Abstract
Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A bynow common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing, this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upperlevel problem to assume access to exact solutions of the lowerlevel problem, which is practically infeasible. In this work we propose to solve these problems using inexact derivativefree optimization algorithms which never require exact lowerlevel problem solutions, but instead assume access to approximate solutions with controllable accuracy, which is achievable in practice. We prove global convergence and a worstcase complexity bound for our approach. We test our proposed framework on ROF denoising and learning MRI sampling patterns. Dynamically adjusting the lowerlevel accuracy yields learned parameters with similar reconstruction quality as highaccuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).
1 Introduction
Variational regularization techniques are dominant in the field of mathematical imaging. For example, when solving a linear inverse problem \(Ax = y\), variational regularization can be posed as the solution to
Here, the data fidelity \(\mathcal {D}\) is usually chosen related to the assumed noise model of the data y and the regularizer \(\mathcal {R}\) models our a priori knowledge of the unknown solution. Many options have been proposed in the literature, see, for instance, [1, 8, 14, 29, 44] and references therein. An important parameter for any variational regularization technique is the regularization parameter \(\alpha \). While some theoretical results and heuristic choices have been proposed in the literature, see, e.g., [8, 23] and references therein or the Lcurve criterion [27], the appropriate choice of the regularization parameter in a practical setting remains an open problem. Similarly, other parameters in (1) have to be chosen by the user, such as smoothing of the total variation [14], the hyperparameter for total generalized variation [9] or the sampling pattern in magnetic resonance imaging (MRI), see, e.g., [25, 45, 46].
Instead of using heuristics for choosing all of these parameters, here we are interested in finding these from data. A bynow common strategy to learn parameters of a variational regularization model from data is bilevel learning, see, e.g., [4, 21, 28, 30, 38, 39, 45] and references in [1] . Given labeled data \((x_i, y_i)_{i=1,\ldots ,n}\) we find parameters \(\theta \in \Theta \subset \mathbb R^m\) by solving the upperlevel problem
where \(\hat{x}_i(\theta ) \in X\subset \mathbb R^d\) aims to recover the true data \(x_i\) by solving the lowerlevel problems
The lowerlevel objective \(\Phi _{i,\theta }\) could be of the form \(\Phi _{i,\theta }(x) = \mathcal {D}(Ax, y_i) + \theta \mathcal {R}(x)\) as in (1), but we will not restrict ourselves to this special case. In general \(\Phi _{i,\theta }\) will depend on the data \(y_i\).
In many situations, it is possible to acquire suitable data \((x_i, y_i)_{i=1,\ldots ,n}\). For image denoising, we may take any ground truth images \(x_i\) and add artificial noise to generate \(y_i\). Alternatively, if we aim to learn a sampling pattern (such as for learning MRI sampling patterns, which we consider in this work), then \(x_i\) can be any fully sampled image. The same also holds for problems such as image compression, where again \(x_i\) is any ground truth image. In both these cases, \(y_i\) is subsampled information from \(x_i\) (depending on \(\theta \)) from which the remaining information is reconstructed to get \(\hat{x}_i(\theta )\).
While mathematically appealing, this nested optimization problem is computationally very difficult to handle since even the evaluation of the upperlevel problem (2) requires the exact solution of the lowerlevel problems (3). This requirement is practically infeasible, and common algorithms in the literature compute the lowerlevel solution only to some accuracy, thereby losing any theoretical performance guarantees, see, e.g., [21, 39, 45]. One reason for needing exact solutions is to compute the gradient of the upperlevel objective using the implicit function theorem [45], which we address by using upperlevel solvers which do not require gradient computations.
In this work we propose to solve these problems using inexact derivativefree optimization (DFO) algorithms which never require exact solutions to the lowerlevel problem while still yielding convergence guarantees. Moreover, by dynamically adjusting the accuracy we gain a significant computational speedup compared to using a fixed accuracy for all lowerlevel solves. The proposed framework is tested on two problems: learning regularization parameters for ROF denoising and learning the sampling pattern in MRI.
We contrast our approach to [30], which develops a semismooth Newton method to solve the full bilevel optimality conditions. In [30] the upper and lowerlevel problems are of specific structure, and exact solutions of the (possibly very large) Newton system are required. Separately, the approach in [38] replaces the lowerlevel problem with finitely many iterations of some algorithm and solves this perturbed problem exactly. Our formulation is very general, and all approximations are controlled to guarantee convergence to the solution of the original variational problem.
Aim: Use inexact computations of \(\hat{x}_i(\theta )\) within a derivativefree upperlevel solver, which makes (2) computationally tractable, while retaining convergence guarantees.
1.1 DerivativeFree Optimization
Derivativefree optimization methods—that is, optimization methods that do not require access to the derivatives of the objective (and/or constraints)—have grown in popularity in recent years and are particularly suited to settings where the objective is computationally expensive to evaluate and/or noisy; we refer the reader to [2, 18] for background on DFO and examples of applications and to [32] for a comprehensive survey of recent work. The use of DFO for algorithm tuning has previously been considered in a general framework [3], and in the specific case of hyperparameter tuning for neural networks in [31].
Here, we are interested in the particular setting of learning for variational methods (2), which has also been considered in [39] where a new DFO algorithm based on discrete gradients has been proposed. In [39] it was assumed that the lowerlevel problem can be solved exactly such that the bilevel problem can be reduced to a single nonconvex optimization problem. In the present work we lift this stringent assumption.
In this paper we focus on DFO methods which are adapted to nonlinear leastsquares problems as analyzed in [13, 47]. These methods are called model based in that they construct a model approximating the objective at each iteration, locally minimize the model to select the next iterate, and update the model with new objective information. Our work also connects to [19], which considers modelbased bilevel optimization where both the lower and upperlevel problems are solved in a derivativefree manner; particular attention is given here to reusing evaluations of the (assumed expensive) lowerlevel objective at nearby upperlevel parameters, to make lowerlevel model construction simpler.
Our approach for bilevel DFO is based on dynamicaccuracy (derivativebased) trustregion methods [17, Chapter 10.6]. In these approaches, we use the measures of convergence (e.g., trustregion radius, model gradient) to determine a suitable level of accuracy with which to evaluate the objective; we start with lowaccuracy requirements, and increase the required accuracy as we converge to a solution. In a DFO context, this framework is the basis of [19], and a similar approach was considered in [15] in the context of analyzing protein structures. This framework has also been recently extended in a derivativebased context to higherorder regularization methods [7, 26]. We also note that there has been some work on multilevel and multifidelity models (in both a DFO and derivativebased context), where an expensive objective can be approximated by surrogates which are cheaper to evaluate [11, 34].
1.2 Contributions
There are a number of novel aspects to this work. Our use of DFO for bilevel learning means our upperlevel solver genuinely expects inexact lowerlevel solutions. We give worstcase complexity theory for our algorithm both in terms of upperlevel iterations and computational work from the lowerlevel problems. Our numerical results on ROF denoising and a new framework for learning MRI sampling patterns demonstrate our approach is substantially faster—up to 100 times faster—than the same DFO approach with highaccuracy lowerlevel solutions, while achieving the same quality solutions. More details on the different aspects of our contributions are given below.
1.2.1 Dynamic Accuracy DFO Algorithm for Bilevel Learning
As noted in [45], bilevel learning can require very highaccuracy solutions to the lowerlevel problem. We avoid this via the introduction of a dynamic accuracy modelbased DFO algorithm. In this setting, the upperlevel solver dynamically changes the required accuracy for lowerlevel problem minimizers, where less accuracy is required in earlier phases of the upperlevel optimization. The proposed algorithm is similar to [19], but adapted to the nonlinear leastsquares case and allowing derivativebased methods to solve the lowerlevel problem. Our theoretical results extend the convergence results of [19] to include derivativebased lowerlevel solvers and a leastsquares structure, as well as adding a worstcase complexity analysis in a style similar to [13] (which is also not present in the derivativebased convergence theory in [17]). This analysis gives bounds on the number of iterations of the upperlevel solver required to reach a given optimality, which we then extend to bound the total computational effort required for the lowerlevel problem solves. There is increasing interest, but comparatively fewer works, which explicitly bound the total computational effort of nonconvex optimization methods; see [42] for NewtonCG methods and references therein. We provide a preliminary argument that our computational effort bounds are tight with regards to the desired upperlevel solution accuracy, although we delegate a complete proof to future work.
1.2.2 Robustness
We observe in all our results using several lowerlevel solvers (gradient descent and FISTA) for a variety of applications that the proposed upperlevel DFO algorithm converges to similar objective values and minimizers. We also present numerical results for denoising showing that the learned parameters are robust to initialization of the upperlevel solver despite the upperlevel problem being likely nonconvex. Together, these results suggest that this framework is a robust approach for bilevel learning.
1.2.3 Efficiency
Bilevel learning with a DFO algorithm was previously considered [39], but there a different DFO method based on discrete gradients was used, and was applied to nonsmooth problems with exact lowerlevel evaluations. In [39], only up to two parameters were learned, whereas here we demonstrate our approach is capable of learning many more. Our numerical results include examples with up to 64 parameters.
We demonstrate that the dynamic accuracy DFO achieves comparable or better objective values than the fixed accuracy variants and final reconstructions of comparable quality. However, our approach is able to achieve this with a dramatically reduced computational load, in some cases up to 100 times less work than the fixed accuracy variants.
1.2.4 New Framework for Learning MRI Sampling
We introduce a new framework to learn the sampling pattern in MRI based on bilevel learning. Our idea is inspired by the image inpainting model of [16]. Compared to other algorithms to learn the sampling pattern in MRI based on firstorder methods [45], the proposed approach seems to be much more robust to initialization and choice of solver for the lowerlevel problem. As with the denoising examples, our dynamic accuracy DFO achieves the same upperlevel objective values and final reconstructions as fixed accuracy variants but with substantial reductions in computational work.
1.2.5 Regularization Parameter Choice Rule with Machine Learning
Our numerical results suggest that the bilevel framework can learn regularization parameter choice rule which yields a convergent regularization method in the sense of [29, 44], indicating for the first time that machine learning can be used to learn mathematically sound regularization methods.
1.3 Structure
In Sect. 2 we describe problems where the lowerlevel model (1) applies and describe how to efficiently attain a given accuracy level using standard firstorder methods. Then, in Sect. 3 we introduce the dynamic accuracy DFO algorithm and present our global convergence and worstcase complexity bounds. Finally, our numerical experiments are described in Sect. 4.
1.4 Notation
Throughout, we let \(\Vert \cdot \Vert \) denote the Euclidean norm of a vector in \(\mathbb {R}^n\) and the operator 2norm of a matrix in \(\mathbb {R}^{m\times n}\). We also define the weighted (semi)norm \(\Vert x\Vert ^2_S := x^T Sx\) for a symmetric and positive (semi)definite matrix S. The gradient of a scalarvalued function \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) is denoted by \(\nabla f:\mathbb {R}^n\rightarrow \mathbb {R}^n\), and the derivative of a vectorvalued function \(f : \mathbb R^n \rightarrow \mathbb R^m\) is denoted by \(\partial f : \mathbb R^n \rightarrow \mathbb R^{n \times m}, (\partial f)_{i,j} = \partial _i f_j\) where \(\partial _i f_j\) denotes the partial derivative of \(f_j\) with respect to the ith coordinate. If f is a function of two variables x and y, then \(\partial _x f\) denotes the derivative with respect to x.
1.5 Software
Our implementation of the DFO algorithm and all numerical testing code will be made public upon acceptance.
2 LowerLevel Problem
In order to have sufficient control over the accuracy of the solution to (3) we will assume that \(\Phi _{i,\theta }\) are \(L_i\)smooth and \(\mu _i\)strongly convex, see definitions below.
Definition 1
(Smoothness) A function \(f : \mathbb R^n \rightarrow \mathbb R\) is Lsmooth if it is differentiable and its derivative is Lipschitz continuous with constant \(L>0\), i.e., for all \(x, y \in \mathbb R^n\) we have \(\Vert \nabla f(x)  \nabla f(y)\Vert \le L \Vert x  y\Vert \).
Definition 2
(Strong Convexity) A function \(f : \mathbb R^n \rightarrow \mathbb R\) is \(\mu \)strongly convex for \(\mu >0\) if \(f  \frac{\mu }{2}\Vert \cdot \Vert ^2\) is convex.
Moreover, when the lowerlevel problem is strictly convex and smooth, with \(\Phi _i(x, \theta ) := \Phi _{i,\theta }(x)\) we can equivalently describe the minimizer of \(\Phi _{i,\theta }\) by
Smoothness properties of \(\hat{x}_i\) follow from the implicit function theorem and its generalizations if \(\Phi _i\) is smooth and regular enough.
Assumption 1
We assume that for all \(i = 1, \ldots , n\) the following statements hold.

