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preconditioned_stochastic_gradient_descent.py
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preconditioned_stochastic_gradient_descent.py
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"""
Created in May, 2018
Pytorch functions for preconditioned SGD
@author: XILIN LI, lixilinx@gmail.com
Updated in Dec, 2020:
Wrapped Kronecker product preconditioner for easy use: the code will select the proper Kronecker product
preconditioner based on the formats of input left and right preconditioners.
Added torch.jit.script decorator by default.
Updates in 2022:
Added low-rank approximation (LRA) and XMat preconditioners.
Wrapped LRA, XMat and Newton preconditioners as classes for easy use.
Updates in 2023:
Added gradient whitening preconditioner.
Replaced matrix norm lower bound max(abs(A)) with sqrt(max(max_i sum_j a_ij^2, max_j sum_i a_ij^2)).
Initialize Q to ((v^T*v)/(h^T*h))^(1/4)*I if the initial scale of Q is set to None.
Wrapped affine family as a class.
Updates in 2024 Jan:
Further tightened lower bound of a matrix spectral norm (see norm_lower_bound).
Updates in 2024 Mar:
By default, the 2nd (previously 1st) order derivative info is used to normalized the step size for preconditioner update.
For class Newton optimizer, also providing a choice for keeping inv(Q) via matrix inverse rank-2 update.
Update rule for a triangular Q is modified to approximately match that on GL(n, R).
Functional usage of PSGD is to be deprecated, and not updated.
Updates in 2024 Aug:
Reversing triu01 back to triu.
QR approximation via triu01, i.e.,
[I + A]_R = I + triu(A) + triu(A, 1)
is fairly accurate when ||A|| < 0.25, but causes regressions for large lr_preconditioner.
Impacted classes: Affine and Newton.
"""
import torch
def norm_lower_bound(A):
"""
Returns a cheap lower bound for the spectral norm of A.
Numerical results on random matrices with a wide range of distributions and sizes suggest,
norm(A) <= sqrt(2) * norm_lower_bound(A)
Looks to be a very tight lower bound.
"""
max_abs = torch.max(torch.abs(A)) # used to normalize A to avoid numerically under- or over-flow
if max_abs > 0:
A = A/max_abs
aa = torch.real(A * A.conj())
value0, i = torch.max(torch.sum(aa, dim=0), 0)
value1, j = torch.max(torch.sum(aa, dim=1), 0)
if value0 > value1:
x = A[:, i].conj() @ A
# We must have norm(x) > 0 since norm(x) >= value0 > value1 >= 0
# Also, avoid expression norm(x*A^H)/norm(x) as x*A^H could under/over flow
return max_abs * torch.linalg.vector_norm((x / torch.linalg.vector_norm(x)) @ A.H)
else:
x = A @ A[j].conj()
# normx = torch.linalg.vector_norm(x)
# if normx > 0:
# # Again, avoid expression norm(A^H*x)/norm(x) as A^H*x could under/over flow
# return max_abs * torch.linalg.vector_norm(A.H @ (x / normx))
# else: # A = 0
# return normx
return max_abs * torch.linalg.vector_norm(A.H @ (x / torch.linalg.vector_norm(x)))
else: # must have A=0
return max_abs
def woodbury_identity_(invA, U, V):
# implements the Woodbury identity,
#
# inv(A + U*V) = inv(A) - inv(A)*U*inv(I + V*inv(A)*U)*V*inv(A)
#
# with inplace update of invA.
#
# Note that using the Woodbury identity multiple times could accumulate numerical erros.
invAU = invA @ U
VinvAU = V @ invAU
I = torch.eye(VinvAU.shape[0], dtype=VinvAU.dtype, device=VinvAU.device)
invA.sub_( invAU @ torch.linalg.solve(I + VinvAU, V@invA) )
def triu01(A):
# it is useful as for a small A, the R of QR decomposition qr(I + A) is about I + triu(A, 0) + triu(A, 1)
return torch.triu(A, diagonal=0) + torch.triu(A, diagonal=1)
###############################################################################
@torch.jit.script
def update_precond_dense(Q, dxs, dgs, step=0.01, _tiny=1.2e-38):
# type: (Tensor, List[Tensor], List[Tensor], float, float) -> Tensor
"""
update dense preconditioner P = Q^T*Q
Q: Cholesky factor of preconditioner with positive diagonal entries
dxs: list of perturbations of parameters
dgs: list of perturbations of gradients
step: update step size normalized to range [0, 1]
_tiny: an offset to avoid division by zero
"""
dx = torch.cat([torch.reshape(x, [-1, 1]) for x in dxs])
dg = torch.cat([torch.reshape(g, [-1, 1]) for g in dgs])
a = Q.mm(dg)
#b = torch.triangular_solve(dx, Q, upper=True, transpose=True)[0]
b = torch.linalg.solve_triangular(Q.t(), dx, upper=False)
grad = torch.triu(a.mm(a.t()) - b.mm(b.t()))
# step0 = step/(grad.abs().max() + _tiny)
step0 = step/(norm_lower_bound(grad) + _tiny)
return Q - step0*grad.mm(Q)
@torch.jit.script
def precond_grad_dense(Q, grads):
# type: (Tensor, List[Tensor]) -> List[Tensor]
"""
return preconditioned gradient using dense preconditioner
Q: Cholesky factor of preconditioner
grads: list of gradients
"""
grad = [torch.reshape(g, [-1, 1]) for g in grads]
lens = [g.shape[0] for g in grad]
grad = torch.cat(grad)
grad = Q.t().mm(Q.mm(grad))
pre_grads = []
idx = 0
for i in range(len(grads)):
pre_grads.append(torch.reshape(grad[idx : idx + lens[i]], grads[i].shape))
idx = idx + lens[i]
return pre_grads
###############################################################################
def update_precond_kron(Ql, Qr, dX, dG, step=0.01, _tiny=1.2e-38):
"""
Update Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql)
Either Ql or Qr can be sparse, and the code can choose the right update rule.
