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bound_recurjac.py
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bound_recurjac.py
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## bound_recurjac.py
##
## Implementation of RecurJac bound
##
## RecurJac is an efficient recursive algorithm for element-wise bounding Jacobian matrix of
## neural networks with general activation functions
##
## 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 os
from numba import njit, prange
import numpy as np
import inspect
from activation_functions import *
functions_constructed = False
grad_wrapper_constructed = False
grad_relu_wrapper = """
@njit(cache=False)
def get_grad_bounds_wrapper(lb, ub):
return relu_grad_bounds(lb, ub, alpha = {})
"""
grad_sigmoid_wrapper = """
@njit(cache=False)
def get_grad_bounds_wrapper(lb, ub):
return sigmoid_grad_bounds(lb, ub, grad_f = {})
"""
# upper bound of sigmoid familiy's gradient, given a input range lb and ub
# returns: lower and upper bound of gradient in that region
@njit
def sigmoid_grad_bounds(lb, ub, grad_f):
# allows a small tolerance
assert lb <= ub + 1e-5
if lb >= 0:
# always positive
return grad_f(ub), grad_f(lb)
if ub <= 0:
return grad_f(lb), grad_f(ub)
# lb < 0, ub > 0
return grad_f(max(-lb, ub)), grad_f(0)
# upper bound of ReLU familiy's gradient, given a input range lb and ub
# alpha is slope on the negative side. alpha > 0 for leaky-ReLU
# returns: lower and upper bound of gradient in that region
@njit
def relu_grad_bounds(lb, ub, alpha):
# allows a small tolerance
assert lb <= ub + 1e-5
assert alpha >= 0
if lb >= 0:
return 1, 1
if ub <= 0:
return alpha, alpha
else:
return alpha, 1
# compute the bound of the last two layers
# W2 \in [c, M2], W1 \in [M2, M1]
# LB and UB are \in [M2]
# l, u \in [c, M1]
# r \in [c], k \in [M1], i \in [M2]
# output shape is [c, M1]
@njit(parallel=True)
def recurjac_bound_2layer(W2, W1, grad_UB, grad_LB):
# this is the delta, and l <=0, u >= 0
l = np.zeros((W2.shape[0], W1.shape[1]), dtype=W2.dtype)
u = np.zeros_like(l)
for r in prange(W2.shape[0]):
for k in range(W1.shape[1]):
# i is the index of neurons in the middle (M2)
for i in range(W2.shape[1]):
prod = W2[r,i] * W1[i,k]
if prod > 0:
u[r,k] += grad_UB[i] * prod
l[r,k] += grad_LB[i] * prod
else:
l[r,k] += grad_UB[i] * prod
u[r,k] += grad_LB[i] * prod
return l, u
@njit(parallel=True)
def recurjac_bound_next_with_grad_bounds(prev_l, prev_u, W1, UB, LB):
# l, u in shape(num_output_class, num_neurons_in_this_layer)
l = np.zeros((prev_l.shape[0], W1.shape[1]), dtype = W1.dtype)
u = np.zeros_like(l)
# r is dimension for delta.shape[0]
for r in prange(prev_l.shape[0]):
for i in range(W1.shape[0]):
grad_LB, grad_UB = get_grad_bounds(LB[i], UB[i])
for k in range(W1.shape[1]):
if W1[i,k] > 0:
if prev_u[r,i] > 0:
u[r,k] += grad_UB * W1[i,k] * prev_u[r,i]
else:
u[r,k] += grad_LB * W1[i,k] * prev_u[r,i]
if prev_l[r,i] < 0:
l[r,k] += grad_UB * W1[i,k] * prev_l[r,i]
else:
l[r,k] += grad_LB * W1[i,k] * prev_l[r,i]
else:
if prev_l[r,i] > 0:
u[r,k] += grad_LB * W1[i,k] * prev_l[r,i]
else:
u[r,k] += grad_UB * W1[i,k] * prev_l[r,i]
if prev_u[r,i] < 0:
l[r,k] += grad_LB * W1[i,k] * prev_u[r,i]
else:
l[r,k] += grad_UB * W1[i,k] * prev_u[r,i]
return l, u
# prev_l, prev_u are the bounds for previous layers
# prev_l, prev_u \in [c, M2], W1 \in [M2, M1]
# r \in [c], k \in [M1], i \in [M2]
# output in [c, M1]
@njit(parallel=True)
def recurjac_bound_next(prev_l, prev_u, W1, grad_UB, grad_LB):
