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No-Regret Language Generation

Jiaxun Li, Vinod Raman, Ambuj Tewari

arXiv:2610.23641v1cs.LGstat.ML

TL;DR

Finite-last-mistake guarantees can be too rigid for generation, which motivates controlling mistakes uniformly over finite horizons. The paper characterizes realizable and agnostic no-regret through mistake-forcing dimensions and shows that both settings admit a broad range of sublinear rates.

  • Problem

    A finite last mistake can be too rigid because a generator may make occasional mistakes indefinitely even as their frequency tends to zero.

    Source: §1 · p. 3

  • Method

    The paper introduces tree-based mistake-forcing dimensions that characterize finite-horizon minimax complexity in realizable and agnostic generation.

    Source: §1.1 · p. 5

  • Results

    Up to constant factors, essentially every admissible nondecreasing profile can occur as the optimal realizable mistake rate and agnostic regret rate.

    Source: §1.1 · p. 6

  • Takeaways & Limitations

    Sublinear finite-horizon mistake bounds can support anytime no-regret generation even when no finite total-mistake bound exists.

    Source: §7 · p. 22

    The paper imposes no computational constraints, and evaluating its trajectory-tree characterizations for concrete classes may be difficult.

    Source: §7 · p. 23

Abstract

from arXiv · show

We introduce no-regret generation, which requires a generator to make a sublinear number of mistakes uniformly over all demonstration sequences and compatible target languages. We also formulate an agnostic variant that allows arbitrary demonstrations that may not be compatible with any languages. For both settings, we introduce tree-based mistake-forcing dimensions that exactly characterize finite-horizon minimax complexity and establish necessary and sufficient conditions for no-regret generation. Our characterizations reveal a rich spectrum of generation rates: up to constant rescaling, every admissible nondecreasing profile is attainable in both the realizable and agnostic settings. Finally, we connect generation complexity to online prediction: for language classes of finite Littlestone dimension \(d\), we obtain realizable mistake bounds of \(O(d\log(1+T/d))\) and agnostic regret bounds of \(O(\sqrt{dT\log(1+T/d)})\). The realizable bound is optimal up to universal constants.

1 Introduction

The paper asks for finite-horizon guarantees that hold uniformly over compatible targets, rather than generation guarantees whose convergence rate can depend on the target and enumeration. It introduces mistake-forcing dimensions to characterize no-regret generation, its range of possible rates, and its quantitative connection to online prediction.

1 Introduction

  • A compatible target can force T mistakes at horizon T even when the class is generatable in the limit, motivating uniform finite-horizon control.No-regret generation instead allows mistakes to continue as long as their frequency vanishes.Source: §1 · pp. 3–4

1.1 Main contributions

  • The paper introduces realizable and agnostic no-regret generation, with agnostic regret comparing generated errors against demonstration errors for each language.When a language contains every demonstration, that regret reduces to the realizable mistake count.Source: §1 · pp. 4–5
  • Mistake-forcing dimensions exactly characterize optimal finite-horizon mistakes, uniform total-mistake bounds, and agnostic regret, yielding necessary and sufficient no-regret conditions.The realizable and agnostic criteria are sublinear growth of their respective dimensions; uniform mistake-bounded generation corresponds to a finite uniform dimension.Source: §1.1 · p. 5
  • Up to constant factors, essentially every admissible nondecreasing sublinear profile can occur as the optimal rate in both realizable and agnostic generation.The attainable profiles include logarithmic, polynomial, and arbitrarily slow sublinear rates.Source: §1.1 · pp. 5–6
  • Finite Littlestone dimension connects online prediction to generation: such classes are both realizably and agnostically no-regret generatable.The paper frames this as a quantitative comparison despite generators never observing whether their outputs are valid.Source: §1.1 · p. 6, §1.2 · p. 7

1.2 Related work

  • Prior generation work studies validity, breadth, density, noise, or resource limits; this paper instead focuses on how mistakes accumulate over time.Its agnostic model compares output loss with demonstration loss under each candidate language, without a target or noise budget.Source: §1.2 · pp. 6–7

2 Preliminaries

The preliminaries define the language classes, generation game, and success criteria used to study finite-horizon generation. They distinguish compatible demonstrations from arbitrary sequences and define realizable mistakes and agnostic regret accordingly.