dX: perturbation of (matrix) parameter
dG: perturbation of (matrix) gradient
step: update step size
_tiny: an offset to avoid division by zero
"""
m, n = Ql.shape
p, q = Qr.shape
if m==n: # left is dense
if p==q: #(dense, dense) format
return _update_precond_dense_dense(Ql, Qr, dX, dG, step, _tiny)
elif p==2: # (dense, normalization) format
return _update_precond_norm_dense(Qr, Ql, dX.t(), dG.t(), step, _tiny)[::-1]
elif p==1: # (dense, scaling) format
return _update_precond_dense_scale(Ql, Qr, dX, dG, step, _tiny)
else:
raise Exception('Unknown Kronecker product preconditioner')
elif m==2: # left is normalization
if p==q: # (normalization, dense) format
return _update_precond_norm_dense(Ql, Qr, dX, dG, step, _tiny)
elif p==1: # (normalization, scaling) format
return _update_precond_norm_scale(Ql, Qr, dX, dG, step, _tiny)
else:
raise Exception('Unknown Kronecker product preconditioner')
elif m==1: # left is scaling
if p==q: # (scaling, dense) format
return _update_precond_dense_scale(Qr, Ql, dX.t(), dG.t(), step, _tiny)[::-1]
elif p==2: # (scaling, normalization) format
return _update_precond_norm_scale(Qr, Ql, dX.t(), dG.t(), step, _tiny)[::-1]
else:
raise Exception('Unknown Kronecker product preconditioner')
else:
raise Exception('Unknown Kronecker product preconditioner')
def precond_grad_kron(Ql, Qr, Grad):
"""
return preconditioned gradient using Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql)
Either Ql or Qr can be sparse, and the code can choose the right way to precondition the gradient
Grad: (matrix) gradient
"""
m, n = Ql.shape
p, q = Qr.shape
if m==n: # left is dense
if p==q: #(dense, dense) format
return _precond_grad_dense_dense(Ql, Qr, Grad)
elif p==2: # (dense, normalization) format
return _precond_grad_norm_dense(Qr, Ql, Grad.t()).t()
elif p==1: # (dense, scaling) format
return _precond_grad_dense_scale(Ql, Qr, Grad)
else:
raise Exception('Unknown Kronecker product preconditioner')
elif m==2: # left is normalization
if p==q: # (normalization, dense) format
return _precond_grad_norm_dense(Ql, Qr, Grad)
elif p==1: # (normalization, scaling) format
return _precond_grad_norm_scale(Ql, Qr, Grad)
else:
raise Exception('Unknown Kronecker product preconditioner')
elif m==1: # left is scaling
if p==q: # (scaling, dense) format
return _precond_grad_dense_scale(Qr, Ql, Grad.t()).t()
elif p==2: # (scaling, normalization) format
return _precond_grad_norm_scale(Qr, Ql, Grad.t()).t()
else:
raise Exception('Unknown Kronecker product preconditioner')
else:
raise Exception('Unknown Kronecker product preconditioner')
###############################################################################
@torch.jit.script
def _update_precond_dense_dense(Ql, Qr, dX, dG, step=0.01, _tiny=1.2e-38):
# type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]
"""
update Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql)
Ql: (left side) Cholesky factor of preconditioner with positive diagonal entries
Qr: (right side) Cholesky factor of preconditioner with positive diagonal entries
dX: perturbation of (matrix) parameter
dG: perturbation of (matrix) gradient
step: update step size normalized to range [0, 1]
_tiny: an offset to avoid division by zero
"""
max_l = torch.max(torch.diag(Ql))
max_r = torch.max(torch.diag(Qr))
rho = torch.sqrt(max_l/max_r)
Ql /= rho
Qr *= rho
#A = Ql.mm( dG.mm( Qr.t() ) )
#Bt = torch.triangular_solve((torch.triangular_solve(dX.t(), Qr, upper=True, transpose=True))[0].t(),
# Ql, upper=True, transpose=True)[0]
A = torch.linalg.multi_dot([Ql, dG, Qr.t()])
Bt = torch.linalg.solve_triangular(Ql.t(), torch.linalg.solve_triangular(Qr, dX, upper=True, left=False), upper=False)
grad1 = torch.triu(A.mm(A.t()) - Bt.mm(Bt.t()))
grad2 = torch.triu(A.t().mm(A) - Bt.t().mm(Bt))
# step1 = step/(torch.max(torch.abs(grad1)) + _tiny)
# step2 = step/(torch.max(torch.abs(grad2)) + _tiny)
step1 = step/(norm_lower_bound(grad1) + _tiny)
step2 = step/(norm_lower_bound(grad2) + _tiny)
return Ql - step1*grad1.mm(Ql), Qr - step2*grad2.mm(Qr)
@torch.jit.script
def _precond_grad_dense_dense(Ql, Qr, Grad):
# type: (Tensor, Tensor, Tensor) -> Tensor
"""
return preconditioned gradient using Kronecker product preconditioner
Ql: (left side) Cholesky factor of preconditioner
Qr: (right side) Cholesky factor of preconditioner