# l, u in shape(num_output_class, num_neurons_in_this_layer)
l = np.zeros((prev_l.shape[0], W1.shape[1]), dtype = W1.dtype)
u = np.zeros_like(l)
# r is dimension for delta.shape[0]
for r in prange(prev_l.shape[0]):
for i in range(W1.shape[0]):
for k in range(W1.shape[1]):
if W1[i,k] > 0:
if prev_u[r,i] > 0:
u[r,k] += grad_UB[i] * W1[i,k] * prev_u[r,i]
else:
u[r,k] += grad_LB[i] * W1[i,k] * prev_u[r,i]
if prev_l[r,i] < 0:
l[r,k] += grad_UB[i] * W1[i,k] * prev_l[r,i]
else:
l[r,k] += grad_LB[i] * W1[i,k] * prev_l[r,i]
else:
if prev_l[r,i] > 0:
u[r,k] += grad_LB[i] * W1[i,k] * prev_l[r,i]
else:
u[r,k] += grad_UB[i] * W1[i,k] * prev_l[r,i]
if prev_u[r,i] < 0:
l[r,k] += grad_LB[i] * W1[i,k] * prev_u[r,i]
else:
l[r,k] += grad_UB[i] * W1[i,k] * prev_u[r,i]
return l, u
# improved recursive bound, handles 1 row and 1 column only (used for backtracking)
# w1 is a row vector
# select_column: only compute a specific column, this is the column for prev_l and prev_u
# neuron_masks: only handle selected neurons
# outputs are two scalars, l and u
@njit
def recurjac_bound_forward_improved_1x1(weights, w1, all_l, all_u, layer, gradUBs, gradLBs, select_column):
l = 0
u = 0
grad_UB = gradUBs[layer]
grad_LB = gradLBs[layer]
# print("recursion layer", layer)
# handle special case where we only have the last 2 layers to bound
if layer == 0:
for i in range(w1.shape[0]):
prod = w1[i] * weights[0][i,select_column]
if prod > 0:
u += grad_UB[i] * prod ##!!UB
l += grad_LB[i] * prod ##!!LB
else:
l += grad_UB[i] * prod ##!!LB
u += grad_LB[i] * prod ##!!UB
# print("layer is 0,", l.shape)
return l, u
# l, u in shape(num_output_class, num_neurons_in_this_layer)
prev_l = all_l[layer]
prev_u = all_u[layer]
# r is dimension for delta.shape[0]
r = select_column
new_row_u = np.zeros_like(w1) ##!!UB
new_row_l = np.zeros_like(w1) ##!!LB
cu = 0
cl = 0
ce = 0
# for each neuron
for i in range(w1.shape[0]):
# new row with activation function considered
if w1[i] > 0:
# this element is always negative, can propagate
if prev_u[i,r] <= 0:
cu += 1
new_row_u[i] = w1[i] * grad_LB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
# this element is always positive, can propagate
elif prev_l[i,r] >= 0:
cl += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_LB[i] ##!!LB
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
else:
# otherwise, we consider the worst case bound
# where prev_u[i,r] > 0 AND prev_l[i,r] < 0
u += grad_UB[i] * w1[i] * prev_u[i,r] ##!!UB
l += grad_UB[i] * w1[i] * prev_l[i,r] ##!!LB
else:
# this element is always negative, can propagate
if prev_u[i,r] <= 0:
cu += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_LB[i] ##!!LB
# this element is always positive, can propagate
elif prev_l[i,r] >= 0:
cl += 1
new_row_u[i] = w1[i] * grad_LB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
# no difference between upper and lower bounds
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
else:
# otherwise, we consider the worst case bound
# where prev_u[i,r] > 0 AND prev_l[i,r] < 0
u += grad_UB[i] * w1[i] * prev_l[i,r] ##!!UB
l += grad_UB[i] * w1[i] * prev_u[i,r] ##!!LB
if cu + cl + ce > 0:
new_w_u = np.dot(np.expand_dims(new_row_u, axis=0), weights[layer]) ##!!UB
new_w_l = np.dot(np.expand_dims(new_row_l, axis=0), weights[layer]) ##!!LB
# now compute the bound for this 1-row vector only, with the previous layer
_, u_kr = recurjac_bound_forward_improved_1x1_UB(weights, new_w_u[0], all_l, all_u, layer - 1, gradUBs, gradLBs, r) ##!!UB
