Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Strong convexity for an extended-real objective
Definition
Let be a proper convex extended-real function (The extended real line , its order, and the arithmetic that is left undefined). For , say that is -strongly convex if for every and , This definition permits to be extended-real-valued outside its effective domain. The endpoint cases are identities and are not included in the strict range for .
Depends on
Used by
- The assigned SAGA Lyapunov factor fails under standard initialization Counterexample
- A regularized logistic finite sum with an exact minimizer Example
- Strong convexity makes a differentiable gradient strongly monotone Lemma
- Expected linear contraction of an SVRG epoch Theorem
- Last-iterate SGD rate for smooth strongly convex objectives Theorem
- Projected SGD converges almost surely on a strongly convex objective Theorem
- SAGA constant-step linear contraction with a memory-capped factor Theorem
- SAGA expected linear contraction with a memory-scaled step size Theorem
- Strong convexity gives linear proximal-gradient and restarted-FISTA rates Theorem
Dependency tree · two levels
24 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, First-Order Methods in Optimization, SIAM, 2017, Chapter 5 (standard reference, not scraped)
- Mark Schmidt, Nicolas Le Roux, and Francis Bach, Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization, §3 (standard reference, not scraped)