the Jacobian for the linearized inputs and output. If one of the two is a tuple, then the Jacobian will be a tuple of Tensors. If both of them are tuples, then the Jacobian will be a tuple of tuple of Tensors where ``Jacobian[i][j]`` will contain the Jacobian of the ``i``\th output and ``j``\th input and will have as size the concatenation of the sizes of the corresponding output and the corresponding input and will have same dtype and device as the corresponding input. Example: >>> def exp_reducer(x): ... return x.exp().sum(dim=1) >>> inputs = torch.rand(2, 2) >>> jacobian(exp_reducer, inputs) tensor([[[1.4917, 2.4352], [0.0000, 0.0000]], [[0.0000, 0.0000], [2.4369, 2.3799]]]) >>> jacobian(exp_reducer, inputs, create_graph=True) tensor([[[1.4917, 2.4352], [0.0000, 0.0000]], [[0.0000, 0.0000], [2.4369, 2.3799]]], grad_fn=<ViewBackward>) >>> def exp_adder(x, y): ... return 2 * x.exp() + 3 * y >>> inputs = (torch.rand(2), torch.rand(2)) >>> jacobian(exp_adder, inputs) (tensor([[2.8052, 0.0000], [0.0000, 3.3963]]), tensor([[3., 0.], [0., 3.]])) """withtorch.enable_grad():is_inputs_tuple,inputs=_as_tuple(inputs,"inputs","jacobian")inputs=_grad_preprocess(inputs,create_graph=create_graph,need_graph=True)outputs=func(*inputs)is_outputs_tuple,outputs=_as_tuple(outputs,"outputs of the user-provided function","jacobian")_check_requires_grad(outputs,"outputs",strict=strict)ifvectorize:ifstrict:raiseRuntimeError('torch.autograd.functional.jacobian: `strict=True` ''and `vectorized=True` are not supported together. ''Please either set `strict=False` or ''`vectorize=False`.')# NOTE: [Computing jacobian with vmap and grad for multiple outputs]## Let's consider f(x) = (x**2, x.sum()) and let x = torch.randn(3).# It turns out we can compute the jacobian of this function with a single# call to autograd.grad by using vmap over the correct grad_outputs.## Firstly, one way to compute the jacobian is to stack x**2 and x.sum()# into a 4D vector. E.g., use g(x) = torch.stack([x**2, x.sum()])## To get the first row of the jacobian, we call# >>> autograd.grad(g(x), x, grad_outputs=torch.tensor([1, 0, 0, 0]))# To get the 2nd row of the jacobian, we call# >>> autograd.grad(g(x), x, grad_outputs=torch.tensor([0, 1, 0, 0]))# and so on.## Using vmap, we can vectorize all 4 of these computations into one by# passing the standard basis for R^4 as the grad_output.# vmap(partial(autograd.grad, g(x), x))(torch.eye(4)).## Now, how do we compute the jacobian *without stacking the output*?# We can just split the standard basis across the outputs. So to# compute the jacobian of f(x), we'd use# >>> autograd.grad(f(x), x, grad_outputs=_construct_standard_basis_for(...))# The grad_outputs looks like the following:# ( torch.tensor([[1, 0, 0],# [0, 1, 0],# [0, 0, 1],# [0, 0, 0]]),# torch.tensor([[0],# [0],# [0],# [1]]) )## But we're not done yet!# >>> vmap(partial(autograd.grad(f(x), x, grad_outputs=...)))# returns a Tensor of shape [4, 3]. We have to remember to split the# jacobian of shape [4, 3] into two:# - one of shape [3, 3] for the first output# - one of shape [ 3] for the second output# Step 1: Construct grad_outputs by splitting the standard basisoutput_numels=tuple(output.numel()foroutputinoutputs)grad_outputs=_construct_standard_basis_for(outputs,output_numels)flat_outputs=tuple(output.reshape(-1)foroutputinoutputs)# Step 2: Call vmap + autograd.graddefvjp(grad_output):vj=list(_autograd_grad(flat_outputs,inputs,grad_output,create_graph=create_graph))forel_idx,vj_elinenumerate(vj):ifvj_elisnotNone:continuevj[el_idx]=torch.zeros_like(inputs[el_idx])returntuple(vj)jacobians_of_flat_output=_vmap(vjp)(grad_outputs)# Step 3: The returned jacobian is one big tensor per input. In this step,# we split each Tensor by output.jacobian_input_output=[]forjac,input_iinzip(jacobians_of_flat_output,inputs):jacobian_input_i_output=[]forjac,output_jinzip(jac.split(output_numels,dim=0),outputs):jacobian_input_i_output_j=jac.view(output_j.shape+input_i.shape)jacobian_input_i_output.append(jacobian_input_i_output_j)jacobian_input_output.append(jacobian_input_i_output)# Step 4: Right now, `jacobian` is a List[List[Tensor]].