l_kr, _ = recurjac_bound_forward_improved_1x1_LB(weights, new_w_l[0], all_l, all_u, layer - 1, gradUBs, gradLBs, r) ##!!LB
u += u_kr ##!!UB
l += l_kr ##!!LB
return l, u
# improved recursive bound, handles 1 row and 1 column only (used for backtracking)
# w1 is a column vector, but passed as a 1-d vector
# select_row: only compute a specific row, this is the row for prev_l and prev_u
# outputs are two scalars, l and u
@njit
def recurjac_bound_backward_improved_1x1(weights, w1, all_l, all_u, layer, gradUBs, gradLBs, select_row):
l = 0
u = 0
grad_UB = gradUBs[layer]
grad_LB = gradLBs[layer]
# print(layer, grad_UB, grad_LB, w1.shape)
# print("recursion layer", layer)
# handle special case where we only have the first 2 layers to bound
if layer == len(weights) - 2:
for i in range(w1.shape[0]):
prod = w1[i] * weights[-1][select_row,i]
# print(w1[i], weights[-1][select_row,i], grad_UB[i])
if prod > 0:
u += grad_UB[i] * prod ##!!UB
l += grad_LB[i] * prod ##!!LB
else:
l += grad_UB[i] * prod ##!!LB
u += grad_LB[i] * prod ##!!UB
# print("last layer", layer, l, u)
return l, u
# l, u in shape(num_output_class, num_neurons_in_this_layer)
prev_l = all_l[layer+1]
prev_u = all_u[layer+1]
# weights of next layer
next_W = weights[layer+1]
# r is dimension for delta.shape[0]
k = select_row
new_row_u = np.zeros_like(w1) ##!!UB
new_row_l = np.zeros_like(w1) ##!!LB
cu = 0
cl = 0
ce = 0
# for each neuron
for i in range(prev_l.shape[1]):
# new row with activation function considered
if w1[i] > 0:
# this element is always negative, can propagate
if prev_u[k,i] <= 0:
cu += 1
new_row_u[i] = w1[i] * grad_LB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
# this element is always positive, can propagate
elif prev_l[k,i] >= 0:
cl += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_LB[i] ##!!LB
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
else:
# otherwise, we consider the worst case bound
# where prev_u[k,i] > 0 AND prev_l[k,i] < 0
u += grad_UB[i] * w1[i] * prev_u[k,i] ##!!UB
l += grad_UB[i] * w1[i] * prev_l[k,i] ##!!LB
else:
# this element is always negative, can propagate
if prev_u[k,i] <= 0:
cu += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_LB[i] ##!!LB
# this element is always positive, can propagate
elif prev_l[k,i] >= 0:
cl += 1
new_row_u[i] = w1[i] * grad_LB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
# no difference between upper and lower bounds
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = w1[i] * grad_UB[i] ##!!UB
new_row_l[i] = w1[i] * grad_UB[i] ##!!LB
else:
# otherwise, we consider the worst case bound
# where prev_u[k,i] > 0 AND prev_l[k,i] < 0
u += grad_UB[i] * w1[i] * prev_l[k,i] ##!!UB
l += grad_UB[i] * w1[i] * prev_u[k,i] ##!!LB
if cu + cl + ce > 0:
# new_w_u and new_w_l is a column
new_w_u = np.dot(next_W, np.expand_dims(new_row_u, axis=1)) ##!!UB
new_w_l = np.dot(next_W, np.expand_dims(new_row_l, axis=1)) ##!!LB
# print("layer", layer, l, u)
# now compute the bound for this 1-row vector only, with the previous layer
_, u_kr = recurjac_bound_backward_improved_1x1_UB(weights, new_w_u.reshape(-1), all_l, all_u, layer + 1, gradUBs, gradLBs, k) ##!!UB
l_kr, _ = recurjac_bound_backward_improved_1x1_LB(weights, new_w_l.reshape(-1), all_l, all_u, layer + 1, gradUBs, gradLBs, k) ##!!LB
u += u_kr ##!!UB
l += l_kr ##!!LB
# print("done layer", layer, l, u)
return l, u
