标题： 压缩计算：通用与问题的玩具模型中的密集电路 显示英文标题

标题： Compressed Computation: Dense Circuits in a Toy Model of the Universal-AND Problem

摘要： 神经网络能够实现叠加——表示的特征数量超过维度数量。 最近的研究考虑了计算而非存储的类似概念，并提出了理论构造。 但迄今为止，很少有研究探讨这些电路是否可以在实践中被学习。 在本工作中，我们研究了一个用于通用-AND问题的玩具模型，该模型计算所有$m\choose 2$对$m$稀疏输入的AND。 决定非线性激活数量的隐藏维度受到限制，迫使模型找到一种计算高效的电路，称为压缩计算。 我们发现训练过程找到了一个简单的解决方案，该方案不对应于理论构造。 它完全密集——每个神经元都对每个输出做出贡献。 该解决方案自然地随维度扩展，以误差率为代价换取神经元效率。 它对稀疏性的变化和其他关键参数同样具有鲁棒性，并且可以自然地扩展到其他布尔操作和布尔电路。 我们详细解释了找到的解决方案，并计算了为什么在低稀疏性下它比理论构造更高效。 我们的发现揭示了模型喜欢形成的电路类型以及叠加表示的灵活性。 这有助于更广泛地理解网络电路和可解释性。 显示英文摘要

摘要： Neural networks are capable of superposition -- representing more features than there are dimensions. Recent work considers the analogous concept for computation instead of storage, proposing theoretical constructions. But there has been little investigation into whether these circuits can be learned in practice. In this work, we investigate a toy model for the Universal-AND problem which computes the AND of all $m\choose 2$ pairs of $m$ sparse inputs. The hidden dimension that determines the number of non-linear activations is restricted to pressure the model to find a compute-efficient circuit, called compressed computation. We find that the training process finds a simple solution that does not correspond to theoretical constructions. It is fully dense -- every neuron contributes to every output. The solution circuit naturally scales with dimension, trading off error rates for neuron efficiency. It is similarly robust to changes in sparsity and other key parameters, and extends naturally to other boolean operations and boolean circuits. We explain the found solution in detail and compute why it is more efficient than the theoretical constructions at low sparsity. Our findings shed light on the types of circuits that models like to form and the flexibility of the superposition representation. This contributes to a broader understanding of network circuitry and interpretability.