def testMaxLearningRate(self, train_shape, network, out_logits,
fn_and_kernel):
key = stateless_uniform(shape=[2],
seed=[0, 0],
minval=None,
maxval=None,
dtype=tf.int32)
keys = tf_random_split(key)
key = keys[0]
split = keys[1]
if len(train_shape) == 2:
train_shape = (train_shape[0] * 5, train_shape[1] * 10)
else:
train_shape = (16, 8, 8, 3)
x_train = np.asarray(normal(train_shape, seed=split))
keys = tf_random_split(key)
key = keys[0]
split = keys[1]
y_train = np.asarray(
stateless_uniform(shape=(train_shape[0], out_logits),
seed=split,
minval=0,
maxval=1) < 0.5, np.float32)
# Regress to an MSE loss.
loss = lambda params, x: 0.5 * np.mean((f(params, x) - y_train)**2)
grad_loss = jit(grad(loss))
def get_loss(opt_state):
return loss(get_params(opt_state), x_train)
steps = 20
for lr_factor in [0.5, 3.]:
params, f, ntk = fn_and_kernel(key, train_shape[1:], network,
out_logits)
g_dd = ntk(x_train, None, 'ntk')
step_size = predict.max_learning_rate(
g_dd, y_train_size=y_train.size) * lr_factor
opt_init, opt_update, get_params = optimizers.sgd(step_size)
opt_state = opt_init(params)
init_loss = get_loss(opt_state)
for i in range(steps):
params = get_params(opt_state)
opt_state = opt_update(i, grad_loss(params, x_train),
opt_state)
trained_loss = get_loss(opt_state)
loss_ratio = trained_loss / (init_loss + 1e-12)
if lr_factor == 3.:
if not math.isnan(loss_ratio):
self.assertGreater(loss_ratio, 10.)
else:
self.assertLess(loss_ratio, 0.1)
def main(unused_argv):
# Build data pipelines.
print('Loading data.')
x_train, y_train, x_test, y_test = \
datasets.get_dataset('mnist', FLAGS.train_size, FLAGS.test_size)
# Build the network
init_fn, apply_fn, _ = stax.serial(stax.Dense(512, 1., 0.05), stax.Erf(),
stax.Dense(10, 1., 0.05))
key = stateless_uniform(shape=[2],
seed=[0, 0],
minval=None,
maxval=None,
dtype=tf.int32)
_, params = init_fn(key, (1, 784))
# Create and initialize an optimizer.
opt_init, opt_apply, get_params = optimizers.sgd(FLAGS.learning_rate)
state = opt_init(params)
# Create an mse loss function and a gradient function.
loss = lambda fx, y_hat: 0.5 * np.mean((fx - y_hat)**2)
grad_loss = jit(grad(lambda params, x, y: loss(apply_fn(params, x), y)))
# Create an MSE predictor to solve the NTK equation in function space.
ntk = nt.batch(nt.empirical_ntk_fn(apply_fn), batch_size=4, device_count=0)
g_dd = ntk(x_train, None, params)
g_td = ntk(x_test, x_train, params)
predictor = nt.predict.gradient_descent_mse(g_dd, y_train)
# Get initial values of the network in function space.
fx_train = apply_fn(params, x_train)
fx_test = apply_fn(params, x_test)
# Train the network.
train_steps = int(FLAGS.train_time // FLAGS.learning_rate)
print('Training for {} steps'.format(train_steps))
for i in range(train_steps):
params = get_params(state)
state = opt_apply(i, grad_loss(params, x_train, y_train), state)
# Get predictions from analytic computation.
print('Computing analytic prediction.')
fx_train, fx_test = predictor(FLAGS.train_time, fx_train, fx_test, g_td)
# Print out summary data comparing the linear / nonlinear model.
util.print_summary('train', y_train, apply_fn(params, x_train), fx_train,
loss)
util.print_summary('test', y_test, apply_fn(params, x_test), fx_test, loss)
def testTrainedEnsemblePredCov(self, train_shape, test_shape, network,
out_logits):
training_steps = 1000
learning_rate = 0.1
ensemble_size = 1024
init_fn, apply_fn, kernel_fn = stax.serial(
stax.Dense(128, W_std=1.2, b_std=0.05), stax.Erf(),
stax.Dense(out_logits, W_std=1.2, b_std=0.05))
opt_init, opt_update, get_params = optimizers.sgd(learning_rate)
opt_update = jit(opt_update)
key, x_test, x_train, y_train = self._get_inputs(
out_logits, test_shape, train_shape)
predict_fn_mse_ens = predict.gradient_descent_mse_ensemble(
kernel_fn,
x_train,
y_train,
learning_rate=learning_rate,
diag_reg=0.)
