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An Extended Frank-Wolfe Method with "In-Face" Directions, and its Application to Low-Rank Matrix Completion

Robert M. Freund, Paul Grigas, Rahul Mazumder

arXiv:1511.02204v1math.OCstat.COstat.ML

TL;DR

Matrix completion seeks to estimate unobserved entries from partial, possibly noisy data, often assuming the underlying matrix is low-rank. This paper extends Frank-Wolfe with in-face directions and rules for choosing between those and regular steps. It reports faster computation of nearly optimal, very low-rank solutions than Frank-Wolfe and its away-step variant.

  • Problem

    Matrix completion seeks to estimate unobserved entries from partial, possibly noisy data, and is ill-posed without structural assumptions such as low rank.

    Source: §1 · p. 2

  • Method

    The extended Frank-Wolfe method uses descent directions within the current minimal face and selects between these in-face steps and regular Frank-Wolfe steps.

    Source: §1 · p. 4, §1.1 · p. 4

  • Results

    The method reports significant speed-ups over Frank-Wolfe and its traditional away-step variant when computing nearly optimal, very low-rank solutions on artificial and real datasets.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The method seeks to balance near-optimality with low-rank structure by promoting iterates on low-dimensional faces.

    Source: §1 · p. 4

    Favoring low-dimensional faces by setting γ1 and γ2 to low values comes at a price in computational guarantees.

    Source: §2.2 · p. 12

Abstract

from arXiv · show

Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. This is accomplished by a new approach to generating ``in-face" directions at each iteration, as well as through new choice rules for selecting between in-face and ``regular" Frank-Wolfe steps. Our framework for generating in-face directions generalizes the notion of away-steps introduced by Wolfe. In particular, the in-face directions always keep the next iterate within the minimal face containing the current iterate. We present computational guarantees for the new method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply the new method to the matrix completion problem, where low-dimensional faces correspond to low-rank matrices. We present computational results that demonstrate the effectiveness of our methodological approach at producing nearly-optimal solutions of very low rank. On both artificial and real datasets, we demonstrate significant speed-ups in computing very low-rank nearly-optimal solutions as compared to either the Frank-Wolfe method or its traditional away-step variant.

1 Introduction

Matrix completion must infer unobserved entries from partial, possibly noisy data, commonly under a low-rank assumption. The paper extends Frank-Wolfe with in-face directions to seek low-rank, near-optimal iterates, and reports computational advantages over existing methods.

1 Introduction

  • Because recovering unobserved entries is ill-posed without structural assumptions, low rank is a common assumption for estimating the underlying matrix.The rank-constrained formulation is combinatorially hard, motivating the nuclear norm as a proxy for rank.Source: §1 · p. 2
  • Frank-Wolfe updates use rank-one solutions to their linear subproblems, yielding a rank bound that trades off data fidelity with low-rank structure.This favorable structure follows from the nuclear norm ball and the additive form of the updates.Source: §1 · pp. 3–4
  • In-face directions keep iterates within their current minimal face, while regular Frank-Wolfe steps remain available when in-face directions do not improve the objective enough.The method targets low-dimensional faces, which correspond to low-rank matrices in the nuclear norm ball, and its guarantees maintain O(c/k) convergence while promoting low-rank iterates.Source: §1 · p. 4, §1.1 · p. 4

1.1 Organization/Results

  • Across simulated instances and MovieLens10M, In-Face Extended Frank-Wolfe versions show computational advantages, delivering low-rank solutions at low run times to compute a target optimality gap; one version beats existing methods by a factor of 10 or more.Source: §1.1 · p. 5

1.2 Notation

  • The notation section defines the nuclear norm as the sum of a matrix’s nonzero singular values and the nuclear norm ball as a radius-δ neighborhood under that norm.Source: §1.2 · p. 5

2 Frank-Wolfe Method, Away Steps, and In-Face Steps

The section reviews Frank–Wolfe and away-step methods, then introduces an extension that balances objective progress with iterates on low-dimensional faces. Its in-face steps operate within the current minimal face and, when fully optimized over that face, alternate with regular Frank–Wolfe steps.

