README.md

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license: apple-amlr
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<h1 align="center"> Reversal Blessing: Thinking Backward May Outpace Thinking Forward in Multi-choice Questions </h1>

<p align="center">
<a href="https://arxiv.org/abs/2502.18435">📃 Paper</a>
•
<a href="https://machinelearning.apple.com" >📝 Blog</a>
</p>

This model card accompanies the research paper, [Reversal Blessing: Thinking Backward May Outpace Thinking Forward in Multi-choice Questions](https://arxiv.org/abs/2502.18435).

Here we release 4 models' checkpoints trained with simulation data described in out paper Section 4. 
Please follow our github README to download and evaluate these models.


<div align="center">

### Results of the Controlled Simulation Study of 4-Digits Multiplication

| || **Forward X** | || **Reverse X** | |
|:--|:--:|:--:|:--:|:--:|:--:|:--:|
| | **L2R** | **R2L(m,n)** | **R2L(m)** | **R2L** | **L2R(m,n)** | **L2R(n)** |
| **Test Accuracy (%)** | **99.81±0.15** | 59.71±1.99 | 60.93±0.88 | **100±0** | 97.82±0.35 | 99.85±0.10 |
| **Train Accuracy (%)** | **99.76±0.15** | 59.03±1.66 | 61.22±1.12 | **100±0** | 97.90±0.42 | 99.98±0.04 |
| **Test Cond. Ent. (nats)** | 0.06 | 1.18 | 0.08 | 0 | 0.84 | 0.01 |
| **Train Cond. Ent. (nats)** | 0.06 | 1.17 | 0.08 | 0 | 0.83 | 0.01 |
| **Theo. Cond. Ent. (nats)** | 0 | 1.49 | 0 | 0 | 1.49 | 0 |
| **Training loss** | **0.86** | 0.94 | 0.94 | **0.86** | 0.94 | 0.94 |

</div>