EL MEDIO AMBIENTE COMO BIEN JURÍDICO UNICO

Now that we have designed the EdMC optimization scheme, we can begin the numerical analysis by comparing the EdMC results, observed on finite size single-problem instances, with the theoretical predictions obtained via replica calculations. We therefore ran EdMC on a series of samples at size

N = 201 and α = 0.6. The scoping increment was determined by using the

relationship between the coupling parameter γ and the resulting polarization in the magnetizations, γ = tanh−1_{(p), and by implementing a linear increment} in p (p ∈ [0.4, 0.9], in steps of 0.1).

In order to study the behavior of the free entropy F (˜x, γ) as a function of γ, we carried on with each EdMC simulation even when a solution was already found. The inverse temperature was both set to ∞, in a greedy version of the algorithm, and to a finite value, slowly incremented in an annealing procedure with an exponential rate of 1.01 (applied every 10 accepted moves).

Fig. 5.3 A. Probability of error on a pattern (cf. figure 5.2) B. Local entropy (cf. figure 4.6A). See text for details on the procedure. For the version with cooling, 700 pattern sets were tested for each value of γ. For the y = ∞ version, 2000 samples were used. Error bars represent standard deviation estimates of the mean values.

The chosen stopping criterion was the consecutive rejection of 5N consecutive move proposals.

As we can see in figure 5.3, the recorded values of the local entropy S and of the error probability per pattern, as a function of the overlap S, are in good agreement with the theoretical analysis. The observed qualitative behavior is the same: the error rate goes to zero at S → 1 and the entropy is positive until

S = 1, as it should be in a dense cluster. We note that, in the more accurate

version with an annealing in the inverse temperature y, the gap between the theoretical values and the measured ones is partially closed. The remaining discrepancy could be due to:

• finite size effects, since N = 201 is rather small;

• inaccuracy of the Monte Carlo sampling, which can be handled by lowering the cooling rate for y;

• inaccuracy of the theoretical curves due to additional RSB effects. With the chosen settings, we note that the average number of errors per pattern set is almost always less than 2 for all points plotted in figure 5.3A, and that a zero energy configuration was always reached by EdMC. Also note that, in the plots, the noise recorded in the averages is only ascribable to the tails of the error distribution, while the modes and the medians are always found at 0.

Fig. 5.4 Typical trajectories of standard Simulated Annealing (red curve, right) and Entropy-driven Monte Carlo (blue curve, left), for N = 801, α = 0.3. Notice the logarithmic scale in the x axis. EdMC is run at 0 temperature with fixed

γ = tanh−1(0.6), SA is started at y0= 1 and run with a cooling rate of fy = 1.001

for each 103 accepted moves, to ensure convergence to a solution.

We can now test the efficacy of our method as a solver, by comparing its performance, at various problem sizes N and different values of α, with a standard energetic MCMC. EdMC, in fact, shows a remarkable ability in retrieving solutions, even in a greedy zero temperature setting, with a relatively small (O (N)) number of required MC steps. In the same setting the standard MCMC would get immediately trapped in a local minimum, even at small N. Instead, in order to find a solution also with the energetic MCMC, we employed a Simulated Annealing (SA) with initial inverse temperature y0 = 1, increased by a factor fy, every 103 accepted moves (the cooling rate fy was optimized for each problem instance).

In figure 5.4 we show a comparison between a two typical trajectories, exemplifying the difference between standard SA and EdMC (at y = ∞ with fixed γ = tanh−1_{(0.6)) on the very same instance: the number of required} accepted moves, in order to reach a solution with EdMC, is of 4 or 5 orders of magnitude smaller than the ones in the energetic SA. This highlights the qualitatively different smoothness of the landscape explored by EdMC.

Fig. 5.5 Number of iterations required to reach 0 energy in log-log scale, as a function of the problem size N . A: Simulated Annealing at α = 0.3, B: EdMC at α = 0.3 (bottom) and α = 0.6 (top). See text for the details of the procedure. Notice the difference in the y axes scales. For both methods, 100 samples were tested for each value of N . Color shades reflect data density. Empty circles and squares represent medians, error bars span the 5-th to 95-th percentile interval. The dashed lines are fitted curves: the SA points are fitted by an exponential curve exp (a + bN ) with a = 8.63 ± 0.06, b = (8.79 ± 0.08) · 10−3; the EdMC points are fitted by two polynomial curves aNb with a = 0.54 ± 0.04, b = 1.23 ± 0.01 for α = 0.3, and with

a = 0.14 ± 0.02, b = 1.74 ± 0.02 for α = 0.6.

More importantly, a scaling analysis with the size of the network N (varied between between 201 and 1601), shows two radically different behaviors between the two MCMC strategies (as pictured in figure 5.4):

• the behavior of the standard SA is clearly exponential at α = 0.3, and no solution can be found at α = 0.6 already at N = 201.

• in the case of EdMC (panel B in the figure), instead, the gathered data can be fit by polynomial curves, with a scaling of ∼ N1.23 _{for α = 0.3,} and ∼ N1.74 _{for α = 0.6.}

• Even in the simple α = 0.3 instances, a difference of several orders of magnitude was recorded in the required number of iterations in the two cases.

A detailed description of these numerical experiments can be found in the original paper ([2]).