def update(opt_state, step_size):
_, update_fun, get_params = optimizers.sgd(step_size)
x = get_params(opt_state)
g = grad(loss)(x)
return update_fun(0, g, opt_state)
def test_grad(self, with_jit=False): x = np.float32(3.) res = _maybe_jit(with_jit, jax.grad(jax2tf.call_tf(tf.math.sin)))(x) self.assertAllClose(np.cos(x), res)
def update(params, opt_state, problem):
g = jax.grad(prediction_loss)(params, problem)
updates, opt_state = opt_update(g, opt_state)
return optax.apply_updates(params, updates), opt_state
def test_gradient_log_normalizer(
generator: Generator,
distribution_info: DistributionInfo[Any, Any, Any]) -> None:
"""
Tests that the gradient log-normalizer evaluates to the same as the gradient of the
log-normalizer.
"""
# pylint: disable=too-many-locals, disable=protected-access
cls = type(distribution_info.nat_parameter_generator(generator, shape=()))
original_ln = cls._original_log_normalizer
original_gln = jit(grad(cls._original_log_normalizer))
optimized_ln = cls.log_normalizer
optimized_gln = jit(grad(optimized_ln))
for _ in range(20):
nat_parameters = distribution_info.nat_parameter_generator(generator,
shape=())
kw_nat_parameters = nat_parameters.fixed_parameters_mapping()
exp_parameters = nat_parameters.to_exp() # Regular transformation.
nat_cls = type(nat_parameters)
ep_cls = type(exp_parameters)
# Original GLN.
original_nat_parameters = original_gln(nat_parameters)
f_original_nat_parameters = original_nat_parameters.flattened()
original_exp_parameters = ep_cls.unflattened(f_original_nat_parameters,
**kw_nat_parameters)
# Optimized GLN.
optimized_nat_parameters = optimized_gln(nat_parameters)
f_optimized_nat_parameters = optimized_nat_parameters.flattened()
optimized_exp_parameters = ep_cls.unflattened(
f_optimized_nat_parameters, **kw_nat_parameters)
# Test primal evaluation.
assert_jax_allclose(exp_parameters, original_exp_parameters, rtol=1e-5)
assert_jax_allclose(exp_parameters,
optimized_exp_parameters,
rtol=1e-5)
# Test JVP.
ones_like_nat_parameters = nat_cls(
**{
name: jnp.zeros_like(value)
for name, value in
nat_parameters.fixed_parameters_mapping().items()
}, **{
name: jnp.ones_like(value)
for name, value in nat_parameters.parameters_name_value()
})
original_gradients = jvp(original_ln, (nat_parameters, ),
(ones_like_nat_parameters, ))
optimized_gradients = jvp(optimized_ln, (nat_parameters, ),
(ones_like_nat_parameters, ))
assert_allclose(original_gradients, optimized_gradients, rtol=1.5e-5)
# Test VJP.
original_ln_of_nat, original_vjp = vjp(original_ln, nat_parameters)
original_gln_of_nat, = original_vjp(1.0)
optimized_ln_of_nat, optimized_vjp = vjp(optimized_ln, nat_parameters)
optimized_gln_of_nat, = optimized_vjp(1.0)
assert_jax_allclose(original_ln_of_nat, optimized_ln_of_nat, rtol=1e-5)
for name, original_value in original_gln_of_nat.parameters_name_value(
):
optimized_value = getattr(optimized_gln_of_nat, name)
assert_jax_allclose(original_value, optimized_value, rtol=1e-5)
def update(params, opt_state, x, y_true, dropout_key):
grads = grad(mean_cross_entropy)(params, x, y_true, dropout_key)
updates, opt_state = opt.update(grads, opt_state, params)
params = optax.apply_updates(params, updates)
return params, opt_state
def testIssue1789(self):
def f(x):
return random.gamma(random.PRNGKey(0), x)
grad(lambda x: jnp.sum(vmap(f)(x)))(jnp.ones(2))
def main(unused_argv):
numpts = 7
key = random.PRNGKey(0)
eye = jnp.eye(numpts)
def cov_map(cov_func, xs, xs2=None):
"""Compute a covariance matrix from a covariance function and data points.
Args:
cov_func: callable function, maps pairs of data points to scalars.
xs: array of data points, stacked along the leading dimension.
Returns:
A 2d array `a` such that `a[i, j] = cov_func(xs[i], xs[j])`.
"""
if xs2 is None:
return vmap(lambda x: vmap(lambda y: cov_func(x, y))(xs))(xs)
else:
return vmap(lambda x: vmap(lambda y: cov_func(x, y))(xs))(xs2).T
def softplus(x):
return jnp.logaddexp(x, 0.)
