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FISTA energy estimate-sequence recurrence

Statement

Let $F = f + g$ satisfy the exact FISTA hypotheses of Proximal-gradient, FISTA, and fixed-period restarted FISTA, and suppose $x_{*}$ minimizes $F$ . Use its iterates $x_{k}, y_{k}, t_{k}$ and set $t_{0} = 0$ , $u_{0} = x_{0} - x_{*}$ , and $u_{k} = t_{k} x_{k} - (t_{k} - 1) x_{k - 1} - x_{*} (k \geq 1) .$ Then for every $k \geq 1$ , $\frac{2 t_{k}^{2}}{L} (F (x_{k}) - F (x_{*})) + ∥ u_{k} ∥_{2}^{2} \leq \frac{2 t_{k - 1}^{2}}{L} (F (x_{k - 1}) - F (x_{*})) + ∥ u_{k - 1} ∥_{2}^{2},$ where the $k = 1$ right-hand objective term is interpreted as zero.

Facts & Assumptions

Given: The exact FISTA recurrence, composite objective and minimizer in the Statement.

[F1]

Proximal-gradient, FISTA, and fixed-period restarted FISTA: $t_{1} = 1$ , $y_{1} = x_{0}$ , $x_{k} = T_{L} (y_{k})$ , $t_{k + 1} = (1 + \sqrt{1 + 4 t_{k}^{2}}) / 2$ , the consequent identity $t_{k + 1} (t_{k + 1} - 1) = t_{k}^{2}$ , 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 proximal-gradient estimate: $F (z) \geq F (T_{L} (y)) + (L / 2) ∥ T_{L} (y) - y ∥_{2}^{2} + L ⟨ y - z, T_{L} (y) - y ⟩$ for every $z$ .

Proof

1.1F1giveninductionalgebra

The recurrence [F1] gives $t_{k} \geq 1$ for every $k$ : it starts at $1$ and $t_{k + 1} \geq t_{k} + 1 / 2$ . Thus $z_{k} = (1 - 1 / t_{k}) x_{k - 1} + (1 / t_{k}) x_{*}$ is a convex combination (and $z_{1} = x_{*}$ ). The same recurrence gives $t_{k} (y_{k} - z_{k}) = u_{k - 1}$ , where at $k = 1$ this reads $y_{1} - x_{*} = u_{0}$ . Also $u_{k} = t_{k} (x_{k} - z_{k}) = u_{k - 1} + t_{k} (x_{k} - y_{k})$ .

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] with $y = y_{k}$ , $z = z_{k}$ , and $x_{k} = T_{L} (y_{k})$ . The upper bound for $F (z_{k}) - F (x_{k})$ and its lower bound from [F2], multiplied by $2 t_{k}^{2} / L$ , imply $\frac{2 t_{k}^{2}}{L} (F (x_{k}) - F (x_{*})) \leq \frac{2 t_{k} (t_{k} - 1)}{L} (F (x_{k - 1}) - F (x_{*})) - t_{k}^{2} ∥ x_{k} - y_{k} ∥_{2}^{2} - 2 t_{k}^{2} ⟨ y_{k} - z_{k}, x_{k} - y_{k} ⟩ .$ Using [F1] to replace $t_{k} (t_{k} - 1)$ by $t_{k - 1}^{2}$ , and the two vector identities of step 1.1 to replace the last two terms by $∥ u_{k - 1} ∥_{2}^{2} - ∥ u_{k} ∥_{2}^{2}$ , gives the claimed recurrence for $k \geq 2$ . At $k = 1$ , $z_{1} = x_{*}$ and [F2] directly gives $F (x_{1}) - F (x_{*}) \leq (L / 2) (∥ x_{0} - x_{*} ∥_{2}^{2} - ∥ x_{1} - x_{*} ∥_{2}^{2})$ ; since $t_{1} = 1$ and $u_{1} = x_{1} - x_{*}$ , this is the claimed base case without using $F (x_{0})$ .

Depends on

Used by

FISTA has an O(1/k²) objective bound Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Amir Beck and Marc Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, Lemma 4.1 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, §4.3 (standard reference, not scraped)