nn/Jacobian.lua at singleton · dominikgrewe/nn

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nn.Jacobian = {}

function nn.Jacobian.backward(module, input, param, dparam)

local doparam = 0

if param then

doparam = 1

end

param = param or input

-- output deriv

module:forward(input)

local dout = module.output.new():resizeAs(module.output)

-- 1D view

local sdout = module.output.new(dout:storage(),1,dout:nElement())

-- jacobian matrix to calculate

local jacobian = torch.Tensor(param:nElement(),dout:nElement()):zero()

for i=1,sdout:nElement() do

dout:zero()

sdout[i] = 1

module:zeroGradParameters()

local din = module:updateGradInput(input, dout)

module:accGradParameters(input, dout)

if doparam == 1 then

jacobian:select(2,i):copy(dparam)

else

jacobian:select(2,i):copy(din)

end

return jacobian

end

function nn.Jacobian.backwardUpdate(module, input, param)

-- output deriv

module:forward(input)

local dout = module.output.new():resizeAs(module.output)

-- 1D view

local sdout = module.output.new(dout:storage(),1,dout:nElement())

-- jacobian matrix to calculate

local jacobian = torch.Tensor(param:nElement(),dout:nElement()):zero()

-- original param

local params = module:parameters()

local origparams = {}

for j=1,#params do

table.insert(origparams, params[j]:clone())

end

for i=1,sdout:nElement() do

for j=1,#params do

params[j]:copy(origparams[j])

end

dout:zero()

sdout[i] = 1

module:updateGradInput(input, dout)

module:accUpdateGradParameters(input, dout, 1)

jacobian:select(2,i):copy(param)

end

for j=1,#params do

params[j]:copy(origparams[j])

end

return jacobian

end

function nn.Jacobian.forward(module, input, param, perturbation)

param = param or input

-- perturbation amount

perturbation = perturbation or 1e-6

-- 1D view of input

--local tst = param:storage()

local sin = param.new(param):resize(param:nElement())--param.new(tst,1,tst:size())

-- jacobian matrix to calculate

local jacobian = torch.Tensor():resize(param:nElement(),module:forward(input):nElement())

local outa = torch.Tensor(jacobian:size(2))

local outb = torch.Tensor(jacobian:size(2))

for i=1,sin:nElement() do

local orig = sin[i]

sin[i] = orig - perturbation

outa:copy(module:forward(input))

sin[i] = orig + perturbation

outb:copy(module:forward(input))

sin[i] = orig

outb:add(-1,outa):div(2*perturbation)

jacobian:select(1,i):copy(outb)

end

return jacobian

end

function nn.Jacobian.backwardDiagHessian(module, input, diagHessianParamName)

-- Compute the second derivatives (diagonal Hessian elements)

-- by backpropagation (using the code from hessian.lua).

-- This function computes the diagonal Hessian elements of the following function:

-- F(x_1, x_2, ..., x_n) = y_1^2/2 + y_2^2/2 + ... + y_m^2/2,

-- where

-- x_1, ..., x_n are the input values and parameters of the given module,

-- y_1, ..., y_m are the output values of the given module.

-- All x_i and y_i values are scalars here. In other words,

-- x_1, ..., x_n denote the scalar elements of the module input tensor,

-- the scalar elements of module.weight,

-- and the scalar elements of module.bias;

-- y_1, ..., y_m are the scalar elements of the module output tensor.

-- The diagonal Hessian elements of F are computed with respect to

-- the module input values and parameters (x_1, .., x_n).

-- The function F is chosen for its convenient properties:

-- dF / dy_i = y_i,

-- d^2F / dy_i^2 = 1.

-- In other words, the diagonal Hessian elements of F with respect

-- to the module OUTPUT values (y_1, ... y_m) are equal to 1.

-- Because of that, computing the diagonal Hessian elements of F

-- with respect to the module INPUT values and PARAMETERS (x_1, ..., x_n)

-- can be done by calling updateDiagHessianInput() and accDiagHessianParameters()

-- using a tensor of ones as diagHessianOutput.

module:forward(input)

local diagHessianOutput = module.output.new():resizeAs(module.output):fill(1)

module.diagHessianWeight:zero()

module.diagHessianBias:zero()

module:updateDiagHessianInput(input, diagHessianOutput)

module:accDiagHessianParameters(input, diagHessianOutput)

return module[diagHessianParamName]

end

function nn.Jacobian.linearModuleDiagHessian(module, input, gradParamName)

-- Compute the second derivatives (diagonal Hessian elements)

-- from the first derivatives for the given module

-- (without using the code from hessian.lua).

-- The given module is assumed to be linear with respect to its inputs and weights

-- (like nn.Linear, nn.SpatialConvolution, etc.)

