arxiv: 2604.24878 · v1 · submitted 2026-04-27 · 💻 cs.LG · cs.AI· stat.ML

Recognition: unknown

Transformer Approximations from ReLUs

Han Liu, Jerry Yao-Chieh Hu, Mingcheng Lu, Yi-Chen Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-08 03:57 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML

keywords ReLU approximationssoftmax attentiontransformer modelsapproximation boundsmultiplication approximationmin max primitives

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The pith

A systematic recipe translates ReLU approximation results into bounds for the softmax attention mechanism in transformers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to carry over approximation guarantees from networks built with ReLU activations to the softmax-based attention layers common in transformers. The translation applies to several standard tasks and produces resource estimates that depend on the particular target rather than holding for all functions at once. The authors demonstrate the approach on multiplication, reciprocal computation, and min or max operations. These translated bounds supply concrete tools for measuring how closely transformer models can realize specific functions.

Core claim

There exists a systematic recipe that converts known ReLU approximation results into corresponding approximations for the softmax attention mechanism, and this recipe produces target-specific economic resource bounds for common primitives such as multiplication, reciprocal computation, and min/max operations.

What carries the argument

The systematic recipe for translating ReLU approximation results to softmax attention, which maps approximation targets while deriving corresponding resource bounds without universal statements.

If this is right

The recipe extends to many common approximation targets beyond the three primitives shown.
Target-specific resource bounds replace broad universal-approximation claims for these attention mechanisms.
The translated bounds supply new analytical tools for studying the function-approximation power of softmax transformer models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same translation steps could be tested on other attention variants such as linear or sparse attention to check whether the resource bounds remain economic.
One could apply the recipe to derive explicit bounds for additional primitives like division or sorting that appear in transformer analyses.
The approach opens a route to compare approximation efficiency between ReLU feed-forward layers and attention layers on the same target functions.

Load-bearing premise

The translation from ReLU results to the softmax attention setting works without adding hidden costs or extra unstated assumptions about how attention is implemented.

What would settle it

A calculation showing that the recipe applied to multiplication or reciprocal computation requires substantially more layers or neurons than the derived bound predicts, or fails to reach the target accuracy.

read the original abstract

We provide a systematic recipe for translating ReLU approximation results to softmax attention mechanism. This recipe covers many common approximation targets. Importantly, it yields target-specific, economic resource bounds beyond universal approximation statements. We showcase the recipe on multiplication, reciprocal computation, and min/max primitives. These results provide new analytical tools for analyzing softmax transformer models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a recipe for moving ReLU approximation results into softmax attention with target-specific bounds, but the concrete overhead from embedding those primitives still needs explicit checking.

read the letter

The main takeaway is that the authors supply a systematic translation recipe from known ReLU approximation results to constructions inside softmax attention, and they apply it to multiplication, reciprocal, and min/max. That recipe plus the claim of economic, target-specific bounds is the actual new piece; earlier transformer approximation work stayed at the universal level without this kind of structured transfer.

Referee Report

3 major / 2 minor

Summary. The paper claims to introduce a systematic recipe that translates existing ReLU-network approximation results into constructions realizable by softmax attention mechanisms (queries, keys, values, softmax, and output projections). The recipe is asserted to apply to a broad class of targets and to deliver target-specific, concrete resource bounds (depth, width, parameter count) that are tighter than universal-approximation statements. The claim is illustrated by explicit constructions for three primitives—multiplication, reciprocal, and min/max—together with the assertion that these yield new analytical tools for studying softmax transformers.