# improved recursive bound
# prev_l, prev_u are the bounds for previous layers
# prev_l, prev_u \in [c, M2], W1 \in [M2, M1]
# r \in [c], k \in [M1], i \in [M2]
# select_column: only compute a specific column
# output in [c, M1]
@njit(parallel=False)
def recurjac_bound_forward_improved(weights, W1, all_l, all_u, layer, gradUBs, gradLBs):
# l, u in shape(num_output_class, num_neurons_in_this_layer)
prev_l = all_l[layer]
prev_u = all_u[layer]
l = np.zeros((W1.shape[0], prev_l.shape[1]), dtype = W1.dtype)
u = np.zeros_like(l)
# weight of next layer is W1
grad_UB = gradUBs[layer]
grad_LB = gradLBs[layer]
# weights of previous layer
prev_W = weights[layer]
# r is dimension for delta.shape[0]
for k in prange(W1.shape[0]):
for r in range(prev_l.shape[1]):
new_row_u = np.zeros_like(W1[k,:])
new_row_l = np.zeros_like(W1[k,:])
cu = 0
cl = 0
ce = 0
for i in range(W1.shape[1]):
# new row with activation function considered
if W1[k,i] > 0:
# this element is always negative, can propagate
if prev_u[i,r] <= 0:
cu += 1
new_row_u[i] = W1[k,i] * grad_LB[i]
new_row_l[i] = W1[k,i] * grad_UB[i]
# this element is always positive, can propagate
elif prev_l[i,r] >= 0:
cl += 1
new_row_u[i] = W1[k,i] * grad_UB[i]
new_row_l[i] = W1[k,i] * grad_LB[i]
# no difference between upper and lower bounds
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = W1[k,i] * grad_UB[i]
new_row_l[i] = W1[k,i] * grad_UB[i]
else:
# otherwise, we consider the worst case bound
# where prev_u[i,r] > 0 AND prev_l[i,r] < 0
u[k,r] += grad_UB[i] * W1[k,i] * prev_u[i,r]
l[k,r] += grad_UB[i] * W1[k,i] * prev_l[i,r]
else:
# this element is always negative, can propagate
if prev_u[i,r] <= 0:
cu += 1
new_row_u[i] = W1[k,i] * grad_UB[i]
new_row_l[i] = W1[k,i] * grad_LB[i]
# this element is always positive, can propagate
elif prev_l[i,r] >= 0:
cl += 1
new_row_u[i] = W1[k,i] * grad_LB[i]
new_row_l[i] = W1[k,i] * grad_UB[i]
# no difference between upper and lower bounds
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = W1[k,i] * grad_UB[i]
new_row_l[i] = W1[k,i] * grad_UB[i]
else:
# otherwise, we consider the worst case bound
# where prev_u[i,r] > 0 AND prev_l[i,r] < 0
u[k,r] += grad_UB[i] * W1[k,i] * prev_l[i,r]
l[k,r] += grad_UB[i] * W1[k,i] * prev_u[i,r]
# for each k and r, we need to form a new row vector, as the contents of the vector
# depends on the sign of the k-th row and W and bounds of the r-th column of Y
# after forming new_row_u and new_row_l, merge it with previous layer weights
# print("layer", layer, "k", k, "r", r, "cu", cu, "cl", cl)
if cu + cl + ce > 0:
new_w_u = np.dot(np.expand_dims(new_row_u, axis=0), prev_W)
new_w_l = np.dot(np.expand_dims(new_row_l, axis=0), prev_W)
# now compute the bound for this 1-row vector only, with the previous layer
_, u_kr = recurjac_bound_forward_improved_1x1_UB(weights, new_w_u[0], all_l, all_u, layer - 1, gradUBs, gradLBs, r)
l_kr, _ = recurjac_bound_forward_improved_1x1_LB(weights, new_w_l[0], all_l, all_u, layer - 1, gradUBs, gradLBs, r)
u[k,r] += u_kr
l[k,r] += l_kr
return l, u
# improved recursive bound
# prev_l, prev_u are the bounds for previous layers
# prev_l, prev_u \in [c, M2], W1 \in [M2, M1]
# r \in [c], k \in [M1], i \in [M2]
# select_column: only compute a specific column
# output in [c, M1]
@njit(parallel=False)
def recurjac_bound_backward_improved(weights, W1, all_l, all_u, layer, gradUBs, gradLBs):
# l, u in shape(num_output_class, num_neurons_in_this_layer)
prev_l = all_l[layer+1]
prev_u = all_u[layer+1]
l = np.zeros((prev_l.shape[0], W1.shape[1]), dtype = W1.dtype)
