# dW[i] = gradient of W[i]
dW = [None]*(n+1)
pre_dW = [None]*(n+1) # preconditioned dW
pre_dW_stable = [None]*(n+1) # preconditioned stable dW
cov_A = [None]*(n+1) # covariance of activations[i]
cov_B2 = [None]*(n+1) # covariance of synthetic backprops[i]
vars_svd_A = [None]*(n+1)
vars_svd_B2 = [None]*(n+1)
for i in range(1,n+1):
cov_A[i] = init_var(A[i]@t(A[i])/dsize, "cov_A%d"%(i,))
cov_B2[i] = init_var(B2[i]@t(B2[i])/dsize, "cov_B2%d"%(i,))
vars_svd_A[i] = u.SvdWrapper(cov_A[i],"svd_A_%d"%(i,))
vars_svd_B2[i] = u.SvdWrapper(cov_B2[i],"svd_B2_%d"%(i,))
if use_tikhonov:
whitened_A = u.regularized_inverse2(vars_svd_A[i],L=Lambda) @ A[i]
else:
whitened_A = u.pseudo_inverse2(vars_svd_A[i]) @ A[i]
if use_tikhonov:
whitened_B2 = u.regularized_inverse2(vars_svd_B2[i],L=Lambda) @ B[i]
else:
whitened_B2 = u.pseudo_inverse2(vars_svd_B2[i]) @ B[i]
whitened_A_stable = u.pseudo_inverse_sqrt2(vars_svd_A[i]) @ A[i]
whitened_B2_stable = u.pseudo_inverse_sqrt2(vars_svd_B2[i]) @ B[i]
pre_dW[i] = (whitened_B2 @ t(whitened_A))/dsize
pre_dW_stable[i] = (whitened_B2_stable @ t(whitened_A_stable))/dsize
dW[i] = (B[i] @ t(A[i]))/dsize
# Loss function
reconstruction = u.L2(err) / (2 * dsize)
sparsity = beta * tf.reduce_sum(kl(rho, rho_hat))
def main():
np.random.seed(0)
tf.set_random_seed(0)
dtype = np.float32
# 64-bit doesn't help much, search for 64-bit in
# https://www.wolframcloud.com/objects/5f297f41-30f7-4b1b-972c-cac8d1f8d8e4
u.default_dtype = dtype
machine_epsilon = np.finfo(dtype).eps # 1e-7 or 1e-16
train_images = load_MNIST.load_MNIST_images('data/train-images-idx3-ubyte')
dsize = 10000
patches = train_images[:, :dsize]
fs = [dsize, 28 * 28, 196, 28 * 28]
# values from deeplearning.stanford.edu/wiki/index.php/UFLDL_Tutorial
X0 = patches
lambda_ = 3e-3
rho = tf.constant(0.1, dtype=dtype)
beta = 3
W0f = W_uniform(fs[2], fs[3])
def f(i):
return fs[i + 1] # W[i] has shape f[i] x f[i-1]
dsize = f(-1)
n = len(fs) - 2
# helper to create variables with numpy or TF initial value
init_dict = {} # {var_placeholder: init_value}
vard = {} # {var: util.VarInfo}
def init_var(val, name, trainable=False, noinit=False):
if isinstance(val, tf.Tensor):
collections = [] if noinit else None
var = tf.Variable(val, name=name, collections=collections)
else:
val = np.array(val)
assert u.is_numeric, "Unknown type"
holder = tf.placeholder(dtype,
shape=val.shape,
name=name + "_holder")
var = tf.Variable(holder, name=name, trainable=trainable)
init_dict[holder] = val
var_p = tf.placeholder(var.dtype, var.shape)
var_setter = var.assign(var_p)
vard[var] = u.VarInfo(var_setter, var_p)
return var
lr = init_var(0.2, "lr")
if purely_linear: # need lower LR without sigmoids
lr = init_var(.02, "lr")
Wf = init_var(W0f, "Wf", True)
Wf_copy = init_var(W0f, "Wf_copy", True)
W = u.unflatten(Wf, fs[1:]) # perftodo: this creates transposes
X = init_var(X0, "X")
W.insert(0, X)
def sigmoid(x):
if not purely_linear:
return tf.sigmoid(x)
else:
return tf.identity(x)
def d_sigmoid(y):
if not purely_linear:
return y * (1 - y)
else:
return 1
def kl(x, y):
return x * tf.log(x / y) + (1 - x) * tf.log((1 - x) / (1 - y))
def d_kl(x, y):
return (1 - x) / (1 - y) - x / y
# A[i] = activations needed to compute gradient of W[i]
# A[n+1] = network output
A = [None] * (n + 2)