2 Frank-Wolfe Method, Away Steps, and In-Face Steps

  • Standard Frank–Wolfe iterations typically incur their main computational burden in the linear optimization subproblem; with exact line search or the stated quadratic-approximation rule, the objective decreases monotonically and the method has a computational guarantee.The accumulated bound Bk is a lower bound on the optimum and can also support step-size strategies and termination criteria.Source: §2 · pp. 6–7

2.1 Away Steps

  • Away steps choose a direction within the current minimal face, but require the ability to characterize that face; the convergence analysis does not require the away point to be an extreme point or a linear-optimization solution.The authors use this observation to motivate a different approach to generating and analyzing away steps.Source: §2.1 · pp. 7–8
  • Away-step methods are not affine-invariant: an atomic representation can yield smaller minimal faces than the original feasible set, changing the away-step subproblem and potentially its computational cost.The authors illustrate this mismatch with a polytope and its atomic simplex representation, and state that using the original set can save computation in matrix completion.Source: §2.1 · p. 9, Figure 1 · p. 9

2.2 An “In-Face” Extended Frank-Wolfe Method

  • The in-face extension targets both objective progress and low-dimensional minimal faces, which correspond to low-rank iterates in matrix completion and can reduce computation.Its directions keep the next candidate within the current minimal face; the method selects a boundary step, another step in that face, or a regular Frank–Wolfe step according to decrease criteria.Source: §2.2 · pp. 9–10
  • The method can use away-step or full-optimization directions within the current face, and unlike the away-step method it solves the large subproblem over the full feasible set only when needed.A regular Frank–Wolfe step within the face is another option, but the authors identify away-step and full-optimization strategies as their main interests.Source: §2.2 · p. 11
  • Lower γ1 and γ2 make face-selection criteria easier to satisfy, favoring lower-dimensional faces or staying within the current face, but this comes at a cost in computational guarantees.Setting γ1 = 0 accepts a lower-dimensional-face candidate when its objective value is no worse than the current iterate’s.Source: §2.2 · pp. 11–12
  • Choosing smaller values of γ1 and γ2 can have a detrimental effect on algorithm progress in terms of the objective-function optimality gap, while larger values ensure better convergence guarantees.Source: §2.2 · p. 12
  • After a full optimization over the current minimal face, the next iteration must take a regular Frank-Wolfe step because no valid in-face descent direction remains.In this case γ1 and γ2 are unnecessary, and alternating full face optimizations with regular steps is distinct from the fully corrective Frank-Wolfe variant, except on the lifted problem where they coincide.Source: §2.2 · p. 13
  • The alternating full-face-optimization method guarantees f(xk)-f* ≤ f(xk)-Bk ≤ 1/(1/(f(x0)-B0)+k/(2L̄D̄^2)) < 2L̄D̄^2/k.Each iteration consists of a full optimization over the current face and a regular Frank-Wolfe step, with step size chosen by exact line search or the quadratic approximation rule.Source: §2.2 · p. 13

3 Solving Matrix Completion Problems using the In-Face Extended Frank-Wolfe Method

For nuclear-norm-regularized matrix completion, the method represents iterates through thin SVDs and uses the nuclear-norm ball’s face structure to choose in-face or regular Frank-Wolfe directions. Its implementation and rank accounting make explicit how these choices affect computation and iterate rank.

3 Solving Matrix Completion Problems using the In-Face Extended Frank-Wolfe Method

  • The matrix-completion objective has gradient (Z − X) restricted to observed entries, and the nuclear-norm ball has diameter 2δ.The section sets the Lipschitz constant to 1 under the nuclear norm.Source: §3 · pp. 13–14

3.1 Characterization of faces of the nuclear norm ball

  • On the nuclear-norm ball’s boundary, a rank-r iterate’s minimal face is represented by U M V^T, with M positive semidefinite and trace δ.For an interior point, the minimal face is the entire ball.Source: §3.1 · p. 14

3.3 Strategies and computation of the in-face direction Dk

  • An in-face step stays within the current minimal face, while a regular Frank-Wolfe step uses a rank-one solution of the linear subproblem.The regular-step solution comes from the leading singular vectors of the current gradient.Source: §3.2 · p. 14, §3.3 · p. 15
  • The away-step strategy chooses an in-face direction from a linear optimization problem; direct face optimization instead targets an exact or approximate objective minimizer on that face.Direct face optimization is considered sensible when the current iterate has low rank.Source: §3.3 · pp. 15–16

3.4 Computing the maximal step-size αstop

  • The maximal in-face step size is found by preserving membership in the current face; for interior iterates, the method approximately finds the feasible step size by binary search.For a boundary iterate, the step-size condition is reformulated using its thin SVD and the in-face direction.Source: §3.4 · p. 16

