Interacting Dreaming Neural Networks

We work on a statistical mechanics problem in the canonical formalism with temperature $T=\beta^{-1}$. The components of the quenched memory patterns $\bm{\xi}=\{\xi_{i}^{\mu}\}_{\mu=1\cdots p,i=1\cdots N}$, are drawn independently from a uniform Bernoulli distribution. The Hamiltonian of an isolated agent is [8]

where $t\geq 0$ quantifies the length of the dreaming process. $\bm{C}$ is the $p\times p$ matrix with elements $C_{\mu\nu}=\frac{1}{N}\sum_{i=1}^{N}\xi_{i}^{\mu}\xi_{i}^{\nu}$, $\sigma_{i}=\pm 1$ is an Ising variable. For $t=0$ the Hopfield Hebbian Hamiltonian is recovered and as $t\rightarrow\infty$, $t^{-1}$ acts as a Tikhonov regularizer, leading to the pseudo-inverse model [17][14]. The two member group Hamiltonian is

$\epsilon$ is the coupling constant of the interaction, the sums over $i,j$ run from $1$ to $N$, the sums over $\mu,\nu$ from $1$ to $p$. We study this problem in equilibrium or offline with quenched disorder. A related problem, the inverse problem of learning online a spin glass Hamiltonian, was studied in [15].

The first two terms of the Hamiltonian in expression 2.1 represent two distinct neural network agents, the third term connects them.

This interaction is similar to dreaming since it changes the couplings from $J^{(1)}_{ij}\rightarrow J_{ij}^{(1)}+\frac{2\epsilon}{N}S_{i}S_{j}$ and $J^{(2)}_{ij}\rightarrow J_{ij}^{(2)}+\frac{2\epsilon}{N}\sigma_{i}\sigma_{j}$, i.e. they pick up a Hebbian contribution from the other agent. However, we emphasize the fact that they are different, as in the interaction all states are included, and not only the stables ones, as it is done in the unlearning algorithm.

For $\epsilon=0$, this is the model of reinforcement and removal model [8]; and with $\epsilon\neq 0$ and $p=1$, the infinite range Askhin-Teller model[13].

Without any loss, we take $t_{1}\leq t_{2}$ and refer to the neural network agents as NN1 and NN2 respectively.

The free-energy is obtained using the replica method, where the Hamiltonian is a sum of the individual replica Hamiltonians. The calculation follows roughly those for the single agent case by Fachechi et al. [8], except for a few important steps where it is necessary to be careful. The expression of the free energy and the self-consistent equations for the order parameters appear in [1]. Since we are interested in the memory retrieval phases, we have not looked in detail into the spin-glass phase, hence, in order to understand global features of the five dimensional phase diagram ($t_{1},t_{2},\epsilon,\alpha,\beta$), the important order parameters for replica $a$ are $h^{a}=\langle\langle\frac{1}{N}\sum_{i=1}^{N}\sigma_{i}^{a}S_{i}^{a}\rangle\rangle$, $m_{1}^{a}=\langle\langle\frac{1}{N}\sum_{i=1}^{N}\xi^{1}_{i}\sigma_{i}^{a}\rangle\rangle$ and $m_{2}^{a}=\langle\langle\frac{1}{N}\sum_{i=1}^{N}\xi_{i}^{1}S_{i}^{a}\rangle\rangle$, where the double brackets represent the thermal and quenched average over the patterns. We will also consider the changes induced by the interaction, i.e. the difference between the order parameters for $\epsilon>0$ and $\epsilon=0$, for a more careful identification of the different phases. Other order parameters of the form $\Delta^{\rho}$, $r^{\rho}$, $Q^{\rho}$ and $q^{\rho}$ are related to the overlap between the $\rho$ elements, where $\rho$ is an index that refers to either agent $\sigma$, or $S$ or to their overlap $\sigma S$.

We assume the following replica symmetry ansatz:

where

and $r_{ab}$ is the auxiliary variable of $q_{ab}$, so it has a similar physical meaning.