2.1 Language classes

  • A language is represented by a binary hypothesis on a countable example space, and the UUS condition ensures another unseen positive example remains after any finite sequence of distinct positive demonstrations.The positive version space consists of hypotheses that label every observed example positive.Source: §2.1 · p. 7

2.2 Generation game and trajectories

  • In each round, a deterministic anytime generator outputs an example not previously demonstrated, then the adversary reveals a fresh demonstration; demonstrations are pairwise distinct.The generator moves first and may reuse an output until it is revealed as a demonstration.Source: §2.2 · pp. 7–8

2.3 Realizable and agnostic generation

  • Realizable plays require a language containing all demonstrations, whereas agnostic plays allow demonstrations that no language in the class jointly realizes.Agnostic regret compares generated examples outside a language with demonstrations outside that same language; when it realizes all demonstrations, regret equals the mistake count.Source: §2.3 · p. 9

2.4 Success criteria

  • Realizable and agnostic no-regret generation require one deterministic anytime generator whose respective cumulative loss grows sublinearly, uniformly over the indicated targets and sequences.A finite uniform mistake bound implies realizable no-regret, and agnostic no-regret implies realizable no-regret; the converse implications need not hold.Source: §2.4 · p. 10

3 Realizable No-Regret Generation

The section introduces mistake-forcing dimensions for adversarial generation and uses them to characterize realizable no-regret generation, uniform mistake bounds, and finite-horizon complexity. It also shows that realizable finite-horizon mistake profiles span a broad range of sublinear rates.

3.1 Mistake-forcing dimension

  • Mistake-forcing trees model generator-first, adversary-response interactions; horizon-T and well-founded versions distinguish common-time from path-dependent finite stopping guarantees.Source: §3.1 · pp. 10–12

3.2 Characterizations of realizable no-regret generation

  • For nonempty UUS classes, realizable no-regret generation is characterized by MFT(H)=o(T), while finite MFunif(H) exactly captures a uniform finite total-mistake bound.Source: §3.2 · pp. 12–13

3.3 Finite-horizon versus uniform mistake bounds

  • Finite-horizon guarantees need not yield a uniform total-mistake bound: the gap can be arbitrarily large, with mistakes accumulating across finite blocks.For each fixed horizon, the construction can make at most one mistake, while an anytime generator may accumulate mistakes across blocks.Source: §3.3 · pp. 13–14

3.4 The spectrum of realizable no-regret rates

  • Up to constant rescaling, realizable generation admits essentially every admissible sublinear finite-horizon mistake profile.Source: §3.4 · pp. 14–15

4 Agnostic No-Regret Generation

Agnostic generation compares generated mistakes with each language’s mistakes on the same trajectory, so regret can rise or fall. Signed mistake-forcing trees characterize its finite-horizon complexity and no-regret behavior, while possible rates range from matching realizable rates to linear separation.

4.1 Signed mistake-forcing dimension

  • A common finite horizon is essential because agnostic regret can increase or decrease over time, unlike the nondecreasing realizable mistake count.For this reason, the paper does not introduce a well-founded analogue of the signed dimension.Source: §4.1 · p. 16
  • Signed mistake-forcing trees exactly characterize finite-horizon minimax agnostic regret using signed patterns that track invalid generated examples and invalid demonstrations.The signed score subtracts invalid demonstrations from invalid generated examples; when demonstrations are all valid, it reduces to the ordinary mistake count.Source: §4.1 · pp. 15–16, §4.2 · p. 16

4.2 Characterizations of agnostic no-regret generation

  • The signed dimension gives the optimal worst-case regret at each horizon, and its profile characterizes agnostic no-regret generation.The optimal horizon-specific generator may depend on T, but this does not create a gap: a single deterministic anytime generator can satisfy the guarantee simultaneously across horizons.Source: §4.2 · p. 16

4.3 Realizable versus agnostic generation

  • Agnostic no-regret generation implies realizable no-regret generation, but the converse can fail: Hpair has constant realizable mistakes and regret at least T/3.When a comparator contains every demonstration, its demonstration loss is zero, so signed regret reduces to the generator’s realizable mistake count.Source: §4.3 · p. 17

4.4 The spectrum of agnostic no-regret rates

  • Agnostic profiles can range from matching realizable rates to linear separation, and essentially arbitrary sublinear profiles are attainable up to constant rescaling.The paper also identifies examples including logarithmic and power-law profiles, with 0 < α < 1 and β > 0.Source: §4.4 · pp. 17–18

5 Comparison with Other Generation Criteria

The paper compares generation criteria, clarifies how agnostic guarantees transfer to noisy generation, and distinguishes the guarantees available for finite classes from those implied by countability. Its examples show that countability alone need not yield sublinear mistake or regret profiles.

