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Understanding Addition in Transformers
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Reverse-engineers a one-layer, three-head Transformer trained on n-digit integer addition, showing it processes digits in parallel streams with left-to-right staging and explaining its failure on rare carry-cascade cases.
Problem
Mechanistic understanding of how a small Transformer performs multi-digit integer addition, including why it is accurate on most questions but fails on rare carry-cascade cases.
Approach
Introduces a mathematical framework of necessary addition subtasks (BaseAdd, MakeCarry, MakeSum, UseCarry, UseSum) used only for analysis, and combines training-loss decomposition, attention-pattern inspection, and ablation to reverse-engineer the model's left-to-right, digit-parallel algorithm.
Key results
- Final training loss of 0.009 for 5-digit, 0.011 for 10-digit and 0.031 for 15-digit addition models.
- The model computes each answer in 6 steps (rows 11 to 16) for 5-digit addition, with three attention heads giving a double-staircase pattern.
- Mean-ablating head 1 raises BaseAdd loss from about 0.05 to 4.7 (or 3.7 in the text), identifying it as the BaseAdd head; the model fails on cascading UseSum cases such as 445+555.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the Base Add computation performed by Head 1 (ablation) | 1-layer Transformer · 5-digit addition, Base Add questions | average loss: 4.712 -> 0.021 | Helped |
| Add the function of Head 0 (ablation) | 1-layer Transformer · 5-digit addition, Base Add questions | average loss: 0.016 -> 0.021 | Helped |
| Add the function of Head 2 (ablation) | 1-layer Transformer · 5-digit addition, Base Add questions | average loss: 0.062 -> 0.021 | Helped |
| Use the calculations at token positions 11–16 (ablation) | 1-layer Transformer · 5-digit addition | average loss: 0.322, 0.497, 0.604, 0.921, 1.181, 1.021 when ablated | Helped |
| Avoid relying on token positions 0–10 and 17 for the addition calculation (ablation) | 1-layer Transformer · 5-digit addition | average loss: 0.070 | No clear effect |
| Use three attention heads rather than two or four (ablation) | 1-layer Transformer · 5-digit addition | attention-task separation | Helped |
| Use 2 or 4 attention heads instead of 3 (ablation) | 1-layer Transformer · 5-digit addition | interpretability | Hurt |
| Implement the cascading Use Sum 9 cases accurately | 1-layer Transformer · addition with Use Sum 9 cascades | answer correctness | Helped |
| Change the random seed, optimization algorithm, and answer format while retaining a 1-layer model | 1-layer Transformer · 5-digit addition | algorithm behavior | No clear effect |
Surprising findings
- The model starts computing answers later than it could, and it processes digits left to right, which is the opposite of the human right-to-left carry procedure, causing failures on carry cascades.
- Digit 4 and the leading answer digits are learned faster than middle digits, using more attention-head compute than the other digits.
Limitations
- Analysis is limited to a one-layer, small Transformer and does not establish the mechanisms in larger multi-layer models.
- The MLP key-value memory claim is not proven experimentally, and the paper admits some details (such as A5/A4 algorithm) are unresolved.
- Failure on rare cases such as 999,999,999 + 1 is noted but not explained.
Experimental setup
Datasets: synthetic n-digit addition questions (5, 10, 15 digit)
Benchmarks: 5-digit addition test questions 10- and 15-digit addition
Metrics: negative log likelihood loss answer accuracy loss after ablation
Builds on: TransformerLens mathematical framework for transformer circuits
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