def follow_hyperplane(LAYER,
start_point,
known_T,
known_A,
known_B,
history=[],
MAX_POINTS=1e3,
only_need_positive=False):
"""
This is the ugly algorithm that will let us recover sign for expansive networks.
Assumes we have extracted up to layer K-1 correctly, and layer K up to sign.
start_point is a neuron on layer K+1
known_T is the transformation that computes up to layer K-1, with
known_A and known_B being the layer K matrix up to sign.
We're going to come up with a bunch of different inputs,
each of which has the same critical point held constant at zero.
"""
def choose_new_direction_from_minimize(previous_axis):
"""
Given the current point which is at a critical point of the next
layer neuron, compute which direction we should travel to continue
with finding more points on this hyperplane.
Our goal is going to be to pick a direction that lets us explore
a new part of the space we haven't seen before.
"""
print("Choose a new direction to travel in")
if len(history) == 0:
which_to_change = 0
new_perp_dir = perp_dir
new_start_point = start_point
initial_signs = get_polytope_at(known_T, known_A, known_B,
start_point)
# If we're in the 1 region of the polytope then we try to make it smaller
# otherwise make it bigger
fn = min if initial_signs[0] == 1 else max
else:
neuron_values = np.array([x[1] for x in history])
neuron_positive_count = np.sum(neuron_values > 1, axis=0)
neuron_negative_count = np.sum(neuron_values < -1, axis=0)
mean_plus_neuron_value = neuron_positive_count / (
neuron_positive_count + neuron_negative_count + 1)
mean_minus_neuron_value = neuron_negative_count / (
neuron_positive_count + neuron_negative_count + 1)
# we want to find values that are consistently 0 or 1
# So map 0 -> 0 and 1 -> 0 and the middle to higher values
if only_need_positive:
neuron_consistency = mean_plus_neuron_value
else:
neuron_consistency = mean_plus_neuron_value * mean_minus_neuron_value
# Print out how much progress we've made.
# This estimate is probably worse than Windows 95's estimated time remaining.
# At least it's monotonic. Be thankful for that.
print("Progress",
"%.1f" % int(np.mean(neuron_consistency != 0) * 100) + "%")
print("Counts on each side of each neuron")
print(neuron_positive_count)
print(neuron_negative_count)
# Choose the smallest value, which is the most consistent
which_to_change = np.argmin(neuron_consistency)
print("Try to explore the other side of neuron", which_to_change)
if which_to_change != previous_axis:
if previous_axis is not None and neuron_consistency[
previous_axis] == neuron_consistency[which_to_change]:
# If the previous thing we were working towards has the same value as this one
# the don't change our mind and just keep going at that one
# (almost always--sometimes we can get stuck, let us get unstuck)
which_to_change = previous_axis
new_start_point = start_point
new_perp_dir = perp_dir
else:
valid_axes = np.where(
neuron_consistency ==
neuron_consistency[which_to_change])[0]
best = (np.inf, None, None)
for _, potential_hidden_vector, potential_point in history[
-1:]:
for potential_axis in valid_axes:
value = potential_hidden_vector[potential_axis]
if np.abs(value) < best[0]:
best = (np.abs(value), potential_axis,
potential_point)
_, which_to_change, new_start_point = best
new_perp_dir = perp_dir
else:
new_start_point = start_point
new_perp_dir = perp_dir
# If we're in the 1 region of the polytope then we try to make it smaller
# otherwise make it bigger
fn = min if neuron_positive_count[
which_to_change] > neuron_negative_count[
which_to_change] else max
arg_fn = np.argmin if neuron_positive_count[
which_to_change] > neuron_negative_count[
which_to_change] else np.argmax
print("Changing", which_to_change, 'to flip sides because mean is',
mean_plus_neuron_value[which_to_change])
val = matmul(known_T.forward(new_start_point, with_relu=True), known_A,
known_B)[which_to_change]
initial_signs = get_polytope_at(known_T, known_A, known_B,
new_start_point)
