`Bandits.ETC.prob_arm_mul_eq_le`🔗

Lemma

From the authors

The probability that at time K * m the ETC algorithm chooses arm a is at most exp(- m * Δ_a^2 / (4 * σ2)).

Types

Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.

Given

K : ℕNeZero K
m : ℕ
ν : ProbabilityTheory.Kernel (Fin K) ℝA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel νA kernel is a Markov kernel if every measure in its image is a probability measure.
P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsProbabilityMeasure PA measure μ is called a probability measure if μ univ = 1.
O : ℕ → Ω → Unit
A : ℕ → Ω → Fin K
R : ℕ → Ω → ℝ
σ2 : NNReal
a : Fin K

Assuming

h : Learning.IsAlgEnvSeq O A R (etcAlgorithm K m) (Learning.stationaryEnv ν) PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.
hν : ∀ (a : Fin K), ProbabilityTheory.HasSubgaussianMGF (fun x => x - ∫ (x : ℝ), id x ∂ν a) σ2 (ν a)A random variable X has a sub-Gaussian moment-generating function with parameter c with respect to a measure μ if for all t : ℝ, exp (t * X) is μ-integrable and the moment-generating funct…
hm : m ≠ 0

Then

P.real {ω | A (K * m) ω = a} ≤ Real.exp (-↑m * gap ν a ^ 2 / (4 * ↑σ2))

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

NeZero : {R : Type u_1} → [Zero R] → R → Prop

A type-class version of `n ≠ 0`.

Nat : Type

The natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.

ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A kernel is a Markov kernel if every measure in its image is a probability measure.

ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)

A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.

Fin : ℕ → Type

Natural numbers less than some upper bound.

In particular, a `Fin n` is a natural number `i` with the constraint that `i < n`. It is the
canonical type with `n` elements.

Real : Type

The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers.

MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called a probability measure if `μ univ = 1`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Unit : Type

The canonical type with one element. This element is written `()`.

`Unit` has a number of uses:
* It can be used to model control flow that returns from a function call without providing other
  information.
* Monadic actions that return `Unit` have side effects without computing values.
* In polymorphic types, it can be used to indicate that no data is to be stored in a particular
  field.

NNReal : Type

Nonnegative real numbers, denoted as `ℝ≥0` within the NNReal namespace

Learning.IsAlgEnvSeq : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {Ω : Type u_4} →
        {m𝓞 : MeasurableSpace 𝓞} →
          {m𝓐 : MeasurableSpace 𝓐} →
            {m𝓨 : MeasurableSpace 𝓨} →
              {mΩ : MeasurableSpace Ω} →
                (ℕ → Ω → 𝓞) →
                  (ℕ → Ω → 𝓐) →
                    (ℕ → Ω → 𝓨) →
                      Learning.Algorithm 𝓞 𝓐 𝓨 →…

An algorithm-environment sequence: a sequence of observations, actions and feedbacks generated
by an algorithm interacting with an environment.

Go to its page

Bandits.etcAlgorithm : (K : ℕ) → [NeZero K] → ℕ → Learning.Algorithm Unit (Fin K) ℝ

The Explore-Then-Commit algorithm: deterministic algorithm that chooses the next arm according
to `ETC.nextArm`.

Go to its page

Learning.stationaryEnv : {𝓐 : Type u_2} →
  {𝓨 : Type u_3} →
    {m𝓐 : MeasurableSpace 𝓐} →
      {m𝓨 : MeasurableSpace 𝓨} →
        (ν : ProbabilityTheory.Kernel 𝓐 𝓨) → [ProbabilityTheory.IsMarkovKernel ν] → Learning.Environment Unit 𝓐 𝓨

A stationary environment without observations, in which the distribution of the next feedback
depends only on the last action.

Go to its page

ProbabilityTheory.HasSubgaussianMGF : {Ω : Type u_1} →
  {mΩ : MeasurableSpace Ω} →
    (Ω → ℝ) → NNReal → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasSubgaussianMGF._auto_1 → Prop

A random variable `X` has a sub-Gaussian moment-generating function with parameter `c`
with respect to a measure `μ` if for all `t : ℝ`, `exp (t * X)` is `μ`-integrable and
the moment-generating function of `X` is bounded by `exp (c * t ^ 2 / 2)` for all `t : ℝ`.
This implies in particular that `X` has expectation 0.

This is equivalent to `Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())`,
as proved in `HasSubgaussianMGF_iff_kernel`.
Properties about sub-Gaussian moment-generating functions should be proved first for
`Kernel.HasSubgaussianMGF` when possible.

HSub.hSub : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HSub α β γ] → α → β → γ

`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).