# A[0] is just for shape checks, assert fail on run
# tf.assert always fails because of static assert
# fail_node = tf.assert_equal(1, 0, message="too huge")
fail_node = tf.Print(0, [0], "fail, this must never run")
with tf.control_dependencies([fail_node]):
A[0] = u.Identity(dsize, dtype=dtype)
A[1] = W[0]
for i in range(1, n + 1):
A[i + 1] = sigmoid(W[i] @ A[i])
# reconstruction error and sparsity error
err = (A[3] - A[1])
rho_hat = tf.reduce_sum(A[2], axis=1, keep_dims=True) / dsize
# B[i] = backprops needed to compute gradient of W[i]
# B2[i] = backprops from sampled labels needed for natural gradient
B = [None] * (n + 1)
B2 = [None] * (n + 1)
B[n] = err * d_sigmoid(A[n + 1])
sampled_labels_live = tf.random_normal((f(n), f(-1)), dtype=dtype, seed=0)
sampled_labels = init_var(sampled_labels_live,
"sampled_labels",
noinit=True)
B2[n] = sampled_labels * d_sigmoid(A[n + 1])
for i in range(n - 1, -1, -1):
backprop = t(W[i + 1]) @ B[i + 1]
backprop2 = t(W[i + 1]) @ B2[i + 1]
if i == 1 and not drop_sparsity:
backprop += beta * d_kl(rho, rho_hat)
backprop2 += beta * d_kl(rho, rho_hat)
B[i] = backprop * d_sigmoid(A[i + 1])
B2[i] = backprop2 * d_sigmoid(A[i + 1])
# dW[i] = gradient of W[i]
dW = [None] * (n + 1)
pre_dW = [None] * (n + 1) # preconditioned dW
pre_dW_stable = [None] * (n + 1) # preconditioned stable dW
cov_A = [None] * (n + 1) # covariance of activations[i]
cov_B2 = [None] * (n + 1) # covariance of synthetic backprops[i]
vars_svd_A = [None] * (n + 1)
vars_svd_B2 = [None] * (n + 1)
for i in range(1, n + 1):
cov_A[i] = init_var(A[i] @ t(A[i]) / dsize, "cov_A%d" % (i, ))
cov_B2[i] = init_var(B2[i] @ t(B2[i]) / dsize, "cov_B2%d" % (i, ))
vars_svd_A[i] = u.SvdWrapper(cov_A[i], "svd_A_%d" % (i, ))
vars_svd_B2[i] = u.SvdWrapper(cov_B2[i], "svd_B2_%d" % (i, ))
if use_tikhonov:
whitened_A = u.regularized_inverse2(vars_svd_A[i], L=Lambda) @ A[i]
else:
whitened_A = u.pseudo_inverse2(vars_svd_A[i]) @ A[i]
if use_tikhonov:
whitened_B2 = u.regularized_inverse2(vars_svd_B2[i],
L=Lambda) @ B[i]
else:
whitened_B2 = u.pseudo_inverse2(vars_svd_B2[i]) @ B[i]
whitened_A_stable = u.pseudo_inverse_sqrt2(vars_svd_A[i]) @ A[i]
whitened_B2_stable = u.pseudo_inverse_sqrt2(vars_svd_B2[i]) @ B[i]
pre_dW[i] = (whitened_B2 @ t(whitened_A)) / dsize
pre_dW_stable[i] = (whitened_B2_stable @ t(whitened_A_stable)) / dsize
dW[i] = (B[i] @ t(A[i])) / dsize
# Loss function
reconstruction = u.L2(err) / (2 * dsize)
sparsity = beta * tf.reduce_sum(kl(rho, rho_hat))
L2 = (lambda_ / 2) * (u.L2(W[1]) + u.L2(W[1]))
loss = reconstruction
if not drop_l2:
loss = loss + L2
if not drop_sparsity:
loss = loss + sparsity
grad_live = u.flatten(dW[1:])
pre_grad_live = u.flatten(pre_dW[1:]) # fisher preconditioned gradient
pre_grad_stable_live = u.flatten(
pre_dW_stable[1:]) # sqrt fisher preconditioned grad
grad = init_var(grad_live, "grad")
pre_grad = init_var(pre_grad_live, "pre_grad")
pre_grad_stable = init_var(pre_grad_stable_live, "pre_grad_stable")
update_params_op = Wf.assign(Wf - lr * pre_grad).op
update_params_stable_op = Wf.assign(Wf - lr * pre_grad_stable).op
save_params_op = Wf_copy.assign(Wf).op
pre_grad_dot_grad = tf.reduce_sum(pre_grad * grad)
pre_grad_stable_dot_grad = tf.reduce_sum(pre_grad * grad)
grad_norm = tf.reduce_sum(grad * grad)
pre_grad_norm = u.L2(pre_grad)
pre_grad_stable_norm = u.L2(pre_grad_stable)
def dump_svd_info(step):
"""Dump singular values and gradient values in those coordinates."""
