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The concentration game: Bayesian updating, regret, and information

Akshay Balsubramani

arXiv:2608.18061v1cs.LGcs.GTmath.PRmath.ST

TL;DR

The paper addresses fragmented formalisms connecting Bayesian updating, exponential-weights regret, and concentration. It formulates a sequential zero-sum concentration game whose Bellman equalizer yields Bayesian/Gibbs updating and an exact regret decomposition at every horizon.

  • Problem

    Bayesian updating, exponential-weights regret, concentration, and repeated-game analysis use separate formalisms despite sharing information-accounting structure.

    Source: §1 · p. 1

  • Method

    The paper models these phenomena as a sequential zero-sum game with relative-entropy comparator regularization, centered RCGF information budgets, and Bellman-equalizing Gibbs/Bayes strategies.

    Source: §1 · p. 1, §2.4 · p. 6

  • Results

    The decomposition holds with equality at every horizon, comparator, and predictable schedule, splitting regret into per-round information loss, retempering drift, and comparator information.

    Source: §1.2 · p. 3

  • Takeaways & Limitations

    The framework unifies Bayesian, online-learning, game-theoretic, and concentration primitives while treating variance and bounded-range quantities as relaxations of an exact information ledger.

    Source: §1 · p. 1, §11 · p. 23

    The exact value remains open for the per-round-capped game over multiple rounds, including the case K = 2 beyond T = 1.

    Source: §11.3 · p. 25

Abstract

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We give a two-player zero-sum repeated game between a learner and nature whose value identity generates Bayesian updating and an exact accounting of exponential-weights regret at once, and supplies the comparator-class variational form that a wide class of concentration phenomena share. The terminal payoff is the most a comparator can gain at fixed relative entropy from the prior, and the one-step constraint is an information budget on nature's move under the learner's mixed action. With the learner's move otherwise unrestricted, Gibbs/Bayes weights emerge as its unique Bellman equalizer -- the mixed action that makes the per-round loss independent of which direction nature moves -- with log-partition functions playing the role of value functions. The regret decomposes exactly into three parts: a per-round information loss reflecting the variation in observed outcomes, an additive retempering drift that accounts exactly for any change of measurement scale between rounds, and the information the comparator carries relative to the prior. The variance and bounded-range proxies that drive standard regret bounds are looser relaxations of this decomposition, which holds generally and governs them all. Both players' strategies are read off from the decomposition term by term, and repeated play yields an information-theoretic ledger of self-play in place of the usual quadratic-variation surrogate. The same comparator-class geometry accounts for the classical large-deviation bounds, and methods across bandits, posterior sampling, aggregation, and boosting are specializations of the one regret decomposition.

1 Introduction

The paper presents Bayesian updating, exponential-weights regret, concentration, and related methods as manifestations of one sequential zero-sum concentration game. Its exact regret ledger separates intrinsic-time information loss, scale-change drift, and terminal comparator information, with standard bounds arising as relaxations.

1 Introduction

  • Bayesian updating, exponential weights, concentration bounds, repeated-game equilibria, Thompson sampling, log-pooling, and AdaBoost arise as specializations of one sequential game value identity.Bayes/exponential-weights updating is the equilibrium strategy, the moment-generating function is the per-round loss, and the log-partition function is the value.Source: §1 · p. 1

1.1 The game and the central identity

  • Cumulative regret decomposes exactly into intrinsic-time information loss, retempering drift from changing measurement scales, and terminal relative-entropy transport from prior to comparator.The identity holds as a complete ledger with no remainder; constant measurement scale makes the retempering drift zero.Source: §1.1 · pp. 2–3
  • The concentration game constrains nature by centered per-round RCGF budgets and pays the largest terminal comparator gain within a relative-entropy ball around the prior.The learner may choose any action distribution; the Gibbs tilt parameterizes its move, while the scale determines the information meaning of each move.Source: §1.1 · pp. 1, 3

1.2 Main results

  • The decomposition holds with equality at every horizon, for every comparator, and for every predictable schedule, while Gibbs play equalizes each round and nature saturates its budget in the worst comparator direction.The scale-change drift is a closed-form prior-to-posterior relative entropy and vanishes at fixed scale.Source: §1.1 · p. 2, §1.2 · p. 3
  • Variance and bounded-range regret quantities are looser relaxations of the centered RCGF, whose small-scale limit yields quadratic-variation bounds; under cumulative information budget V, the constant-scale value is Γ/η + ηV.The square-root envelope 2√ΓV is attained from a finite horizon onward, while sharp application-specific rates still require bounding the per-round RCGF.Source: §1.2 · pp. 3–4

2 Coordinates, the terminal payoff, and the per-round primitive

This section formulates the game in centered exponential-family coordinates, defines a relative-entropy terminal payoff, and identifies the centered RCGF as nature’s one-round information primitive. Gibbs tilts simultaneously describe optimal terminal comparators, robust-Bayes posteriors, and the exact concentration mechanism underlying variance and range relaxations.

2 Coordinates, the terminal payoff, and the per-round primitive

  • Unrestricted learner play is bijectively represented by Bayes/exponential-weights coordinates, while only relative score differences matter and the scale couples play to nature’s information budget.The sufficient statistic may be losses, offsets, importance-weighted estimates, residuals, margins, or another chosen score sequence; no ambient true loss is required.Source: §2 · p. 5, §2.1 · p. 5

2.1 Centered excess-score coordinates

  • The centered-coordinate representation removes the global-bias direction: each move has learner mean zero, and cumulative regret depends only on relative performance.The centered state is built from z_t(i)=μ_t−c_t(i), with ⟨p_t,z_t⟩=0.Source: §2.1 · p. 5

2.2 The terminal concentration functional

  • The terminal functional L_Γ(S) is the worst comparator payoff over distributions within KL divergence Γ of the prior, interpolating from the prior mean at Γ=0 to the best action once the simplex is reachable.Its dual is an exact Cramér–Chernoff variational identity: optimizing the prior log-moment generating function over scale yields the largest displacement permitted by the rate budget, with looseness arising only from fixing a nonoptimal scale.Source: §2.2 · pp. 5–6

2.3 Robust Bayes, maximum entropy, and exponential-family structure

  • The Gibbs tilt ρ_η,S is simultaneously the terminal optimizer, the robust-Bayes maximum-entropy distribution matching its exposed score mean, and the Bellman equalizer along a one-dimensional exponential family.The intrinsic-time density Q_t(c) is the one-step analogue of Fisher information.Source: §2.2 · p. 6, §2.3 · p. 6