# Note, writing out the vectorized form of the identity
# ||x-y||^2 = <x-y,x-y> = ||x||^2 + ||y||^2 - 2<x,y>
# for computing squared distances would be more efficient (but less succinct).
def exp_quadratic(x1, x2):
return jnp.exp(-jnp.sum((x1 - x2)**2))
def gp(params, x, y, xtest=None, compute_marginal_likelihood=False):
noise = softplus(params['noise'])
amp = softplus(params['amplitude'])
ls = softplus(params['lengthscale'])
ymean = jnp.mean(y)
y = y - ymean
x = x / ls
train_cov = amp*cov_map(exp_quadratic, x) + eye * (noise + 1e-6)
chol = scipy.linalg.cholesky(train_cov, lower=True)
kinvy = scipy.linalg.solve_triangular(
chol.T, scipy.linalg.solve_triangular(chol, y, lower=True))
if compute_marginal_likelihood:
log2pi = jnp.log(2. * 3.1415)
ml = jnp.sum(
-0.5 * jnp.dot(y.T, kinvy) -
jnp.sum(jnp.log(jnp.diag(chol))) -
(numpts / 2.) * log2pi)
ml -= jnp.sum(-0.5 * jnp.log(2 * 3.1415) - jnp.log(amp)**2) # lognormal prior
return -ml
if xtest is not None:
xtest = xtest / ls
cross_cov = amp*cov_map(exp_quadratic, x, xtest)
mu = jnp.dot(cross_cov.T, kinvy) + ymean
v = scipy.linalg.solve_triangular(chol, cross_cov, lower=True)
var = (amp * cov_map(exp_quadratic, xtest) - jnp.dot(v.T, v))
return mu, var
marginal_likelihood = partial(gp, compute_marginal_likelihood=True)
predict = partial(gp, compute_marginal_likelihood=False)
grad_fun = jit(grad(marginal_likelihood))
# Covariance hyperparameters to be learned
params = {"amplitude": jnp.zeros((1, 1)),
"noise": jnp.zeros((1, 1)) - 5.,
"lengthscale": jnp.zeros((1, 1))}
momentums = dict([(k, p * 0.) for k, p in params.items()])
scales = dict([(k, p * 0. + 1.) for k, p in params.items()])
lr = 0.01 # Learning rate
def train_step(params, momentums, scales, x, y):
grads = grad_fun(params, x, y)
for k in (params):
momentums[k] = 0.9 * momentums[k] + 0.1 * grads[k][0]
scales[k] = 0.9 * scales[k] + 0.1 * grads[k][0]**2
params[k] -= lr * momentums[k]/jnp.sqrt(scales[k] + 1e-5)
return params, momentums, scales
# Create a really simple toy 1D function
y_fun = lambda x: jnp.sin(x) + 0.1 * random.normal(key, shape=(x.shape[0], 1))
x = (random.uniform(key, shape=(numpts, 1)) * 4.) + 1
y = y_fun(x)
xtest = jnp.linspace(0, 6., 200)[:, None]
for i in range(1000):
params, momentums, scales = train_step(params, momentums, scales, x, y)
if i % 50 == 0:
ml = marginal_likelihood(params, x, y)
print("Step: %d, neg marginal likelihood: %f" % (i, ml))
print(params)
mu, var = predict(params, x, y, xtest)
std = jnp.sqrt(jnp.diag(var))
plt.plot(x, y, "k.")
plt.plot(xtest, mu)
plt.fill_between(xtest.flatten(),
mu.flatten() - std * 2, mu.flatten() + std * 2)
def test_grad_and_aux_nested(self):
def f(x):
g, aux = grad(lambda x: (x**3, [x**3]), has_aux=True)(x)
return aux[0]
f2 = lambda x: x**3
self.assertEqual(grad(f)(4.), grad(f2)(4.))
self.assertEqual(jit(grad(f))(4.), grad(f2)(4.))
self.assertEqual(jit(grad(jit(f)))(4.), grad(f2)(4.))
def f(x):
g, aux = grad(lambda x: (x**3, [x**3]), has_aux=True)(x)
return aux[0] * np.sin(x)
f2 = lambda x: x**3 * np.sin(x)
self.assertEqual(grad(f)(4.), grad(f2)(4.))
self.assertEqual(jit(grad(f))(4.), grad(f2)(4.))
self.assertEqual(jit(grad(jit(f)))(4.), grad(f2)(4.))
def grads_real(params, x):
r = jax.grad(lambda pars, v: f_real_flat_scalar(pars, v).real)(params, x)
i = jax.grad(lambda pars, v: f_real_flat_scalar(pars, v).imag)(params, x)
return jax.lax.complex(r, i)
def compute_gradients_jax(w, X_test, y_test):
print("Starting JAX demo")
grad_jax = jax.grad(NLL)(w, (X_test, y_test))
print("grad {}".format(grad_jax))
return grad_jax
eq = dfdx_vect(params, x, t, bc1) + dfdt_vect(params, x, t,
bc1) - (3 * x) - t
bc1_res = f_vect(params, x - x, t, bc1) - bc1
return np.mean(eq**2) + np.mean(bc1_res**2)
##########
## Main ##
##########
key = random.PRNGKey(0)
params = random.normal(key, shape=(401, ))
#-- Setting up the functions and derivatives --#
dfdx = grad(f, 1)
dfdt = grad(f, 2)
f_vect = vmap(f, (None, 0, 0, 0))
dfdx_vect = vmap(dfdx, (None, 0, 0, 0))
dfdt_vect = vmap(dfdt, (None, 0, 0, 0))
grad_loss = jit(grad(loss, 0))
#-- Defining the domain of x, t, and bc1 --#
x_values = np.linspace(-1, 1, num=40)
t_values = np.linspace(-1, 1, num=40)
bc1_values = np.linspace(1, 4, num=4)
x = []
t = []
bc1 = []
def body_fun(i, opt_state):
elbo_rng, data_rng = random.split(random.fold_in(rng, i))
batch = binarize_batch(data_rng, i, train_images)
loss = lambda params: -elbo(elbo_rng, params, batch) / batch_size
g = grad(loss)(get_params(opt_state))
return opt_update(i, g, opt_state)
def initialize(self, n, m, h=64):
"""
Description: Randomly initialize the RNN.