1.
Convexity: For all \(\theta \in \Theta \) the functions \(\Phi _{i,\theta }\) are \(\mu _i\)strongly convex.

2.
Smoothness in x: For all \(\theta \in \Theta \) the functions \(\Phi _{i,\theta }\) are \(L_i\)smooth.

3.
Smoothness in \((x, \theta )\): The derivatives \(\partial _x \Phi _i: X\times \Theta \rightarrow X\) and \(\partial _{xx} \Phi _i: X\times \Theta \rightarrow X^2\) exist and are continuous.
Theorem 1
Under Assumption 1 the function \(\hat{x}_i(\theta ) := \arg \min _x \Phi _i(x, \theta )\) is
 1.:

welldefined
 2.:

locally Lipschitz
 3.:

continuously differentiable and
Proof
Ad (1) Finite and convex functions are continuous [41, Corollary 2.36]. It is easy to show that \(\mu \)strongly convex functions are coercive. Then, the existence and uniqueness follow from classical theorems, e.g., [10, Theorem 6.31]. Ad (2) This statement follows directly from [40, Theorem 2.1]. Ad (3) This follows directly from the classical inverse function theorem, see, e.g., [22, Theorem 3.5.1]. \(\square \)
2.1 Examples
A relevant case of the model introduced above is the parameter tuning for linear inverse problems, which can be solved via the variational regularization model
where \({\text {TV}}(x) := \sum _{j=1}^m \Vert \widehat{\nabla } x(j)\Vert \) denotes the discretized total variation, e.g., \(\widehat{\nabla } x(j)\) is the finite forward difference discretization of the spatial gradient of x at pixel j. However, we note that (5) does not satisfy Assumption 1.
To ensure Assumption 1 holds, we instead use \(\Vert x\Vert \approx \sqrt{\Vert x\Vert ^2 + \nu ^2}\), to approximate problem (5) by a smooth and strongly convex problem of the form
with the smoothed total variation given by \({\text {TV}}_{\nu (\theta )}(x) := \sum _{j=1}^m \sqrt{\Vert \widehat{\nabla } x(j)\Vert ^2 + \nu (\theta )^2}\). Here we already introduced the notation that various parts of the problem may depend on a vector of parameters \(\theta \) which usually needs to be selected manually. We will learn these parameters using the bilevel framework. For simplicity denote \(A_\theta := A(\theta )\), \(S_\theta := S(\theta )\), \(\alpha _\theta := \alpha (\theta )\), \(\nu _\theta := \nu (\theta )\) and \(\xi _\theta := \xi (\theta )\). Note that \(\Phi _{i,\theta }\) in (6) is \(L_i\)smooth and \(\mu _i\)strongly convex with
where \(\lambda _{{\text {min}}}(A_\theta ^* S_\theta A_\theta )\) denotes the smallest eigenvalue of \(A_\theta ^* S_\theta A_\theta \) and \(A_\theta ^*\) is the adjoint of \(A_\theta \).
We now describe two specific problems we will use in our numerical results. They both choose a specific form for (3) which aims to find a minimizer \(\hat{x}_i(\theta )\) which (approximately) recovers the data \(x_i\), and so both use (2) as the upperlevel problem.
2.1.1 Total VariationBased Denoising
A particular problem we consider is a smoothed version of the ROF model [43], i.e., \(A_\theta = I, S_\theta = I\). Then, (6) simplifies to
which is \(L_i\)smooth and \(\mu _i\)strongly convex with constants as in (7) with \(\Vert A_\theta ^* S_\theta A_\theta \Vert = \Vert I\Vert = 1\) and \(\lambda _{{\text {min}}}(A_\theta ^* S_\theta A_\theta ) = \lambda _{{\text {min}}}(I) = 1\). In our numerical examples we will consider two cases. First, we will just learn the regularization parameter \(\alpha \) given manually set \(\nu \) and \(\xi \). Second, we will learn all three parameters \(\alpha , \nu \) and \(\xi \).
2.1.2 Undersampled MRI Reconstruction
Another problem we consider is the reconstruction from undersampled MRI data, see, e.g., [33], which can be phrased as (6) with \(A_\theta = F\) where F is the discrete Fourier transform and \(S_\theta = {\text {diag}}(s), s \in [0, 1]^d\). Then, (6) simplifies to
which is \(L_i\)smooth and \(\mu _i\)strongly convex with constants as in (7) with \(\Vert A_\theta ^* S_\theta A_\theta \Vert \le 1\) and \(\lambda _{{\text {min}}}(A_\theta ^* S_\theta A_\theta ) \ge 0\). The sampling coefficients \(s_j\) indicate the relevance of a sampling location. The data term (9) can be rewritten as
Most commonly the values s are binary and manually chosen. Here we aim to use bilevel learning to find a sparse s such that the images \(x_i\) can be reconstructed well from sparse samples of \(y_i\). This approach was first proposed in [45].
2.2 Example Training Data
Throughout this paper, we will consider training data of artificially generated 1D images. Each ground truth image \(x_i\) is randomly generated piecewiseconstant function. For a desired image size N, we select values \(C_i\in [N/4,3N/4]\) and \(R_i\in [N/8,N/4]\) from a uniform distribution. We then define \(x_i\in \mathbb {R}^N\) by
That is, each \(x_i\) is zero except for a single randomly generated subinterval of length \(2 R_i\) centered around \(C_i\) where it takes the value 1.
We then construct our \(y_i\) by taking the signal to be reconstructed and adding Gaussian noise. Specifically, for the image denoising problem we take
where \(\sigma >0\) and \(\omega _i\in \mathbb {R}^N\) is randomly drawn vector of i.i.d. standard Gaussians. For the MRI sampling problem, we take
where \(\sigma >0\) and \(\omega _i\in \mathbb {C}^N\) is a randomly drawn vector with real and imaginary parts both standard Gaussians.
In Fig. 1 we plot an example collection of pairs \((x_i,y_i)\) for the image denoising problem with \(N=256\), and in Fig. 2 we plot the solution to (8) for the first of these \((x_i,y_i)\) pairs for a variety of choices for the parameters \(\alpha _{\theta },\epsilon _{\theta },\eta _{\theta }\).
2.3 Approximate Solutions
2.3.1 Gradient Descent
For simplicity we drop the dependence on i for the remainder of this section.
The lowerlevel problem (3) can be solved with gradient descent (GD) which converges linearly for Lsmooth and \(\mu \)strongly convex problems. One can show (e.g., [14]) that GD
with \(\tau = 1/L\), converges linearly to the unique solution \(x^*\) of (3). More precisely, for all \(k \in \mathbb N\) we have [5, Theorem 10.29]
Moreover, if one has a good estimate of the strong convexity constant \(\mu \), then it is better to choose \(\tau =2/(L+\mu )\), which gives an improved linear rate [36, Theorem 2.1.15]
2.3.2 FISTA
Similarly, we can use FISTA [6] to approximately solve the lowerlevel problem. FISTA applied to a smooth objective with convex constraints is a modification of [35] and can be formulated as the iteration
where \(q:=\tau \mu \), and we choose \(\tau =1/L\) and \(t_0=0\) [14, Algorithm 5]. We then achieve linear convergence with [14, Theorem 4.10]
and so, since \(\Phi (x^k)\Phi (x^*)\ge (\mu /2)\Vert x^kx^*\Vert ^2\) from \(\mu \)strong convexity, we get
2.3.3 Ensuring Accuracy Requirements
We will need to be able to solve the lowerlevel problem to sufficient accuracy that we can guarantee \(\Vert x^kx^*\Vert ^2 \le \epsilon \), for a suitable accuracy \(\epsilon >0\). We can guarantee this accuracy by ensuring we terminate with k sufficiently large, given an estimate \(\Vert x^0x^*\Vert ^2\), using the a priori bounds (15) or (19). A simple alternative is to use the a posteriori bound \(\Vert xx^*\Vert \le \Vert \nabla \Phi (x)\Vert /\mu \) for all x (a consequence of [6, Theorem 5.24(iii)]), and terminate once
To compare these two options, we consider two test problems: (i) a version of Nesterov’s quadratic [36, Section 2.1.4] in \(\mathbb {R}^{10}\), and (ii) 1D image denoising. Nesterov’s quadratic is defined as
for \(x\in \mathbb {R}^{10}\), with \(\widetilde{\mu }=1\) and \(Q=100\), which is \(\mu \)strongly convex and Lsmooth for \(\mu \approx 3\) and \(L\approx 98\); we apply no constraints, \(X=\mathbb {R}^{10}\).
We also consider a 1D denoising problem as in (8) with randomly generated data \(y\in \mathbb {R}^N\) (with \(N=100\) pixels) as per Section 2.2, \(\alpha =0.3\), \(\nu =\xi =10^{3}\), and \(x^*\) estimated by running \(10^4\) iterations of FISTA. Here, the problem is \(\mu \)strongly convex and Lsmooth with \(\mu \approx 1\) and \(L\approx 1,201\). We estimate the true solution \(x^*\) by running FISTA for 10,000 iterations (which gives an upper bound estimate \(\Vert x^kx^*\Vert ^2\le 3\mathrm {e}{26}\) from (20)).
In Fig. 3, we compare the true error \(\Vert x^kx^*\Vert ^2\) against the a priori linear convergence bounds (15) or (19) with the true value of \(\Vert x^0x^*\Vert ^2\), and the a posteriori gradient bound (20). In both cases, the gradientbased bound (20) provides a much tighter estimate of the error, particularly for highaccuracy requirements. Thus, in our numerical results, we terminate the lowerlevel solver as soon (20) is achieved for our desired tolerance. The gradientbased bound has the additional advantage of not requiring an a priori estimate of \(\Vert x^0x^*\Vert \). For comparison, in our results below we will also consider terminating GD/FISTA after a fixed number of iterations.
3 Dynamic Accuracy DFO Algorithm
3.1 DFO Background