2.4 The one-scale centered RCGF

  • The centered RCGF Q_η(p,z) is the unique nonnegative information quantity making the Bellman increment exact, and the constraint Q_η(p,z)≤q caps nature’s concentration at the chosen scale.It is the η^-2-rescaled centered CGF, approaches 1/2 Var_p(z) at small scale, captures worst-tail behavior at large scale, and yields the bounded-range bound Q_η(p,z)≤(b−a)^2/8; variance and range are relaxations, not the primitive.Source: §2.4 · pp. 6–7
  • The same RCGF admits coincidence-budget, robust-Bayes, and relative-entropy readings, recurs across comparator, bandit, and posterior-sampling constructions, and remains estimable with bounded per-draw influence when centered scores are bounded.Its tilted-variance representation establishes nonnegativity, with equality exactly when z is p-almost surely constant.Source: §2.4 · p. 7

3 The fixed-scale concentration game

At fixed scale, a single log-partition potential yields the Bellman equalizer, exact transport accounting, and classical variance/range relaxations. The section also distinguishes the Bellman envelope from the exact one-round minimax value, which can be strictly smaller.

3.1 Nature’s feasible set and the Bellman equalizer

  • Gibbs weights equalize each feasible round: the log-partition increment is exactly ηQ_η(p,z), capped by nature’s information budget ηq, so the potential telescopes.The resulting Bellman relaxation becomes tight against budget-saturating replies, although it need not equal the game value for a committed scale.Source: §3.1 · pp. 7–8
  • Under a cumulative budget, optimizing the Bellman bound gives a sub-Gaussian profile 2√Γq, while the fixed-scale bound can become exact after a finite horizon.At T=1, Γ/η + ηq is minimized at η=√(Γ/q), attaining 2√Γq; short-horizon slack can disappear under cumulative play.Source: §3.1 · p. 8

3.2 One-step transport identity

  • The fixed-scale transport identity decomposes comparator gain into the learner’s centered RCGF information loss and the drop in terminal relative entropy to the comparator.This is the constant-scale case because retempering drift vanishes, leaving intrinsic-time loss plus terminal transport.Source: §3.2 · p. 8, Summing over a fixed-scale trajectory… · p. 9

3.3 Variance and range as upper bounds on the centered RCGF

  • Variance and range bounds upper-bound the centered RCGF, yielding looser second-order and Hoeffding-style regret envelopes while discarding tail-shape information.Neither proxy uniformly dominates the other, and retaining the remainder exposes explicit slack from the posterior mismatch and proxy gap.Source: §3.3 · p. 9
  • The learner observes Q_η at only its chosen scale, so intrinsic time depends on the schedule: small η is variance-like, while large η reveals tail behavior.For a symmetric binary move, Q(η)=log cosh(η)/η^2 decreases to zero as η grows, even though ηQ(η) increases.Source: §3.3 · p. 9

3.4 The exact one-round value

  • The Bellman envelope Γ/η + ηq can be strictly above the exact one-round value, including for two experts and for sufficiently large comparator budgets at every K.Gibbs play equalizes at every state but need not be minimax; non-uniform priors and skewed states produce over-tilting relative to exact minimax play.Source: §3.1 · p. 8, §3.4 · p. 9

4 Changing scale: retempering, allocation, and active constraints

Changing scale replaces one fixed Bellman potential with round-dependent potentials, whose switching cost is an exact retempering drift. The section develops cumulative-budget, self-tuned, and active-constraint formulations that preserve exact information accounting while recovering square-root and classical second-order bounds as relaxations.

4.1 From per-round caps to a cumulative budget

  • With a cumulative nature budget, the learner selects the Gibbs scale each round while nature allocates difficulty unevenly; the constant-scale value becomes horizon-free after a finite threshold.A per-round cap is recovered by setting V = Tβ, whereas concentration in one round creates an edge term; computed instances have T0 = 2 for K ≤ 5 and T0 = 3 near saturation.Source: §4.1 · pp. 9–10

4.2 The exact variable-temperature decomposition

  • The exact variable-temperature identity decomposes pathwise regret into centered RCGF losses, retempering drift, and terminal relative-entropy transport, without a quadratic-variation surrogate.For non-increasing schedules, the drift satisfies D_T ≤ 0; at fixed temperature it vanishes, and the cumulative mix loss becomes a terminal log-partition function, the negative log Bayes marginal likelihood [25].Source: §4.2 · p. 11

4.3 The second-order equalizer and nature’s allocation

  • The second-order equalizer chooses a predictable scale from revealed intrinsic time, matches the known-V square-root rate pathwise up to edge terms, and needs neither V nor the horizon.Its non-increasing schedule makes retempering beneficial, while the exact RCGF remains tight for heavy-tailed centered moves where variance relaxations are loose; discrete initialization and single-jump edge costs are necessary.Source: §4.3 · pp. 11–12

4.4 The active-constraint game and pressure targets

  • The active-constraint game reverses move order: nature reveals its centered move, and the learner line-searches for the scale making a chosen pressure target active, with concentration and boosting sharing this equilibrium.The target is the per-round loss or RCGF, while the scale is its dual variable; quadratic-variation penalties arise as the small-scale limit, and the two game formulations agree on common terminal trajectories.Source: §4.4 · pp. 12–13
  • Against the worst comparator, the three horizon-dependent ledger terms telescope to a function of the realized terminal state, so schedule effects reduce to which terminal states nature can reach.This terminal collapse gives the same comparator-worst regret for the two dual games along any common score trajectory and reduces their value comparison to reachability.Source: §4.4 · p. 13

5 Comparator classes, concentration, and luckiness

The paper presents comparator classes, sufficient statistics, event restrictions, luckiness, and confidence regions as terminal or statistic-level slices of one concentration game. This shared geometry recovers quantile and tracking regret, classical large deviations, adaptive low-noise guarantees, and confidence trajectories.