train = (x_train, y_train)
ensemble_key = tf_random_split(key, ensemble_size)
loss = jit(lambda params, x, y: 0.5 * np.mean(
(apply_fn(params, x) - y)**2))
grad_loss = jit(lambda state, x, y: grad(loss)
(get_params(state), x, y))
def train_network(key):
_, params = init_fn(key, (-1, ) + train_shape[1:])
opt_state = opt_init(params)
for i in range(training_steps):
opt_state = opt_update(i, grad_loss(opt_state, *train),
opt_state)
return get_params(opt_state)
params = vmap(train_network)(ensemble_key)
rtol = 0.08
for x in [None, 'x_test']:
with self.subTest(x=x):
x = x if x is None else x_test
x_fin = x_train if x is None else x_test
ensemble_fx = vmap(apply_fn, (0, None))(params, x_fin)
mean_emp = np.mean(ensemble_fx, axis=0)
mean_subtracted = ensemble_fx - mean_emp
cov_emp = np.einsum(
'ijk,ilk->jl',
mean_subtracted,
mean_subtracted,
optimize=True) / (mean_subtracted.shape[0] *
mean_subtracted.shape[-1])
ntk = predict_fn_mse_ens(training_steps,
x,
'ntk',
compute_cov=True)
self._assertAllClose(mean_emp, ntk.mean, rtol)
self._assertAllClose(cov_emp, ntk.covariance, rtol)
def testNTKGDPrediction(self, train_shape, test_shape, network, out_logits,
fn_and_kernel, momentum, learning_rate, t, loss):
key, x_test, x_train, y_train = self._get_inputs(
out_logits, test_shape, train_shape)
params, f, ntk = fn_and_kernel(key, train_shape[1:], network,
out_logits)
g_dd = ntk(x_train, None, 'ntk')
g_td = ntk(x_test, x_train, 'ntk')
# Regress to an MSE loss.
loss_fn = lambda y, y_hat: 0.5 * np.mean((y - y_hat)**2)
grad_loss = jit(grad(lambda params, x: loss_fn(f(params, x), y_train)))
trace_axes = () if g_dd.ndim == 4 else (-1, )
if loss == 'mse_analytic':
if momentum is not None:
raise absltest.SkipTest(momentum)
predictor = predict.gradient_descent_mse(
g_dd,
y_train,
learning_rate=learning_rate,
trace_axes=trace_axes)
elif loss == 'mse':
predictor = predict.gradient_descent(loss_fn,
g_dd,
y_train,
learning_rate=learning_rate,
momentum=momentum,
trace_axes=trace_axes)
else:
raise NotImplementedError(loss)
predictor = jit(predictor)
fx_train_0 = f(params, x_train)
fx_test_0 = f(params, x_test)
self._test_zero_time(predictor, fx_train_0, fx_test_0, g_td, momentum)
self._test_multi_step(predictor, fx_train_0, fx_test_0, g_td, momentum)
if loss == 'mse_analytic':
self._test_inf_time(predictor, fx_train_0, fx_test_0, g_td,
y_train)
if momentum is None:
opt_init, opt_update, get_params = optimizers.sgd(learning_rate)
else:
opt_init, opt_update, get_params = optimizers.momentum(
learning_rate, momentum)
opt_state = opt_init(params)
for i in range(t):
params = get_params(opt_state)
opt_state = opt_update(i, grad_loss(params, x_train), opt_state)
params = get_params(opt_state)
fx_train_nn, fx_test_nn = f(params, x_train), f(params, x_test)
fx_train_t, fx_test_t = predictor(t, fx_train_0, fx_test_0, g_td)
self.assertAllClose(fx_train_nn, fx_train_t, rtol=RTOL, atol=ATOL)
self.assertAllClose(fx_test_nn, fx_test_t, rtol=RTOL, atol=ATOL)
def main(unused_argv):
# Build data and .
print('Loading data.')
