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from torch.autograd import Function
@torch.no_grad()
def alt_qp_solve_batched(
P, q, A, b, G, h,
rho=1.0, eps=1e-6, max_iter=5000, reg=1e-7
):
device, dtype = q.device, q.dtype
B, n = q.shape
rho_t = torch.as_tensor(rho, device=device, dtype=dtype)
m = A.shape[1] # A: (B,m,n)
p = G.shape[1] # G: (B,p,n)
x = torch.zeros((B, n), device=device, dtype=dtype)
s = torch.zeros((B, p), device=device, dtype=dtype)
lam = torch.zeros((B, m), device=device, dtype=dtype)
nu = torch.zeros((B, p), device=device, dtype=dtype)
I = torch.eye(n, device=device, dtype=dtype).expand(B, n, n)
if m > 0:
AtA = A.transpose(-1, -2) @ A
Atb = (A.transpose(-1, -2) @ b.unsqueeze(-1)).squeeze(-1)
else:
AtA = torch.zeros((B, n, n), device=device, dtype=dtype)
Atb = torch.zeros((B, n), device=device, dtype=dtype)
if p > 0:
GtG = G.transpose(-1, -2) @ G
else:
GtG = torch.zeros((B, n, n), device=device, dtype=dtype)
H = P + rho_t * (AtA + GtG) + reg * I
L = torch.linalg.cholesky(H)
def solve_H(rhs_2d): # (B,n)
return torch.cholesky_solve(rhs_2d.unsqueeze(-1), L).squeeze(-1)
alive = torch.ones((B,), device=device, dtype=torch.bool)
for _ in range(max_iter):
if not alive.any():
break
x0 = x
s0 = s
rhs = q.clone()
if m > 0:
rhs = rhs + (A.transpose(-1, -2) @ lam.unsqueeze(-1)).squeeze(-1) - rho_t * Atb
if p > 0:
rhs = rhs + (G.transpose(-1, -2) @ nu.unsqueeze(-1)).squeeze(-1)
rhs = rhs + rho_t * (G.transpose(-1, -2) @ (s - h).unsqueeze(-1)).squeeze(-1)
x = solve_H(-rhs)
if m > 0:
ax = (A @ x.unsqueeze(-1)).squeeze(-1) # (B,m)
lam = lam + rho_t * (ax - b)
else:
ax = None
if p > 0:
gx = (G @ x.unsqueeze(-1)).squeeze(-1) # (B,p)
s = torch.relu(-(nu / rho_t) - (gx - h))
nu = nu + rho_t * (gx + s - h)
else:
gx = None
dx = torch.linalg.norm(x - x0, dim=-1)
tol_x = eps * (1.0 + torch.linalg.norm(x0, dim=-1))
conv_x = dx <= tol_x
conv_pri = torch.ones((B,), device=device, dtype=torch.bool)
if m > 0:
r_eq = torch.linalg.norm(ax - b, dim=-1)
axn = torch.linalg.norm(ax, dim=-1)
bn = torch.linalg.norm(b, dim=-1)
tol_eq = eps * (1.0 + torch.maximum(axn, bn))
conv_pri = conv_pri & (r_eq <= tol_eq)
if p > 0:
r_in = torch.linalg.norm(gx + s - h, dim=-1)
gsn = torch.linalg.norm(gx + s, dim=-1)
hn = torch.linalg.norm(h, dim=-1)
tol_in = eps * (1.0 + torch.maximum(gsn, hn))
conv_pri = conv_pri & (r_in <= tol_in)
ds = torch.linalg.norm(s - s0, dim=-1)
tol_s = eps * (1.0 + torch.linalg.norm(s0, dim=-1))
conv_s = ds <= tol_s
else:
conv_s = torch.ones((B,), device=device, dtype=torch.bool)
conv = conv_x & conv_pri & conv_s
alive = alive & (~conv)
return x, lam, nu, s
def _adjoint_and_param_grads_single(
P, q, A, b, G, h, x, lam, nu, active_mask, g, reg=1e-9, sym_P=True
):
device, dtype = g.device, g.dtype