-- This function computes the diagonal Hessian elements of the following function:

-- F(x_1, x_2, ..., x_n) = y_1^2/2 + y_2^2/2 + ... + y_m^2/2.

-- (See the the comment for nn.Jacobian.backwardDiagHessian() for explanation.)

-- The first derivatives of F with respect to

-- the module inputs and parameters (x_1, ..., x_n) are:

-- dF / dx_i = \sum_k (dF / dy_k) (dy_k / dx_i).

-- The second derivatives are:

-- d^2F / dx_i = \sum_k [(d^2F / dy_k^2) (dy_k / dx_i)^2 + (dF / dy_k) (d^2y_k / dx_i^2)].

-- The second derivatives of F with respect to the module outputs (y_1, ..., y_m)

-- are equal to 1, so:

-- d^2F / dx_i = \sum_k [(dy_k / dx_i)^2 + (dF / dy_k) (d^2y_k / dx_i^2)].

1084A

-- Assuming the linearity of module outputs (y_1, ..., y_m)

-- with respect to module inputs and parameters (x_1, ..., x_n),

-- we have (d^2y_k / dx_i^2) = 0,

-- and the expression finally becomes:

-- d^2F / dx_i = \sum_k (dy_k / dx_i)^2.

-- The first derivatives (dy_k / dx_i) are computed by normal backpropagation,

-- using updateGradInput() and accGradParameters().

local gradParam = module[gradParamName]

local diagHessian = gradParam.new():resize(gradParam:nElement()):zero()

module:forward(input)

local gradOutput = module.output.new():resizeAs(module.output)

local gradOutput1D = gradOutput:view(gradOutput:nElement())

for i=1,gradOutput:nElement() do

gradOutput1D:zero()

gradOutput1D[i] = 1

module.gradWeight:zero()

module.gradBias:zero()

module:updateGradInput(input, gradOutput)

module:accGradParameters(input, gradOutput)

diagHessian:addcmul(gradParam, gradParam)

end

return diagHessian

end

function nn.Jacobian.forwardUpdate(module, input, param, perturbation)

-- perturbation amount

perturbation = perturbation or 1e-6

-- 1D view of input

--local tst = param:storage()

local sin = param.new(param):resize(param:nElement())--param.new(tst,1,tst:size())

-- jacobian matrix to calculate

local jacobian = torch.Tensor():resize(param:nElement(),module:forward(input):nElement())

local outa = torch.Tensor(jacobian:size(2))

local outb = torch.Tensor(jacobian:size(2))

for i=1,sin:nElement() do

local orig = sin[i]

sin[i] = orig - perturbation

outa:copy(module:forward(input))

sin[i] = orig + perturbation

outb:copy(module:forward(input))

sin[i] = orig

outb:add(-1,outa):div(2*perturbation)

jacobian:select(1,i):copy(outb)

jacobian:select(1,i):mul(-1)

jacobian:select(1,i):add(sin[i])

end

return jacobian

end

function nn.Jacobian.testJacobian(module, input, minval, maxval, perturbation)

minval = minval or -2

maxval = maxval or 2

local inrange = maxval - minval

input:copy(torch.rand(input:nElement()):mul(inrange):add(minval))

local jac_fprop = nn.Jacobian.forward(module, input, input, perturbation)

local jac_bprop = nn.Jacobian.backward(module, input)

local error = jac_fprop-jac_bprop

return error:abs():max()

end

function nn.Jacobian.testJacobianParameters(module, input, param, dparam, minval, maxval, perturbation)

minval = minval or -2

maxval = maxval or 2

local inrange = maxval - minval

input:copy(torch.rand(input:nElement()):mul(inrange):add(minval))

param:copy(torch.rand(param:nElement()):mul(inrange):add(minval))

local jac_bprop = nn.Jacobian.backward(module, input, param, dparam)

local jac_fprop = nn.Jacobian.forward(module, input, param, perturbation)

local error = jac_fprop - jac_bprop

return error:abs():max()

end

function nn.Jacobian.testJacobianUpdateParameters(module, input, param, minval, maxval, perturbation)

minval = minval or -2

maxval = maxval or 2

local inrange = maxval - minval

input:copy(torch.rand(input:nElement()):mul(inrange):add(minval))

param:copy(torch.rand(param:nElement()):mul(inrange):add(minval))

local params_bprop = nn.Jacobian.backwardUpdate(module, input, param)

local params_fprop = nn.Jacobian.forwardUpdate(module, input, param, perturbation)

local error = params_fprop - params_bprop

return error:abs():max()

end

function nn.Jacobian.testDiagHessian(module, input, gradParamName, diagHessianParamName, minval, maxval)