x_train, y_train, x_test, y_test = datasets.get_dataset('mnist',
permute_train=True)
# Build the network
init_fn, f, _ = stax.serial(stax.Dense(512, 1., 0.05), stax.Erf(),
stax.Dense(10, 1., 0.05))
key = stateless_uniform(shape=[2],
seed=[0, 0],
minval=None,
maxval=None,
dtype=tf.int32)
_, params = init_fn(key, (1, 784))
# Linearize the network about its initial parameters.
f_lin = nt.linearize(f, params)
# Create and initialize an optimizer for both f and f_lin.
opt_init, opt_apply, get_params = momentum(FLAGS.learning_rate, 0.9)
# opt_init, opt_apply, get_params = optimizers.sgd(FLAGS.learning_rate)
opt_apply = jit(opt_apply)
state = opt_init(params)
state_lin = opt_init(params)
# momentum = MomentumOptimizer(learning_rate=FLAGS.learning_rate, momentum=0.9)
# momentum_lin = MomentumOptimizer(learning_rate=FLAGS.learning_rate, momentum=0.9)
# Create a cross-entropy loss function.
loss = lambda fx, y_hat: -np.mean(log_softmax(fx) * y_hat)
# Specialize the loss function to compute gradients for both linearized and
# full networks.
grad_loss = jit(grad(lambda params, x, y: loss(f(params, x), y)))
grad_loss_lin = jit(grad(lambda params, x, y: loss(f_lin(params, x), y)))
# Train the network.
print('Training.')
print('Epoch\tLoss\tLinearized Loss')
print('------------------------------------------')
epoch = 0
steps_per_epoch = 50000 // FLAGS.batch_size
for i, (x, y) in enumerate(
datasets.minibatch(x_train, y_train, FLAGS.batch_size,
FLAGS.train_epochs)):
params = get_params(state)
state = opt_apply(i, grad_loss(params, x, y), state)
params_lin = get_params(state_lin)
state_lin = opt_apply(i, grad_loss_lin(params_lin, x, y), state_lin)
# x = np.asarray(x)
# y = np.asarray(y)
# momentum.apply_gradients((grad_loss(params, x, y), params))
# momentum.apply_gradients((grad_loss_lin(params_lin, x, y), params_lin))
if i % steps_per_epoch == 0:
print('{}\t{}\t{}'.format(epoch, loss(f(params, x), y),
loss(f_lin(params_lin, x), y)))
epoch += 1
# Print out summary data comparing the linear / nonlinear model.
x, y = x_train[:10000], y_train[:10000]
util.print_summary('train', y, f(params, x), f_lin(params_lin, x), loss)
util.print_summary('test', y_test, f(params, x_test),
f_lin(params_lin, x_test), loss)
def gradient_descent(
loss: Callable[[np.ndarray, np.ndarray], float],
k_train_train: np.ndarray,
y_train: np.ndarray,
learning_rate: float = 1.,
momentum: float = None,
trace_axes: Axes = (-1, )
) -> Callable[[
ArrayOrScalar, Union[ArrayOrScalar,
ODEState], ArrayOrScalar, Optional[np.ndarray]
], Union[np.ndarray, Tuple[np.ndarray, np.ndarray], ODEState]]:
r"""Predicts the outcome of function space training using gradient descent.
Uses an ODE solver. If `momentum != None`, solves a continuous-time version of
gradient descent with momentum (note: this case uses standard momentum as
opposed to Nesterov momentum).
Solves the function space ODE for [momentum] gradient descent with a given
`loss` (detailed in [*]) given a Neural Tangent Kernel[s] over the dataset[s]
at arbitrary time[s] (step[s]) `t`. Note that for gradient descent
`absolute_time = learning_rate * t` and the scales of the learning rate and
query step[s] `t` are interchangeable. However, the momentum gradient descent
ODE is solved in the units of `learning_rate**0.5`, and therefore
`absolute_time = learning_rate**0.5 * t`, hence the `learning_rate` and
training time[s] (step[s]) `t` scales are not interchangeable.