# The outer List corresponds to the number of inputs,# the inner List corresponds to the number of outputs.# We need to exchange the order of these and convert to tuples# before returning.jacobian_output_input=tuple(zip(*jacobian_input_output))jacobian_output_input=_grad_postprocess(jacobian_output_input,create_graph)return_tuple_postprocess(jacobian_output_input,(is_outputs_tuple,is_inputs_tuple))jacobian:Tuple[torch.Tensor,...]=tuple()fori,outinenumerate(outputs):# mypy complains that expression and variable have different types due to the empty listjac_i:Tuple[List[torch.Tensor]]=tuple([]for_inrange(len(inputs)))# type: ignore[assignment]forjinrange(out.nelement()):vj=_autograd_grad((out.reshape(-1)[j],),inputs,retain_graph=True,create_graph=create_graph)forel_idx,(jac_i_el,vj_el,inp_el)inenumerate(zip(jac_i,vj,inputs)):ifvj_elisnotNone:ifstrictandcreate_graphandnotvj_el.requires_grad:msg=("The jacobian of the user-provided function is ""independent of input {}. This is not allowed in ""strict mode when create_graph=True.".format(i))raiseRuntimeError(msg)jac_i_el.append(vj_el)else:ifstrict:msg=("Output {} of the user-provided function is ""independent of input {}. This is not allowed in ""strict mode.".format(i,el_idx))raiseRuntimeError(msg)jac_i_el.append(torch.zeros_like(inp_el))jacobian+=(tuple(torch.stack(jac_i_el,dim=0).view(out.size()+inputs[el_idx].size())for(el_idx,jac_i_el)inenumerate(jac_i)),)jacobian=_grad_postprocess(jacobian,create_graph)return_tuple_postprocess(jacobian,(is_outputs_tuple,is_inputs_tuple))
[docs]defhessian(func,inputs,create_graph=False,strict=False,vectorize=False):r"""Function that computes the Hessian of a given scalar function. Args: func (function): a Python function that takes Tensor inputs and returns a Tensor with a single element. inputs (tuple of Tensors or Tensor): inputs to the function ``func``. create_graph (bool, optional): If ``True``, the Hessian will be computed in a differentiable manner. Note that when ``strict`` is ``False``, the result can not require gradients or be disconnected from the inputs. Defaults to ``False``. strict (bool, optional): If ``True``, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If ``False``, we return a Tensor of zeros as the hessian for said inputs, which is the expected mathematical value. Defaults to ``False``. vectorize (bool, optional): This feature is experimental, please use at your own risk. When computing the hessian, usually we invoke ``autograd.grad`` once per row of the hessian. If this flag is ``True``, we use the vmap prototype feature as the backend to vectorize calls to ``autograd.grad`` so we only invoke it once instead of once per row. This should lead to performance improvements in many use cases, however, due to this feature being incomplete, there may be performance cliffs. Please use `torch._C._debug_only_display_vmap_fallback_warnings(True)` to show any performance warnings and file us issues if warnings exist for your use case. Defaults to ``False``. Returns: Hessian (Tensor or a tuple of tuple of Tensors): if there is a single input, this will be a single Tensor containing the Hessian for the input. If it is a tuple, then the Hessian will be a tuple of tuples where ``Hessian[i][j]`` will contain the Hessian of the ``i``\th input and ``j``\th input with size the sum of the size of the ``i``\th input plus the size of the ``j``\th input. ``Hessian[i][j]`` will have the same dtype and device as the corresponding ``i``\th input. Example: >>> def pow_reducer(x): ... return x.pow(3).sum() >>> inputs = torch.rand(2, 2) >>> hessian(pow_reducer, inputs) tensor([[[[5.2265, 0.0000], [0.0000, 0.0000]], [[0.0000, 4.8221], [0.0000, 0.0000]]], [[[0.0000, 0.0000], [1.9456, 0.0000]], [[0.0000, 0.0000], [0.0000, 3.2550]]]]) >>> hessian(pow_reducer, inputs, create_graph=True) tensor([[[[5.2265, 0.0000], [0.0000, 0.0000]], [[0.0000, 4.8221], [0.0000, 0.0000]]], [[[0.0000, 0.0000], [1.9456, 0.0000]], [[0.0000, 0.0000], [0.0000, 3.2550]]]], grad_fn=<ViewBackward>) >>> def pow_adder_reducer(x, y): ... return (2 * x.pow(2) + 3 * y.pow(2)).sum() >>> inputs = (torch.rand(2), torch.rand(2)) >>> hessian(pow_adder_reducer, inputs) ((tensor([[4., 0.], [0., 4.]]), tensor([[0., 0.], [0., 0.]])), (tensor([[0., 0.], [0., 0.]]), tensor([[6., 0.], [0., 6.]]))) """is_inputs_tuple,inputs=_as_tuple(inputs,"inputs","hessian")defensure_single_output_function(*inp):out=func(*inp)is_out_tuple,t_out=_as_tuple(out,"outputs of the user-provided function","hessian")_check_requires_grad(t_out,"outputs",strict=strict)ifis_out_tupleornotisinstance(out,torch.Tensor):raiseRuntimeError("The function given to hessian should return a single Tensor")ifout.nelement()!