# Now we're going to figure out what direction makes this biggest/smallest
# this doesn't take any queries
# There's probably an analytical way to do this.
# But thinking is hard. Just try 1000 random angles.
# There are no queries involved in this process.
choices = []
for _ in range(1000):
random_dir = np.random.normal(size=DIM)
perp_component = np.dot(random_dir, new_perp_dir) / (np.dot(
new_perp_dir, new_perp_dir)) * new_perp_dir
parallel_dir = random_dir - perp_component
# This is the direction we're going to travel in.
go_direction = parallel_dir / np.sum(parallel_dir**2)**.5
try:
a_bit_further, high = binary_search_towards(
known_T, known_A, known_B, new_start_point, initial_signs,
go_direction)
except AcceptableFailure:
continue
if a_bit_further is None:
continue
# choose a direction that makes the Kth value go down by the most
val = matmul(
known_T.forward(a_bit_further[np.newaxis, :], with_relu=True),
known_A, known_B)[0][which_to_change]
#print('\t', val, high)
choices.append([val, new_start_point + high * go_direction])
best_value, multiple_intersection_point = fn(choices,
key=lambda x: x[0])
print('Value', best_value)
return new_start_point, multiple_intersection_point, which_to_change
###################################################
### Actual code to do the sign recovery starts. ###
###################################################
start_box_step = 0
points_on_plane = []
if CHEATING:
layer = np.abs(
cheat_get_inner_layers(np.array(start_point))[LAYER + 1])
print("Layer", layer)
which_is_zero = np.argmin(layer)
current_change_axis = 0
while True:
print("\n\n")
print("-----" * 10)
if CHEATING:
layer = np.abs(
cheat_get_inner_layers(np.array(start_point))[LAYER + 1])
#print('layer',LAYER+1, layer)
#print('all inner layers')
#for e in cheat_get_inner_layers(np.array(start_point)):
# print(e)
which_is_zero_2 = np.argmin(np.abs(layer))
if which_is_zero_2 != which_is_zero:
print("STARTED WITH", which_is_zero, "NOW IS", which_is_zero_2)
print(layer)
raise
# Keep track of where we've been, so we can go to new places.
which_polytope = get_polytope_at(known_T, known_A, known_B,
start_point, False) # [-1 1 -1]
hidden_vector = get_hidden_at(known_T, known_A, known_B, LAYER,
start_point, False)
sign_at_init = sign_to_int(which_polytope) # 0b010 -> 2
print("Number of collected points", len(points_on_plane))
if len(points_on_plane) > MAX_POINTS:
return points_on_plane, False
neuron_values = np.array([x[1] for x in history])
neuron_positive_count = np.sum(neuron_values > 1, axis=0)
neuron_negative_count = np.sum(neuron_values < -1, axis=0)
if (np.all(neuron_positive_count > 0) and np.all(neuron_negative_count > 0)) or \
(only_need_positive and np.all(neuron_positive_count > 0)):
print("Have all the points we need (1)")
print(query_count)
print(neuron_positive_count)
print(neuron_negative_count)
neuron_values = np.array([
get_hidden_at(known_T, known_A, known_B, LAYER, x, False)
for x in points_on_plane
])
neuron_positive_count = np.sum(neuron_values > 1, axis=0)
neuron_negative_count = np.sum(neuron_values < -1, axis=0)
print(neuron_positive_count)
print(neuron_negative_count)
return points_on_plane, True
# 1. find a way to move along the hyperplane by computing the normal
# direction using the ratios function. Then find a parallel direction.
try:
#perp_dir = get_ratios([start_point], [range(DIM)], eps=1e-4)[0].flatten()
perp_dir = get_ratios_lstsq(0, [start_point], [range(DIM)],
KnownT([], []),
eps=1e-5)[0].flatten()
except AcceptableFailure:
print(
"Failed to compute ratio at start point. Something very bad happened."