u = np.zeros_like(l)
# weight of next layer is W1
grad_UB = gradUBs[layer]
grad_LB = gradLBs[layer]
# weights of next layer
next_W = weights[layer+1]
# r is dimension for delta.shape[0]
for k in prange(prev_l.shape[0]):
for r in range(W1.shape[1]):
new_row_u = np.zeros_like(W1[:,r])
new_row_l = np.zeros_like(W1[:,r])
cu = 0
cl = 0
ce = 0
for i in range(prev_l.shape[1]):
# new row with activation function considered
if W1[i,r] > 0:
# this element is always negative, can propagate
if prev_u[k,i] <= 0:
cu += 1
new_row_u[i] = W1[i,r] * grad_LB[i]
new_row_l[i] = W1[i,r] * grad_UB[i]
# this element is always positive, can propagate
elif prev_l[k,i] >= 0:
cl += 1
new_row_u[i] = W1[i,r] * grad_UB[i]
new_row_l[i] = W1[i,r] * grad_LB[i]
# no difference between upper and lower bounds
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = W1[i,r] * grad_UB[i]
new_row_l[i] = W1[i,r] * grad_UB[i]
else:
# otherwise, we consider the worst case bound
# where prev_u[k,i] > 0 AND prev_l[k,i] < 0
u[k,r] += grad_UB[i] * W1[i,r] * prev_u[k,i]
l[k,r] += grad_UB[i] * W1[i,r] * prev_l[k,i]
else:
# this element is always negative, can propagate
if prev_u[k,i] <= 0:
cu += 1
new_row_u[i] = W1[i,r] * grad_UB[i]
new_row_l[i] = W1[i,r] * grad_LB[i]
# this element is always positive, can propagate
elif prev_l[k,i] >= 0:
cl += 1
new_row_u[i] = W1[i,r] * grad_LB[i]
new_row_l[i] = W1[i,r] * grad_UB[i]
# no difference between upper and lower bounds
elif grad_UB[i] == grad_LB[i]:
ce += 1
new_row_u[i] = W1[i,r] * grad_UB[i]
new_row_l[i] = W1[i,r] * grad_UB[i]
else:
# otherwise, we consider the worst case bound
# where prev_u[k,i] > 0 AND prev_l[k,i] < 0
u[k,r] += grad_UB[i] * W1[i,r] * prev_l[k,i]
l[k,r] += grad_UB[i] * W1[i,r] * prev_u[k,i]
# for each k and r, we need to form a new row vector, as the contents of the vector
# depends on the sign of the k-th row and W and bounds of the r-th column of Y
# after forming new_row_u and new_row_l, merge it with previous layer weights
# print("layer", layer, "k", k, "r", r, "cu", cu, "cl", cl)
if cu + cl + ce > 0:
# new_w_u and new_w_l is a column
new_w_u = np.dot(next_W, np.expand_dims(new_row_u, axis=1))
new_w_l = np.dot(next_W, np.expand_dims(new_row_l, axis=1))
# now compute the bound for this 1-row vector only, with the previous layer
_, u_kr = recurjac_bound_backward_improved_1x1_UB(weights, new_w_u.reshape(-1), all_l, all_u, layer + 1, gradUBs, gradLBs, k)
l_kr, _ = recurjac_bound_backward_improved_1x1_LB(weights, new_w_l.reshape(-1), all_l, all_u, layer + 1, gradUBs, gradLBs, k)
u[k,r] += u_kr
l[k,r] += l_kr
return l, u
# an optimized bound for the sum of absolute value in the first layer (computed at the end)
# last_l, last_u in shape [c,M1]
# prev_l, prev_u in shape [c,M2]
# last_W in shape [M2, M1]
# first_UB, first_LB in [M2]
def last_layer_abs_sum(weights, all_l, all_u, gradUBs, gradLBs, direction, shift):
if direction == -1:
last_l = all_l[0]
last_u = all_u[0]
else:
last_l = all_l[-1]
last_u = all_u[-1]
first_W = weights[0]
# add a new weights at the end
s = np.empty(last_l.shape[0], dtype = last_l.dtype)
# get the set where the Jacobian is always positive or negative
for i in range(last_l.shape[0]):
pos_idx = last_l[i] > 0
neg_idx = last_u[i] < 0
# print(sum(pos_idx), sum(neg_idx))
uns_idx = np.logical_not(np.logical_or(pos_idx, neg_idx))
# terms with unknown sign, use worst case bound
unsure_term = np.sum(np.maximum(np.abs(last_l[i][uns_idx]), np.abs(last_u[i][uns_idx])))