[*] https://arxiv.org/abs/1902.06720
Example:
>>> from neural_tangents import empirical_ntk_fn
>>> from neural_tangents import predict
>>>
>>> t = 1e-7
>>> learning_rate = 1e-2
>>> momentum = 0.9
>>>
>>> kernel_fn = empirical_ntk_fn(f)
>>> k_test_train = kernel_fn(x_test, x_train, params)
>>>
>>> from jax.experimental import stax
>>> cross_entropy = lambda fx, y_hat: -np.mean(stax.logsoftmax(fx) * y_hat)
>>> predict_fn = predict.gradient_descent(cross_entropy, k_train_train,
>>> y_train, learning_rate, momentum)
>>>
>>> fx_train_0 = f(params, x_train)
>>> fx_test_0 = f(params, x_test)
>>>
>>> fx_train_t, fx_test_t = predict_fn(t, fx_train_0, fx_test_0,
>>> k_test_train)
Args:
loss:
a loss function whose signature is `loss(f(x_train), y_train)`. Note:
the loss function should treat the batch and output dimensions
symmetrically.
k_train_train:
kernel on the training data. Must have the shape of
`zip(y_train.shape, y_train.shape)` with `trace_axes` absent.
y_train:
targets for the training data.
learning_rate:
learning rate, step size.
momentum:
momentum scalar.
trace_axes:
`f(x_train)` axes such that `k_train_train` lacks these pairs of
dimensions and is to be interpreted as :math:`\Theta \otimes I`, i.e.
block-diagonal along `trace_axes`. These can can be specified either to
save space and compute, or to even improve approximation accuracy of the
infinite-width or infinite-samples limit, since in in these limits the
covariance along channel / feature / logit axes indeed converges to a
constant-diagonal matrix. However, if you target linearized dynamics of a
specific finite-width network, `trace_axes=()` will yield most accurate
result.
Returns:
A function that returns output train [and test] set[s] predictions at
time[s] `t`.
"""
_, odd, _, _ = _get_axes(k_train_train)
trace_axes = utils.canonicalize_axis(trace_axes, y_train)
non_t_axes = tuple(a for a in range(y_train.ndim) if a not in trace_axes)
last_t_axes = range(-len(trace_axes), 0)
dtype = k_train_train.dtype
grad_loss = grad(lambda fx: loss(fx, y_train))
if momentum is not None:
learning_rate **= 0.5
momentum = (momentum - 1.0) / learning_rate
def get_state_0(fx_train_or_state_0, fx_test_0, fx_test_shape):
if isinstance(fx_train_or_state_0, ODEState):
fx_train_0 = fx_train_or_state_0.fx_train
fx_test_0 = fx_train_or_state_0.fx_test
qx_train_0 = fx_train_or_state_0.qx_train
qx_test_0 = fx_train_or_state_0.qx_test
else:
fx_train_0 = fx_train_or_state_0
qx_train_0 = qx_test_0 = None
if fx_train_0 is None:
fx_train_0 = np.zeros_like(y_train, dtype)
else:
fx_train_0 = np.broadcast_to(fx_train_0, y_train.shape)
if fx_test_0 is not None:
fx_test_0 = np.broadcast_to(fx_test_0, fx_test_shape)
if momentum is None:
if qx_train_0 is not None or qx_test_0 is not None:
raise ValueError('Got passed momentum state variables, while '
'`momentum is None`.')