)
return points_on_plane, False
# Record these points.
history.append((which_polytope, hidden_vector, np.copy(start_point)))
# We can't just pick any parallel direction. If we did, then we would
# not end up covering much of the input space.
# Instead, we're going to figure out which layer-1 hyperplanes are "visible"
# from the current point. Then we're going to try and go reach all of them.
# This is the point at which the first and second layers intersect.
start_point, multiple_intersection_point, new_change_axis = choose_new_direction_from_minimize(
current_change_axis)
if new_change_axis != current_change_axis:
start_point, multiple_intersection_point, current_change_axis = choose_new_direction_from_minimize(
None)
#if CHEATING:
# print("INIT MULTIPLE", cheat_get_inner_layers(multiple_intersection_point))
# Refine the direction we're going to travel in---stay numerically stable.
towards_multiple_direction = multiple_intersection_point - start_point
step_distance = np.sum(towards_multiple_direction**2)**.5
print("Distance we need to step:", step_distance)
if step_distance > 1 or True:
mid_point = 1e-4 * towards_multiple_direction / np.sum(
towards_multiple_direction**2)**.5 + start_point
random_dir = np.random.normal(size=DIM)
mid_points = do_better_sweep(mid_point,
perp_dir / np.sum(perp_dir**2)**.5,
low=-1e-3,
high=1e-3,
known_T=known_T)
if len(mid_points) > 0:
mid_point = mid_points[np.argmin(
np.sum((mid_point - mid_points)**2, axis=1))]
towards_multiple_direction = mid_point - start_point
towards_multiple_direction = towards_multiple_direction / np.sum(
towards_multiple_direction**2)**.5
initial_signs = get_polytope_at(known_T, known_A, known_B,
start_point)
_, high = binary_search_towards(known_T, known_A, known_B,
start_point, initial_signs,
towards_multiple_direction)
multiple_intersection_point = towards_multiple_direction * high + start_point
# Find the angle of the next hyperplane
# First, take random steps away from the intersection point
# Then run the search algorithm to find some intersections
# what we find will either be a layer-1 or layer-2 intersection.
print("Now try to find the continuation direction")
success = None
while success is None:
if start_box_step < 0:
start_box_step = 0
print("VERY BAD FAILURE")
print("Choose a new random point to start from")
which_point = np.random.randint(0, len(history))
start_point = history[which_point][2]
print("New point is", which_point)
current_change_axis = np.random.randint(0, sizes[LAYER + 1])
print("New axis to change", current_change_axis)
break
print("\tStart the box step with size", start_box_step)
try:
success, camefrom, stepsize = find_plane_angle(
known_T, known_A, known_B, multiple_intersection_point,
sign_at_init, start_box_step)
except AcceptableFailure:
# Go back to the top and try with a new start point
print("\tOkay we need to try with a new start point")
start_box_step = -10
start_box_step -= 2
if success is None:
continue
val = matmul(
known_T.forward(multiple_intersection_point, with_relu=True),
known_A, known_B)[new_change_axis]
print("Value at multiple:", val)
val = matmul(known_T.forward(success, with_relu=True), known_A,
known_B)[new_change_axis]
print("Value at success:", val)
if stepsize < 10:
new_move_direction = success - multiple_intersection_point
# We don't want to be right next to the multiple intersection point.
# So let's binary search to find how far away we can go while remaining in this polytope.
# Then we'll go half as far as we can maximally go.
initial_signs = get_polytope_at(known_T, known_A, known_B, success)
print("polytope at initial", sign_to_int(initial_signs))
low = 0
high = 1
while high - low > 1e-2:
mid = (high + low) / 2
query_point = multiple_intersection_point + mid * new_move_direction
next_signs = get_polytope_at(known_T, known_A, known_B,
query_point)
print(
"polytope at", mid, sign_to_int(next_signs), "%x" %
(sign_to_int(next_signs) ^ sign_to_int(initial_signs)))
if initial_signs == next_signs:
low = mid
else:
high = mid
print("GO TO", mid)
success = multiple_intersection_point + (mid /
2) * new_move_direction
val = matmul(known_T.forward(success, with_relu=True), known_A,
known_B)[new_change_axis]
print("Value at moved success:", val)
print("Adding the points to the set of known good points")
points_on_plane.append(start_point)
if camefrom is not None:
points_on_plane.append(camefrom)
#print("Old start point", start_point)
#print("Set to success", success)
start_point = success
start_box_step = max(stepsize - 1, 0)
return points_on_plane, False
def find_plane_angle(known_T,
known_A,
known_B,
multiple_intersection_point,
sign_at_init,
init_step,
exponential_base=1.5):
"""
Given an input that's at the multiple intersection point, figure out how
to continue along the path after it bends.
