How can I possibly learn on that thing I have just become aware I do?

My editorial on You Tube

I keep working on the application of neural networks to simulate the workings of collective intelligence in humans. I am currently macheting my way through the model proposed by de Vincenzo et al in their article entitled ‘Mimicking the collective intelligence of human groups as an optimization tool for complex problems’ (2018[1]). In the spirit of my own research, I am trying to use optimization tools for a slightly different purpose, that is for simulating the way things are done. It usually means that I relax some assumptions which come along with said optimization tools, and I just watch what happens.

Vincenzo et al propose a model of artificial intelligence, which combines a classical perceptron, such as the one I have already discussed on this blog (see « More vigilant than sigmoid », for example) with a component of deep learning based on the observable divergences in decisions. In that model, social agents strive to minimize their divergences and to achieve relative consensus. Mathematically, it means that each decision is characterized by a fitness function, i.e. a function of mathematical distance from other decisions made in the same population.

I take the tensors I have already been working with, namely the input tensor T_I = {LCOE_R, LCOE_NR, K_R, K_NR, I_R, I_NR, P_A;R, P_A;NR, P_B;R, P_B;NR} and the output tensor is T_O = {Q_R/N; Q_NR/N}. Once again, consult « More vigilant than sigmoid » as for the meaning of those variables. In the spirit of the model presented by Vincenzo et al, I assume that each variable in my tensors is a decision. Thus, for example, P_A;R, i.e. the basic price of energy from renewable sources, which small consumers are charged with, is the tangible outcome of a collective decision. Same for the levelized cost of electricity from renewable sources, the LCOE_R, etc. For each i-th variable x_i in T_I and T_O, I calculate its relative fitness to the overall universe of decisions, as the average of itself, and of its Euclidean distances to other decisions. It looks like:

V(x_i) = (1/N)*{x_i + [(x_i – x_i;1)²]^0,5 + [(x_i – x_i;2)²]^0,5 + … + [(x_i – x_i;K)²]^0,5}

…where N is the total number of variables in my tensors, and K = N – 1.

In a next step, I can calculate the average of averages, thus to sum up all the individual V(x_i)’s and divide that total by N. That average V*(x) = (1/N) * [V(x₁) + V(x₂) + … + V(x_N)] is the measure of aggregate divergence between individual variables considered as decisions.

Now, I imagine two populations: one who actively learns from the observed divergence of decisions, and another one who doesn’t really. The former is represented with a perceptron that feeds back the observable V(x_i)’s into consecutive experimental rounds. Still, it is just feeding that V(x_i) back into the loop, without any a priori ideas about it. The latter is more or less what it already is: it just yields those V(x_i)’s but does not do much about them.

I needed a bit of thinking as for how exactly should that feeding back of fitness function look like. In the algorithm I finally came up with, it looks differently for the input variables on the one hand, and for the output ones. You might remember, from the reading of « More vigilant than sigmoid », that my perceptron, in its basic version, learns by estimating local errors observed in the last round of experimentation, and then adding those local errors to the values of input variables, just to make them roll once again through the neural activation function (sigmoid or hyperbolic tangent), and see what happens.

As I upgrade my perceptron with the estimation of fitness function V(x_i), I ask: who estimates the fitness function? What kind of question is that? Well, a basic one. I have that neural network, right? It is supposed to be intelligent, right? I add a function of intelligence, namely that of estimating the fitness function. Who is doing the estimation: my supposedly intelligent network or some other intelligent entity? If it is an external intelligence, mine, for a start, it just estimates V(x_i), sits on its couch, and watches the perceptron struggling through the meanders of attempts to be intelligent. In such a case, the fitness function is like sweat generated by a body. The body sweats but does not have any way of using the sweat produced.

Now, if the V(x_i) is to be used for learning, the perceptron is precisely the incumbent intelligent structure supposed to use it. I see two basic ways for the perceptron to do that. First of all, the input neuron of my perceptron can capture the local fitness functions on input variables and add them, as additional information, to the previously used values of input variables. Second of all, the second hidden neuron can add the local fitness functions, observed on output variables, to the exponent of the neural activation function.