# now, use W_new to replace the first layer weight, and recompute the bound!
if np.sum(uns_idx) != last_l.shape[1]:
# merge the columns of first_W; after merging, its dimension becomes [M2, t], t < M1
W_new = np.empty((first_W.shape[0], ), dtype = first_W.dtype)
for r in range(first_W.shape[0]):
W_new[r] = np.sum(first_W[r][pos_idx]) - np.sum(first_W[r][neg_idx])
if direction == -1:
_, tu = recurjac_bound_backward_improved_1x1_UB(weights, W_new, all_l, all_u, 0, gradUBs, gradLBs, i)
s[i] = tu + unsure_term
else:
if shift == 0:
# replace the first layer weight
weights_new = list(weights)
# add a column vector to last
weights_new[0] = np.expand_dims(W_new, axis=1)
# we need to refill all_l and all_u as weights[0].shape[1] changed!
all_l_t = []
all_u_t = []
for j in range(len(weights_new)):
all_l_t.append(np.zeros(shape = (weights_new[j].shape[0], weights_new[0].shape[1]), dtype = weights_new[j].dtype))
all_u_t.append(np.zeros_like(all_l_t[-1]))
# for forward computation, we have to start from scratch
tl, tu = recurjac_bound(tuple(weights_new), tuple(gradUBs), tuple(gradLBs), tuple(all_l_t), tuple(all_u_t), direction)
s[i] = tu[0,0] + unsure_term
elif shift == 1:
# shifted by 1; we computed layer 1 and 2 in backward mode, so we have Y(2) that can be directly used
tl, tu = recurjac_bound_next(all_l[-2], all_u[-2], np.expand_dims(W_new, axis=1), gradUBs[0], gradLBs[0])
s[i] = tu[0,0] + unsure_term
# for debugging
naive_bound = np.sum(np.maximum(np.abs(last_l[i]), np.abs(last_u[i])))
saved = last_l.shape[1] - np.sum(uns_idx)
print("i =", i," naive_bound =", naive_bound, " improved_bound =", s[i], "saved =", saved)
else:
print("i =", i, "saved = 0")
s[i] = unsure_term
return s
def construct_function(fn, suffix, mask):
lines = inspect.getsource(fn)
lines = lines.replace("def "+fn.__name__, "def "+fn.__name__+"_"+suffix)
lines = lines.splitlines()
for i in range(len(lines)):
if lines[i].endswith("##!!"+mask):
start = len(lines[i]) - len(lines[i].lstrip())
lines[i] = lines[i][:start] + "# " + lines[i][start:]
lines = "\n".join(lines)
code = compile(lines, __file__, 'exec')
exec(code, globals())
def compile_recurjac_bounds(activation, leaky_slope):
global grad_wrapper_constructed
if not grad_wrapper_constructed:
# construct functions for gradient bounds dynamically
if activation == "relu":
code = compile(grad_relu_wrapper.format(0.0), __file__, 'exec')
exec(code, globals())
elif activation == "leaky":
code = compile(grad_relu_wrapper.format(leaky_slope), __file__, 'exec')
exec(code, globals())
elif activation == "tanh" or activation == "sigmoid" or activation == "arctan":
code = compile(grad_sigmoid_wrapper.format("act_"+activation+"_d"), __file__, 'exec')
exec(code, globals())
else:
raise(ValueError("Unknown activation function"))
grad_wrapper_constructed = True
exec("get_grad_bounds = get_grad_bounds_wrapper", globals())
# for network with M layers, LBs dimension is M-1 (not including last layer)
def recurjac_bound_wrapper(weights, UBs, LBs, numlayer, norm, separated_bounds = False, direction = -1, shift = 1):
global functions_constructed
if not functions_constructed:
construct_function(recurjac_bound_backward_improved_1x1, "UB", "LB")
construct_function(recurjac_bound_backward_improved_1x1, "LB", "UB")
construct_function(recurjac_bound_forward_improved_1x1, "UB", "LB")
