Most learning algorithms in machine learning rely on gradient descent to adjust model parameters, and a growing literature in computational neuroscience leverages these ideas to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying assumption: the choice of distance for synaptic changes (i.e. the geometry of synaptic plasticity). Gradient descent assumes that the distance is Euclidean, but many other distances are possible, and there is no reason that biology necessarily uses Euclidean geometry. Here, using the theoretical tools provided by mirror descent, we show that, regardless of the loss being minimized, the distribution of synaptic weights will depend on the geometry of synaptic plasticity. We use these results to show that experimentally-observed log-normal weight distributions found in several brain areas are not consistent with standard gradient descent (i.e. a Euclidean geometry), but rather with non-Euclidean distances. Finally, we show that it should be possible to experimentally test for different synaptic geometries by comparing synaptic weight distributions before and after learning. Overall, this work shows that the current paradigm in theoretical work on synaptic plasticity that assumes Euclidean synaptic geometry may be misguided and that it should be possible to experimentally determine the true geometry of synaptic plasticity in the brain.

翻译：摘要：机器学习中的大多数学习算法依赖于梯度下降来调整模型参数，而在计算神经科学领域中，越来越多的文献借鉴这些思想来研究大脑中的突触可塑性。然而，这些工作绝大多数忽略了一个关键的基本假设：突触变化的距离选择（即突触可塑性的几何结构）。梯度下降假设距离是欧几里得式的，但许多其他可能的距离也存在，并且没有理由认为生物学必然采用欧几里得几何。在这里，我们利用镜像下降提供的理论工具证明，无论被最小化的损失函数如何，突触权重的分布都将取决于突触可塑性的几何结构。我们利用这些结果表明，在几个脑区中实验观察到的对数正态权重分布不符合标准梯度下降（即欧几里得几何），而是与非欧几里得距离一致。最后，我们证明，通过比较学习前后的突触权重分布，应该能够通过实验检测不同的突触几何结构。总体而言，这项工作表明，当前关于突触可塑性的理论工作假设欧几里得突触几何的范式可能存在误导，并且应有可能通过实验确定大脑中突触可塑性的真实几何结构。