Since evaluating \(\hat{x}_i(\theta )\) in the upperlevel problem (2) is only possible with some error (it is computed by running an iterative process), it is not straightforward or cheap to evaluate \(\partial \hat{x}_i(\theta )\). Hence for solving (2) we turn to DFO techniques, and specifically consider those which exploit the nonlinear leastsquares problem structure. In this section we outline a modelbased DFO method for nonlinear leastsquares problems [13], a trustregion method based on the classical (derivativebased) Gauss–Newton method [37, Chapter 10]. However, these approaches are based on having access to exact function evaluations, and so we augment this with a standard approach for dynamic accuracy trustregion methods [17, Chapter 10.6]; this was previously considered for general modelbased DFO methods in [19].
Here, we write the upperlevel problem (2) in the general form
where \(r_i(\theta ):= \Vert \hat{x}_i(\theta )x_i\Vert \) and \(r(\theta ):=[r_1(\theta ), \ldots , r_n(\theta )]^T\). Without loss of generality, we do not include a regularization term \(\mathcal {J}(\theta )\); we can incorporate this term by defining \(r_{n+1}(\theta ):=\sqrt{\mathcal {J}(\theta )}\) and then taking \(r(\theta ):=[r_1(\theta ), \ldots , r_{n+1}(\theta )]^T\), for instance.
The upperlevel objective (22) assumes access to exact evaluations of the lowerlevel objective \(r_i(\theta )\), which is not achievable in practice. We therefore assume we only have access to inaccurate evaluations \(\widetilde{x}_i(\theta )\approx \hat{x}_i(\theta )\), giving \(\widetilde{r}_i(\theta ):= \Vert \widetilde{x}_i(\theta )x_i\Vert \), \(\widetilde{r}(\theta ):=[\widetilde{r}_1(\theta ), \ldots , \widetilde{r}_n(\theta )]^T\), and \(\widetilde{f}(\theta ):=\frac{1}{n}\Vert \widetilde{r}(\theta )\Vert ^2\).
Our overall algorithmic framework is based on trustregion methods, where at each iteration k we construct a model \(m^k\) for the objective which we hope is accurate in a neighborhood of our current iterate \(\theta ^k\). Simultaneously we maintain a trustregion radius \(\Delta ^k>0\), which tracks the size of the neighborhood of \(\theta ^k\) where we expect \(m^k\) to be accurate. Our next iterate is determined by minimizing the model \(m^k\) within a ball of size \(\Delta ^k\) around \(\theta ^k\).
Usually \(m^k\) is taken to be a quadratic function (e.g., a secondorder Taylor series for f about \(\theta ^k\)). However, here we use the leastsquares problem structure (22) and construct a linear model
where \(\widetilde{r}(\theta ^k)\) is our approximate evaluation of \(r(\theta ^k)\) and \(J^k\in \mathbb {R}^{n\times d}\) is a matrix approximating \(\partial r(\theta ^k)^T\). We construct \(J^k\) by interpolation: we maintain an interpolation set \(z^0,\ldots ,z^d\in \mathbb {R}^d\) (where \(z^0:=\theta ^k\) at each iteration k) and choose \(J^k\) so that
This condition ensures that our linear model \(M^k\) exactly interpolates \(\widetilde{r}\) at our interpolation points \(z^t\) (i.e., the second approximation in (23) is exact for each \(s=z^t\theta ^k\)). We can therefore find \(J^k\) by solving the \(d\times d\) linear system (with n righthand sides):
for all \(i=1,\ldots ,n\), where \(g^k_i\in \mathbb {R}^d\) is the ith row of \(J^k\). The model \(M^k\) gives a natural quadratic model for the full objective f:
where \(g^k:= \frac{2}{n} (J^k)^T \widetilde{r}(\theta ^k)\) and \(H^k:=\frac{2}{n}(J^k)^T J^k\). We compute a tentative step \(s^k\) as a(n approximate) minimizer of the trustregion subproblem
There are a variety of efficient algorithms for computing \(s^k\) [17, Chapter 7]. Finally, we evaluate \(\widetilde{f}(\theta ^k+s^k)\) and decide whether to accept or reject the step (i.e., set \(\theta ^{k+1}=\theta ^k+s^k\) or \(\theta ^{k+1}=\theta ^k\)) depending on the ratio
Although we would like to accept/reject using \(\rho ^k\), in reality we only observe the approximation
and so we use this instead.
This gives us the key components of a standard trustregion algorithm. We have two extra considerations in our context: the accuracy of our derivativefree model (26) and the lack of exact evaluations of the objective.
Firstly, we require a procedure to verify if our model (26) is sufficiently accurate inside the trustregion, and if not, modify the model to ensure its accuracy. We discuss this in Section 3.2. The notion of ‘sufficiently accurate’ we use here is that \(m^k\) is as good an approximation to f as a firstorder Taylor series (up to constant factors), which we call ‘fully linear.’^{Footnote 1}
Definition 3
(Fully linear model) The model \(m^k\) (26) is a fully linear model for \(f(\theta )\) in \(B(\theta ^k,\Delta ^k)\) if there exist constants \(\kappa _\mathrm{ef},\kappa _\mathrm{eg}>0\) (independent of \(\theta ^k\) and \(\Delta ^k\)) such that
for all \(\Vert s\Vert \le \Delta ^k\).
Secondly, we handle the inaccuracy in objective evaluations by ensuring \(\widetilde{f}(\theta ^k)\) and \(\widetilde{f}(\theta ^k+s^k)\) are evaluated to a sufficiently high accuracy when we compute \(\widetilde{\rho }^k\) (29). Specifically, suppose we know that \(\widetilde{f}(\theta ^k)f(\theta ^k)\le \delta ^k\) and \(\widetilde{f}(\theta ^k+s^k)f(\theta ^k+s^k)\le \delta ^k_+\) for some accuracies \(\delta ^k\) and \(\delta ^k_+\). Throughout, we use \(\delta ^k\) and \(\delta ^k_+\) to refer to the accuracies with which \(\widetilde{f}(\theta ^k)\) and \(\widetilde{f}(\theta ^k+s^k)\) have been evaluated, in the sense above. Before we compute \(\widetilde{\rho }^k\), we first ensure that
where \(\eta _1^{\prime }>0\) is an algorithm parameter. We achieve this by running the lowerlevel solver for a sufficiently large number of iterations.
The full upperlevel algorithm is given in Algorithm 1; it is similar to the approach in [19], the DFO method [18, Algorithm 10.1]—adapted for the leastsquares problem structure—and the (derivativebased) dynamic accuracy trustregion method [17, Algorithm 10.6.1].
Our main convergence result is the below.
Theorem 2
Suppose Assumptions 2 and 3 hold. Then if
with \(\kappa _\mathrm{eg}\) and \(\Delta _{\min }\) given by Lemmas 1 and 8, respectively, then \(\min _{k=0,\ldots ,K}\Vert \nabla f(\theta ^k)\Vert < \epsilon \).
We summarize Theorem 2 as follows, noting that the iteration and evaluation counts match the standard results for modelbased DFO (e.g., [13, 24]).
Corollary 1
Suppose the assumptions of Theorem 2 hold. Then, Algorithm 1 is globally convergent, i.e.,
Also, if \(\epsilon \in (0,1]\), then the number of iterations before \(\Vert \nabla f(\theta ^k)\Vert <\epsilon \) for the first time is at most \(\mathcal {O}(\kappa ^3\epsilon ^{2})\) and the number of evaluations of \(\widetilde{r}(\theta )\) is at most \(\mathcal {O}(d\kappa ^3\epsilon ^{2})\), where \(\kappa :=\max (\kappa _\mathrm{ef}, \kappa _\mathrm{eg}, \kappa _H)\).^{Footnote 2}
We note that since \(\theta ^k \in \mathcal {B}\) and \(\mathcal {B}\) is bounded from Assumption 2 (and closed by continuity of f), then by Corollary 1 and compactness there exists a subsequence of iterates \(\{\theta _{k_i}\}_{i\in \mathbb {N}}\) which converges to a stationary point of f. However, there are relatively few results which prove convergence of the full sequence of iterates for nonconvex trustregion methods (see [17, Theorem 10.13] for a restricted result in the derivativebased context).
3.2 Guaranteeing Model Accuracy
As described above, we need a process to ensure that \(m^k\) (26) is a fully linear model for f inside the trust region \(B(\theta ^k,\Delta ^k)\). For this, we need to consider the geometry of the interpolation set.
Definition 4
The Lagrange polynomials of the interpolation set \(\{z^0,z^1,\ldots ,z^d\}\) are the linear polynomials \(\ell _t\), \(t=0,\ldots ,d\) such that \(\ell _t(z^s)=\delta _{s,t}\) for all \(s,t=0,\ldots ,d\).
The Lagrange polynomials of \(\{z^0,\ldots ,z^d\}\) exist and are unique whenever the matrix in (25) is invertible. The required notion of ‘good geometry’ is given by the below definition (where small \(\Lambda \) indicates better geometry).
Definition 5
(\(\Lambda \)poisedness) For \(\Lambda >0\), the interpolation set \(\{z^0,\ldots ,z^d\}\) is \(\Lambda \)poised in \(B(\theta ^k,\Delta ^k)\) if \(\ell _t(\theta ^k+s)\le \Lambda \) for all \(t=0,\ldots ,d\) and all \(\Vert s\Vert \le \Delta ^k\).
The below result confirms that, provided our interpolation set has sufficiently good geometry, and our evaluations \(\widetilde{r}(\theta ^k)\) and \(\widetilde{r}(y^t)\) are sufficiently accurate, our interpolation models are fully linear.
Assumption 2
The extended level set
is bounded, and \(r(\theta )\) is continuously differentiable and \(\partial r(\theta )\) is Lipschitz continuous with constant \(L_J\) in \(\mathcal {B}\).
In particular, Assumption 2 implies that \(r(\theta )\) and \(\partial r(\theta )\) are uniformly bounded in the same region—that is, \(\Vert r(\theta )\Vert \le r_{\max }\) and \(\Vert \partial r(\theta )\Vert \le J_{\max }\) for all \(\theta \in \mathcal {B}\)—and f (22) is Lsmooth in \(\mathcal {B}\) [13, Lemma 3.2].
