Title: Computable universal online learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: universal learning

computable learners

online learning

computational learning theory

agnostic learning
Abstract: Understanding when learning is possible is a fundamental task in the theory of machine learning. However, many characterizations known from the literature deal with abstract learning as a mathematical object and ignore the crucial question: when can learning be implemented as a computer program? We address this question for universal online learning, a generalist theoretical model of online binary classification, recently characterized by Bousquet et al. (STOC 2021). In this model, there is no hypothesis fixed in advance; instead, Adversary—playing the role of Nature—can change their mind as long as local consistency with the given class of hypotheses is maintained. We require Learner to achieve a finite number of mistakes while using a strategy that can be implemented as a computer program. We show that universal online learning does not imply computable universal online learning, even if the class of hypotheses is relatively easy from a computability-theoretic perspective. We then study the agnostic variant of computable universal online learning and provide an exact characterization of classes that are learnable in this sense. We also consider a variant of proper universal online learning and show exactly when it is possible. Together, our results give a more realistic perspective on the existing theory of online binary classification and the related problem of inductive inference.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis submission studies universal learning from a computability perspective. Universal Learning is a theoretical model of online binary classification where an adaptive Adversary can change the target concept (rather than fixing it ahead of time). Prior work (e.g., Bousquet et al. 2021) characterized universal learnability. This paper adds the constraint that the Learner must be implementable as a computer program. In this case, no characterization was known. The submission also studies proper computable universal learning.\n\nThe submission shows:\n- Universal learnability does not imply computable universal learnability, even for recursively enumerably representable (RER) classes.\n- A characterization for when a hypothesis class is computably learnable in the agnostic setting (here, the Adversary can choose any sequence, not just one belonging to the class).\n- A characterization for when proper computable learning is possible.\n- Separations that indicate that some classes are computably universally learnable but not computably agnostically learnable, and some RER classes are computably universally learnable but not by any computable proper learner.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- I find the paper to be written pretty clearly, even as someone who has not studied universal learning in depth.\n- The paper makes progress on and in some cases (e.g., characterizations) fills what stands as a gap in the literature on the topic.\n- The paper's topic is fundamental: what are the limits of universal learning when we require that the learner be implementable as a program?\n\n**Weaknesses**\n- The paper feels much better aligned with theoretical computer science venues such as COLT, STOC/FOCS, and even CCC (computational complexity conference). While the connection to \"learning\" is clear, in my opinion, the paper doesn't demonstrate why these specific computability results will inspire future research in the typical scope of Neurips (relative to COLT, STOC/FOCS, CCC).\n- The paper motivates the question it studies (we should be able to implement algorithms by a program), but it does not sufficiently motivate why answering this kind of question with computability theory is appropriate.\n- It is hard to interpret example 3 (LLMs and universal learning) since it is stated \"At a high level of abstraction.\"\n\n**Overall:**\nI understand that computability is an established, classical approach to complexity. And I also agree that initial motivation is good, which is that universal learning should be studied with attention to the fact that learning algorithms, for all intents and purposes, should be implemented as a computer program. However, I'm not convinced that characterizing computability (in the sense defined) of universal learning actually answers this question in a practically meaningful way.\n\nQUESTIONS:\n1. How does randomized sampling on LLMs fit into example 3, if at all?\n2. Proposition 1 shows that even DR (decidably representable) classes might not be computably universally learnable. Is there some intuition on what aspects of a DR class leads to the noncomputability of a universal learner?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work investigates when learners that achieve objectives such as universal learning (i.e., making only finitely many mistakes in the limit) can be made computable in the online binary classification setting. Specifically, it asks: for which hypothesis classes $\\mathcal{H}$ does there exist a computable online learner that makes only finitely many mistakes on every realizable example sequence $(z_t) = (x_t, y_t)$ such that for any $t \\in \\mathbb{N}$, there exists a hypothesis $h \\in \\mathcal{H}$ such that $y_s = h(x_s)$ for all $s \\le t$? Note that this definition allows the realizing hypothesis $h_t$ to vary with the finite horizon $t$, a condition the authors call local consistency of the labels with $\\mathcal{H}$.