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MAINT: Skip the check for theta <= 1 in the PDHG #1637

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MAINT: Skip the check for theta <= 1 in the PDHG #1637

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26 changes: 17 additions & 9 deletions odl/solvers/nonsmooth/primal_dual_hybrid_gradient.py
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Original file line number Diff line number Diff line change
Expand Up @@ -70,7 +70,7 @@ def pdhg(x, f, g, L, niter, tau=None, sigma=None, **kwargs):
callback : callable, optional
Function called with the current iterate after each iteration.
theta : float, optional
Relaxation parameter, required to fulfill ``0 <= theta <= 1``.
Relaxation parameter, required to fulfill ``theta >= 0``.
Default: 1
gamma_primal : non-negative float, optional
Acceleration parameter. If not ``None``, it overrides ``theta`` and
Expand Down Expand Up @@ -105,16 +105,16 @@ def pdhg(x, f, g, L, niter, tau=None, sigma=None, **kwargs):

where the formal conditions are that :math:`L` is an operator
between Hilbert spaces :math:`X` and :math:`Y`.
Further, :math:`f : X \rightarrow [0, +\infty]` and
:math:`g : Y \rightarrow [0, +\infty]` are proper, convex,
Further, :math:`f : X \rightarrow (-\infty, +\infty]` and
:math:`g : Y \rightarrow (-\infty, +\infty]` are proper, convex,
lower-semicontinuous functionals.

Convergence is only guaranteed if :math:`L` is linear, :math:`X, Y`
are finite dimensional and the step lengths :math:`\sigma` and
:math:`\tau` satisfy
Convergence is only guaranteed if :math:`L` is linear,
:math:`\theta > 1/2`, and the step lengths :math:`\sigma` and :math:`\tau`
satisfy

.. math::
\tau \sigma \|L\|^2 < 1
\tau \sigma \|L\|^2 < \frac{4}{1 + 2 \theta}

where :math:`\|L\|` is the operator norm of :math:`L`.

Expand Down Expand Up @@ -147,6 +147,9 @@ def pdhg(x, f, g, L, niter, tau=None, sigma=None, **kwargs):
The non-linear case is analyzed in `[Val2014]
<https://doi.org/10.1088/0266-5611/30/5/055012>`_.

The conditions on :math:`\tau` and :math:`sigma` are according to
`[BUG2023] <https://arxiv.org/abs/2309.03998v1>`_.

See Also
--------
odl.solvers.nonsmooth.douglas_rachford.douglas_rachford_pd :
Expand All @@ -158,6 +161,11 @@ def pdhg(x, f, g, L, niter, tau=None, sigma=None, **kwargs):

References
----------
[BUG2023] Banert, S, Upadhyaya M, and Giselsson, P.
*The Chambolle--Pock method converges weakly with :math:`\theta > 1/2`
and :math:`\tau \sigma \|L\|^2 < 4/(1+2\theta)`*. arXiv preprint
2309.03998v1 [math.OC] (2023).

[CP2011a] Chambolle, A and Pock, T. *A First-Order
Primal-Dual Algorithm for Convex Problems with Applications to
Imaging*. Journal of Mathematical Imaging and Vision, 40 (2011),
Expand Down Expand Up @@ -203,8 +211,8 @@ def pdhg(x, f, g, L, niter, tau=None, sigma=None, **kwargs):
# Relaxation parameter
theta = kwargs.pop('theta', 1)
theta, theta_in = float(theta), theta
if not 0 <= theta <= 1:
raise ValueError('`theta` {} not in [0, 1]'
if not theta >= 0:
raise ValueError('`theta` must be non-negative, got {}'
''.format(theta_in))

# Acceleration parameters
Expand Down