a OSic@sddlZddlmZmZddlmZddlmZddZ d,dd Z d d Z d d Z ddZ ddZddZd-ddZddZd.ddZd/ddZeejdfeedfeejdfddd Zd0d!d"Zd1d$d%Zd2d&d'Zd3d(d)Zd4d*d+ZdS)5N)TupleList) forward_ad)_vmapcCs*t|tr|St|tr t|S|fSdSN) isinstancetuplelist)xr U/var/www/html/django/DPS/env/lib/python3.9/site-packages/torch/autograd/functional.py_as_tuple_nochecks   rc Cs|dur|durt|Sd}t|ts0|f}d}t|D]J\}}t|tjs8|rjtd|||t|q8td|||t|q8||fS)NTFzgThe {} given to {} must be either a Tensor or a tuple of Tensors but the value at index {} has type {}.z^The {} given to {} must be either a Tensor or a tuple of Tensors but the given {} has type {}.) rrr enumeratetorchTensor TypeErrorformattype)inparg_namefn_nameZ is_inp_tupleielr r r _as_tuples     rcCsVt|trFt|dksJ|ds4tdd|D}|dsR|d}n |sR|d}|S)Nrcss|]}|dVqdS)rNr ).0rr r r /z%_tuple_postprocess..r)rr len)resZ to_unpackr r r _tuple_postprocess&s  r!cCs\g}|D]J}|r>|jr>|js.|||qR||q|||qt|Sr) requires_grad is_sparseappendview_asclonedetachrequires_grad_r )inputs create_graph need_graphr rr r r _grad_preprocess7s r,csFt|dtjr,s&tdd|DS|Sntfdd|DSdS)Nrcss|]}|VqdSr)r'rrr r r rSrz$_grad_postprocess..c3s|]}t|VqdSr)_grad_postprocessr-r*r r rWr)rrrr )r)r*r r/r r.Ns r.cCst|t|kr6|r.tdt|t|ntdtt||D]H\}\}}||krDd}|rrd|}td|||qDdS)Nz8v is a tuple of invalid length: should be {} but got {}.z+The given v should contain a single Tensor.z Entry {} in z.{}v has invalid size: should be {} but got {}.)r RuntimeErrorrrzipsize)votherZis_other_tupleidxZel_vZel_otherprependr r r _validate_vYs r8cCs|sdS|dvrtdt|D]z\}}|dur>td||js |dkr\td|q |dkrttd|q |dkrtd |q td |q dS) N)outputs grad_inputsjacobianhessianz*Invalid input_type to _check_requires_gradhThe output of the user-provided function is independent of input {}. 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The outputs must be computed in a differentiable manner from the input when running in strict mode.)r1rrr")r) input_typestrictrrr r r _check_requires_gradks2r@Fc Cst|tsJ|dur"dt|}t|ts0Jt|t|ksDJt}t}t||D]*\}} |durZ|jrZ||f7}|| f7}qZt|dkrdt|Stjj|||d|||dSdS)NrrT) allow_unusedr* retain_graphis_grads_batched)rr rr2r"rautogradgrad) r9r) grad_outputsr*rBrCZ new_outputsZnew_grad_outputsoutgrad_outr r r _autograd_grads"     rIcCs|dvrtd|t}t|D]\}}|dur|r|dkrPtd|n>|dkrhtd|n&|dkrtd|ntd |t||}n4|r|r|jsd |vrtd |ntd |||f7}q$|S) N)back back_trick double_backdouble_back_trickz-Invalid stage argument '{}' to _fill_in_zerosrJr=rKzThe gradient with respect to the input is independent of entry {} in the grad_outputs when using the double backward trick to compute forward mode gradients. This is not allowed in strict mode.rLzjThe jacobian of the user-provided function is independent of input {}. This is not allowed in strict mode.zThe hessian of the user-provided function is independent of entry {} in the grad_jacobian. This is not allowed in strict mode as it prevents from using the double backward trick to replace forward mode AD.doubleThe jacobian of the user-provided function is independent of input {}. This is not allowed in strict mode when create_graph=True.zThe hessian of the user-provided function is independent of input {}. This is not allowed in strict mode when create_graph=True.)r1rr rr zeros_liker")gradsrefsr?r*stager rZgrads_ir r r _fill_in_zeross@ rTc CsHtt|dd\}}t||dd}||}t|dd\}}t|d|d|durt|d d\}}t||d d}t|||n$t|d ks|d d krtd Wdn1s0Y|rdnt } t | 0t ||||d} t | |||d} Wdn1s0Yt ||}t | |} t||t| |fS)av 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=), 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=)) >>> 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.]))) r)vjpTr*r+%outputs of the user-provided functionr9r?Nr4FrrzjThe vector v can only be None if the user-provided function returns a single Tensor with a single element.r/rJ)r enable_gradrr,r@r8rnelementr1is_grad_enabledset_grad_enabledrIrTr.r!) funcr)r4r*r?is_inputs_tupler9is_outputs_tuple_rYgrad_resrUr r r rUs&; & 0  rUc Cstt|dd\}}t||dd}|dur\t|dd\}}t||dd}t|||n$t|dksx|d dkrtd ||}t|d d\}}t|d |d t dd|D} t ||| dd} t| d|d Wdn1s0Y|rFt0t | | ||d} t | |||d} Wdn1s:0Yn t | | ||d} t | |||d} t ||}t | |} t ||t | |fS)a 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. Note: ``autograd.functional.jvp`` computes the jvp by using the backward of the backward (sometimes called the double backwards trick). This is not the most performant way of computing the jvp. Please consider using `functorch's jvp `_ or the :ref:`low-level forward-mode AD API ` instead. 