The first line of (2) is not hard to justify, since there are no reasons to expect that any replica is privileged.

The Ansatz related to $q_{ab}^{\rho}$ and $r_{ab}^{\rho}$ for $\rho=\sigma S$ deserves a comment. It represents the idea that the correlations between the agents will be different for the same replica than for different replicas, because the interaction is not between different replicas, but within the same, so we expect a greater intrareplica similarity of the agents than the interreplica similarity.

A homogeneous Ansatz ($q^{\sigma S}_{ab}=q^{\sigma S},r^{\sigma S}_{ab}=r^{\sigma S}\forall a,b$), leads to the puzzling result that the free energy does not depend on $q^{\sigma S}$, which indicates that this is not the right Ansatz. This can be seen from the fact that the dependence of $Q^{\sigma S}$ and $q^{\sigma S}$ on the free energy occurs via the differences $Q^{\sigma S}-q^{\sigma S}$ and $R^{\sigma S}-r^{\sigma S}$, so using a homogeneous Ansatz would lead to no dependence on $q^{\sigma S}$.

It is important to recognize that $q_{ab}$ does not have the exact same meaning as the original Edwards-Anderson parameter. Its additional complex part inside the average leads to different values, and in particular, $q^{\sigma S}$ is different from $h$ for $t_{1},t_{2}\neq 0$, despite having some similarity. It also explains why already in the original article [8] the usual choice $q_{aa}=1$ was not used. Still, this parameter gives us a measure of the correlations in the system, and can still be used to determine the existence of the spin-glass state.

We emphasize that despite using the same Ansatz for different $\rho$’s, the justification behind them is distinct. The replica symmetric free energy depends on 9 independent order parameters ($m_{1},m_{2}$, $h$, $Q^{\sigma}$, $Q^{S}$, $Q^{\sigma S}$, $q^{\sigma}$, $q^{S}$, $q^{\sigma S}$). We also have 6 dependent variables ($r^{\sigma}$, $r^{S}$, $r^{\sigma S}$, $\Delta^{\sigma}$, $\Delta^{S}$, $\Delta^{\sigma S}$), which can be fully eliminated from the equations of state.

Table 1 shows the different phases, obtained from the possible regimes of $m_{1},m_{2},h$ and, in addition, a measure of the benefit or damage due to the interaction, given by

Before entering into details in section 2.3, we give a rough description of the types of phases where the system may be found. The insufficient phase occurs when NN2 is in the retrieval phase and NN1 is not and $\epsilon$ is small enough such that no relevant changes in the learning occurs. As the interaction increases, it can transition to the student-professor phase, where the performance of only NN1 is enhanced significantly. Further increase of $\epsilon$ can result in the mutualism phase, where both neural networks have noticeable increases in their capacity. The disordered phase includes the paramagnetic and spin glass phases, where the interaction cannot lead to interesting results. It is not hard to detect the spin glass phase, since for $h=0$, the role of $\epsilon$ vanishes, hence the line of the spin glass phase is simply the same obtained in [2].

The indifferent phase occurs where, even without interaction, both neural networks could already process information in an adequate manner. The reinforced delusion phase indicates the creation of an alternate ordered state, where the agents, despite having no overlap with the memory patterns, are similar to each other. From their own perspective, the agents are in a retrieval ferromagnetic phase. For a third party, the individual agents cannot process information, and seem to be in a disordered phase. The interaction still preserves the glassy state, so it is still a spin glass phase. The last phase is the amensalism, where while NN1 is not significantly affected by the interaction, NN2 ends up losing the understanding of the situation, replicating an interaction of amensalism.

We analyze numerically the self-consistent equations obtained from the extreme conditions of the free energy. Three types of 2-dimensional cuts are used to describe the five dimensional parameter space, yielding phase diagrams in the: (i) $\alpha,T_{c}$, (ii) $\epsilon,\alpha_{c}$ and (iii) $\epsilon,T_{c}$ planes, shown respectively in panels 1, 2 and 3. In all diagrams, since $t_{2}\geq t_{1}$, we have $m_{2}\geq m_{1}$.