=1:raiseRuntimeError("The Tensor returned by the function given to hessian should contain a single element")returnout.squeeze()defjac_func(*inp):jac=jacobian(ensure_single_output_function,inp,create_graph=True)_check_requires_grad(jac,"jacobian",strict=strict)returnjacres=jacobian(jac_func,inputs,create_graph=create_graph,strict=strict,vectorize=vectorize)return_tuple_postprocess(res,(is_inputs_tuple,is_inputs_tuple))
[docs]defvhp(func,inputs,v=None,create_graph=False,strict=False):r"""Function that computes the dot product between a vector ``v`` and the Hessian of a given scalar function at the point given by the inputs. Args: func (function): a Python function that takes Tensor inputs and returns a Tensor with a single element. inputs (tuple of Tensors or Tensor): inputs to the function ``func``. v (tuple of Tensors or Tensor): The vector for which the vector Hessian product is computed. Must be the same size as the input of ``func``. This argument is optional when ``func``'s input contains a single element and (if it is not provided) will be set as a Tensor containing a single ``1``. create_graph (bool, optional): If ``True``, both the output and result will be computed in a differentiable way. Note that when ``strict`` is ``False``, the result can not require gradients or be disconnected from the inputs. Defaults to ``False``. strict (bool, optional): If ``True``, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If ``False``, we return a Tensor of zeros as the vhp for said inputs, which is the expected mathematical value. Defaults to ``False``. Returns: output (tuple): tuple with: func_output (tuple of Tensors or Tensor): output of ``func(inputs)`` vhp (tuple of Tensors or Tensor): result of the dot product with the same shape as the inputs. Example: >>> def pow_reducer(x): ... return x.pow(3).sum() >>> inputs = torch.rand(2, 2) >>> v = torch.ones(2, 2) >>> vhp(pow_reducer, inputs, v) (tensor(0.5591), tensor([[1.0689, 1.2431], [3.0989, 4.4456]])) >>> vhp(pow_reducer, inputs, v, create_graph=True) (tensor(0.5591, grad_fn=<SumBackward0>), tensor([[1.0689, 1.2431], [3.0989, 4.4456]], grad_fn=<MulBackward0>)) >>> def pow_adder_reducer(x, y): ... return (2 * x.pow(2) + 3 * y.pow(2)).sum() >>> inputs = (torch.rand(2), torch.rand(2)) >>> v = (torch.zeros(2), torch.ones(2)) >>> vhp(pow_adder_reducer, inputs, v) (tensor(4.8053), (tensor([0., 0.]), tensor([6., 6.]))) """withtorch.enable_grad():is_inputs_tuple,inputs=_as_tuple(inputs,"inputs","vhp")inputs=_grad_preprocess(inputs,create_graph=create_graph,need_graph=True)ifvisnotNone:_,v=_as_tuple(v,"v","vhp")v=_grad_preprocess(v,create_graph=create_graph,need_graph=False)_validate_v(v,inputs,is_inputs_tuple)else:iflen(inputs)!=1orinputs[0].nelement()!=1:raiseRuntimeError("The vector v can only be None if the input to the user-provided function ""is a single Tensor with a single element.")outputs=func(*inputs)is_outputs_tuple,outputs=_as_tuple(outputs,"outputs of the user-provided function","vhp")_check_requires_grad(outputs,"outputs",strict=strict)ifis_outputs_tupleornotisinstance(outputs[0],torch.Tensor):raiseRuntimeError("The function given to vhp should return a single Tensor")ifoutputs[0].nelement()!=1:raiseRuntimeError("The Tensor returned by the function given to vhp should contain a single element")jac=_autograd_grad(outputs,inputs,create_graph=True)_check_requires_grad(jac,"jacobian",strict=strict)enable_grad=Trueifcreate_graphelsetorch.is_grad_enabled()withtorch.set_grad_enabled(enable_grad):grad_res=_autograd_grad(jac,inputs,v,create_graph=create_graph)vhp=_fill_in_zeros(grad_res,inputs,strict,create_graph,"double_back")outputs=_grad_postprocess(outputs,create_graph)vhp=_grad_postprocess(vhp,create_graph)return_tuple_postprocess(outputs,is_outputs_tuple),_tuple_postprocess(vhp,is_inputs_tuple)
