{
  "abstract": "We propose a continuous learning formulation for $T$ non-forgetting tasks with the spherical parameter space $\\mathbb{S}^{d-1}(\\sqrt{d})$ in asymptotic regime $d\\rightarrow \\infty$. Each task $i$ has an optimum $\\bmΘ_i$, minimizing its corresponding loss function $R_i(\\bm{w})$ on sphere. We reduce each task $i$ to an acceptable loss region modeled by a geodesic ball $B_{\\bmΘ_i}(r_i(d))$ on $\\mathbb{S}^{d-1}(\\sqrt{d})$ centered at $\\bmΘ_i$ with radius $r_i(d)$. To model task dependencies (similarities), task optima $\\bmΘ_i$ are distributed by a Neyman-Scott clustered point process $Φ$. Cluster centers $\\{\\bm{Y}_j\\}_{j=1}^{N_p}$ form a homogeneous Poisson point process on sphere, while intra-cluster task optima follow a von Mises-Fisher distribution with mean $\\bm{Y}_j$ and concentration $κ$. Each task $t\\in \\{1, ..., T\\}$ learning algorithm is considered to be a channel $Q_{t}(\\mathrm{d}\\bm{w}_t | \\bm{w}_{t-1}, \\mathcal{K}_{t-1}(\\bmθ_{1:t-1}), \\bmθ_t)$, where $\\bm{W}_t$ is the learned parameter of task $t$ and $\\mathcal{K}_{t-1}(\\bmΘ_{1:t-1})=\\bigcap_{i=1}^{t-1}B_{\\bmΘ_i}(r_i(d))$. Since in continual learning, the learner does not need to reach $\\bmΘ_t$ exactly to successfully learn task $t$, to avoid forgetting as well as reliable learning, it only needs for the learned parameter vector $\\bm{W}_{t}$ to enter $B_{\\bmΘ_t}(r_t(d)) \\cap \\mathcal{K}_{t-1}(\\bmΘ_{1:t-1})$. Using this model, we derive an upper bound on $T$ ensuring $\\mathbb{P}[\\mathcal{K}_{T}(\\bmΘ_{1:T}) \\neq \\emptyset] \\rightarrow 1$ as $d\\rightarrow \\infty$ and $κ/d \\rightarrow α$, where $α>0$ is a constant. Finally, for $T$ tasks sampled uniformly from $Φ$, we derive an upper bound for total information theoretic capacity under the non-forgetting constraint for each task.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17453",
  "author_total": 1,
  "authors": [
    {
      "name": "Nazanin Mirhosseini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/en82m17x20jxa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17453",
  "fulltext_url": null,
  "id": "en82m17x20jxa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17453",
  "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"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Domain Adaptation and Few-Shot Learning"
    ]
  ],
  "openalex": "W7221752588",
  "pdf_url": "https://arxiv.org/pdf/2610.17453",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/en82m17x20jxa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17453",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17453"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Effect of Spherical Parameter Space on Number of Non-Forgetting Tasks and Capacity of Continual Learning",
  "title_text": "Effect of Spherical Parameter Space on Number of Non-Forgetting Tasks and Capacity of Continual Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/en82m17x20jxa",
  "venue": "arXiv",
  "versions": []
}