def test_010_softmax():
"""Test Case for sigmoid
"""
u = ACTIVATION_DIFF_ACCEPTANCE_VALUE
P = softmax(np.array([2.44756739, 2.13945115]).astype(TYPE_FLOAT))
E = np.array([0.57642539, 0.42357461]).astype(TYPE_FLOAT)
assert np.all(np.abs(P - E) < u)
for _ in range(NUM_MAX_TEST_TIMES):
N: int = np.random.randint(1, NUM_MAX_BATCH_SIZE)
M: int = np.random.randint(2, NUM_MAX_NODES)
X = MAX_ACTIVATION_VALUE * np.random.randn(N, M).astype(TYPE_FLOAT)
np.all(np.isfinite(softmax(X)))
def predict(self, x):
w1, w2 = self.params['w1'], self.params['w2']
b1, b2 = self.params['b1'], self.params['b2']
a1 = np.dot(x, w1) + b1
z1 = sigmoid(a1)
a2 = np.dot(z1, w2) + b2
y = softmax(a2)
return y
def predict(network, x):
W1, W2, W3 = network['W1'], network['W2'], network['W3']
b1, b2, b3 = network['b1'], network['b2'], network['b3']
a1 = np.dot(x, W1) + b1
z1 = sigmoid(a1)
a2 = np.dot(z1, W2) + b2
z2 = sigmoid(a2)
a3 = np.dot(z2, W3) + b3
y = softmax(a3)
return y
def forward(self, x, t):
self.t = t
self.y = softmax(x)
# one-hot vector to
if self.t.size == self.y.size:
self.t = self.t.argmax(axis=1)
loss = cross_entropy_error(self.y, self.t)
return loss
def grad(self, x, t):
w1, w2 = self.params['w1'], self.params['w2']
b1, b2 = self.params['b1'], self.params['b2']
grads = {}
# forward
a1 = np.dot(x, w1) + b1
z1 = sigmoid(a1)
a2 = np.dot(z1, w2) + b2
y = softmax(a2)
# backward
dy = (y - t) / x.shape[0] # softmax with entropy loss's gradient, dL/dy
grads['w2'] = np.dot(z1.T, dy)
grads['b2'] = np.sum(dy, axis=0)
da1 = np.dot(dy, w2.T)
dz1 = (1.0 - sigmoid(a1)) * sigmoid(a1) * da1 # sigmoid's gradient
grads['w1'] = np.dot(x.T, dz1)
grads['b1'] = np.sum(dz1, axis=0)
return grads
def predict(self, X):
"""
Responsibility:
Generate a concrete prediction 0/1 for binary classification
and an index for categorical classification
For binary classification where M=1, 1/True if Xi > 0 else 0.
For categorical classification, argmax(X, axis=1) that identifies
the class that gives the max probability.
Args:
X: scores
Returns:
Predictions
"""
assert isinstance(X, np.ndarray) and X.dtype == TYPE_FLOAT, \
f"Only np array of type {TYPE_FLOAT} is accepted"
if X.ndim <= 1:
X = np.array(X).reshape(1, -1)
if self._log_loss_function == sigmoid_cross_entropy_log_loss:
return (X > 0).astype(TYPE_LABEL)
else:
return np.argmax(softmax(X), axis=1).astype(TYPE_LABEL)
def test_020_cross_entropy_log_loss_1d(caplog):
"""
Objective:
Test the categorical log loss values for P in 1 dimension.
Constraints:
1. The numerical gradient gn = (-t * logarithm(p+h) + t * logarithm(p-h)) / 2h.
2. The numerical gradient gn is within +/- u within the analytical g = -T/P.
P: Probabilities from softmax of shape (M,)
M: Number of nodes in the cross_entropy_log_loss layer.
T: Labels
Note:
log(P=1) -> 0
dlog(x)/dx = 1/x
"""
def f(P: np.ndarray, T: np.ndarray):
return np.sum(cross_entropy_log_loss(P, T))