Grad: (matrix) gradient
"""
#return torch.chain_matmul(Ql.t(), Ql, Grad, Qr.t(), Qr)
return torch.linalg.multi_dot([Ql.t(), Ql, Grad, Qr.t(), Qr])
###############################################################################
# (normalization, dense) format Kronecker product preconditioner
@torch.jit.script
def _update_precond_norm_dense(ql, Qr, dX, dG, step=0.01, _tiny=1.2e-38):
# type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]
"""
update (normalization, dense) Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql), where
dX and dG have shape (M, N)
ql has shape (2, M)
Qr has shape (N, N)
ql[0] is the diagonal part of Ql
ql[1,0:-1] is the last column of Ql, excluding the last entry
dX is perturbation of (matrix) parameter
dG is perturbation of (matrix) gradient
step: update step size normalized to range [0, 1]
_tiny: an offset to avoid division by zero
"""
# make sure that Ql and Qr have similar dynamic range
max_l = torch.max(ql[0])
max_r = torch.max(torch.diag(Qr))
rho = torch.sqrt(max_l/max_r)
ql /= rho
Qr *= rho
# refer to https://arxiv.org/abs/1512.04202 for details
A = ql[0:1].t()*dG + ql[1:].t().mm( dG[-1:] ) # Ql*dG
A = A.mm(Qr.t())
Bt = dX/ql[0:1].t()
Bt[-1:] -= (ql[1:]/(ql[0:1]*ql[0,-1])).mm(dX)
#Bt = torch.triangular_solve(Bt.t(), Qr, upper=True, transpose=True)[0].t()
Bt = torch.linalg.solve_triangular(Qr, Bt, upper=True, left=False)
grad1_diag = torch.sum(A*A, dim=1) - torch.sum(Bt*Bt, dim=1)
grad1_bias = A[:-1].mm(A[-1:].t()) - Bt[:-1].mm(Bt[-1:].t())
grad1_bias = torch.cat([torch.squeeze(grad1_bias), grad1_bias.new_zeros(1)])
step1 = step/(torch.max(torch.max(torch.abs(grad1_diag)),
torch.max(torch.abs(grad1_bias))) + _tiny)
new_ql0 = ql[0] - step1*grad1_diag*ql[0]
new_ql1 = ql[1] - step1*(grad1_diag*ql[1] + ql[0,-1]*grad1_bias)
grad2 = torch.triu(A.t().mm(A) - Bt.t().mm(Bt))
# step2 = step/(torch.max(torch.abs(grad2)) + _tiny)
step2 = step/(norm_lower_bound(grad2) + _tiny)
return torch.stack((new_ql0, new_ql1)), Qr - step2*grad2.mm(Qr)
@torch.jit.script
def _precond_grad_norm_dense(ql, Qr, Grad):
# type: (Tensor, Tensor, Tensor) -> Tensor
"""
return preconditioned gradient using (normalization, dense) Kronecker product preconditioner
Suppose Grad has shape (M, N)
ql[0] is the diagonal part of Ql
ql[1, 0:-1] is the last column of Ql, excluding the last entry
Qr: shape (N, N), Cholesky factor of right preconditioner
Grad: (matrix) gradient
"""
preG = ql[0:1].t()*Grad + ql[1:].t().mm(Grad[-1:]) # Ql*Grad
#preG = torch.chain_matmul(preG, Qr.t(), Qr)
preG = torch.linalg.multi_dot([preG, Qr.t(), Qr])
add_last_row = ql[1:].mm(preG) # use it to modify the last row
preG *= ql[0:1].t()
preG[-1:] += add_last_row
return preG
###############################################################################
# (normalization, scaling) Kronecker product preconditioner
# the left one is a normalization preconditioner; the right one is a scaling preconditioner
@torch.jit.script
def _update_precond_norm_scale(ql, qr, dX, dG, step=0.01, _tiny=1.2e-38):
# type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]
"""
update (normalization, scaling) preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql), where
dX and dG have shape (M, N)
ql has shape (2, M)
qr has shape (1, N)
ql[0] is the diagonal part of Ql
ql[1, 0:-1] is the last column of Ql, excluding the last entry
qr is the diagonal part of Qr
dX is perturbation of (matrix) parameter
dG is perturbation of (matrix) gradient
step: update step size
_tiny: an offset to avoid division by zero
"""
# make sure that Ql and Qr have similar dynamic range
max_l = torch.max(ql[0])
max_r = torch.max(qr) # qr always is positive
rho = torch.sqrt(max_l/max_r)
ql /= rho
qr *= rho
# refer to https://arxiv.org/abs/1512.04202 for details
A = ql[0:1].t()*dG + ql[1:].t().mm( dG[-1:] ) # Ql*dG
A *= qr # Ql*dG*Qr
Bt = dX/ql[0:1].t()
Bt[-1:] -= (ql[1:]/(ql[0:1]*ql[0,-1])).mm(dX)
Bt /= qr # Ql^(-T)*dX*Qr^(-1)
grad1_diag = torch.sum(A*A, dim=1) - torch.sum(Bt*Bt, dim=1)
grad1_bias = A[:-1].mm(A[-1:].t()) - Bt[:-1].mm(Bt[-1:].t())
grad1_bias = torch.cat([torch.squeeze(grad1_bias), grad1_bias.new_zeros(1)])
step1 = step/(torch.max(torch.max(torch.abs(grad1_diag)),
torch.max(torch.abs(grad1_bias))) + _tiny)
