Memorisation and forgetting in a learning Hopfield neural network: bifurcation mechanisms, attractors and basins

TL;DR

This work comprehensively analyse mechanisms of memory formation in an 81-neuron Hopfield network undergoing Hebbian learning by revealing bifurcations leading to formation and destruction of attractors and their basin boundaries, and demonstrates mechanisms of catastrophic forgetting.

Abstract

Despite explosive expansion of artificial intelligence based on artificial neural networks (ANNs), these are employed as "black boxes'', as it is unclear how, during learning, they form memories or develop unwanted features, including spurious memories and catastrophic forgetting. Much research is available on isolated aspects of learning ANNs, but due to their high dimensionality and non-linearity, their comprehensive analysis remains a challenge. In ANNs, knowledge is thought to reside in connection weights or in attractor basins, but these two paradigms are not linked explicitly. Here we comprehensively analyse mechanisms of memory formation in an 81-neuron Hopfield network undergoing Hebbian learning by revealing bifurcations leading to formation and destruction of attractors and their basin boundaries. We show that, by affecting evolution of connection weights, the applied stimuli induce a pitchfork and then a cascade of saddle-node bifurcations creating new attractors with their basins that can code true or spurious memories, and an abrupt disappearance of old memories (catastrophic forgetting). With successful learning, new categories are represented by the basins of newly born point attractors, and their boundaries by the stable manifolds of new saddles. With this, memorisation and forgetting represent two manifestations of the same mechanism. Our strategy to analyse high-dimensional learning ANNs is universal and applicable to recurrent ANNs of any form. The demonstrated mechanisms of memory formation and of catastrophic forgetting shed light on the operation of a wider class of recurrent ANNs and could aid the development of approaches to mitigate their flaws.

Figures (17)

• Figure 1: Illustration of construction of input signals $I_{i}(t)$ to the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}) with $N$$=$$81$ from training Set 1 of vectors $\mathbf{I}^k$ ($k$$=$$1,$$\ldots$$,6$) given in Tabs. \ref{['supp-tab:tab1-1']}--\ref{['supp-tab:tab1-2']} of Supplementary Note. Vertical green dashed line indicates the time $t_s$ during which a single vector $\mathbf{I}^k$ is applied to the NN. Red dashed line indicates one full period of stimulus $\mathbf{I}(t)$ (duration of "training epoch"), which is equal to $6t_s$. Panels (a), (b) and (c) show inputs $I_{i}(t)$ to the $1^{\textrm{st}}$, $2^{\textrm{nd}}$ and $81^{\textrm{st}}$ neurons, respectively.
• Figure 2: Evolution of connection weights $\omega_{ij}(t)$ (solid lines in various colours) during learning by the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}) with $N$$=$$81$, $A$$=$$30$, $B_{ij}$$=$$B$$=$$300$$(\forall i,j)$, $g$$=$$0.3$ and $\lambda$$=$$1.4$. Panels show learning from two different training sets (see Tabs. \ref{['supp-tab:tab1-1']}--\ref{['supp-tab:tab1-2']}): (a) Set 1 and (b) Set 2. Vertical dashed lines mark four different stages of learning corresponding to times $t$ given in brackets, at which instantaneous weights $\omega_{ij}$ are collected to reveal memories formed, as illustrated in Fig. \ref{['EJNB_NN_bif_mech_appeal_final2:fig:fig3.1']}.
• Figure 3: Memories formed at various stages of learning by the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}) with $N$$=$$81$, $A$$=$$30$, $B_{ij}$$=$$B$$=$$300$$(\forall i,j)$, $g$$=$$0.3$ and $\lambda$$=$$1.4$. Panels illustrate stages of learning from two different training sets (see Tabs. \ref{['supp-tab:tab1-1']}--\ref{['supp-tab:tab1-2']}): (a)--(d) Set 1 and (e)--(h) Set 2. Panels display projections onto the $(x_1,x_2)$-plane of phase portraits of the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq3.1']}) with sets of fixed values of $\omega_{ij}$ taken at various stages of learning corresponding to times $t$ of (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}) indicated above each panel, also marked in Fig. \ref{['EJNB_NN_bif_mech_appeal_final2:fig:omega_t']}. Each panel shows attractors associated with true memories of input vectors $\mathbf{I}^k$ (black diamonds), attractors representing spurious memories (red crosses) not associated with any $\mathbf{I}^k$, phase trajectories originating from six vectors $\mathbf{I}^k$ and their vicinities (Type 1 and 2 ICs, dashed and solid lines with one colour corresponding to one $\mathbf{I}^k$), and phase trajectories launched from random ICs (Type 3 ICs, grey dashed lines).
• Figure 4: One-parameter bifurcation diagram illustrating memory formation in early stages of learning by the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}) from training Set 1 (Tabs. \ref{['supp-tab:tab1-1']}--\ref{['supp-tab:tab1-2']}) with $N$$=$$81$. Panels show $x_1$-coordinates of fixed points of (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq3.1']}) (solid lines) as functions of control parameter $t$, which coincides with time $t$ in (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}). Stability of fixed points is indicated by the line colour: stable point, i.e. attractor potentially associable with memory (black), "useful" saddle point with a single positive eigenvalue (blue), and other saddle points (red). Dots along each branch indicate values of $t$ at which the respective fixed point was numerically found and analysed; the dots are connected by interpolating cubic splines. Translucent circles mark bifurcation points: pitchfork at $t$$=$$6.5$ (pink), saddle-node at $t$$=$$22.8783$ and at $t$$\approx$$35.9$ (green). All saddle-node bifurcations occur in pairs due to symmetry in (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq3.1']}). Panel (a) shows the bifurcation diagram for $t \in [5,38]$; (b) is a close-up of the rectangular selection in (a). All bifurcations involving attractors are additionally illustrated by phase portraits in Fig. \ref{['EJNB_NN_bif_mech_appeal_final2:fig:Pitchfork']}.
• Figure 5: Illustration of the first four bifurcations producing new potential memories in the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}) with $N$$=$$81$ learning from training Set 1, compare with Fig. \ref{['EJNB_NN_bif_mech_appeal_final2:fig:81N_Bif_Diag']}. Panels show projections of fixed points on the $(x_1,x_2)$-plane of the NN (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq3.1']}) before (first row) and after (second row) the respective bifurcations, and provide the value of parameter $t$. Fixed points are marked as: attractors potentially associable with memories (black boxes), "useful" saddle points with one positive eigenvalue (white diamonds), and all other saddle or unstable points (grey triangles). Yellow arrows indicate the flow of time $t$ in (\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.1']})--(\ref{['EJNB_NN_bif_mech_appeal_final2:eqn:eq2.4']}). Bifurcations illustrated are: (a)--(b) pitchfork bifurcation at $t$$=$$6.5$, which destabilises the point $\mathbf{0}$ and produces two new attractors; (c)--(d) a pair of saddle-node bifurcations at $t$$=$$22.8783$ creating two new attractors (compare with Fig. \ref{['EJNB_NN_bif_mech_appeal_final2:fig:81N_Bif_Diag']}(a)), and (e)--(f) two pairs of saddle-node bifurcations at $t$$\approx$$35.9$ creating four new attractors (compare with Fig. \ref{['EJNB_NN_bif_mech_appeal_final2:fig:81N_Bif_Diag']}(b)).