MeasureTheory.integral : {α : Type u_6} →
  {G : Type u_7} →
    [inst : NormedAddCommGroup G] → [NormedSpace ℝ G] → {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → G) → G

The Bochner integral

id : {α : Sort u} → α → α

The identity function. `id` takes an implicit argument `α : Sort u`
(a type in any universe), and an argument `a : α`, and returns `a`.

Although this may look like a useless function, one application of the identity
function is to explicitly put a type on an expression. If `e` has type `T`,
and `T'` is definitionally equal to `T`, then `@id T' e` typechecks, and Lean
knows that this expression has type `T'` rather than `T`. This can make a
difference for typeclass inference, since `T` and `T'` may have different
typeclass instances on them. `show T' from e` is sugar for an `@id T' e`
expression.

Ne : {α : Sort u} → α → α → Prop

`a ≠ b`, or `Ne a b` is defined as `¬ (a = b)` or `a = b → False`,
and asserts that `a` and `b` are not equal.


Conventions for notations in identifiers:

 * The recommended spelling of `≠` in identifiers is `ne`.

LE.le : {α : Type u} → [self : LE α] → α → α → Prop

The less-equal relation: `x ≤ y` 

Conventions for notations in identifiers:

 * The recommended spelling of `≤` in identifiers is `le`.

MeasureTheory.Measure.real : {α : Type u_5} → {m : MeasurableSpace α} → MeasureTheory.Measure α → Set α → ℝ

The real-valued version of a measure. Maps infinite measure sets to zero. Use as `μ.real s`.
The API is developed in `Mathlib/MeasureTheory/Measure/Real.lean`.

Set.ofPred : {α : Type u} → (α → Prop) → Set α

Turn a predicate `p : α → Prop` into a set, also written as `{x | p x}`

HMul.hMul : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HMul α β γ] → α → β → γ

`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.

Eq : {α : Sort u_1} → α → α → Prop

The equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.

Real.exp : ℝ → ℝ

The real exponential function, defined as the real part of the complex exponential

Neg.neg : {α : Type u} → [self : Neg α] → α → α

`-a` computes the negative or opposite of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator).

Bandits.gap : {𝓐 : Type u_2} → {m𝓐 : MeasurableSpace 𝓐} → ProbabilityTheory.Kernel 𝓐 ℝ → 𝓐 → ℝ

Gap of an action `a`: difference between the highest mean of the actions and the mean of `a`.

Go to its page

HPow.hPow : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HPow α β γ] → α → β → γ

`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`.

HDiv.hDiv : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HDiv α β γ] → α → β → γ

`a / b` computes the result of dividing `a` by `b`.
The meaning of this notation is type-dependent.
* For most types like `Nat`, `Int`, `Rat`, `Real`, `a / 0` is defined to be `0`.
* For `Nat`, `a / b` rounds downwards.
* For `Int`, `a / b` rounds downwards if `b` is positive or upwards if `b` is negative.
  It is implemented as `Int.ediv`, the unique function satisfying
  `a % b + b * (a / b) = a` and `0 ≤ a % b < natAbs b` for `b ≠ 0`.
  Other rounding conventions are available using the functions
  `Int.fdiv` (floor rounding) and `Int.tdiv` (truncation rounding).
* For `Float`, `a / 0` follows the IEEE 754 semantics for division,
  usually resulting in `inf` or `nan`. 

Conventions for notations in identifiers:

 * The recommended spelling of `/` in identifiers is `div`.

Code

lemma prob_arm_mul_eq_le (h : IsAlgEnvSeq O A R (etcAlgorithm K m) (stationaryEnv ν) P)
    (hν : ∀ a, HasSubgaussianMGF (fun x ↦ x - (ν a)[id]) σ2 (ν a)) (a : Fin K)
    (hm : m ≠ 0) :
    P.real {ω | A (K * m) ω = a} ≤ Real.exp (- (m : ℝ) * gap ν a ^ 2 / (4 * σ2))

Proof

by
  have h_pos : 0 < K * m := Nat.mul_pos (Nat.pos_of_neZero K) hm.bot_lt
  have h_le : P.real {ω | A (K * m) ω = a}
      ≤ P.real {ω | sumRewards A R (bestArm ν) (K * m) ω ≤ sumRewards A R a (K * m) ω} := by
    simp_rw [measureReal_def]
    gcongr 1
    · simp
    refine measure_mono_ae ?_
    exact sumRewards_bestArm_le_of_arm_mul_eq h a hm
  exact h_le.trans (probReal_sumRewards_le_sumRewards_le h hν a)

Meaning last changed in v4.34.0-rc2-90-gdde3322 (2026-09-16), the 9th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 21 project declarations, 128 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Bandits.ETC.prob_arm_mul_eq_le🔗

`Bandits.ETC.prob_arm_mul_eq_le`🔗