from typing import Literal, Optional, Union
import casadi as cs
import numpy as np
import numpy.typing as npt
from ..core.parameters import LearnableParametersDict
from ..core.schedulers import Scheduler
from .gradient_based_optimizer import GradientBasedOptimizer, LrType
[docs]
class RMSprop(GradientBasedOptimizer[LrType]):
r"""RMSprop optimizer, based on :cite:`hinton_neural_2012` and
:class:`torch.optim.RMSprop`.
While there is an abundance of resources to understand the RMSprop optimizer, how do
we address the constraints on the learnable parameters? Similarly to
:class:`mpcrl.optim.GradientDescent`, once the update rule has calculated the change
:math:`g` in the parameters (in an unconstrained fashion), we force the constraints
via the Quadratic Programming (QP) problem
.. math::
\begin{aligned}
\min_{\Delta\theta} & \quad \frac{1}{2} \lVert \Delta\theta \rVert_2^2
+ \alpha g^\top \Delta\theta \\
\text{s.t.} & \quad \theta_{\text{lower}} \leq \theta
+ \Delta\theta \leq \theta_{\text{upper}}
\end{aligned}
Parameters
----------
learning_rate : float or array or :class:`mpcrl.core.schedulers.Scheduler`
The learning rate of the optimizer. It can be:
- a float, in case the learning rate must stay constant and is the same for all
learnable parameters
- an array, in case the learning rate must stay constant but is different for
each parameter (should have the same size as the number of learnable
parameters)
- a :class:`mpcrl.core.schedulers.Scheduler`, in case the learning rate can vary
during the learning process (usually, it is set to decay). See the ``hook``
argument for more details on when this scheduler is stepped.
alpha : float, optional
A positive float that specifies the decay rate of the running average of the
gradient. By default, it is set to ``0.99``.
eps : float, optional
Term added to the denominator to improve numerical stability. By default, it
is set to ``1e-8``.
weight_decay : float, optional
A positive float that specifies the decay of the learnable parameters in the
form of an L2 regularization term. By default, it is set to ``0.0``, so no
decay/regularization takes place.
momentum : float, optional
A positive float that specifies the momentum factor. By default, it is set
to ``0.0``, so no momentum is used.
centered : bool, optional
If ``True``, compute the centered RMSProp, i.e., the gradient is normalized by
an estimation of its variance.
hook : {"on_update", "on_episode_end", "on_timestep_end"}, optional
Specifies when to step the optimizer's learning rate's scheduler to decay
its value. This allows to vary the rate over the learning iterations. The
options are:
- ``"on_update"`` steps the learning rate after each agent's update
- ``"on_episode_end"`` steps the learning rate after each episode's end
- ``"on_timestep_end"`` steps the learning rate after each env's timestep.
By default, ``"on_update"`` is selected.
max_percentage_update : float, optional
A positive float that specifies the maximum percentage change the learnable
parameters can experience in each update. For example,
``max_percentage_update=0.5`` means that the parameters can be updated by up
to 50% of their current value. By default, it is set to ``+inf``.
If specified, the update becomes constrained and has to be solved as a QP, which
is inevitably slower than its unconstrained counterpart (a linear system).
bound_consistency : bool, optional
A boolean that, if ``True``, forces the learnable parameters to lie in their
bounds when updated. This is done via :func:`numpy.clip`. Only beneficial if
numerical issues arise during updates, e.g., due to the QP solver not being able
to guarantee bounds.
"""
_order = 1
_hessian_sparsity = "diag"
# In RMSprop, hessian is at most diagonal, i.e., in case we have constraints
def __init__(
self,
learning_rate: Union[LrType, Scheduler[LrType]],
alpha: float = 0.99,
eps: float = 1e-8,
weight_decay: float = 0.0,
momentum: float = 0.0,
centered: bool = False,
hook: Literal["on_update", "on_episode_end", "on_timestep_end"] = "on_update",
max_percentage_update: float = float("+inf"),
bound_consistency: bool = False,
) -> None:
super().__init__(learning_rate, hook, max_percentage_update, bound_consistency)
self.weight_decay = weight_decay
self.alpha = alpha
self.eps = eps
self.momentum = momentum
self.centered = centered
[docs]
def set_learnable_parameters(self, pars: LearnableParametersDict) -> None:
super().set_learnable_parameters(pars)
# initialize also running averages
n = pars.size
self._square_avg = np.zeros(n, dtype=float)
self._grad_avg = np.zeros(n, dtype=float) if self.centered else None
self._momentum_buf = np.zeros(n, dtype=float) if self.momentum > 0.0 else None
def _first_order_update(
self, gradient: npt.NDArray[np.floating]
) -> tuple[npt.NDArray[np.floating], Optional[str]]:
theta = self.learnable_parameters.value
# compute candidate update
weight_decay = self.weight_decay
lr = self.lr_scheduler.value
if weight_decay > 0.0:
gradient = gradient + weight_decay * theta
dtheta, self._square_avg, self._grad_avg, self._momentum_buf = _rmsprop(
gradient,
self._square_avg,
lr,
self.alpha,
self.eps,
self.centered,
self._grad_avg,
self.momentum,
self._momentum_buf,
)
# if unconstrained, apply the update directly; otherwise, solve the QP
solver = self._update_solver
if solver is None:
return theta + dtheta, None
lbx, ubx = self._get_update_bounds(theta)
sol = solver(h=cs.DM.eye(theta.shape[0]), g=-dtheta, lbx=lbx, ubx=ubx)
dtheta = np.asarray(sol["x"].elements())
stats = solver.stats()
return theta + dtheta, None if stats["success"] else stats["return_status"]
def _rmsprop(
grad: np.ndarray,
square_avg: np.ndarray,
lr: LrType,
alpha: float,
eps: float,
centered: bool,
grad_avg: Optional[np.ndarray],
momentum: float,
momentum_buffer: Optional[np.ndarray],
) -> tuple[np.ndarray, np.ndarray, Optional[np.ndarray], Optional[np.ndarray]]:
"""Computes the update's change according to Adam algorithm."""
square_avg = alpha * square_avg + (1 - alpha) * np.square(grad)
if centered:
grad_avg = alpha * grad_avg + (1 - alpha) * grad
avg = np.sqrt(square_avg - np.square(grad_avg))
else:
avg = np.sqrt(square_avg)
avg += eps
if momentum > 0.0:
momentum_buffer = momentum * momentum_buffer + grad / avg
dtheta = -lr * momentum_buffer
else:
dtheta = -lr * grad / avg
return dtheta, square_avg, grad_avg, momentum_buffer
