Abstract
This paper presents two inexact composite gradient methods, one inner accelerated and another doubly accelerated, for solving a class of nonconvex spectral composite optimization problems. More specifically, the objective function for these problems is of the form \(f_{1}+f_{2}+h\), where \(f_{1}\) and \(f_{2}\) are differentiable nonconvex matrix functions with Lipschitz continuous gradients, \(h\) is a proper closed convex matrix function, and both \(f_{2}\) and \(h\) can be expressed as functions that operate on the singular values of their inputs. The methods essentially use an accelerated composite gradient method to solve a sequence of proximal subproblems involving the linear approximation of \(f_{1}\) and the singular value functions underlying \(f_{2}\) and \(h\). Unlike other composite gradient-based methods, the proposed methods take advantage of both the composite and spectral structure underlying the objective function in order to efficiently generate their solutions. Numerical experiments are presented to demonstrate the practicality of these methods on a set of real-world and randomly generated spectral optimization problems.
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The authors would like to thank the two anonymous referees and the associate editor for their insightful comments on earlier drafts of this paper.
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The works of these authors were partially supported by ONR Grant N00014-18-1-2077, AFOSR Grant FA9550-22-1-0088, NSERC Grant PGSD3-516700-2018, and the IDEaS-TRIAD Fellowship (NSF Grant CCF-1740776). The first author has been supported by the US Department of Energy (DOE) and UT-Battelle, LLC, under contract DE-AC05-00OR22725 and also supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration.
Appendices
A Technical bounds
The result below presents a basic property of the composite gradient step.
Proposition 19
Let \(h\in \overline{\mathrm{Conv}}\ (\mathcal{Z})\), \(z\in \mathrm{dom}\,h\), and g be a differentiable function on \(\mathrm{dom}\,h\) which satisfies \(g(u)-\ell _{g}(u;z)\le L\Vert u-z\Vert ^{2}/2\) for some \(L\ge 0\) and every \(u\in \mathrm{dom}\,g\). Moreover, define
Then, it holds that
Proof
Using the definition of \({\hat{z}}\), the fact that \(\ell _{g}(\cdot ;z)+h(\cdot )+L\Vert \cdot -z\Vert ^{2}/2\) is L-strongly convex, and the assumed bound \(g(u)-\ell _{g}(u;z)\le L\Vert u-z\Vert ^{2}/2\) at \(u={\hat{z}}\), we have
\(\square\)
B R-ACG algorithm
This section presents technical results related to the R-ACG algorithm.
The first set of results describes some basic properties of the generated iterates.
Proposition 20
If \(\psi _{s}\) is \(\mu\)–strongly convex, then the following statements hold:
-
(a)
\(z_{j}^{c}=\mathrm{argmin}_{u\in \mathcal{Z}}\left\{ B_{j}\Gamma _{j}(u)+\Vert u-z_{0}^{c}\Vert ^{2}/2\right\}\);
-
(b)
\(\Gamma _{j}\le \psi\) and \(B_{j}\psi (z_{j})\le \inf _{u\in \mathcal{Z}}\left\{ B_{j}\Gamma _{j}(u)+\Vert u-z_{0}^{c}\Vert ^{2}/2\right\}\);
-
(c)
\(\eta _{j}\ge 0\) and \(r_{j}\in {\partial }_{\eta _{j}}\left( \psi -\mu \Vert \cdot -z_{j}\Vert ^{2}/2\right) (z_{j})\);
-
(d)
it holds that
$$\begin{aligned} \left( \frac{1}{1+\mu B_{j}}\right) \Vert B_{j}r_{j}+z_{j}-z_{0}\Vert ^{2}+2B_{j}\eta _{j}\le \Vert z_{j}-z_{0}\Vert ^{2} \end{aligned}$$
Proof
(a) See [14, Proposition 1].
(b) See [14, Proposition 1(b)].
(c) The optimality of \(z_{j}^{c}\) in part (a), the \(\mu\)-strong convexity of \(\Gamma _{j}\), and the definition of \(r_{j}\) imply that
Using the above inclusion, the definition of \(\eta _{j}\), the fact that \(\Gamma _{j}-\mu \Vert \cdot \Vert ^{2}/2\) is affine, and part (b), we now conclude that
for every \(z\in \mathrm{dom}\,\psi _{n}\), which is exactly the desired inclusion. The fact that \(\eta _{j}\ge 0\) follows from the above inequality with \(z=z_{j}\).
(d) It follows from parts (a)–(b) and the definition of \(\eta _{j}\) that
Multiplying both sides of the above inequality by \(2B_{j}\) yields the desired conclusion. \(\square\)
The next result presents the general iteration complexity of the algorithm, i.e. Proposition 2(a).