5 Comparator classes, concentration, and luckiness

  • Changing comparator classes, sufficient statistics, or terminal events turns the same game into quantile and tracking regret, side information, optimism, concentration, luckiness, and confidence sets.These cases are terminal slices packaged by the terminal payoff LΓ.Source: §5 · p. 13

5.1 Changing comparator classes

  • Quantile regret follows by choosing a hindsight set A of good experts, with KL(νA∥π) = log(K/|A|) under uniform priors.A dyadic controller makes the guarantee simultaneous over quantiles, using O(log log K) workers and an additive controller complexity under the Theorem 4.4 envelope.Source: §5.1 · p. 13
  • Dynamic comparator paths add an exact comparator-path transport term, which becomes a total-variation control for losses in [0, 1].The transport term is exact before bounding, so switching and tracking regret arise directly from the same calculus.Source: §5.1 · p. 13

5.2 The sufficient statistic as a design choice

  • Choosing the sufficient statistic changes the game’s state, intrinsic time, and move constraint, recovering optimism, side information, specialists, sparsity, and bounded-influence robustness.Predictable baselines isolate residual losses, while confidence weighting, variance corrections, positive-part excess losses, and Catoni or Winsorization transforms provide further instances [16],.Source: §5.2 · pp. 13–14

5.3 Event-restricted terminal games, Gibbs conditioning, and Sanov

  • Event-restricted games use the Gibbs tilt of the conditioned prior, making the event cost −log π(A) and recovering Chernoff, Cramér, Gibbs conditioning, and Sanov through one comparator-class variational derivation.Only the final conversion from rate function to probability statement differs across these classical results.Source: §5.3 · p. 14

5.4 Comparator-centered luckiness

  • Comparator-centered luckiness yields constant-in-T expected regret under a low-noise condition, while the no-gap regime with accumulating means remains open.For finite K with a strict off-face mean gap, tied minimum-mean comparators are handled separately; comparators above the minimum are bounded by a linear margin.Source: §5.4 · p. 15

5.5 Confidence regions and confidence trajectories

  • Confidence regions are relative-entropy transport super-level sets, with anytime-versus-fixed-horizon differences governed by comparator-specific drawdown and scale drift.At point masses they recover posterior-tail sets; horizon knowledge removes drift, whereas plug-in or mixed scales incur additional width.Source: §5.5 · pp. 15–16

6 Repeated games, equilibria, and the information ledger

Repeated play turns the concentration-game decomposition into an exact information ledger: equilibrium error equals the relevant players’ ledger entries, while classical variance-based rates arise by relaxing those entries. Intrinsic-time regret matching can therefore adapt to realized play, yielding faster rates when play stabilizes, subject to convergence conditions.

6 Repeated games, equilibria, and the information ledger

  • Equilibrium error is exactly a sum of player ledgers in zero-sum games and the largest player ledger in general-sum games, rather than merely being bounded by them.The identities expose which information terms classical guarantees discard and introduce no quadratic-variation surrogate.Source: §6 · p. 16, §6.2 · p. 17, §6.3 · p. 18

6.1 Repeated zero-sum play and intrinsic-time regret matching

  • Intrinsic-time regret matching applies the retempered concentration game to centered regret vectors, preserving the ledger decomposition and its square-root envelope while remaining scale-free in the unclipped regime.Scaling losses by a > 0 scales intrinsic time by a^2 and the learning rate by 1/a, so the schedule needs no prior loss-scale knowledge.Source: §6.1 · p. 16

6.2 An exact information ledger: two-player duality gap

  • In two-player zero-sum self-play, the exact duality-gap ledger can be strictly faster than the worst-case rate: when intrinsic time is O(log T), the expected gap is O(√log T/T).This improvement is driven by realized play and drift gains, but the equality is specific to this algorithmic family and is not a minimax lower bound.Source: §6.2 · p. 17

6.3 General-sum play and equilibrium consequences

  • For general-sum games, the exact per-player ledgers identify CCE distance with the largest unilateral deviation ledger, and convergence to a strict pure Nash profile yields O(1/T), faster than the classical Θ(1/√T).The faster rate is conditional: the schedule does not establish convergence, and cyclic or interior-equilibrium play can keep intrinsic time growing.Source: §6.3 · p. 18

7 Partial information: bandits, graphs, context, and robust scores

Partial information preserves the intrinsic-time game on estimated scores while adding observation, exploration, and bias terms. The same framework covers bandits, feedback graphs, contextual wrappers, and testing exponents through estimator-specific scores and observation geometry.

7 Partial information: bandits, graphs, context, and robust scores

  • Partial information replaces full loss-vector feedback with separate play and sampling laws, estimated scores, and explicit observation, exploration, and predictable-bias terms.The learner plays Gibbs weights from the estimated state, samples from a potentially different law, observes one loss, and is constrained through the conditional expected estimated RCGF.Source: §7 · p. 19

7.1 Generic bandit decomposition

  • The generic decomposition equals the estimated-score full-information ledger plus play/estimation martingales, exploration cost, and predictable bias; feedback graphs enter through observation geometry, bounded by the independence number [3].IPS removes bias while recording exploration cost, whereas EXP3-IX, control variates, and feedback graphs correspond to alternative estimated-score choices.Source: §7.1 · p. 19

7.2 An identity for the estimated RCGF

  • The estimated RCGF has an explicit true-loss and observation-geometry form, reducing at small scales to propensity-inflated variance and yielding the O(√KT log K) relaxation when µt = pt.The exact minimax value remains the corresponding explicit optimization; solving it generally is the full-information exact-value problem, while K = 2 over one round is solved by Proposition A.6.Source: §7.2 · p. 19

7.3 Contextual wrappers and identification scores

  • Contextual wrappers lift policy classes into experts with importance-weighted losses, while log-likelihood-ratio scores recover Chernoff tilts, Chernoff information, and MAP identification exponents [8].The policy-wrapper bound uses exact constants and comparator complexity from aggregate prior mass over loss-identical copies; testing exponents equal edge-restricted multi-way coincidence radii.Source: §7.3 · p. 20

8 Thompson sampling and prior-posterior-ratio martingales

Randomized play yields two martingale interpretations: Thompson sampling preserves the deterministic concentration-game value in expectation up to a mean-zero sampling martingale, while Bayes updating becomes a prior-posterior-ratio test supermartingale. The resulting entropy ledger bounds comparator margins by the prior information budget and connects Gibbs play to anytime-valid inference.