# caplog.set_level(logging.DEBUG, logger=Logger.name)
h: TYPE_FLOAT = OFFSET_DELTA
u: TYPE_FLOAT = GRADIENT_DIFF_ACCEPTANCE_VALUE
# --------------------------------------------------------------------------------
# For (P, T): P[index] = True/1, OHE label T[index] = 1 where
# P=[0,0,0,...,1,...0], T = [0,0,0,...1,...0]. T[i] == 1
#
# Do not forget the Jacobian shape is (N,) and calculate each element.
# 1. For T=1, loss L = -log(Pi) = 0 and dL/dP=(1/Pi)= -1 is expected.
# 2. For T=0, Loss L = (-log(0+offset+h)-log(0+offset-h)) / 2h = 0 is expected.
# --------------------------------------------------------------------------------
M: TYPE_INT = np.random.randint(2, NUM_MAX_NODES)
index: TYPE_INT = TYPE_INT(np.random.randint(
0, M)) # Position of the true label in P
P1 = np.zeros(M, dtype=TYPE_FLOAT)
P1[index] = TYPE_FLOAT(1.0)
T1 = np.zeros(M, dtype=TYPE_LABEL)
T1[index] = TYPE_LABEL(1)
# Analytica correct gradient for P=1, T=1
AG = np.zeros_like(P1, dtype=TYPE_FLOAT)
AG[index] = TYPE_FLOAT(-1) # dL/dP = -1
EGN1 = np.zeros_like(P1, dtype=TYPE_FLOAT) # Expected numerical gradient
EGN1[index] = (-1 * logarithm(TYPE_FLOAT(1.0 + h)) + TYPE_FLOAT(1) *
logarithm(TYPE_FLOAT(1.0 - h))) / TYPE_FLOAT(2 * h)
assert np.all(np.abs(EGN1-AG) < u), \
"Expected EGN-1<%s but %s\nEGN=\n%s" % (u, (EGN1-AG), EGN1)
GN1 = numerical_jacobian(partial(f, T=T1), P1)
assert np.all(np.abs(GN1-AG) < u), \
"Expected GN-1<%s but %s\nGN=\n%s" % (u, (GN1-AG), GN1)
# The numerical gradient gn = (-t * logarithm(p+h) + t * logarithm(p-h)) / 2h
assert GN1.shape == EGN1.shape
assert np.all(np.abs(EGN1-GN1) < u), \
"Expected GN1==EGN1 but GN1-EGN1=\n%sP=\n%s\nT=%s\nEGN=\n%s\nGN=\n%s\n" \
% (np.abs(GN1-EGN1), P1, T1, EGN1, GN1)
# The numerical gradient gn is within +/- u within the analytical g = -T/P
G1 = np.zeros_like(P1, dtype=TYPE_FLOAT)
G1[T1 == 1] = -1 * (T1[index] / P1[index])
# G1[T1 != 0] = 0
check.equal(np.all(np.abs(G1 - GN1) < u), True,
"G1-GN1 %s\n" % np.abs(G1 - GN1))
# --------------------------------------------------------------------------------
# For (P, T): P[index] = np uniform(), index label T=index
# --------------------------------------------------------------------------------
for _ in range(NUM_MAX_TEST_TIMES):
M = np.random.randint(2, NUM_MAX_NODES) # M > 1
T2 = TYPE_LABEL(np.random.randint(0, M)) # location of the truth
P2 = np.zeros(M, dtype=TYPE_FLOAT)
while not (x := TYPE_FLOAT(
np.random.uniform(low=-BOUNDARY_SIGMOID,
high=BOUNDARY_SIGMOID))):
pass
p = softmax(x)
P2[T2] = p
# --------------------------------------------------------------------------------
# The Jacobian G shape is the same with P.shape.
# G:[0, 0, ...,g, 0, ...] where Gi is numerical gradient close to -1/(1+k).
# --------------------------------------------------------------------------------
N2 = np.zeros_like(P2, dtype=TYPE_FLOAT)
N2[T2] = TYPE_FLOAT(-1) * (logarithm(p + h) -
logarithm(p - h)) / TYPE_FLOAT(2 * h)
N2 = numerical_jacobian(partial(f, T=T2), P2)
# The numerical gradient gn = (-t * logarithm(p+h) + t * logarithm(p-h)) / 2h
assert N2.shape == N2.shape
assert np.all(np.abs(N2-N2) < u), \
f"Delta expected to be < {u} but \n{np.abs(N2-N2)}"
G2 = np.zeros_like(P2, dtype=TYPE_FLOAT)
G2[T2] = -1 / p
# The numerical gradient gn is within +/- u within the analytical g = -T/P
check.equal(np.all(np.abs(G2 - N2) < u), True,
"G2-N2 %s\n" % np.abs(G2 - N2))
def test_020_cross_entropy_log_loss_2d(caplog):
"""
Objective:
Test case for cross_entropy_log_loss(X, T) for X:shape(N,M), T:shape(N,)
Expected:
"""
def f(P: np.ndarray, T: np.ndarray):
"""Loss function"""
# For P.ndim==2 of shape (N, M), cross_entropy_log_loss() returns (N,).
# Each of which has the loss for P[n].
# If divided by P.shape[0] or N, the loss gets 1/N, which is wrong.
# This is not a gradient function but a loss function.
# return np.sum(cross_entropy_log_loss(P, T)) / P.shape[0]
return np.sum(cross_entropy_log_loss(P, T))
# caplog.set_level(logging.DEBUG, logger=Logger.name)
h: TYPE_FLOAT = OFFSET_DELTA
u: TYPE_FLOAT = GRADIENT_DIFF_ACCEPTANCE_VALUE
for _ in range(NUM_MAX_TEST_TIMES):