5.1 Comparison of realizable generation criteria

  • The implication diagram for realizable generation criteria is complete: every arrow is strict, and criteria without a directed path between them are incomparable.The criteria differ in what aspect of mistakes they control and whether guarantees hold uniformly or separately across targets and demonstration sequences.Source: §5.1 · pp. 18–19

5.2 Noisy generation versus agnostic generation

  • An agnostic regret guarantee transfers to noisy generation: sublinear regret plus a sublinear number of invalid demonstrations yields sublinear invalid generated examples relative to the target.The noisy model constrains demonstrations outside a target language, whereas the agnostic model compares against every language in H without requiring a compatible target.Source: §5.2 · p. 19

5.3 Finite and countable classes

  • Finite classes have uniform realizable mistake bounds and sublinear agnostic regret, but countability alone guarantees neither sublinear profile.The countable class Hcof is non-uniformly generatable yet has both realizable and agnostic profiles equal to T, so neither no-regret guarantee holds.Source: §5.3 · pp. 19–20

6 From Prediction to Generation

The paper relates online prediction complexity to language-generation complexity, using Littlestone dimension to bound realizable mistakes and agnostic regret. It also establishes that the realizable bound is sharp up to universal constants, while finite VC dimension alone does not guarantee no-regret generation.

6 From Prediction to Generation

  • Finite Littlestone dimension controls both realizable and agnostic generation complexity, connecting prediction complexity to generation despite their different feedback.The section frames the connection against prior work showing that online learnability and generatability are qualitatively incomparable.Source: §6 · p. 20, §6.1 · p. 20

6.1 Online prediction and Littlestone dimension

  • Littlestone dimension measures the greatest depth of a binary instance tree whose every edge-label path is realized by a hypothesis in H.In online prediction, the learner predicts a label before observing the true label; the realizable and agnostic settings differ in whether labels share a target in H.Source: §6.1 · p. 20

6.2 Upper bounds from finite Littlestone dimension

  • Finite-Littlestone upper bounds yield sublinear realizable mistakes and agnostic regret, so every UUS class with finite Littlestone dimension is no-regret generatable in both settings.For agnostic generation, the proof controls the difference between the language’s mistakes on demonstrations and its mistakes on generated examples.Source: §6.2 · pp. 21–22
  • Finite VC dimension is insufficient: a countable UUS class with VCdim(H) = 1 has MFT(H) = SMFT(H) = T for every T ≥ 1.Thus VC dimension one still permits linear generation regret.Source: §6.2 · p. 22

6.3 Sharpness in the realizable setting

  • For every integer d ≥ 2, a countable UUS class demonstrates that the realizable upper bound is optimal up to universal constants as a function of d and T.The construction combines independent finite blocks. No corresponding sharp lower bound is known for agnostic generation at fixed finite Littlestone dimension.Source: §6.3 · p. 22

7 Conclusion and Future Work

The conclusion separates mistake counts from when mistakes stop, and distinguishes realizable performance from agnostic regret. It identifies unresolved questions about evaluating the characterizations, agnostic bounds, and computational efficiency.

7 Conclusion and Future Work

  • A class can have at most one mistake at each fixed horizon yet no finite total-mistake bound for any single anytime generator; sublinear finite-horizon bounds still suffice for anytime no-regret.A constant realizable mistake bound also need not imply sublinear agnostic regret.Source: §7 · p. 22
  • Open questions include evaluating trajectory-tree characterizations for concrete classes, finding optimal agnostic bounds in d and T, and achieving minimax guarantees efficiently.The present results impose no computational constraints.Source: §7 · p. 23

AI Disclosure

The authors report that ChatGPT 5.6 Sol contributed to technical ideas and writing, while the authors take responsibility for the paper’s content.

