{
  "abstract": "Dynamic regret measures an online learner against a moving comparator, and standard analyses assume the learner may move freely at every round. We study what happens when it may not: the learner is given a hard budget $S$ on its total movement $\\sum_t \\|x_t - x_{t-1}\\|$. Our main result is that a movement budget creates a feasibility barrier. On a domain of diameter $D$ with $G$-Lipschitz convex losses, for every comparator path length $P_T \\in [D, DT]$ we exhibit a deterministic adversary that forces $R_d = Ω(GDT)$ on every budget-feasible algorithm whenever $S \\le c P_T$ for a universal constant $c>0$. This barrier is caused by the budget rather than by a volatile environment: for fixed $D$, it applies when $P_T = o(T)$, where the unconstrained minimax rate $Θ(G\\sqrt{D(D+P_T)T})$ is sublinear, so the same instances are learnable with an unbounded budget and unlearnable without one. Conversely, projected online gradient descent with a budget-aware step size is budget-feasible and attains $o(GDT)$ once $S = ω(D + P_T)$. Thus the critical movement scale is $D+P_T$: linear regret is unavoidable below a sufficiently small constant multiple of $P_T$, while sublinear regret is achievable when $S/(D+P_T)\\to\\infty$ in the learnable regime. The construction rests on a distinction we make explicit: what makes a comparator expensive to track under a movement budget is the distance it forces the learner to travel, not its path length. Above the threshold we give a budget-dependent upper bound and identify the tight minimax rate in this region as an open problem rather than claiming a product law. Experiments support the predicted transition location and the $1/S$ decay.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02812",
  "author_total": 1,
  "authors": [
    {
      "name": "Ridham Patel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55cp4387qatkw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02812",
  "fulltext_url": null,
  "id": "55cp4387qatkw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02812",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.SY"
    ],
    [
      "arxiv",
      "eess.SY"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02812",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55cp4387qatkw.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02812",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02812"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "When Is Tracking Affordable? A Movement-Budget Feasibility Threshold for Dynamic Regret",
  "title_text": "When Is Tracking Affordable? A Movement-Budget Feasibility Threshold for Dynamic Regret",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/55cp4387qatkw",
  "venue": "arXiv",
  "versions": []
}