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

TL;DR

This paper develops a theoretical framework to quantify how allocating a fixed budget of learnable weights across neural networks affects expressivity, using a teacher-student benchmark and a match probability MP(A,m). It shows sharp, often dichotomous expressivity results for linear models (LRNNs and LFFNs) and demonstrates that distributing learnable weights more evenly across layers, rows, or columns generally increases expressive power; nonlinearities (shallow ReLU) further encourage distributed allocations. The results are supported by analytical theorems and numerical experiments, including a MNIST case, highlighting practical implications for learning with resource constraints. The work provides a principled lens for understanding how sparse or targeted weight updates influence a network's capacity to replicate complex mappings, with potential relevance to neuroscience and efficient learning in AI systems.

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.

Figures (3)

• Figure 1: Schema of the student-teacher setup. The match probability (MP) estimates the expressive power of a student with an allocation $\mathcal{A}_i$ of its learnable weights by measuring its ability to match the teacher.
• Figure 2: a. Estimation of MP for allocations in the recurrent weights of LRNN with $d=4$ for different sizes of the hidden state, $n$. Note that MP increases with $n$. b. Same as (a), but with fixed $d/n=\frac{1}{4}$. In this case, $\frac{r}{Tb} = \frac{n}{4}$, which means that allocations using $\frac{1}{4}$ of the rows for every $n$ follow \ref{['thm:lrnn sufficient conditions']}, and are thus maximal. Allocations using less rows are minimal due to \ref{['thm:lrnn necessary conditions']}. Note that MP approaches 1 as $n$ increases. All experiments ran with second order optimization methods. See \ref{['sec:experiments']} for full details.
• Figure 3: Estimation of MP for FFNs. MP increases when distributing the weights. a. Estimation of MP for allocations in 3-layer linear FFNs network, for different sizes of the intermediate layer denoted as $n$. b. Allocation for shallow ReLU network for different sizes of the hidden layer size. All experiments ran with second order optimization methods. c. Allocations in a shallow ReLU network with n=1000 on structured data (MNIST). The MP when the allocation uses more then 40 rows was constant 1. See \ref{['sec:experiments']} for full details.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 4.1
• Definition A.1