AI Disclosure

  • ChatGPT 5.6 Sol assisted with the paper’s technical ideas and writing; the authors take full responsibility for its content.Source: AI Disclosure · p. 23

A.1 Proofs for Section 3.2

These appendix proofs develop the finite-horizon and anytime-generator arguments behind the paper’s realizable characterizations, then give a block-based construction for uniform mistake bounds. The proofs use safe choices to preserve non-forcing trajectories and epoch schedules to turn horizon-specific guarantees into anytime guarantees.

A.1 Proofs for Section 3.2

  • The proof of the finite-horizon lower bound follows a forcing tree for T rounds, then selects a feasible mistake pattern and a witness language at its leaf.The resulting play has at least the forced number of mistakes.Source: §A.1 · p. 25
  • A forcing tree supplies a lower-bound strategy against any generator, while safe outputs preserve a non-forcing invariant and yield an anytime generator attaining the uniform mistake bound.The proof extends a finite adversarial path to an infinite realizable sequence using fresh positive examples.Source: §A.1 · pp. 26–27
  • A horizon-dependent mistake allowance can be converted into one anytime generator by running horizon-specific generators in epochs, with only a constant-factor increase in the bound.The epoch budgets double, so the accumulated cost of completed epochs is bounded by the scale of the latest epoch.Source: §A.1 · p. 28

A.2 Proofs for Section 3.3

  • The block construction gives countable UUS classes Hk with uniform mistake bound k, and H∞ with no finite uniform mistake bound.Each language excludes at most one point per finite block; at a fixed horizon the generator can stay in a sufficiently large block, but an anytime generator must eventually move between blocks.Source: §A.2 · pp. 30–31

A.3 Proofs for Section 3.4

The section constructs countable UUS language classes with specified mistake-forcing profiles. For rescaling-stable profiles, the construction attains the profile up to constant factors and includes several logarithmic and polynomial rates.

A.3 Proofs for Section 3.4

  • The construction produces a countable UUS class Hf for each qualifying profile f.Its languages exclude finite sets E with at most kn = f(sn) excluded points in each block Bn.Source: §A.3 · pp. 31–32
  • The block-and-reserve construction uses blocks of size 4n with capacity kn = f(4n), bounding mistakes within a block while allowing an adversary to force them.The upper-bound generator stays in a sufficiently large block; the lower-bound argument fills that block’s reserve.Source: §A.3 · p. 32
  • MFT(Hf) = Θ(f(T)) when f is stable under constant rescaling.The upper and lower bounds use f(8T) and f(⌊T/4⌋), respectively.Source: §A.3 · p. 33
  • The realizable mistake-forcing profile can have order (log(1 + T))^β, T^α, T^α(log(1 + T))^β, or T(log(1 + T))^β.Here α ∈ (0, 1) and β > 0.Source: §A.3 · p. 33

B.1 Proofs for Section 4.2

The section proves that signed mistake-forcing trees characterize finite-horizon agnostic regret, then uses the characterization to connect no-regret generation to sublinear profiles. It also compares realizable and agnostic complexity and gives a class illustrating their different behavior.

B.1 Proofs for Section 4.2

  • For every nonempty UUS class and horizon T, the minimax agnostic regret equals SMFT(H).The safe generator constructed in the proof attains this minimum.Source: §B.1 · pp. 33–34
  • Agnostic no-regret generation is characterized by SMFT(H) = o(T).The theorem also gives one deterministic anytime generator with regret at most 4b(T) when SMFT(H) ≤ b(T) for every T.Source: §B.1 · pp. 35, 37
  • Fresh horizon-dependent generators are combined across epochs, with zero-regret virtual rounds handling earlier demonstrations and incomplete epochs.Signed regret can decrease, so a horizon guarantee alone does not control every epoch prefix; padding the virtual play preserves its regret while reaching the generator’s guaranteed horizon.Source: §B.1 · pp. 36–37

B.2 Proofs for Section 4.3

  • For every nonempty UUS class, MFT(H) ≤ SMFT(H).The proof maps an ordinary mistake-forcing tree to a signed one by using the pattern (I, ∅).Source: §B.2 · p. 37
  • The pair-language construction gives a realizable anytime generator whose only possible mistakes are its first two outputs.After those rounds, it outputs fresh pair points known to belong to the target language.Source: §B.2 · p. 38
  • For the same pair-language construction, the signed lower-bound argument targets regret of at least T/3 for every deterministic horizon-T generator.It constructs demonstrations with at most 2T/3 invalid points for a comparator while making every generated point invalid.Source: §B.2 · pp. 37–38

B.3 Proofs for Section 4.4

The section constructs agnostic classes with specified signed mistake-forcing profiles and records the generation criteria used to compare limiting notions. The block construction attains rescaling-stable profiles up to constant factors.

