item_id stringlengths 26 54 | paper_id stringlengths 10 22 | target_name stringlengths 4 33 | source stringclasses 2
values | nl_statement stringlengths 78 1.64k | context listlengths 1 16 ⌀ | nl_notes stringlengths 0 1.77k | lean stringlengths 395 9.09k | target_names listlengths 1 4 | topic_domain stringclasses 6
values | topic_subtopic stringclasses 16
values | holes listlengths 1 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|
v10-duchi09a::theorem6::A-stmt | v10-duchi09a | theorem6 | Research-level | Theorem 6 Suppose that $\| \pmb{w}_{t}- \pmb{w}^{\star}\| \leq D$ holds at every iteration, and that G is an upper bound on the norms of the subgradient sets $\partial f_{t}$ and $\partial r$. Fix an arbitrary scalar $c > 0$. Then FoBoS run with step sizes $\eta_{t}= c / \sqrt{t}$ has a regret bound of the form
$$
R_{... | [
{
"name": "Setup and notation: objective f + r, convex and nonnegative (Section 1)",
"statement": "We first fix the notation used throughout the paper. Scalars are written as lower case letters and vectors as bold lower case letters, e.g. w. For two vectors u and v, $\\langle{\\pmb u},{\\pmb v}\\rangle$ is ... | import Mathlib
namespace ResearchLevel
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
def HasSubgradientAt (f : E → ℝ) (g x : E) : Prop :=
∀ y, f y ≥ f x + @inner ℝ _ _ g (y - x)
def IsFoBoSRun (f r : E → ℝ) (w gf gr : ℕ → E) (η ηhalf : ℕ → ℝ) : Prop :=
(∀ t, HasSubg... | [
"theorem6",
"theorem8",
"proposition11"
] | Online Learning & RL | Online Learning & Regret | [
{
"start": 1269,
"end": 1888,
"gold": "(f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G c : ℝ)\n (wstar : E)\n (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n (hrun : IsFoBoSOnlineRun f r w gf gr η)\... | |
v10-duchi09a::theorem8::A-stmt | v10-duchi09a | theorem8 | Research-level | Theorem 8 As in Theorem 6, suppose that $\| \pmb{w}_{t}- \pmb{w}^{\star}\| \leq D$ and that G bounds $\partial f_{t}$ and $\partial r$ from above. In addition, suppose that for every t the function $f_{t}+ r$ is H-strongly convex on the region $\| \pmb{w}- \pmb{w}^{\star}\| \leq D$. Then, with the step sizes $\eta_{t}=... | [
{
"name": "Setup and notation: objective f + r, convex and nonnegative (Section 1)",
"statement": "We first fix the notation used throughout the paper. Scalars are written as lower case letters and vectors as bold lower case letters, e.g. w. For two vectors u and v, $\\langle{\\pmb u},{\\pmb v}\\rangle$ is ... | import Mathlib
namespace ResearchLevel
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
def HasSubgradientAt (f : E → ℝ) (g x : E) : Prop :=
∀ y, f y ≥ f x + @inner ℝ _ _ g (y - x)
def IsFoBoSRun (f r : E → ℝ) (w gf gr : ℕ → E) (η ηhalf : ℕ → ℝ) : Prop :=
(∀ t, HasSubg... | [
"theorem6",
"theorem8",
"proposition11"
] | Online Learning & RL | Online Learning & Regret | [
{
"start": 1925,
"end": 2636,
"gold": "(f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G H : ℝ)\n (wstar : E)\n (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n (hrun : IsFoBoSOnlineRun f r w gf gr η)\... | |
v10-duchi09a::proposition11::A-stmt | v10-duchi09a | proposition11 | Research-level | Proposition 11 For $t=1,\ldots,T$, solve one after another the T self-similar optimization problems
$$
\mathcal{P}.1:~{\pmb w}_{t}=\operatorname*{argmin}_{{\pmb w}}\frac{1}{2}\|{\pmb w}-{\pmb w}_{t-1}\|^{2}+\lambda_{t}\|{\pmb w}\|_{q}\enspace .\tag{28}
$$
and let $\pmb{w}_{T}$ be the vector obtained at the end. Denot... | [
{
"name": "Setup and notation: objective f + r, convex and nonnegative (Section 1)",
"statement": "We first fix the notation used throughout the paper. Scalars are written as lower case letters and vectors as bold lower case letters, e.g. w. For two vectors u and v, $\\langle{\\pmb u},{\\pmb v}\\rangle$ is ... | import Mathlib
namespace ResearchLevel
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
def HasSubgradientAt (f : E → ℝ) (g x : E) : Prop :=
∀ y, f y ≥ f x + @inner ℝ _ _ g (y - x)
def IsFoBoSRun (f r : E → ℝ) (w gf gr : ℕ → E) (η ηhalf : ℕ → ℝ) : Prop :=
(∀ t, HasSubg... | [
"theorem6",
"theorem8",
"proposition11"
] | Optimization | Convex Analysis & Duality | [
{
"start": 2678,
"end": 3066,
"gold": "{n : ℕ} (T : ℕ) (hT : 0 < T) (w0 wstar : Fin n → ℝ) (w : ℕ → Fin n → ℝ)\n (lam : ℕ → ℝ) (hlam : ∀ t ∈ Finset.Icc 1 T, 0 ≤ lam t)\n (hw0 : w 0 = w0)\n (N : (Fin n → ℝ) → ℝ)\n (hN : N = l1Norm ∨ N = l2Norm ∨ N = linftyNorm)\n (hseq : ∀ t ∈ Finset.Icc 1... | |
v23-20-411::lemma5_oblivious::A-stmt | v23-20-411 | lemma5_oblivious | Research-level | Lemma 5 (Weighted-Regret Bound for EXP3). Let $\{\eta_{t}\}$ be a non-increasing step-size sequence such that $\eta_{t}\leq\frac 12e^{-2}$ for all t. Let $\left\{l_{t}^{\left(i\right)}\right\}$ be a cost sequence such that $l_{t}^{(i)}\in [0,1]$ for every $t,i.$ Let $\{d_{t}\}$ be a delay sequence such that the cost fr... | [
