Pith Number

pith:KDYMTDIC

pith:2026:KDYMTDICUY4DU6BCLQNQ6VXH35

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When and Why is Optimistic Multiplicative Weights Slow? The Geometry of Energy Dissipation

Anas Barakat, Andre Wibisono, Antonios Varvitsiotis, Georgios Piliouras, John Lazarsfeld

Optimistic multiplicative weights updates converge linearly in KL divergence to unique interior Nash equilibria in zero-sum games.

arxiv:2605.13242 v1 · 2026-05-13 · cs.GT · cs.LG

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Claims

C1strongest claim

we prove over the dual iterates that energy is dissipative, and by establishing tight bounds on the magnitude of dissipation, our analysis quantifies the geometric bottlenecks that arise when the corresponding primal iterates are close to the simplex boundary. This further translates into a new linear last-iterate convergence rate in KL divergence on games with a unique and interior Nash equilibrium... we prove this dependence is optimal.

C2weakest assumption

The central analysis rests on viewing OMWU dual iterates as optimistic skew-gradient descent with respect to a specific energy function whose dissipation can be tightly bounded; if this modeling choice does not capture the dominant dynamics or if the energy function is not sufficiently dissipative under the stated conditions, the linear rate and optimality claims would not hold.

C3one line summary

OMWU achieves linear last-iterate convergence in KL divergence for unique interior Nash equilibria with optimal game-constant dependence due to quantified energy dissipation, while uniform best-iterate rates exhibit constant lower bounds in KL and TV but improved O(T^{-1/2}) duality-gap rates in 2x2

References

34 extracted · 34 resolved · 0 Pith anchors

[1] For this, fix w ∈ ri(W )

[2] Further recall from Proposition D.3 that ∇F is invariant to constant shifts, meaning for any s ∈ S that ∇F(v + s) = ∇F(v) = w

[3] For this, first observe that the property Z + S = Rm+n is equivalent to establishing equality between the orthogonal complements of the two sets

[4] Thus over the linear subspace Z ⊂ Rm+n, the first-order optimality conditions for F give (see, e.g., Boyd and Vandenberghe (2004), Sec 2004

[5] Thus fixing z⋆ ∈ argminz∈Z F(z) from Step (1), then also zτ = z⋆ + τJw ⋆ ∈ argminz∈Z F(z) for any τ ∈ R

Receipt and verification

First computed	2026-05-18T02:44:49.496855Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

50f0c98d02a6383a78225c1b0f56e7df56c94f0b27224468828875d3b0937683

Aliases

arxiv: 2605.13242 · arxiv_version: 2605.13242v1 · doi: 10.48550/arxiv.2605.13242 · pith_short_12: KDYMTDICUY4D · pith_short_16: KDYMTDICUY4DU6BC · pith_short_8: KDYMTDIC

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/KDYMTDICUY4DU6BCLQNQ6VXH35 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 50f0c98d02a6383a78225c1b0f56e7df56c94f0b27224468828875d3b0937683

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "1c767019ae4a9f095970a9c290a36d16c449a309123e43b71a1249f90d9adca3",
    "cross_cats_sorted": [
      "cs.LG"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.GT",
    "submitted_at": "2026-05-13T09:27:10Z",
    "title_canon_sha256": "ed5a30d41dee0ef363614ba2decc1bb82cd76581f3ebb25fd7a7e014b4658298"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13242",
    "kind": "arxiv",
    "version": 1
  }
}