from typing import Literal, Optional, Union
import casadi as cs
import numpy as np
import numpy.typing as npt
from ..core.schedulers import Scheduler
from .gradient_based_optimizer import GradientBasedOptimizer, LrType
[docs]
class GradientDescent(GradientBasedOptimizer[LrType]):
r"""First-order Gradient descent optimizer, based on
:cite:`sutskever_importance_2013` and :class:`torch.optim.SGD`.
In its basic formulation, this optimizer updates the parameters as
.. math:: \theta \gets \theta - \alpha g,
where :math:`\theta` are the learnable parameters, :math:`\alpha` is the learning
rate (could be extended to the case this is a vector of rates), and :math:`g` is the
gradient of the loss function w.r.t. the parameters. If momentum or weight decay are
used, the gradient :math:`g` is modified before using it, but the update rule
remains the same. However, when considering a constrained parameter space, we need
to solve a Quadratic Programming (QP) problem to ensure the parameters stay within
their bounds. For gradient descent, the QP problem is
.. 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}
followed by the update :math:`\theta \gets \theta + \Delta\theta`.
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.
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.
dampening : float, optional
A positive float that specifies the dampening factor for the momentum. By
default, it is set to ``0.0``, so no dampening is used.
nesterov : bool, optional
A boolean that specifies whether to use Nesterov momentum. By default, it is
set to ``False``.
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 GD, the hessian is at most diagonal, i.e., in case we have constraints.
def __init__(
self,
learning_rate: Union[LrType, Scheduler[LrType]],
weight_decay: float = 0.0,
momentum: float = 0.0,
dampening: float = 0.0,
nesterov: 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.momentum = momentum
self.dampening = dampening
self.nesterov = nesterov
self._momentum_buffer = 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
dtheta, self._momentum_buffer = _gd(
theta,
gradient,
self._momentum_buffer,
self.weight_decay,
self.momentum,
self.lr_scheduler.value,
self.dampening,
self.nesterov,
)
# 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 _gd(
theta: np.ndarray,
g: np.ndarray,
momentum_buffer: Optional[np.ndarray],
weight_decay: float,
momentum: float,
lr: LrType,
dampening: float,
nesterov: bool,
) -> tuple[np.ndarray, Optional[np.ndarray]]:
"""Computes the update's change according to the gradient descent algorithm."""
if weight_decay > 0.0:
g += weight_decay * theta
if momentum > 0.0:
if momentum_buffer is None:
momentum_buffer = g.copy()
else:
momentum_buffer = momentum * momentum_buffer + (1 - dampening) * g
if nesterov:
g += momentum * momentum_buffer
else:
g = momentum_buffer
dtheta = -lr * g
return dtheta, momentum_buffer
