Frank–Wolfe and friends: a journey into projection-free first-order optimization methods

Bomze, Immanuel M.; Rinaldi, Francesco; Zeffiro, Damiano

doi:10.1007/s10288-021-00493-y

Cited by 17 publications

(3 citation statements)

References 63 publications

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“…FW is demonstrated as an effective method for optimization over simplex [23] or spactrahedron domains [41]. We refer to [48] for convergence analysis of FW and a detailed discussion on its applications, and to [13] for a review on recent advances in FW.…”

Section: Related Workmentioning

confidence: 99%

Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization

Yurtsever¹,

Birdal²,

Golyanik³

2022

Preprint

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We present a hybrid classical-quantum framework based on the Frank-Wolfe algorithm, Q-FW, for solving quadratic, linearly-constrained, binary optimization problems on quantum annealers (QA). The computational premise of quantum computers has cultivated the re-design of various existing vision problems into quantum-friendly forms. Experimental QA realisations can solve a particular non-convex problem known as the quadratic unconstrained binary optimization (QUBO). Yet a naive-QUBO cannot take into account the restrictions on the parameters. To introduce additional structure in the parameter space, researchers have crafted ad-hoc solutions incorporating (linear) constraints in the form of regularizers. However, this comes at the expense of a hyper-parameter, balancing the impact of regularization. To date, a true constrained solver of quadratic binary optimization (QBO) problems has lacked. Q-FW first reformulates constrained-QBO as a copositive program (CP), then employs Frank-Wolfe iterations to solve CP while satisfying linear (in)equality constraints. This procedure unrolls the original constrained-QBO into a set of unconstrained QUBOs all of which are solved, in a sequel, on a QA. We use D-Wave Advantage QA to conduct synthetic and real experiments on two important computer vision problems, graph matching and permutation synchronization, which demonstrate that our approach is effective in alleviating the need for an explicit regularization coefficient.

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Section: Related Workmentioning

confidence: 99%

Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization

Yurtsever¹,

Birdal²,

Golyanik³

2022

Preprint

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“…Such a strategy might however be costly even when the projection is performed over some structured sets like, e.g., the flow polytope, the nuclear-norm ball, the Birkhoff polytope, the permutahedron (see, e.g., [18]). This is the reason why, in recent years, projection-free methods (see, e.g., [13,21,25]) have been massively used when dealing with those structured constraints.…”

Section: Introductionmentioning

confidence: 99%

Projection free methods on product domains

Bomze¹,

Rinaldi²,

Zeffiro³

2023

Preprint

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Projection-free block-coordinate methods avoid high computational cost per iteration and at the same time exploit the particular problem structure of product domains. Frank-Wolfelike approaches rank among the most popular ones of this type. However, as observed in the literature, there was a gap between the classical Frank-Wolfe theory and the block-coordinate case. Moreover, most of previous research concentrated on convex objectives. This study now deals also with the non-convex case and reduces above-mentioned theory gap, in combining a new, fully developed convergence theory with novel active set identification results which ensure that inherent sparsity of solutions can be exploited in an efficient way. Preliminary numerical experiments seem to justify our approach and also show promising results for obtaining global solutions in the non-convex case.

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“…In this work, we tackle the general membership problem for the local polytope via methods from the field of constrained convex optimisation, where this is known as the approximate Carathéodory problem [16,17]. Specifically, we rephrase the distance algorithm previously credited to Gilbert as the original Frank-Wolfe algorithm [18,19] (see [20,21] for recent reviews) to leverage the improvements brought to this algorithm over the last decade. Combined with refinements of the proof in [14], this allows us to improve on the bounds for the nonlocality threshold of the two-qubit Werner states, thereby reducing the corresponding range by about 40%.…”

mentioning

confidence: 99%