8.1 Thompson sampling as the same game plus a martingale

  • Thompson sampling matches exponential weights’ deterministic concentration-game value in expectation, with all realized-path randomness represented by a single mean-zero sampling martingale.The posterior-sampling strategy is the unique equalizer at the game value; any alternative learner weights strictly over-drive the Gibbs increment and loses.Source: §8 · p. 20, §8.1 · p. 20
  • The sampling martingale is a genuine realized-path cost: variance-clock concentration gives anytime bounds, while worst-case regret matches exponential weights up to an iterated-logarithm factor and vanishes in expectation under low noise.No scoring rule removes the path fluctuation; Ville’s inequality supplies uniform control, and the sharp high-probability scale differs from the in-expectation optimum.Source: §8.1 · p. 20

8.2 The prior-posterior-ratio martingale

  • For every fixed comparator ν, Bayes updating obeys KL(ν∥p_t+1) − KL(ν∥p_t) = η_t(⟨ν,c_t⟩−m_t(η_t)), so total margin is capped by KL(ν∥π) ≤ Γ.Under Thompson sampling, the sampled log-weight has nonpositive conditional drift and its excess is the prior-posterior-ratio martingale, linking the game to anytime-valid testing.Source: §8.2 · p. 21

9 Boosting and the pressure game

The paper identifies boosting with a pressure game on signed margins, where examples are actions, weak learners supply nature’s moves, and comparator reweightings control margin guarantees. This yields an exact exponential-loss ledger that recovers AdaBoost and sharpens classical training-error bounds.

9 Boosting and the pressure game

  • Boosting is a pressure game on signed margins: the booster acts over training examples, weak learners provide nature’s moves, and comparators reweight examples.The uniform prior is u_N, while the ε-quantile margin uses the comparator uniform on the hardest ⌈εN⌉ examples.Source: §9 · p. 21

9.1 From regret to the exponential-loss terminal game

  • The ε-quantile margin falls short of twice the average edge by at most 2√(log(1/ε)/T) plus O(log(1/ε)/T) worst-case edge terms.The bound improves when the realized intrinsic time of the example-loss sequence is smaller.Source: §9.1 · p. 21
  • Training error, margin tails, and quantile margins are governed by one terminal concentration functional, with exponential training loss arising as an exact mixed-coincidence identity.A Chernoff step also bounds the fraction of examples with normalized margin at most θ by exp(θA_T) times the exponential loss.Source: §9.1 · pp. 21–22

9.2 Pressure-target boosting and AdaBoost

  • For binary weak hypotheses, the pressure-target coefficient is AdaBoost’s classical α⋆, while convexity recovers c_err_T ≤ e^-2γ^2T as a corollary.The exact ledger is stronger because it needs no uniform lower bound on the edge and retains the per-round shortfall discarded by the classical bound.Source: §9.2 · p. 22
  • Replacing signed scores with positive-part shortfalls yields sparse example weights without changing the game, assigning zero cost to examples already above target.This is the boosting analogue of a positive-part sufficient-statistic reduction.Source: §9.2 · p. 22

10 Related work

The paper frames its concentration game as a constrained minimax value unifying exponential-weights regret, Bayesian updating, and concentration, while connecting to game-theoretic probability, adaptive regret, e-processes, and related learning frameworks. Its distinctive contribution is deriving information budgets, retempering drift, and comparator transport directly from the Bellman/log-partition identity rather than imposing them externally.

Minimax and drifting games

  • The fixed-scale concentration game extends drifting-games minimax methodology: log-partition potentials and centered RCGF increments solve backward induction in closed form, yielding standard external- and swap-regret consequences.The effective state is (p_t, V_t−1), and variance and range games arise as relaxations rather than primitives.Source: §10 · p. 22

Game-theoretic probability

  • Unlike game-theoretic probability’s Skeptic capital primitive, this framework starts from comparator-class regret and obtains a Gibbs equalizing equilibrium, with both views coinciding through the prior-posterior-ratio martingale.In the time-uniform regime, the log-partition value and testing supermartingale describe the same process.Source: §10 · p. 22

Adaptive learning and concentration

  • The framework unifies adaptive regret, PAC-Bayes, e-processes, bandits, information-directed sampling, mixability, and sequential likelihood mixing, while making explicit that comparator transport B_T(ν) and per-round loss η_tQ_t are the regret-to-tail conversion factor.It also identifies reverse information projection as the extremal property underlying log-optimality and treats retempering drift as a distinct closed-form term.Source: §10 · pp. 22–23

Sharp concentration accounting

  • The paper relates its exact terminal accounting to sharpened Markov, Chebyshev, Chernoff, and Ville bounds, including explicit anytime-valid slack decompositions and comparator-path identities for dynamic comparators.The cited lines include randomization and exchangeability improvements and concurrent overshoot, predictable-loss, and residual-survival decompositions [23].Source: §10 · p. 23

Optimization, Bayes, and boosting

  • The concentration game reverses the direction of Fenchel-game optimization and natural-gradient variational Bayes, while its pressure-target view connects to classical and entropy-regularized boosting.Those programs derive games or algorithms from optimization or Bayesian primitives; here the constrained min-max value is primitive.Source: §10 · p. 23

Budgeted adversaries

  • Unlike budgeted-adversary analyses, the round-budget allocation follows from the RCGF/log-partition Bellman structure: P_tQ_t ≤ V is sufficient, front-loading is forced, and variance bounds relax the identity.The resulting cumulative intrinsic time is V_T(c), with the same √ΓV_T leading constant up to the stated relaxation.Source: §10 · p. 23

11 Discussion

The discussion presents a common entropic game structure underlying the framework’s settings and neighboring interpretations, while identifying limits on simultaneous adaptation and exact-value analysis. It also emphasizes higher-cumulant information and several open problems beyond the solved cases.

11.1 Common structure across settings; the scaling-time question

  • All settings share an RCGF budget Q_t(·) ≤ β_t, with differences only in observations, action spaces, sufficient statistics, and protocol components.Table 2 compares which game components vary across boxed variants.Source: §11.1 · p. 24, Table 2 · p. 24
  • The fixed-budget PAC-Bayes scaling-time problem is solved for exact cumulant intrinsic time, but simultaneous adaptivity in comparator complexity and intrinsic time remains open under lower-bound barriers [46].A logarithmic meta-controller provides simultaneous quantile guarantees in the fixed-budget setting.Source: §11.1 · p. 24