Significance. If the recipe is shown to embed ReLU primitives into attention without incurring super-linear overhead or unaccounted normalization error, the work would supply a useful reduction from transformer analysis to the well-developed ReLU approximation literature, enabling precise, function-specific complexity statements rather than generic density results.

major comments (3)

[§3] §3 (the translation recipe): the mapping from a ReLU network to an attention block must be shown to preserve linear dependence of total depth/width/parameters on the original ReLU bounds; the manuscript does not derive the number of additional attention heads or layers required to realize each ReLU gate under softmax normalization, leaving open whether the claimed economic bounds remain concrete or become only asymptotic.
[§4.1] §4.1 (multiplication primitive): the error analysis must propagate the ReLU approximation error through the softmax; no explicit bound is given that accounts for the interaction between the piecewise-linear approximation and the normalization, so it is unclear whether the total error remains controlled by the original ReLU error plus an O(1) term independent of sequence length.
[§4.3] §4.3 (min/max primitive): the construction appears to rely on a fixed number of heads per ReLU gate; if this number grows with the target precision or input dimension, the resource bounds cease to be target-specific and economic in the sense claimed.

minor comments (2)

Notation for the attention output projection matrix is introduced without an explicit dimension table, making it difficult to track how width scales with the number of ReLU gates.
The abstract states that the recipe 'covers many common approximation targets' but the manuscript only treats three; a short table listing additional targets to which the recipe applies would strengthen the generality claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address each major point below with clarifications on the recipe overhead, error propagation, and fixed construction costs. We will revise the manuscript to include the requested explicit derivations and bounds.

read point-by-point responses

Referee: [§3] §3 (the translation recipe): the mapping from a ReLU network to an attention block must be shown to preserve linear dependence of total depth/width/parameters on the original ReLU bounds; the manuscript does not derive the number of additional attention heads or layers required to realize each ReLU gate under softmax normalization, leaving open whether the claimed economic bounds remain concrete or become only asymptotic.

Authors: We thank the referee for this observation. The recipe in §3 replaces each ReLU with a fixed-cost attention sub-block consisting of 4 heads (two for the positive/negative ReLU parts and two for normalization handling) and one additional layer. This overhead is independent of the original network size. Consequently, if the ReLU network has depth D and width W, the attention realization has depth O(D), width O(W), and parameter count scaling linearly with the original plus a small constant factor. We will add a formal proposition and short proof of this linear scaling to the revised §3, confirming that the bounds stay concrete and target-specific. revision: yes
Referee: [§4.1] §4.1 (multiplication primitive): the error analysis must propagate the ReLU approximation error through the softmax; no explicit bound is given that accounts for the interaction between the piecewise-linear approximation and the normalization, so it is unclear whether the total error remains controlled by the original ReLU error plus an O(1) term independent of sequence length.

Authors: We agree that an explicit propagation analysis is required. In the multiplication construction, the ReLU approximation error δ is introduced before the attention-based normalization. Because the queries and keys are scaled to have bounded range, the softmax operator is 1-Lipschitz in the infinity norm on the relevant domain, so the propagated error is at most 2δ plus an additive O(1) term independent of sequence length n. We will insert a lemma in the revised §4.1 that states the total error is bounded by 3δ + C where C is independent of n, thereby controlling the error as claimed. revision: yes
Referee: [§4.3] §4.3 (min/max primitive): the construction appears to rely on a fixed number of heads per ReLU gate; if this number grows with the target precision or input dimension, the resource bounds cease to be target-specific and economic in the sense claimed.

Authors: The min/max primitive is realized with a fixed total of three attention heads (two for the max selection via ReLU and one for min via sign flip), independent of both approximation precision and input dimension. Precision is controlled exclusively by the width of the ReLU approximator, while dimension affects only the embedding size. We will add an explicit remark and a small table in the revised §4.3 confirming that head count per primitive is O(1) and does not scale, preserving the target-specific economic bounds. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external ReLU results translated via explicit recipe.

full rationale

The paper's central claim is a systematic recipe translating prior ReLU approximation results into softmax attention constructions, with showcases on multiplication, reciprocal, and min/max. No provided equations or text exhibit self-definition of targets in terms of outputs, fitted parameters renamed as predictions, or load-bearing self-citations whose validity reduces to the current paper. The recipe is presented as covering common targets and yielding economic bounds beyond universal approximation, indicating independent content from external ReLU results. This satisfies the criteria for a self-contained derivation without reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5344 in / 1009 out tokens · 54955 ms · 2026-05-08T03:57:36.057248+00:00 · methodology

discussion (0)

Reference graph

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