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bound_crown_quad.py
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bound_crown_quad.py
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## bound_crown_quad.py
##
## Implementation of CROWN-quad bound
##
## Copyright (C) 2018, Huan Zhang <[email protected]> and contributors
##
## This program is licenced under the BSD 2-Clause License,
## contained in the LICENCE file in this directory.
## See CREDITS for a list of contributors.
##
import numpy as np
from numba import jit
import activation_functions_quad as quad_fit
@jit(nopython=True)
def proj_li(x, i_lb, i_ub):
return np.minimum(np.maximum(x, i_lb), i_ub)
@jit(nopython=True)
def proj_l2(x, x0, eps):
delta = x - x0
norm = np.linalg.norm(delta)
if norm > eps:
delta /= (norm / eps)
return x0 + delta
@jit(nopython=True)
def proj_l1(x_real, x0, q):
x = x_real - x0
# sort the array in descending order
s = np.sort(np.abs(x))[::-1]
cs = np.cumsum(s)
r = np.arange(1, len(s)+1)
s2 = np.empty_like(s)
s2[:-1] = s[1:]
s2[-1] = 0
t = cs - r * s2
ndxs = np.nonzero(t >= q + 1e-8)[0]
if len(ndxs):
ndx = ndxs[0]
thresh = np.float32((cs[ndx] - q) / (ndx+1))
x = x * (np.float32(1) - thresh / np.maximum( np.abs(x), thresh))
return x + x0
# quadratic bounds, two layers only
# currently we only use the bounds of the first two layers
@jit(nopython=True)
def crown_quad_bound(Ws,bs,UBs,LBs,neuron_state,nlayer,x0,eps,p_np,verbose=False):
assert nlayer >= 2
assert nlayer == len(Ws) == len(bs) == len(UBs) == len(LBs) == (len(neuron_state) + 1)
# for 2-layer, we can use other norms (for input perturbation x)
assert p_np == np.inf or nlayer == 2
x0 = x0.astype(np.float32)
# we only consider the last two layers
W2 = Ws[nlayer-1]
W1 = Ws[nlayer-2]
b2 = bs[nlayer-1]
b1 = bs[nlayer-2]
# upper and lower bounds of hidden layer
h_ub = UBs[nlayer-1]
h_lb = LBs[nlayer-1]
# upper and lower bounds of input layer
if nlayer == 2:
i_ub = UBs[nlayer-2]
i_lb = LBs[nlayer-2]
else:
# UBs and LBs are pre-relu values, apply ReLU here
i_ub = np.maximum(UBs[nlayer-2], 0)
i_lb = np.maximum(LBs[nlayer-2], 0)
UB_final = np.empty_like(b2)
LB_final = np.empty_like(b2)
# indices for unsure neurons
idx_unsure = np.nonzero(neuron_state[nlayer - 2] == 0)[0]
idx_active = np.nonzero(neuron_state[nlayer - 2] == 1)[0]
# form linear term slope
alpha = neuron_state[nlayer - 2].astype(np.float32)
np.maximum(alpha, 0, alpha)
alpha[idx_unsure] = UBs[nlayer-1][idx_unsure]/(UBs[nlayer-1][idx_unsure] - LBs[nlayer-1][idx_unsure])
# quadratic bound parameters
a = np.empty_like(b1)
b = np.empty_like(b1)
# fit best quadratic lower bound
for r in idx_unsure:
a[r], b[r], _ = quad_fit.get_best_lower_quad(h_ub[r], h_lb[r])
for j in range(len(b2)):
# positive entries in j-th row, unsure neurons
pos = np.nonzero(W2[j][idx_unsure] > 0)[0]
# negative entries in j-th row, unsure neurons
neg = np.nonzero(W2[j][idx_unsure] < 0)[0]