I explain. I am a perceptron. I start my adventure with two tensors: input T_I = {LCOE_R, LCOE_NR, K_R, K_NR, I_R, I_NR, P_A;R, P_A;NR, P_B;R, P_B;NR} and output T_O = {Q_R/N; Q_NR/N}. The initial values I start with are slightly modified in comparison to what was being processed in « More vigilant than sigmoid ». I assume that the initial market of renewable energies – thus most variables of quantity with ‘R’ in subscript – is quasi inexistent. More specifically, Q_R/N = 0,01 and  Q_NR/N = 0,99 in output variables, whilst in the input tensor I have capital invested in capacity I_R = 0,46 (thus a readiness to go and generate from renewables), and yet the crowdfunding flow K is K_R = 0,01 for renewables and K_NR = 0,09 for non-renewables. If you want, it is a sector of renewable energies which is sort of ready to fire off but hasn’t done anything yet in that department. All in all, I start with: LCOE_R = 0,26; LCOE_NR = 0,48; K_R = 0,01; K_NR = 0,09; I_R = 0,46; I_NR = 0,99; P_A;R = 0,71; P_A;NR = 0,46; P_B;R = 0,20; P_B;NR = 0,37; Q_R/N = 0,01; and Q_NR/N = 0,99.

Being a pure perceptron, I am dumb as f**k. I can learn by pure experimentation. I have ambitions, though, to be smarter, thus to add some deep learning to my repertoire. I estimate the relative mutual fitness of my variables according to the V(x_i) formula given earlier, as arithmetical average of each variable separately and its Euclidean distance to others. With the initial values as given, I observe: V(LCOE_R; t₀) = 0,302691788; V(LCOE_NR; t₀) = 0,310267104; V(K_R; t₀) = 0,410347388; V(K_NR; t₀) = 0,363680721; V(I_R ; t₀) = 0,300647174; V(I_NR ; t₀) = 0,652537097; V(P_A;R ; t₀) = 0,441356844 ; V(P_A;NR ; t₀) = 0,300683099 ; V(P_B;R ; t₀) = 0,316248176 ; V(P_B;NR ; t₀) = 0,293252713 ; V(Q_R/N ; t₀) = 0,410347388 ; and V(Q_NR/N ; t₀) = 0,570485945. All that stuff put together into an overall fitness estimation is like average V*(x; t₀) = 0,389378787.

I ask myself: what happens to that fitness function when as I process information with my two alternative neural functions, the sigmoid or the hyperbolic tangent. I jump to experimental round 1500, thus to t₁₅₀₀, and I watch. With the sigmoid, I have V(LCOE_R; t₁₅₀₀) =  0,359529289 ; V(LCOE_NR; t₁₅₀₀) =  0,367104605; V(K_R; t₁₅₀₀) =  0,467184889; V(K_NR; t₁₅₀₀) = 0,420518222; V(I_R ; t₁₅₀₀) =  0,357484675; V(I_NR ; t₁₅₀₀) =  0,709374598; V(P_A;R ; t₁₅₀₀) =  0,498194345; V(P_A;NR ; t₁₅₀₀) =  0,3575206; V(P_B;R ; t₁₅₀₀) =  0,373085677; V(P_B;NR ; t₁₅₀₀) =  0,350090214; V(Q_R/N ; t₁₅₀₀) =  0,467184889; and V(Q_NR/N ; t₁₅₀₀) = 0,570485945, with average V*(x; t₁₅₀₀) =  0,441479829.

Hmm, interesting. Working my way through intelligent cognition with a sigmoid, after 1500 rounds of experimentation, I have somehow decreased the mutual fitness of decisions I make through individual variables. Those V(x_i)’s have changed. Now, let’s see what it gives when I do the same with the hyperbolic tangent: V(LCOE_R; t₁₅₀₀) =   0,347752478; V(LCOE_NR; t₁₅₀₀) =  0,317803169; V(K_R; t₁₅₀₀) =   0,496752021; V(K_NR; t₁₅₀₀) = 0,436752021; V(I_R ; t₁₅₀₀) =  0,312040791; V(I_NR ; t₁₅₀₀) =  0,575690006; V(P_A;R ; t₁₅₀₀) =  0,411438698; V(P_A;NR ; t₁₅₀₀) =  0,312052766; V(P_B;R ; t₁₅₀₀) = 0,370346458; V(P_B;NR ; t₁₅₀₀) = 0,319435252; V(Q_R/N ; t₁₅₀₀) =  0,496752021; and V(Q_NR/N ; t₁₅₀₀) = 0,570485945, with average V*(x; t₁₅₀₀) =0,413941802.