n = P.shape[0]
m = A.shape[0]
p = G.shape[0]
if p == 0 or active_mask.numel() == 0 or active_mask.sum() == 0:
Ga = G.new_zeros((0, n))
nu_a = nu.new_zeros((0,))
ha = h.new_zeros((0,))
else:
Ga = G[active_mask]
nu_a = nu[active_mask]
ha = h[active_mask]
ka = Ga.shape[0]
K = torch.zeros((n + m + ka, n + m + ka), device=device, dtype=dtype)
K[:n, :n] = P + reg * torch.eye(n, device=device, dtype=dtype)
if m > 0:
K[:n, n:n+m] = A.T
K[n:n+m, :n] = A
if ka > 0:
K[:n, n+m:] = Ga.T
K[n+m:, :n] = Ga
rhs = torch.zeros((n + m + ka,), device=device, dtype=dtype)
rhs[:n] = g
sol = torch.linalg.solve(K, rhs)
u = sol[:n]
v = sol[n:n+m] if m > 0 else sol.new_zeros((0,))
w = sol[n+m:] if ka > 0 else sol.new_zeros((0,))
grad_q = -u
grad_P = -torch.outer(u, x)
if sym_P:
grad_P = 0.5 * (grad_P + grad_P.T)
if m > 0:
grad_b = v
grad_A = -(torch.outer(v, x) + torch.outer(lam, u))
else:
grad_b = None
grad_A = None
if p > 0:
grad_h = h.new_zeros((p,))
grad_G = G.new_zeros((p, n))
if ka > 0:
grad_h[active_mask] = w
grad_Ga = -(torch.outer(w, x) + torch.outer(nu_a, u))
grad_G[active_mask] = grad_Ga
else:
grad_h = None
grad_G = None
return grad_P, grad_q, grad_G, grad_h, grad_A, grad_b
def AltDiffLayer(
eps=1e-5,
max_iter=2500,
rho=1.0,
reg_fwd=1e-5,
active_tol=1e-6,
reg_bwd=1e-5,
sym_P=True,
):
class L(Function):
@staticmethod
def forward(ctx, P, q, G=None, h=None, A=None, b=None):
unbatched = (q.dim() == 1)
P_in, q_in, A_in, b_in, G_in, h_in = P, q, A, b, G, h
def ensure_batch(x, want_dim):
if x is None:
return None
return x.unsqueeze(0) if x.dim() == want_dim else x
P = ensure_batch(P, 2)
q = ensure_batch(q, 1)
B, n = q.shape
A = ensure_batch(A, 2)
b = ensure_batch(b, 1)
G = ensure_batch(G, 2)
h = ensure_batch(h, 1)
if A is None or A.numel() == 0:
A = torch.empty((1, 0, n), device=q.device, dtype=q.dtype)
if b is None or b.numel() == 0:
b = torch.empty((1, 0), device=q.device, dtype=q.dtype)
if G is None or G.numel() == 0:
G = torch.empty((1, 0, n), device=q.device, dtype=q.dtype)
if h is None or h.numel() == 0:
h = torch.empty((1, 0), device=q.device, dtype=q.dtype)
P_shared = (P.shape[0] == 1 and B > 1)
A_shared = (A.shape[0] == 1 and B > 1)
b_shared = (b.shape[0] == 1 and B > 1)
G_shared = (G.shape[0] == 1 and B > 1)
h_shared = (h.shape[0] == 1 and B > 1)
if P_shared: P = P.expand(B, -1, -1)
if A_shared: A = A.expand(B, -1, -1)
if b_shared: b = b.expand(B, -1)
if G_shared: G = G.expand(B, -1, -1)
if h_shared: h = h.expand(B, -1)
P_d, q_d, A_d, b_d, G_d, h_d = P.detach(), q.detach(), A.detach(), b.detach(), G.detach(), h.detach()
x, lam, nu, s = alt_qp_solve_batched(
P_d, q_d, A_d, b_d, G_d, h_d,
rho=rho, eps=eps, max_iter=max_iter, reg=reg_fwd
)
if s.numel() == 0:
active = torch.zeros((B, 0), device=q.device, dtype=torch.bool)
