{
  "abstract": "A parallel sampler can choose at random which coordinates to generate together. For a target $p$ and the output law $q_U$ of a schedule $U$ drawn afresh for each output, we compare the schedule-averaged divergence $\\mathbb{E}_U D_{\\mathrm{KL}}(p\\|q_U)$ with the output-law divergence $D_{\\mathrm{KL}}(p\\|\\mathbb{E}_U q_U)$, in nats. Rounds commit independent draws from exact conditional marginals frozen at the round's start, each output's schedule is drawn before sampling from a finite mixture of schedules, and every scheduled round is charged. For the uniform distribution on binary strings of even length $n$ with $n/2$ ones, the least expected number of rounds at a fixed tolerance $\\varepsilon>0$ is $Θ_\\varepsilon((\\log n)^2)$ under the first criterion but $(e^{-\\varepsilon}+o(1))\\log_2\\ln n$ under the second, and the double-logarithmic order is attainable with the same round count on every run. For this target, a count-state recursion gives the exact least output divergence at every round cap over all causal randomized policies. The separation extends to weighted fixed-count targets with bounded weight ratios and quota fractions bounded away from zero and one, without permutation symmetry, and to the full-support Beta$(1,1)$--Bernoulli target, where the sharp expected cost under the output-law criterion is $(e^{-\\varepsilon}/2+o(1))\\log_2\\log_2n$, attained with rare $n$-round runs. Prior work implies the order of the schedule-averaged cost on the balanced target and its $O_\\varepsilon((\\log n)^2)$ upper bounds on both extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13924",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianyi Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4k32g45ccxkgk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13924",
  "fulltext_url": null,
  "id": "4k32g45ccxkgk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13924",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7221532418",
  "pdf_url": "https://arxiv.org/pdf/2610.13924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4k32g45ccxkgk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13924"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2613|Markov Chains and Monte Carlo Methods",
      "subfield": "statistics"
    }
  ],
  "title": "KL Before and After Mixing: A Round-Complexity Separation in Parallel Sampling",
  "title_text": "KL Before and After Mixing: A Round-Complexity Separation in Parallel Sampling",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/4k32g45ccxkgk",
  "venue": "arXiv",
  "versions": []
}