B.3 Proofs for Section 4.4

  • The agnostic upper bound adapts the finite-block generator: distinct outputs in one block incur regret no greater than the number of mistakes.The comparator’s loss on demonstrations is nonnegative, so discarding it can only increase regret.Source: §B.3 · p. 40
  • SMFT(Hf) = Θ(f(T)) when f is stable under constant rescaling.The proof combines lower and upper bounds for the constructed class.Source: §B.3 · pp. 39–40
  • Agnostic signed mistake-forcing profiles include (log(1 + T))^β, T^α, T^α(log(1 + T))^β, and T(log(1 + T))^β.Here α ∈ (0, 1) and β > 0.Source: §B.3 · p. 40

C.1 Definitions of the generation criteria

  • The criteria distinguish uniform generation, non-uniform generation, generation in the limit, and zero-rate generation in the limit by their quantifiers over targets, plays, and cutoffs.The definitions use a shared generator; the cutoff may be global, target-specific, or target-and-enumeration-specific, while zero-rate generation requires a vanishing mistake rate.Source: §C.1 · p. 41
  • Realizable no-regret generation requires cumulative mistakes MT(G, h; x1:T) = o(T), while uniform mistake-bounded generation imposes a fixed bound B.Both criteria are stated for every target and every infinite play realizable by that target.Source: §C.1 · p. 41

C.2 Proofs for Section 5.1

This section develops proofs and tools for studying generation criteria and hypothesis-class complexity, including criterion implications and separations, regret transfer under imperfect demonstrations, finite-class bounds, and adaptive binary-tree measures. It also bounds weighted sequential leaf capacity by induction using restrictions on the Littlestone dimensions of the subclasses at each split.

C.2 Proofs for Section 5.1

  • U ⇒ M ⇒ R ⇒ Z and U ⇒ N ⇒ L ⇒ Z, with M ⇒ L; these implications are complete, so the criteria are strict or incomparable exactly as the directed paths indicate.An implication means every nonempty UUS class with the first property has the second; absent paths are witnessed by classes separating the properties.Source: §C.2 · p. 42
  • Hrow has a uniform two-mistake bound but is not non-uniformly generatable, while Htr allows at most one mistake at every horizon yet is not generatable in the limit.These constructions establish separations used to show that N ⇒ L and L ⇒ Z are strict, and that M and N, N and R, and R and L are incomparable.Source: §C.2 · pp. 43–44

C.3 Proofs for Section 5.2

  • When r(T) = o(T), a generator with regret bounded by r(T) produces o(T) invalid examples if the demonstrations have o(T) invalid examples relative to the same target.The result transfers sublinear regret under imperfect demonstrations into sublinear generation mistakes relative to that target.Source: §C.3 · pp. 44–45

C.4 Proofs for Section 5.3

  • For finite H with |H| = N, the proof derives realizable mistakes bounded by ⌊log N⌋ and an agnostic bound involving √(8T log N) for every horizon T ≥ 1.The realizable bound uses an anytime generator; the agnostic argument combines a regret bound with the finite-horizon characterization.Source: §C.4 · p. 45
  • The countable UUS cofinite class Hcof is non-uniformly generatable but has MFT(Hcof) = T, ruling out both realizable and agnostic no-regret generation.The proof constructs a realizable length-T play on which an arbitrary horizon-T generator makes T mistakes.Source: §C.4 · p. 46

D.1 Weighted sequential leaf capacity

  • Weighted sequential leaf capacity is the maximum sum of leafwise maximum hypothesis weights over adaptive binary trees of depth n; with unit weights, it counts the maximum number of nonempty leaves.At each node, a query splits hypotheses by label, and later queries may depend on the branch.Source: §D.1 · p. 47
  • At any split z, both subclasses have dimension at most d but cannot both have dimension d; induction and Pascal’s identity bound depth-n capacity, and taking the supremum over z completes the proof.The argument applies the induction hypothesis to both depth-(n −1) capacities.Source: §D.1 · p. 48

D.2 Proof of the realizable upper bound

The proofs control capacity across two tree levels per generation round, then convert finite-horizon bounds into anytime guarantees for realizable mistakes and agnostic regret.