else:
active = (s <= active_tol)
ctx.save_for_backward(P_d, q_d, A_d, b_d, G_d, h_d, x, lam, nu, active)
ctx.unbatched = unbatched
ctx.shared_flags = (P_shared, A_shared, b_shared, G_shared, h_shared)
ctx.orig_none = (P_in is None, q_in is None, G_in is None, h_in is None, A_in is None, b_in is None)
ctx.orig_shapes = (
getattr(P_in, "shape", None),
getattr(q_in, "shape", None),
getattr(G_in, "shape", None),
getattr(h_in, "shape", None),
getattr(A_in, "shape", None),
getattr(b_in, "shape", None),
)
ctx.sym_P = sym_P
return x[0] if unbatched else x
@staticmethod
def backward(ctx, grad_output):
P, q, A, b, G, h, x, lam, nu, active = ctx.saved_tensors
P_shared, A_shared, b_shared, G_shared, h_shared = ctx.shared_flags
unbatched = ctx.unbatched
sym_P = ctx.sym_P
g = grad_output.unsqueeze(0) if grad_output.dim() == 1 else grad_output
B, n = g.shape
grad_P = torch.zeros_like(P)
grad_q = torch.zeros_like(q)
grad_A = torch.zeros_like(A) if A.numel() > 0 else None
grad_b = torch.zeros_like(b) if b.numel() > 0 else None
grad_G = torch.zeros_like(G) if G.numel() > 0 else None
grad_h = torch.zeros_like(h) if h.numel() > 0 else None
for i in range(B):
act_i = active[i] if active.numel() > 0 else active.new_zeros((0,), dtype=torch.bool)
Pi = P[i]
qi = q[i]
Ai = A[i] if A.numel() > 0 else A.new_zeros((0, n))
bi = b[i] if b.numel() > 0 else b.new_zeros((0,))
Gi = G[i] if G.numel() > 0 else G.new_zeros((0, n))
hi = h[i] if h.numel() > 0 else h.new_zeros((0,))
xi = x[i]
lami = lam[i] if lam.numel() > 0 else lam.new_zeros((0,))
nui = nu[i] if nu.numel() > 0 else nu.new_zeros((0,))
dP_i, dq_i, dG_i, dh_i, dA_i, db_i = _adjoint_and_param_grads_single(
Pi, qi, Ai, bi, Gi, hi, xi, lami, nui, act_i, g[i],
reg=reg_bwd, sym_P=sym_P
)
grad_P[i] = dP_i
grad_q[i] = dq_i
if grad_A is not None and dA_i is not None:
grad_A[i] = dA_i
if grad_b is not None and db_i is not None:
grad_b[i] = db_i
if grad_G is not None and dG_i is not None:
grad_G[i] = dG_i
if grad_h is not None and dh_i is not None:
grad_h[i] = dh_i
if P_shared:
grad_P = grad_P.sum(dim=0, keepdim=True)
if A_shared and grad_A is not None:
grad_A = grad_A.sum(dim=0, keepdim=True)
if b_shared and grad_b is not None:
grad_b = grad_b.sum(dim=0, keepdim=True)
if G_shared and grad_G is not None:
grad_G = grad_G.sum(dim=0, keepdim=True)
if h_shared and grad_h is not None:
grad_h = grad_h.sum(dim=0, keepdim=True)
if unbatched:
grad_P = grad_P[0]
grad_q = grad_q[0]
if grad_A is not None: grad_A = grad_A[0]
if grad_b is not None: grad_b = grad_b[0]
if grad_G is not None: grad_G = grad_G[0]
if grad_h is not None: grad_h = grad_h[0]
return grad_P, grad_q, grad_G, grad_h, grad_A, grad_b
return L.apply
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