# --------------------------------------------------------------------------------
# [2D test case]
# P:(N, M) is probability matrix where Pnm = p, 0 <=n<N-1, 0<=m<M-1
# T:(N,) is index label where Tn=m is label as integer k.g. m=3 for 3rd label.
# Pnm = log(P[i][j])
# L = -log(p), -> dlog(P)/dP -> -1 / (p)
#
# Keep p value away from 0. As p gets close to 0, the log(p+/-h) gets large e.g
# -11.512925464970229, hence log(p+/-h) / 2h explodes.
# --------------------------------------------------------------------------------
while not (x := TYPE_FLOAT(
np.random.uniform(low=-BOUNDARY_SIGMOID,
high=BOUNDARY_SIGMOID))):
pass
p = softmax(x)
N = np.random.randint(1, NUM_MAX_BATCH_SIZE)
M = np.random.randint(2, NUM_MAX_NODES)
# label index, not OHE
T = np.random.randint(0, M, N).astype(
TYPE_LABEL) # N rows of labels, max label value is M-1
P = np.zeros((N, M)).astype(TYPE_FLOAT)
P[range(N), # Set p at random row position
T] = p
E = np.zeros_like(P).astype(TYPE_FLOAT)
E[range(N), # Set p at random row position
T] = (TYPE_FLOAT(-1) * logarithm(p + h) +
TYPE_FLOAT(1) * logarithm(p - h)) / (TYPE_FLOAT(2) * h)
G = numerical_jacobian(partial(f, T=T), P)
assert E.shape == G.shape, \
f"Jacobian shape is expected to be {E.shape} but {G.shape}."
assert np.all(np.abs(E-G) < u), \
f"Delta expected to be < {u} but \n{np.abs(E-G)}"
A = np.zeros_like(P).astype(TYPE_FLOAT)
A[range(N), # Set p at random row position
T] = -1 / p
check.equal(np.all(np.abs(A - G) < u), True,
"A-G %s\n" % np.abs(A - G))
for _ in range(NUM_MAX_TEST_TIMES):
# --------------------------------------------------------------------------------
# [1D test case]
# P:[0, 0, ..., 1, 0, ...] where Pi = 1
# T:[0, 0, ..., 1, 0, ...] is OHE label where Ti=1
# sum(-t * log(p+k)) -> log(1+k)
# dlog(P+k)/dP -> -1 / (1+k)
# --------------------------------------------------------------------------------
M = np.random.randint(2, NUM_MAX_NODES) # M > 1
index = np.random.randint(0, M) # location of the truth
while not (x := TYPE_FLOAT(
np.random.uniform(low=-BOUNDARY_SIGMOID,
high=BOUNDARY_SIGMOID))):
pass
p = softmax(x)
P3 = np.zeros(M, dtype=TYPE_FLOAT)
P3[index] = p
T3 = np.zeros(M).astype(TYPE_LABEL) # OHE index
T3[index] = TYPE_LABEL(1)
# --------------------------------------------------------------------------------
# The Jacobian G shape is the same with P.shape.
# --------------------------------------------------------------------------------
N3 = np.zeros_like(P3, dtype=TYPE_FLOAT)
N3[index] = TYPE_FLOAT(-1 * logarithm(p + h) +
1 * logarithm(p - h)) / TYPE_FLOAT(2 * h)
N3 = numerical_jacobian(partial(f, T=T3), P3)
assert N3.shape == N3.shape
assert np.all(np.abs(N3-N3) < u), \
f"Delta expected to be < {u} but \n{np.abs(N3-N3)}"
def forward(self, x, t):
self.t = t
self.y = softmax(x)
self.loss = cross_entropy_error(self.y, self.t)
return self.loss
def forward(self, x, **kwargs):
self.y = kwargs['y']
self.y_hat = softmax(x)
self.loss = cross_entropy_error(self.y, self.y_hat)
return self.loss
def disabled_test_040_softmax_log_loss_2d(caplog):
"""
TODO: Disabled as need to redesign numerical_jacobian for 32 bit floating.
Objective:
Verify the forward path constraints:
1. Layer output L/loss is np.sum(softmax_cross_entropy_log_loss) / N.
2. gradient_numerical() == numerical_jacobian(objective, X).
Verify the backward path constraints:
1. Analytical gradient G: gradient() == (P-1)/N
2. Analytical gradient G is close to GN: gradient_numerical().
"""
caplog.set_level(logging.DEBUG)
# --------------------------------------------------------------------------------
# Instantiate a CrossEntropyLogLoss layer
# --------------------------------------------------------------------------------
name = "test_040_softmax_log_loss_2d_ohe"
profiler = cProfile.Profile()
profiler.enable()
for _ in range(NUM_MAX_TEST_TIMES):
N: int = np.random.randint(1, NUM_MAX_BATCH_SIZE)
M: int = np.random.randint(2, NUM_MAX_NODES) # number of node > 1
_layer = layer.CrossEntropyLogLoss(
name=name,
num_nodes=M,
log_loss_function=softmax_cross_entropy_log_loss,
log_level=logging.DEBUG)
# ================================================================================
# Layer forward path
# ================================================================================
X = np.random.randn(N, M).astype(TYPE_FLOAT)
T = np.zeros_like(X, dtype=TYPE_LABEL) # OHE labels.