\n\nPrevious work has provided a characterization of universally learnable hypothesis classes. However, the question of whether learners achieving the finite-mistakes objective can be made computable has remained unaddressed until now. In this sense, the present work fills a fundamental gap by focusing on the algorithmic implementability of the online learner, emphasizing that a learning \"procedure\" must be an algorithm (in the Turing sense), not merely an explicit mathematical construction.\n\nThe main contributions are several. First, the authors show that even if $\\mathcal{H}$ is recursively enumerably representable (RER), this alone does not guarantee the existence of a computable universal learner. Undeterred by the failure of this natural sufficient condition, the authors develop a deeper characterization, which falls short of their initial goal but still yields significant and interesting results. \n\nIt is shown that $\\mathcal{H}$ admits a *total* computable universal learner if and only if the topological closure of $\\mathcal{H}$ is contained in an RER set. Note that “computable” alone refers to partial computability, meaning the learner may be undefined or fail to halt on some inputs in its domain. Hence, the existence of a total computable learner is a strictly stronger assumption than merely requiring computable universal learning. Moreover, the aforementioned conditions are equivalent to the existence of a computable agnostic learner for $\\mathcal{H}$, where a computable agnostic learner is a randomized, almost-total learner that achieves sublinear regret relative to $\\mathcal{H}$ on any example sequence, not just those realizable by $\\mathcal{H}$. A complete characterization of computable universal learning, which was perhaps the original inspiration for this work, remains an open problem.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1. **Addresses a significant and often overlooked gap in the field.** The majority of works in learning theory, aside from a recent line of remarkable contributions, leave the issue of computability entirely unaddressed. These works often incorrectly refer to explicit mathematical procedures as “algorithms,” a term with a precise technical meaning established by Gödel, Church, and Turing, without formally considering their computability. I therefore find it greatly satisfying that this work rigorously tackles the computability question in the context of online learning.\n2. **Proof techniques are particularly illustrative.** The authors identify hypotheses on the input domain $\\mathbb{N}$ with elements of Cantor space, which is a complete metric space (w.r.t. the Baire metric on infinite sequences) and hence a well-defined topological space. This identification enables a clean mathematical formulation and facilitates the use of well-established tools from analysis.\n3. **Excellent exposition.** I believe this level of clarity is a significant contribution on its own, given the technical nature of the notions and proof techniques involved in computability. Minor clarifications could further improve it, especially since computability is highly sensitive to precise specification of the objects involved (see **Questions** for details).\n\n### Weaknesses\n1. **Setup is clean but lacks a connection to standard hypothesis classes.** For example, which reasonable subsets of linear threshold functions (LTFs) over $\\mathbb{S}^{d-1}$ are RER? This gap could leave the exposition feeling incomplete. The authors’ statement,\n> ... in this paper we limit our attention to universal learning of hypotheses defined on $\\mathbb{N}$. This might seem limiting but this choice is guided by practical considerations. Indeed, all practical machine learning happens on computers, which inherently work with countable domains, such as images or binary strings. From the point of view of computability, learnability on natural numbers is sufficiently general to model any domain relevant to practical applications.\n>\nreads more like a deflection than a rigorous argument, as it does not directly address how standard hypothesis classes like LTFs can be formalized within this framework. While it is reasonable to view focusing on $\\mathbb{N}$ as a necessary first step, it should also be recognized that many standard hypothesis classes, particularly those defined over continuous domains with real-valued parameters like threshold functions on [0,1], are technically not RER and thus fall outside the current formalism.\\\nExtending this computability framework to encompass more standard hypothesis classes, such as LTFs defined on unit spheres in $\\mathbb{R}^d$, remains an important connection to establish and will require careful, meticulous formalization, even if it does not involve fundamentally new ideas.\n    \n2. Some aspects of the setup and proofs are unclear, as detailed in the **Questions** section.\n\nQUESTIONS:\n1. **Are the authors assuming that each hypothesis in the given class is a total computable function?** Identifying $h$ with an infinite binary sequence in Cantor space appears to require that every entry of the sequence is defined. Moreover, in parts of the proof where the consistency of $h$ with the current finite history of examples is checked, it seems crucial to ensure that a defined (halted) value for $h(x_s)$ exists for all $s \\le t$.\n\n2. **Related to the above issue, Lemma 5 seems to require that every expert is a total function.** If an active expert's prediction is undefined on feature x_t, then we cannot compute the exponential weights update for time t+1.