The impact of allocation strategies in subset learning on the expressive power of neural networks

TL;DR

This paper investigates how different strategies for allocating a fixed number of learnable weights in neural networks affect their expressive capacity, highlighting that more widespread allocation generally improves expressivity.

Contribution

It provides a theoretical framework and benchmark for understanding how weight allocation impacts neural network expressivity, including heuristic principles and empirical validation.

Findings

Widespread weight allocation enhances network expressivity.

Optimal allocations depend on network architecture and task.

Heuristic principles can estimate expressivity of suboptimal allocations.

Abstract

In traditional machine learning, models are defined by a set of parameters, which are optimized to perform specific tasks. In neural networks, these parameters correspond to the synaptic weights. However, in reality, it is often infeasible to control or update all weights. This challenge is not limited to artificial networks but extends to biological networks, such as the brain, where the extent of distributed synaptic weight modification during learning remains unclear. Motivated by these insights, we theoretically investigate how different allocations of a fixed number of learnable weights influence the capacity of neural networks. Using a teacher-student setup, we introduce a benchmark to quantify the expressivity associated with each allocation. We establish conditions under which allocations have maximal or minimal expressive power in linear recurrent neural networks and linear multi-layer feedforward networks. For suboptimal allocations, we propose heuristic principles to estimate their expressivity. These principles extend to shallow ReLU networks as well. Finally, we validate our theoretical findings with empirical experiments. Our results emphasize the critical role of strategically distributing learnable weights across the network, showing that a more widespread allocation generally enhances the network's expressive power.