Args:
n (int): Input dimension.
m (int): Observation/output dimension.
h (int): Default value 64. Hidden dimension of RNN.
Returns:
The first value in the time-series
"""
self.T = 0
self.initialized = True
self.n, self.m, self.h = n, m, h
glorot_init = stax.glorot(
) # returns a function that initializes weights
self.W_h = glorot_init(generate_key(), (h, h))
self.W_u = glorot_init(generate_key(), (h, n))
self.W_out = glorot_init(generate_key(), (m, h))
self.b_h = np.zeros(h)
self.hid = np.zeros(h)
self.rollout_controller = None
self.target = jax.random.uniform(generate_key(),
shape=(self.m, ),
minval=-1,
maxval=1)
'''
def _step(x, hid):
next_hid = np.tanh(np.dot(self.W_h, hid) + np.dot(self.W_x, x) + self.b_h)
y = np.dot(self.W_out, next_hid)
return (next_hid, y)'''
def _dynamics(hid, u):
next_hid = np.tanh(
np.dot(self.W_h, hid) + np.dot(self.W_u, u) + self.b_h)
y = np.dot(self.W_out, next_hid)
return (next_hid, y)
# self._step = jax.jit(_step)
self._dynamics = jax.jit(_dynamics)
self._loss = lambda x, u: (self.target - self._dynamics(x, u))**2
# stack the jacobians of environment dynamics gradient
jacobian = jax.jacrev(self._dynamics, argnums=(0, 1))
self._dynamics_jacobian = jax.jit(
lambda x, u: np.hstack(jacobian(x, u)))
# stack the gradients of environment loss
loss_grad = jax.grad(self._loss, argnums=(0, 1))
self._loss_grad = jax.jit(lambda x, u: np.hstack(loss_grad(x, u)))
# block the hessian of environment loss
block_hessian = lambda A: np.vstack(
[np.hstack([A[0][0], A[0][1]]),
np.hstack([A[1][0], A[1][1]])])
hessian = jax.hessian(self._loss, argnums=(0, 1))
self._loss_hessian = jax.jit(lambda x, u: block_hessian(hessian(x, u)))
def _rollout(act, dyn, x_0, T):
def f(x, i):
u = act(x)
x_next = dyn(x, u)
return x_next, np.hstack((x, u))
_, trajectory = jax.lax.scan(f, x_0, np.arange(T))
return trajectory
self._rollout = jax.jit(_rollout, static_argnums=(0, 1, 3))
return np.dot(self.W_out, self.hid)
def step(i, opt_state):
p = get_params(opt_state)
g = grad(self.negative_log_evidence)(p)
return opt_update(i, g, opt_state)
def solve_sdp_dual_simple(verif_instance, key=None, opt=None, num_steps=10000,
eval_every=1000, verbose=False,
use_exact_eig_eval=True, use_exact_eig_train=False,
n_iter_lanczos=100,
kappa_reg_weight=None, kappa_zero_after=None,
device_type=None):
"""Compute verified lower bound via dual of SDP relaxation.
Args:
verif_instance: a utils.SdpDualVerifInstance
key: jax.random.PRNGKey, used for Lanczos
opt: an optax.GradientTransformation instance, the optimizer.
If None, defaults to Adam with learning rate 1e-3.
num_steps: int, the number of outer loop optimization steps
eval_every: int, frequency of running evaluation step
verbose: bool, enables verbose logging
use_exact_eig_eval: bool, whether to use exact eigendecomposition instead of
Lanczos when computing evaluation loss
use_exact_eig_train: bool, whether to use exact eigendecomposition instead
of Lanczos during training
n_iter_lanczos: int, number of Lanczos iterations
kappa_reg_weight: float, adds a penalty of sum(abs(kappa_{1:N})) to loss,
which regularizes kappa_{1:N} towards zero. Default None is disabled.
kappa_zero_after: int, clamps kappa_{1:N} to zero after ``kappa_zero_after``
steps. Default None is disabled.
device_type: string, used to clamp to a particular hardware device. Default
None uses JAX default device placement
Returns:
A pair. The first element is a float, the final dual loss, which forms a
valid upper bound on the objective specified by ``verif_instance``. The
second element is a dict containing various debug info.
