# HQS

### *class* deepinv.optim.HQS(data_fidelity=None, prior=None, lambda_reg=1.0, stepsize=1.0, g_param=None, sigma_denoiser=None, max_iter=100, crit_conv='residual', thres_conv=1e-5, early_stop=False, custom_metrics=None, custom_init=None, g_first=False, unfold=False, trainable_params=None, DEQ=None, anderson_acceleration=None, cost_fn=None, params_algo=None, \*\*kwargs)

Bases: [`BaseOptim`](https://deepinv.org/api/stubs/deepinv.optim.BaseOptim.html.md#deepinv.optim.BaseOptim)

Half-Quadratic Splitting (HQS) module for solving the problem

$$
\begin{equation}
\label{eq:min_prob}
\tag{1}
\underset{x}{\arg\min} \quad  \datafid{x}{y} + \lambda \reg{x},
\end{equation}

$$

where $\datafid{x}{y}$ is the data-fidelity term, $\reg{x}$ is the regularization term.
If the attribute `g_first` is set to False (by default), the HQS iterations are given by

$$
u_{k} &= \operatorname{prox}_{\gamma f}(x_k) \\
x_{k+1} &= \operatorname{prox}_{\sigma \lambda \regname}(u_k).

$$

If the attribute `g_first` is set to True, the functions $f$ and $\regname$ are inverted in the previous iteration.
The HQS iterations are defined in the iterator class [`deepinv.optim.optim_iterators.HQSIteration`](https://deepinv.org/api/stubs/deepinv.optim.optim_iterators.HQSIteration.html.md#deepinv.optim.optim_iterators.HQSIteration).
For using early stopping or stepsize backtracking, see the documentation of the [`deepinv.optim.BaseOptim`](https://deepinv.org/api/stubs/deepinv.optim.BaseOptim.html.md#deepinv.optim.BaseOptim) class.

If the attribute `unfold` is set to `True`, the algorithm is unfolded and the parameters of the algorithm are trainable.
By default, all the algorithm parameters are trainable : the stepsize $\gamma$, the regularization parameter $\lambda$, the prior parameter.
Use the `trainable_params` argument to adjust the list of trainable parameters.
Note also that by default, if the prior has trainable parameters (e.g. a neural network denoiser), these parameters are learnable by default.
If the model is used for inference only, use the `with torch.no_grad():` context when calling the model in order to avoid unnecessary gradient computations.

* **Parameters:**
  * **data_fidelity** ([*list*](https://docs.python.org/3.12/library/stdtypes.html#list) *,* [*deepinv.optim.DataFidelity*](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity)) – data-fidelity term $\datafid{x}{y}$.
    Either a single instance (same data-fidelity for each iteration) or a list of instances of
    [`deepinv.optim.DataFidelity`](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity) (distinct data fidelity for each iteration). Default: `None` corresponding to $\datafid{x}{y} = 0$.
  * **prior** ([*list*](https://docs.python.org/3.12/library/stdtypes.html#list) *,* [*deepinv.optim.Prior*](https://deepinv.org/api/stubs/deepinv.optim.Prior.html.md#deepinv.optim.Prior)) – regularization prior $\reg{x}$.
    Either a single instance (same prior for each iteration) or a list of instances of
    [`deepinv.optim.Prior`](https://deepinv.org/api/stubs/deepinv.optim.Prior.html.md#deepinv.optim.Prior) (distinct prior for each iteration). Default: `None` corresponding to $\reg{x} = 0$.
  * **lambda_reg** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – regularization parameter $\lambda$. Default: `1.0`.
  * **stepsize** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – stepsize parameter $\gamma$. Default: `1.0`.
  * **g_param** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – parameter of the prior function. For example the noise level for a denoising prior. Default: `None`.
  * **sigma_denoiser** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – same as `g_param`. If both `g_param` and `sigma_denoiser` are provided, `g_param` is used. Default: `None`.
  * **max_iter** ([*int*](https://docs.python.org/3.12/library/functions.html#int)) – maximum number of iterations of the optimization algorithm. Default: `100`.
  * **crit_conv** ([*str*](https://docs.python.org/3.12/library/stdtypes.html#str)) – convergence criterion to be used for claiming convergence, either `"residual"` (residual
    of the iterate norm) or `"cost"` (on the cost function). Default: `"residual"`
  * **thres_conv** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – convergence threshold for the chosen convergence criterion. Default: `1e-5`.
  * **early_stop** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to stop the algorithm as soon as the convergence criterion is met. Default: `False`.
  * **custom_metrics** ([*dict*](https://docs.python.org/3.12/library/stdtypes.html#dict)) – dictionary of custom metric functions to be computed along the iterations. The keys of the dictionary are the names of the metrics, and the values are functions that take as input the current and previous iterates, and return a scalar value. Default: `None`.
  * **custom_init** (*Callable*) – 

    Custom initialization of the algorithm.
    The callable function `custom_init(y, physics)` takes as input the measurement $y$ and the physics `physics` and returns the initialization in the form of either:
    - a tuple $(x_0, z_0)$ (where `x_0` and `z_0` are the initial primal and dual variables),
    - a torch.Tensor $x_0$ (if no dual variables $z_0$ are used), or
    - a dictionary of the form `X = {'est': (x_0, z_0)}`.