new_ql0 = ql[0] - step1*grad1_diag*ql[0]
new_ql1 = ql[1] - step1*(grad1_diag*ql[1] + ql[0,-1]*grad1_bias)
grad2 = torch.sum(A*A, dim=0, keepdim=True) - torch.sum(Bt*Bt, dim=0, keepdim=True)
step2 = step/(torch.max(torch.abs(grad2)) + _tiny)
return torch.stack((new_ql0, new_ql1)), qr - step2*grad2*qr
@torch.jit.script
def _precond_grad_norm_scale(ql, qr, Grad):
# type: (Tensor, Tensor, Tensor) -> Tensor
"""
return preconditioned gradient using (normalization, scaling) Kronecker product preconditioner
Suppose Grad has shape (M, N)
ql has shape (2, M)
qr has shape (1, N)
ql[0] is the diagonal part of Ql
ql[1, 0:-1] is the last column of Ql, excluding the last entry
qr is the diagonal part of Qr
Grad: (matrix) gradient
"""
preG = ql[0:1].t()*Grad + ql[1:].t().mm(Grad[-1:]) # Ql*Grad
preG *= (qr*qr) # Ql*Grad*Qr^T*Qr
add_last_row = ql[1:].mm(preG) # use it to modify the last row
preG *= ql[0:1].t()
preG[-1:] += add_last_row
return preG
###############################################################################
@torch.jit.script
def _update_precond_dense_scale(Ql, qr, dX, dG, step=0.01, _tiny=1.2e-38):
# type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]
"""
update (dense, scaling) preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql), where
dX and dG have shape (M, N)
Ql has shape (M, M)
qr has shape (1, N)
qr is the diagonal part of Qr
dX is perturbation of (matrix) parameter
dG is perturbation of (matrix) gradient
step: update step size
_tiny: an offset to avoid division by zero
"""
max_l = torch.max(torch.diag(Ql))
max_r = torch.max(qr)
rho = torch.sqrt(max_l/max_r)
Ql /= rho
qr *= rho
A = Ql.mm( dG*qr )
#Bt = torch.triangular_solve(dX/qr, Ql, upper=True, transpose=True)[0]
Bt = torch.linalg.solve_triangular(Ql.t(), dX/qr, upper=False)
grad1 = torch.triu(A.mm(A.t()) - Bt.mm(Bt.t()))
grad2 = torch.sum(A*A, dim=0, keepdim=True) - torch.sum(Bt*Bt, dim=0, keepdim=True)
# step1 = step/(torch.max(torch.abs(grad1)) + _tiny)
step1 = step/(norm_lower_bound(grad1) + _tiny)
step2 = step/(torch.max(torch.abs(grad2)) + _tiny)
return Ql - step1*grad1.mm(Ql), qr - step2*grad2*qr
@torch.jit.script
def _precond_grad_dense_scale(Ql, qr, Grad):
# type: (Tensor, Tensor, Tensor) -> Tensor
"""
return preconditioned gradient using (dense, scaling) Kronecker product preconditioner
Suppose Grad has shape (M, N)
Ql: shape (M, M), (left side) Cholesky factor of preconditioner
qr: shape (1, N), defines a diagonal matrix for output feature scaling
Grad: (matrix) gradient
"""
#return torch.chain_matmul(Ql.t(), Ql, Grad*(qr*qr))
return torch.linalg.multi_dot([Ql.t(), Ql, Grad*(qr*qr)])
###############################################################################
@torch.jit.script
def update_precond_splu(L12, l3, U12, u3, dxs, dgs, step=0.01, _tiny=1.2e-38):
# type: (Tensor,Tensor,Tensor,Tensor, List[Tensor],List[Tensor], float,float) -> Tuple[Tensor,Tensor,Tensor,Tensor]
"""
update sparse LU preconditioner P = Q^T*Q, where
Q = L*U,
L12 = [L1; L2]
U12 = [U1, U2]
L = [L1, 0; L2, diag(l3)]
U = [U1, U2; 0, diag(u3)]
l3 and u3 are column vectors
dxs: a list of random perturbation on parameters
dgs: a list of resultant perturbation on gradients
step: update step size normalized to range [0, 1]
_tiny: an offset to avoid division by zero
"""
# make sure that L and U have similar dynamic range
max_l = torch.max(torch.max(torch.diag(L12)), torch.max(l3))
max_u = torch.max(torch.max(torch.diag(U12)), torch.max(u3))
rho = torch.sqrt(max_l/max_u)
L12 /= rho
l3 /= rho
U12 *= rho
u3 *= rho
# extract the blocks
r = U12.shape[0]
L1 = L12[:r]
L2 = L12[r:]
U1 = U12[:, :r]
U2 = U12[:, r:]
dx = torch.cat([torch.reshape(x, [-1, 1]) for x in dxs]) # a tall column vector
dg = torch.cat([torch.reshape(g, [-1, 1]) for g in dgs]) # a tall column vector
# U*dg
Ug1 = U1.mm(dg[:r]) + U2.mm(dg[r:])
Ug2 = u3*dg[r:]
# Q*dg
Qg1 = L1.mm(Ug1)
Qg2 = L2.mm(Ug1) + l3*Ug2
# inv(U^T)*dx
#iUtx1 = torch.triangular_solve(dx[:r], U1, upper=True, transpose=True)[0]
iUtx1 = torch.linalg.solve_triangular(U1.t(), dx[:r], upper=False)
iUtx2 = (dx[r:] - U2.t().mm(iUtx1))/u3
# inv(Q^T)*dx
iQtx2 = iUtx2/l3
#iQtx1 = torch.triangular_solve(iUtx1 - L2.t().mm(iQtx2), L1, upper=False, transpose=True)[0]
iQtx1 = torch.linalg.solve_triangular(L1.t(), iUtx1 - L2.t().mm(iQtx2), upper=True)
