Convergence of difference inclusions via a diameter criterion
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914565316411392 |
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| author | Lai, Lexiao Song, Mingzhi |
| author_facet | Lai, Lexiao Song, Mingzhi |
| contents | We study discrete dynamics governed by a difference inclusion whose increment is the sum of a selection from a set-valued map and a noise term. For any bounded realization, convergence follows once the inter-iterate diameter is controlled by the variation of a continuous potential. The limit point is then critical for a scaled outer limit of the update map. To certify this diameter criterion, we develop a stratified descent framework: we project iterates onto a suitable stratification and track a potential that decreases up to a summable error. Combining the diameter criterion with a diameter estimate obtained from this framework yields convergence of common first-order optimization methods under step sizes of order $1/k$. The guarantees cover inexact and stochastic subgradient methods, as well as the momentum method, for locally Lipschitz objectives definable in polynomially bounded o-minimal structures. Our arguments are entirely discrete, with no appeal to continuous-time approximations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14345 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of difference inclusions via a diameter criterion Lai, Lexiao Song, Mingzhi Optimization and Control 65K10, 39A05, 03C64 We study discrete dynamics governed by a difference inclusion whose increment is the sum of a selection from a set-valued map and a noise term. For any bounded realization, convergence follows once the inter-iterate diameter is controlled by the variation of a continuous potential. The limit point is then critical for a scaled outer limit of the update map. To certify this diameter criterion, we develop a stratified descent framework: we project iterates onto a suitable stratification and track a potential that decreases up to a summable error. Combining the diameter criterion with a diameter estimate obtained from this framework yields convergence of common first-order optimization methods under step sizes of order $1/k$. The guarantees cover inexact and stochastic subgradient methods, as well as the momentum method, for locally Lipschitz objectives definable in polynomially bounded o-minimal structures. Our arguments are entirely discrete, with no appeal to continuous-time approximations. |
| title | Convergence of difference inclusions via a diameter criterion |
| topic | Optimization and Control 65K10, 39A05, 03C64 |
| url | https://arxiv.org/abs/2605.14345 |