    Note that custom initialization can also be directly defined via the `init` argument in the `forward` method.

    If `None` (default value), the algorithm is initialized with the adjoint $A^{\top}y$ when the adjoint is defined,
    and with the observation `y` if the adjoint is not defined. Default: `None`.
  * **g_first** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to perform the proximal step on $\reg{x}$ before that on $\datafid{x}{y}$, or the opposite. Default: `False`.
  * **unfold** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to unfold the algorithm or not. Default: `False`.
  * **trainable_params** ([*list*](https://docs.python.org/3.12/library/stdtypes.html#list)) – list of HQS parameters to be trained if `unfold` is True. To choose between `["lambda", "stepsize", "g_param"]`. Default: None, which means that all parameters are trainable if `unfold` is True. For no trainable parameters, set to an empty list.
  * **DEQ** ([*deepinv.optim.DEQConfig*](https://deepinv.org/api/stubs/deepinv.optim.DEQConfig.html.md#deepinv.optim.DEQConfig) *,* [*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – Configuration for a Deep Equilibrium (DEQ) unfolding strategy.
    DEQ algorithms are virtually unrolled infinitely, leveraging the implicit function theorem.
    If `None` (default) or `False`, DEQ is disabled and the algorithm runs a standard finite number of iterations.
    Otherwise, `DEQ` must be an instance of [`deepinv.optim.DEQConfig`](https://deepinv.org/api/stubs/deepinv.optim.DEQConfig.html.md#deepinv.optim.DEQConfig), which defines the parameters
    for forward and backward equilibrium-based implicit differentiation.
    If `True`, the default `DEQConfig` is used.
  * **anderson_acceleration** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – Configure Anderson acceleration for the fixed-point iterations.
    If `None` (default) or `False`, Anderson acceleration is disabled.
    Otherwise, `anderson_acceleration` must be an instance of [`deepinv.optim.AndersonAccelerationConfig`](https://deepinv.org/api/stubs/deepinv.optim.AndersonAccelerationConfig.html.md#deepinv.optim.AndersonAccelerationConfig), which defines the parameters for Anderson acceleration.
    If `True`, the default `AndersonAccelerationConfig` is used.
  * **cost_fn** (*Callable*) – Custom user input cost function.
    `cost_fn(x, data_fidelity, prior, cur_params, y, physics)` takes as input
    the current primal variable ([`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)), the current data-fidelity ([`deepinv.optim.DataFidelity`](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity)),
    the current prior ([`deepinv.optim.Prior`](https://deepinv.org/api/stubs/deepinv.optim.Prior.html.md#deepinv.optim.Prior)), the current parameters (dict), and the measurement ([`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)).
    Default: `None`.
  * **params_algo** ([*dict*](https://docs.python.org/3.12/library/stdtypes.html#dict)) – optionally, directly provide the HQS parameters in a dictionary. This will overwrite the parameters in the arguments `stepsize`, `lambda_reg` and `g_param`.

<a id="sphx-glr-backref-deepinv-optim-hqs"></a>

## Examples using `HQS`:

<div class="sphx-glr-thumbnails">
<!-- thumbnail-parent-div-open --><div class="sphx-glr-thumbcontainer" tooltip="This example shows how to use the DPIR method to solve a PnP image deblurring problem. The DPIR method is described in zhang2021plug. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 3929-3938).">![](auto_examples/plug-and-play/images/thumb/sphx_glr_demo_PnP_DPIR_deblur_thumb.png)

[DPIR method for PnP image deblurring.](https://deepinv.org/auto_examples/plug-and-play/demo_PnP_DPIR_deblur.html.md)

  <div class="sphx-glr-thumbnail-title">DPIR method for PnP image deblurring.</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Some unfolded architectures rely on a least-squares solver to compute the proximal step w.r.t. the data-fidelity term (e.g., ADMM or HQS):  ">![](auto_examples/unfolded/images/thumb/sphx_glr_demo_unfolded_constant_memory_thumb.png)

[Reducing the memory and computational complexity of unfolded network training](https://deepinv.org/auto_examples/unfolded/demo_unfolded_constant_memory.html.md)

  <div class="sphx-glr-thumbnail-title">Reducing the memory and computational complexity of unfolded network training</div>
</div>
<!-- thumbnail-parent-div-close --></div>