`Learning.Algorithm.comap`🔗

Definition

From the authors

The algorithm with observations in 𝓞' and feedbacks in 𝓨' obtained from alg : Algorithm 𝓞 𝓐 𝓨 by transforming the pair (past rounds, current observation) by F n at each round n before applying the policy of alg.

This is the primitive transport operation on algorithms: Algorithm.comapObs and Algorithm.comapFeedback are the special cases in which F n is a round-wise map of the observation and of the feedback.

Types

𝓞 : Type u_1m𝓞 : MeasurableSpace 𝓞A measurable space is a space equipped with a σ-algebra.
𝓞' : Type u_2m𝓞' : MeasurableSpace 𝓞'
𝓐 : Type u_4m𝓐 : MeasurableSpace 𝓐
𝓨 : Type u_7m𝓨 : MeasurableSpace 𝓨
𝓨' : Type u_8m𝓨' : MeasurableSpace 𝓨'

Given

alg : Algorithm 𝓞 𝓐 𝓨A stochastic, sequential algorithm.
F : (n : ℕ) → Hist 𝓞' 𝓐 𝓨' n × 𝓞' → Hist 𝓞 𝓐 𝓨 n × 𝓞

Assuming

hF : ∀ (n : ℕ), Measurable (F n)A function f between measurable spaces is measurable if the preimage of every measurable set is measurable.

Result

Algorithm 𝓞' 𝓐 𝓨'

Body

{ policy := fun n => (alg.policy n).comap (F n) ⋯, isMarkovKernel_policy := ⋯ }

MeasurableSpace : Type u_6 → Type u_6

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

Learning.Algorithm : (𝓞 : Type u_5) →
  (𝓐 : Type u_6) →
    (𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → Type (max (max u_5 u_6) u_7)

A stochastic, sequential algorithm.
At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives
feedback in `𝓨`. The action is a random function of the past rounds and the current observation.

Go to its page

Prod : Type u → Type v → Type (max u v)

The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`.

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.

Learning.Hist : Type u_5 → Type u_6 → Type u_7 → ℕ → Type (max (max u_7 u_6) u_5)

History of `n` complete rounds; `n = 0` is the empty history.

Go to its page

Measurable : {α : Type u_1} → {β : Type u_2} → [MeasurableSpace α] → [MeasurableSpace β] → (α → β) → Prop

A function `f` between measurable spaces is measurable if the preimage of every
measurable set is measurable.

ProbabilityTheory.Kernel.comap : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} →
        {γ : Type u_4} →
          {mγ : MeasurableSpace γ} →
            ProbabilityTheory.Kernel α β → (g : γ → α) → Measurable g → ProbabilityTheory.Kernel γ β

Pullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`.
We include measurability in the assumptions instead of using junk values
to make sure that typeclass inference can infer that the `comap` of a Markov kernel
is again a Markov kernel.

Code

def Algorithm.comap (alg : Algorithm 𝓞 𝓐 𝓨)
    (F : (n : ℕ) → Hist 𝓞' 𝓐 𝓨' n × 𝓞' → Hist 𝓞 𝓐 𝓨 n × 𝓞) (hF : ∀ n, Measurable (F n)) :
    Algorithm 𝓞' 𝓐 𝓨' where
  policy n := (alg.policy n).comap (F n) (hF n)

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).

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: 3 project declarations, 10 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.

Learning.Algorithm.comap🔗

`Learning.Algorithm.comap`🔗