"""
assert device_type in (None, 'cpu', 'gpu'), 'invalid device_type'
assert isinstance(verif_instance, utils.SdpDualVerifInstance), 'invalid type'
key = key if key is not None else jax.random.PRNGKey(0)
opt = opt if opt is not None else optax.adam(1e3)
dual_vars = jax.tree_map(
lambda s: None if s is None else jnp.zeros(s), verif_instance.dual_shapes)
dual_vars = init_duals_ibp(verif_instance, dual_vars)
# Define loss function
def loss(dual_vars, exact=use_exact_eig_train):
return _loss(dual_vars, exact)
@functools.partial(jax.jit, static_argnums=(1,), backend=device_type)
def _loss(dual_var, exact):
loss_val, step_info = dual_fun(
verif_instance, dual_var, key, n_iter=n_iter_lanczos, exact=exact,
include_info=True)
step_info['loss_val'] = loss_val
return loss_val, step_info
# Define a compiled update step
grad = jax.jit(jax.grad(loss, has_aux=True), backend=device_type)
@functools.partial(jax.jit, backend=device_type)
def grad_step(params, opt_state):
g, info = grad(params)
updates, new_opt_state = opt.update(g, opt_state)
new_params = optax.apply_updates(params, updates)
return new_params, new_opt_state, info
# Optimize parameters in a loop
opt_state = opt.init(dual_vars)
info = collections.defaultdict(list)
loss_log = []
best_loss = 1e9
# Main loop
for i in range(num_steps):
dual_vars, opt_state, step_info = grad_step(dual_vars, opt_state)
loss_val = step_info['loss_val']
print(f'Iter {i}: Loss {loss_val}')
best_loss = min(best_loss, loss_val)
loss_log.append(loss_val)
# Regularization of kappa
if kappa_reg_weight is not None and kappa_reg_weight >= 0:
onehot = jax.nn.one_hot([0], dual_vars[-1].shape[1])
mask = jnp.ones_like(onehot) - onehot
dual_vars[-1] -= mask * kappa_reg_weight
if (kappa_zero_after is not None and kappa_zero_after >= 0 and
i > kappa_zero_after):
onehot = jax.nn.one_hot([0], dual_vars[-1].shape[1])
dual_vars[-1] *= onehot
dual_vars = project_duals(dual_vars, verif_instance.dual_types)
if i % eval_every == 0:
dual_val, _ = loss(dual_vars, exact=use_exact_eig_eval)
info['steps'].append(i)
info['loss_vals'].append(float(dual_val))
if verbose:
print(f'Dual iter {i}: Train loss: {loss_val} Loss {dual_val}')
final_loss = float(loss(dual_vars, exact=use_exact_eig_eval)[0])
info['final_dual_vars'] = dual_vars
info['final_loss'] = final_loss
info['loss_log'] = loss_log
info['best_train_loss'] = best_loss
return final_loss, info
def test_grad_and_aux_basic(self):
g, aux = grad(lambda x: (x**3, [x**2]), has_aux=True)(3.)
self.assertAllClose(g, grad(lambda x: x**3)(3.), check_dtypes=True)
self.assertAllClose(aux, [9.], check_dtypes=True)
def solve_sdp_dual(verif_instance, key=None, opt=None, num_steps=10000,
verbose=False, eval_every=1000, use_exact_eig_eval=True,
use_exact_eig_train=False, n_iter_lanczos=30, scl=-1.0,
lr_init=1e-3, steps_per_anneal=100, anneal_factor=1.0,
num_anneals=3, opt_name='adam', gd_momentum=0.9,
add_diagnostic_stats=False,
opt_multiplier_fn=None, init_dual_vars=None,
init_opt_state=None, opt_dual_vars=None,
kappa_reg_weight=None, kappa_zero_after=None,
device_type=None, save_best_k=1):
# pylint: disable=g-doc-return-or-yield, g-doc-args
"""Compute verified lower bound via dual of SDP relaxation.
NOTE: This method exposes many hyperparameter options, and the method
signature is subject to change. We instead suggest using
``solve_sdp_dual_simple`` instead if you need a stable interface.
"""
# NB: Whereas the rest of the code in this library is fairly top-down
# readable, avoids excessive `if` statements, tries to make the code look
# like the formalism, etc, this is not the case for this method.
# This is essentially the outer loop, and includes all the debugging/logging/
# optimization tricks we need to get/debug good results.
#
# NB: Time profiling: On toy VerifInstances, JIT compilation dominates time
# cost: JIT compilation takes ~12s, then we do ~3000 steps/sec.
assert device_type in (None, 'cpu', 'gpu'), 'invalid device_type'
assert isinstance(verif_instance, utils.SdpDualVerifInstance), 'invalid type'
key = key if key is not None else jax.random.PRNGKey(0)
dual_vars = jax.tree_map(
lambda s: None if s is None else jnp.zeros(s), verif_instance.dual_shapes)
dual_vars = init_duals_ibp(verif_instance, dual_vars)
if init_dual_vars is not None:
# Casting, here for Colab. Essentially same as `dual_vars = init_dual_vars`
dual_vars = utils.structure_like(init_dual_vars, dual_vars)
if opt_dual_vars is not None:
opt_dual_vars = utils.structure_like(opt_dual_vars, dual_vars)
# Create optimizer
if opt is None:
if (isinstance(steps_per_anneal, float) or
isinstance(steps_per_anneal, int)):
anneal_steps = [steps_per_anneal*(i+1) for i in range(num_anneals)]
else:
anneal_steps = np.cumsum(steps_per_anneal)
anneal_steps = jnp.array(anneal_steps)
def lr_schedule(t):
cur_epoch = jnp.minimum(num_anneals, jnp.sum(t > anneal_steps))
return lr_init * jnp.float_power(anneal_factor, cur_epoch)
opt_class = getattr(optax, opt_name)
base_opt = (opt_class(1., momentum=gd_momentum) if opt_name == 'sgd' else
opt_class(1.))
