On the Markov Property of Neural Algorithmic Reasoning: Analyses and Methods

TL;DR

This work presents the ForgetNet, which does not use historical embeddings and thus is consistent with the Markov nature of algorithmic reasoning tasks, and introduces G-ForgetNet, which uses a gating mechanism to allow for the selective integration of historical embeddings.

Abstract

Neural algorithmic reasoning is an emerging research direction that endows neural networks with the ability to mimic algorithmic executions step-by-step. A common paradigm in existing designs involves the use of historical embeddings in predicting the results of future execution steps. Our observation in this work is that such historical dependence intrinsically contradicts the Markov nature of algorithmic reasoning tasks. Based on this motivation, we present our ForgetNet, which does not use historical embeddings and thus is consistent with the Markov nature of the tasks. To address challenges in training ForgetNet at early stages, we further introduce G-ForgetNet, which uses a gating mechanism to allow for the selective integration of historical embeddings. Such an enhanced capability provides valuable computational pathways during the model's early training phase. Our extensive experiments, based on the CLRS-30 algorithmic reasoning benchmark, demonstrate that both ForgetNet and G-ForgetNet achieve better generalization capability than existing methods. Furthermore, we investigate the behavior of the gating mechanism, highlighting its degree of alignment with our intuitions and its effectiveness for robust performance.

Figures (10)

• Figure 1: An illustration of (a) the baseline, (b) ForgetNet, and (c) G-ForgetNet methods. $\mathcal{E}$, $\mathcal{P}$, and $\mathcal{D}$ represent the encoder, processor, and decoder module, respectively.
• Figure 2: Illustration of the execution steps in insertion sort. The top row represents the intermediate states, while the bottom row shows the corresponding partially sorted lists. At a specific step, the present state, denoted as the hints, includes the current order (the black pointers), the recently inserted element (the green pointer), and the current iterator (the blue pointer). The present state can fully determine the next intermediate state. The figure is adapted from velivckovic2022clrs.
• Figure 3: Comparison between ForgetNet and the baseline. Reported results are the average of 10 runs with random seeds. Numerical results can be found in Table \ref{['tab:three_model_comparison']}.
• Figure 4: Training curves for the baseline, ForgetNet, and G-ForgetNet methods on the Floyd-Warshall task. The shaded region indicates the standard deviation. Figure is smoothed for clarity.
• Figure 5: Average L2 norm value throughout the training process on the Floyd-Warshall task. The shaded region indicates the standard deviation.