11.2 The identities in neighboring fields

  • The framework’s identities connect retempering drift, stochastic control, free energy, bandit estimator tradeoffs, and classical redundancy–capacity, while the entropic ledger generalizes across these readings.EXP3-IX trades smaller estimation bias for a larger martingale term, whereas Tsallis-INF trades smaller entropic information for larger sample-dependent drift.Source: §11.2 · p. 24
  • Higher cumulants κ≥3 capture gains discarded by variance and range proxies; changing sufficient statistics, bounded-influence transforms, and optimism can reduce the resulting relaxation remainder.The remainder has the closed form given in Proposition A.23.Source: §11.2 · p. 24
  • For strictly convex Legendre potentials, the gradient ∇W(S) is both the Bellman equalizer and maximum-(−Ψ)-entropy distribution, while Shore–Johnson axioms uniquely select centered RCGF inference [69].The invariance of this inference rule from the equalizer postulate alone is outside the paper’s scope.Source: §11.2 · pp. 24–25

11.3 Exact values and rates

  • Exact values remain open for the common one-round refinement and per-round-capped multi-round games, although cumulative-budget values stabilize from finite horizon T0 and varying scale changed computed values by under 0.1%.The K=2 per-round-capped case has a state-dependent reachable-set recursion with no elementary closed form beyond T=1.Source: §11.3 · p. 25
  • Dyadic and continuous-mixture meta-controllers restore complexity adaptivity with lower-order overhead, but one-copy parameter-free adaptation without an additive term is ruled out by model-selection lower bounds [46].The continuous-mixture construction achieves O(log log VT) overhead while retaining the leading constant.Source: §11.3 · p. 25

A Further results … A.7.3 Why the ledger supplies no converse

This section develops the RCGF framework’s exact gates, scale schedules, variational concentration results, and applications to comparators, equilibrium, and self-play, including its reparameterization, certificates, and chain characterizations. It also delineates the limits of fixed-scale exactness and regret-to-equilibrium converses, including the role of intrinsic time and strict pure equilibria.

A.1.1 The exponential-family parameterization costs no generality

  • The exponential-family map is bijective onto interior simplex play, with centered states encoding posterior-to-prior log-ratios and scale changes rescaling the required state.A round’s move is the consecutive-posterior log-ratio at constant scale.Source: §A.1.1 · p. 26

A.1.2 The exponential family traced by the inverse temperature

  • The RCGF forms a one-dimensional exponential family whose curvature is Fisher information, while its intrinsic-time density measures the relative-entropy cost of incorporating observations.The appendix connects this interpretation to adaptive Bayes and SafeBayes [7].Source: §A.1.2 · p. 26

A.1.3 The moment expansion behind the RCGF

  • The centered RCGF simultaneously encodes all moments, the constrained min-max value, and the posterior information increment η^2Qη(p,z)=KL(p∥p+).Variance and range proxies arise by truncating the moment expansion, whereas the cumulative identity is Σ_t KL(p_t∥p_t+1)/η_t.Source: §A.1.3 · p. 26

A.1.4 Influence of the centered CGF under a contaminated play

  • Under contaminated play, centered-CGF influence is bounded independently of the number of experts, linearly in score-range depth and exponentially in score height.For z(i)∈[a,b], the bounds are −(1+ηb)≤ξ_η(j)≤e^(ηb)−1−ηa.Source: §A.1.4 · p. 27

A.2.1 Why the value-to-go is a certificate

  • The Bayes normalizer is a Bellman certificate because its one-step decrement is nonnegative on feasible moves and vanishes at nature’s budget-saturating reply.A precommitted scale is generally loose because the terminal dual optimizer is unknown until the final state.Source: §A.2.1 · p. 27

A.2.3 The Fenchel dual: from constrained to regularized

  • Fenchel duality identifies prior-anchored FTRL with Gibbs updates and distinguishes a global prior-based constraint from a local, distribution-dependent RCGF constraint.The local constraint conditions one-round losses on the current posterior, while the global constraint controls the prior-based log-partition function.Source: §A.2.3 · p. 28

A.3 The exact one-round value

  • Exact one-round analysis shows saturation at Γmax, closed forms for saturated budgets and two experts, but multi-round minimax play can over-tilt relative to Gibbs and lacks an elementary closed form.For K=5, η=1, q=0.1, the saturated-budget value is 0.8178 versus envelope log 5+0.1≈1.7094; at T=2, the exact value is 0.6486.Source: §A.3 · pp. 28–30

A.4.3 The two-step gate in closed form

  • The two-step gate is exact in closed form, but it fails near Γmax: T0 jumps from 2 to 3 at Γ = 0.6479, and the gap factor does not persist.The measured T = 1 value is 0.06% below 2 at that scale, 2.0% below at Γ = 1/2, and 0% below for every T ≥ T0.Source: §A.4.3 · p. 31

A.4.4 What a varying scale is worth

  • Predictable scale variation lowers the value by under a tenth of a percent at computed budgets, whereas an unclipped adaptive rule lets nature make the value unbounded.At Γ = 1/2 the reduction is 6×10^-4, and at Γ = 1/5 it is under 10^-6; clipping ηt at 1 prevents the opening-move exploit.Source: §A.4.4 · p. 31

A.4.5 The intrinsic-time loss as a Riemann sum of a falling scale

  • The falling-scale schedule is a left-endpoint Riemann sum that upper-bounds its intrinsic-time integral, with only a largest-increment discretization gap.The bracketing becomes tight whenever no single round carries a macroscopic share of intrinsic time, and the schedule keeps the loss-to-time ratio pinned along the path.Source: §A.4.5 · p. 32

A.5 Classical bounds from the game

  • The game recovers classical concentration: Chernoff’s exponent is the comparator’s minimum relative entropy, Cramér’s rate is exact, and Sanov’s exponent is the same variational dual.Gibbs tilting attains the interior Cramér rate, while finite-state Sanov follows from type counting and the comparator-class dual.Source: §A.5.1 · p. 32, §A.5.2 · p. 32, §A.5.4 · p. 33

A.5.3 Gibbs conditioning and the conditional limit

  • On faces, conditioning selects the I-projection exactly; for general convex constraints, the conditional law converges to the unique I-projection through the Pythagorean inequality.The game interprets this limit as η ↓ 0, when the optimizer’s Gibbs tilt relaxes onto the I-projection.Source: §A.5.3 · p. 33

A.6.1 The sufficient-statistic catalogue; A.6.2 Comparators on and off the minimal-mean face; A.7.1 Opponent models and the robust center; A.7.2 The regret-to-equilibrium conversions

  • The applications catalogue preserves the ledger under alternative sufficient statistics, reduces diffuse comparators to best experts under stated low-noise conditions, and links regret types to CCE and CE.Opponent-model priors combine geometrically into a robust center; external regret implies CCE, while swap regret implies correlated equilibrium.Source: §A.6.1 · p. 33, §A.6.2 · p. 34