T[np.arange(N), np.random.randint(0, M, N)] = int(1)
# log_loss function require (X, T) in X(N, M), and T(N, M) in index label format.
X, T = transform_X_T(X, T)
_layer.T = T
Logger.debug("%s: X is \n%s\nT is \n%s", name, X, T)
# --------------------------------------------------------------------------------
# Expected analytical gradient EG = (dX/dL) = (A-T)/N
# --------------------------------------------------------------------------------
A = softmax(X)
EG = np.copy(A)
EG[np.arange(N), T] -= TYPE_FLOAT(
1) # Shape(N,), subtract from elements for T=1 only
EG /= TYPE_FLOAT(N)
# --------------------------------------------------------------------------------
# Total loss Z = np.sum(J)/N
# Expected loss EL = -sum(T*log(_A))
# (J, P) = softmax_cross_entropy_log_loss(X, T) and J:shape(N,) where J:shape(N,)
# is loss for each input and P is activation by sigmoid(X).
# --------------------------------------------------------------------------------
L = _layer.function(X)
J, P = softmax_cross_entropy_log_loss(X, T)
EL = np.array(-np.sum(logarithm(A[np.arange(N), T])) / N,
dtype=TYPE_FLOAT)
# Constraint: A == P as they are sigmoid(X)
assert np.all(np.abs(A-P) < ACTIVATION_DIFF_ACCEPTANCE_VALUE), \
f"Need A==P==sigmoid(X) but A=\n{A}\n P=\n{P}\n(A-P)=\n{(A-P)}\n"
# Constraint: Log loss layer output L == sum(J) from the log loss function
Z = np.array(np.sum(J) / N, dtype=TYPE_FLOAT)
assert np.array_equal(L, Z), \
f"Need log loss layer output L == sum(J) but L=\n{L}\nZ=\n{Z}."
# Constraint: L/loss is close to expected loss EL.
assert np.all(np.abs(EL-L) < LOSS_DIFF_ACCEPTANCE_VALUE), \
"Need EL close to L but \nEL=\n{EL}\nL=\n{L}\n"
# constraint: gradient_numerical() == numerical_jacobian(objective, X)
# TODO: compare the diff to accommodate numerical errors.
GN = _layer.gradient_numerical() # [dL/dX] from the layer
def objective(x):
"""Function to calculate the scalar loss L for cross entropy log loss"""
j, p = softmax_cross_entropy_log_loss(x, T)
return np.array(np.sum(j) / N, dtype=TYPE_FLOAT)
EGN = numerical_jacobian(objective, X) # Expected numerical dL/dX
assert np.array_equal(GN[0], EGN), \
f"GN[0]==EGN expected but GN[0] is \n%s\n EGN is \n%s\n" % (GN[0], EGN)
# ================================================================================
# Layer backward path
# ================================================================================
# constraint: Analytical gradient G: gradient() == EG == (P-1)/N.
dY = TYPE_FLOAT(1)
G = _layer.gradient(dY)
assert np.all(np.abs(G-EG) <= GRADIENT_DIFF_ACCEPTANCE_VALUE), \
f"Layer gradient dL/dX \n{G} \nneeds to be \n{EG}."
# constraint: Analytical gradient G is close to GN: gradient_numerical().
assert \
np.all(np.abs(G - GN[0]) <= GRADIENT_DIFF_ACCEPTANCE_VALUE) or \
np.all(np.abs(G - GN[0]) <= np.abs(GRADIENT_DIFF_ACCEPTANCE_RATIO * GN[0])), \
f"dX is \n{G}\nGN[0] is \n{GN[0]}\nRatio * GN[0] is \n{GRADIENT_DIFF_ACCEPTANCE_RATIO * GN[0]}.\n"
# constraint: Gradient g of the log loss layer needs -1 < g < 1
# abs(P-T) = abs(sigmoid(X)-T) cannot be > 1.
assert np.all(np.abs(G) < 1), \
f"Log loss layer gradient cannot be < -1 nor > 1 but\n{G}"
assert np.all(np.abs(GN[0]) < (1+GRADIENT_DIFF_ACCEPTANCE_RATIO)), \
f"Log loss layer gradient cannot be < -1 nor > 1 but\n{GN[0]}"
profiler.disable()
profiler.print_stats(sort="cumtime")
def forward(self, x):
self.out = softmax(x)
return self.out
def validate_relu_neuron_training(matmul: Matmul,
activation: ReLU,
loss: CrossEntropyLogLoss,
X: np.ndarray,
T: np.ndarray,
num_epochs: int = 100,
test_numerical_gradient: bool = False,
callback: Callable = None):
activation.objective = loss.function
matmul.objective = compose(activation.function, loss.function)
objective = compose(matmul.function, matmul.objective)
num_no_progress: int = 0 # how many time when loss L not decreased.
history: List[np.ndarray] = []
loss.T = T
for i in range(num_epochs):
L = objective(X)
N = X.shape[0]
P = softmax(relu(np.matmul(matmul.X, matmul.W.T)))
EDA = expected_gradient_from_log_loss(P=P, T=T, N=N)
# ********************************************************************************
# Constraint: Expected gradients must match actual
# ********************************************************************************
validate_relu_neuron_round_trip(matmul=matmul,
activation=activation,
X=X,
dA=EDA)
# --------------------------------------------------------------------------------
# gradient descent and get the analytical dL/dX, dL/dW
# --------------------------------------------------------------------------------
previous_W = copy.deepcopy(matmul.W)
matmul.update() # dL/dX, dL/dW
# ********************************************************************************
# Constraint. W in the matmul has been updated by the gradient descent.
# ********************************************************************************
Logger.debug("W after is \n%s", matmul.W)
if np.array_equal(previous_W, matmul.W):
Logger.warning("W has not been updated")
# ********************************************************************************
# Constraint: Objective/Loss L(Yn+1) after gradient descent < L(Yn)
# ********************************************************************************
if i > 0 and L >= history[-1]:
Logger.warning(
"Iteration [%i]: Loss[%s] has not improved from the previous [%s] for %s times.",
i, L, history[-1], num_no_progress + 1)