\n\n3. **Proof of Theorem 5, (3) => (2) is unclear. In particular, in lines 310-314 and 324-327.** I do not see the necessity of introducing a new probability measure $P'$. Given the $[0,1]$-valued forecasts produced by the multiplicative weights update (MWU) algorithm, which the authors denote as the randomized no-regret algorithm $L$, the goal is to construct a deterministic online learner $L'$. The authors achieve this by simulating all random advice sequences at round $t$ until all but a $1/(k+1)^2$ fraction have halted, and taking a majority decision. However, the majority decision yields a deterministic value with no residual randomness because the random advice sequences are simulated according to a fixed (potentially dovetailed) enumeration to ensure that $L'$ is deterministic. Therefore, the introduction of the probability measure $P'$ seems unnecessary and is confusing to me.\\\n\\\nThe quantity of interest seems to be the gap between the expected 0/1 loss of the randomized prediction $\\hat{y}_t$, output by $L$, and the majority decision $\\bar{y}_t$, a deterministic quantity once the enumeration of random advice sequences is fixed, computed from all but a $\\delta_k$ fraction of the random advice sequences on which $L$ halts. Using the inequality $(\\bar{y}_t \\neq y_t) \\le (\\bar{y}_t \\neq \\hat{y}_t) + (\\hat{y}_t \\neq y_t)$, where $y_t$ is the observed label, and then taking the expectation over $\\hat{y}_t$, appears sufficient to analyze the regret gap.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nRecently, it has been shown that PAC learning might be undecidable, and that learnability cannot be determined within ZFC. In the model of computational PAC learning, where algorithms must computably output predictors, the main idea is that the VC dimension no longer characterizes learnability. Instead, a \"computational\" version of the VC dimension, known as the effective VC dimension, takes its place. Similar results have been shown for online learning and the Littlestone dimension.\n\nThis paper studies the computational version of universal learning, a model introduced by Bousquet et al., STOC 2021. This model lies between classical universal consistency and PAC learning. Unlike PAC learning, the sample complexity in this model depends on unspecified constants that themselves depend on the (fixed but unknown) data distribution.\n\nMy main concern is that the learning model described in the paper does not actually correspond to universal learning, see below.\n\nSTRENGTHS AND WEAKNESSES:\nI believe there is considerable confusion in the paper regarding the definition and scope of the universal learning setting as introduced by Bousquet et al. (STOC '21). In particular, the authors refer to this model as an online learning setting, which is not accurate. For example:\n\nAbstract: \"We address this question for universal learning, a generalist theoretical model of online binary classification, recently characterized by Bousquet et al. (STOC´21).\"\nThe model proposed by Bousquet et al. is not an online learning model. Although online learning techniques are used within the algorithm, the setting itself is batch, not sequential.\n\nLines 30–31: \"This setting aligns with real-world scenarios where uniform bounds are not always available and where data distributions may shift.\"\nThis is misleading. The distribution in this setting is fixed in advance and does not shift. The key feature is that the sample complexity may depend on the distribution (via unknown constants), but the distribution itself does not change with the sample size.\n\n The paper claims to study “agnostic computable universal learning,” but the characterization of even the simpler model of agnostic universal learning is still unresolved. Without a full understanding of the base model, it is unclear how one can meaningfully study a more complex, computability-based variant.\n\nDefinition 1: This is neither the standard definition of universal learning nor a rigorous formalization. \n\nAdditional Comments:\n\nLine 20 (Introduction): It is unclear what relevance multi-armed bandits have to the discussion in this paper.\n\nLine 28: \"Adversary fixes one true function H in advance.\"  There seems to be a typo here. \n\nDefinition 2: The definition of the agnostic setting appears to be incorrect. It is not clear what learning model is being formalized here.\n\nFinally, I must note that some sections may have been generated or heavily by an LLM, especially the introduction.\n\nQUESTIONS:\nMy main concern is about the definitions of the learning model, see above.\n\nLIMITATIONS:\nLimitations are addressed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce computability requirements to the framework of universal learning and show:\n- That there exists a recursively enumerable class that is universally learnable but not computably universally learnable.\n- A characterization of agnostic computable universal learning\n- A characterization of a variant that requires algorithms to be proper\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- Novel set-up: computability has not yet been considered for universal learning, of interest to the learning theory community\n- A variety of results are shown, with very few lose ends\n- The technical weight of the proofs seems adequate for NeurIPS\n\n## Weaknesses\n- There isn't much context/review/explanations about universal learning, which is relatively recent (2021). I would guess most of the NeurIPS readership (even learning theory folks) would not be familiar enough with these concepts for the level of explanations given here. If the authors intend on submitting in a conference like NeurIPS, more explanations should be given on this set-up.