A.7.3 Why the ledger supplies no converse

  • The reverse implication is not available, and deciding the T = o(T) claim remains undecided because a lower bound is required but the envelope cannot supply it at that constant.At the headline C, the residual coefficient 2C − C^-1 is exactly zero.Source: §A.7.3 · p. 35
  • Strict pure Nash convergence makes off-equilibrium mass decay geometrically, freezes the scale positively, bounds intrinsic time, and gives d(σT, CCE) = O(1/T).Theorem A.13 requires a stability gap but no assumption on the rate of approach; this improves over the O(√log T/T) rate from merely logarithmic intrinsic time.Source: §A.7.3 · pp. 35–36
  • Whether the realized rate matches the ledger’s bound is undecided, including when T is super-logarithmic under the stated schedule.The super-logarithmic case is described as a convergence-of-dynamics question the paper does not settle [51].Source: §A.7.3 · p. 36
  • A cyclic loss sequence that keeps p(k)t far from any equalizer yields V(k)T = Θ(T), so the ledger recovers the classical O(1/T) bound.Along this trajectory, the ledger is an equality.Source: §A.7.3 · p. 36
  • The ledger reproduces each player’s exact unilateral deviation gain, making d(σT, CCE) one exploitability rather than the Nash duality gap.The instances separate through realized intrinsic time alone.Source: §A.7.3 · p. 36

A.8 Further results from Section 7

Section 7’s extensions give exact pathwise bandit decompositions, closed-form information costs, comparator-complexity reductions, and a testing interpretation of the concentration game. These identities unify exploration, feedback geometry, policy duplication, and Chernoff exponents within the same Bellman framework.

Generic bandit decomposition

  • The generic pathwise bandit decomposition separates sampled regret into estimation and predictable-bias martingales, bias, exploration, comparator, and full-information terms.Both martingale terms are explicit, while the final three terms form the full-information game on estimated scores.Source: §A.8 · p. 36

Exact per-round value

  • The estimated per-round RCGF has an exact finite-sum form in true losses and observation geometry, yielding propensity inflation for plain bandits and observation-probability geometry for feedback graphs.Its small-scale limit is the observation-inflated variance, and the identity holds at every scale η_t.Source: §A.8 · p. 36

Proper-wrapper duplication

  • Duplicating loss-identical policy experts pools their prior mass, reducing comparator complexity exactly to log 1/˜π(G), with the reduction equal to the duplicated-to-original mass log-ratio.In the explicit-exploration wrapper, bias vanishes, exploration costs at most γT, and the comparator term uses the duplication-adjusted mass rather than log |Π|.Source: §A.8 · p. 37

Pairwise testing exponents

  • Pairwise testing exponents are Bellman potentials: the Gibbs equalizer is the Chernoff tilt, scale optimization yields Chernoff information, and Γ(a) equals the MAP identification exponent.At the optimizing tilt, the score is neutral in nature’s direction, and Γ(a) is the edge-restricted coincidence radius of the arm-induced laws.Source: §A.8 · p. 37

Algorithmic specializations

  • Estimator-specific bandit methods differ only through decomposition terms: exploration enters Ξ_T, implicit exploration changes conditional bias, offsets alter intrinsic-time and jump terms, and graph topology enters through observation ratios.For feedback graphs, undirected observation ratios are bounded by the independence number α(G_t); stochastic gaps can make realized intrinsic time stop accumulating for good actions.Source: §A.8 · p. 38

A.9 Further results from Section 8 … B.1 Weighted entropy and multiscale experts

The appendices extend the concentration-game framework to sampled play, boosting and pooling identities, exact relaxation errors, residual finite-horizon values, and alternative geometries. They show posterior sampling is the exact expected-game equalizer, while realized sampling fluctuations and geometry-specific transport determine the remaining limitations and generalizations.

A.9 Further results from Section 8

  • Posterior sampling is the unique exact equalizer for the expected sampled concentration game, attaining the deterministic value, while any non-posterior sampling law strictly loses when some budget is positive.The expected play excess vanishes under every predictable sampling law, but the Gibbs sampler uniquely equalizes nature’s direction and saturates the value; all sampling laws coincide only when every budget is zero.Source: §A.9 · pp. 38–39
  • Realized posterior sampling inevitably retains a comparator-independent mean-zero martingale, and no payoff genuinely depending on sampled play can eliminate its per-round sampling variance.For high-probability objectives, the variance optimizer generally differs from posterior sampling: a Gaussian approximation yields a quantile-minimizing law with β* ≠ 1.Source: §A.9 · p. 39

A.10.1 The quantile-margin shortfall in full

  • The pressure ledger recovers AdaBoost’s normalizer-minimizing coefficient and classical exponential-loss rate, while bounding quantile-margin shortfall by 2 log(1/ε)/T plus additive O(log(1/ε)/T) edge terms.For binary weak hypotheses, the pressure-optimal coefficient is α*_t = 1/2 log((1−ε_t)/ε_t), and the exact ledger precedes the usual variance and range relaxations.Source: §A.10.1 · pp. 39–40

A.11.1 The identities read in neighboring languages

  • The neighboring-language identities interpret retempering drift as a log-partition derivative and connect committed and reactive games to dual stochastic control on an information manifold.The framework also organizes algorithmic relaxations by rate, comparator complexity, statistics, observations, budgets, and retempering, with bandit methods trading estimation, martingale, and entropic-information terms.Source: §A.11.1 · p. 40

A.11.2 The exact-value residual at two experts

  • For two experts, the one-round value has a closed arcosh form, but the multi-round exact value requires a state-dependent reachable-set recursion and generally lacks an elementary closed form.The finite-horizon supremum remains finite because each centered RCGF is second order in nature’s step, while the recursion’s optimum over-tilts the Gibbs iterate.Source: §A.11.2 · p. 40

A.12 Logarithmic pooling as the same coincidence game

  • Online logarithmic pooling is the same coincidence game on a predictive-distribution simplex, yielding a benchmark stronger than the best convex combination through an explicit nonnegative coincidence discount.The pooling weights can themselves be learned with the same entropic game.Source: §A.12 · p. 41

A.13 Exact relaxation remainders

  • Exact relaxation remainders quantify the nonnegative Hoeffding slack incurred when replacing the true centered RCGF with variance or bounded-range proxy potentials.The proposition expresses these remainders through cumulants and centered coincidence rates, identifying the algorithm’s exact trajectory-wise gain from retaining the RCGF.Source: §A.13 · p. 41

B Other geometries and domains; B.1 Weighted entropy and multiscale experts

  • The same Gibbs equalizer and exact concentration bookkeeping extend beyond the simplex to weighted entropy, continuum priors, and compact convex domains, with multiscale experts using expert-specific transport divergences.Weighted entropy makes each expert’s regret scale with its loss range, while tree-conditioned refinements also support dynamic comparators in metrical task systems.Source: B Other geometries and domains · p. 41, §B.1 · pp. 41–42

B.2 Continuum priors and forgetting mixtures · B.3 Continuous-action online convex optimization

The paper extends its concentration geometry to continuum hyperparameters, turning forgetting-rate priors into mixtures of memory kernels and yielding multi-prior PAC-Bayes penalties. It also transfers the framework to continuous-action online convex optimization, obtaining density-regret and point-comparator bounds while noting dimension dependence as a limitation.