We note that \(\mathcal {B}\) in Assumption 2 is bounded whenever the regularizer \(\mathcal {J}\) is coercive, such as in Sect. 4.4. This may also be replaced by the weaker assumption that r and \(\partial r\) are uniformly bounded on \(\mathcal {B}\) (and \(\mathcal {B}\) need not be bounded) [13, Assumption 3.1], and there are theoretical results which give this for some inverse problems in image restoration [20]. In our numerical experiments, we enforce upper and lower bounds on \(\theta \), which also yields the uniform boundedness of r and \(\partial r\). Also, we note that if \(r_i(\theta )=\Vert \hat{x}_i(\theta )x_i\Vert \) is not itself Lsmooth, we can instead treat each entry of \(\hat{x}_i(\theta )x_i\) as a separate term in (22).
Lemma 1
Suppose Assumption 2 holds and \(\Delta ^k\le \Delta _{\max }\). If the interpolation set \(\{z^0:=\theta ^k,z^1,\ldots ,z^d\}\) is \(\Lambda \)poised in \(B(\theta ^k,\Delta ^k)\) and for each evaluation \(t=0,\ldots ,d\) and each \(i=1,\ldots ,n\) we have
for some \(c>0\), then the corresponding models \(M^k\) (23) and \(m^k\) (26) are fully linear models for \(r(\theta )\) and \(f(\theta )\), respectively.
Proof
This is a straightforward extension of [13, Lemma 3.3], noting that
and so (38) gives \(\Vert \widetilde{r}(z^t)  r(z^t)\Vert \le c \sqrt{n} (\Delta ^k)^2\) for all \(t=0,\ldots ,d\).
We conclude by noting that for any \(\Lambda >1\) there are algorithms available to determine if a set is \(\Lambda \)poised, and if it is not, change some interpolation points to make it so; details may be found in [18, Chapter 6], for instance.
3.3 LowerLevel Objective Evaluations
We now consider the accuracy requirements that Algorithm 1 imposes on our lowerlevel objective evaluations. In particular, we require the ability to satisfy (32), which imposes requirements on the error in the calculated \(\widetilde{f}\), rather than the lowerlevel evaluations \(\widetilde{r}\). The connection between errors in \(\widetilde{r}\) and \(\widetilde{f}\) is given by the below result.
Lemma 2
Suppose we compute \(\widetilde{x}_i(\theta )\) satisfying \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert \le \delta _x\) for all \(i=1,\ldots ,n\). Then, we have
Moreover, if \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert \le \sqrt{\widetilde{f}(\theta )+\delta _f}  \sqrt{\widetilde{f}(\theta )}\) for \(i=1,\ldots ,n\), then \(\widetilde{f}(\theta )f(\theta )\le \delta _f\).
Proof
Letting \(\epsilon (\theta ):=\widetilde{r}(\theta )r(\theta )\), we have
and hence
and the first part of (40) follows since \(\Vert \epsilon (\theta )\Vert _{\infty }\le \delta _x\) from (39). The second part of (40) follows from an identical argument but writing \(\widetilde{f}(\theta )=\frac{1}{n}\Vert r(\theta )+\epsilon (\theta )\Vert ^2\), and the final conclusion follows immediately from the first part of (40). \(\square \)
We construct these bounds to rely mostly on \(\widetilde{f}(\theta )\), since this is the value which is observed by the algorithm (rather than the true value \(f(\theta )\)). From the concavity of \(\sqrt{\cdot }\), if \(\widetilde{f}(\theta )\) is larger, then \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert \) must be smaller to achieve the same \(\delta _f\).
Lastly, we note the key reason why we require (32): it guarantees that our estimate \(\widetilde{\rho }^k\) of \(\rho ^k\) is not too inaccurate.
Lemma 3
Suppose \(\widetilde{f}(\theta ^k)f(\theta ^k)\le \delta ^k\) and \(\widetilde{f}(\theta ^k+s^k)f(\theta ^k+s^k)\le \delta ^k_+\). If (32) holds, then \(\widetilde{\rho }^k\rho ^k \le 2\eta _1^{\prime }\).
Proof
Follows immediately from (29) and (28); see [17, Section 10.6.1]. \(\square \)
3.4 Convergence and WorstCase Complexity
We now prove the global convergence of Algorithm 1 and analyze its worstcase complexity (i.e., the number of iterations required to achieve \(\Vert \nabla f(\theta ^k)\Vert \le \epsilon \) for the first time).
Assumption 3
The computed trustregion step \(s^k\) satisfies
and there exists \(\kappa _H\ge 1\) such that \(\Vert H^k\Vert + 1 \le \kappa _H\) for all k.
Assumption 3 is standard and the condition (45) easy to achieve in practice [17, Chapter 6.3].
Firstly, we must show that the inner loops for the criticality and accuracy phases terminate. We begin with the criticality phase and then consider the accuracy phase.
Lemma 4
([13], Lemma B.1]) Suppose Assumption 2 holds and \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon >0\). Then, the criticality phase terminates in finite time with
where \(\Delta ^k_\mathrm{init}\) is the value of \(\Delta ^k\) before the criticality phase begins.
Lemma 5
([13], Lemma 3.7]) Suppose Assumption 2 holds. Then, in all iterations we have \(\Vert g^k\Vert \ge \min (\epsilon , \Delta ^k)\). Also, if \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon >0\), then \(\Vert g^k\Vert \ge \epsilon /(\kappa _\mathrm{eg}+ 1) > 0\).
We note that our presentation of the criticality phase here can be made more general by allowing \(\Vert g^k\Vert \ge \epsilon _C \ne \epsilon \) as the entry test, setting \(\Delta ^k\) to \(\omega ^i \Delta ^k\) for some \(\omega \in (0,1)\) possibly different to \(\gamma _\mathrm{dec}\), and having an exit test \(\Delta ^k \le \mu \Vert g^k\Vert \) for some \(\mu >0\). All the below results hold under these assumptions, with modifications as per [13].
Lemma 6
If Assumptions 2 and 3 hold and \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon >0\), then the accuracy phase terminates in finite time (i.e., line 10 of Algorithm 1 is eventually called)
Proof
From Lemma 5 we have \(\Vert g^k\Vert \ge \epsilon /(\kappa _\mathrm{eg}+ 1)\), and the result then follows from [17, Lemma 10.6.1]. \(\square \)
We now collect some key preliminary results required to establish complexity bounds.
Lemma 7
Suppose Assumptions 2 and 3 hold, \(m^k\) is fully linear in \(B(\theta ^k,\Delta ^k)\) and
then \(\widetilde{\rho }^k\ge \eta _2\).
Proof
We compute
Since \(\Delta ^k \le \Vert g^k\Vert /\kappa _H\), from Assumption 3 we have
From this and full linearity, we get
and so \(\rho ^k\ge \eta _2+2\eta _1^{\prime }\); hence, \(\widetilde{\rho }^k\ge \eta _2\) from Lemma 3. \(\square \)
Lemma 8
Suppose Assumptions 2 and 3 hold. Suppose \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon \) for all \(k=0,\ldots ,k_{\epsilon }\) and some \(\epsilon \in (0,1)\). Then, for all \(k\le k_{\epsilon }\),
Proof
As above, we let \(\Delta ^k_\mathrm{init}\) and \(m^k_\mathrm{init}\) denote the values of \(\Delta ^k\) and \(m^k\) before the criticality phase (i.e., \(\Delta ^k_\mathrm{init}=\Delta ^k\) and \(m^k_\mathrm{init}=m^k\) if the criticality phase is not called). From Lemma 5, we know \(\Vert g^k\Vert \ge \epsilon /(\kappa _\mathrm{eg}+1)\) for all \(k\le k_{\epsilon }\). Suppose by contradiction \(k\le k_{\epsilon }\) is the first iteration such that \(\Delta ^k<\Delta _{\min }\). Then from Lemma 4,
and so \(\Delta ^k\ge \Delta ^k_\mathrm{init}\); hence, \(\Delta ^k=\Delta ^k_\mathrm{init}\). That is, either the criticality phase is not called, or terminates with \(i=0\) (in this case, the model \(m^k\) is formed simply by making \(m^k_\mathrm{init}\) fully linear in \(B(\theta ^k,\Delta ^k)=B(\theta ^k,\Delta ^k_\mathrm{init})\)).
If the accuracy phase loop occurs, we go back to the criticality phase, which can potentially happen multiple times. However, since the only change is that \(\widetilde{r}(\theta ^k)\) is evaluated to higher accuracy, incorporating this information into the model \(m^k\) can never destroy full linearity. Hence, after the accuracy phase, by the same reasoning as above, either one iteration of the criticality phase occurs (i.e., \(m^k\) is made fully linear) or it is not called. If the accuracy phase is called multiple times and the criticality phase occurs multiple times, all times except the first have no effect (since the accuracy phase can never destroy full linearity). Thus, \(\Delta ^k\) is unchanged by the accuracy phase.
Since \(\Delta _{\min }<\Delta ^0_\mathrm{init}\), we have \(k\ge 1\). As k is the first iteration such that \(\Delta ^k<\Delta _{\min }\) and \(\Delta ^k=\Delta ^k_\mathrm{init}\), we must have \(\Delta ^k_\mathrm{init}=\gamma _\mathrm{dec}\Delta ^{k1}\) (as this is the only other way \(\Delta ^k\) can be reduced). Therefore, \(\Delta ^{k1}=\Delta ^k/\gamma _\mathrm{dec}<\Delta _{\min }/\gamma _\mathrm{dec}\), and so
We then have \(\Delta ^{k1} \le c_0\epsilon /(\kappa _\mathrm{eg}+1) \le c_0\Vert g^{k1}\Vert \), and so by () either \(\widetilde{\rho }^k\ge \eta _2\) or \(m^{k1}\) is not fully linear. Either way, we set \(\Delta ^k_\mathrm{init}\ge \Delta ^{k1}\) in (). This contradicts \(\Delta ^k_\mathrm{init}=\gamma _\mathrm{dec}\Delta ^{k1}\) above, and we are done. \(\square \)
We now bound the number of iterations of each type. Specifically, we suppose that \(k_{\epsilon }+1\) is the first k such that \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon \). Then, we define the sets of iterations:

\(\mathcal {S}_\epsilon \) is the set of iterations \(k\in \{0,\ldots ,k_{\epsilon }\}\) which are ‘successful,’ i.e., \(\widetilde{\rho }^k\ge \eta _2\), or \(\widetilde{\rho }^k\ge \eta _1\) and \(m^k\) is fully linear in \(B(\theta ^k,\Delta ^k)\).

\(\mathcal {M}_\epsilon \) is the set of iterations \(k\in \{0,\ldots ,k_{\epsilon }\}\) which are ‘modelimproving,’ i.e., \(\widetilde{\rho }^k<\eta _2\) and \(m^k\) is not fully linear in \(B(\theta ^k,\Delta ^k)\).

\(\mathcal {U}_\epsilon \) is the set of iterations \(k\in \{0,\ldots ,k_{\epsilon }\}\) which are ‘unsuccessful,’ i.e., \(\widetilde{\rho }^k<\eta _1\) and \(m^k\) is fully linear in \(B(\theta ^k,\Delta ^k)\).
These three sets form a partition of \(\{0,\ldots ,k_{\epsilon }\}\).
Proposition 1
Suppose Assumptions 2 and 3 hold. Then,
Proof
By definition of \(k_{\epsilon }\), \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon \) for all \(k\le k_{\epsilon }\) and so Lemma 5 and Lemma 8 give \(\Vert g^k\Vert \ge \epsilon /(\kappa _\mathrm{eg}+1)\) and \(\Delta ^k\ge \Delta _{\min }\) for all \(k\le k_{\epsilon }\), respectively. For any \(k\le k_{\epsilon }\) we have
by definition of \(\rho ^k\) and Assumption 3. If \(k\in \mathcal {S}_{\epsilon }\), we know \(\widetilde{\rho _k}\ge \eta _1\), which implies \(\rho ^k\ge \eta _12\eta _1^{\prime }>0\) from Lemma 3. Therefore,
for all \(k\in \mathcal {S}_{\epsilon }\), where the last line follows since \(\Delta _{\min } < c_0\epsilon /(\kappa _\mathrm{eg}+1) \le \epsilon /[\kappa _H(\kappa _\mathrm{eg}+1)]\) by definition of \(\Delta _{\min }\) (52) and \(c_0\) (47).
The iterate \(\theta ^k\) is only changed on successful iterations (i.e., \(\theta ^{k+1}=\theta ^k\) for all \(k\notin \mathcal {S}_{\epsilon }\)). Thus, as \(f(\theta )\ge 0\) from the leastsquares structure (22), we get
and the result follows. \(\square \)
We are now in a position to prove our main results.
Proof of Theorem 2
To derive a contradiction, suppose that \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon \) for all \(k\in \{0,\ldots ,K\}\), and so \(\Vert g^k\Vert \ge \epsilon /(\kappa _\mathrm{eg}+1)\) and \(\Delta ^k\ge \Delta _{\min }\) by Lemma 5 and Lemma 8, respectively. Since \(K\le k_{\epsilon }\) by definition of \(k_{\epsilon }\), we will try to construct an upper bound on \(k_{\epsilon }\). We already have an upper bound on \(\mathcal {S}_{\epsilon }\) from Proposition 1.
If \(k\in \mathcal {S}_{\epsilon }\), we set \(\Delta ^{k+1}\le \gamma _\mathrm{inc}\Delta ^k\). Similarly, if \(k\in \mathcal {U}_{\epsilon }\) we set \(\Delta ^{k+1}=\gamma _\mathrm{dec}\Delta ^k\). Thus,
That is, \(\Delta _{\min }/\Delta ^0 \le \gamma _\mathrm{inc}^{\mathcal {S}_{\epsilon }} \gamma _\mathrm{dec}^{\mathcal {U}_{\epsilon }}\), and so
noting we have changed \(\Delta _{\min }/\Delta ^0<1\) to \(\Delta ^0/\Delta _{\min }>1\) and used \(\log \gamma _\mathrm{dec} < 0\), so all terms in (65) are positive. Now, the next iteration after a modelimproving iteration cannot be modelimproving (as the resulting model is fully linear), giving
If we combine (65) and (66) with \(k_{\epsilon }\le \mathcal {S}_{\epsilon } + \mathcal {M}_{\epsilon } + \mathcal {U}_{\epsilon }\), we get
which, given the bound on \(\mathcal {S}_{\epsilon }\) (55) means \(K\le k_{\epsilon }\) is bounded above by the righthand side of (35), a contradiction. \(\square \)
Proof of Corollary 1
The iteration bound follows directly from Theorem 2, noting that \(\Delta _{\min }=\mathcal {O}(\kappa ^{2} \epsilon )\). This also implies that \(\liminf _{k\rightarrow \infty } \Vert \nabla f(\theta ^k)\Vert =0\) and so (36) holds from the same argument as in [18, Theorem 10.13] without modification.
For the evaluation bound, we also need to count the number of inner iterations of the criticality phase. Suppose \(\Vert \nabla f(\theta ^k)\Vert <\epsilon \) for \(k=0,\ldots ,k_{\epsilon }\). Similar to the above, we define: (a) \(\mathcal {C}^{M}_{\epsilon }\) to be the number of criticality phase iterations corresponding to the first iteration of \(i=0\) where \(m^k\) was not already fully linear, in iterations \(0,\ldots ,k_{\epsilon }\) and (b) \(\mathcal {C}^{U}_{\epsilon }\) to be the number of criticality phase iterations corresponding to all other iterations \(i>0\) (where \(\Delta ^k\) is reduced and \(m^k\) is made fully linear) in iterations \(0,\ldots ,k_{\epsilon }\).
From Lemma 8 we have \(\Delta ^k\ge \Delta _{\min }\) for all \(k\le k_{\epsilon }\). We note that \(\Delta ^k\) is reduced by a factor \(\gamma _\mathrm{dec}\) for every iteration of the criticality phase in \(\mathcal {C}^{U}_{\epsilon }\). Thus by a more careful reasoning as we used to reach (65), we conclude
Also, after every iteration k in which the first iteration of criticality phase makes \(m^k\) fully linear, we have either a (very) successful or unsuccessful step, not a modelimproving step. From the same reasoning as in Lemma 8, the accuracy phase can only cause at most one more step criticality phase in which \(m^k\) is made fully linear, regardless of how many times it is called.^{Footnote 3} Thus,
Combining (70) and (71) with (65) and (66), we conclude that the number of times we make \(m^k\) fully linear is
where the second inequality follows from Proposition 1.
If \(\epsilon <1\), we conclude that the number of times we make \(m^k\) fully linear before \(\Vert \nabla f(\theta ^k)\Vert <\epsilon \) for the first time is the same as the number of iterations, \(\mathcal {O}(\kappa ^3\epsilon ^{2})\). Since each iteration requires one new objective evaluation (at \(\theta ^k+s^k\)) and each time we make \(m^k\) fully linear requires at most \(\mathcal {O}(d)\) objective evaluations (corresponding to replacing the entire interpolation set), we get the stated evaluation complexity bound. \(\square \)
3.5 Estimating the LowerLevel Work
We have from Corollary 1 that we can achieve \(\Vert \nabla f(\theta ^k)\Vert <\epsilon \) in \(\mathcal {O}(\epsilon ^{2})\) evaluations of \(\widetilde{r}(\theta )\). In this section, we use the fact that evaluations of \(\widetilde{r}(\theta )\) come from finitely terminating a linearly convergent procedure (i.e., strongly convex optimization) to estimate the total work required in the lowerlevel problem. This is particularly relevant in an imaging context, where the lowerlevel problem can be large scale and poorly conditioned; this can be the dominant cost of Algorithm 1.
