[RemyDegenne]

Amends the Bandits roadmap so that its stochastic, adversarial, online-learning and linear layers ask for the theory at the generality of the literature. The cases they asked for before (Gaussian and Bernoulli arms, oblivious tables with a known horizon, a fixed learning rate, invertible designs) become instances. Companion to #10, the Identification roadmap amendment. Bandits now consumes three of its layers: Layer 1.2 (exponential families and their Cramér–Chernoff rate function), Layer 4.1 (the run-to-stream transfer) and Layer 4.2 (conditional MGF domination and its supermartingales).

Why. KL-UCB, Thompson sampling, d_inf and asymptotic optimality are stated in the literature for one-parameter exponential families [GC11, Section 4; CGMMS13; KKM13; BK96]. The roadmap asked for them only for Bernoulli or Gaussian arms. As a result LMLExplore has two parallel UCB algorithms and proofs (ucbF, klucb) and computes d_inf case by case. Adversarial bandits are analysed against adaptive adversaries, with anytime learning rates [BCB12, Section 3; Neu15], but the roadmap fixed an oblivious table and a horizon-tuned rate. It also specified Exp3, Exp3-IX and Exp4 as three algorithms, although Exp4-IX contains all three. FTRL and mirror descent are developed with time-varying regularizers and local norms [Ora, Chapters 6–7]. The roadmap asked only for a fixed potential, fixed rate and dual-norm stability, which cannot handle importance-weighted estimates. It also excluded online convex optimization, a one-line reduction through subgradients. Finally, "the inverse on the span" of the design layers was never named, and LMLExplore's Ring.inverse is 0 on singular operators.

Changes (only LMLRoadmaps/Bandits/):