/ X : multiple intersection point
......../.. ---- : layer N hyperplane
. / . | : layer N+1 hyperplane that bends
. / .
--------X-----------
. | .
. | .
.....|.....
|
|
We need to make sure to bend, and not turn onto the layer N hyperplane.
To do this we will draw a box around the X and intersect with the planes
and determine the four coordinates. Then draw another box twice as big.
The first layer plane will be the two points at a consistent angle.
The second layer plane will have an inconsistent angle.
Choose the inconsistent angle plane, and make sure we move to a new
polytope and don't just go backwards to where we've already bene.
"""
success = None
camefrom = None
prev_iter_intersections = []
while True:
x_dir_base = np.sign(np.random.normal(size=DIM)) / DIM**.5
y_dir_base = np.sign(np.random.normal(size=DIM)) / DIM**.5
# When the input dimension is odd we can't have two orthogonal
# vectors from {-1,1}^DIM
if np.abs(np.dot(x_dir_base, y_dir_base)) <= DIM % 2:
break
MAX = 35
start = [10] if init_step > 10 else []
for stepsize in start + list(range(init_step, MAX)):
print("\tTry stepping away", stepsize)
x_dir = x_dir_base * (exponential_base**(stepsize - 10))
y_dir = y_dir_base * (exponential_base**(stepsize - 10))
# Draw the box as shown in the diagram above, and compute where
# the critical points are.
top = do_better_sweep(multiple_intersection_point + x_dir,
y_dir,
-1,
1,
known_T=known_T)
bot = do_better_sweep(multiple_intersection_point - x_dir,
y_dir,
-1,
1,
known_T=known_T)
left = do_better_sweep(multiple_intersection_point + y_dir,
x_dir,
-1,
1,
known_T=known_T)
right = do_better_sweep(multiple_intersection_point - y_dir,
x_dir,
-1,
1,
known_T=known_T)
intersections = top + bot + left + right
# If we only have two critical points, and we're taking a big step,
# then something is seriously messed up.
# This is not an acceptable error. Just abort out and let's try to
# do the whole thing again.
if len(intersections) == 2 and stepsize >= 10:
raise AcceptableFailure()
if CHEATING:
print("\tHAVE BOX INTERSECT COUNT", len(intersections))
print("\t", len(left), len(right), len(top), len(bot))
if (len(intersections) == 0 and stepsize >
15): # or (len(intersections) == 3 and stepsize > 5):
# Probably we're in just a constant flat 0 region
# At this point we're basically dead in the water.
# Just fail up and try again.
print("\tIt looks like we're in a flat region, raise failure")
raise AcceptableFailure()
# If we somehow went from almost no critical points to more than 4,
# then we've really messed up.
# Just fail out and let's hope next time it doesn't happen.
if len(intersections) > 4 and len(prev_iter_intersections) < 2:
print("\tWe didn't get enough inner points")
if exponential_base == 1.2:
print("\tIt didn't work a second time")
return None, None, 0
else:
print("\tTry with smaller step")
return find_plane_angle(known_T,
known_A,
known_B,
multiple_intersection_point,
sign_at_init,
init_step,
exponential_base=1.2)