Proposition 2
Suppose Assumptions 2 and 3 hold and \(\Vert \nabla f(\theta ^k)\Vert \ge \epsilon \) for all \(k=0,\ldots ,k_{\epsilon }\) and some \(\epsilon \in (0,1]\). Then, for every objective evaluation in iterations \(k\le k_{\epsilon }\) it suffices to guarantee that \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert = \mathcal {O}(\epsilon ^2)\) for all \(i=1,\ldots ,n\).
Proof
For all \(k\le k_{\epsilon }\) we have \(\Vert g^k\Vert \ge \epsilon /(\kappa _\mathrm{eg}+1)\) and \(\Delta ^k\ge \Delta _{\min }\) by Lemmas 5 and 8, respectively. There are two places where we require upper bounds on \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert \) in our objective evaluations: ensuring \(\widetilde{f}(\theta ^k)\) and \(\widetilde{f}(\theta ^k+s^k)\) satisfy (32) and ensuring our model is fully linear using Lemma 1.
In the first case, we note that
by Assumption 3 and using \(\Delta _{\min } < c_0\epsilon /(\kappa _\mathrm{eg}+1) \le \epsilon /[\kappa _H(\kappa _\mathrm{eg}+1)]\) by definition of \(\Delta _{\min }\) (52) and \(c_0\) (47). Therefore, to ensure (32) it suffices to guarantee
From Lemma 2, specifically the second part of (40), this means to achieve (32) it suffices to guarantee
for all \(i=1,\ldots ,n\), where \(\theta \in \cup _{k\le k_{\epsilon }} \{\theta ^k, \theta ^k+s^k\}\). From Assumption 2 we have \(f(\theta )\le f_{\max } := r_{\max }^2/n\), and so from the fundamental theorem of calculus we have
Since \(\epsilon <1\), \(\delta _f^{\min }\) is bounded above by a constant and so \(\sqrt{f_{\max }+\delta _f^{\min }}\) is bounded above. Thus, (32) is achieved provided \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert = \mathcal {O}(\delta _f^{\min })\) for all \(i=1,\ldots ,n\).
For the second case (ensuring full linearity), we need to guarantee (38) holds. This is achieved provided \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert = \mathcal {O}(\Delta _{\min }^2)\) for all \(i=1,\ldots ,n\). The result then follows by noting \(\delta _f^{\min }=\mathcal {O}(\epsilon \Delta _{\min })\) and \(\Delta _{\min }=\mathcal {O}(\epsilon )\). \(\square \)
Corollary 1 and Proposition 2 say that to ensure \(\Vert \nabla f(\theta ^k)\Vert <\epsilon \) for some k, we have to perform \(\mathcal {O}(d\kappa ^3\epsilon ^{2})\) upperlevel objective evaluations, each requiring accuracy at most \(\Vert \widetilde{x}_i(\theta )\hat{x}_i(\theta )\Vert = \mathcal {O}(\epsilon ^2)\) for all i. Since our lowerlevel evaluations correspond to using GD/FISTA to solve a strongly convex problem, the computational cost of each upperlevel evaluation is \(\mathcal {O}(n\log (\epsilon ^{2}))\) provided we have reasonable initial iterates. From this, we conclude that the total computational cost before achieving \(\Vert \nabla f(\theta ^k)\Vert <\epsilon \) is at most \(\mathcal {O}(\epsilon ^{2}\log (\epsilon ^{1}))\) iterations of the lowerlevel algorithm. However, this is a conservative approach to estimating the cost: many of the iterations correspond to \(\Vert \nabla f(\theta ^k)\Vert \gg \epsilon \), and so the work required for these is less. This suggests the question: can we more carefully estimate the work required at different accuracy levels to prove a lower \(\epsilon \)dependence on the total work? We now argue that this is not possible without further information about asymptotic convergence rates (e.g., local convergence theory). For simplicity we drop all constants and \(\mathcal {O}(\cdot )\) notation in the below.
Suppose we count the work required to achieve progressively higher accuracy levels \(1\ge \epsilon _0>\epsilon _1>\cdots >\epsilon _N:=\epsilon \) for some desired accuracy \(\epsilon \ll 1\). Since each \(\epsilon _i<1\), we assume that we require \(\epsilon _i^{2}\) evaluations to achieve accuracy \(\epsilon _i\), where each evaluation requires \(\log (\epsilon _i^{1})\) computational work. We may choose \(\epsilon _0<1\), since the cost to achieve accuracy \(\epsilon _0\) is fixed (i.e., independent of our desired accuracy \(\epsilon \)), so does not affect our asymptotic bounds. Counting the total lowerlevel problem work—which we denote \(W(\epsilon )\)—in this way, we get
The second term of (80) corresponds to a right Riemann sum approximating \(\int _{\epsilon _0^{2}}^{\epsilon _N^{2}}\log (\sqrt{x})\mathrm{d}x\). Since \(x\rightarrow \log (\sqrt{x})=\log (x)/2\) is strictly increasing, the right Riemann sum overestimates the integral; hence,
independent of our choices of \(\epsilon _1,\ldots ,\epsilon _{N1}\). That is, as \(\epsilon \rightarrow 0\), we have \(W(\epsilon )\sim \epsilon ^{2}\log (\epsilon ^{1})\), so our naïve estimate is tight.
We further note that this naïve bound applies more generally. Suppose the work required for a single evaluation of the lowerlevel objective to accuracy \(\epsilon \) is \(w(\epsilon ^{2})\ge 0\) (e.g., \(w(x)=\log (x)/2\) above). Assuming w is increasing (i.e., higher accuracy evaluations require more work), we get, similar to the above,
Since w is increasing and nonnegative, by
the naïve work bound \(W(\epsilon )\sim \epsilon ^{2}w(\epsilon ^{2})\) holds provided \(w(x) = \mathcal {O}(w(x/2))\) as \(x\rightarrow \infty \); that is, w(x) does not increase too quickly. This holds in a variety of cases, such as w(x) bounded, concave or polynomial (but not if w(x) grows exponentially). In particular, this holds for \(w(x)\sim \log (x)/2\) as above, and \(w(x)\sim x^{1/2}\) and \(w(x)\sim x\), which correspond to the work required (via standard sublinear complexity bounds) if the lowerlevel problem is a strongly convex, convex or nonconvex optimization problem, respectively.
4 Numerical Results
4.1 UpperLevel Solver (DFOLS)
We implement the dynamic accuracy algorithm (Algorithm 1) in DFOLS [12], an opensource Python package which solves nonlinear leastsquares problems subject to bound constraints using modelbased DFO.^{Footnote 4} As described in [12], DFOLS has a number of modifications compared to the theoretical algorithm Algorithm 1. The most notable modifications here are that DFOLS:

