DYNAMICAL EQUATIONS WITH BOTTOM-UP SELF-ORGANIZING PROPERTIES LEARN ACCURATE DYNAMICAL HIERARCHIES WITHOUT ANY LOSS FUNCTION

Abstract

Self-organization is ubiquitous in nature and mind. However, machine learning and theories of cognition still barely touch the subject. The hurdle is that general patterns are difficult to define in terms of dynamical equations and designing a system that could learn by reordering itself is still to be seen. Here, we propose a learning system, where patterns are defined within the realm of nonlinear dynamics with positive and negative feedback loops, allowing attractor-repeller pairs to emerge for each pattern observed. Experiments reveal that such a system can map temporal to spatial correlation, enabling hierarchical structures to be learned from sequential data. The results are accurate enough to surpass stateof-the-art unsupervised learning algorithms in seven out of eight experiments as well as two real-world problems. Interestingly, the dynamic nature of the system makes it inherently adaptive, giving rise to phenomena similar to phase transitions in chemistry/thermodynamics when the input structure changes. Thus, the work here sheds light on how self-organization can allow for pattern recognition and hints at how intelligent behavior might emerge from simple dynamic equations without any objective/loss function.

1. INTRODUCTION

Self-organization is present in diverse scientific fields, from biology (Misteli, 2007; Deglincerti et al., 2016; Sasai, 2013) to neuroscience (Linsker, 1988; Tognoli & Kelso, 2014; Imam & L. Finlay, 2020; Schoner & Kelso, 1988) , chemistry (Montalti et al., 2017; Lehn, 2002a; b) and physics (Haken, 1975; Wickman & Korley, 1998; Tersoff et al., 1996; Haken, 1977) . It shows how order can arise intrinsically from a system. It is a set of interactions that allows for the emergence of patterns and is responsible for complex behavior from simple interactions (Kauffman et al., 1993; Haken, 1977) . Albeit the ubiquitous presence of self-organization in nature and in the brain, it is unknown how self-organization can lead to intelligence. For this reason, theories of intelligence rarely use the concept in their development. The free energy principle (Friston, 2010; 2009) and reinforcement learning paradigms (Sutton & Barto, 2018; Mnih et al., 2015; Schrittwieser et al., 2020) define a top-down view of learning based on objectives that are satisfied locally or globally. However, from a bottom-up perspective, it is still barely understood how Hebbian learning (Hebb, 2005; Magee & Johnston, 1997) and other neuron behaviors allow for top-down theories of intelligence to emerge. In fact, there is strong evidence the brain does not behave as a computer but as a more self-organizing system (GRAY, 1987; Eckhorn et al., 1988) . In this paper, we show how the learning of patterns can be achieved by Hebbian and anti-Hebbian learning dynamics, linking between Hebbian learning and top-down theories of intelligence (Hebb, 2005) . The recent success of machine learning, similar to the current theories of intelligence, is mostly given to optimization-based deep learning algorithms. While deep learning utilizes optimization and loss functions (objective functions) to learn the model's parameters and improve in the task at hand, self-organization existence in machine learning is mostly limited to Self-Organizing Map (SOM) variations (Kohonen, 1982; Chang et al., 2020; Reker et al., 2014) . Such SOMs are only employed in clustering and dimensional reduction tasks, as they lack the ability to find patterns in data required for further processing and acting on the environment. Here, inspired by many successful modelings of neuron behaviors based on dynamical equations composed of attractor dynamics (Tognoli & Kelso, 2014; Wills et al., 2005; Spalla et al., 2021; Ooi et al., 2018) , we show how a system of dynamical equations can give rise to order and represent patterns. Our proposed system is arguably more biologically plausible, and it is also shown to be more accurate and adaptive than state-of-the-art unsupervised algorithms. In fact, it sets up a foundation for a new paradigm in machine learning solely based on self-organization from dynamical equations, namely Self-Organizing Dynamical Equations, which are inherently accurate and adaptive. We propose Hierarchical Temporal Spatial Feature Map (TSFMap), a learning system implementing the Self-Organizing Dynamical Equations paradigm. It creates a space in which distances in it reflect the temporal correlation between input variables. A simple clustering in this self-organized space reveals that the representation learned is very accurate. Adaptation comes from the fact that the proposed system, Hierarchical TSFMap, couples its internal dynamics with the input, resulting in patterns encoded as emergent attractor-repellers at equilibrium. Consequently, alterations in the underlying structure of the problem result in different equilibrium with new attractor-repellers, triggering an inherent adaptation when the problem changes. Interestingly, structural changes in the environment cause in Hierarchical TSFMap a phenomenon very similar to phase transition observed in thermodynamics, and chemistry, among other areas (Fig. 1 ). In this paper, Hierarchical TSFMap is evaluated in one of the hardest types of patterns, e.g., recognition of dynamical and imbalanced hierarchical patterns present in sequential data. The problem of learning the hierarchical relationships from sequential input is a challenging unsolved one (Uddén et al., 2020) . This becomes even harder when the problem structure is dynamic, e.g., variable correlations change over time. Since any information can be serialized, the pattern recognition over sequences is a general one that can be applied ubiquitously to any type of serialized data. Albeit the difficulty of the task, Hierarchical TSFMap provides, perhaps surprisingly, near-optimal solutions to more than half of the problems. Lastly, we have demonstrated that Hierarchical TSFMap can extract hierarchical structures from sequential data generated from two real-world networks: (1) Zachary's karate club network and (2) Lusseau's bottlenose dolphin social network.

2. RELATED WORK

A recent work (Vasconcellos Vargas & Asabuki, 2021 ) demonstrated how a self-organizing system called SyncMap, can learn features from sequences using dynamical equations alone (e.g., without any type of optimization). Here we go beyond this work on simple chunks to show how dynamical equations that self-organize compose a paradigm and can be used to deal with challenging hierarchical structures and imbalanced problems. In fact, the experiments suggest that Hierarchical TSFMap can deal with dynamical variations of the problems with little difficulty.



Figure 1: Hierarchical TSFMap's phase transition. A phenomenon similar to phase transition takes place in the proposed algorithm when the underlying structure of the problem changes. (a) Lines indicate the relative distance for all weight pairs. (b) The average rate of change for all weight pairs' distances (a). Random initialized weights start to form patterns with respect to the input and enter an equilibrium state. Once the problem's data structure is altered, Hierarchical TSFMap automatically adapts its weights. Subsequently, weights enter another equilibrium state.