# L^T*Q*dg
LtQg1 = L1.t().mm(Qg1) + L2.t().mm(Qg2)
LtQg2 = l3*Qg2
# P*dg
Pg1 = U1.t().mm(LtQg1)
Pg2 = U2.t().mm(LtQg1) + u3*LtQg2
# inv(L)*inv(Q^T)*dx
#iLiQtx1 = torch.triangular_solve(iQtx1, L1, upper=False)[0]
iLiQtx1 = torch.linalg.solve_triangular(L1, iQtx1, upper=False)
iLiQtx2 = (iQtx2 - L2.mm(iLiQtx1))/l3
# inv(P)*dx
iPx2 = iLiQtx2/u3
#iPx1 = torch.triangular_solve(iLiQtx1 - U2.mm(iPx2), U1, upper=True)[0]
iPx1 = torch.linalg.solve_triangular(U1, iLiQtx1 - U2.mm(iPx2), upper=True)
# update L
grad1 = Qg1.mm(Qg1.t()) - iQtx1.mm(iQtx1.t())
grad1 = torch.tril(grad1)
grad2 = Qg2.mm(Qg1.t()) - iQtx2.mm(iQtx1.t())
grad3 = Qg2*Qg2 - iQtx2*iQtx2
# max_abs_grad = torch.max(torch.abs(grad1))
# max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad2)))
# max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad3)))
# step0 = step/(max_abs_grad + _tiny)
step0 = step/(torch.maximum(norm_lower_bound(torch.cat([grad1, grad2], 0)), torch.max(torch.abs(grad3))) + _tiny)
newL1 = L1 - step0*grad1.mm(L1)
newL2 = L2 - step0*grad2.mm(L1) - step0*grad3*L2
newl3 = l3 - step0*grad3*l3
# update U
grad1 = Pg1.mm(dg[:r].t()) - dx[:r].mm(iPx1.t())
grad1 = torch.triu(grad1)
grad2 = Pg1.mm(dg[r:].t()) - dx[:r].mm(iPx2.t())
grad3 = Pg2*dg[r:] - dx[r:]*iPx2
# max_abs_grad = torch.max(torch.abs(grad1))
# max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad2)))
# max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad3)))
# step0 = step/(max_abs_grad + _tiny)
step0 = step/(torch.maximum(norm_lower_bound(torch.cat([grad1, grad2], 1)), torch.max(torch.abs(grad3))) + _tiny)
newU1 = U1 - U1.mm(step0*grad1)
newU2 = U2 - U1.mm(step0*grad2) - step0*grad3.t()*U2
newu3 = u3 - step0*grad3*u3
return torch.cat([newL1, newL2], dim=0), newl3, torch.cat([newU1, newU2], dim=1), newu3
@torch.jit.script
def precond_grad_splu(L12, l3, U12, u3, grads):
# type: (Tensor,Tensor,Tensor,Tensor, List[Tensor]) -> List[Tensor]
"""
return preconditioned gradient with sparse LU preconditioner
where P = Q^T*Q,
Q = L*U,
L12 = [L1; L2]
U12 = [U1, U2]
L = [L1, 0; L2, diag(l3)]
U = [U1, U2; 0, diag(u3)]
l3 and u3 are column vectors
grads: a list of gradients to be preconditioned
"""
grad = [torch.reshape(g, [-1, 1]) for g in grads] # a list of column vector
lens = [g.shape[0] for g in grad] # length of each column vector
grad = torch.cat(grad) # a tall column vector
r = U12.shape[0]
L1 = L12[:r]
L2 = L12[r:]
U1 = U12[:, :r]
U2 = U12[:, r:]
# U*g
Ug1 = U1.mm(grad[:r]) + U2.mm(grad[r:])
Ug2 = u3*grad[r:]
# Q*g
Qg1 = L1.mm(Ug1)
Qg2 = L2.mm(Ug1) + l3*Ug2
# L^T*Q*g
LtQg1 = L1.t().mm(Qg1) + L2.t().mm(Qg2)
LtQg2 = l3*Qg2
# P*g
pre_grad = torch.cat([U1.t().mm(LtQg1),
U2.t().mm(LtQg1) + u3*LtQg2])
pre_grads = [] # restore pre_grad to its original shapes
idx = 0
for i in range(len(grads)):
pre_grads.append(torch.reshape(pre_grad[idx : idx + lens[i]], grads[i].shape))
idx = idx + lens[i]
return pre_grads
##############################################################################
#
# The low-rank approximation (LRA) preconditioner is defined as
#
# Q = (I + U*V')*diag(d)
#
# which, after reparameterization, is equivalent to form
#
# diag(d) + U*V'
#
# UVd as an alias of LRA due to the form of this preconditioner.
#
#@torch.jit.script
def IpUVtmatvec(U, V, x):
# type: (Tensor, Tensor, Tensor) -> Tensor
"""
Returns (I + U*V')*x. All variables are either matrices or column vectors.
"""
return x + U.mm(V.t().mm(x))
# def IpUVtsolve(U, V, x):
# """
# Returns inv(I + U*V')*x. All variables are either matrices or column vectors.
# """
# VtU = V.t().mm(U)
# I = torch.eye(VtU.size(dim=0), dtype=VtU.dtype, device=VtU.device)
# return x - U.mm(torch.linalg.solve(I + VtU, V.t().mm(x))) # torch.solve is slow
# def norm_UVt(U, V):
# """
# Returns ||U*V'||_fro = sqrt(tr(U'*U*V'*V)) = sqrt(sum((U'*U)*(V'*V)))
# """
# return torch.sqrt(torch.abs(torch.sum( (U.t().mm(U))*(V.t().mm(V)) )))
#@torch.jit.script
def update_precond_UVd_math_(U, V, d, v, h, step, step_normalizer, tiny):
# type: (Tensor, Tensor, Tensor, Tensor, Tensor, float, str, float) -> None
"""
Update preconditioner Q = (I + U*V')*diag(d) with (vector, Hessian-vector product) = (v, h).
State variables U, V and d are updated inplace.
U, V, d, v, and h are either matrices or column vectors.
