Learning Implicitly with Noisy Data in Linear Arithmetic

Robust learning in expressive languages with real-world data continues to be a challenging task. Numerous conventional methods appeal to heuristics without any assurances of robustness. While probably approximately correct (PAC) Semantics offers strong guarantees, learning explicit representations is not tractable, even in propositional logic. However, recent work on so-called “implicit" learning has shown tremendous promise in terms of obtaining polynomial-time results for fragments of first-order logic. In this work, we extend implicit learning in PAC-Semantics to handle noisy data in the form of intervals and threshold uncertainty in the language of linear arithmetic. We prove that our extended framework keeps the existing polynomial-time complexity guarantees. Furthermore, we provide the first empirical investigation of this hitherto purely theoretical framework. Using benchmark problems, we show that our implicit approach to learning optimal linear programming objective constraints significantly outperforms an explicit approach in practice.

Threshold and Group Size Uncertainty in Common-pool Resources: An Experimental Study

Threshold common-pool resources (TCPRs), such as fisheries or groundwater reserves, face irreversible damage if harvesting exceeds a sustainability threshold. Uncertainty about the threshold for sustainable use or the number of resource users can exacerbate the overharvesting problem. Policy makers may therefore seek to reduce threshold or group size uncertainty in TCPRs. Overall, we find that reducing threshold and group size uncertainty (moving from high to low uncertainty) increases expected earnings from the resource. However, complete elimination of group size uncertainty reduces expected earnings. Furthermore, the impact of group size uncertainty on earnings varies by the level of threshold uncertainty. Moving from high to low group size uncertainty increases earnings at low levels of threshold uncertainty but not at high levels of threshold uncertainty. Taken together, we find that reducing threshold uncertainty is beneficial while tackling group size uncertainty requires a more nuanced approach, highlighting the importance of a joint analysis.

DEVELOPMENT OF A SELF-ADJUSTING METHOD FOR CALCULATING RECURRENT DIAGRAMS IN A SPACE WITH A SCALAR PRODUCT

A self-adjusting method for calculating recurrence diagrams has been developed. The proposed method is aimed at overcoming the metric-threshold uncertainty inherent in the known methods for calculating recurrence diagrams. The method provides invariance to the nature of the measured data, and also allows to display the recurrence of states, adequate to real systems of various fields. A new scientific result consists in the theoretical justification of the method for calculating recurrence diagrams, which is capable of overcoming the existing metric-threshold uncertainty of known methods on the basis of self-adjusting by measurements by improving the topology of the metric space. The topology is improved due to the additional introduction of the scalar product of state vectors into the operation space. This allowed to develop a self-adjusting method for calculating recurrence diagrams with increased accuracy and adequacy of the display of recurrence states of real systems. Moreover, the method has a relatively low computational complexity, providing invariance with respect to the nature of the irregularity of measurements. Verification of the proposed method was carried out on the basis of experimental measurements of concentrations of gas pollutants of atmospheric air for a typical industrial city. The main gas pollutants of the atmosphere are formaldehyde, ammonia and nitrogen dioxide, caused by stationary and mobile sources of urban pollution. The obtained experimental verification results confirm the increased accuracy and adequacy of the display of the recurrence of atmospheric pollution states, as well as the invariance of the method with respect to the nature of the irregularity of measurements. It has been established that the accuracy of the method is influenced by the a priori boundary angular dimensions of the recurrence cone. It was shown that with a decrease in the boundary angular dimensions of the recurrence cone, the accuracy of the recurrence mapping of the real states of dynamical systems in the calculated diagrams increases. It was experimentally established that the accuracy and adequacy of the mapping of the recurrence states of real dynamical systems acceptable for applications is provided for a boundary angular size of the recurrence cone of 10° or less.