# unsure neurons, corresponding to positive entries in W2
idx_unsure_pos = idx_unsure[pos]
# unsure neurons, corresponding to negative entries in W2
idx_unsure_neg = idx_unsure[neg]
# form the diag matrix for quadratic term
diag_ub = np.zeros_like(b1)
diag_lb = np.zeros_like(b1)
# all diagnal elements are guaranteed negative
diag_ub[idx_unsure_neg] = W2[j][idx_unsure_neg] * a[idx_unsure_neg]
diag_lb[idx_unsure_pos] = W2[j][idx_unsure_pos] * a[idx_unsure_pos]
# form the vector for linear term
lamb_ub = np.zeros_like(b1)
lamb_lb = np.zeros_like(b1)
lamb_ub[idx_active] = W2[j][idx_active]
lamb_ub[idx_unsure_pos] = W2[j][idx_unsure_pos]*alpha[idx_unsure_pos]
lamb_ub[idx_unsure_neg] = W2[j][idx_unsure_neg]*b[idx_unsure_neg]
lamb_lb[idx_active] = W2[j][idx_active]
lamb_lb[idx_unsure_neg] = W2[j][idx_unsure_neg]*alpha[idx_unsure_neg]
lamb_lb[idx_unsure_pos] = W2[j][idx_unsure_pos]*b[idx_unsure_pos]
# form the constant term
const_ub = -np.dot(W2[j][idx_unsure_pos] * alpha[idx_unsure_pos], h_lb[idx_unsure_pos])
const_lb = -np.dot(W2[j][idx_unsure_neg] * alpha[idx_unsure_neg], h_lb[idx_unsure_neg])
const_ub += b2[j]
const_lb += b2[j]
# expand to previous layer
# add more constant terms
const_ub += np.dot(b1, diag_ub * b1) + np.dot(lamb_ub, b1)
const_lb += np.dot(b1, diag_lb * b1) + np.dot(lamb_lb, b1)
# quadratic term
# Q_ub = np.dot(W1.T * diag_ub, W1)
# Q_lb = np.dot(W1.T * diag_lb, W1)
# form only the non-zero part
diag_ub_part = diag_ub[idx_unsure_neg[a[idx_unsure_neg] != 0]]
Q_half_ub = (W1[idx_unsure_neg[a[idx_unsure_neg] != 0]].T * np.sqrt(-diag_ub_part)).T
diag_lb_part = diag_lb[idx_unsure_pos[a[idx_unsure_pos] != 0]]
Q_half_lb = (W1[idx_unsure_pos[a[idx_unsure_pos] != 0]].T * np.sqrt(diag_lb_part)).T
# linear term
B_ub = 2 * np.dot(W1.T * diag_ub, b1) + np.dot(W1.T, lamb_ub)
B_lb = 2 * np.dot(W1.T * diag_lb, b1) + np.dot(W1.T, lamb_lb)
# print("number of components: ub, lb", np.sum(a[idx_unsure_neg] != 0), np.sum(a[idx_unsure_pos] != 0))
# print("Q.shape, B.shape:", Q_ub.shape, B_ub.shape)
# print("Q_half.shape, diag_part.shape", Q_half_ub.shape, diag_ub_part.shape)
# w, v = np.linalg.eig(Q_ub)
# print(w[:20])
# print("const_ub, const_lb:", const_ub, const_lb)
x = (i_lb + i_ub)
x /= 2.0
f_qp_lb = lambda x : np.dot(x, np.dot(Q_half_lb.T, np.dot(Q_half_lb, x))) + np.dot(B_lb, x) + const_lb
# f_qp_lb = lambda x : np.dot(x, np.dot(Q_lb, x)) + np.dot(B_lb, x) + const_lb
# obj = np.dot(x, np.dot(Q_lb, x)) + np.dot(B_lb, x) + const_lb
grad_lb = lambda x : np.dot(Q_half_lb.T, np.dot(Q_half_lb, x)) + B_lb
# grad_lb = lambda x : np.dot(Q_lb, x) + B_lb
# grad = np.dot(Q_lb, x) + B_lb
# proj = lambda x : np.minimum(np.maximum(x, i_lb), i_ub)
# for L1 norm, use a larger eta
eta_init = np.float32(1.0)
if p_np == 1:
eta_init = np.float32(10.0)
q_np = 1
if p_np == 1:
q_np = np.inf
if p_np == 2:
q_np = 2
# no quadratic term, no need to run
if len(Q_half_lb) == 0:
LB_final[j] = -np.linalg.norm(B_lb, q_np) * eps + np.dot(B_lb, x) + const_lb
converged_lb = True
else:
last_obj = np.inf
eta = eta_init
obj = f_qp_lb(x)
converged_lb = False
for i in range(200):
if abs(last_obj - obj) < 1e-4:
if verbose:
print("solver terminated, obj = ", obj)
converged_lb = True
break
cur_grad = grad_lb(x)
if p_np == np.inf:
proj_grad = proj_li(x - cur_grad, i_lb, i_ub)
elif p_np == 2:
proj_grad = proj_l2(x - cur_grad, x0, eps)
elif p_np == 1:
proj_grad = proj_l1(x - cur_grad, x0, eps)
proj_grad -= x
grad_norm = np.linalg.norm(proj_grad)
eta = eta_init
# line search
while eta > 1e-5:
if p_np == np.inf:
new_grad = proj_li(x - eta * cur_grad, i_lb, i_ub)
elif p_np == 2:
new_grad = proj_l2(x - eta * cur_grad, x0, eps)
elif p_np == 1:
new_grad = proj_l1(x - eta * cur_grad, x0, eps)
new_grad_norm = np.dot(new_grad - x, proj_grad)
if not (f_qp_lb(new_grad) > obj - np.float32(0.1) * eta * new_grad_norm):
break
eta *= np.float32(0.5)
x -= eta * cur_grad
if p_np == np.inf:
x = proj_li(x, i_lb, i_ub)
elif p_np == 2:
x = proj_l2(x, x0, eps)
elif p_np == 1:
x = proj_l1(x, x0, eps)
last_obj = obj
obj = f_qp_lb(x)
if verbose:
print("iter", i, "eta", eta, "obj", obj, "norm", np.linalg.norm(x - x0, p_np))
LB_final[j] = obj
x = (i_lb + i_ub)
x /= 2.0
f_qp_ub = lambda x : -np.dot(x, np.dot(Q_half_ub.T, np.dot(Q_half_ub, x))) + np.dot(B_ub, x) + const_ub
# obj = np.dot(x, np.dot(Q_ub, x)) + np.dot(B_ub, x) + const_ub
grad_ub = lambda x : -np.dot(Q_half_ub.T, np.dot(Q_half_ub, x)) + B_ub
# grad = np.dot(Q_ub, x) + B_ub
# no quadratic term, no need to run
if len(Q_half_ub) == 0:
UB_final[j] = np.linalg.norm(B_ub, q_np) * eps + np.dot(B_ub, x) + const_ub
converged_ub = True
else:
last_obj = -np.inf
eta = eta_init
obj = f_qp_ub(x)
converged_ub = True
for i in range(200):
if abs(last_obj - obj) < 1e-4:
if verbose:
print("solver terminated, obj = ", obj)
converged_ub = True
break
cur_grad = grad_ub(x)
if p_np == np.inf:
proj_grad = proj_li(x - cur_grad, i_lb, i_ub)
elif p_np == 2:
proj_grad = proj_l2(x - cur_grad, x0, eps)
elif p_np == 1:
proj_grad = proj_l1(x - cur_grad, x0, eps)
proj_grad -= x
grad_norm = np.linalg.norm(proj_grad)
eta = eta_init
# line search
while eta > 1e-5:
if p_np == np.inf:
new_grad = proj_li(x + eta * cur_grad, i_lb, i_ub)
elif p_np == 2:
new_grad = proj_l2(x + eta * cur_grad, x0, eps)
elif p_np == 1:
new_grad = proj_l1(x + eta * cur_grad, x0, eps)
new_grad_norm = np.dot(new_grad - x, proj_grad)
if not (f_qp_ub(new_grad) < obj + np.float32(0.1) * eta * new_grad_norm):
break
eta *= np.float32(0.5)
x += eta * cur_grad
if p_np == np.inf:
x = proj_li(x, i_lb, i_ub)
elif p_np == 2:
x = proj_l2(x, x0, eps)
elif p_np == 1:
x = proj_l1(x, x0, eps)
last_obj = obj
obj = f_qp_ub(x)
if verbose:
x = proj_l1(x, x0, eps)
print("iter", i, "eta", eta, "obj", obj, "norm", np.linalg.norm(x - x0, p_np))