Well, it is becoming more and more interesting. Being a dumb perceptron, I can, nevertheless, create two different states of mutual fitness between my decisions, depending on the kind of neural function I use. I want to have a bird’s eye view on the whole thing. How can a perceptron have a bird’s eye view of anything? Simple: it rents a drone. How can a perceptron rent a drone? Well, how smart do you have to be to rent a drone? Anyway, it gives something like the graph below:

Wow! So this is what I do, as a perceptron, and what I haven’t been aware so far? Amazing. When I think in sigmoid, I sort of consistently increase the relative distance between my decisions, i.e. I decrease their mutual fitness. The sigmoid, that function which sorts of calms down any local disturbance, leads to making a decision-making process like less coherent, more prone to embracing a little chaos. The hyperbolic tangent thinking is different. It occasionally sort of stretches across a broader spectrum of fitness in decisions, but as soon as it does so, it seems being afraid of its own actions, and returns to the initial level of V*(x). Please, note that as a perceptron, I am almost alive, and I produce slightly different outcomes in each instance of myself. The point is that in the line corresponding to hyperbolic tangent, the comb-like pattern of small oscillations can stretch and move from instance to instance. Still, it keeps the general form of a comb.

OK, so this is what I do, and now I ask myself: how can I possibly learn on that thing I have just become aware I do? As a perceptron, endowed with this precise logical structure, I can do one thing with information: I can arithmetically add it to my input. Still, having some ambitions for evolving, I attempt to change my logical structure, and I risk myself into incorporating somehow the observable V(x_i) into my neural activation function. Thus, the first thing I do with that new learning is to top the values of input variables with local fitness functions observed in the previous round of experimenting. I am doing it already with local errors observed in outcome variables, so why not doubling the dose of learning? Anyway, it goes like: x_i(t₀) = x_i(t-1) + e(x_i; t_-1) + V(x_i; t_-1). It looks interesting, but I am still using just a fraction of information about myself, i.e. just that about input variables. Here is where I start being really ambitious. In the equation of the sigmoid function, I change s = 1 / [1 + exp(∑x_i*W_i)] into s = 1 / [1 + exp(∑x_i*W_i + V(T_o)], where V(T_o) stands for local fitness functions observed in output  variables. I do the same by analogy in my version based on hyperbolic tangent. The th = [exp(2*∑x_i*w_i)-1] / [exp(2*∑x_i*w_i) + 1] turns into th = {exp[2*∑x_i*w_i + V(T_o)] -1} / {exp[2*∑x_i*w_i + V(T_o)] + 1}. I do what I know how to do, i.e. adding information from fresh observation, and I apply it to change the structure of my neural function.

All those ambitious changes in myself, put together, change my pattern of learing as shown in the graph below:

When I think sigmoid, the fact of feeding back my own fitness function does not change much. It makes the learning curve a bit steeper in the early experimental rounds, and makes it asymptotic to a little lower threshold in the last rounds, as compared to learning without feedback on V(x_i). Yet, it is the same old sigmoid, with just its sleeves ironed. On the other hand, the hyperbolic tangent thinking changes significantly. What used to look like a comb, without feedback, now looks much more aggressive, like a plough on steroids. There is something like a complex cycle of learning on the internal cohesion of decisions made. Generally, feeding back the observable V(x_i) increases the finally achieved cohesion in decisions, and, in the same time, it reduces the cumulative error gathered by the perceptron. With that type of feedback, the cumulative error of the sigmoid, which normally hits around 2,2 in this case, falls to like 0,8. With hyperbolic tangent, cumulative errors which used to be 0,6 ÷ 0,8 without feedback, fall to 0,1 ÷ 0,4 with feedback on V(x_i).

The (provisional) piece of wisdom I can have as my takeaway is twofold. Firstly, whatever I do, a large chunk of perceptual learning leads to a bit less cohesion in my decisions. As I learn by experience, I allow myself more divergence in decisions. Secondly, looping on that divergence, and including it explicitly in my pattern of learning leads to relatively more cohesion at the end of the day. Still, more cohesion has a price – less learning.

I am consistently delivering good, almost new science to my readers, and love doing it, and I am working on crowdfunding this activity of mine. As we talk business plans, I remind you that you can download, from the library of my blog, the business plan I prepared for my semi-scientific project Befund  (and you can access the French version as well). You can also get a free e-copy of my book ‘Capitalism and Political Power’ You can support my research by donating directly, any amount you consider appropriate, to my PayPal account. You can also consider going to my Patreon page and become my patron. If you decide so, I will be grateful for suggesting me two things that Patreon suggests me to suggest you. Firstly, what kind of reward would you expect in exchange of supporting me? Secondly, what kind of phases would you like to see in the development of my research, and of the corresponding educational tools?

[1] De Vincenzo, I., Massari, G. F., Giannoccaro, I., Carbone, G., & Grigolini, P. (2018). Mimicking the collective intelligence of human groups as an optimization tool for complex problems. Chaos, Solitons & Fractals, 110, 259-266.