else:
qx_train_0 = (np.zeros_like(y_train, dtype) if qx_train_0 is None
else np.broadcast_to(qx_train_0, y_train.shape))
qx_test_0 = (None if fx_test_0 is None else
(np.zeros(fx_test_shape, dtype) if qx_test_0 is None
else np.broadcast_to(qx_test_0, fx_test_shape)))
return ODEState(fx_train_0, fx_test_0, qx_train_0, qx_test_0) # pytype: disable=wrong-arg-count
def get_dstate_dt(k_test_train):
def dstate_dt(state_t: ODEState, unused_t) -> ODEState:
fx_train_t, fx_test_t, qx_train_t, qx_test_t = (state_t.fx_train,
state_t.fx_test,
state_t.qx_train,
state_t.qx_test)
dy_df_t = grad_loss(fx_train_t)
fx_train_t = -np.moveaxis(
np.tensordot(k_train_train, dy_df_t,
(odd, non_t_axes)), last_t_axes, trace_axes)
if fx_test_t is not None:
fx_test_t = -np.moveaxis(
np.tensordot(k_test_train, dy_df_t,
(odd, non_t_axes)), last_t_axes, trace_axes)
if momentum is None:
return ODEState(fx_train_t, fx_test_t) # pytype: disable=wrong-arg-count
fx_train_t += momentum * qx_train_t
if qx_test_t is not None:
fx_test_t += momentum * qx_test_t
return ODEState(qx_train_t, qx_test_t, fx_train_t, fx_test_t) # pytype: disable=wrong-arg-count
return dstate_dt
def predict_fn(
t: ArrayOrScalar = None,
fx_train_or_state_0: Union[ArrayOrScalar, ODEState] = 0.,
fx_test_0: ArrayOrScalar = None,
k_test_train: np.ndarray = None
) -> Union[np.ndarray, Tuple[np.ndarray, np.ndarray], ODEState]:
"""Return output predictions on train [and test] set[s] at time[s] `t`.
Args:
t:
a scalar or array of scalars of any shape in strictly increasing order.
`t=None` is equivalent to `t=np.inf` and may not converge. Equivalent of
training steps (but can be fractional).
fx_train_or_state_0:
either (a) output of the network at `t == 0` on the training set or (b)
complete ODE state (`predict.ODEState`). Pass an ODE state if you want
to operate on the full ODE state instead of output variables only
(useful for inspecting auxiliary variables or resuming an optimizer with
auxiliary variables from a specific state. Note that only
`momentum != None` optimizer currently has auxiliary variables. To
initialize an ODE state from scratch, call
`predict.ODEState(fx_train_0, fx_test_0)`. If an ODE state is passed, an
ODE state is returned. `fx_train_0=None` means to not compute
predictions on the training set.
fx_test_0:
output of the network at `t == 0` on the test set. `fx_test_0=None`
means to not compute predictions on the test set.
k_test_train:
kernel relating test data with training data. Must have the shape of
`zip(y_test.shape, y_train.shape)` with `trace_axes` absent. Pass
`k_test_train=None` if you only need predictions on the training set.
Returns:
`fx_train_t` or `(fx_train_t, fx_test_t)` if `fx_test_0 != None` with
potentially additional leading time dimensions matching `t.shape`.
Alternatively can return an `ODEState` at time[s] `t`.
Raises:
ValueError: if `fx_test_0` is not `None`, but `k_test_train` is `None`.
"""
_check_inputs(fx_train_or_state_0, fx_test_0, k_test_train)
t = np.array(t if t is not None else np.inf, dtype) * learning_rate
t_shape = t.shape
t = t.reshape((-1, ))
# ODE solver requires `t[0]` to be the time where `fx_train_0` [and
# `fx_test_0`] are evaluated, but also a strictly increasing sequence of
# timesteps, so we always temporarily append an [almost] `0` at the start.
identity = lambda x: x
t0 = np.where(t[0] == 0, np.full((1, ), -1e-24, t.dtype),
np.zeros((1, ), t.dtype))
t = np.concatenate([t0, t])
# Solve the ODE.
fx_test_shape = _get_fx_test_shape(y_train, k_test_train, trace_axes)
state_0 = get_state_0(fx_train_or_state_0, fx_test_0, fx_test_shape)
state_t = ode.odeint(get_dstate_dt(k_test_train), state_0, t)
# Remove the added `t0`.
trim = lambda x: x[1:].reshape(t_shape + x.shape[1:])
trim_tree = lambda tree: tree_map(trim, tree)
state_t = trim_tree(state_t)
# `ODEState` -> `ODEState`
if isinstance(fx_train_or_state_0, ODEState):
return state_t
# `np.ndarray` -> `np.ndarray`
fx_train_t, fx_test_t = state_t.fx_train, state_t.fx_test
if fx_train_or_state_0 is not None and fx_test_0 is None:
return fx_train_t
if fx_test_0 is not None and fx_train_or_state_0 is None:
return fx_test_t
return fx_train_t, fx_test_t
return predict_fn