# --------------------------------------------------------------------------------
# Reduce the learning rate can make the situation worse.
# When reduced the lr every time L >= history, the (L >= history) became successive
# and eventually exceeded 50 successive non-improvement ending in failure.
# Keep the learning rate make the L>=history more frequent but still up to 3
# successive events, and the training still kept progressing.
# --------------------------------------------------------------------------------
num_no_progress += 1
if num_no_progress > 5:
matmul.lr = matmul.lr * 0.95
if num_no_progress > 50:
Logger.error(
"The training has no progress more than %s times.",
num_no_progress)
break
else:
num_no_progress = 0
history.append(L)
if callback:
callback(W=matmul.W)
return history
def test_030_objective_methods_1d_ohe():
"""
Objective:
Verify the forward path constraints:
1. Layer output L/loss is np.sum(cross_entropy_log_loss(softmax(X), T)) / N.
2. gradient_numerical() == numerical Jacobian numerical_jacobian(O, X).
Verify the backward path constraints:
1. Analytical gradient G: gradient() == (P-1)/N
2. Analytical gradient G is close to GN: gradient_numerical().
Expected:
Initialization detects the access to the non-initialized parameters and fails.
For X.ndim > 0, the layer transform X into 2D so as to use the numpy tuple-
like indexing:
P[
(0,3),
(2,4)
]
Hence, the shape of GN, G are 2D.
"""
# --------------------------------------------------------------------------------
# Instantiate a CrossEntropyLogLoss layer
# --------------------------------------------------------------------------------
name = "test_030_objective_methods_1d_ohe"
N = 1
for _ in range(NUM_MAX_TEST_TIMES):
M: int = np.random.randint(2, NUM_MAX_NODES)
assert M >= 2, "Softmax is for multi label classification. "\
" Use Sigmoid for binary classification."
_layer = layer.CrossEntropyLogLoss(name=name,
num_nodes=M,
log_level=logging.DEBUG)
# ================================================================================
# Layer forward path
# ================================================================================
X = np.random.randn(M).astype(TYPE_FLOAT)
T = np.zeros_like(X, dtype=TYPE_LABEL) # OHE labels.
T[np.random.randint(0, M)] = TYPE_LABEL(1)
_layer.T = T
P = softmax(X)
EG = ((P - T) / N).reshape(1, -1).astype(
TYPE_FLOAT) # Expected analytical gradient dL/dX = (P-T)/N
Logger.debug("%s: X is \n%s\nT is %s\nP is %s\nEG is %s\n", name, X, T,
P, EG)
# --------------------------------------------------------------------------------
# constraint: L/loss == np.sum(cross_entropy_log_loss(softmax(X), T)) / N.
# --------------------------------------------------------------------------------
L = _layer.function(X)
Z = np.array(np.sum(cross_entropy_log_loss(softmax(X), T)),
dtype=TYPE_FLOAT) / TYPE_FLOAT(N)
assert np.array_equal(
L, Z), f"SoftmaxLogLoss output should be {L} but {Z}."
# --------------------------------------------------------------------------------
# constraint: gradient_numerical() == numerical Jacobian numerical_jacobian(O, X)
# Use a dummy _layer for the objective function because using the "_layer"
# updates the X, Y which can interfere the independence of the _layer.
# --------------------------------------------------------------------------------
GN = _layer.gradient_numerical() # [dL/dX] from the _layer
# --------------------------------------------------------------------------------
# Cannot use CrossEntropyLogLoss.function() to simulate the objective function L.
# because it causes applying transform_X_T multiple times.
# Because internally transform_X_T(X, T) has transformed T into the index label
# in 1D with with length 1 by "T = T.reshape(-1)".
# Then providing X in 1D into "dummy.function(x)" re-run "transform_X_T(X, T)"
# again. The (X.ndim == T.ndim ==1) as an input and T must be OHE label for such
# combination and T.shape == P.shape must be true for OHE labels.
# However, T has been converted into the index format already by transform_X_T
# (applying transform_X_T multiple times) and (T.shape=(1,1), X.shape=(1, > 1)
# that violates the (X.shape == T.shape) constraint.
# --------------------------------------------------------------------------------
# dummy = CrossEntropyLogLoss(
# name="dummy",
# num_nodes=M,
# log_level=logging.DEBUG
# )
# dummy.T = T
# dummy.objective = objective
# dummy.function(X)
# --------------------------------------------------------------------------------
# O = lambda x: dummy.objective(dummy.function(x)) # Objective function
O = lambda x: np.sum(cross_entropy_log_loss(softmax(x), T),
dtype=TYPE_FLOAT) / TYPE_FLOAT(N)
# --------------------------------------------------------------------------------
EGN = numerical_jacobian(O, X).reshape(1,
-1) # Expected numerical dL/dX
assert np.array_equal(GN[0], EGN), \
f"Layer gradient_numerical GN \n{GN} \nneeds to be \n{EGN}."