for i in range(1, n + 1):
svd = vars_svd_A[i]
s0, u0, v0 = sess.run([svd.s, svd.u, svd.v])
util.dump(s0, "A_%d_%d" % (i, step))
A0 = A[i].eval()
At0 = v0.T @ A0
util.dump(A0 @ A0.T, "Acov_%d_%d" % (i, step))
util.dump(At0 @ At0.T, "Atcov_%d_%d" % (i, step))
util.dump(s0, "As_%d_%d" % (i, step))
for i in range(1, n + 1):
svd = vars_svd_B2[i]
s0, u0, v0 = sess.run([svd.s, svd.u, svd.v])
util.dump(s0, "B2_%d_%d" % (i, step))
B0 = B[i].eval()
Bt0 = v0.T @ B0
util.dump(B0 @ B0.T, "Bcov_%d_%d" % (i, step))
util.dump(Bt0 @ Bt0.T, "Btcov_%d_%d" % (i, step))
util.dump(s0, "Bs_%d_%d" % (i, step))
def advance_batch():
sess.run(sampled_labels.initializer) # new labels for next call
def update_covariances():
ops_A = [cov_A[i].initializer for i in range(1, n + 1)]
ops_B2 = [cov_B2[i].initializer for i in range(1, n + 1)]
sess.run(ops_A + ops_B2)
def update_svds():
if whitening_mode > 1:
vars_svd_A[2].update()
if whitening_mode > 2:
vars_svd_B2[2].update()
if whitening_mode > 3:
vars_svd_B2[1].update()
def init_svds():
"""Initialize our SVD to identity matrices."""
ops = []
for i in range(1, n + 1):
ops.extend(vars_svd_A[i].init_ops)
ops.extend(vars_svd_B2[i].init_ops)
sess = tf.get_default_session()
sess.run(ops)
init_op = tf.global_variables_initializer()
# tf.get_default_graph().finalize()
from tensorflow.core.protobuf import rewriter_config_pb2
rewrite_options = rewriter_config_pb2.RewriterConfig(
disable_model_pruning=True,
constant_folding=rewriter_config_pb2.RewriterConfig.OFF,
memory_optimization=rewriter_config_pb2.RewriterConfig.MANUAL)
optimizer_options = tf.OptimizerOptions(opt_level=tf.OptimizerOptions.L0)
graph_options = tf.GraphOptions(optimizer_options=optimizer_options,
rewrite_options=rewrite_options)
config = tf.ConfigProto(graph_options=graph_options)
#sess = tf.Session(config=config)
sess = tf.InteractiveSession(config=config)
sess.run(Wf.initializer, feed_dict=init_dict)
sess.run(X.initializer, feed_dict=init_dict)
advance_batch()
update_covariances()
init_svds()
sess.run(init_op, feed_dict=init_dict) # initialize everything else
print("Running training.")
u.reset_time()
step_lengths = [] # keep track of learning rates
losses = []
ratios = [] # actual loss decrease / expected decrease
grad_norms = []
pre_grad_norms = [] # preconditioned grad norm squared
pre_grad_stable_norms = [] # sqrt preconditioned grad norms squared
target_delta_list = [] # predicted decrease linear approximation
target_delta2_list = [] # predicted decrease quadratic appromation
actual_delta_list = [] # actual decrease
# adaptive line search parameters
alpha = 0.3 # acceptable fraction of predicted decrease
beta = 0.8 # how much to shrink when violation
growth_rate = 1.05 # how much to grow when too conservative
def update_cov_A(i):
sess.run(cov_A[i].initializer)
def update_cov_B2(i):
sess.run(cov_B2[i].initializer)
# only update whitening matrix of input activations in the beginning
if whitening_mode > 0:
vars_svd_A[1].update()
# compute t(delta).H.delta/2
def hessian_quadratic(delta):
# update_covariances()
W = u.unflatten(delta, fs[1:])
W.insert(0, None)
total = 0
for l in range(1, n + 1):
decrement = tf.trace(t(W[l]) @ cov_B2[l] @ W[l] @ cov_A[l])
total += decrement
return (total / 2).eval()
# compute t(delta).H^-1.delta/2
def hessian_quadratic_inv(delta):
# update_covariances()
W = u.unflatten(delta, fs[1:])
W.insert(0, None)
total = 0
for l in range(1, n + 1):
invB2 = u.pseudo_inverse2(vars_svd_B2[l])
invA = u.pseudo_inverse2(vars_svd_A[l])
decrement = tf.trace(t(W[l]) @ invB2 @ W[l] @ invA)
total += decrement