B.2 Continuum priors and forgetting mixtures

  • A prior over forgetting rates is exactly a prior over memory kernels, bringing hyperparameter averaging inside the concentration geometry rather than treating it as outer model selection.This is presented as the continuum version of mixed coincidence.Source: §B.2 · p. 42
  • Geometric pooling across expert classes or hyperparameter settings produces an exact geometric mixture of their action distributions.The pooled distribution is defined across a hyperparameter continuum.Source: §B.2 · p. 42
  • The same pooling yields a multi-prior PAC-Bayes variational formula with penalty Σ_w α_w KL(ν∥π_w) − C_α(π_1:W), where C_α is the coincidence term.The construction applies whenever the original variational formula depends on the prior only through KL(ν∥π), with potential uses in domain adaptation, meta-learning, and transfer.Source: §B.2 · p. 42

B.3 Continuous-action online convex optimization

  • On compact convex S ⊂ R^d, exponential densities p_t(dx) ∝ e^−η_tF_{t−1}(x)π(dx) extend the same chain from sums to integrals.The resulting density regret is evaluated against posterior measures ν absolutely continuous with respect to π.Source: §B.3 · p. 42
  • Convexity of the losses converts density regret into ordinary online convex optimization regret for the barycenters x_t.This provides the continuous-action link between posterior-measure analysis and standard comparator regret.Source: §B.3 · p. 42
  • For an interior point comparator and uniform prior, geometric shrinking gives KL(ν_{ε,x⋆}∥π) = log(1/ε), while the regret is bounded by log(1/ε) + Q*_oco.With ε = 1/T and C = 1, the bound becomes d log T + Q*_oco.Source: §B.3 · p. 42
  • The resulting rate Reg_T(x⋆) = O(√dT log T) is mainly structural, since projection-based and dual-averaging methods achieve dimension-independent rates using L2 geometry instead.The analysis also uses Q*_oco_T ≤ T/8.Source: §B.3 · p. 42

C Continuous-time limit and Itô-level treatment of the retempering drift … D.2 Proofs from Section 3

This section develops the continuous-limit and Itô-level treatment of retempering drift and supplies proofs for the paper’s main identities, bounds, and concentration results. It also establishes the matching lower bound, characterizes when the upper bound is strict, and quantifies the benchmark ratio κ(Γ), while clarifying the role of nonuniform priors in Gibbs play.

C Continuous-time limit and Itô-level treatment of the retempering drift

  • The retempering drift converges to a relative-entropy path integral, with warming favorable and the small-η integrand reducing to 1/2 Var_π(¯C(s)).The drift is a Riemann–Stieltjes integral of prior-to-posterior relative entropy, and its schedule optimization has a deterministic optimal-control interpretation.Source: C Continuous-time limit and Itô-level… · p. 43
  • The Itô treatment makes the retempering–drift duality exact: the log-partition potential has a closed differential, its loss and scale legs telescope by endpoints, and Gibbs play is the unique equalizer.Finite-variation scales eliminate scale-side quadratic and covariation corrections, leaving a pure Riemann–Stieltjes drift leg and carrying the duality to the Itô level.Source: C Continuous-time limit and Itô-level… · pp. 43–44

D Proofs

  • The proofs establish the two-sided per-round-RCGF envelope, show its gap is exactly the necessary edge terms, and derive fixed-time, anytime, and almost-sure iterated-logarithm concentration forms.The almost-sure mixture argument requires support on the sub-gamma range λ ∈(0, 3), where the exponential process is a supermartingale.Source: D Proofs · p. 44

D.1 Proofs from Section 2

  • The Section 2 proofs derive the Gibbs variational and comparator dual forms, uniqueness of the Gibbs optimizer, and the variance and bounded-range limits for the information budget.The budget satisfies Q_η(p,z) → 1/2 Var_p(z) as η ↓0 and Q_η(p,z) ≤ (b−a)^2/8 when z lies in [a,b].Source: §D.1 · pp. 44–45

D.2 Proofs from Section 3

  • The Section 3 proofs show that Gibbs play saturates the Bellman increment under the information budget, telescopes the potential bound, and yields the equalizing game value.They also establish the fixed-scale comparator KL telescoping identity and the endpoint behavior when the relative-entropy constraint is vacuous.Source: §D.2 · pp. 45–46
  • The lower-bound proof uses Proposition A.3 with Proposition A.5’s response rule, while Theorem 3.2 supplies the relaxation and is not generally minimax under a nonuniform prior.The equalizer is minimax at S = 0 under a uniform prior, but not under a general one.Source: §D.2 · p. 46
  • The bound in (A.16) is strict for every finite scale except η^2q = t⋆ when Γ < log 2; for Γ ≥ log 2, equality is only approached as the scale tends to infinity.For Γ < log 2, the unique equality scale is t⋆ = −(1/2) log((1 + g(Γ))/(1 − g(Γ))).Source: §D.2 · pp. 46–47
  • At η = ηB, the ratio of the value to 2Γ/ηB is κ(Γ) = g(Γ) arcosh(e^Γ)/(2Γ), independent of q and strictly below 1.The ratio approaches 1 as Γ → 0+; at η = ηB, the scale is not the equality scale.Source: §D.2 · p. 47

D.3 Proofs from Section 4 · D.4 Proofs from Section 5

The proofs in Section 4 establish the exact regret telescope, retempering drift, envelope bounds, and Gibbs-chain characterizations underlying the paper’s information-theoretic analysis. Section 5 specializes these identities to constrained comparators, variance envelopes, and stochastic low-noise guarantees, including constant-in-time regret results.