UB_final[j] = obj
print(j+1, "/", len(b2), "solved, size", len(diag_ub_part), len(diag_lb_part), ", converged", converged_lb and converged_ub)
return UB_final, LB_final
# quadratic bounds, two layers only
# currently we only use the bounds of the last two layers
@jit(nopython=True)
def get_layer_bound_quad_both(Ws,bs,UBs,LBs,neuron_state,nlayer,x0,eps,p_n):
assert nlayer >= 2
assert nlayer == len(Ws) == len(bs) == len(UBs) == len(LBs) == (len(neuron_state) + 1)
# we only consider the last two layers
W2 = Ws[nlayer-1]
W1 = Ws[nlayer-2]
b2 = bs[nlayer-1]
b1 = bs[nlayer-2]
# upper and lower bounds of hidden layer
h_ub = UBs[nlayer-1]
h_lb = LBs[nlayer-1]
# upper and lower bounds of input layer
if nlayer == 2:
i_ub = UBs[nlayer-2]
i_lb = LBs[nlayer-2]
else:
# UBs and LBs are pre-relu values, apply ReLU here
i_ub = np.maximum(UBs[nlayer-2], 0)
i_lb = np.maximum(LBs[nlayer-2], 0)
UB_final = np.empty_like(b2)
LB_final = np.empty_like(b2)
# indices for unsure neurons
idx_unsure = np.nonzero(neuron_state[nlayer - 2] == 0)[0]
idx_active = np.nonzero(neuron_state[nlayer - 2] == 1)[0]
# form linear term slope
alpha = neuron_state[nlayer - 2].astype(np.float32)
np.maximum(alpha, 0, alpha)
alpha[idx_unsure] = UBs[nlayer-1][idx_unsure]/(UBs[nlayer-1][idx_unsure] - LBs[nlayer-1][idx_unsure])
# quadratic bound parameters
a_low = np.empty_like(b1)
b_low = np.empty_like(b1)
a_high = np.empty_like(b1)
b_high = np.empty_like(b1)
c_high = np.empty_like(b1)
# fit best quadratic lower bound
for r in idx_unsure:
a_r, b_r, c_r = quad_fit.get_best_lower_quad(h_ub[r], h_lb[r])
a_low[r] = a_r
b_low[r] = b_r
a_r, b_r, c_r = quad_fit.get_best_upper_quad(h_ub[r], h_lb[r])
a_high[r] = a_r
b_high[r] = b_r
c_high[r] = c_r
for j in range(len(b2)):
# positive entries in j-th row, unsure neurons
pos = np.nonzero(W2[j][idx_unsure] > 0)[0]
# negative entries in j-th row, unsure neurons
neg = np.nonzero(W2[j][idx_unsure] < 0)[0]
# unsure neurons, corresponding to positive entries in W2
idx_unsure_pos = idx_unsure[pos]
# unsure neurons, corresponding to negative entries in W2
idx_unsure_neg = idx_unsure[neg]
# form the diag matrix for quadratic term
diag_ub = np.zeros_like(b1)
diag_lb = np.zeros_like(b1)
# all diagnal elements are guaranteed negative
diag_ub[idx_unsure_neg] = W2[j][idx_unsure_neg] * a_low[idx_unsure_neg]
diag_ub[idx_unsure_pos] = W2[j][idx_unsure_pos] * a_high[idx_unsure_pos]
diag_lb[idx_unsure_pos] = W2[j][idx_unsure_pos] * a_low[idx_unsure_pos]
diag_lb[idx_unsure_neg] = W2[j][idx_unsure_neg] * a_high[idx_unsure_neg]
# form the vector for linear term
lamb_ub = np.zeros_like(b1)
lamb_lb = np.zeros_like(b1)
lamb_ub[idx_active] = W2[j][idx_active]
lamb_ub[idx_unsure_pos] = W2[j][idx_unsure_pos]*b_high[idx_unsure_pos]
lamb_ub[idx_unsure_neg] = W2[j][idx_unsure_neg]*b_low[idx_unsure_neg]
lamb_lb[idx_active] = W2[j][idx_active]
lamb_lb[idx_unsure_neg] = W2[j][idx_unsure_neg]*b_high[idx_unsure_neg]
