data/archive.jsonl

{"additional_fields":[],"background":"Viering, Mey and Loog posed the question at COLT 2019, with examples where monotonicity holds and examples where it fails. The failure cases are not pathological: empirical risk minimisation over simple classes can exhibit non-monotone learning curves, and \"more data can hurt\" is observed in practice.","contributions":[],"dates":{"created_at":"2026-09-07T23:45:31.857Z","published_at":"2026-09-07T23:45:35.197Z","solved_at":null,"updated_at":"2026-09-07T23:45:31.857Z"},"difficulty":"moderate","field":{"main":"machine learning","name":"Learning theory","original_discipline":"machine-learning","slug":"learning-theory","sub":"learning theory"},"formalizations":[],"horizon":"sharp","id":"problem:monotonicity-of-learning","lean4_proof":null,"lean4_proof_status":"not-recorded","lean4_statement":null,"lean4_target":null,"linked_problem_ids":[],"metadata_json":"{\"database_id\":\"1cb6c5a9-9a69-49df-9b00-cb0f62fc3624\",\"format_issues\":null,\"number\":42}","msc2020":[],"partial_results":"- Bousquet, Daniely, Kaplan, Mansour, Moran and Stemmer (COLT 2022) showed that any learning algorithm can be transformed into a monotone one with essentially the same rate, resolving the general existence question affirmatively.\n- What remains is the algorithm-specific question: characterise the distributions and classes for which standard procedures — ERM, regularised ERM, cross-validated selection — are themselves monotone, without a wrapper.\n- Non-monotone behaviour of the double-descent kind is a distinct phenomenon and is not explained by these results.","posed_by":"Tom Viering, Alexander Mey, Marco Loog","posed_year":2019,"prize_amount_usd":null,"prize_sponsor":null,"problem_statement":"Let $A$ be a learning algorithm and let $\\mathcal{E}_n$ denote its expected risk when trained on $n$ i.i.d. samples. Is it true that $\\mathcal{E}_{n+1} \\le \\mathcal{E}_n$ for all $n$ — that is, does adding one more training example never hurt in expectation? If not in general, for which algorithms and distributions does monotonicity hold?","proof":null,"proof_status":"not-recorded","provenance":{"evidence_label":null,"has_full_text":null,"imported_at":null,"note":null,"original_file":null,"sha256":null,"source_record":null,"status_as_of":null},"quality_flags":[],"record_kind":"archived_problem","references":[{"citation":"T. Viering, A. Mey, M. Loog, Open Problem: Monotonicity of Learning, PMLR 99:3198–3201, 2019.","kind":"paper","note":null,"sort_order":0,"url":"https://proceedings.mlr.press/v99/viering19a.html"},{"citation":"O. Bousquet, A. Daniely, H. Kaplan, Y. Mansour, S. Moran, U. Stemmer, Monotone Learning, COLT 2022.","kind":"paper","note":null,"sort_order":1,"url":"https://arxiv.org/abs/2202.05246"}],"review_state":"draft","schema_version":"1.0.0","slug":"monotonicity-of-learning","snapshot_at":"2026-10-10T16:07:49.246Z","source":{"arxiv_id":null,"citation":"T. Viering, A. Mey, M. Loog, Proceedings of the 32nd Conference on Learning Theory, PMLR 99:3198–3201, 2019.","doi":null,"key":"colt","license":"Author-retained copyright; abstracts linked, statements paraphrased","name":"Open Problem: Monotonicity of Learning","notes":"Since 2003 the Conference on Learning Theory has published short peer-reviewed papers whose entire purpose is to pose a problem. Titles begin with 'Open Problem:' and are indexed in the annual PMLR volume, which makes the track the single most machine-readable open-problem source in machine learning.","type":"conference-open-problem","url":"https://proceedings.mlr.press/v99/viering19a.html"},"source_status_json":null,"status":"partially-solved","tags":["colt-open-problem","erm","learning-curves","sample-complexity"],"teaser":"COLT 2019 open problem, resolved in general form at COLT 2022 — but the question for natural algorithms such as ERM stays open.","title":"Is learning monotone in the sample size?","why_it_matters":null}