D.3 Proofs from Section 4

  • Theorem 4.3 follows by applying the one-step transport identity round by round and telescoping relative entropy, with retempering producing an additive drift term D_T that is nonpositive for nonincreasing scales.The proof converts the terminal partition term into the comparator transport term and uses the variational form of A_t(η) to establish monotonicity in η.Source: §D.3 · p. 47
  • At constant scale, the constrained regret supremum separates into a transport budget and comparator information, yielding the upper bound η^2V + Γ, with equality exactly when both caps are attained.The second cap is attained if and only if the terminal distribution has relative entropy Γ from the prior.Source: §D.3 · p. 47
  • Gibbs-chain constructions show that feasible centered nature moves can realize the relevant transport totals, whose achievable values form intervals and whose minimum decreases to zero as the number of rounds grows.This establishes the chain-based attainability argument needed for the corresponding bound, including full use of the information budget on a suitable chain.Source: §D.3 · pp. 47–48
  • The intrinsic-time schedule yields matching regret envelopes with unavoidable edge terms: clipping creates the initialization overhead C^2Γ, while a single large jump forces the Q⋆_T term.Both lower-envelope obstructions are witnessed by one-round sequences, so neither overhead can generally be removed.Source: §D.3 · p. 48
  • The active-scale proof identifies the exact per-round coefficient κ_t=b_t/η_t and shows that, at small scale, κ_t = 1/2 Var_{p_t}(z_t) + O(η_t), recovering the variance proxy as a local expansion.The active-scale equation also preserves the exact centered transport identity before approximation.Source: §D.3 · pp. 48–49
  • Variable-scale Gibbs proofs cancel all interior potential terms, leaving endpoint contributions and identifying the constrained regret with the terminal log-partition variational value L_Γ(S_T).The endpoint calculation uses the terminal Gibbs distribution and the comparator supremum over KL(ν∥π) ≤ Γ.Source: §D.3 · p. 49

D.4 Proofs from Section 5

  • Section 5 derives the event-restricted Gibbs variational formula by decomposing KL through the conditional prior π(· | A), with the optimizer given by the corresponding Gibbs tilt.The decomposition separates the cost of restricting support to A from the conditional relative-entropy cost.Source: §D.4 · pp. 49–50
  • For losses in [0,1] and η_t ∈ (0,1], the exponential term satisfies Q_t(c) ≤ (e − 2) Var_{p_t}(c), preserving the variance-based bound.Source: §D.4 · p. 50

D.5 Proofs from Section 6 … D.8 Proofs from Section 9

These proofs develop the paper’s information-theoretic regret framework across sampled, composite-loss, and Gibbs-sampling settings, establishing exact duality, entropy-budget, and martingale identities. They also derive concentration and margin bounds and characterize log-odds dynamics, comparator complexity, benchmark invariance, and pooled log-loss discounts.

D.5 Proofs from Section 6

  • The Section 6 proofs derive exact duality-gap and CCE characterizations from centered-regret decompositions, with pure comparators admissible when their KL complexities fit the budgets.The resulting equalities use no quadratic-variation surrogate; under smaller budgets, only the restricted comparator classes are controlled.Source: §D.5 · p. 52
  • The Section 6 dynamics are analyzed in log-odds coordinates, where strict gaps force geometric deviation decay and finite cumulative regret without assuming bounded intrinsic time.The argument shows the transient length scales as 1/∆, while convergence of the coupled dynamics to a strict profile remains open.Source: §D.5 · pp. 52–53

D.6 Proofs from Section 7

  • The Section 7 proofs regroup sampled composite-loss regret into estimated full-information regret, bias, martingale, and observation terms, with conditional unbiasedness isolating the stochastic component.For inverse-propensity scores, the centered RCGF admits an exact variance identity and a second-order expansion in ηt.Source: §D.6 · pp. 53–54
  • For loss-identical expert copies, the Section 7 proofs replace a designated comparator by the renormalized prior on its equivalence class, reducing complexity to −log of the class’s aggregate prior mass.The benchmark is unchanged because every comparator in the class has the same cumulative estimated score.Source: §D.6 · p. 54

D.7 Proofs from Section 8

  • The Section 8 proofs decompose sampling regret into a full-information term plus a martingale, then apply Freedman and Ville inequalities using the conditional variance clock for fixed-time and anytime bounds.A Gamma-mixture argument supplies the law-of-the-iterated-logarithm refinement.Source: §D.7 · pp. 54–55
  • The mixture-supermartingale argument yields the stated law-of-the-iterated-logarithm bound M_T^sam proportional to sqrt(V_T^sam log log V_T^sam) almost surely for large T.Nonnegative supermartingales, Gaussian-tail calculations, and Ville’s inequality supply the proof.Source: §D.7 · p. 55
  • The Bayesian update telescopes relative entropy into the information-budget bound sum_t η_t(m_t−⟨ν,c_t⟩) ≤ KL(ν∥π) ≤ Γ.Under Thompson sampling, the posterior log-weight residual is a martingale difference, while the prior–posterior ratio exponentiates to an e-process under the usual hypotheses.Source: §D.7 · p. 55
  • The Gibbs sampler uniquely attains the game value, while any non-Gibbs learner permits nature to exceed the benchmark through a feasible direction.At the Gibbs law, nature is indifferent across budget-saturating directions; the equalizer identity makes the value attainable and the minimizer unique.Source: §D.7 · pp. 55–56
  • Sampling randomness generally creates per-round variance, and under a Gaussian martingale approximation the upper-quantile minimizer can use β* ≠ 1 to reduce that variance.This is only a leading-order approximation, not an exact quantile identity; the expected payoff remains stationary at β = 1.Source: §D.7 · p. 56

D.8 Proofs from Section 9

  • Exponential weights satisfy the Donsker–Varadhan identity −log L_exp_T = inf_ν{⟨ν,M_T⟩ + KL(ν∥u_N)}, yielding the stated margin-tail bound by Chernoff’s method.The identity follows by rewriting the exponential update in relative-entropy form and minimizing over the comparator distribution.Source: §D.8 · p. 56
  • The pooled log loss is bounded by weighted expert log loss with a nonnegative coincidence discount, while binary pressure optimization gives the stated γ_t-dependent bound.The discount vanishes exactly when all non-null experts have identical predictive distributions, not merely when they agree on the observed outcome.Source: §D.8 · p. 56

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