

![0_image_0.png](0_image_0.png)

Figure 4: The result for a Boltzmann machine ring of an arbitrary number of nodes. All thresholds and weights have the same value. In the left panel the gap is shown between the upper and lower bound of the mean of the nodes.  The algorithm converges exponentially to the final bounding values as soon as it is close enough.

Thus the distance to its asymptotic value is proportional to α n , where n is the nmber of iterations. The parameter α is shown in the right panel.
fixed point the following holds for any i:

$$p_{+}\left(\,s_{i}\,\right)=\operatorname*{max}_{q\left(\,s_{i-1}\,s_{i+1}\,\right)}p\left(\,s_{i}\,\right|s_{i-1}\,s_{i+1}\,\right)\,q\left(\,s_{i-1}\,s_{i+1}\,\right)$$
$$\left(15\right)$$

uder the constraints

$q\left(s_{i\,-\,1}\right)\leq p_{+}\left(\,s_{i\,-\,1}\right)=p_{+}\left(\,s_{i}\right)$  $q\left(s_{i\,+\,1}\right)\leq p_{+}\left(\,s_{i\,+\,1}\right)=p_{+}\left(\,s_{i}\right)$  $q\left(s_{i\,-\,1}\right)\geq p_{-}\left(\,s_{i\,-\,1}\right)=p_{-}\left(\,s_{i}\right)$  $q\left(s_{i\,+\,1}\right)\geq p_{-}\left(\,s_{i\,+\,1}\right)=p_{-}\left(\,s_{i}\right)$
$$i\,+\,w\,s_{i}\,s_{i}+1\,\Big)$$
$$\begin{array}{l}{(\,1\,6\,)}\\ {(\,1\,7\,)}\\ {(\,1\,8\,)}\\ {(\,1\,9\,)}\end{array}$$
and similarly for the lower bound. The conditional probability in Equation 15 is given by

$$p\left(\left.\mathbf{s}_{i}\right|\mathbf{s}\right)$$

p ( s i | s i − 1 s i +1 ) ∞ exp ( θ s i  + w s i − 1 s i  + w s i s i +1 )

$\left(20\right)$. 
From these equations one can derive the fixed point for p+ ( si ) and p= ( si ). It turns out, however, that it is easier to determine the fixed point of the upper and lower bound on the mean, hence p + (si = 1) - p_ (si = -1) and p_ (si = 1) - p+ (si = -1). But this has no effect on the principle results as tightness and convergence speed.

In Figure 4a the difference between the upper and lower bound on the means is shown for various choices of the weight and threshold. As we saw before tight bounds can be obtained for small weights and somewhat larger, but positive weights, whereas negative weights result in rather poor, but still non-trivial bounds. It should be mentioned, however, that these results correspond to the choice of the smallest clusters ( | Smar | = 1 and | S,ep | = 2)