Nature inspired algorithms with randomized hypercomputational perspective

Highlights

• Two closely related nature inspired algorithms are evaluated extensively on a function for which nothing can be proved.

• Results show that both of them are similar.

• The case of randomized hypercomputation is debunked.

• Three theses related to nature inspired algorithms and hypercomputation are proposed.

• Antithesis to belief, these theses are meant to be proved (or disproved) and not only meant as theses.

Abstract

The intention of this paper is to empirically compare two closely related (and considered novel metaphor based) nature inspired algorithms and then to use these comparisons and other ideas to shed some light on randomized hypercomputation. The Bat and the Novel Bat algorithms are two of the recently created nature inspired algorithms with the claim of novelty in both of them as both of them uses different randomization, but how novel or similar are they in reality remains hazy especially from the practical perspective. Therefore, here I compare both of them on a data dependent, unexplainable real world classification function. Particularly, I create two classification machines, one derived by BA and another derived by the NBA, using weighted liner loss twin support vector machine and compare it with other trailblazing classifiers on the real world UCI machine learning data sets. These machines are, especially compared with each other also. The results show that the formulated machines perform better with respect to other classifiers on more than 80% of comparisons but are extraordinarily similar between themselves, as they agree with each other up to even four decimal places in almost 100% of cases. Moreover, these optimization algorithms form a perfect example of partially random machines which are claimed to be hypercomputational. But, here it can be shown that unless one has an access to the uncomputable input and uses it intelligently one cannot hypercompute.

Introduction

The No-Free-Lunch theorems are foundational results in supervised machine learning and optimization theories. The No-Free-Lunch theorems for supervised machine learning (NFLM) [1], [2], [3], [4] basically says that the entire supervised machine learning algorithms are the same with respect to all of the datasets, while the No-Free-Lunch theorems for optimization (NFLO) says that the entire optimization algorithms are just the same with respect to all of the (objective) functions [1], [5]. These theorems have a few conditions with them which are generally satisfied with respect to the Turing machine model. Despite these two foundational results experts are motivated to produce new optimization and supervised machine learning algorithms for which it is not possible to prove any proposed properties, for e.g. algorithm’s generalization power [1], [2], [3], [4], [5].

Motivated by the last line of the previous paragraph in this paper, I first evaluate two of the most “celebrated” and considered “novel” optimization algorithms, the bat algorithm (BA) [6] and the novel bat algorithm (NBA) [7] extensively (and especially) with each other. The special function on which they are tested is data dependent, unexplainable [44], binary classification function from supervised machine learning “weighted liner loss twin support vector machine” (WLTSVM) [8]. Moreover, it is interesting to know that the problem of classification in general is NP-complete. The role model of many learning algorithms “support vector machine” SVM [9], [10] is one in which the decision of a class of an unknown data is based on support vectors (with respect to training data) which are used to create two parallel supporting hyperplanes, while the decision hyperplane is in the center of the two supporting hyperplanes. Then, the decision of a class depends on the proximity of the unknown data to the supporting hyperplanes (or on which side of the decision hyperplane (with respect to the supporting hyperplanes) the data lays). In WLTSVM, the condition that the two supporting hyperplanes must be parallel is relaxed.

Here, it is interesting to know that optimization experts use randomization to solve wide varieties of problems for e.g. scheduling [46], routing [45], instance selection [47], information retrieval [48] etc. The previous line is because of the fact that optimization is omnipresent. But, there are many questions in the minds of experts that are needed to be addressed. Then, here WLTSVM is combined with BA and NBA to produce “weighted liner loss bat twin support vector machine” (WLBTSVM) and “weighted liner loss novel-bat twin support vector machine” $(\mathrm{W}\mathrm{L}{\mathrm{B}}_{\mathrm{N}}\mathrm{T}\mathrm{S}\mathrm{V}\mathrm{M})$ respectively. These two are tested extensively on the real world, UCI machine learning datasets. While it is worth noting that the parameter optimization problem is NP-Complete [1]. In this regard, this paper answers the following:

• What can be said (theoretically and especially empirically) for NFLO following (e.g. grid) and violating (e.g. BA/NBA) algorithms, for a function (here WLTSVM) that is completely “blind”, i.e. for which nothing is known, when compared collectively?

• What can be said (theoretically and especially empirically) for NFLO violating algorithms, for a function that is completely “blind” when compared among themselves?

• Given the set of all of the algorithms can NFLO induces some structure on them and in what sense?

• Are there any (provable) differences between the BA and NBA?

• If some condition(s) of NFLO is violated, then what can be projected for such similar condition violating algorithms?

• Are all of the nature inspired algorithms same?

Moreover secondly, and apart from all this BA and the NBA are good examples of partially random machines (PRMs) [12] because of randomization in them, while it is postulated that such PRMs are hypercomputational in some cases. Here, hypercomputation is the extension of the concept of computation. Computation is defined by Turing Machines (TMs) which are nothing more than sets of well defined instructions. In this sense hypercomputation is the study of those machines that can go beyond the power of conventional TMs [12], [13], [14], [15], [16]. There are many forms of hypercomputation proposed in the literature for e.g.:

• Relativistic hypercomputation [16], [17], [18], [19]

• Biological hypercomputation [20]

• Randomized hypercomputation [12]

• Quantum hypercomputation [21], [22], [23] etc.

That said, it is worth pointing out that the role model of hypercomputation is Turing’s O-machines (O for Oracle) and the concept of hypercomputation has never been proved or refuted completely, although there are various arguments in favor of and against it [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. The situation of this paper provides an excellent opportunity to empirically have some evidences against randomized hypercomputation. Randomized hypercomputation can be called by the name of evolutionary hypercomputation as evolution is the best possible real world candidate for it. In this regard, this paper answers the following:

• How randomization creates an opportunity for hypercomputation? Is it harnessable? If not, then what it lacks?

• Discrete physical systems are directed by the laws of nature, can they hypercompute?

• Can some Markov process hypercompute?

• Empirically what can be said about the Church-Turing thesis?

• What the results of this paper say about all these issues?

• Are nature inspired algorithms hypercomputational?

Then, the plan of the presentation here is as follows: section 2 presents the background from WLTSVM, BA, NBA, NFLO, NFLM to hypercomputation. Section 3 presents the construction of WLBTSVM and $\mathrm{W}\mathrm{L}{\mathrm{B}}_{\mathrm{N}}\mathrm{T}\mathrm{S}\mathrm{V}\mathrm{M}$ while section 4 (and APPENDIX-A) describes the results of the conducted experiments. Section 5 introduces the NFLO hierarchy and compares BA and NBA in the light of the available results (and available theory in APPENDIX-B). Then, section 6 is decisional on randomized hypercomputation while section 7 concludes this paper with future work.