{
"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)",
"statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbo... | ### Lemma 5
NOTE (Lemma 5, oblivious case) — The Lean adds a hypothesis (`a_fresh`) that the paper never states as an assumption of Lemma 5 or Algorithm 3, but uses inside the proof of the oblivious case of Lemma 5 (appendix Section 13.2, headed "Proof of Lemma 6" in the paper): each arm a_t is drawn with fresh randomn... | import Mathlib
namespace ResearchLevel
open scoped Classical
open MeasureTheory Filter Topology
def missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t)
noncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t ∨ ... | [
"lemma5_oblivious",
"lemma5_adaptive",
"theorem6_regretBound",
"lemma6_noWeightedRegret"
] | Online Learning & RL | Bandits | [
{
"start": 2424,
"end": 3388,
"gold": "{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n (η : ℕ → ℝ) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n (hη_bound :... |
v23-20-411::lemma5_adaptive::A-stmt | v23-20-411 | lemma5_adaptive | Research-level | Lemma 5 (Weighted-Regret Bound for EXP3). Let $\{\eta_{t}\}$ be a non-increasing step-size sequence such that $\eta_{t}\leq\frac 12e^{-2}$ for all t. Let $\left\{l_{t}^{\left(i\right)}\right\}$ be a cost sequence such that $l_{t}^{(i)}\in [0,1]$ for every $t,i.$ Let $\{d_{t}\}$ be a delay sequence such that the cost fr... | [
{
"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)",
"statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbo... | ### Lemma 5
NOTE (Lemma 5, adaptive case) — The Lean adds a hypothesis (`a_law_adaptive`) that the paper never states as an assumption of Lemma 5 or Algorithm 3, but uses inside the proof of Lemma 11 ("given 𝒢_t, a_t = i with probability p_t^(i)"), which the proof of the adaptive case of Lemma 5 applies (appendix Sect... | import Mathlib
namespace ResearchLevel
open scoped Classical
open MeasureTheory Filter Topology
def missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t)
noncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t ∨ ... | [
"lemma5_oblivious",
"lemma5_adaptive",
"theorem6_regretBound",
"lemma6_noWeightedRegret"
] | Online Learning & RL | Bandits | [
{
"start": 3428,
"end": 4421,
"gold": "{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n (hη_boun... |
v23-20-411::theorem6_regretBound::A-stmt | v23-20-411 | theorem6_regretBound | Research-level | Theorem 6. Let $\left\{l_{t}^{\left(i\right)}\right\}$ be a cost sequence such that $l_{t}^{(i)}\in [0,1]$ for every $t,i$ Let $\{d_{t}\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . Define the set $\mathcal{M}=$ $\{t|t+d_{t}>T,t\in [1,T]\}$ of all samples that are not receive... | [
{
"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)",
"statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbo... | ### Theorem 6
NOTE (Theorem 6) — As in the oblivious case of Lemma 5, the Lean assumes (`a_fresh`) that each arm a_t is drawn with fresh randomness: for t ≤ t' < t + d_t, the conditional law of a_t given 𝓕_{t'}, together with the arms of the rounds whose feedback arrives before that of round t, is still p_t. The paper... | import Mathlib
namespace ResearchLevel
open scoped Classical
open MeasureTheory Filter Topology
def missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t)
noncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t ∨ ... | [
"lemma5_oblivious",
"lemma5_adaptive",
"theorem6_regretBound",
"lemma6_noWeightedRegret"
] | Online Learning & RL | Bandits | [
{
"start": 4466,
"end": 5519,
"gold": ":\n ∃ C : ℝ, 0 < C ∧\n ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n (K T : ℕ) (hK : 0 < K) (hT : 0 < T) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ)\n (hd : ∀ t ≥ 1, 1 ≤ d t)\n (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1... |
v23-20-411::lemma6_noWeightedRegret::A-stmt | v23-20-411 | lemma6_noWeightedRegret | Research-level | Lemma 6. EXP3 (Algorithm 3 with $\gamma_{t}= \eta_{t}$ for all t, against an adaptive adversary) with a non-increasing and positive step-size sequence $\{\eta_{t}\}$ such that $\eta_{t}\leq\frac 12e^{-2}$ for all t has no weighted-regret with respect to the sequence of delays $\{d_{t}\}$ and $\{\eta_{t}\}$ as the weigh... | [
{
"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)",
"statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbo... | ### Lemma 6
Lemma 6 concludes that EXP3 "has no weighted-regret", which Definition 2 / eq. (2) defines as the TWO-SIDED limit `lim_{T->inf} E{ sum eta_t (l_t(a_t) - l_t(a*)) / sum eta_t } = 0`.