Proof of Proposition 2(a)
Let \(\ell\) be the first iteration where
and suppose that the R-ACG has not stopped with failure before iteration \(\ell\). We show that it must stop with success at the end of the \(\ell ^{\mathrm{th}}\) iteration. Combining the triangle inequality, the successful check in step 3 of the method, (64), and the relation \((a+b)^{2}\le 2a^{2}+2b^{2}\) for all \(a,b\in {\mathbb {R}},\) we first have that
and hence the method must terminate at the \(\ell ^{\mathrm{th}}\) iteration. We now bound \(\ell\) based on the requirement in (64). Solving for the quadratic in \(B_{\ell }\) in the first bound of (64), it is easy to see that \(B_{\ell }\ge 4\mu K_{\theta }^{2}+2K_{\theta }\) implies (64). On the other hand, for the second condition in (64), it is immediate that \(B_{\ell }\ge 2K_{\theta }^{2}\) implies (64). In view of (18) and the previous two bounds, it follows that
implies (64). Using the bound \(\log (1+t)\ge t/(1+t)\) for \(t\ge 0\) and the above bound on \(\ell\), it is straightforward to see that \(\ell\) is on the same order of magnitude as in (19). \(\square\)
C Refined ICG points
This appendix presents technical results related to the refined points of the ICG methods.
The result below proves Lemma 3 from the main body of the paper.
Proof of Lemma 3
(a) Using Proposition 1(a), the definition of \({\hat{v}}\), and the definitions of \(\psi _{s}\) and \(\psi _{n}\) in (28), we have that
(b) Using assumption (A3), Proposition 1(b), the choice of M in (28), and the fact that \(\Delta _{\mu }(y_{r};y_{},v)\le \varepsilon\), we first observe that
Using now (65), the choice of M in (28), Proposition 1(c) with \(L(\cdot ,\cdot )=\lambda L_{2}(\cdot ,\cdot )\), the fact that \(\sigma \le 1\), and the definition of \(C_{\lambda }(\cdot ,\cdot )\), we conclude that
\(\square\)
D Spectral functions
This section presents some results about spectral functions as well as the proof of Propositions 6. It is assumed that the reader is familiar with the key quantities given in Sect. 4.1 (e.g., see (40) and (41)).
We first state two well-known results [2, 11] about spectral functions.
Lemma 21
Let \(\Psi =\Psi ^{\mathcal{V}}\circ \sigma\) for some absolutely symmetric function \(\Psi ^{\mathcal{V}}:{\mathbb {R}}^{r}\mapsto {\mathbb {R}}\). Then, the following properties hold:
-
(a)
\(\Psi ^{*}=(\Psi ^{\mathcal{V}}\circ \sigma )^{*}=(\Psi ^{\mathcal{V}})^{*}\circ \sigma\);
-
(b)
\(\nabla \Psi =(\nabla \Psi ^{\mathcal{V}})\circ \sigma\);
Lemma 22
Let \((\Psi ,\Psi ^{\mathcal{V}})\) be as in Lemma 21, the pair \((S,{Z}_{})\in \mathcal{Z}\times \mathrm{dom}\,\Psi\) be fixed, and the decomposition \(S=P[\mathrm{dg}\,\sigma (S)]Q^{*}\) be an SVD of S, for some \((P,Q)\in \mathcal{U}^{m}\times \mathcal{U}^{n}\). If \(\Psi \in \overline{\mathrm{Conv}}\ {\mathbb {R}}^{m\times n}\) and \(\Psi ^{\mathcal{V}}\in \overline{\mathrm{Conv}}\ {\mathbb {R}}^{r}\), then for every \(M>0\), we have
We now present a new result about spectral functions.
Theorem 23
Let \((\Psi ,\Psi ^{\mathcal{V}})\) be as in Lemma 21 and the point \({Z}_{}\in {\mathbb {R}}^{m\times n}\) be such that \(\sigma ({Z}_{})\in \mathrm{dom}\,\Psi ^{\mathcal{V}}\). Then for every \(\varepsilon \ge 0\), we have \(S\in {\partial }_{\varepsilon }\Psi ({Z}_{})\) if and only if \(\sigma (S)\in {\partial }_{\varepsilon (S)}\Psi ^{\mathcal{V}}(\sigma ({Z}_{}))\), where
Moreover, if S and Z have a simultaneous SVD, then \(\varepsilon (S)=\varepsilon\).
Proof
Using Lemma 21(a), (66), and the well-known fact that \(S\in {\partial }_{\varepsilon }\Psi ({Z}_{})\) if and only if \(\varepsilon \ge \Psi ({Z}_{})+\Psi ^{*}(S)-\left\langle {Z}_{},S\right\rangle\), we have that \(S\in {\partial }_{\varepsilon }\Psi ({Z}_{})\) if and only if
or, equivalently, \(\sigma (S)\in {\partial }_{\varepsilon (S)}\Psi ^{\mathcal{V}}(\sigma ({Z}_{}))\) and \(\varepsilon (S)\ge 0\). To show that the existence of a simultaneous SVD of S and Z implies \(\varepsilon (S)=\varepsilon\) it suffices to show that \(\langle \sigma (S),\sigma ({Z}_{})\rangle =\langle S,{Z}_{}\rangle\). Indeed, if \(S=P[\mathrm{dg}\,\sigma (S)]Q^{*}\) and \({Z}_{}=P[\mathrm{dg}\,\sigma ({Z}_{})]Q^{*}\), for some \((P,Q)\in \mathcal{U}^{m}\times \mathcal{U}^{n}\), then we have
\(\square\)
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Kong, W., Monteiro, R.D.C. Accelerated inexact composite gradient methods for nonconvex spectral optimization problems. Comput Optim Appl 82, 673–715 (2022). https://doi.org/10.1007/s10589-022-00377-9
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DOI: https://doi.org/10.1007/s10589-022-00377-9