"""
# balance the numerical dynamic ranges of U and V; optional
if torch.rand([]) < 0.01:
normU = torch.linalg.vector_norm(U)
normV = torch.linalg.vector_norm(V)
rho = torch.sqrt(normU/normV)
U.div_(rho)
V.mul_(rho)
Qh = IpUVtmatvec(U, V, d*h)
Ph = d*IpUVtmatvec(V, U, Qh)
# invQtv = IpUVtsolve(V, U, v/d)
# invPv = IpUVtsolve(U, V, invQtv)/d
VtU = V.t().mm(U)
I = torch.eye(VtU.size(dim=0), dtype=VtU.dtype, device=VtU.device)
IpVtU = I + VtU
invQtv = v/d
# torch's linalg.solve is slow for small matrix
# invQtv = invQtv - V.mm(torch.linalg.solve(IpVtU.t(), U.t().mm(invQtv)))
# invPv = invQtv - U.mm(torch.linalg.solve(IpVtU, V.t().mm(invQtv)))
LU, pivots = torch.linalg.lu_factor(IpVtU)
invQtv = invQtv - V.mm(torch.linalg.lu_solve(LU, pivots, U.t().mm(invQtv), adjoint=True))
invPv = invQtv - U.mm(torch.linalg.lu_solve(LU, pivots, V.t().mm(invQtv)))
invPv = invPv/d
nablaD = Ph*h - v*invPv
if step_normalizer == '2nd':
mu = step*torch.min(torch.rsqrt(Ph*Ph + v*v + tiny)*torch.rsqrt(h*h + invPv*invPv + tiny)) # two seperate rsqrt's to avoid underflow
else:
mu = step/(torch.max(torch.abs(nablaD)) + tiny)
# d = d - mu*d*nablaD
d.sub_(mu*d*nablaD)
# update either U or V, not both at the same time
a, b = Qh, invQtv
if torch.rand([]) < 0.5:
# nablaU = Qh.mm(Qh.t().mm(V)) - invQtv.mm(invQtv.t().mm(V))
# mu = step/(norm_UVt(nablaU, V) + _tiny)
# U = U - mu*(nablaU + nablaU.mm(V.t().mm(U)))
atV = a.t().mm(V)
btV = b.t().mm(V)
atVVt = atV.mm(V.t())
btVVt = btV.mm(V.t())
if step_normalizer == '2nd':
mu = step/( torch.linalg.vector_norm(a)*torch.linalg.vector_norm(atVVt)
+torch.linalg.vector_norm(b)*torch.linalg.vector_norm(btVVt) + tiny)
else: # '1st'
norm = torch.sqrt(torch.abs( (a.t().mm(a))*(atVVt.mm(atVVt.t())) # abs to avoid sqrt(-0.0)
+(b.t().mm(b))*(btVVt.mm(btVVt.t()))
-2*(a.t().mm(b))*(atVVt.mm(btVVt.t())) ))
mu = step/(norm + tiny)
# U = U - mu*( a.mm(atV.mm(IpVtU))
# -b.mm(btV.mm(IpVtU)) )
U.sub_(mu*( a.mm(atV.mm(IpVtU))
-b.mm(btV.mm(IpVtU)) ))
else:
# nablaV = Qh.mm(Qh.t().mm(U)) - invQtv.mm(invQtv.t().mm(U))
# mu = step/(norm_UVt(U, nablaV) + _tiny)
# V = V - mu*(nablaV + V.mm(U.t().mm(nablaV)))
atU = a.t().mm(U)
btU = b.t().mm(U)
UUta = U.mm(atU.t())
UUtb = U.mm(btU.t())
if step_normalizer == '2nd':
mu = step/( torch.linalg.vector_norm(a)*torch.linalg.vector_norm(UUta)
+torch.linalg.vector_norm(b)*torch.linalg.vector_norm(UUtb) + tiny)
else: # '1st'
norm = torch.sqrt(torch.abs( (UUta.t().mm(UUta))*(a.t().mm(a)) # abs to avoid sqrt(-0.0)
+(UUtb.t().mm(UUtb))*(b.t().mm(b))
-2*(UUta.t().mm(UUtb))*(a.t().mm(b)) ))
mu = step/(norm + tiny)
# V = V - mu*( (a + V.mm(atU.t())).mm(atU)
# -(b + V.mm(btU.t())).mm(btU) )
V.sub_(mu*( (a + V.mm(atU.t())).mm(atU)
-(b + V.mm(btU.t())).mm(btU) ))
# return [U, V, d]
#@torch.jit.script
def precond_grad_UVd_math(U, V, d, g):
# type: (Tensor, Tensor, Tensor, Tensor) -> Tensor
"""
Preconditioning gradient g with Q = (I + U*V')*diag(d).
All variables here are either matrices or column vectors.
"""
g = IpUVtmatvec(U, V, d*g)
g = d*IpUVtmatvec(V, U, g)
return g
class LRA:
"""
Implements the low-rank approximation (LRA, UVd as an alias) preconditioner, Q = (I + U*V')*diag(d), as a class.
Args for initialization:
params_with_grad: a list of parameters or variables requiring gradients;
rank_of_approximation: rank of approximation, i.e., max rank of U or V, with 0 for diagonal preconditioner;
preconditioner_init_scale: initial scale of Q, or roughly, Q = preconditioner_init_scale*eye(), with None for automatical setting;
lr_params: normalized learning rate for parameters in range [0, 1];
lr_preconditioner: normalized learning rate for preconditioner in range [0, 1];
momentum: momentum factor in range [0,1);
grad_clip_max_norm: maximum allowable gradient norm after clipping, None for no clipping;
preconditioner_update_probability: probability on updating Q, 1 for updating at every step, and 0 for never, i.e., SGD when Q=I;
step_normalizer: '1st' for normalizing lr_preconditioner with 1st order derivative info, and '2nd' for normalizing with 2nd derivative info;
exact_hessian_vector_product: True for exact Hessian-vector product via 2nd derivative,
and False for approximate one via the finite difference method;
preconditioner_type: "Newton" or "whitening", see https://arxiv.org/abs/1809.10232 for the Newton and (empirical) Fisher types.
Notes:
Note 1: The Hessian-vector product can be approximated using the finite difference method by setting
exact_hessian_vector_product = False when the 2nd derivatives is not available.
In this case, make sure that the closure produces the same outputs given the same inputs,
except for numerical errors due to non-deterministic behaviors.
Random numbers, if any, used inside the closure should be generated starting from the same state, where the rng state can be
read and set by, e.g., `torch.cuda.get_rng_state' and `torch.cuda.set_rng_state', respectively.
Note 2: Momentum here is the moving average of gradient so that its setting is decoupled from the learning rate.
This is necessary as the learning rate in PSGD is normalized.
Note 3: `torch.linalg.solve' is called twice in function `update_precond_UVd_math_'.
Certain solver could be orders of magnitude faster than others, especially for small matrices
(see https://drive.google.com/file/d/1CTNx1q67_py87jn-0OI-vSLcsM1K7VsM/view, Table 2).
Considering replace it with faster ones if the default solver is too slow.
Note 4: Currently, no support of sparse and mixed-precision gradients.
Half precision (bfloat16) works well except that torch.linalg.solve (v2.2) requires casting bfloat16 to float32.
