Untapped Potential in Self-Optimization of Hopfield Networks: The Creativity of Unsupervised Learning

TL;DR

This work argues that creativity can emerge in a minimal Hopfield network through the Self-Optimization (SO) mode, where unsupervised Hebbian learning continuously reshapes the energy landscape to bias toward better solutions. By treating the learning process as a generative, goal-directed activity, the model exhibits creative products that are novel and appropriate, with their occurrence exceeding chance levels. The study introduces a formal framework that links agency, learning, and creativity, and shows that varying the learning rate and the number of resets yields four distinct regimes, including a creative regime that combines novelty and value. The findings offer a biologically plausible, mathematically tractable platform to study creativity in artificial life and cognitive systems, with implications for open-ended learning, scalable network design, and the analysis of creative processes in minimal agents.

Abstract

The Self-Optimization (SO) model can be considered as the third operational mode of the classical Hopfield Network, leveraging the power of associative memory to enhance optimization performance. Moreover, it has been argued to express characteristics of minimal agency, which renders it useful for the study of artificial life. In this article, we draw attention to another facet of the SO model: its capacity for creativity. Drawing on creativity studies, we argue that the model satisfies the necessary and sufficient conditions of a creative process. Moreover, we show that learning is needed to find creative outcomes above chance probability. Furthermore, we demonstrate that modifying the learning parameters in the SO model gives rise to four different regimes that can account for both creative products and inconclusive outcomes, thus providing a framework for studying and understanding the emergence of creative behaviors in artificial systems that learn.

Figures (9)

• Figure 1: Dynamics of the three operational modes of , distinguished via the rigidness of the landscape metaphor. In all three modes, the initial landscape at $t_{0}$ is defined by the initial weight matrix $\mathbf{W}_{0}$, and the hands holding the two balls represent two possible initial states of the system (here visualized by the two horizontal degrees of freedom). (a) In the classical , the weight matrix $\mathbf{W}_{0}$ does not change and as a result the landscape is static from $t_{0}$ to $t_{f}$. The minimum that the system will converge to highly depends on the initial conditions of the system. (b) In the model, the dynamics depend on the learned weight matrix $\mathbf{W}_{\mathrm{L}}$, that is constantly updated and as a consequence the landscape is constantly changing. With many consecutive Hopfield optimizations, the energy landscape is adjusted to deepen and widen the minima such that at some point, $t_{f}$, only one global attractor is left (second column). Regardless of the initial conditions, the system will converge to that state. Crucially, depending on the specifics of the initial weight matrix and the choice of learning rate $\alpha$, the final global attractor may or may not be the initial global minimum (third column). In the figure, ground shading indicates a static landscape, while represents a changing landscape.
• Figure 2: An illustration of the weight matrix and the effect of learning. It shows a symmetric modular connectivity weight matrix, Eq. (\ref{['eq:modular_W']}), (a) before and (b) after learning ($\alpha=5\mathrm{e}{-7}$) for a system of size $N=100$. Initial weights have 20 modules of size $k=5$, intra-module weights $p$ set at random to either 1 or -1, and inter-module weights set at random to either 0.1 or -0.1.
• Figure 3: Dynamics of the model with and without learning. (a) Without learning. This is equivalent to the dynamics of a regular Hopfield network, where the system will converge to various minima according to the initial state. (b) With learning, Eq. (\ref{['eq:w_update']}). The arrow indicates converges to a single lower energy attractor for all initial random states. On both plots, the energy is computed using Eq. (\ref{['eq:E_W']}) for the chosen initial weight matrix $\mathbf{W}_{\mathrm{0}}$ in Fig. \ref{['fig:W-init']}, and the state at each step is updated asynchronously according to Eq. (\ref{['eq:state_update']}). The plots show the energy for 50 different initial random states each.
• Figure 4: Four regimes of learning outcomes. (a) When $\alpha$ is too low, the outcome is neither novel or appropriate. It's not novel because the distribution remains unchanged over time, nor appropriate because it hasn't converge. (b) When $\alpha$ is too high, the outcome is novel (converging to a previously unvisited state), but not appropriate as that state's energy is higher than before learning. (c) When $\alpha$ is intermediate, the typical case is an appropriate outcome, but not novel, and finally (d) for some $\alpha$, the outcome is both novel and appropriate, i.e., creative. In each plot, the points represent the energy at the end of convergence (e.g., the energies at step=1000 in Fig. \ref{['fig:E_conv']}), for a set before learning (resets 1--1000, dark blue), during learning (1001--2000, red), and after learning (2001-3000, light blue).
• Figure 5: Statistical assessment of creativity in the model: Before Learning (BL) vs After Learning (AL). (a) Energy distribution BL, $p_{\rm BL}(E)$. The blue curve represents a Poisson fit to the data. The yellow, orange, and red dashed lines mark one, two, and three standard deviations ($\sigma_{\rm{BL}}$) from the mean, $\mu_{\rm BL}$, respectively, (i.e., $\mu_{\rm BL} \pm \epsilon$). (b) Energy distribution AL, $p_{\rm{AL},\alpha}(E)$, spread on a logarithmic scale. Dashed lines as in (a). The corresponding solid lines show the probability $p_{\rm AL,\epsilon,\alpha}$ for the AL energy to be lower than $\mu_{\rm BL} - \epsilon$. The black shaded area shows the seed averaged width $\sigma_{\rm{AL}}$ of the AL energy distribution indicating convergence of the system through learning. (c) Novelty $\mathfrak{n}(E)$ and value $\mathfrak{v}(E)$ as defined by Eqs. (\ref{['eq:novelty']}),(\ref{['eq:value']}). (d) Four regimes of learning outcomes (roughly indicated by dashed grey lines), from left to right: Not novel, not appropriate; novel, appropriate; not novel, appropriate; novel, not appropriate. Novelty (blue) and value (orange) are computed by integrating over all the learning outcomes for each learning rate. Appropriateness (blue dots) is the product of value and convergence (green). For a detailed mathematical description see Sec. \ref{['subsec:above-chance']} and Sec. \ref{['subsec:novelty-SO']}.

Theorems & Definitions (2)

• Definition 1: Creative product
• Definition 2: Creative process