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Exact tiled attention and memory-aware differentiation — Examples

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

None yet.

4 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

One-key tiled attention

Example

Take $n = d = v = 1$ , $Q = [2]$ , $K = [3]$ , $V = [7]$ , scale $c = 1$ , block size $b = 1$ , and allowed-key set $A_{1} = {1}$ . The tiled output is $Y = [7]$ . For upstream scalar $G = [4]$ , the exact reverse output is $d Q = [0]$ , $d K = [0]$ , and $d V = [4]$ .

Example references

Tiled forward and reverse procedures

Exactness of the forward and reverse outputs

Facts & Assumptions

Given: The scalar inputs and procedures in the example statement.

[F1]

Exact tiled softmax attention with block-recomputed vector-Jacobian products: for one allowed key the forward summary is copied as the row state, and the reverse updates use $p_{i j}$ and $γ_{i j}$ .

[F2]

Tiled attention and its recomputed reverse pass are exact: the reverse procedure returns the derivative of the scalarized output for each fixed upstream value.

Verification

1.1F1givenalgebra

The only score is $s_{11} = 2 \cdot 3 = 6$ . Its summary is $(m, ℓ, z) = (6, 1, 7)$ , so the output is $z / ℓ = 7$ .

2.1F1F2step 1.1algebra∎

The probability is $p_{11} = 1$ , the scalar $δ_{1} = G_{1} Y_{1} = 28$ , and $γ_{11} = 1 (4 \cdot 7 - 28) = 0$ . The reverse updates therefore give $d Q = 0$ , $d K = 0$ , and $d V = p_{11} G_{1} = 4$ .

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

An empty first masked key block

Example

Take $n = 2$ , $d = v = 1$ , $Q = K = [\begin{array}{c} 0 \\ 0 \end{array}]$ , $V = [\begin{array}{c} 3 \\ 5 \end{array}]$ , scale $c = 1$ , block size $b = 1$ , and allowed-key sets $A_{1} = {2}$ and $A_{2} = {1, 2}$ . The first key block is empty for query row $1$ but nonempty for query row $2$ . The tiled output is $Y = [\begin{array}{c} 5 \\ 4 \end{array}]$ .

Example references

Empty-block and nonempty-block handling

Exactness for every nonempty allowed row

Facts & Assumptions

Given: The two-row inputs, fixed allowed-key sets, and block size in the example statement.

[F1]

Exact tiled softmax attention with block-recomputed vector-Jacobian products: each row begins in state $⊥$ ; an empty allowed block is skipped, while the first nonempty block summary is copied.

[F2]

Tiled attention and its recomputed reverse pass are exact: a nonempty row's final weighted summary divided by its positive normalizer is its exact attention output.

Verification

1.1F1givenalgebra

For query row $1$ , the first block has $J_{1} \cap A_{1} = \emptyset$ , so the state remains $⊥$ . The second block contains key $2$ , whose score is $0$ and value is $5$ , giving summary $(0, 1, 5)$ and output $5$ .

2.1F1F2step 1.1algebra∎

For query row $2$ , both blocks are allowed and have score $0$ . The first summary is $(0, 1, 3)$ and merging the second gives $(0, 2, 8)$ , so the output is $8 / 2 = 4$ . Both allowed-key sets are nonempty, so these are defined rows of the claimed attention map.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

Two merge orders give the same attention summary

Example

Take $n = 2$ , $d = v = 1$ , both rows of $Q$ equal to $[1]$ , $K_{1} = [0]$ , $K_{2} = [\ln 2]$ , $V_{1} = [0]$ , $V_{2} = [3]$ , scale $c = 1$ , block size $b = 1$ , and $A_{1} = A_{2} = {1, 2}$ . For either query, merging the summaries of blocks ${1}$ then ${2}$ , or ${2}$ then ${1}$ , gives $(m, ℓ, z) = (\ln 2, 3 / 2, 3)$ and output $2$ .

Example references

Block summaries and the merge operation

Summary invariant and order-independent exact output

Facts & Assumptions

Given: The two query rows, two key/value rows, and allowed sets in the example statement.

[F1]

Stable summary of an attention key block: a singleton block summary uses its score maximum, shifted exponential sum, and shifted value sum; a merge rescales both blocks to their common maximum.

[F2]

Tiled attention and its recomputed reverse pass are exact: the completed summary divided by its normalizer is the rowwise masked attention output.

Verification

1.1F1givenalgebra

The two scores are $s_{1} = 0$ and $s_{2} = \ln 2$ . Thus $Σ ({1}) = (0, 1, 0)$ and $Σ ({2}) = (\ln 2, 1, 3)$ .

2.1F1F2step 1.1algebra∎

Merging in either order gives maximum $\ln 2$ , normalizer $e^{- \ln 2} + 1 = 3 / 2$ , and weighted sum $3$ . The output is $3 / (3 / 2) = 2$ in either order.

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A different dropout mask changes the reverse result

Example

Consider the optional inverted-dropout extension with dropout rate $ρ = 1 / 2$ , in which a fixed realization $ξ_{j} \in {0, 1}$ multiplies attention weight $p_{j}$ by $ξ_{j} / (1 - ρ)$ . Take $n = 2$ , $d = v = 1$ , $Q = K = [\begin{array}{c} 0 \\ 0 \end{array}]$ , $V = [\begin{array}{c} 2 \\ 0 \end{array}]$ , $c = 1$ , and let both query rows allow both keys. For upstream matrix $G = [\begin{array}{c} 1 \\ 0 \end{array}]$ , the forward realization $ξ = (1, 0)$ has row- $1$ output $2$ and full value gradient $(d V_{1}, d V_{2}) = (1, 0)$ . Recomputing with the different realization $ξ^{'} = (0, 1)$ would give value gradient $(0, 1)$ for the same upstream matrix, so it is not the reverse derivative of the forward realization.

Example references

The base procedure and its same-input recomputation convention

Facts & Assumptions

Given: The optional dropout rule, the fixed inputs, and the two dropout realizations stated in the example.

[F1]

Exact tiled softmax attention with block-recomputed vector-Jacobian products: the base procedure has no dropout; its reverse pass uses the same fixed attention mask and recomputed scores.

Verification

1.1F1givenalgebra

All scores are zero, so the undropped probabilities are $(1 / 2, 1 / 2)$ . Under $ξ = (1, 0)$ and $ρ = 1 / 2$ , the dropped weights are $(1, 0)$ , hence query row $1$ has output $2$ and derivative with respect to $(V_{1}, V_{2})$ equal to $(1, 0)$ . Since $G_{2} = 0$ , query row $2$ contributes zero to the full value gradient, which is therefore $(1, 0)$ .

2.1F1step 1.1algebra∎

Under $ξ^{'} = (0, 1)$ the dropped weights are $(0, 1)$ , so the recomputed value derivative is $(0, 1)$ . This differs from the derivative of the fixed forward realization in step 1.1, confirming that this extension must replay $ξ$ .

Exact tiled attention and memory-aware differentiation — Examples

2 · Logical flowchart

3 · Definitions, theorems and proofs

4 · Examples, counterexamples and false statements

One-key tiled attention

Example

Example references

Facts & Assumptions

Verification

An empty first masked key block

Example

Example references

Facts & Assumptions

Verification

Two merge orders give the same attention summary

Example

Example references

Facts & Assumptions

Verification

A different dropout mask changes the reverse result

Example

Example references

Facts & Assumptions

Verification

Sources