construct_function(recurjac_bound_forward_improved_1x1, "LB", "UB")
functions_constructed = True
assert shift >= 0
assert shift <= 1
if direction == +1:
if shift == 1:
print('forward bounding Lipschitz constant from layer {}, shape is'.format(shift+1), weights[shift].shape)
# first, bound from layer shift, forward to the last layer
# assume layer 2 is the input layer
l, u, all_l, all_u, gradUBs, gradLBs = recurjac_bound_launcher(weights[shift:], UBs[shift:], LBs[shift:], numlayer - 1, norm, direction = +1)
# then manually bound the first layer
gradLBs = [np.empty_like(LBs[0])] + gradLBs
gradUBs = [np.empty_like(UBs[0])] + gradUBs
for r in prange(UBs[0].shape[0]):
gradLBs[0][r], gradUBs[0][r] = get_grad_bounds(LBs[0][r], UBs[0][r])
l, u = recurjac_bound_next_with_grad_bounds(l, u, weights[0], UBs[0], LBs[0])
# update all_l and all_u
# TODO: currently last l and u are placed at last element
all_l.append(l)
all_u.append(u)
else:
print('forward bounding Lipschitz constant, shape is', weights[0].shape)
# only 3 layers, directly bound
l, u, all_l, all_u, gradUBs, gradLBs = recurjac_bound_launcher(weights, UBs, LBs, numlayer, norm, direction = +1)
elif direction == -1:
print('Backward bounding Lipschitz constant, shape is', weights[-1].shape)
l, u, all_l, all_u, gradUBs, gradLBs = recurjac_bound_launcher(weights, UBs, LBs, numlayer, norm, direction = -1)
# count unsure elements
p1 = l < 0
p2 = u > 0
uns = np.logical_and(p1, p2)
n_uns = np.sum(uns)
print('Jacobian shape', l.shape)
# print(u)
print('unsure elements:', n_uns)
return get_final_bound(weights, l, u, all_l, all_u, gradUBs, gradLBs, direction, shift, norm, separated_bounds), n_uns
@njit
def form_grad_bounds(UBs, LBs, gradUBs, gradLBs):
# form gradient upper and lower bounds
for i in range(len(UBs)):
gradLB = gradLBs[i]
gradUB = gradUBs[i]
LB = LBs[i]
UB = UBs[i]
for r in range(UB.shape[0]):
gradLB[r], gradUB[r] = get_grad_bounds(LB[r], UB[r])
def recurjac_bound_launcher(weights, UBs, LBs, numlayer, norm, direction):
# pre-allocate arrays for lower and upper bounds
all_l = []
all_u = []
if direction == -1:
for i in range(numlayer):
all_l.append(np.zeros(shape = (weights[numlayer-1].shape[0], weights[i].shape[1]), dtype = weights[i].dtype))
all_u.append(np.zeros_like(all_l[-1]))
elif direction == 1:
for i in range(numlayer):
all_l.append(np.zeros(shape = (weights[i].shape[0], weights[0].shape[1]), dtype = weights[i].dtype))
all_u.append(np.zeros_like(all_l[-1]))
# pre-allocate arrays for gradient lower and upper bounds
gradUBs = []
gradLBs = []
for i in range(len(UBs)):
gradUBs.append(np.empty_like(UBs[i]))
gradLBs.append(np.empty_like(LBs[i]))
# form gradient upper and lower bounds
form_grad_bounds(tuple(UBs), tuple(LBs), tuple(gradUBs), tuple(gradLBs))
# compute the bound
l, u = recurjac_bound(weights, tuple(gradUBs), tuple(gradLBs), tuple(all_l), tuple(all_u), direction)
return l, u, all_l, all_u, gradUBs, gradLBs
# if the output dimension is c, when separated is True, output is c individual bounds
# when separated is False, output is a scalar and it is the Lipschitz constant for the norm of the output
# direct -1: from last to first layers; direction 1: from first to last layers
@njit(parallel=True)
def recurjac_bound(weights, gradUBs, gradLBs, all_l, all_u, direction):
print('This is RecurJac Jacobian bound!')