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=), tensor([6.3090, 4.6742, 7.9114, 8.2106], grad_fn=)) >>> 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.])) r)jvpTrVNr4FrrrThe vector v can only be None if the input to the user-provided function is a single Tensor with a single element.rWr9rXcss|]}tj|ddVqdST)r"NrrP)rrGr r r ryrzjvp..r/r:rK)rrYrr,r8rrZr1r@r rIrTr.r!) r]r)r4r*r?r^r`r9r_rFr:rarbr r r rb(s0= , 2  rb.)tensors tensor_numelsreturncs|t|t|ksJt|dks$Jt|tfddt||D}d}t||D] \}}||d||8}qV|S)Nrc3s|]\}}||VqdSr) new_zeros)rtensorZ tensor_numelZ total_numelr r rsz0_construct_standard_basis_for..r)rsumr r2diagonalfill_)rfrgchunksZdiag_start_idxchunknumelr rkr _construct_standard_basis_fors  rrcs|r tdtdd\}g|rtddD}t|}fdd}t||}\} } g} t|| D]p\} } g}t| j|dd D]D\} }| jgtd | j dR tg| j |j }| |q| |qrt | | |fStd dS) Nztorch.autograd.functional.jacobian: `strict=True` and `strategy="forward-mode"` are not supported together (yet). Please either set `strict=False` or `strategy="reverse-mode"`.r)r;css|]}|VqdSrrq)rinputr r r rrz_jacfwd..c sttddt|D}t|d\}}|g}g}|D]@}t|\}}|||durz||qJ|t|qJ|t|WdS1s0YdS)Ncss$|]\}}t|||VqdSr)fwAD make_dualr%)rrttangentr r r rsz'_jacfwd..jvp..r9) ru dual_levelr r2rr$ unpack_dualrrP) tangents dual_inputsZ_is_outputs_tuple dual_outputsZjvZ primal_outsZdual_outprimalrwr]r)Z output_infor r rbs      z_jacfwd..jvprdimrzlComputing Jacobian using forward-AD or forward-over-reverse Hessian isonly implemented for `vectorize=True`.)r1rr rrrr2splitpermuterangendimreshapeshaper$r!NotImplementedError)r]r)r? vectorizer^Z input_numelsrzrbZoutputs_before_splitr_r9jacobian_input_outputjacZoutput_iZjacobian_output_i_outputZinput_jjacobian_input_i_output_jr r~r _jacfwds*    r reverse-modec s|dvsJd|dkr2r$tdt|||Sthtdd\}tdd|}t|d d\}}t|d |d |rp|rtd td d|Dt |} tdd|Dfdd} | | } g} t | D]R\} }g}t | j dd|D]&\} }| |j |j }||q | |qtt | }t|}t|||fWdSt}t|D]\}tddttD}tD]}td|fdd}tt ||D]v\}\}}}|dur&|rr|jsd|}t|||n*|r@d||}t||t|qܐq|tfddt|Df7}q~t|}t|||fWdS1s0YdS)aFunction 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 consider using `functorch's jacrev or jacfwd `_ instead if you are looking for something less experimental and more performant. When computing the jacobian, usually we invoke ``autograd.grad`` once per row of the jacobian. If this flag is ``True``, we perform only a single ``autograd.grad`` call with ``batched_grad=True`` which uses the vmap prototype feature. Though this should lead to performance improvements in many cases, because this feature is still experimental, there may be performance cliffs. See :func:`torch.autograd.grad`'s ``batched_grad`` parameter for more information. strategy (str, optional): Set to ``"forward-mode"`` or ``"reverse-mode"`` to determine whether the Jacobian will be computed with forward or reverse mode AD. Currently, ``"forward-mode"`` requires ``vectorized=True``. Defaults to ``"reverse-mode"``. If ``func`` has more outputs than inputs, ``"forward-mode"`` tends to be more performant. Otherwise, prefer to use ``"reverse-mode"``. 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. If strategy is ``forward-mode``, the dtype will be that of the output; otherwise, the 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=) >>> 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.]]))  