{"additional_fields":[{"main":"machine learning","name":"Optimization","original_discipline":"machine-learning","slug":"optimization","sub":"optimization"}],"background":"Posed by Francesco Orabona and Dávid Pál at COLT 2016. Both adaptivity properties were individually well understood; the difficulty is that the standard constructions for one assume knowledge of the quantity the other adapts to.","contributions":[],"dates":{"created_at":"2026-09-07T23:45:31.857Z","published_at":"2026-09-07T23:45:35.204Z","solved_at":null,"updated_at":"2026-09-07T23:45:31.857Z"},"difficulty":"hard","field":{"main":"machine learning","name":"Learning theory","original_discipline":"machine-learning","slug":"learning-theory","sub":"learning theory"},"formalizations":[],"horizon":"sharp","id":"problem:parameter-free-and-scale-free-online-learning","lean4_proof":null,"lean4_proof_status":"not-recorded","lean4_statement":null,"lean4_target":null,"linked_problem_ids":[],"metadata_json":"{\"database_id\":\"f5183685-1481-4030-8152-d11ba90bafb6\",\"format_issues\":null,\"number\":43}","msc2020":[],"partial_results":"- Cutkosky and Boahen (COLT 2017) showed that in the unbounded-domain setting there is a genuine price for simultaneous adaptivity, giving both algorithms and lower bounds.\n- Mhammedi and Koolen (COLT 2020) achieved comparator-norm and Lipschitz adaptivity together under conditions that cover many cases of interest.\n- The precise trade-off in the most general setting, and matching lower bounds, are not fully settled.","posed_by":"Francesco Orabona, Dávid Pál","posed_year":2016,"prize_amount_usd":null,"prize_sponsor":null,"problem_statement":"In online linear optimisation, *parameter-free* algorithms achieve regret $\\tilde{O}(\\lVert u\\rVert\\sqrt{T})$ against every comparator $u$ without knowing $\\lVert u \\rVert$ in advance. *Scale-free* algorithms achieve regret scaling with $\\max_{t\\le T}\\lVert \\ell_t\\rVert$ without knowing a bound on the loss vectors in advance.\n\nIs there an algorithm that is both — with regret $\\tilde{O}\\!\\left(\\lVert u\\rVert \\max_{t \\le T}\\lVert \\ell_t \\rVert \\sqrt{T}\\right)$, knowing neither the comparator norm nor the scale of the losses?","proof":null,"proof_status":"not-recorded","provenance":{"evidence_label":null,"has_full_text":null,"imported_at":null,"note":null,"original_file":null,"sha256":null,"source_record":null,"status_as_of":null},"quality_flags":[],"record_kind":"archived_problem","references":[],"review_state":"draft","schema_version":"1.0.0","slug":"parameter-free-and-scale-free-online-learning","snapshot_at":"2026-10-10T16:07:49.246Z","source":{"arxiv_id":null,"citation":"F. Orabona, D. Pál, 29th Annual Conference on Learning Theory, PMLR 49:1659–1664, 2016.","doi":null,"key":"colt","license":"Author-retained copyright; abstracts linked, statements paraphrased","name":"Open Problem: Parameter-Free and Scale-Free Online Algorithms","notes":"Since 2003 the Conference on Learning Theory has published short peer-reviewed papers whose entire purpose is to pose a problem. Titles begin with 'Open Problem:' and are indexed in the annual PMLR volume, which makes the track the single most machine-readable open-problem source in machine learning.","type":"conference-open-problem","url":"https://proceedings.mlr.press/v49/orabona16.html"},"source_status_json":null,"status":"partially-solved","tags":["adaptivity","colt-open-problem","online-learning","regret"],"teaser":"COLT 2016 open problem: adapt to the comparator norm and to the loss scale at the same time, with no prior knowledge of either.","title":"Algorithms that are simultaneously parameter-free and scale-free","why_it_matters":"Every hyperparameter that an online algorithm requires is a hyperparameter a practitioner must tune. Parameter-free and scale-free adaptivity together would remove the two most common ones, and the techniques developed for it — coin betting, potential functions from martingale arguments — have propagated into optimisation more broadly."}