[docs]defhvp(func,inputs,v=None,create_graph=False,strict=False):r"""Function that computes the dot product between the Hessian of a given scalar function and a vector ``v`` at the point given by the inputs. Args: func (function): a Python function that takes Tensor inputs and returns a Tensor with a single element. inputs (tuple of Tensors or Tensor): inputs to the function ``func``. v (tuple of Tensors or Tensor): The vector for which the Hessian vector product is computed. Must be the same size as the input of ``func``. This argument is optional when ``func``'s input contains a single element and (if it is not provided) will be set as a Tensor containing a single ``1``. create_graph (bool, optional): If ``True``, both the output and result will be computed in a differentiable way. Note that when ``strict`` is ``False``, the result can not require gradients or be disconnected from the inputs. Defaults to ``False``. strict (bool, optional): If ``True``, an error will be raised when we detect that there exists an input such that all the outputs are independent of it. If ``False``, we return a Tensor of zeros as the hvp for said inputs, which is the expected mathematical value. Defaults to ``False``. Returns: output (tuple): tuple with: func_output (tuple of Tensors or Tensor): output of ``func(inputs)`` hvp (tuple of Tensors or Tensor): result of the dot product with the same shape as the inputs. Example: >>> def pow_reducer(x): ... return x.pow(3).sum() >>> inputs = torch.rand(2, 2) >>> v = torch.ones(2, 2) >>> hvp(pow_reducer, inputs, v) (tensor(0.1448), tensor([[2.0239, 1.6456], [2.4988, 1.4310]])) >>> hvp(pow_reducer, inputs, v, create_graph=True) (tensor(0.1448, grad_fn=<SumBackward0>), tensor([[2.0239, 1.6456], [2.4988, 1.4310]], grad_fn=<MulBackward0>)) >>> def pow_adder_reducer(x, y): ... return (2 * x.pow(2) + 3 * y.pow(2)).sum() >>> inputs = (torch.rand(2), torch.rand(2)) >>> v = (torch.zeros(2), torch.ones(2)) >>> hvp(pow_adder_reducer, inputs, v) (tensor(2.3030), (tensor([0., 0.]), tensor([6., 6.]))) Note: This function is significantly slower than `vhp` due to backward mode AD constraints. If your functions is twice continuously differentiable, then hvp = vhp.t(). So if you know that your function satisfies this condition, you should use vhp instead that is much faster with the current implementation. """withtorch.enable_grad():is_inputs_tuple,inputs=_as_tuple(inputs,"inputs","hvp")inputs=_grad_preprocess(inputs,create_graph=create_graph,need_graph=True)ifvisnotNone:_,v=_as_tuple(v,"v","hvp")v=_grad_preprocess(v,create_graph=create_graph,need_graph=False)_validate_v(v,inputs,is_inputs_tuple)else:iflen(inputs)!=1orinputs[0].nelement()!=1:raiseRuntimeError("The vector v can only be None if the input to the user-provided function ""is a single Tensor with a single element.")outputs=func(*inputs)is_outputs_tuple,outputs=_as_tuple(outputs,"outputs of the user-provided function","hvp")_check_requires_grad(outputs,"outputs",strict=strict)ifis_outputs_tupleornotisinstance(outputs[0],torch.Tensor):raiseRuntimeError("The function given to hvp should return a single Tensor")ifoutputs[0].nelement()!=1:raiseRuntimeError("The Tensor returned by the function given to hvp should contain a single element")jac=_autograd_grad(outputs,inputs,create_graph=True)_check_requires_grad(jac,"jacobian",strict=strict)grad_jac=tuple(torch.zeros_like(inp,requires_grad=True)forinpininputs)double_back=_autograd_grad(jac,inputs,grad_jac,create_graph=True)_check_requires_grad(jac,"hessian",strict=strict)enable_grad=Trueifcreate_graphelsetorch.is_grad_enabled()withtorch.set_grad_enabled(enable_grad):grad_res=_autograd_grad(double_back,grad_jac,v,create_graph=create_graph)hvp=_fill_in_zeros(grad_res,inputs,strict,create_graph,"double_back_trick")outputs=_grad_postprocess(outputs,create_graph)hvp=_grad_postprocess(hvp,create_graph)return_tuple_postprocess(outputs,is_outputs_tuple),_tuple_postprocess(hvp,is_inputs_tuple)
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