Note 5: lr_params, lr_preconditioner, momentum, grad_clip_max_norm, preconditioner_update_probability, step_normalizer,
and exact_hessian_vector_product (bool) all can be reset on the fly.
"""
def __init__(self, params_with_grad, rank_of_approximation:int=10, preconditioner_init_scale=None,
lr_params=0.01, lr_preconditioner=None, momentum=0.0,
grad_clip_max_norm=None, preconditioner_update_probability=1.0,
step_normalizer='2nd',
exact_hessian_vector_product:bool=True, preconditioner_type="Newton"):
# mutable members
self.lr_params = lr_params
if lr_preconditioner is None:
if step_normalizer == '2nd':
self.lr_preconditioner = 0.1
else:
self.lr_preconditioner = 0.01
else:
self.lr_preconditioner = lr_preconditioner
self.momentum = momentum if (0<momentum<1) else 0.0
self.grad_clip_max_norm = grad_clip_max_norm
self.preconditioner_update_probability = preconditioner_update_probability
self.exact_hessian_vector_product = exact_hessian_vector_product
self.step_normalizer = step_normalizer
# protected members
params_with_grad = [params_with_grad,] if isinstance(params_with_grad, torch.Tensor) else params_with_grad
self._params_with_grad = [param for param in params_with_grad if param.requires_grad] # double check requires_grad flag
dtype, device = self._params_with_grad[0].dtype, self._params_with_grad[0].device
self._tiny = torch.finfo(dtype).tiny
self._delta_param_scale = torch.finfo(dtype).eps**0.5
self._param_sizes = [torch.numel(param) for param in self._params_with_grad]
self._param_cumsizes = torch.cumsum(torch.tensor(self._param_sizes), 0)
num_params = self._param_cumsizes[-1]
# check rank_of_approximation
if rank_of_approximation <= 0:
print("Hint: the Xmat preconditioner may be more efficinet in this case.")
if 2*rank_of_approximation + 1 >= num_params:
print("Hint: the Newton preconditioner may be more efficient in this case.")
# +10 to 1) avoid /0; 2) make sure that norm(U*V') << 1 even when rank_of_approximation=1
self._U = torch.randn(num_params, rank_of_approximation, dtype=dtype, device=device) / (num_params*(rank_of_approximation+10))**0.5
self._V = torch.randn(num_params, rank_of_approximation, dtype=dtype, device=device) / (num_params*(rank_of_approximation+10))**0.5
if preconditioner_init_scale is None:
self._d = None # set it on the fly
else:
self._d = torch.ones(num_params, 1, dtype=dtype, device=device) * preconditioner_init_scale
self._m = None # momentum buffer
self._preconditioner_type = preconditioner_type
@torch.no_grad()
def step(self, closure):
"""
Performs a single step of PSGD with the low-rank approximation (LRA, or UVd) preconditioner, i.e.,
updating the trainable parameters once, and returning what closure returns.
Args:
closure (callable): a (stateless) closure that evaluates the function of self._params_with_grad,
and returns the loss, or an iterable with the first one being loss.
Random numbers, if any, used inside the closure should be generated starting
from the same rng state if exact_hessian_vector_product=False and preconditioner_type="Newton".
"""
if (self._preconditioner_type=="Newton") and ((torch.rand([]) < self.preconditioner_update_probability) or (self._d is None)):
# evaluates gradients, Hessian-vector product, and updates the preconditioner
if self.exact_hessian_vector_product:
# exact Hessian-vector product
with torch.enable_grad():
closure_returns = closure()
loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]
grads = torch.autograd.grad(loss, self._params_with_grad, create_graph=True)
vs = [torch.randn_like(param) for param in self._params_with_grad]
Hvs = torch.autograd.grad(grads, self._params_with_grad, vs)
else:
# approximate Hessian-vector product via finite-difference formulae. Use it with cautions.
with torch.enable_grad():
closure_returns = closure()
loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]
grads = torch.autograd.grad(loss, self._params_with_grad)
vs = [self._delta_param_scale * torch.randn_like(param) for param in self._params_with_grad]
[param.add_(v) for (param, v) in zip(self._params_with_grad, vs)]
with torch.enable_grad():
perturbed_returns = closure()
perturbed_loss = perturbed_returns if isinstance(perturbed_returns, torch.Tensor) else perturbed_returns[0]
perturbed_grads = torch.autograd.grad(perturbed_loss, self._params_with_grad)
Hvs = [perturbed_g - g for (perturbed_g, g) in zip(perturbed_grads, grads)]
# update preconditioner
v = torch.cat([torch.reshape(v, [-1, 1]) for v in vs]) # column vector
h = torch.cat([torch.reshape(h, [-1, 1]) for h in Hvs]) # column vector
# set self._d if it is None
if self._d is None:
self._d = (torch.sum(v*v)/torch.sum(h*h))**0.25 * torch.ones_like(v)
# update self._U, _V and _d
update_precond_UVd_math_(self._U, self._V, self._d, v, h, self.lr_preconditioner, self.step_normalizer, self._tiny)
# if self.exact_hessian_vector_product:
# update_precond_UVd_math_(self._U, self._V, self._d,
# v[:,None], h[:,None], step=self.lr_preconditioner, tiny=self._tiny)
# else: # compensate the levels of v and h; helpful to reduce numerical errors in half-precision training
# update_precond_UVd_math_(self._U, self._V, self._d,
# v[:,None]/self._delta_param_scale, h[:,None]/self._delta_param_scale, step=self.lr_preconditioner, tiny=self._tiny)
else:
# only evaluates the gradients
with torch.enable_grad():
closure_returns = closure()
loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]
grads = torch.autograd.grad(loss, self._params_with_grad)
vs = None # no vs and Hvs
# cat grads
grad = torch.cat([torch.reshape(g, [-1, 1]) for g in grads]) # column vector
# update preconditioner here if it is the whitening type
if (self._preconditioner_type!="Newton") and ((torch.rand([]) < self.preconditioner_update_probability) or (self._d is None)):
if self._d is None:
self._d = (len(grad)/torch.sum(grad*grad))**0.25 * torch.ones_like(grad)
# update the preconditioner whitening the gradients
v = torch.randn_like(grad)
update_precond_UVd_math_(self._U, self._V, self._d, v, grad, self.lr_preconditioner, self.step_normalizer, self._tiny)
# preconditioned gradients; momentum is optional
if self.momentum > 0:
if self._m is None:
self._m = (1 - self.momentum)*grad
else:
self._m.mul_(self.momentum).add_(grad, alpha=1 - self.momentum)
pre_grad = precond_grad_UVd_math(self._U, self._V, self._d, self._m)
else:
self._m = None # clean the buffer when momentum is set to zero
pre_grad = precond_grad_UVd_math(self._U, self._V, self._d, grad)
# gradient clipping is optional
if self.grad_clip_max_norm is None:
lr = self.lr_params
else:
grad_norm = torch.linalg.vector_norm(pre_grad) + self._tiny
lr = self.lr_params * min(self.grad_clip_max_norm/grad_norm, 1.0)
# update the parameters
if self.exact_hessian_vector_product or (vs is None) or (self._preconditioner_type!="Newton"):
delta = lr * pre_grad
else: # in this case, do not forget to remove the perturbation on parameters
delta = lr * pre_grad + v
# -delta
[param.subtract_(delta[j - i:j].view_as(param))
for (param, i, j) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes)]
# return whatever closure returns
return closure_returns
# UVd as an alias
UVd = LRA
################## end of LRA/UVd preconditioner #################################