{"additional_fields":[{"main":"statistics","name":"High-dimensional statistics","original_discipline":"statistics","slug":"high-dimensional-statistics","sub":"high dimensional statistics"}],"background":"Posed by Bingshan Hu and Nishant Mehta at COLT 2024. Differential privacy typically costs an additive term whose form depends on the problem structure; here even the shape of the correct dependence on $\\varepsilon$, $K$ and $\\Delta_{\\min}$ is not settled.","contributions":[],"dates":{"created_at":"2026-09-07T23:45:31.857Z","published_at":"2026-09-07T23:45:35.218Z","solved_at":null,"updated_at":"2026-09-07T23:45:31.857Z"},"difficulty":"moderate","field":{"main":"machine learning","name":"Learning theory","original_discipline":"machine-learning","slug":"learning-theory","sub":"learning theory"},"formalizations":[],"horizon":"sharp","id":"problem:private-decision-theoretic-online-learning-rates","lean4_proof":null,"lean4_proof_status":"not-recorded","lean4_statement":null,"lean4_target":null,"linked_problem_ids":[],"metadata_json":"{\"database_id\":\"606260e0-94b6-494d-803e-0124134b5de6\",\"format_issues\":null,\"number\":45}","msc2020":[],"partial_results":null,"posed_by":"Bingshan Hu, Nishant A. Mehta","posed_year":2024,"prize_amount_usd":null,"prize_sponsor":null,"problem_statement":"Consider stochastic decision-theoretic online learning with $K$ actions over $T$ rounds, with minimum suboptimality gap $\\Delta_{\\min}$. Without privacy, the optimal gap-dependent pseudo-regret is\n$$\\Theta\\!\\left(\\frac{\\log K}{\\Delta_{\\min}}\\right).$$\nDetermine the optimal gap-dependent pseudo-regret under $\\varepsilon$-differential privacy.","proof":null,"proof_status":"not-recorded","provenance":{"evidence_label":null,"has_full_text":null,"imported_at":null,"note":null,"original_file":null,"sha256":null,"source_record":null,"status_as_of":null},"quality_flags":[],"record_kind":"archived_problem","references":[],"review_state":"draft","schema_version":"1.0.0","slug":"private-decision-theoretic-online-learning-rates","snapshot_at":"2026-10-10T16:07:49.246Z","source":{"arxiv_id":null,"citation":"B. Hu, N. A. Mehta, Proceedings of the 37th Conference on Learning Theory, PMLR 247:5330–5334, 2024.","doi":null,"key":"colt","license":"Author-retained copyright; abstracts linked, statements paraphrased","name":"Open Problem: Optimal Rates for Stochastic Decision-Theoretic Online Learning Under Differential Privacy","notes":"Since 2003 the Conference on Learning Theory has published short peer-reviewed papers whose entire purpose is to pose a problem. Titles begin with 'Open Problem:' and are indexed in the annual PMLR volume, which makes the track the single most machine-readable open-problem source in machine learning.","type":"conference-open-problem","url":"https://proceedings.mlr.press/v247/hu24a.html"},"source_status_json":null,"status":"open","tags":["colt-open-problem","differential-privacy","online-learning","regret"],"teaser":"COLT 2024 open problem: the non-private rate is known exactly. Add ε-differential privacy and the right answer is unknown.","title":"Optimal gap-dependent rates for stochastic decision-theoretic online learning under differential privacy","why_it_matters":"Decision-theoretic online learning is the base case for a large family of private online algorithms. Getting the gap-dependent rate right determines whether privacy is essentially free in the easy regime, as it is for several batch problems, or fundamentally costly."}