Allows for bound constraints (and internally scales variables so that the feasible region is [0, 1] for all variables);

Does not implement a criticality phase;

Uses a simplified modelimproving step;

Maintains two trustregion radii to avoid decreasing \(\Delta ^k\) too quickly;

Implements a ‘safety phase,’ which treats iterations with short steps \(\Vert s^k\Vert \ll \Delta ^k\) similarly to unsuccessful iterations.
More discussion on DFOLS can be found in [12, 13].
Here, we use DFOLS v1.1.1, modified for the dynamic accuracy framework as described above. When determining the accuracy level for a given evaluation, we require accuracy level \(\delta _x=10(\Delta ^k)^2\) for all evaluations (c.f. Lemma 1), and also (32) when checking objective decrease (29).
4.2 Application: 1D Image Denoising
In this section, we consider the application of DFOLS to the problem of learning the regularization and smoothing parameters for the image denoising model (8) as described in Section 2.1.1. We use training data constructed using the method described in Sect. 2.2 with \(N=256\) and \(\sigma =0.1\).
1parameter case The simplest example we consider is the 1parameter case, where we only wish to learn \(\alpha \) in (8). We fix \(\nu = \xi =10^{3}\) and use a training set of \(n=10\) randomly generated images. We choose \(\alpha = 10^\theta \), optimize over \(\theta \) within bounds \(\theta \in [7,7]\) with starting value \(\theta ^0=0\). We do not regularize this problem, i.e., \(\mathcal {J}=0\).
3parameter case We also consider the more complex problem of learning three parameters for the denoising problem (namely \(\alpha \), \(\nu \) and \(\xi \)). We choose to penalize a large condition number of the lowerlevel problem, thus promoting efficient solution of the lowerlevel problem after training. To be precise we choose
where L and \(\mu \) are the smoothness and strong convexity constants given in Sect. 2.1.1.
The problem is solved using the parameterization \(\alpha = 10^{\theta _1}, \nu = 10^{\theta _2}\) and \(\xi = 10^{\theta _3}\). Here, we use a training set of \(n=20\) randomly generated images, and optimize over \(\theta \in [7,7] \times [7,0]^2\). Our default starting value is \(\theta ^0 = (0, 1, 1)\) and our default choice of upperlevel regularization parameter is \(\beta =10^{6}\).
Solver settings We run DFOLS with a budget of 20 and 100 evaluations of the upperlevel objective f for the 1 and 3parameter cases, respectively, and with \(\rho _\mathrm{end}=10^{6}\) in both cases. We compare the dynamic accuracy variant of DFOLS (given by Algorithm 1) against two variants of DFOLS (as originally implemented in [12]):

1.
Lowaccuracy evaluations: each value \(\hat{x}_i\) received by DFOLS is inaccurately estimated via a fixed number of iterations of GD/FISTA; we use 1,000 iterations of GD and 200 iterations of FISTA.

