An Information-Theoretic Definition for Open-Ended Learning
Wanqiao Xu, Yifan Zhu, Benjamin Van Roy
Abstract
A growing body of work points to the great promise of AI systems that can continually expand their capabilities as they operate in an open-ended environment. But yet there is no coherent definition of open-endedness or theory about how an agent ought to explore an open-ended environment. We introduce an information-theoretic definition based on a new concept -- the -- which quantifies the information required to attain each level of expected reward. We consider an environment to be open-ended if an agent can attain linear growth in the bit-equivalent. We establish that classical bandit environments are not open-ended and formulate a bandit environment that is. We also introduce an algorithm that achieves open-ended learning in this environment.
AI Impact Assessments
(1 models)Scientific Impact Assessment
Core Contribution
This paper addresses a fundamental gap in the open-ended learning literature: the absence of a rigorous, quantitative definition of what makes an environment "open-ended." The authors propose an information-theoretic framework built on the bit-equivalent — the minimum mutual information about the environment parameter θ required to achieve a given level of expected reward. An environment is deemed open-ended if some agent can achieve linear growth in the average bit-equivalent over time, meaning the agent must continually acquire useful information to sustain performance improvement.
This is a genuinely novel conceptual contribution. Prior definitions (Sigaud et al., 2024; Hughes et al., 2024) focused on novelty and learnability of artifacts from an observer's perspective, which captures behavioral diversity but not whether information acquisition translates into capability improvement. The bit-equivalent directly ties information to performance, making it a more operationally meaningful criterion.
Methodological Rigor
The paper is mathematically rigorous and well-structured. The proofs are clean and complete. Key results include:
1. Lemma 4 elegantly connects information gain to the bit-equivalent via the data processing inequality, establishing that sublinear information gain is sufficient (but not necessary) for non-open-endedness.
2. Theorems 5-9 systematically rule out classical bandit environments — finite-armed, finite-dimensional linear, Gaussian process, and infinite-armed bandits with i.i.d. means — as open-ended. The distinction between environments with sublinear information gain (trivially non-open-ended) and those with linear information gain but bounded bit-equivalent (Theorems 8, 9) is particularly insightful, demonstrating that raw information acquisition ≠ useful information acquisition.
3. Theorem 14 constructively shows open-ended learning via truncated Thompson sampling (TTS) in the insatiable linear bandit, with a carefully designed epoch-based truncation schedule. The proof leverages existing finite-dimensional TS regret bounds and builds quadratic cumulative reward growth, implying linear bit-equivalent growth.
4. Theorem 15 provides a matching upper bound, showing the TTS rate is optimal.
The negative results on classical Thompson sampling (Theorems 11-12) and fixed truncation (Theorem 13) are well-crafted, showing that naïve approaches fail in qualitatively different ways — TS produces invalid actions, while fixed truncation bounds the information extractable.
One subtle strength: Lemma 18's proof uses the Donsker-Varadhan variational principle and Gaussian KL lower bounds to establish that reward growth necessitates proportional information, which is the linchpin connecting cumulative reward to the bit-equivalent.
Potential Impact
Theoretical impact: This paper provides the first formal framework for reasoning about open-endedness that is grounded in information theory and operational performance. This could catalyze a rigorous theory of open-ended learning, analogous to how regret definitions shaped bandit theory. The bit-equivalent concept could extend beyond bandits to MDPs, multi-agent systems, and evolutionary frameworks.
Practical implications: While the current analysis is restricted to bandits, the conceptual insight — that open-ended agents should pursue sequences of learning targets of increasing complexity — has direct relevance to curriculum learning, progressive neural network training, and self-improving AI systems. The connection to satisficing Thompson sampling and rate-distortion theory opens algorithmic design avenues.
Broader influence: The paper's classification of what is *not* open-ended is arguably as valuable as the positive results. Establishing that infinite action sets and unbounded rewards are insufficient for open-endedness challenges common intuitions and redirects research toward structural properties (like correlated arms and non-summable spectral tails) that enable sustained information acquisition.
Timeliness & Relevance
This work is extremely timely. With the explosion of interest in self-improving AI agents (foundation models, autonomous scientific discovery, open-ended code generation), the field urgently needs rigorous definitions to distinguish genuine open-ended capability growth from superficial novelty generation. The paper directly responds to position papers (Hughes et al., 2024) calling for formal definitions and fills a recognized theoretical vacuum.
Strengths
1. Conceptual clarity: The bit-equivalent elegantly separates "how much reward" from "how much useful information is needed for that reward," capturing the essence of open-endedness.
2. Comprehensive negative results: The systematic exclusion of classical environments provides a clear taxonomy and prevents trivial claims of open-endedness.
3. Constructive positive result: The insatiable linear bandit and TTS algorithm demonstrate the definition is achievable, not vacuous.
4. Tight bounds: The matching Ω(T) lower and O(T) upper bounds on the bit-equivalent show the definition is well-calibrated in this setting.
5. The logistic bandit variant (Theorem 21) preempts the objection that open-endedness requires unbounded rewards.
Limitations
1. Bandit-only scope: The restriction to bandit environments is significant. Real open-ended environments involve state, sequential decision-making, and non-stationarity. The authors acknowledge this but provide no roadmap for extension.
2. Constructed environment: The insatiable linear bandit, while mathematically elegant, is artificial. It's unclear whether natural environments exhibit the spectral properties (non-summable eigenvalues) required for open-endedness under this definition.
3. Algorithm design fragility: TTS requires a prescribed truncation schedule — essentially a curriculum — undermining claims of autonomous open-ended learning. The authors acknowledge this limitation.