{"additional_fields":[],"background":"Li and Chapelle posed this at COLT 2012 after empirical studies showed Thompson sampling outperforming UCB on real problems while lacking any theory. The algorithm dates to Thompson's 1933 paper on clinical trials.","contributions":[],"dates":{"created_at":"2026-09-07T23:45:31.857Z","published_at":"2026-09-07T23:45:35.210Z","solved_at":null,"updated_at":"2026-09-07T23:45:31.857Z"},"difficulty":"moderate","field":{"main":"machine learning","name":"Learning theory","original_discipline":"machine-learning","slug":"learning-theory","sub":"learning theory"},"formalizations":[],"horizon":"sharp","id":"problem:thompson-sampling-regret-bounds","lean4_proof":null,"lean4_proof_status":"not-recorded","lean4_statement":null,"lean4_target":null,"linked_problem_ids":[],"metadata_json":"{\"database_id\":\"e058fb1a-ba68-4feb-bd89-2445376303fd\",\"format_issues\":null,\"number\":44}","msc2020":[],"partial_results":"- Agrawal and Goyal (2012, 2013) proved the first finite-time problem-dependent bounds and a near-minimax $O(\\sqrt{NT\\log T})$ bound.\n- Kaufmann, Korda and Munos proved asymptotic optimality for Bernoulli rewards, matching the Lai–Robbins lower bound.\n- Russo and Van Roy developed the information-theoretic analysis that extends the guarantees to structured and contextual settings.\n- The entry is retained as a worked example of a resolved open problem: the record of what was unknown in 2012 is exactly what makes the resolution legible.","posed_by":"Lihong Li, Olivier Chapelle","posed_year":2012,"prize_amount_usd":null,"prize_sponsor":null,"problem_statement":"Thompson sampling maintains a posterior over arm parameters and pulls the arm that is optimal under a posterior sample. At the time of posing, no finite-time regret bound was known. Prove a problem-dependent regret bound of the form $O\\!\\left(\\sum_{i:\\Delta_i>0} \\frac{\\log T}{\\Delta_i}\\right)$, and determine whether the algorithm is minimax optimal.","proof":null,"proof_status":"not-recorded","provenance":{"evidence_label":null,"has_full_text":null,"imported_at":null,"note":null,"original_file":null,"sha256":null,"source_record":null,"status_as_of":null},"quality_flags":[],"record_kind":"archived_problem","references":[{"citation":"S. Agrawal, N. Goyal, Analysis of Thompson sampling for the multi-armed bandit problem, COLT 2012.","kind":"paper","note":null,"sort_order":0,"url":null},{"citation":"E. Kaufmann, N. Korda, R. Munos, Thompson sampling: an asymptotically optimal finite-time analysis, ALT 2012.","kind":"paper","note":null,"sort_order":1,"url":null}],"review_state":"draft","schema_version":"1.0.0","slug":"thompson-sampling-regret-bounds","snapshot_at":"2026-10-10T16:07:49.246Z","source":{"arxiv_id":null,"citation":"L. Li, O. Chapelle, Proceedings of the 25th Annual Conference on Learning Theory, PMLR 23:43.1–43.3, 2012.","doi":null,"key":"colt","license":"Author-retained copyright; abstracts linked, statements paraphrased","name":"Open Problem: Regret Bounds for Thompson Sampling","notes":"Since 2003 the Conference on Learning Theory has published short peer-reviewed papers whose entire purpose is to pose a problem. Titles begin with 'Open Problem:' and are indexed in the annual PMLR volume, which makes the track the single most machine-readable open-problem source in machine learning.","type":"conference-open-problem","url":"https://proceedings.mlr.press/v23/li12.html"},"source_status_json":null,"status":"solved","tags":["bandits","colt-open-problem","regret","resolved","thompson-sampling"],"teaser":"Posed at COLT 2012 for an algorithm from 1933 that worked well and had no analysis. Now resolved.","title":"Regret bounds for Thompson sampling","why_it_matters":null}