2.
Highaccuracy evaluations: each value \(z_i\) received by DFOLS is estimated using 10,000 iterations of GD or 2,000 iterations of FISTA.
We estimate \(\delta _f\) in the plots below by taking \(\delta _r\) to be the maximum estimate of \(\Vert \hat{x}_i(\theta )x_i\Vert \) for each \(i=1,\ldots ,n\). When running the lowerlevel solvers, our starting point is the final reconstruction from the previous upperlevel evaluation, which we hope is a good estimate of the solution.
1parameter denoising results In Fig. 4 we compare the six algorithm variants (low, high and dynamic accuracy versions of both GD and FISTA) on the 1parameter denoising problem. Firstly in Figs. 4a, b, we show the best upperlevel objective value observed against ‘computational cost,’ measured as the total GD/FISTA iterations performed (over all upperlevel evaluations). For each variant, we plot the value \(\widetilde{f}(\theta )\) and the uncertainty range \(\widetilde{f}(\theta )\pm \delta _f\) associated with that evaluation. In Fig. 4c we show the best \(\alpha _{\theta }\) found against the same measure of computational cost.
We see that both lowaccuracy variants do not converge to the optimal \(\theta \). Both highaccuracy variants converge to the same objective value and \(\theta \), but take much more computational effort to do this. Indeed, we did not know a priori how many GD/FISTA iterations would be required to achieve convergence. By contrast, both dynamic accuracy variants find the optimal \(\theta \) without any tuning.
Moreover, dynamic accuracy FISTA converges faster than highaccuracy FISTA, but the reverse is true for GD. In Fig. 4d we show the cumulative number of GD/FISTA iterations performed after each evaluation of the upperlevel objective. We see that the reason for dynamic accuracy GD converging slower than highaccuracy GD is that the initial upperlevel evaluations require many GD iterations; the same behavior is seen in dynamic accuracy FISTA, but to a lesser degree. This behavior is entirely determined by our (arbitrary) choices of \(\theta ^0\) and \(\Delta ^0\). We also note that the number of GD/FISTA iterations required by the dynamic accuracy variants after the initial phase is much lower than both the fixed accuracy variants. The difference between the GD and FISTA behavior in Fig. 4d is based on how the initial dynamic accuracy requirements compare to the chosen number of highaccuracy iterations (10,000 GD or 2,000 FISTA). Finally, in Fig. 5 we show the reconstructions achieved using the \(\alpha _{\theta }\) found by dynamic accuracy FISTA. All reconstructions are close to the ground truth, with a small loss of contrast.
To further understand the impact of the initial evaluations and the robustness of our framework, in Fig. 4 we run the same problem with different choices \(\theta ^0\in \{2,1,1\}\) (where \(\theta ^0=0\) before). In Fig. 6 we show best \(\alpha _{\theta }\) found for a given computational effort for these choices. When \(\theta ^0>0\), the lowerlevel problem starts more illconditioned, and so the first upperlevel evaluations for the dynamic accuracy variants require more GD/FISTA iterations. However, when \(\theta ^0<0\), we initially have a wellconditioned lowerlevel problem, and so the dynamic accuracy variants require many fewer GD/FISTA iterations initially, and they converge at the same or a faster rate than the highaccuracy variants.
These results also demonstrate that the dynamic accuracy variants give a final regularization parameter which is robust to the choice of \(\theta ^0\). In Fig. 7 we plot the final learned \(\alpha _{\theta }\) value compared to the initial choice of \(\alpha _{\theta }\) for all variants. The lowaccuracy variants do not reach a consistent minimizer for different starting values, but the dynamic and highaccuracy variants both reach the same minimizer for all starting points. Thus although our upperlevel problem is nonconvex, we see that our dynamic accuracy approach can produce solutions which are robust to the choice of starting point.
3parameter denoising results Next, we consider the 3parameter (\(\alpha _{\theta }\), \(\nu _{\theta }\) and \(\xi _{\theta }\)) denoising problem.
As shown in Fig. 8, both dynamic accuracy variants (GD and FISTA) achieve the best objective value at least one order of magnitude faster than the corresponding low and highaccuracy variants. We note that (for instance) 200 FISTA iterations was insufficient to achieve convergence in the 1parameter case, but converges here. By contrast, aside from the substantial speedup in the 3parameter case, our approach converges in both cases without needing to select the computational effort in advance.
The final reconstructions achieved by the optimal parameters for dynamic accuracy FISTA are shown in Fig. 9. We note that all variants produced very similar reconstructions (since they converged to similar parameter values), and that all training images are recovered with high accuracy.
Next, we consider the effect of the upperlevel regularization parameter \(\beta \). If the smaller \(\beta \) value of \(10^{8}\) is chosen, all variants converge to slightly smaller values of \(\nu _{\theta }\) and \(\xi _{\theta }\) as the original \(\beta =10^{6}\), but produce reconstructions of a similar quality. However, increasing the value of \(\beta \) yields parameters which give noticeably worse reconstructions. The reconstructions for \(\beta =10^{4}\) are shown in Fig. 10.
We conclude by demonstrating in Fig. 11 that, aside from reducing our upperlevel objective , the parameters found by DFOLS do in fact progressively improve the quality of the reconstructions. The figure shows the reconstructions of one training image achieved by the best parameters found (by the dynamic accuracy FISTA variant) after a given number of upperlevel objective evaluations. We see a clear improvement in the quality of the reconstruction as the upperlevel optimization progresses.
4.3 Application: 2D Denoising
Next, we demonstrate the performance of dynamic accuracy DFOLS on the same 3parameter denoising problem from Sect. 4.2, but applied to 2D images. Our training data are the 25 images from the Kodak dataset.^{Footnote 5} We select the central \(256\times 256\)pixel region of each image, convert to monochrome and add Gaussian noise \(N(0,\sigma ^2)\) with \(\sigma =0.1\) to each pixel independently. We run DFOLS for 200 upperlevel evaluations with \(\rho _\mathrm{end}=10^{6}\). Unlike Sect. 4.2, we find that there is no need to regularize the upperlevel problem with the condition number of the lowerlevel problem (i.e., \(\mathcal {J}(\theta )=0\) for these results).
The resulting objective decrease, final parameter values and cumulative lowerlevel iterations are shown in Fig. 12. All variants achieve the same (upperlevel) objective value and parameter \(\alpha _{\theta }\), but the dynamic accuracy variants achieve this with substantially fewer GD/FISTA iterations compared to the low and highaccuracy variants. Interestingly, despite all variants achieving the same upperlevel objective value, they do not reach a consistent choice for \(\nu _{\theta }\) and \(\xi _{\theta }\).
In Fig. 13 we show the reconstructions achieved by the dynamic accuracy FISTA variant for three of the training images. We see highquality reconstructions in each case, where the piecewiseconstant reconstructions favored by TV regularization are evident.
Lastly, we study the impact of changing the noise level \(\sigma \) on the calibrated total variational regularization parameter \(\alpha _{\theta }\). We run DFOLS with dynamic accuracy FISTA for 200 upperlevel evaluations on the same training data, but corrupted with noise level \(\sigma \) ranging from \(10^{1}\) (as above) to \(10^{8}\), see Fig. 14. We see that as \(\sigma \rightarrow 0\), so does \(\alpha _{\theta }\) and \(\sigma ^2/\alpha _\theta \). Note that this is a common assumption on the parameter choice rule in regularization theory to yield a convergent regularization method [29, 44]. It is remarkable that the learned optimal parameter also has this property.
4.4 Application: Learning MRI Sampling Patterns
Lastly, we turn our attention to the problem of learning MRI sampling patterns. In this case, our lowerlevel problem is (6) with \(A(\theta )=F\), where F is the Fourier transform, and \(S(\theta )\) is a nonnegative diagonal sampling matrix. Following [16], we aim to find sampling parameters \(\theta \in [0,1]^d\) corresponding to the weight associated to each Fourier mode, our sampling matrix is defined as
The resulting lowerlevel problem is \(\mu \)strongly convex and Lsmooth as per (7) with \(\Vert A_\theta ^* S_\theta A_\theta \Vert = \Vert S(\theta )\Vert = \max _i \theta _i/(1\theta _i)\) and \(\lambda _{{\text {min}}}(A_\theta ^* S_\theta A_\theta ) = \min _i \theta _i/(1\theta _i)\).
For our testing, we fix the regularization and smoothness parameters \(\alpha =0.01\), \(\nu =0.01\) and \(\xi =10^{4}\) in (6). We use \(n=10\) training images constructed using the method described in Sect. 2.2 with \(N=64\) and \(\sigma =0.05\). Lastly, we add a penalty to our upperlevel objective to encourage sparse sampling patterns: \(\mathcal {J}(\theta ):=\beta \Vert \theta \Vert _1\), where we take \(\beta =0.1\). To fit the leastsquares structure (22), we rewrite this term as \(\mathcal {J}(\theta )=(\sqrt{\beta \Vert \theta \Vert _1})^2\). To ensure that \(S(\theta )\) remains finite and \(\mathcal {J}(\theta )\) remains Lsmooth, we restrict \(0.001 \le \theta _i \le 0.99\).
We run DFOLS with a budget of 3000 evaluations of the upperlevel objective and \(\rho _\mathrm{end}=10^{6}\). As in Sect. 4.2, we compare dynamic accuracy DFOLS against (fixed accuracy) DFOLS with low and highaccuracy evaluations given by a 1,000 and 10,000 iterations of GD or 200 and 1,000 iterations of FISTA.
With our \(\ell _1\) penalty on \(\theta \), we expect DFOLS to find a solution where many entries of \(\theta \) are at their lower bound \(\theta _i=0.001\). Our final sampling pattern is chosen by using the corresponding \(\theta _i\) if \(\theta _i>0.001\); otherwise, we set that Fourier mode weight to zero.
In Fig. 15 we show the objective decrease achieved by each variant and the cumulative lowerlevel work required by each variant. All variants except lowaccuracy GD achieve the best objective value with low uncertainty. However, as above, the dynamic accuracy variants achieve this value significantly earlier than the fixed accuracy variants, largely as a result of needing much fewer GD/FISTA iterations in the (lower accuracy) early upperlevel evaluations. In particular dynamic accuracy GD reaches the minimum objective value about 100 times faster than highaccuracy GD. We note that FISTA with 200 iterations ends up requiring fewer lowerlevel iterations after a large number of upperlevel evaluations, but the dynamic accuracy variant achieves is minimum objective value sooner.
We show the final pattern of sampled Fourier coefficients (after thresholding) in Fig. 16. Of the five variants which found the best objective value, all reached a similar set of ‘active’ coefficients \(\theta _i>0.001\) with broadly similar values for \(\theta _i\) at all frequencies. For demonstration purposes we plot the reconstructions corresponding to the coefficients from the ‘dynamic FISTA’ variant in Fig. 17 (the reconstructions of the other variants were all similar). All the training images are reconstructed to high accuracy, with only a small loss of contrast near the jumps.
5 Conclusion
We introduce a dynamic accuracy modelbased DFO algorithm for solving bilevel learning problems. This approach allows us to learn potentially large numbers of parameters, and allowing inexact upperlevel objective evaluations with which we dramatically reduce the lowerlevel computational effort required, particularly in the early phases of the algorithm. Compared to fixed accuracy DFO methods, we often achieve better upperlevel objective values and lowaccuracy methods, and similar objective values as highaccuracy methods but with much less work: in some cases up to 100 times faster. These observations can be made for both lowerlevel solvers GD and FISTA, with different fixed accuracy requirements, for ROF denoising and learning MRI sampling patterns. Thus, the proposed approach is robust in practice, computationally efficient and backed by convergence and worstcase complexity guarantees. Although the upperlevel problem is nonconvex, our numerics do not suggest that convergence to nonglobal minima is a point for concern here.
Future work in this area includes relaxing the smoothness and/or strong convexity assumptions on the lowerlevel problem (making the upperlevel problem less theoretically tractable). Our theoretical analysis would benefit from a full proof that our worstcase complexity bound on the lowerlevel computational work is tight. Another approach for tackling bilevel learning problems would be to consider gradientbased methods which allow inexact gradient information. Lastly, bilevel learning appears to compute a regularization parameter choice strategy which yields a convergent regularization method. Further investigation is required to back these numerical results by sound mathematical theory.
Notes
If f is Lsmooth, then the Taylor series \(m^k (s)=f(\theta ^k)+\nabla f(\theta ^k)^T s\) is fully linear with \(\kappa _\mathrm{ef}=L/2\) and \(\kappa _\mathrm{eg}=L\) for all \(\Delta ^k\).
If we have to evaluate \(\widetilde{r}(\theta )\) at different accuracy levels as part of the accuracy phase, we count this as one evaluation, since we continue solving the corresponding lowerlevel problem from the solution from the previous, lower accuracy evaluation.
Of course, there may be many more initial steps of the criticality phase in which \(m^k\) is already fully linear, but no work is required in this case.
Available at https://github.com/numericalalgorithmsgroup/dfols.
Available from http://www.cs.albany.edu/~xypan/research/snr/Kodak.html.
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MJE acknowledges support from the EPSRC (EP/S026045/1, EP/T026693/1), the Faraday Institution (EP/T007745/1) and the Leverhulme Trust (ECF2019478).
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Ehrhardt, M.J., Roberts, L. Inexact DerivativeFree Optimization for Bilevel Learning. J Math Imaging Vis 63, 580–600 (2021). https://doi.org/10.1007/s10851021010208
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DOI: https://doi.org/10.1007/s10851021010208