\n- Likewise, there is a lack of clarity in the beginning of Section 2.1. Most readers will be familiar with online learning, but, at first glance, lines 88-92 do not make it clear what is the difference with the more common set-up of online learning and this one: even in (uniform) online learning the adversary is allowed the flexibility to change the target as long as it is consistent with the history. What differentiates uniform and universal online learning is that, in the former, the number of mistakes has to be uniformly bounded (there exists a learner $\\mathcal A$ and integer $d\\in\\mathbb N$ such that *for any* $h\\in \\mathcal H$ , $\\mathcal A$ makes at most $d$ mistakes) vs in the uniform online learning, the number of mistakes is simply required to be finite (so, there is an algorithm $\\mathcal A$ such that for any $h\\in\\mathcal H$ there exists $d_h\\in\\mathbb N$ such that $\\mathcal A$ makes at most $d_h$ mistakes). This is distinction between set-ups is not clear at all, and would be confusing for most readers who are only/mostly familiar with uniform online learning.\n- I am not very familiar with universal learning, but my understanding is that there is a distribution at hand (in a PAC-like set-up, but where the learning rate can also be distribution-dependent), and that the universal online learning results are tools to take back to this main set-up. In particular, the error in universal learning depends on the distribution at hand (as in p.2-3 of Bousquet et al., 2021), and a central concern in universal learning is that of learning rates (not really studied here). Here it seems the paper is mostly concerned about the online version of universal learning? In any case, the authors do not provide any explanations for the departure from the universal learning set-up of Bousquet et al. (2021)\n\t- An attempt was made for Definition 1 to be readable/intuitive, but this was at the cost of clarity and precision, which is not ideal considering that the whole paper is based on this set up. Please include appropriate mathematical objects ( $\\mathcal H$ , $X$ , labels $y_t$ ) and quantifiers. The authors should also reference the work of (Bousquet et al, 2021) in the definition (and the appropriate works in any subsequent definition)\n- There are basic definitions missing:\n\t- For example, a computable learner not only halts on every sample, but also output functions that are themselves total computable (see Definition 8 in Argarwal et al., 2020). Most NeurIPS readers will *not* be familiar with computable learning frameworks, and some more effort should be made in the write-up to introduce these concepts, and also attribute the definitions to the works they originally belong to.\n- The authors state many literature results without references:\n\t- l.127-129: Delle Rose et al., 2025.\n\t- l.184: not sure where this comes from. Definition 2 should read agnostic online learning I believe.  In any case, universal learning depends on the learning rates, so you have classes that are universally learnable but at arbitrarily slow rates, and thus not PAC learnable (implying not online learnable). So I am not sure this assertion is true -- but it is hard to check without a proper reference.\n- Overall, the writing lacks clarity, precision, rigour and the necessary definitions/ideas to get the point through to a NeurIPS audience (including learning theory folks), which has been a big factor in my score. I am a bit confused by the various set-ups and the difference with the Bousquet et al. (2021) paper (which I read briefly to prepare for this review).\n\n## A note on writing:\n- Learner vs learner (l.139): it would be better to have different names. Why not simply \"strategy\"?\n- The writing could be improved (e.g., l.169 \"In agnostic learning, it only makes sense when the Learner has a strategy defined on all possible inputs, including non-realizable ones\", awkward phrasing, l.184 \"in standard theory\" - of what?, etc., and there are many instances where an article is missing before the noun, e.g., l.13 \"We then study agnostic variant of\" -> \"We then study the agnostic variant of\"; l. 25 \"a strategy that guarantees uniform bound on mistakes\" -> \"a strategy that guarantees a uniform bound on mistakes\", etc.)\n- In Section 4, it would be worth restating the theorems being proved for readability (right now, the reader has to go back and forth)\n- I am also generally not a fan of anthropomorphizing learners/ adversaries (even less so with only \"he\" for all parties). Perhaps the authors would consider changing \"he/his to \"it/its\"\n- While not crucial, it would also be helpful to have definitions of DR, RER, closure of $\\mathcal H$ , and general computability definitions in proper definition environments for readability.\n\nQUESTIONS:\nCould you comment on the set-up/my confusion between the universal learning set-up of Bousquet et al. (2021) and the universal vs uniform online learning set-up mentioned in the weakness section? It is a bit difficult to evaluate the rest of the paper without that issue clarified. \n\nCould you give more intuition/details on your definition of proper learning, as it is not really standard? \n\nThere are also proper/improper separation results in standard CPAC learning (Delle Rose et al., 2023) -- how do your results compare?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}