numlayer = len(weights)
assert numlayer >= 2
assert direction == +1 or direction == -1
# gradXB[i] represent the bounds between i+1 and i+2 layer
# all_l[i] represent Y related to layer i+1
if direction == -1:
# form L and U from the last layer, they are just the last layer weight
all_l[-1][:] = weights[-1]
all_u[-1][:] = weights[-1]
l, u = recurjac_bound_2layer(weights[-1], weights[-2], gradUBs[-1], gradLBs[-1])
# L and U for numlayer-1
all_l[-2][:] = l
all_u[-2][:] = u
print("layer", numlayer - 2, "shape", l.shape)
# for layers other than the last two layers
# layer 2, 1, 0, etc
for i in list(range(numlayer - 2))[::-1]:
# get bound for layer i+2 (Y) and i+1 (W)
l, u = recurjac_bound_backward_improved(weights, weights[i], all_l, all_u, i, gradUBs, gradLBs)
all_l[i][:] = l
all_u[i][:] = u
print("layer", i, "shape", l.shape)
# get the final upper and lower bound
l = all_l[0]
u = all_u[0]
elif direction == 1:
# precreated L and U for each layer
# form L and U from the first layer
all_l[0][:] = weights[0]
all_u[0][:] = weights[0]
# layer 1 and 2
l, u = recurjac_bound_2layer(weights[1], weights[0], gradUBs[0], gradLBs[0])
all_l[1][:] = l
all_u[1][:] = u
# we need to transpose l and U to reuse recurjac_bound_next()
print("layer", 0, "shape", all_l[1].shape)
# for layers other than the last two layers
# layer 2 and 3, 3 and 4, etc
for i in list(range(1, numlayer - 1)):
# note that W is transposed
# get bound for layer i+1 (Y) and i+2 (W)
l, u = recurjac_bound_forward_improved(weights, weights[i+1], all_l, all_u, i, gradUBs, gradLBs)
all_l[i + 1][:] = l
all_u[i + 1][:] = u
print("layer", i, "shape", all_l[i+1].shape)
# get the final upper and lower bound
l = all_l[-1]
u = all_u[-1]
return l, u
def get_final_bound(weights, l, u, all_l, all_u, gradUBs, gradLBs, direction, shift, norm, separated_bounds = False):
M = np.maximum(np.abs(l), np.abs(u))
if separated_bounds:
# We look each row individually
if norm == 1:
# Linf norm (max) of each row
ret = np.empty((M.shape[0],), dtype = M.dtype)
for i in range(M.shape[0]):
ret[i] = np.max(M[i])
return ret
elif norm == 2:
# L2 norm of each row
return np.sqrt(np.sum(M**2, axis = 1))
elif norm == np.inf: # q_n = inf, return Li norm of max component
# L1 norm of each row
# return np.sum(M, axis = 1)
return last_layer_abs_sum(weights, all_l, all_u, gradUBs, gradLBs, direction, shift)
else:
# look for the matrix induced norm
# we need to unify return type for numba, so we return an array
ret = np.empty((1,), dtype = M.dtype)
if norm == 1:
# L1 induced norm of matrix M
ret[0] = np.linalg.norm(M, norm)
elif norm == 2:
# L2 induced norm of matrix M
ret[0] = np.linalg.norm(M, norm)
elif norm == np.inf:
# Linf induced norm of M
# Linf induced norm is the max column abs sum
# ret[0] = np.linalg.norm(M, norm)
ret[0] = np.max(last_layer_abs_sum(weights, all_l, all_u, gradUBs, gradLBs, direction, shift))
return ret