opt = optax.chain(base_opt, optax.scale_by_schedule(lr_schedule))
if opt_multiplier_fn:
# NB: Interface very specific to tree.map_structure_with_path
# Example: opt_multiplier_fn=lambda path: 0.1 if 'lam' in path else 1.0
opt_multipliers = tree.map_structure_with_path(
lambda path, v: opt_multiplier_fn(path), dual_vars)
opt = optax.chain(base_opt, optax.scale_by_schedule(lr_schedule),
utils.scale_by_variable_opt(opt_multipliers))
else:
opt = optax.chain(base_opt, optax.scale_by_schedule(lr_schedule))
# Define loss function
def loss(dual_vars, loss_scl=scl, exact=use_exact_eig_train):
return _loss(dual_vars, loss_scl, exact)
@functools.partial(jax.jit, static_argnums=(1, 2), backend=device_type)
def _loss(dual_var, loss_scl, exact):
loss_val, step_info = dual_fun(
verif_instance, dual_var, key, n_iter=n_iter_lanczos, exact=exact,
scl=loss_scl, include_info=True)
step_info['loss_val'] = loss_val
return loss_val, step_info
# Define a compiled update step
grad = jax.jit(jax.grad(loss, has_aux=True), backend=device_type)
@functools.partial(jax.jit, backend=device_type)
def grad_step(params, opt_state):
g, info = grad(params)
updates, new_opt_state = opt.update(g, opt_state)
new_params = optax.apply_updates(params, updates)
info['g'] = g
info['updates'] = updates
return new_params, new_opt_state, info
# Optimize parameters in a loop
opt_state = opt.init(dual_vars)
if init_opt_state:
opt_state = utils.structure_like(init_opt_state, opt_state)
info = collections.defaultdict(list)
loss_log = []
store_best = []
recent_eig_vecs = collections.deque(maxlen=10)
best_loss = 1e9
last_H = None
start_i = 0
# Main loop
for i in range(start_i, num_steps):
dual_vars_prev = dual_vars
dual_vars, opt_state, step_info = grad_step(dual_vars, opt_state)
loss_val = step_info['loss_val']
print(f'Iter {i}: Loss {loss_val}')
best_loss = min(best_loss, loss_val)
if add_diagnostic_stats:
info['dual_vars'].append(dual_vars_prev)
eig_vec = step_info['eig_vec']
cosine_sims = []
for prev_eig_vec in recent_eig_vecs:
denom = jnp.sqrt(jnp.linalg.norm(eig_vec)*jnp.linalg.norm(prev_eig_vec))
eig_sim = jnp.sum(prev_eig_vec * eig_vec) / denom
cosine_sims.append(abs(float(eig_sim)))
info['c_lambda'].append(float(step_info['c_lambda']))
info['past_10_cosine_sims'].append(np.array(cosine_sims))
info['g'].append(step_info['g'])
info['updates'].append(step_info['updates'])
if use_exact_eig_train:
# The info is for -H, so to get smallest for H, take negative of max
eig_vals = -step_info['eig_info'][0][-1:-20:-1]
cur_H = step_info['eig_info'][2]
diff_H = 0 if last_H is None else np.linalg.norm(cur_H - last_H)
last_H = cur_H
info['diff_H'].append(float(diff_H))
info['smallest_20_eig_vals'].append(eig_vals)
recent_eig_vecs.appendleft(eig_vec)
loss_log.append(loss_val)
if len(store_best) < save_best_k:
store_best.append((loss_val, dual_vars_prev))
store_best.sort(key=lambda x: x[0])
elif loss_val < store_best[-1][0]:
store_best[-1] = (loss_val, dual_vars_prev)
store_best.sort(key=lambda x: x[0])
# Regularization of kappa
if kappa_reg_weight is not None and kappa_reg_weight >= 0:
onehot = jax.nn.one_hot([0], dual_vars[-1].shape[1])
mask = jnp.ones_like(onehot) - onehot
dual_vars[-1] -= mask * kappa_reg_weight
if (kappa_zero_after is not None and kappa_zero_after >= 0 and
i > kappa_zero_after):
onehot = jax.nn.one_hot([0], dual_vars[-1].shape[1])
dual_vars[-1] *= onehot
dual_vars = project_duals(dual_vars, verif_instance.dual_types)
if opt_dual_vars:
distance_to_opt = jax.tree_multimap(lambda x, y: jnp.linalg.norm(x - y),
dual_vars, opt_dual_vars)
info['distance_to_opt'].append(distance_to_opt)
if i % eval_every == 0:
dual_val, _ = loss(dual_vars, loss_scl=-1, exact=use_exact_eig_eval)
info['steps'].append(i)
info['loss_vals'].append(float(dual_val))
if verbose:
print(f'Dual iter {i}: Train loss: {loss_val} Loss {dual_val}')
final_loss = float(loss(dual_vars, loss_scl=-1, exact=use_exact_eig_eval)[0])
info['final_dual_vars'] = dual_vars
info['final_opt_state'] = opt_state
info['final_loss'] = final_loss
info['loss_log'] = loss_log
info['store_best'] = store_best
return final_loss, info
def f(x):
g, aux = grad(lambda x: (x**3, [x**3]), has_aux=True)(x)
return aux[0] * np.sin(x)
def dual_fun(verif_instance, dual_vars, key=None, n_iter=30, scl=-1,
exact=False, dynamic_unroll=True, include_info=False):
# pylint: disable=invalid-name
"""Returns the dual objective value.
Args:
verif_instance: a utils.SdpDualVerifInstance, the verification problem
dual_vars: A list of dual variables at each layer
key: PRNGKey passed to Lanczos
n_iter: Number of Lanczos iterations to use
scl: Inverse temperature in softmax over eigenvalues to smooth optimization
problem (if negative treat as hardmax)
exact: Whether to use exact eigendecomposition instead of Lanczos
dynamic_unroll: bool. Whether to use jax.fori_loop for Lanczos for faster
JIT compilation. Default is False.
include_info: if True, also return an `info` dict of various other
values computed for the objective
Returns:
Either a single float, the dual upper bound, or if ``include_info=True``,
returns a pair, the dual bound and a dict containing debugging info
"""
key = key if key is not None else jax.random.PRNGKey(0)
assert isinstance(verif_instance, utils.SdpDualVerifInstance)
bounds = verif_instance.bounds
layer_sizes = utils.layer_sizes_from_bounds(bounds)
layer_sizes_1d = [np.prod(np.array(i)) for i in layer_sizes]
N = sum(layer_sizes_1d) + 1
info = {}
# Mean activations at each layer
activations_center = [(b.lb + b.ub) / 2 for b in bounds]
# Maximum deviation from mean activations
radius = [(b.ub - b.lb) / 2 for b in bounds]
inner_lagrangian = verif_instance.make_inner_lagrangian(dual_vars)
lagrangian = _make_transformed_lagrangian(
inner_lagrangian, activations_center, radius)
# Construct c_lambda and g_lambda terms
zeros = [jnp.zeros(sz) for sz in layer_sizes]
c_lambda = lagrangian(zeros)
g_lambda = jax.grad(lagrangian)(zeros)
g_lambda = flatten(g_lambda)
info['c_lambda'] = c_lambda
def Hv(v):
"""Hessian-vector product for H_lambda - refer to docstring for `Av()`."""
lag_grad = lambda v2: flatten(jax.grad(lagrangian)(v2))
hv_v = jax.grad(lambda v2: jnp.vdot(lag_grad(v2), v))(zeros)
hv_flat = flatten(hv_v)
return hv_flat
def Av(v):
"""Matrix-vector product.
Args:
v: vector, DeviceArray
Returns:
Av: vector, Device array. A is defined as diag(kappa) - M(lambda) where
M(lambda) = [0, g_lambda';
g_lambda, H_lambda], and these terms correspond to
L~(z) = c_lambda + g_lambda' z + z' H_lambda z
"""
# Expand Mv=[0 g'; g H] [v0;v1] = [g'v1; v0*g + H(v1)] = [Mv0;Mv1]
# Compute Mv0 term
mv_zero = jnp.reshape(jnp.vdot(g_lambda, v[1:]), (1,))
# Compute Mv1 term
mv_rest = Hv(v[1:]) + v[0] * g_lambda
mv = jnp.concatenate([mv_zero, mv_rest], axis=0)
diag_kappa_v = jnp.reshape(dual_vars[-1], mv.shape) * v
av = diag_kappa_v - mv
return jnp.reshape(av, v.shape)
# Construct dual function (dual_vars[-1]=kappa)
if exact:
eig_vec, eig_info = eigenvector_utils.min_eigenvector_exact(
Av, N, scl=scl, report_all=True)
info['eig_info'] = eig_info
else:
eig_vec = eigenvector_utils.min_eigenvector_lanczos(
Av, N, min(N, n_iter), key, scl, dynamic_unroll=dynamic_unroll)
info['eig_vec'] = eig_vec
info['kappa'] = dual_vars[-1]
hess_val = jnp.vdot(eig_vec, Av(eig_vec))/(jnp.vdot(eig_vec, eig_vec))
hess_val = jnp.reshape(hess_val, ())
# Form dual objective
lambda_minus = jnp.minimum(hess_val, 0.)
kappa_hat = jnp.maximum(0, dual_vars[-1] - lambda_minus)
dual_val = c_lambda + 0.5 * jnp.sum(kappa_hat)
if include_info:
return dual_val, info
return dual_val
def update_derivative(i, opt_state, batch, l2reg):
params = get_params(opt_state)
return opt_update(i,
jax.grad(loss, 0)(params, batch, l2reg),
opt_state), params
def Hv(v): """Hessian-vector product for H_lambda - refer to docstring for `Av()`.""" lag_grad = lambda v2: flatten(jax.grad(lagrangian)(v2)) hv_v = jax.grad(lambda v2: jnp.vdot(lag_grad(v2), v))(zeros) hv_flat = flatten(hv_v) return hv_flat
def gradients(scalar, variables):
"""Compute the gradients of a scalar w.r.t to a given list of variables.
Arguments
---------
scalar: :class:`symjax.tensor.base.Tensor`
the variable to differentiate
variables: List or Tuple
the variables used to compute the derivative.
Returns
-------
gradients: Tuple
the sequency of gradients ordered as given in the input variables
Example
-------
.. doctest::
>>> import symjax
>>> w = symjax.tensor.ones(3)
>>> x = symjax.tensor.Variable(2., name='x', dtype='float32')
>>> l = (w ** 2).sum() * x
>>> g = symjax.gradients(l, [w])[0]
>>> f = symjax.function(outputs=g, updates={x:x + 1})
>>> for i in range(2):
... print(f())
[4. 4. 4.]
[6. 6. 6.]
"""
if numpy.prod(scalar.shape) != 1:
raise RuntimeError("the variable to differentiate is not a scalar")
if not isinstance(scalar, t.Tensor):
raise RuntimeError(
"the variable used in gradients should be a Tensor type")
if scalar.shape != ():
scalar = scalar.sum()
if isinstance(variables, t.Tensor):
input_variables = [variables]
input_list = False
else:
input_variables = variables
input_list = True
# get the argnum of the variables that we differentiate one
argnums = list(range(len(input_variables)))
# get all the roots of the scalar, this is needed as otherwise they are not
# as the input of the gradient function and thus a change of
# their value will not change the gradient computation, we also ensure
# uniqueness
input_variables += [
i for i in current_graph().roots(scalar) if i not in input_variables
]
# create a dummy function that is needed for jax to compute a gradient func
# this function is the one that builds the graph of computation from all
# roots
# to the scalar varible s.t. automatic diffenrentiation can be applied
def fn(*args):
return current_graph().get(scalar,
dict(zip(input_variables, list(args))))
# now we obtain the grad function. In fact, Jax returns a function that,
# when it is called, returns the gradient values, this function is then
# used to generate the Tuple of symbolic variables
grad_fn = jax.grad(fn, argnums)
wrap_fn = t.jax_wrap(grad_fn)
if input_list:
return wrap_fn(*input_variables)
else:
return wrap_fn(*input_variables)[0]
def id_func(x):
return lambda u: jnp.dot(jnp.identity(x.shape[0]), u)
def hvp(f, primals, tangents):
return jvp(grad(f), primals, tangents)[1]
def breg_bound(vec, lb=-1.0, ub=1.0, *args, **kwargs):
return jnp.sum((-vec + ub) * jnp.log(-vec + ub) +
(vec - lb) * jnp.log(vec - lb))
DP_bound = jax.grad(breg_bound, 0)
def DP_inv_bound(vec, lb=-1.0, ub=1.0):
return (ub * jnp.exp(vec) + lb) / (1 + jnp.exp(vec))
def D2P_bound(vec, lb=-1.0, ub=1.0):
def out(u):
return jvp(lambda x: DP_bound(x, lb, ub), (vec, ), (u, ))[1]
return out
def inv_D2P_bound(vec, lb=-1.0, ub=1.0):
if len(jnp.shape(vec)) <= 1:
print("The input is", val)
print("The result of Rosenbrock's is ", result)
fun_driver(10)
"""### 3. First look at derivatives: `jax.grad()`
1. https://jax.readthedocs.io/en/latest/jax.html#jax.grad
2. Computes $v\cdot J$ for a function that computes a scalar value ($R^n \rightarrow R$).
3. The seed is internally set to `1.0`
(the shape of the seed must match the primal output).
"""
#Create a function that computes the derivatives.
#This needs to happen only once.
grad_rosenbrock = jax.grad(rosenbrock)
def grad_driver(n):
"""
Input: n array length
Output: Result of Rosenbrock's banana function
"""
#create the input array
val = jnp.full(n, 0.5)
#compute the derivatives
result = grad_rosenbrock(val)
plot_vals(val, grad=result)
print("The input is", val)
def hvp(f, primals, tangents):
return jvp(grad(f), primals, tangents)[1]
import jax.random as random
key = random.PRNGKey(0)
if __name__ == '__main__':
# 4. Minimize for a few steps
initial_radii = np.ones(len(all_elements)) * 0.12
initial_scales = np.ones(len(all_elements)) * 0.85
theta0 = pack(initial_radii, initial_scales)
print('initial theta', theta0)
initial_log_prob = log_prob(theta0)
print('initial log prob', initial_log_prob)
# TODO: potentially jit() this
grad_log_prob = grad(log_prob)
#grad_log_prob = parallel_grad_log_prob
print('initial gradient norm = {}'.format(
np.linalg.norm(grad_log_prob(theta0))))
minimized_theta_fname = os.path.join(
data_path, 'elemental_types_l-bfgs_freesolv_{}.npy'.format(name))
print('minimizing...')