-- Compute the diagonal Hessian elements for the same function in two different ways,

-- then compare the results and return the difference.

minval = minval or -2

maxval = maxval or 2

local inrange = maxval - minval

input:copy(torch.rand(input:nElement()):mul(inrange):add(minval))

module:initDiagHessianParameters()

local h_bprop = nn.Jacobian.backwardDiagHessian(module, input, diagHessianParamName)

local h_linearmodule = nn.Jacobian.linearModuleDiagHessian(module, input, gradParamName)

local error = h_bprop - h_linearmodule

return error:abs():max()

end

function nn.Jacobian.testDiagHessianInput(module, input, minval, maxval)

return nn.Jacobian.testDiagHessian(module, input, 'gradInput', 'diagHessianInput', minval, maxval)

end

function nn.Jacobian.testDiagHessianWeight(module, input, minval, maxval)

return nn.Jacobian.testDiagHessian(module, input, 'gradWeight', 'diagHessianWeight', minval, maxval)

end

function nn.Jacobian.testDiagHessianBias(module, input, minval, maxval)

return nn.Jacobian.testDiagHessian(module, input, 'gradBias', 'diagHessianBias', minval, maxval)

end

function nn.Jacobian.testIO(module,input, minval, maxval)

minval = minval or -2

maxval = maxval or 2

local inrange = maxval - minval

-- run module

module:forward(input)

local go = module.output:clone():copy(torch.rand(module.output:nElement()):mul(inrange):add(minval))

module:zeroGradParameters()

module:updateGradInput(input,go)

module:accGradParameters(input,go)

local fo = module.output:clone()

local bo = module.gradInput:clone()

-- write module

local filename = os.tmpname()

local f = torch.DiskFile(filename, 'w'):binary()

f:writeObject(module)

f:close()

-- read module

local m = torch.DiskFile(filename):binary():readObject()

m:forward(input)

m:zeroGradParameters()

m:updateGradInput(input,go)

m:accGradParameters(input,go)

-- cleanup

os.remove(filename)

local fo2 = m.output:clone()

local bo2 = m.gradInput:clone()

local errf = fo - fo2

local errb = bo - bo2

return errf:abs():max(), errb:abs():max()

end

function nn.Jacobian.testAllUpdate(module, input, weight, gradWeight)

local gradOutput

local lr = torch.uniform(0.1, 1)

local errors = {}

-- accGradParameters

local maccgp = module:clone()

local weightc = maccgp[weight]:clone()

maccgp:forward(input)

gradOutput = torch.rand(maccgp.output:size())

maccgp:zeroGradParameters()

maccgp:updateGradInput(input, gradOutput)

maccgp:accGradParameters(input, gradOutput)

maccgp:updateParameters(lr)

errors["accGradParameters"] = (weightc-maccgp[gradWeight]*lr-maccgp[weight]):norm()

-- accUpdateGradParameters

local maccugp = module:clone()

maccugp:forward(input)

maccugp:updateGradInput(input, gradOutput)

maccugp:accUpdateGradParameters(input, gradOutput, lr)

errors["accUpdateGradParameters"] = (maccugp[weight]-maccgp[weight]):norm()

-- shared, accGradParameters

local macsh1 = module:clone()

local macsh2 = module:clone()

macsh2:share(macsh1, weight)

macsh1:forward(input)

macsh2:forward(input)

macsh1:zeroGradParameters()

macsh2:zeroGradParameters()

macsh1:updateGradInput(input, gradOutput)

macsh2:updateGradInput(input, gradOutput)

macsh1:accGradParameters(input, gradOutput)

macsh2:accGradParameters(input, gradOutput)

macsh1:updateParameters(lr)

macsh2:updateParameters(lr)

local err = (weightc-maccgp[gradWeight]*(lr*2)-macsh1[weight]):norm()

err = err + (weightc-maccgp[gradWeight]*(lr*2)-macsh2[weight]):norm()

errors["accGradParameters [shared]"] = err

-- shared, accUpdateGradParameters

local macshu1 = module:clone()

local macshu2 = module:clone()

macshu2:share(macshu1, weight)

macshu1:forward(input)

macshu2:forward(input)

macshu1:updateGradInput(input, gradOutput)

macshu2:updateGradInput(input, gradOutput)

macshu1:accUpdateGradParameters(input, gradOutput, lr)

macshu2:accUpdateGradParameters(input, gradOutput, lr)

err = (weightc-maccgp[gradWeight]*(lr*2)-macshu1[weight]):norm()

err = err + (weightc-maccgp[gradWeight]*(lr*2)-macshu2[weight]):norm()

errors["accUpdateGradParameters [shared]"] = err

return errors

end

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