def sum_first_n(arr, num):
def body_fun(i, state):
arr, total, _ = state
arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
return (arr, lax.add(total, arr_i), ())
init_val = (arr, 0., ())
_, tot, _ = lax.fori_loop(0, lax.min(arr.shape[0], num), body_fun, init_val)
return tot
def sum_first_n(arr, num):
def body_fun(i, state):
arr, total = state['arr'], state['total']
arr_i = lax.dynamic_index_in_dim(arr, i, 0, False)
return {'arr': arr, 'total': lax.add(total, arr_i)}
init_val = {'arr': arr, 'total': 0.}
out_val = lax.fori_loop(0, lax.min(arr.shape[0], num), body_fun, init_val)
return out_val['total']
def lr_schedule(itr):
"""
Learning rate schedule.
Slowly warm-up with a small learning rate.
"""
iter_frac = lax.min((itr.astype(jnp.float32) + 1.) / lax.max(parse_args.warmup_itrs, 1.), 1.)
_epoch = itr // num_batches
id = lambda x: x
return lax.cond(_epoch < 80, parse_args.lr * iter_frac, id, parse_args.lr / 10, id)
def log_ndtr(x, series_order=3):
r"""Log Normal distribution function.
For details of the Normal distribution function see `ndtr`.
This function calculates :math:`\log(\mathrm{ndtr}(x))` by either calling
:math:`\log(\mathrm{ndtr}(x))` or using an asymptotic series. Specifically:
- For `x > upper_segment`, use the approximation `-ndtr(-x)` based on
:math:`\log(1-x) \approx -x, x \ll 1`.
- For `lower_segment < x <= upper_segment`, use the existing `ndtr` technique
and take a log.
- For `x <= lower_segment`, we use the series approximation of `erf` to compute
the log CDF directly.
The `lower_segment` is set based on the precision of the input:
.. math::
\begin{align}
\mathit{lower\_segment} =&
\ \begin{cases}
-20 & x.\mathrm{dtype}=\mathit{float64} \\
-10 & x.\mathrm{dtype}=\mathit{float32} \\
\end{cases} \\
\mathit{upper\_segment} =&
\ \begin{cases}
8& x.\mathrm{dtype}=\mathit{float64} \\
5& x.\mathrm{dtype}=\mathit{float32} \\
\end{cases}
\end{align}
When `x < lower_segment`, the `ndtr` asymptotic series approximation is:
.. math::
\begin{align}
\mathrm{ndtr}(x) =&\ \mathit{scale} * (1 + \mathit{sum}) + R_N \\
\mathit{scale} =&\ \frac{e^{-0.5 x^2}}{-x \sqrt{2 \pi}} \\
\mathit{sum} =&\ \sum_{n=1}^N {-1}^n (2n-1)!! / (x^2)^n \\
R_N =&\ O(e^{-0.5 x^2} (2N+1)!! / |x|^{2N+3})
\end{align}
where :math:`(2n-1)!! = (2n-1) (2n-3) (2n-5) ... (3) (1)` is a
`double-factorial
<https://en.wikipedia.org/wiki/Double_factorial>`_ operator.
Args:
x: an array of type `float32`, `float64`.
series_order: Positive Python integer. Maximum depth to
evaluate the asymptotic expansion. This is the `N` above.
Returns:
an array with `dtype=x.dtype`.
Raises:
TypeError: if `x.dtype` is not handled.
TypeError: if `series_order` is a not Python `integer.`
ValueError: if `series_order` is not in `[0, 30]`.
"""
if not isinstance(series_order, int):
raise TypeError("series_order must be a Python integer.")
if series_order < 0:
raise ValueError("series_order must be non-negative.")
if series_order > 30:
raise ValueError("series_order must be <= 30.")
x = jnp.asarray(x)
dtype = lax.dtype(x)
if dtype == jnp.float64:
lower_segment = _LOGNDTR_FLOAT64_LOWER
upper_segment = _LOGNDTR_FLOAT64_UPPER
elif dtype == jnp.float32:
lower_segment = _LOGNDTR_FLOAT32_LOWER
upper_segment = _LOGNDTR_FLOAT32_UPPER
else:
raise TypeError("x.dtype={} is not supported.".format(np.dtype(dtype)))
# The basic idea here was ported from:
# https://root.cern.ch/doc/v608/SpecFuncCephesInv_8cxx_source.html
# We copy the main idea, with a few changes
# * For x >> 1, and X ~ Normal(0, 1),
# Log[P[X < x]] = Log[1 - P[X < -x]] approx -P[X < -x],
# which extends the range of validity of this function.
# * We use one fixed series_order for all of 'x', rather than adaptive.
# * Our docstring properly reflects that this is an asymptotic series, not a
# Taylor series. We also provided a correct bound on the remainder.
# * We need to use the max/min in the _log_ndtr_lower arg to avoid nan when
# x=0. This happens even though the branch is unchosen because when x=0
# the gradient of a select involves the calculation 1*dy+0*(-inf)=nan
# regardless of whether dy is finite. Note that the minimum is a NOP if
# the branch is chosen.
return jnp.where(
lax.gt(x, upper_segment),
-_ndtr(-x), # log(1-x) ~= -x, x << 1
jnp.where(lax.gt(x, lower_segment),
lax.log(_ndtr(lax.max(x, lower_segment))),
_log_ndtr_lower(lax.min(x, lower_segment), series_order)))
"""
Define the jet rule for a primitive in terms of a composition of simpler primitives.
"""
jet_rules[prim] = partial(jet, comp)
def_comp(lax.expm1_p, lambda x: lax.exp(x) - 1)
def_comp(lax.log1p_p, lambda x: lax.log(1 + x))
def_comp(lax.sqrt_p, lambda x: x**0.5)
def_comp(lax.rsqrt_p, lambda x: x**-0.5)
def_comp(lax.asinh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) + 1)))
def_comp(lax.acosh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) - 1)))
def_comp(lax.atanh_p, lambda x: 0.5 * lax.log(lax.div(1 + x, 1 - x)))
def_comp(lax.erfc_p, lambda x: 1 - lax.erf(x))
def_comp(lax.rem_p, lambda x, y: x - y * lax.floor(x / y))
def_comp(lax.clamp_p, lambda a, x, b: lax.min(lax.max(a, x), b))
def _erf_inv_rule(primals_in, series_in):
x, = primals_in
series, = series_in
u = [x] + series
primal_out = lax.erf_inv(x)
v = [primal_out] + [None] * len(series)
# derivative on co-domain for caching purposes
deriv_const = np.sqrt(np.pi) / 2.
deriv_y = lambda y: lax.mul(deriv_const, lax.exp(lax.square(y)))
# manually propagate through deriv_y since we don't have lazy evaluation of sensitivities
def linear_warmup(warmup_iters):
return lambda i: lax.min(1., i / warmup_iters)
def _warmup_sched(itr):
itr_frac = lax.min(
(itr.astype(jnp.float32) + 1.) / lax.max(warmup, 1.), 1.)
_epoch = itr // nb
id = lambda x: x
return lax.cond(_epoch < 55, lr * itr_frac, id, lr / 10, id)