forward-moderzExpected strategy to be either "forward-mode" or "reverse-mode". Hint: If your function has more outputs than inputs, "forward-mode" tends to be more performant. Otherwise, prefer to use "reverse-mode".rztorch.autograd.functional.jacobian: `create_graph=True` and `strategy="forward-mode"` are not supported together (yet). Please either set `create_graph=False` or `strategy="reverse-mode"`.r)r;TrVrWr9rXztorch.autograd.functional.jacobian: `strict=True` and `vectorized=True` are not supported together. Please either set `strict=False` or `vectorize=False`.css|]}|VqdSrrsroutputr r r rtrzjacobian..css|]}|dVqdS)N)rrr r r rvrcsbtt|dd}t|D]:\}}|dur0qt|tf|j||<qt|S)NT)r*rC) r rIrrrPexpandrlrr ) grad_outputvjel_idxvj_el)r* flat_outputsr) output_numelsr r rUys *zjacobian..vjprrNcss|] }gVqdSrr )rr`r r r rrr)rBr*rOzgOutput {} of the user-provided function is independent of input {}. This is not allowed in strict mode.c3s6|].\}}tj|dd|VqdS)rrN)rstackviewr3)rrjac_i_el)r)rGr r rs )rrrrYrr,r@r1r rrr2rrrr$r.r!rrrrZrIrr"rrP)r]r)r*r?rstrategyr^r9r_rFrUZjacobians_of_flat_outputrrZinput_iZjacobian_input_i_outputZoutput_jrZjacobian_output_inputr;rZjac_ijrrrrZinp_elmsgr )r*rr)rGrr r;sxJ  -       r;c s`t|dd\}}dvs Jdfddfdd}t||||d }t|||fS) awFunction 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 consider using `functorch `_ instead if you are looking for something less experimental and more performant. 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``. outer_jacobian_strategy (str, optional): The Hessian is computed by computing the Jacobian of a Jacobian. The inner Jacobian is always computed in reverse-mode AD. Setting strategy to ``"forward-mode"`` or ``"reverse-mode"`` determines whether the outer Jacobian will be computed with forward or reverse mode AD. Currently, computing the outer Jacobian in ``"forward-mode"`` requires ``vectorized=True``. Defaults to ``"reverse-mode"``. 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=) >>> 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.]]))) r)r<rz@Expected strategy to be either "forward-mode" or "reverse-mode".csZ|}t|dd\}}t|dd|s6t|tjs>td|dkrRtd|S)NrWr<r9rXz;The function given to hessian should return a single TensorrzTThe Tensor returned by the function given to hessian should contain a single element)rr@rrrr1rZsqueeze)rrGZ is_out_tupleZt_out)r]r?r r ensure_single_output_functions z.hessian..ensure_single_output_functioncs:dkrtdd|D}t|dd}t|dd|S)Nrcss|]}|dVqdS)TN)r()rtr r r r#rz,hessian..jac_func..Tr/r;rX)r r;r@)rr)router_jacobian_strategyr?r r jac_funcs zhessian..jac_func)r*r?rr)rr;r!) r]r)r*r?rrr^rr r )rr]rr?r r<sV   r<c Cstt|dd\}}t||dd}|dur\t|dd\}}t||dd}t|||n$t|dksx|d dkrtd ||}t|d d\}}t|d |d |st |d tj std|d dkrtdt ||dd} t| d|d Wdn1s 0Y|r dnt } t | 0t | |||d} t| |||d} Wdn1sj0Yt||}t| |} t||t| |fS)a 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=), tensor([[1.0689, 1.2431], [3.0989, 4.4456]], grad_fn=)) >>> 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.]))) r)vhpTrVNr4FrrrcrWr9rXz7The function given to vhp should return a single TensorzPThe Tensor returned by the function given to vhp should contain a single elementr/r;rL)rrYrr,r8rrZr1r@rrrIr[r\rTr.r!) r]r)r4r*r?r^r`r9r_rrYrarr r r r-s27 . 0  rcCst,t|dd\}}t||dd}|dur^t|dd\}}t||dd}t|||n$t|dksz|d dkrtd ||}t|d d\}}t|d |d |st |d tj std|d dkrtdt ||dd} t| d|d t dd|D} t | || dd} t| d|d Wdn1s>0Y|rRdnt } t| 0t | | ||d} t| |||d}Wdn1s0Yt||}t||}t||t||fS)a 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=), tensor([[2.0239, 1.6456], [2.4988, 1.4310]], grad_fn=)) >>> 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. r)hvpTrVNr4FrrrcrWr9rXz7The function given to hvp should return a single TensorzPThe Tensor returned by the function given to hvp should contain a single elementr/r;css|]}tj|ddVqdSrdrer-r r r rrzhvp..r<rM)rrYrr,r8rrZr1r@rrrIr r[r\rTr.r!)r]r)r4r*r?r^r`r9r_rZgrad_jacrLrYrarr r r rs8@ . 0  r)NN)NFNF)NFF)NFF)FF)FFFr)FFFr)NFF)NFF)rtypingrrr0rrutorch._vmap_internalsrrrr!r,r.r8r@rIrTrUrbrintrrrr;r<rrr r r r s*     , Y e. 9 T u [