lamb_lb[idx_unsure_pos] = W2[j][idx_unsure_pos]*b_low[idx_unsure_pos]
# form the constant term
const_ub = np.dot(W2[j][idx_unsure_pos], c_high[idx_unsure_pos])
const_lb = np.dot(W2[j][idx_unsure_neg], c_high[idx_unsure_neg])
const_ub += b2[j]
const_lb += b2[j]
# expand to previous layer
# add more constant terms
const_ub += np.dot(b1, diag_ub * b1) + np.dot(lamb_ub, b1)
const_lb += np.dot(b1, diag_lb * b1) + np.dot(lamb_lb, b1)
# quadratic term
Q_ub = np.dot(W1.T * diag_ub, W1)
Q_lb = np.dot(W1.T * diag_lb, W1)
# form only the non-zero part
diag_ub_part = diag_ub[idx_unsure]
Q_half_ub = W1[idx_unsure]
diag_lb_part = diag_lb[idx_unsure]
Q_half_lb = W1[idx_unsure]
# linear term
B_ub = 2 * np.dot(W1.T * diag_ub, b1) + np.dot(W1.T, lamb_ub)
B_lb = 2 * np.dot(W1.T * diag_lb, b1) + np.dot(W1.T, lamb_lb)
# print("number of components: ub, lb", np.sum(a[idx_unsure_neg] != 0), np.sum(a[idx_unsure_pos] != 0))
# print("Q.shape, B.shape:", Q_ub.shape, B_ub.shape)
# print("Q_half.shape, diag_part.shape", Q_half_ub.shape, diag_ub_part.shape)
# w, v = np.linalg.eig(Q_ub)
# print(w[:20])
# print("const_ub, const_lb:", const_ub, const_lb)
x = (i_lb + i_ub)
x /= 2.0
# x = i_lb
f_qp_lb = lambda x : np.dot(x, np.dot(Q_half_lb.T * diag_lb_part, np.dot(Q_half_lb, x))) + np.dot(B_lb, x) + const_lb
# obj = np.dot(x, np.dot(Q_lb, x)) + np.dot(B_lb, x) + const_lb
grad_lb = lambda x : np.dot(Q_half_lb.T * diag_lb_part, np.dot(Q_half_lb, x)) + B_lb
# grad = np.dot(Q_lb, x) + B_lb
proj = lambda x : np.minimum(np.maximum(x, i_lb), i_ub)
last_obj = np.inf
eta = np.float32(1.0)
obj = f_qp_lb(x)
for i in range(200):
if abs(last_obj - obj) < 1e-4:
print("solver terminated, obj = ", obj)
break
cur_grad = grad_lb(x)
grad_norm = np.linalg.norm(cur_grad)
eta = np.float32(1.0)
# line search
while f_qp_lb(proj(x - eta * cur_grad)) > obj - np.float32(0.1) * eta * grad_norm:
eta *= np.float32(0.5)
x -= eta * cur_grad
x = proj(x)
last_obj = obj
obj = f_qp_lb(x)
print("iter ", i, " eta ", eta, " obj ", obj)
LB_final[j] = obj
x = (i_lb + i_ub)
x /= 2.0
# x = i_lb
f_qp_ub = lambda x : -np.dot(x, np.dot(Q_half_ub.T * diag_ub_part, np.dot(Q_half_ub, x))) + np.dot(B_ub, x) + const_ub
# obj = np.dot(x, np.dot(Q_ub, x)) + np.dot(B_ub, x) + const_ub
grad_ub = lambda x : -np.dot(Q_half_ub.T * diag_ub_part, np.dot(Q_half_ub, x)) + B_ub
# grad = np.dot(Q_ub, x) + B_ub
last_obj = -np.inf
eta = np.float32(1.0)
obj = f_qp_ub(x)
for i in range(200):
if abs(last_obj - obj) < 1e-4:
print("solver terminated, obj = ", obj)
break
cur_grad = grad_ub(x)
grad_norm = np.linalg.norm(cur_grad)
eta = np.float32(1.0)
# line search
while f_qp_ub(proj(x + eta * cur_grad)) < obj + np.float32(0.1) * eta * grad_norm:
eta *= np.float32(0.5)
x += eta * cur_grad
x = proj(x)
last_obj = obj
obj = f_qp_ub(x)
print("iter ", i, " eta ", eta, " obj ", obj)
UB_final[j] = obj
return UB_final, LB_final