The ratio lies in [-1,1], so `lim = 0` is equivalent to `limsup <= 0` AND `0 <= liminf`. 'The limit exists' adds nothing else.... | import Mathlib
namespace ResearchLevel
open scoped Classical
open MeasureTheory Filter Topology
def missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t)
noncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=
(Finset.Icc 1 T).filter (fun t => T < t + d t ∨ ... | [
"lemma5_oblivious",
"lemma5_adaptive",
"theorem6_regretBound",
"lemma6_noWeightedRegret"
] | Online Learning & RL | Bandits | [
{
"start": 5567,
"end": 6540,
"gold": "{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n [IsProbabilityMeasure μ] (K : ℕ) (hK : 0 < K)\n (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n (hη_pos : ∀ t ≥ 1, 0 < η t) (hη_dec : ∀ t ≥ 1, η (t + 1) ≤ η t)\n (hη_bound : ∀ t... |
v23-21-1027::theorem3::A-stmt | v23-21-1027 | theorem3 | Research-level | Theorem 3 (Regret bound for dual-stabilized OMD) Assume we have $\eta_{t}\geq \eta_{t + 1}>$ 0 for each $t > 1$ . Define $\gamma_{t}=\eta_{t+1}/\eta_{t}\in(0,1]$ for all $t \geq 1$ . Let $\{f_{t}\}_{t \ge 1}$ be a sequence of convex functions with $f_{t}\colon \mathcal{X}\to \mathbb{R}$ for each $t \geq 1$ . Let $\{x_{... | [
{
"name": "Setup: online convex optimization and regret (Section 3)",
"statement": "We consider the online convex optimization problem with unknown time horizon. For each time step $t\\in\\{1,2,\\ldots\\}$ the algorithm proposes a point $x_{t}$ from a closed convex set ${\\mathcal{X}}\\subseteq \\mathbb{R}^... | import Mathlib
open scoped RealInnerProductSpace BigOperators
namespace ResearchLevel
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
noncomputable def bregman (Φ : E → ℝ) (gΦ : E → E) (x y : E) : ℝ :=
Φ x - Φ y - (@inner ℝ E _ (gΦ y) (x - y))
noncomputable def regret ... | [
"theorem3",
"theorem5",
"theorem8"
] | Online Learning & RL | Online Learning & Regret | [
{
"start": 828,
"end": 2491,
"gold": "(Φ : E → ℝ) (gΦ gΦs : E → E)\n (X : Set E)\n (D : Set E) (hDconv : Convex ℝ D)\n (hXD : X ⊆ closure D)\n (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n (hΦstrict : StrictConvexOn ℝ D Φ)\n (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n (η... | |
v23-21-1027::theorem5::A-stmt | v23-21-1027 | theorem5 | Research-level | Theorem 5 (Regret bound for primal-stabilized OMD) Assume $\eta_{t}\geq \eta_{t + 1}> 0$ for each $t > 1$ . Define $\gamma_{t}=\eta_{t+1}/\eta_{t}\in(0,1]$ for all $t \geq 1$ . Let $\{x_{t}\}_{t \ge 1}$ be the sequence of iterates generated by Algorithm 3. Furthermore, assume that
for all $z \in \mathcal{X}$ , the map... | [
{
"name": "Setup: online convex optimization and regret (Section 3)",
"statement": "We consider the online convex optimization problem with unknown time horizon. For each time step $t\\in\\{1,2,\\ldots\\}$ the algorithm proposes a point $x_{t}$ from a closed convex set ${\\mathcal{X}}\\subseteq \\mathbb{R}^... | import Mathlib
open scoped RealInnerProductSpace BigOperators
namespace ResearchLevel
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
noncomputable def bregman (Φ : E → ℝ) (gΦ : E → E) (x y : E) : ℝ :=
Φ x - Φ y - (@inner ℝ E _ (gΦ y) (x - y))
noncomputable def regret ... | [
"theorem3",
"theorem5",
"theorem8"
] | Online Learning & RL | Online Learning & Regret | [
{
"start": 2528,
"end": 4262,
"gold": "(Φ : E → ℝ) (gΦ gΦs : E → E)\n (X : Set E)\n (D : Set E) (hDconv : Convex ℝ D)\n (hXD : X ⊆ closure D)\n (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n (hΦstrict : StrictConvexOn ℝ D Φ)\n (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n (... | |