{"additional_fields":[],"background":"Posed by Steve Hanneke in the COLT 2021 open-problem track. It is the online analogue of universal consistency in the statistical setting, where Hanneke's earlier work identified exactly which processes admit universally consistent learning. The online version removes the i.i.d. assumption entirely and asks for a learner that adapts to whatever structure the sequence happens to have.","contributions":[],"dates":{"created_at":"2026-09-07T23:45:31.857Z","published_at":"2026-09-07T23:45:35.189Z","solved_at":null,"updated_at":"2026-09-07T23:45:31.857Z"},"difficulty":"hard","field":{"main":"machine learning","name":"Learning theory","original_discipline":"machine-learning","slug":"learning-theory","sub":"learning theory"},"formalizations":[],"horizon":"sharp","id":"problem:universally-consistent-online-learning","lean4_proof":null,"lean4_proof_status":"not-recorded","lean4_statement":null,"lean4_target":null,"linked_problem_ids":[],"metadata_json":"{\"database_id\":\"cde186a4-29cb-4c23-b263-36f67bb8e3d4\",\"format_issues\":null,\"number\":41}","msc2020":[],"partial_results":"- For countable $\\mathcal{X}$ the answer is known to be positive by a straightforward argument.\n- Related work on universal consistency for non-i.i.d. processes gives characterisations in the statistical setting that have no known online counterpart.","posed_by":"Steve Hanneke","posed_year":2021,"prize_amount_usd":null,"prize_sponsor":null,"problem_statement":"Fix an instance space $\\mathcal{X}$ and consider online binary classification against an adversary that fixes a (possibly random) sequence of points $\\mathbb{X} = (x_1, x_2, \\dots)$ in advance and reveals labels $y_t = c^\\star(x_t)$ for an arbitrary target concept $c^\\star : \\mathcal{X}\\to\\{0,1\\}$.\n\nSay that $\\mathbb{X}$ is *learnable* if there exists an algorithm making $o(T)$ mistakes on the first $T$ rounds, for every target concept. The question: is there a single algorithm $A$ that makes $o(T)$ mistakes on every learnable $\\mathbb{X}$, for every target concept?\n\nA secondary question asks whether a specific combinatorial condition on $\\mathbb{X}$ characterises learnability.","proof":null,"proof_status":"not-recorded","provenance":{"evidence_label":null,"has_full_text":null,"imported_at":null,"note":null,"original_file":null,"sha256":null,"source_record":null,"status_as_of":null},"quality_flags":[],"record_kind":"archived_problem","references":[{"citation":"S. Hanneke, Open Problem: Is There an Online Learning Algorithm That Learns Whenever Online Learning Is Possible? PMLR 134:4642–4646, 2021.","kind":"paper","note":null,"sort_order":0,"url":"https://proceedings.mlr.press/v134/open-problem-hanneke21b.html"}],"review_state":"draft","schema_version":"1.0.0","slug":"universally-consistent-online-learning","snapshot_at":"2026-10-10T16:07:49.246Z","source":{"arxiv_id":null,"citation":"S. Hanneke, Proceedings of the 34th Conference on Learning Theory, PMLR 134:4642–4646, 2021.","doi":null,"key":"colt","license":"Author-retained copyright; abstracts linked, statements paraphrased","name":"Open Problem: Is There an Online Learning Algorithm That Learns Whenever Online Learning Is Possible?","notes":"Since 2003 the Conference on Learning Theory has published short peer-reviewed papers whose entire purpose is to pose a problem. Titles begin with 'Open Problem:' and are indexed in the annual PMLR volume, which makes the track the single most machine-readable open-problem source in machine learning.","type":"conference-open-problem","url":"https://proceedings.mlr.press/v134/open-problem-hanneke21b.html"},"source_status_json":null,"status":"open","tags":["colt-open-problem","online-learning","universal-consistency"],"teaser":"COLT 2021 open problem: a single algorithm that makes sublinear mistakes on every sequence on which any algorithm could.","title":"Is there an online learner that succeeds whenever online learning is possible?","why_it_matters":"Most online learning guarantees are stated relative to a hypothesis class of bounded Littlestone dimension. This problem asks for a guarantee relative to the *data sequence* instead, which is closer to how learning systems are actually deployed: nobody knows the right hypothesis class in advance, but the data may still be benign."}