3.5 Initial values, step-sizes, and computational guarantees

  • The rank-one initialization has an optimality gap at most 2δ² above the optimum, providing a starting bound for the method’s computational guarantee.The initialization is a full Frank-Wolfe step from zero.Source: §3.5 · p. 17

3.6 Efficiently Updating the Thin SVD of Zk

  • Storing an iterate’s thin SVD requires mr + r + nr entries, and regular Frank-Wolfe updates can be computed through a scaling plus a rank-one update.For an in-face update, the method eigendecomposes the r × r matrix D + Δ to update the thin SVD.Source: §3.6 · p. 17

3.7 Rank accounting

  • For the four exhaustive step cases, rank decreases by at least one on a lower-dimensional face, stays the same within the face’s relative interior, or increases by at most one on a regular step.The proof starts from a rank-one iterate.Source: §3.5 · p. 17, §3.7 · p. 18

4 Computational Experiments and Results

Across simulated examples and the MovieLens10M dataset, the in-face method’s variants offer favorable combinations of low rank and runtime relative to the benchmark methods. IF-Rank-Strategy is fastest overall, while IF-(0,∞) is especially promising for larger instances.

Figure 2 behavior

  • All methods eventually show slow convergence consistent with the O(1/k) theoretical bound, despite rapid early optimality-gap progress by IF-Optimization and IF-Rank-Strategy.The Figure 2 plots show relative optimality gap against runtime.Source: §4 · p. 20

Small-scale examples

  • Across three small-scale example types averaged over 25 instances, IF-Rank-Strategy is fastest, followed by IF-Optimization and IF-(0,∞), while the latter two maintain relatively low maximum rank.FW-Away-Atomic is described as dramatically ineffective at delivering low-rank solutions.Source: §4 · p. 20

Medium-large examples

  • Across eight medium-large examples, IF-Rank-Strategy, IF-(0,∞), and IF-Optimization offer the best balance of final rank, maximum rank, and runtime; IF-(0,∞) tends toward lower ranks but higher runtimes.IF-Rank-Strategy does not consistently reach low rank on these instances because its refinement phase starts later.Source: §4 · pp. 20–21

Figure 2 behavior

  • In the Figure 2 instance, four methods quickly reach and remain near rank 37, while Frank-Wolfe and FW-Away-Atomic build rank to levels that can be an order of magnitude above the apparent optimum.IF-Rank-Strategy and IF-Optimization reduce rank rapidly after their initial rank growth; Frank-Wolfe and FW-Away-Atomic do so slowly.Source: §4 · p. 20, Figure 2 · p. 21

Large-scale real-data results

  • On the MovieLens10M dataset, IF-(0,∞) reinforces the medium-large results with advantages in final-iterate rank and runtime to the target optimality gap.The comparison tested IF-(0,∞) against Frank-Wolfe.Source: §4 · p. 21

Summary Conclusions

  • IF-Rank-Strategy achieves the best run times, beating existing methods by a factor of 10 or more; IF-(0,∞) gives very low ranks with reasonable run times on larger instances.IF-Optimization generally delivers low rank and low runtime together in the large-scale regime.Source: §4 · p. 21

A Appendix - Remainder of the Proof of Theorem 1

The proof completes Theorem 1 by bounding the next objective gap in two cases, according to the relative sizes of the gap-related quantity G_k and curvature term C_k. In both cases, the gap decreases and the stated reciprocal-gap inequality follows.

A Appendix - Remainder of the Proof of Theorem 1

  • When G_k ≤ C_k, the selected step size is G_k/C_k, yielding nonincreasing objective and gap values and the reciprocal-gap bound.The proof uses r_k as the current objective gap and C_k as a curvature term.Source: A Appendix - Remainder of the Proof of… · pp. 24–25
  • When G_k > C_k, the step is full, and the objective and gap decrease strictly; the proof then establishes the reciprocal-gap inequality for this case.The argument uses the bound G_k > C_k in the descent estimate and considers k = 0 separately from k ≥ 1.Source: A Appendix - Remainder of the Proof of… · p. 25
  • Together, the two cases complete the proof of inequality (28), with the k ≥ 1 case using the bound on the next gap derived from (31).For k = 0, the proof uses a separate inequality chain before applying (30).Source: A Appendix - Remainder of the Proof of… · p. 25

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