# ================================================================================
# Layer backward path
# ================================================================================
# --------------------------------------------------------------------------------
# constraint: Analytical gradient G: gradient() == (P-1)/N.
# --------------------------------------------------------------------------------
dY = TYPE_FLOAT(1)
G = _layer.gradient(dY)
assert np.all(np.abs(G-EG) <= GRADIENT_DIFF_ACCEPTANCE_VALUE), \
f"Layer gradient dL/dX \n{G} \nneeds to be \n{EG} but G-EG \n{np.abs(G-EG)}\n"
# --------------------------------------------------------------------------------
# constraint: Analytical gradient G is close to GN: gradient_numerical().
# --------------------------------------------------------------------------------
assert \
np.all(np.abs(G - GN[0]) <= GRADIENT_DIFF_ACCEPTANCE_VALUE) or \
np.all(np.abs(G-GN[0]) <= np.abs(GRADIENT_DIFF_ACCEPTANCE_RATIO * GN[0])), \
f"dX is \n{G}\nGN[0] is \n{GN[0]}\nRatio * GN[0] is \n{GRADIENT_DIFF_ACCEPTANCE_RATIO * GN[0]}.\n"
def disabled_test_030_objective_methods_2d_ohe():
"""
TODO: Disabled as need to redesign numerical_jacobian for 32 bit floating.
Objective:
Verify the forward path constraints:
1. Layer output L/loss is np.sum(cross_entropy_log_loss(softmax(X), T)) / N.
2. gradient_numerical() == numerical Jacobian numerical_jacobian(O, X).
Verify the backward path constraints:
1. Analytical gradient G: gradient() == (P-1)/N
2. Analytical gradient G is close to GN: gradient_numerical().
Expected:
Initialization detects the access to the non-initialized parameters and fails.
"""
def objective(X: np.ndarray) -> Union[float, np.ndarray]:
"""Dummy objective function to calculate the loss L"""
assert X.ndim == 0, "The output of the log loss should be of shape ()"
return X
# --------------------------------------------------------------------------------
# Instantiate a CrossEntropyLogLoss layer
# --------------------------------------------------------------------------------
name = "test_030_objective_methods_2d_ohe"
for _ in range(NUM_MAX_TEST_TIMES):
N: int = np.random.randint(1, NUM_MAX_BATCH_SIZE)
M: int = np.random.randint(2, NUM_MAX_NODES)
assert M >= 2, "Softmax is for multi label classification. "\
" Use Sigmoid for binary classification."
_layer = layer.CrossEntropyLogLoss(name=name,
num_nodes=M,
log_level=logging.DEBUG)
_layer.objective = objective
# ================================================================================
# Layer forward path
# ================================================================================
X = np.random.randn(N, M).astype(TYPE_FLOAT)
T = np.zeros_like(X, dtype=TYPE_LABEL) # OHE labels.
T[np.arange(N), np.random.randint(0, M, N)] = TYPE_LABEL(1)
_layer.T = T
Logger.debug("%s: X is \n%s\nT is \n%s", name, X, T)
P = softmax(X)
EG = (P - T) / N # Expected analytical gradient dL/dX = (P-T)/N
# --------------------------------------------------------------------------------
# constraint: L/loss == np.sum(cross_entropy_log_loss(softmax(X), T)) / N.
# --------------------------------------------------------------------------------
L = _layer.function(X)
Z = np.array(np.sum(cross_entropy_log_loss(softmax(X), T))) / N
assert np.array_equal(
L, Z), f"SoftmaxLogLoss output should be {L} but {Z}."
# --------------------------------------------------------------------------------
# constraint: gradient_numerical() == numerical Jacobian numerical_jacobian(O, X)
# --------------------------------------------------------------------------------
GN = _layer.gradient_numerical() # [dL/dX] from the _layer
# --------------------------------------------------------------------------------
# DO not use CrossEntropyLogLoss.function() to simulate the objective function for
# the expected GN. See the same part in test_030_objective_methods_1d_ohe().
# --------------------------------------------------------------------------------
# dummy= CrossEntropyLogLoss(
# name=name,
# num_nodes=M,
# log_level=logging.DEBUG
# )
# dummy.T = T
# dummy.objective = objective
# O = lambda x: dummy.objective(dummy.function(x)) # Objective function
O = lambda x: np.sum(cross_entropy_log_loss(softmax(x), T)) / N
# --------------------------------------------------------------------------------
EGN = numerical_jacobian(O, X) # Expected numerical dL/dX
assert np.array_equal(GN[0], EGN), \
f"GN[0]==EGN expected but GN[0] is \n%s\n EGN is \n%s\n" % (GN[0], EGN)
# ================================================================================
# Layer backward path
# ================================================================================
# --------------------------------------------------------------------------------
# constraint: Analytical gradient G: gradient() == (P-1)/N.
# --------------------------------------------------------------------------------
dY = float(1)
G = _layer.gradient(dY)
assert np.all(abs(G-EG) <= GRADIENT_DIFF_ACCEPTANCE_VALUE), \
f"Layer gradient dL/dX \n{G} \nneeds to be \n{EG}."