• Conventions. Adversarial bandits are runs against an arbitrary environment (an adaptive adversary). The mean table m_t(a) is LML's Environment.means. The pseudo-regret is max_a E[Σ (m_t(a) − Y_t)] and the random regret is max_a Σ (m_t(a) − Y_t). Against onlineEvalEnv x _ the pseudo-regret is R_n(x); against stationaryEnv ν it is the expected stochastic regret. A new Learning rates convention fixes rates as nonincreasing sequences η_t, with bounds stated through η_{n−1}. The Generality convention lists the special cases not to reintroduce.
• Dependency tables. Add LML Means.lean, Tau Ceti MoorePenrose.lean, Mathlib Convexity.StdSimplex, the three Identification layers, and the LMLExplore material the general forms extend. Online convex optimization leaves the "Outside" list; bandit convex optimization enters it. AdaHedge is recorded as the Identification roadmap's (Layer 6.1).
• 0.5. Pseudo-regret and random regret for any environment, with their oblivious and stochastic identifications. The environment of a deterministic adaptive adversary. Expert regrets with advice read from the observation. The i.i.d.-table domination is stated precisely, as a mixture of oblivious runs.
• 1.3–1.5. The per-arm transfer cites Identification Layer 4.1. Layer 1.4 becomes "Chernoff bounds in rate-function form": it cites Identification Layers 1.2 and 4.2 (including the supermartingale behind [GC11, Theorem 10]). It keeps the bounded instance here: conditional Bernoulli domination [GC11, Lemma 9], klBer as the Bernoulli rate function, and [LS, Lemma 10.3, Corollary 10.4] with the boundary levels. The [0,1] lemmas of KL-UCB move to Layer 3.2.
• Layer 3: KL-UCB for exponential families.
  • The upper confidence level U_ρ(x, c), through the rate-function extension d̄ of the divergence, so boundary empirical means need no special case.
  • A three-point identity for d̄.
  • [LS, Lemmas 10.7–10.8] for arms dominated by a family member.
  • KL-UCB with one family per arm, and a finite-time bound with explicit family constants plus its eventual form.
  • klucb (Bernoulli) and ucbF (Gaussian) are instances; [LS, Theorem 10.6] and the eventual part of [LS, Theorem 8.1] are corollaries. The explicit constants of Theorem 8.1 keep their own proof.
• 4.4–4.5. d_inf for every exponential family [BK96], with [LS, Table 16.1] as instances. Asymptotic optimality of KL-UCB for families with a common set of means.
• Layer 5. Thompson sampling for exponential families with a prior on the natural parameter, as a perturbed-leader policy. Asymptotic optimality with the Jeffreys prior [KKM13, Theorem 1]. Gaussian Thompson sampling is the Jeffreys instance; the minimax part of [LS, Theorem 36.3] stays.
• Layer 6.
  • FTRL with nondecreasing regularizers [Ora, Lemma 7.1, Corollary 7.6] and mirror descent with nonincreasing rates [Ora, Theorem 6.11].
  • Local-norm stability [LS, Theorem 26.13; Ora, Lemma 7.15], and the exponential-weights local-norm bound for nonnegative losses and anytime rates; softmax_regret_le is its constant case.
  • [LS, Theorem 28.10] for adaptively chosen losses (moved here from Layer 10.2).
  • Online convex optimization through subgradients (LML's HasSubgradientWithinAt), with projected online subgradient descent [Zin03] as the Euclidean instance.
• Layer 7.
  • Runs against adaptive adversaries with feedback in [0,1], with estimators conditional on filtrationObs.
  • One algorithm, Exp4-IX [Neu15]; Exp4, Exp3-IX and Exp3 are defined as its instances, and its expert weights are the exponential weights of Layer 6.4.
  • Pseudo-regret bounds with anytime rates: [LS, Theorems 11.2, 18.1] and [BCB12, Theorem 3.1].
  • High-probability bounds: [LS, Theorem 12.1] and [Neu15, Lemma 1, Theorems 1–2].
• Layer 8.
  • Design operators with any positive regularizer V₀, and the self-normalized bound for positive definite V₀ [AYPS11, Theorem 1].
  • The Moore–Penrose inverse V⁺ (Tau Ceti) for operators that may be singular: least squares for λ ≥ 0 (minimal-norm for λ = 0), the fixed-design bound [LS, (20.2)] on the span, and Kiefer–Wolfowitz on non-spanning sets.
  • Designs are Convexity.StdSimplex ℝ E, with the bridge simplex α = range (⇑·.weights) and the image of designs under a feature map.
• 9.3, 10. Phased elimination uses the unregularized estimator on the span. Layer 10 takes adaptive losses and the pseudo-regret, and deduces [LS, Theorem 28.10] from Layer 6.5.
• Completion checks and references. Updated cross-references (the adversarial upper bound is now 7.3). References are keyed: AYPS11, BCB12, BK96, CGMMS13, GC11, KKM13, Neu15, Ora (cited by its v10 numbering), Zin03.
• Suggested.lean.
  • pseudoRegret and randomRegret (with Environment.means), with their oblivious and stochastic identifications.
  • expFamilyUpper, klucbFamIndex, klucbFam; ucbF and klucb as instances, and the asymptotic regret under MGF domination.
  • dInf_range_tilted, jeffreysPrior, expFamilyTS.
  • ftrl_regret_le_of_monotone.
  • exp4IX, with exp3 as its instance and exp3_pseudoRegret_le for adaptive adversaries and antitone rates.
  • designOp taking V₀, and the self-normalized bound for positive definite V₀.
  • designOperator on Convexity.StdSimplex ℝ E, designOperator_map, and Kiefer–Wolfowitz on the span with LinearMap.moorePenroseInverse.

Points to check.

• 3.4. The constant $3/(2,d_{\rho_{a^}}(\mu^-\varepsilon_2,\mu^*))$ of the underestimation term is derived in the draft, following LMLExplore's existing proof, not taken from a paper. Through Pinsker's inequality, the Bernoulli instance gives the constants of LMLExplore's KLUCB.integral_pullCount_le, $1/(2\varepsilon_1^2)+2/\varepsilon_2^2$.
• Layer 5. The Jeffreys-prior result assumes that the Jeffreys posterior is finite after one pull, for almost every reward sum; [KKM13] leaves this implicit.
• LMLExplore signatures. These change: exp3 becomes the instance of Exp4-IX with a learning-rate sequence, and designOp takes $V_0$.
• Scope. Online convex optimization is now in scope.
• Not included. The [GMS19] fundamental inequality as a second route to the regret lower bounds (its event form is Identification Layer 2.2). KL-UCB++. Per-arm families for Thompson sampling.
• Citations. Orabona is cited with its arXiv v10 numbering.
• Vocabulary. stationaryEnv, which this roadmap already used, is LML's Environment.bandit at the pin LMLExplore uses. It is left for a separate update of all three roadmaps.

Suggested.lean builds.

Drafted with Claude Opus 5.5 at the request of the code owner.

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