return (total / 2).eval()
# do line search, dump values as csv
def line_search(initial_value, direction, step, num_steps):
saved_val = tf.Variable(Wf)
sess.run(saved_val.initializer)
pl = tf.placeholder(dtype, shape=(), name="linesearch_p")
assign_op = Wf.assign(initial_value - direction * step * pl)
vals = []
for i in range(num_steps):
sess.run(assign_op, feed_dict={pl: i})
vals.append(loss.eval())
sess.run(Wf.assign(saved_val)) # restore original value
return vals
for step in range(num_steps):
update_covariances()
if step % whiten_every_n_steps == 0:
update_svds()
sess.run(grad.initializer)
sess.run(pre_grad.initializer)
lr0, loss0 = sess.run([lr, loss])
save_params_op.run()
# regular inverse becomes unstable when grad norm exceeds 1
stabilized_mode = grad_norm.eval() < 1
if stabilized_mode and not use_tikhonov:
update_params_stable_op.run()
else:
update_params_op.run()
loss1 = loss.eval()
advance_batch()
# line search stuff
target_slope = (-pre_grad_dot_grad.eval() if stabilized_mode else
-pre_grad_stable_dot_grad.eval())
target_delta = lr0 * target_slope
target_delta_list.append(target_delta)
# second order prediction of target delta
# TODO: the sign is wrong, debug this
# https://www.wolframcloud.com/objects/8f287f2f-ceb7-42f7-a599-1c03fda18f28
if local_quadratics:
x0 = Wf_copy.eval()
x_opt = x0 - pre_grad.eval()
# computes t(x)@H^-1 @(x)/2
y_opt = loss0 - hessian_quadratic_inv(grad)
# computes t(x)@H @(x)/2
y_expected = hessian_quadratic(Wf - x_opt) + y_opt
target_delta2 = y_expected - loss0
target_delta2_list.append(target_delta2)
actual_delta = loss1 - loss0
actual_slope = actual_delta / lr0
slope_ratio = actual_slope / target_slope # between 0 and 1.01
actual_delta_list.append(actual_delta)
if do_line_search:
vals1 = line_search(Wf_copy, pre_grad, lr / 100, 40)
vals2 = line_search(Wf_copy, grad, lr / 100, 40)
u.dump(vals1, "line1-%d" % (i, ))
u.dump(vals2, "line2-%d" % (i, ))
losses.append(loss0)
step_lengths.append(lr0)
ratios.append(slope_ratio)
grad_norms.append(grad_norm.eval())
pre_grad_norms.append(pre_grad_norm.eval())
pre_grad_stable_norms.append(pre_grad_stable_norm.eval())
if step % report_frequency == 0:
print(
"Step %d loss %.2f, target decrease %.3f, actual decrease, %.3f ratio %.2f grad norm: %.2f pregrad norm: %.2f"
% (step, loss0, target_delta, actual_delta, slope_ratio,
grad_norm.eval(), pre_grad_norm.eval()))
if adaptive_step_frequency and adaptive_step and step > adaptive_step_burn_in:
# shrink if wrong prediction, don't shrink if prediction is tiny
if slope_ratio < alpha and abs(
target_delta) > 1e-6 and adaptive_step:
print("%.2f %.2f %.2f" % (loss0, loss1, slope_ratio))
print(
"Slope optimality %.2f, shrinking learning rate to %.2f" %
(
slope_ratio,
lr0 * beta,
))
sess.run(vard[lr].setter, feed_dict={vard[lr].p: lr0 * beta})
# grow learning rate, slope_ratio .99 worked best for gradient
elif step > 0 and i % 50 == 0 and slope_ratio > 0.90 and adaptive_step:
print("%.2f %.2f %.2f" % (loss0, loss1, slope_ratio))
print("Growing learning rate to %.2f" % (lr0 * growth_rate))
sess.run(vard[lr].setter,
feed_dict={vard[lr].p: lr0 * growth_rate})
u.record_time()
# check against expected loss
if 'Apple' in sys.version:
pass
# u.dump(losses, "kfac_small_final_mac.csv")
targets = np.loadtxt("data/kfac_small_final_mac.csv", delimiter=",")
else:
pass
# u.dump(losses, "kfac_small_final_linux.csv")
targets = np.loadtxt("data/kfac_small_final_linux.csv", delimiter=",")
u.check_equal(targets, losses[:len(targets)], rtol=1e-1)
u.summarize_time()
print("Test passed")