# This is the good, expected code-path.
# We've seen four intersections at least twice before, and now
# we're seeing more than 4.
if (len(intersections) > 4
or stepsize > 20) and len(prev_iter_intersections) >= 2:
next_intersections = np.array(prev_iter_intersections[-1])
intersections = np.array(prev_iter_intersections[-2])
# Let's first figure out what points are responsible for the prior-layer neurons
# being zero, and which are from the current-layer neuron being zero
candidate = []
for i, a in enumerate(intersections):
for j, b in enumerate(intersections):
if i == j: continue
score = np.sum(
((a + b) / 2 - multiple_intersection_point)**2)
a_to_b = b - a
a_to_b /= np.sum(a_to_b**2)**.5
variance = np.std((next_intersections - a) / a_to_b,
axis=1)
best_variance = np.min(variance)
#print(i,j,score, best_variance)
candidate.append((best_variance, i, j))
if sorted(candidate)[3][0] < 1e-8:
# It looks like both lines are linear here
# We can't distinguish what way is the next best way to go.
print("\tFailed the box continuation finding procedure. (1)")
print("\t", candidate)
raise AcceptableFailure()
# Sometimes life is just ugly, and nothing wants to work.
# Just abort.
err, index_0, index_1 = min(candidate)
if err / max(candidate)[0] > 1e-5:
return None, None, 0
prior_layer_near_zero = np.zeros(4, dtype=np.bool)
prior_layer_near_zero[index_0] = True
prior_layer_near_zero[index_1] = True
# Now let's walk through each of these points and check that everything looks sane.
should_fail = False
for critical_point, is_prior_layer_zero in zip(
intersections, prior_layer_near_zero):
vs = known_T.extend_by(
known_A, known_B).get_hidden_layers(critical_point)
#print("VS IS", vs)
#print("Is prior", is_prior_layer_zero)
#if CHEATING:
# print(cheat_get_inner_layers(critical_point))
if is_prior_layer_zero:
# We expect the prior layer to be zero.
if all([np.min(np.abs(x)) > 1e-5 for x in vs]):
# If it looks like it's not actually zero, then brutally fail.
print("\tAbort 1: failed to find a valid box")
should_fail = True
if any([np.min(np.abs(x)) < 1e-10 for x in vs]):
# We expect the prior layer to be zero.
if not is_prior_layer_zero:
# If it looks like it's not actually zero, then brutally fail.
print("\tAbort 2: failed to find a valid box")
should_fail = True
if should_fail:
return None, None, 0
# Done with error checking, life is good here.
# Find the direction that corresponds to the next direction we can move in
# and continue our search from that point.
for critical_point, is_prior_layer_zero in zip(
intersections, prior_layer_near_zero):
sign_at_crit = sign_to_int(
get_polytope_at(known_T, known_A, known_B, critical_point))
print("\tMove to", sign_at_crit, 'versus', sign_at_init,
is_prior_layer_zero)
if not is_prior_layer_zero:
if sign_at_crit != sign_at_init:
success = critical_point
if CHEATING:
print('\tinner at success',
cheat_get_inner_layers(success))
print("\tSucceeded")
else:
camefrom = critical_point
# If we didn't get a solution, then abort out.
# Probably what happened here is that we got more than four points
# on the box but didn't see exactly four points on the box twice before
# this means we should decrease the initial step size and try again.
if success is None:
print("\tFailed the box continuation finding procedure. (2)")
raise AcceptableFailure()
#assert success is not None
break
if len(intersections) == 4:
prev_iter_intersections.append(intersections)
return success, camefrom, min(stepsize, MAX - 3)
def is_on_following_layer(known_T, known_A, known_B, point):
print("Check if the critical point is on the next layer")
def is_on_prior_layer(query):
print("Hidden think", known_T.get_hidden_layers(query))
if CHEATING:
print("Hidden real", cheat_get_inner_layers(query))
if any(
np.min(np.abs(layer)) < 1e-5
for layer in known_T.get_hidden_layers(query)):
return True
next_hidden = known_T.extend_by(known_A, known_B).forward(query)
print(next_hidden)
if np.min(np.abs(next_hidden)) < 1e-4:
return True
return False
if is_on_prior_layer(point):
print("It's not, because it's on an earlier layer")