# --------------------------------------------------------------------------------
# constraint: Analytical gradient G is close to GN: gradient_numerical().
# --------------------------------------------------------------------------------
assert \
np.all(np.abs(G - GN[0]) <= GRADIENT_DIFF_ACCEPTANCE_VALUE) or \
np.all(np.abs(G - GN[0]) <= np.abs(GRADIENT_DIFF_ACCEPTANCE_RATIO * GN[0])), \
f"dX is \n{G}\nGN[0] is \n{GN[0]}\nRatio * GN[0] is \n{GRADIENT_DIFF_ACCEPTANCE_RATIO * GN[0]}.\n"
def train_binary_classifier(N: int,
D: int,
M: int,
X: np.ndarray,
T: np.ndarray,
W: np.ndarray,
log_loss_function: Callable,
optimizer: Optimizer,
num_epochs: int = 100,
test_numerical_gradient: bool = False,
log_level: int = logging.ERROR,
callback: Callable = None):
"""Test case for binary classification with matmul + log loss.
Args:
N: Batch size
D: Number of features
M: Number of nodes. 1 for sigmoid and 2 for softmax
X: train data
T: labels
W: weight
log_loss_function: cross entropy logg loss function
optimizer: Optimizer
num_epochs: Number of epochs to run
test_numerical_gradient: Flag if test the analytical gradient with the numerical one.
log_level: logging level
callback: callback function to invoke at the each epoch end.
"""
name = __name__
assert isinstance(T, np.ndarray) and np.issubdtype(
T.dtype, np.integer) and T.ndim == 1 and T.shape[0] == N
assert isinstance(
X, np.ndarray) and X.dtype == TYPE_FLOAT and X.ndim == 2 and X.shape[
0] == N and X.shape[1] == D
assert isinstance(
W, np.ndarray) and W.dtype == TYPE_FLOAT and W.ndim == 2 and W.shape[
0] == M and W.shape[1] == D + 1
assert num_epochs > 0 and N > 0 and D > 0
assert ((log_loss_function == sigmoid_cross_entropy_log_loss and M == 1) or
(log_loss_function == softmax_cross_entropy_log_loss and M >= 2))
# --------------------------------------------------------------------------------
# Instantiate a CrossEntropyLogLoss layer
# --------------------------------------------------------------------------------
loss = CrossEntropyLogLoss(name="loss",
num_nodes=M,
log_loss_function=log_loss_function,
log_level=log_level)
# --------------------------------------------------------------------------------
# Instantiate a Matmul layer
# --------------------------------------------------------------------------------
matmul = Matmul(name="matmul",
num_nodes=M,
W=W,
optimizer=optimizer,
log_level=log_level)
matmul.objective = loss.function
num_no_progress: int = 0 # how many time when loss L not decreased.
loss.T = T
history: List[np.ndarray] = [loss.function(matmul.function(X))]
for i in range(num_epochs):
# --------------------------------------------------------------------------------
# Layer forward path
# Calculate the matmul output Y=f(X), and get the loss L = objective(Y)
# Test the numerical gradient dL/dX=matmul.gradient_numerical().
# --------------------------------------------------------------------------------
Y = matmul.function(X)
L = loss.function(Y)
if not (i % 50): print(f"iteration {i} Loss {L}")
Logger.info("%s: iteration[%s]. Loss is [%s]", name, i, L)
# --------------------------------------------------------------------------------
# Constraint: 1. Objective/Loss L(Yn+1) after gradient descent < L(Yn)
# --------------------------------------------------------------------------------
if L >= history[-1] and (i % 20) == 1:
Logger.warning(
"Iteration [%i]: Loss[%s] has not improved from the previous [%s].",
i, L, history[-1])
if (num_no_progress := num_no_progress + 1) > 20:
Logger.error(
"The training has no progress more than %s times.",
num_no_progress)
# break
else:
num_no_progress = 0
history.append(L)
# --------------------------------------------------------------------------------
# Expected dL/dW.T = X.T @ dL/dY = X.T @ (P-T) / N, and dL/dX = dL/dY @ W
# P = sigmoid(X) or softmax(X)
# dL/dX = dL/dY * W is to use W BEFORE updating W.
# --------------------------------------------------------------------------------
P = None
if log_loss_function == sigmoid_cross_entropy_log_loss:
# P = sigmoid(np.matmul(X, W.T))
P = sigmoid(np.matmul(matmul.X, matmul.W.T))
P = P - T.reshape(-1, 1) # T(N,) -> T(N,1) to align with P(N,1)
assert P.shape == (
N, 1), "P.shape is %s T.shape is %s" % (P.shape, T.shape)
elif log_loss_function == softmax_cross_entropy_log_loss:
# matmul.X.shape is (N, D+1), matmul.W.T.shape is (D+1, M)
P = softmax(np.matmul(matmul.X, matmul.W.T)) # (N, M)
P[np.arange(N), T] -= 1
EDX = np.matmul(P / N, matmul.W) # (N,M) @ (M, D+1) -> (N, D+1)
EDX = EDX[::, 1:] # Hide the bias -> (N, D)
EDW = np.matmul(matmul.X.T,
P / N).T # ((D+1,N) @ (N, M)).T -> (M, D+1)
# --------------------------------------------------------------------------------
# Layer backward path
# 1. Calculate the analytical gradient dL/dX=matmul.gradient(dL/dY) with a dL/dY.
# 2. Gradient descent to update Wn+1 = Wn - lr * dL/dX.
# --------------------------------------------------------------------------------
before = copy.deepcopy(matmul.W)
dY = loss.gradient(TYPE_FLOAT(1))
dX = matmul.gradient(dY)
# gradient descent and get the analytical gradients dS=[dL/dX, dL/dW]
# dL/dX.shape = (N, D)
# dL/dW.shape = (M, D+1)
dS = matmul.update()
dW = dS[0]
# --------------------------------------------------------------------------------
# Constraint 1. W in the matmul has been updated by the gradient descent.
# --------------------------------------------------------------------------------
Logger.debug("W after is \n%s", matmul.W)
assert not np.array_equal(before, matmul.W), "W has not been updated."