v23-21-1027::theorem8::A-stmt | v23-21-1027 | theorem8 | Research-level | "Theorem 8 (Regret bound for dual averaging) Assume we have $\\eta_{t}\\geq \\eta_{t + 1}> 0$ for ea(...TRUNCATED) | [{"name":"Setup: online convex optimization and regret (Section 3)","statement":"We consider the onl(...TRUNCATED) | "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nvari(...TRUNCATED) | [
"theorem3",
"theorem5",
"theorem8"
] | Online Learning & RL | Online Learning & Regret | [{"start":4299,"end":5968,"gold":"(Φ : E → ℝ) (gΦ gΦs : E → E)\n (X : Set E)\n (D : S(...TRUNCATED) |
End of preview. Expand in Data Studio
MLP-Bench
One JSONL file per task, one JSON object per line.
| file | task | size |
|---|---|---|
forward-construction.jsonl |
Lean proving | 80 rows, 120 targets |
backward-verification.jsonl |
FL verification | 120 items, 60 correct / 60 flawed |
auto-formalization.jsonl |
auto-formalization | 121 items, 120 targets |
The 120 theorems of the first and third file come from 20 JMLR papers (Research-level) and 60
graduate-level results from lecture notes and a textbook (Graduate-level); scores are per
target. The verification items come from ICLR submissions and their public reviews, 26 more
JMLR papers, and the same lecture notes and textbook.
forward-construction.jsonl (Lean proving)
Each row is a Lean file whose target theorems are left as sorry. The task is to prove them.
| field | meaning |
|---|---|
id, source |
row identity; source is Research-level / Graduate-level |
lean |
the Lean file given to the model (imports Mathlib only) |
targets |
[{label, decls, statement, ...}], one entry per target theorem: its label, Lean declaration names and natural-language statement |
nl_proof |
natural-language proof of the row |
context |
the definitions and results the statements and proofs rely on: [{id, name, statement}] |
solution |
gold Lean proof. Never show it to the model |
targets[].topic_domain, targets[].topic_subtopic |
the target's topic. Not shown to the model |
backward-verification.jsonl (FL verification)
Each item has a shared context, 3–5 theorems with numbered natural-language proofs, and a
target. The model decides whether the target's proof is valid; if not, it names the first
wrong step.
| field | meaning |
|---|---|
id |
item id. Never show it: it gives away the label |
context |
the shared setting printed above the theorems, one string |
theorems |
[{local_id, statement, proof: {steps: [{n, claim}]}}], in order (T1..Tn) |
target_local_id |
the theorem under test |
label |
gold: correct / flawed |
error_step |
gold: the first wrong step of the target (null when correct) |
error_class6 |
gold: admissible error classes, most likely first |
error_explanation |
gold: what is wrong at error_step (null when correct) |
topic_domain, topic_subtopic |
the item's topic. Never show it: it predicts the label |
auto-formalization.jsonl (auto-formalization)
One item per target declaration of forward-construction.jsonl. The target's Lean statement is
masked as <<HOLE 1>>; the model fills it in from the natural-language statement.
| field | meaning |
|---|---|
item_id |
<paper_id>::<declaration>::A-stmt |
paper_id, target_name, source |
source row, declaration, Research-level / Graduate-level |
lean |
the original file |
holes |
[{start, end, gold}]: lean[start:end] is the masked statement. gold is the answer; never show it |
nl_statement |
the target's natural-language statement |
context |
the target's context, [{name, statement}] |
nl_notes |
formalization notes on how the Lean statement reads the source |
target_names |
all target declarations of the row |
topic_domain, topic_subtopic |
the target's topic. Not shown to the model |
Licence
CC BY 4.0 (LICENSE). Third-party excerpts are attributed per source in
ATTRIBUTION.json.
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