Order-theoretic models for decision-making: Learning, optimization, complexity and computation

TL;DR

The work investigates how uncertainty, modeled by order relations (notably majorization and related preorders), underpins learning, optimization, and computation across physics-inspired domains. It develops a unified framework linking bounded rationality, fluctuation theorems, and real-valued representations of uncertainty, introducing injective monotones and the Debreu dimension to classify preordered spaces. It proves generalized Jarzynski and Crooks fluctuation theorems for general Markov chains, provides experimental evidence in human sensorimotor adaptation, and clarifies the limits of the maximum entropy principle as a sole representation of uncertainty. A comprehensive theory of computation on uncountable spaces is proposed via order-theoretic constructs, comparing uniform and nonuniform approaches and aligning countability restrictions with Debreu-type density and multi-utility notions. The results offer new insights into thermodynamics-inspired learning, molecular diffusion, and quantum resource theories, while advancing a general, geometry-informed classification of uncertainty-driven transitions and their representations.

Abstract

The study of intelligent systems explains behaviour in terms of economic rationality. This results in an optimization principle involving a function or utility, which states that the system will evolve until the configuration of maximum utility is achieved. Recently, this theory has incorporated constraints, i.e., the optimum is achieved when the utility is maximized while respecting some information-processing constraints. This is reminiscent of thermodynamic systems. As such, the study of intelligent systems has benefited from the tools of thermodynamics. The first aim of this thesis is to clarify the applicability of these results in the study of intelligent systems. We can think of the local transition steps in thermodynamic or intelligent systems as being driven by uncertainty. In fact, the transitions in both systems can be described in terms of majorization. Hence, real-valued uncertainty measures like Shannon entropy are simply a proxy for their more involved behaviour. More in general, real-valued functions are fundamental to study optimization and complexity in the order-theoretic approach to several topics, including economics, thermodynamics, and quantum mechanics. The second aim of this thesis is to improve on this classification. The basic similarity between thermodynamic and intelligent systems is based on an uncertainty notion expressed by a preorder. We can also think of the transitions in the steps of a computational process as a decision-making procedure. In fact, by adding some requirements on the considered order structures, we can build an abstract model of uncertainty reduction that allows to incorporate computability, that is, to distinguish the objects that can be constructed by following a finite set of instructions from those that cannot. The third aim of this thesis is to clarify the requirements on the order structure that allow such a framework.

Figures (17)

• Figure 1: Simple example where the left distribution $p$ is more biased than the right distribution $q$, $p \preceq_U q$, and $q$ is obtained from $p$ by transferring probability mass from the first to the second element in $\Omega$.
• Figure 2: Potential intervals that can be obtained when applying the bisection method on some polynomial with rational coefficients $p: \mathbb R \to \mathbb R$ and using $[q,q']$, consisting of rational numbers $q,q' \in \mathbb{Q}$ such that $p(q)p(q')<0$, as initial interval.
• Figure 3: Simulation of Crooks' fluctuation theorem. A Simulation with 1000 cycles. We include the theoretical prediction (black), the linear regression for the simulated data (red) and the simulated points (green). B Simulation with 20 cycles and 1000 bootstraps. We include the theoretical prediction \ref{['prediction']} (black) and both the mean (red) and the 99 % confidence interval (shaded area) of \ref{['prediction']} after the bootstraps. (Reproduced from hack2022thermodynamic, licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).)
• Figure 4: Mean adaptation. The filled triangles are the mean of the observed angles for both the forward (green) and backward (red) processes. The black line is the optimal deviation (the protocol). Participants that achieve at least $50\%$ mean adaptation are shaded by a green background color. (Reproduced from hack2022thermodynamic, licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).)
• Figure 5: Experimental results for Crooks' fluctuation theorem when the sensorimotor loss behaves as an exponential quadratic error \ref{['utility']}. We include the theoretical prediction of Crooks' fluctuation theorem \ref{['prediction']} (black) and the mean path after 1000 bootstraps of the observed driving error values (red). Participants that achieve at least $50\%$ adaptation (c.f. Figure \ref{['hyste plot']}) are shaded by a green background color. The shaded areas inside the graphs are the 99% confidence intervals which result from bootstrapping. Note that we fit the parameters for each participant according to hack2022thermodynamic. (Reproduced from hack2022thermodynamic, licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).)

Theorems & Definitions (73)

• Remark 1.1: Bounded rationality
• Remark 1.2: Shannon entropy and the uncertainty preorder
• Remark 1.3: Computability and computation
• Theorem 2.1: Jarzynski's equality for Markov chains
• Theorem 2.2: Crooks' fluctuation theorem for Markov chains
• Remark 2.1: Fluctuation theorems with continuous state space
• Definition 3.1: Injective monotones
• Proposition 3.1
• Theorem 3.1: Debreu bridges2013representations
• Definition 3.2: Debreu upper separable preorder