##############################################################################
# An Xmat (X-matrix) preconditioner is defined as
#
# Q = diag(a) + adiag(b)
#
# where adiag means anti-diagonal.
# It's slightly more complicated than a diagonal preconditioner (LRA with rank=0), but may perform better.
#
#@torch.jit.script
def update_precond_Xmat_math_(a, b, v, h, step, step_normalizer, tiny):
# type: (Tensor, Tensor, Tensor, Tensor, float, str, float) -> None
"""
Update preconditioner Q = diag(a) + adiag(b) with (vector, Hessian-vector product) = (v, h).
State variables a and b are updated inplace.
"""
Qh = a*h + b*torch.flip(h, [0])
aflip, bflip = torch.flip(a, [0]), torch.flip(b, [0])
invQtv = (aflip*v - bflip*torch.flip(v, [0]))/(a*aflip - b*bflip)
u, v = Qh*Qh, invQtv*invQtv
# nablaA = Qh*Qh - invQtv*invQtv
nablaA = u - v
nablaB = Qh*torch.flip(Qh, [0]) - invQtv*torch.flip(invQtv, [0])
q, r = divmod(len(nablaB), 2)
if r == 1:
nablaB[q] = 0
if step_normalizer == '2nd':
mu = step/(torch.max(u + v) + tiny)
else:
mu = step/(torch.maximum(torch.max(torch.abs(nablaA)), torch.max(torch.abs(nablaB))) + tiny)
a.sub_(mu*(nablaA*a + nablaB*bflip))
b.sub_(mu*(nablaA*b + nablaB*aflip))
#@torch.jit.script
def precond_grad_Xmat_math(a, b, g):
# type: (Tensor, Tensor, Tensor) -> Tensor
"""
Preconditioning gradient g with Q = diag(a) + adiag(b).
"""
ab = a * b
return (a*a + torch.flip(b*b, [0]))*g + (ab + torch.flip(ab, [0]))*torch.flip(g, [0])
class XMat:
"""
Implements the Xmat preconditioner, Q = diag(a) + adiag(b), as a class.
Args for initialization:
params_with_grad: a list of parameters or variables requiring gradients;
preconditioner_init_scale: initial scale of Q, i.e., Q = preconditioner_init_scale*eye(), with None for automatical setting;
lr_params: normalized learning rate for parameters in range [0, 1];
lr_preconditioner: normalized learning rate for preconditioner in range [0, 1];
momentum: momentum factor in range [0,1);
grad_clip_max_norm: maximum allowable gradient norm after clipping, None for no clipping;
preconditioner_update_probability: probability on updating Q, 1 for updating at every step, and 0 for never, i.e., SGD when Q=I;
step_normalizer: '1st' for normalizing lr_preconditioner with 1st order derivative info, and '2nd' for normalizing with 2nd derivative info;
exact_hessian_vector_product: True for exact Hessian-vector product via 2nd derivative,
and False for approximate one via the finite difference method;
preconditioner_type: "Newton" or "whitening", see https://arxiv.org/abs/1809.10232 for the Newton and (empirical) Fisher types.
Notes:
Note 1: The Hessian-vector product can be approximated using the finite difference method by setting
exact_hessian_vector_product = False when the 2nd derivatives is not available.
In this case, make sure that the closure produces the same outputs given the same inputs,
except for numerical errors due to non-deterministic behaviors.
Random numbers, if any, used inside the closure should be generated starting from the same states, where the rng state can be
read and set by, e.g., `torch.cuda.get_rng_state' and `torch.cuda.set_rng_state', respectively.
Note 2: Momentum here is the moving average of gradient so that its setting is decoupled from the learning rate.
This is necessary as the learning rate in PSGD is normalized.
Note 3: Currently, no support of sparse and mixed-precision gradients.
Note 4: lr_params, lr_preconditioner, momentum, grad_clip_max_norm, preconditioner_update_probability, step_normalizer,
and exact_hessian_vector_product (bool) all can be reset on the fly.
"""
def __init__(self, params_with_grad, preconditioner_init_scale=None,
lr_params=0.01, lr_preconditioner=None, momentum=0.0,
grad_clip_max_norm=None, preconditioner_update_probability=1.0,
step_normalizer='2nd',
exact_hessian_vector_product: bool = True, preconditioner_type="Newton"):
# mutable members
self.lr_params = lr_params
if lr_preconditioner is None: