"""
Schedulers for flow-matching paths.
It is advised to use the `CondOTScheduler` for optimal performance with conditional flow matching.
"""
from abc import ABC, abstractmethod
from dataclasses import dataclass, field
from typing import Union
import jax
import jax.numpy as jnp
from jax import Array
@dataclass
[docs]
class SchedulerOutput:
r"""Represents a sample of a conditional-flow generated probability path.
Attributes:
alpha_t (Array): :math:`\alpha_t`, shape (...).
sigma_t (Array): :math:`\sigma_t`, shape (...).
d_alpha_t (Array): :math:`\frac{\partial}{\partial t}\alpha_t`, shape (...).
d_sigma_t (Array): :math:`\frac{\partial}{\partial t}\sigma_t`, shape (...).
"""
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alpha_t: Array = field(metadata={"help": "alpha_t"})
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sigma_t: Array = field(metadata={"help": "sigma_t"})
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d_alpha_t: Array = field(metadata={"help": "Derivative of alpha_t."})
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d_sigma_t: Array = field(metadata={"help": "Derivative of sigma_t."})
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class Scheduler(ABC):
"""Base Scheduler class."""
@abstractmethod
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def __call__(self, t: Array) -> SchedulerOutput:
r"""
Args:
t (Array): times in [0,1], shape (...).
Returns:
SchedulerOutput: :math:`\alpha_t,\sigma_t,\frac{\partial}{\partial t}\alpha_t,\frac{\partial}{\partial t}\sigma_t`
"""
... # pragma: no cover
@abstractmethod
[docs]
def snr_inverse(self, snr: Array) -> Array:
r"""
Computes :math:`t` from the signal-to-noise ratio :math:`\frac{\alpha_t}{\sigma_t}`.
Args:
snr (Array): The signal-to-noise, shape (...)
Returns:
Array: t, shape (...)
"""
... # pragma: no cover
[docs]
class ConvexScheduler(Scheduler):
@abstractmethod
[docs]
def __call__(self, t: Array) -> SchedulerOutput:
"""Scheduler for convex paths.
Args:
t (Array): times in [0,1], shape (...).
Returns:
SchedulerOutput: :math:`\alpha_t,\sigma_t,\frac{\partial}{\partial t}\alpha_t,\frac{\partial}{\partial t}\sigma_t`
"""
... # pragma: no cover
@abstractmethod
[docs]
def kappa_inverse(self, kappa: Array) -> Array:
"""
Computes :math:`t` from :math:`\kappa_t`.
Args:
kappa (Array): :math:`\kappa`, shape (...)
Returns:
Array: t, shape (...)
"""
... # pragma: no cover
[docs]
def snr_inverse(self, snr: Array) -> Array:
r"""
Computes :math:`t` from the signal-to-noise ratio :math:`\frac{\alpha_t}{\sigma_t}`.
Args:
snr (Array): The signal-to-noise, shape (...)
Returns:
Array: t, shape (...)
"""
kappa_t = snr / (1.0 + snr)
return self.kappa_inverse(kappa=kappa_t)
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class CondOTScheduler(ConvexScheduler):
"""CondOT Scheduler."""
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def __call__(self, t: Array) -> SchedulerOutput:
return SchedulerOutput(
alpha_t=t,
sigma_t=1 - t,
d_alpha_t=jnp.ones_like(t),
d_sigma_t=-jnp.ones_like(t),
)
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def kappa_inverse(self, kappa: Array) -> Array:
return kappa
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class PolynomialConvexScheduler(ConvexScheduler):
"""Polynomial Scheduler."""
def __init__(self, n: Union[float, int]) -> None:
assert isinstance(n, (float, int)), f"`n` must be a float or int. Got {type(n)=}."
assert n > 0, f"`n` must be positive. Got {n=}."
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def __call__(self, t: Array) -> SchedulerOutput:
return SchedulerOutput(
alpha_t=t**self.n,
sigma_t=1 - t**self.n,
d_alpha_t=self.n * (t ** (self.n - 1)),
d_sigma_t=-self.n * (t ** (self.n - 1)),
)
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def kappa_inverse(self, kappa: Array) -> Array:
return jnp.power(kappa, 1.0 / self.n)
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class VPScheduler(Scheduler):
"""Variance Preserving Scheduler."""
def __init__(self, beta_min: float = 0.1, beta_max: float = 20.0) -> None:
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self.beta_min = beta_min
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self.beta_max = beta_max
super().__init__()
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def __call__(self, t: Array) -> SchedulerOutput:
b = self.beta_min
B = self.beta_max
T = 0.5 * (1 - t) ** 2 * (B - b) + (1 - t) * b
dT = -(1 - t) * (B - b) - b
return SchedulerOutput(
alpha_t=jnp.exp(-0.5 * T),
sigma_t=jnp.sqrt(1 - jnp.exp(-T)),
d_alpha_t=-0.5 * dT * jnp.exp(-0.5 * T),
d_sigma_t=0.5 * dT * jnp.exp(-T) / jnp.sqrt(1 - jnp.exp(-T)),
)
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def snr_inverse(self, snr: Array) -> Array:
T = -jnp.log(snr**2 / (snr**2 + 1))
b = self.beta_min
B = self.beta_max
t = 1 - ((-b + jnp.sqrt(b**2 + 2 * (B - b) * T)) / (B - b))
return t
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class LinearVPScheduler(Scheduler):
"""Linear Variance Preserving Scheduler."""
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def __call__(self, t: Array) -> SchedulerOutput:
return SchedulerOutput(
alpha_t=t,
sigma_t=(1 - t**2) ** 0.5,
d_alpha_t=jnp.ones_like(t),
d_sigma_t=-t / (1 - t**2) ** 0.5,
)
[docs]
def snr_inverse(self, snr: Array) -> Array:
return jnp.sqrt(snr**2 / (1 + snr**2))
[docs]
class CosineScheduler(Scheduler):
"""Cosine Scheduler."""
[docs]
def __call__(self, t: Array) -> SchedulerOutput:
return SchedulerOutput(
alpha_t=jnp.sin(jnp.pi / 2 * t),
sigma_t=jnp.cos(jnp.pi / 2 * t),
d_alpha_t=jnp.pi / 2 * jnp.cos(jnp.pi / 2 * t),
d_sigma_t=-jnp.pi / 2 * jnp.sin(jnp.pi / 2 * t),
)
[docs]
def snr_inverse(self, snr: Array) -> Array:
return 2.0 * jnp.arctan(snr) / jnp.pi
