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FISTA energy recurrence with certified proximal errors

Statement

Let $x_{k}, y_{k}, t_{k}$ be the certified inexact FISTA iterates of Certified inexact proximal-gradient and FISTA updates, with errors $η_{k} \geq 0$ , and let $x_{*}$ minimize $F = f + g$ . Set $t_{0} = 0$ , $u_{0} = x_{0} - x_{*}$ , and $u_{k} = t_{k} x_{k} - (t_{k} - 1) x_{k - 1} - x_{*}$ . Then $\frac{2 t_{k}^{2}}{L} (F (x_{k}) - F (x_{*})) + ∥ u_{k} ∥_{2}^{2} \leq \frac{2 t_{k - 1}^{2}}{L} (F (x_{k - 1}) - F (x_{*})) + ∥ u_{k - 1} ∥_{2}^{2} + \frac{2 t_{k}^{2}}{L} η_{k},$ with the $k = 1$ previous objective term interpreted as zero.

Facts & Assumptions

Given: The inexact FISTA recurrence, certified errors and a minimizer in the Statement.

[F1]

Certified inexact proximal-gradient and FISTA updates: $t_{1} = 1$ , $y_{1} = x_{0}$ , $x_{k}$ satisfies the stated certificate, $t_{k + 1} = (1 + \sqrt{1 + 4 t_{k}^{2}}) / 2$ and hence $t_{k + 1} (t_{k + 1} - 1) = t_{k}^{2}$ , and $y_{k} = x_{k - 1} + ((t_{k - 1} - 1) / t_{k}) (x_{k - 1} - x_{k - 2})$ for $k \geq 2$ .

[F2]

One-step estimate for a certified inexact proximal call: $F (z) \geq F (x_{k}) + (L / 2) ∥ x_{k} - y_{k} ∥_{2}^{2} + L ⟨ y_{k} - z, x_{k} - y_{k} ⟩ - η_{k}$ for every $z$ .

Proof

1.1F1giveninductionalgebra

The recurrence [F1] gives $t_{k} \geq 1$ for every $k$ , so $z_{k} = (1 - 1 / t_{k}) x_{k - 1} + (1 / t_{k}) x_{*}$ is a convex combination. It also gives $t_{k} (y_{k} - z_{k}) = u_{k - 1}$ and $u_{k} = u_{k - 1} + t_{k} (x_{k} - y_{k})$ , including $k = 1$ with $z_{1} = x_{*}$ .

2.1F1F2step 1.1givenalgebra∎

For $k \geq 2$ , convexity gives $F (z_{k}) \leq (1 - 1 / t_{k}) F (x_{k - 1}) + (1 / t_{k}) F (x_{*})$ . Apply [F2] at $y_{k}, z_{k}$ ; the same multiplication and norm expansion as in step 1.1 yield $\frac{2 t_{k}^{2}}{L} (F (x_{k}) - F (x_{*})) \leq \frac{2 t_{k} (t_{k} - 1)}{L} (F (x_{k - 1}) - F (x_{*})) + ∥ u_{k - 1} ∥_{2}^{2} - ∥ u_{k} ∥_{2}^{2} + \frac{2 t_{k}^{2}}{L} η_{k} .$ Use $t_{k} (t_{k} - 1) = t_{k - 1}^{2}$ to obtain the stated recurrence. For $k = 1$ , [F2] with $z = x_{*}$ gives the same base inequality directly, and $t_{1} = 1$ leaves the previous objective term zero.

Depends on

Used by

Weighted prox-error summability preserves an O(1/k²) FISTA bound Theorem

Dependency tree · two levels

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Sources

Mark Schmidt, Nicolas Le Roux, and Francis Bach, Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization, §3 and Appendix §6.2 (standard reference, not scraped)