SURE-CR

This package enables tractable evaluation of Stein's Unbiased Risk Estimate on convexly regularized estimators.

For an estimator of the mean of a normally-distributed random vector $y$ with known covariance matrix $\sigma^2 I$ given by

$$ \hat\mu(y) = \mathcal A ~ \text{argmin} \frac{1}{2} \lVert\mathcal A b - y\rVert_2^2 + r(b) $$

where $r: \mathbb R^p \to \mathbb R$ is a convex function and $\mathcal A: \mathbb R^p \to \mathbb R^d$ is a linear operator, this package provides methods to compute Stein's Unbiased Risk Estimate of $\hat\mu$:

$$ SURE(\hat\mu, y) = -n \sigma^2 + \lVert\hat\mu(y) - y\rVert_2^2 + 2 \sigma^2 \nabla \cdot \hat\mu(y). $$

$SURE(\hat\mu, y)$ is a good estimate of the $\ell_2$ risk of $\hat\mu$, especially for high dimensional problems.

Installation

SURE-CR can be installed with pip and conda. By default, the conda instillation does not support using the CVXPYSolver, and the cvxpylayers library will have to be installed seperately to use it.

To install with pip:

bash $ pip install surecr

To install with conda:

bash $ conda install -c stanfordcvxgrp -c conda-forge sure-cr

Examples

The easiest way to start using SURE-CR may be to read the examples:

• LASSO with FISTASolver
  • Shows how to define proximal operator.
• LASSO with CVXPYSolver
• Matrix completion with ADMMSolver
  • Includes defining custom linear operator.
• Robust PCA with ADMMSolver
  • Example of separable regularizers.

Usage

There are three key things in this package:

• The SURE class
• The Solver class and its subclasses CVXPYSolver, FISTASolver, and ADMMSolver
• The prox_lib helper library

The SURE Class

The SURE class has the following API: ```python class SURE: def init(self, variance: float, solver: Solver): ...

def compute(self, y: torch.Tensor, divergence_parameters={}) -> float:
    """
    Computes and returns SURE for the estimator computed by the solver
    at the point y.

    Currently, divergence_parameters can contain the key "m" to indicate
    how many samples to use during the divergence estimation (which
    dominates the runtime at high dimensions). The default is for m to be
    102.

    In the future we may switch to A-Hutch++ and may change what options
    the divergence_parameters specifies.
    """

@property
def solution(self) -> torch.Tensor:
    """
    Returns solver.solve(y) from the last compute call.
    """

def runtimes(self) -> TypedDict('Runtimes', solver=float, divergence=float):
    """
    Returns how long it took for the solver to run and how long it took
    the divergence estimator to run during the last compute call.
    """

```

The Solver class

Most uses of the library should use one of the existing Solver subclasses. They have the following APIs:

The three notable Solver instances provided by this library have the following constructors: ```python class FISTASolver(Solver): def init(self, A: linops.LinearOperator, proxR: Callable[[torch.Tensor, float | torch.Tensor], torch.Tensor], x0: torch.Tensor, device=None, lipschitziterations=20, lipschitzvec=None, *, maxiters=5000, eps=1e-3): """ This solver solves problems of the form with a variant on FISTA: min. 1/2 ||A b - y||_2^2 + r(b) and estimates the mean of y with A b^* where b^* is the optimal b.

    A is a linear operator defined using <https://github.com/cvxgrp/torch_linops>

    prox_R is a differentiable-with-respect-to-its-first-argument function to
        find the optimal point b for a (v, t) pair of
          min. t r(b) + 1/2 ||b - v||_2^2

    x0 is the point where we begin iterations, it must be chosen
        indepentently of y.

    lipschitz_iterations is how many iterations of the power method to use
    to approximate the largest eigenvalue of A^T A

    lipschitz_vec is the vector to start the power method. By default, a
    vector of all 1s is used. If this vector is orthogonal to the largest
    eigenvector of A^T A, this argument is mandatory.

    max_iters, eps control when iterations stop.

    """

class ADMMSolver(Solver): def init(self, A: linops.LinearOperator, proxR: Callable[[torch.Tensor, float | torch.Tensor], torch.Tensor], x0: torch.Tensor, device=None, *, maxiters=1000, epsrel=1e-3, epsabs=1e-6): """ This solver solves problems of the form with a variant on ADMM: min. 1/2 ||A b - y||_2^2 + r(b) and estimates the mean of y with A b^* where b^* is the optimal b.

    A is a linear operator defined using <https://github.com/cvxgrp/torch_linops>

    prox_R is a differentiable-with-respect-to-its-first-argument function to
        find the optimal point b for a (v, t) pair of
          min. t r(b) + 1/2 ||b - v||_2^2

    x0 is the point where we begin iterations, it must be chosen
        indepentently of y.

    max_iters, eps_rel, eps_abs control when iterations stop.
    """

class CVXPYSolver(Solver): def init(self, problem: cp.Problem, yparameter: cp.Parameter, variables: list[cp.Variable], estimate: Callable[[list[torch.Tensor]], torch.Tensor]): """ problem must be a CVXPY problem with a single paremeter, yparameter, and variables y_variable.

    estimate must be function which takes tensors with values for each variable
        and returns the estimate.

    WARNING: This solver has poor performance on large problems, and can
    have undetected poor accuracy on some moderately-sized problems.
    """

```

If you wish to implement, Solver, it has has the following API, where T is any type of the implementation's choice: ```python class Solver:

def solve(self, y: torch.Tensor) -> T:
    """
    Returns intermediate value used to estimate the mean of the distribution
    y is sampled from.
    """

def estimate(self, beta: T) -> torch.Tensor: ...
    """
    Given the output of a solve call, returns the estimate of the mean of the
    distribution y was sampled from.
    """

```

Note that for a given instance s of a solver class, s.estimate(s.solve(y)) must be differentiable via torch's backpropagation.

The prox_lib library

Since FISTASolver and ADMMSolver both require a proximal operator for the regularizer we provide some methods here to help construct proximal operators:

There are also many helper methods in surecr.prox_lib.

• prox_l1_norm(v, t): the $\ell_1$ norm's proximal operator.
• prox_l2_norm(v, t): the $\ell_2$ norm's proximal operator.
• make_scaled_prox_nuc_norm(shape: tuple[int, int], t_scale: float): generates the proximal operator $\text{prox}{r}: \mathbb R^{\mathtt{shape}} \to \mathbb R^{\mathtt{shape}}$ of $b \mapsto \mathtt{tscale} \sumi \sigmai(b)$
• combine_proxs(shape: list[int], proxs: list): if there are two regularizers $r1$, $r2$ such that the regularizer for the problem is given by $r(b, b') = r1(b) + r2(b')$, then this function should be called with ([dim(b), dim(b')], [prox_r_1, prox_r_2]).
• scale_prox(prox, t_scale): takes a proximal operator of $r$, and returns the proximal operator of $\mathtt{t_scale} r$.

Citing

If you use this code in a research project, please cite the associated paper. @article{nobel2022tractable, title={Tractable evalutaion of {S}tein's {U}nbiased {R}isk {E}stimate with convex regularizers}, author={Parth Nobel \and Emmanuel Cand\`es \and Stephen Boyd}, publisher = {arXiv}, year = {2022}, note = {arXiv:2211.05947 [math.ST]}, url = {https://arxiv.org/abs/2211.05947}, }