from scipy.optimize import minimize
loss = lambda theta: -log_prob(theta)
grad_loss = lambda theta: -grad_log_prob(theta)
bounds = [(0.001, 2.0)] * len(theta0)
result = minimize(loss,
def test_grad_tuple_output(self):
jtu.check_raises(
lambda: grad(lambda x: (x, x))(1.0), TypeError,
"Gradient only defined for scalar-output functions. ")
def test_system_derivatives(self):
sdf_file = open("examples/host-acd.mol2").read()
smirnoff = ForceField("test_forcefields/smirnoff99Frosst.offxml")
mol = Chem.MolFromMol2Block(sdf_file,
sanitize=True,
removeHs=False,
cleanupSubstructures=True)
guest_potentials, guest_params, guest_param_groups, guest_conf, guest_masses = forcefield.parameterize(
mol, smirnoff)
ref_nrgs = []
test_nrgs = []
for potential, params in guest_potentials:
jax_potential = potential_map[potential]
if potential == timemachine.lib.custom_ops.HarmonicBond_f64:
jp = functools.partial(jax_potential,
box=None,
bond_idxs=params[0],
param_idxs=params[1])
elif potential == timemachine.lib.custom_ops.HarmonicAngle_f64:
jp = functools.partial(jax_potential,
box=None,
angle_idxs=params[0],
param_idxs=params[1])
elif potential == timemachine.lib.custom_ops.PeriodicTorsion_f64:
jp = functools.partial(jax_potential,
box=None,
torsion_idxs=params[0],
param_idxs=params[1])
elif potential == timemachine.lib.custom_ops.LennardJones_f64:
jp = functools.partial(jax_potential,
box=None,
scale_matrix=params[0],
param_idxs=params[1])
elif potential == timemachine.lib.custom_ops.Electrostatics_f64:
jp = functools.partial(jax_potential,
box=None,
scale_matrix=params[0],
param_idxs=params[1])
else:
raise ValueError("unknown functional form")
test_nrgs.append(potential(*params))
ref_nrgs.append(jp)
def ref_total_nrg(conf, params):
nrgs = []
for p in ref_nrgs:
nrgs.append(p(conf, params))
return jnp.sum(nrgs)
dp_idxs = onp.arange(len(params)).astype(onp.int32)
def test_total_nrg(conf, params):
nrgs = []
for p in test_nrgs:
res = p.derivatives(onp.expand_dims(conf, axis=0), params,
dp_idxs)
nrgs.append(res[0])
return onp.sum(nrgs)
num_atoms = guest_conf.shape[0]
ref_e = ref_total_nrg(guest_conf, guest_params)
test_e = test_total_nrg(guest_conf, guest_params)
onp.testing.assert_almost_equal(ref_e, test_e)
dt = 1e-3
ca = 0.5
cb = onp.random.rand(num_atoms) / 10
cc = onp.zeros(num_atoms)
intg = ReferenceLangevin(dt, ca, cb, cc)
ref_dE_dx_fn = jax.grad(ref_total_nrg, argnums=(0, ))
ref_dE_dx_fn = jax.jit(ref_dE_dx_fn)
def integrate(x_t, v_t, params):
for _ in range(100):
x_t, v_t = intg.step(x_t, v_t, ref_dE_dx_fn(x_t, params)[0])
return x_t, v_t
v0 = onp.random.rand(num_atoms * 3).reshape(num_atoms, 3)
x_f, v_f = integrate(guest_conf, v0, guest_params)
lo = custom_ops.LangevinOptimizer_f64(dt, ca, cb, cc)
ctxt = custom_ops.Context_f64(test_nrgs, lo, guest_params, guest_conf,
v0, dp_idxs)
for _ in range(100):
ctxt.step()
onp.testing.assert_almost_equal(x_f, ctxt.get_x())
onp.testing.assert_almost_equal(v_f, ctxt.get_v())
grad_fn = jax.jacfwd(integrate, argnums=(2))
dx_dp_f, dv_dp_f = grad_fn(guest_conf, v0, guest_params)
dx_dp_f = onp.asarray(onp.transpose(dx_dp_f, (2, 0, 1)))
dv_dp_f = onp.asarray(onp.transpose(dv_dp_f, (2, 0, 1)))
onp.testing.assert_almost_equal(dx_dp_f[dp_idxs], ctxt.get_dx_dp())
onp.testing.assert_almost_equal(dv_dp_f[dp_idxs], ctxt.get_dv_dp())
def test_grad_nonscalar_output(self):
jtu.check_raises(
lambda: grad(lambda x: x)(onp.zeros(3)), TypeError,
"Gradient only defined for scalar-output functions. ")
import jax.numpy as np
from jax import grad, jit, vmap
from functools import partial
def predict(params, inputs):
for W, b in params:
outputs = np.dot(inputs, W) + b
inputs = np.tanh(outputs)
return outputs
def logprob_fun(params, inputs, targets):
preds = predict(params, inputs)
return np.sum((preds - targets)**2)
grad_fun = jit(grad(logprob_fun)) # compiled gradient evaluation function
perex_grads = jit(lambda params, inputs, targets: # fast per-example gradients
vmap(partial(grad_fun, params), inputs, targets))
def test_holomorphic_grad(self):
out = grad(lambda x: np.sin(x), holomorphic=True)(1 + 2j)
expected = 2.0327230070196656 - 3.0518977991518j
self.assertAllClose(out, expected, check_dtypes=False)