if not validate_against_expected_gradient(EDX, dX):
Logger.warning("Expected dL/dX \n%s\nDiff\n%s", EDX, EDX - dX)
if not validate_against_expected_gradient(EDW, dW):
Logger.warning("Expected dL/dW \n%s\nDiff\n%s", EDW, EDW - dW)
if test_numerical_gradient:
# --------------------------------------------------------------------------------
# Numerical gradients gn=[dL/dX, dL/dW]
# dL/dX.shape = (N, D)
# dL/dW.shape = (M, D+1)
# --------------------------------------------------------------------------------
gn = matmul.gradient_numerical()
validate_against_numerical_gradient([dX] + dS, gn, Logger)
if callback:
# if W.shape[1] == 1 else callback(W=np.average(matmul.W, axis=0))
callback(W=matmul.W[0])
def loss(self, x, t):
z = self.predict(x)
y = softmax(z)
loss = cross_entropy_error(y, t)
return loss
def test_010_softmax_cross_entropy_log_loss_2d(caplog):
"""
Objective:
Test case for softmax_cross_entropy_log_loss(X, T) = -T * log(softmax(X))
For the input X of shape (N,M) and T in index format of shape (N,),
calculate the softmax log loss and verify the values are as expected.
Expected:
For P = softmax(X) = exp(-X) / sum(exp(-X))
_P = P[
np.arange(N),
T
] selects the probability p for the correct input x.
Then -log(_P) should be almost same with softmax_cross_entropy_log_loss(X, T).
Almost because finite float precision always has rounding errors.
"""
# caplog.set_level(logging.DEBUG, logger=Logger.name)
u = REFORMULA_DIFF_ACCEPTANCE_VALUE
# --------------------------------------------------------------------------------
# [Test case 01]
# N: Batch size, M: Number of features in X
# X:(N,M)=(1, 2). X=(x0, x1) where x0 == x1 == 0.5 by which softmax(X) generates equal
# probability P=(p0, p1) where p0 == p1.
# Expected:
# softmax(X) generates the same with X.
# softmax_cross_entropy_log_loss(X, T) == -log(0.5)
# --------------------------------------------------------------------------------
X = np.array([[0.5, 0.5]]).astype(TYPE_FLOAT)
T = np.array([1]).astype(TYPE_LABEL)
E = -logarithm(np.array([0.5]).astype(TYPE_FLOAT))
P = softmax(X)
assert np.array_equal(X, P)
J, _ = softmax_cross_entropy_log_loss(X, T)
assert (E.shape == J.shape)
assert np.all(np.abs(E - J) < u), \
"Expected abs(E-J) < %s but \n%s\nE=\n%s\nT=%s\nX=\n%s\nJ=\n%s\n" \
% (u, np.abs(E - J), E, T, X, J)
assert np.all(np.abs(P - _) < u)
# --------------------------------------------------------------------------------
# [Test case 01]
# For X:(N,M)
# --------------------------------------------------------------------------------
for _ in range(NUM_MAX_TEST_TIMES):
# X(N, M), and T(N,) in index label format
N = np.random.randint(1, NUM_MAX_BATCH_SIZE)
M = np.random.randint(2, NUM_MAX_NODES)
X = np.random.randn(N, M).astype(TYPE_FLOAT)
T = np.random.randint(0, M, N).astype(TYPE_LABEL)
Logger.debug("T is %s\nX is \n%s\n", T, X)
# ----------------------------------------------------------------------
# Expected value E = -logarithm(_P)
# ----------------------------------------------------------------------
P = softmax(X)
_P = P[np.arange(
N
), T] # Probability of p for the correct input x, which generates j=-log(p)
E = -logarithm(_P)
# ----------------------------------------------------------------------
# Actual J should be close to E.
# ----------------------------------------------------------------------
J, _ = softmax_cross_entropy_log_loss(X, T)
assert (E.shape == J.shape)
assert np.all(np.abs(E-J) < u), \
"Expected abs(E-J) < %s but \n%s\nE=\n%s\nT=%s\nX=\n%s\nJ=\n%s\n" \
% (u, np.abs(E - J), E, T, X, J)
# ----------------------------------------------------------------------
# L = cross_entropy_log_loss(P, T) should be close to J
# ----------------------------------------------------------------------
L = cross_entropy_log_loss(P, T)
assert (L.shape == J.shape)
assert np.all(np.abs(L-J) < u), \
"Expected abs(L-J) < %s but \n%s\nL=\n%s\nT=%s\nX=\n%s\nJ=\n%s\n" \
% (u, np.abs(E - J), E, T, X, J)