D.2 Proof of the realizable upper bound

  • Each realizable round consumes two capacity-tree levels; the generator's choice keeps remaining capacity from increasing despite the adversary's response.After T rounds, the capacity bound controls the weight of any compatible hypothesis on which the generator made mistakes.Source: §D.2 · pp. 48–49
  • O(d log(1 + T/d)) bounds the realizable mistake count for fixed Littlestone dimension d, and the finite-horizon envelope yields an anytime generator.For fixed d, the bound is O(d log T) = o(T).Source: §D.2 · p. 50

D.3 Proof of the agnostic upper bound

  • For agnostic generation, no comparator is discarded after a demonstration; multiplicative weight updates encode its performance relative to the generator.The two-level capacity argument controls the updated weights and supports the regret bound.Source: §D.3 · pp. 51–53
  • The agnostic proof controls comparator regret without removing hypotheses, then converts its finite-horizon envelope into one anytime generator.Comparator weights increase when the comparator outperforms the generator and decrease when it performs worse.Source: §D.3 · pp. 51, 52, 54

E Proof of Theorem 6.3

This section proves Theorem 6.3 by constructing examples that yield a lower bound proportional to \(d\log(1+T/d)\), then matching it with an upper bound.

E Proof of Theorem 6.3

  • For the constructed class Cm, Littlestone dimension is 1; the finite-horizon characterization yields MFK−1(Cm) ≥m, with at least m mistakes forced within K−1 rounds.A candidate’s weight doubles whenever the generator makes another mistake for that candidate.Source: E Proof of Theorem 6.3 · pp. 54–56
  • Combining q independent copies yields finite UUS classes with Littlestone dimension q and forces qm mistakes within 2q(2m−1) rounds.The product adversary runs the single-component strategy independently in each coordinate and freezes resolved coordinates; each final coordinate contributes at least m mistakes.Source: E Proof of Theorem 6.3 · pp. 57, 61
  • For every d ≥ 2 and T ≥ 3, a countable UUS class of Littlestone dimension d has lower bound max qm over 1 ≤ q ≤ d−1, m ≥ 1 with 1 + 2q(2m−1) ≤ T.The proof places marked copies of the component constructions for all pairs (q, m) into one countable class, letting the adversary choose a scale for the horizon.Source: E Proof of Theorem 6.3 · pp. 54, 60, 61
  • The lower-bound construction yields q m at least a universal constant multiple of d log(1 + T/d), combining two horizon regimes.For 3 ≤ T ≤ 2d − 1, it first gives q m = q at least a constant multiple of the desired quantity.Source: E Proof of Theorem 6.3 · p. 61
  • The matching upper bound follows from Theorem 6.1 and the previously established identity for Ldim(Hsharp).Together, the lower and upper estimates complete the proof.Source: E Proof of Theorem 6.3 · p. 61

F Finite VC Dimension Does Not Imply No-Regret Generation

A countable class of infinite binary-tree paths has VC dimension one, yet an adversary can force every generated example to be invalid. This rules out finite VC dimension alone as a guarantee of no-regret generation.

F Finite VC Dimension Does Not Imply No-Regret Generation

  • Finite VC dimension does not imply either realizable or agnostic no-regret generation.The exhibited countable class has VC dimension one, while the adversary can force a mistake on every round.Source: F Finite VC Dimension Does Not Imply… · p. 62
  • The counterexample class consists of infinite paths in a binary tree, with examples valid precisely when they are prefixes of a path.Its hypotheses are eventually-zero infinite sequences, making the class countable and UUS.Source: F Finite VC Dimension Does Not Imply… · p. 62
  • The adversary extends a demonstrated path so each fresh generator output is incompatible with it, while the demonstrations remain jointly realizable.After T rounds, one eventually-zero path is a single target witnessing all T mistakes.Source: F Finite VC Dimension Does Not Imply… · pp. 62–63
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