4. Definition sensitivity: The choice of linear growth in bit-equivalent as the threshold for open-endedness is somewhat arbitrary. Why not superlinear? The paper doesn't explore robustness to this choice.
5. Single environment model: The definition assumes a fixed (though unknown) θ. Truly open-ended environments may involve non-stationary or adversarial dynamics, which this framework doesn't address.
Overall Assessment
This is a strong foundational theory paper that makes a precise, original contribution to an important and under-formalized area. The work is technically sound, conceptually clear, and well-positioned relative to prior art. Its main limitation is narrow scope (bandits only), but the ideas appear extensible and the framework provides a solid starting point for future theoretical development.
Generated Jun 9, 2026
Comparison History (19)
Paper 2 has higher likely impact due to strong real-world applicability (practical, portable LLM inference megakernel generation), demonstrated end-to-end correctness, extensive empirical validation across architectures, and a reusable statically-checked harness enabling safe agent-driven optimization. Its contributions are timely for LLM deployment and span systems, compilers, GPU programming, and AI tooling. Paper 1 is conceptually novel and potentially foundational, but its impact is less immediately verifiable and appears scoped to a constructed bandit setting; broader adoption depends on subsequent empirical and theoretical development.
Paper 2 is likely higher impact due to its foundational scope: it proposes a formal, information-theoretic definition of “open-endedness” (a widely discussed but weakly defined concept), characterizes when environments are open-ended, and provides a constructive example and algorithm. This can influence multiple subfields (RL theory, continual learning, exploration, AI safety) and set common benchmarks/metrics. Paper 1 is a strong, practical method for long-horizon credit assignment with clear applications, but its contribution is more incremental and narrower to outcome-based RL/agentic LLM training.
Paper 1 introduces a principled mathematical framework (TNOs) that unifies and extends neural operators to topological domains using Discrete Exterior Calculus, with demonstrated empirical improvements on PDE benchmarks. It has immediate practical applications in scientific computing and physics-informed ML, strong methodological rigor, and subsumes existing methods. Paper 2 offers a valuable theoretical contribution defining open-ended learning via information theory, but its scope is narrower (bandit environments), more preliminary, and lacks broad empirical validation. Paper 1's combination of theoretical depth, practical utility, and unifying perspective gives it higher near-term impact.
Paper 1 offers a more foundational contribution: a principled information-theoretic definition of open-ended learning (bit-equivalent), a criterion for open-ended environments, and a constructive example plus algorithm. This kind of formalization can reframe a broad research area and influence multiple subfields (RL theory, exploration, continual/open-ended agents). Paper 2 is timely and likely impactful in practice for LLM post-training, but it is a more incremental algorithmic refinement within an active line (PPO/DPPO-style trust-region regularization). Overall, Paper 1 has higher potential for long-term, cross-field scientific impact.
Paper 1 has higher likely impact due to a concrete, novel causal-intervention framework for diagnosing LLM-agent failures with confidence intervals, addressing an immediate, widely felt tooling gap in deployed agent systems. Its methodological contributions (SCM modeling, do-operator replay, contrastive estimator resolving stochastic confounding, Shapley credit assignment) are operationalizable and validated against ground-truth synthetic SCMs, and it is open-sourced—boosting adoption. Paper 2 is conceptually ambitious and potentially broad, but the abstract indicates more limited empirical grounding and unclear applicability beyond a constructed bandit setting, making near-term impact less certain.
Paper 2 is likely to have higher near-term scientific impact: it proposes a practical, broadly applicable training method (DAIL) that directly targets a major bottleneck in LLM reasoning—leveraging scarce expert data despite distribution mismatch—showing sizable empirical gains and sample efficiency with out-of-domain generalization. This is timely and relevant across NLP, alignment, and applied AI. Paper 1 offers a novel theoretical framing of open-endedness via bit-equivalent and a matching algorithm in a constructed bandit setting, but its immediate applicability and demonstrated breadth are narrower and may require more follow-up to influence practice.
Paper 1 provides a foundational, theoretical framework for 'open-ended learning,' a critical and highly relevant frontier in artificial general intelligence (AGI) research. By establishing a rigorous mathematical definition (the 'bit-equivalent') where none existed, it has the potential to broadly influence future reinforcement learning theories, algorithms, and environment designs. In contrast, Paper 2 presents an applied, domain-specific architecture for time series geo-localization. While methodologically sound and practically useful, its impact is narrower compared to the overarching theoretical contributions of Paper 1.
Paper 2 is likely to have higher scientific impact because it proposes a general, information-theoretic definition of “open-endedness” (bit-equivalent) and connects it to provable growth conditions and algorithms. This is a foundational contribution that can influence multiple areas (RL, exploration, lifelong learning, AI safety/AGI discussions) and is timely given current interest in open-ended agents. Paper 1 is strong and practically valuable for biomolecular modeling efficiency, but it is a more specialized, incremental/distillation-focused advance within an already fast-moving application domain.
Paper 1 solves a critical scalability bottleneck in neural fields, demonstrating massive efficiency gains (42x less memory) and strong performance across diverse domains (vision, 3D, climate). Its immediate practical applicability and cross-disciplinary impact give it an edge over Paper 2's theoretical contributions.
Paper 2 offers a foundational, theoretical framework for open-ended learning by introducing a novel information-theoretic metric. While Paper 1 presents a practical and useful architectural improvement for speech emotion recognition, Paper 2 tackles a broader, fundamental problem in general AI. Foundational definitions and theoretical bounds typically have a wider breadth of impact across diverse AI subfields like reinforcement learning, evolutionary computing, and general agent design, giving it a significantly higher potential for long-term, cross-disciplinary scientific impact.