return False
if CHEATING:
ls = ([np.min(np.abs(x)) for x in cheat_get_inner_layers(point)])
initial_signs = get_polytope_at(known_T, known_A, known_B, point)
normal = get_ratios([point], [range(DIM)], eps=GRAD_EPS)[0].flatten()
normal = normal / np.sum(normal**2)**.5
for tol in range(10):
random_dir = np.random.normal(size=DIM)
perp_component = np.dot(random_dir, normal) / (np.dot(normal,
normal)) * normal
parallel_dir = random_dir - perp_component
go_direction = parallel_dir / np.sum(parallel_dir**2)**.5
_, high = binary_search_towards(known_T, known_A, known_B, point,
initial_signs, go_direction)
if CHEATING:
print(
cheat_get_inner_layers(point +
go_direction * high / 2)[np.argmin(ls)])
point_in_same_polytope = point + (high * .999 - 1e-4) * go_direction
print("high", high)
solutions = do_better_sweep(point_in_same_polytope,
normal,
-1e-4 * high,
1e-4 * high,
known_T=known_T)
if len(solutions) >= 1:
print("Correctly found", len(solutions))
else:
return False
point_in_different_polytope = point + (high * 1.1 +
1e-1) * go_direction
solutions = do_better_sweep(point_in_different_polytope,
normal,
-1e-4 * high,
1e-4 * high,
known_T=known_T)
if len(solutions) == 0:
print("Correctly found", len(solutions))
else:
return False
#print("I THINK IT'S ON THE NEXT LAYER")
if CHEATING:
soln = [np.min(np.abs(x)) for x in cheat_get_inner_layers(point)]
print(soln)
assert np.argmin(soln) == len(known_T.A) + 1
return True
def get_more_points(NUM):
"""
Gather more points. This procedure is really kind of ugly and should probably be fixed.
We want to find points that are near where we expect them to be.
So begin by finding preimages to points that are on the line with gradient descent.
This should be completely possible, because we have d_0 input dimensions but
only want to control one inner layer.
"""
logger.log("Gather some more actual critical points on the plane",
level=Logger.INFO)
stepsize = .1
critical_points = []
while len(critical_points) <= NUM:
logger.log("On this iteration I have ",
len(critical_points),
"critical points on the plane",
level=Logger.INFO)
points = np.random.normal(0, 1e3, size=(
100,
DIM,
))
lr = 10
for step in range(5000):
# Use JaX's built in optimizer to do this.
# We want to adjust the LR so that we get a better solution
# as we optimize. Probably there is a better way to do this,
# but this seems to work just fine.
# No queries involvd here.
if step % 1000 == 0:
lr *= .5
init, opt_update, get_params = jax.experimental.optimizers.adam(
lr)
@jax.jit
def update(i, opt_state, batch):
params = get_params(opt_state)
return opt_update(i, loss_grad(batch, row), opt_state)
opt_state = init(points)
if step % 100 == 0:
ell = loss(points, row)
if CHEATING:
# This isn't cheating, but makes things prettier
print(ell)
if ell < 1e-5:
break
opt_state = update(step, opt_state, points)
points = opt_state.packed_state[0][0]
for point in points:
# For each point, try to see where it actually is.
# First, if optimization failed, then abort.
if loss(point, row) > 1e-5:
continue
if LAYER > 0:
# If wee're on a deeper layer, and if a prior layer is zero, then abort
if min(
np.min(np.abs(x))
for x in known_T.get_hidden_layers(point)) < 1e-4:
logger.log("is on prior", level=Logger.INFO)
continue
# print("Stepsize", stepsize)
tmp = Tracker().query_count
solution = do_better_sweep(offset=point,
low=-stepsize,
high=stepsize,
known_T=known_T)
# print("qs", query_count-tmp)
if len(solution) == 0:
stepsize *= 1.1
elif len(solution) > 1:
stepsize /= 2
elif len(solution) == 1:
stepsize *= 0.98
potential_solution = solution[0]
hiddens = extended_T.get_hidden_layers(potential_solution)
this_hidden_vec = extended_T.forward(potential_solution)
this_hidden = np.min(np.abs(this_hidden_vec))
if min(np.min(np.abs(x)) for x in
this_hidden_vec) > np.abs(this_hidden) * 0.9:
critical_points.append(potential_solution)
else:
logger.log("Reject it", level=Logger.INFO)
logger.log("Finished with a total of",
len(critical_points),
"critical points",
level=Logger.INFO)
return critical_points