In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

[EX Q]: We develop a new semantics for defeasible infer­ ence based on extended probability measures al­ lowed to take infinitesimal values, on the inter­ pretation of defaults as generalized conditional probability constraints and on a preferred-model implementation of entropy-maximization.
[EX A]: Defaults and lnfinitesimals Defeasible Inference by Nonarchimedean Entropy Maximization

[EX Q]: Existing methods for dealing with knowledge updates differ greatly depending on the underlying knowledge representation formalism. When Classical Logic is used, updates are typically performed by manipulating the knowledge base on the model-theoretic level. On the opposite side of the spectrum stand the semantics for updating Answer-Set Programs that need to rely on rule syntax. Yet, a unifying perspective that could embrace both these branches of research is of great importance as it enables a deeper understanding of all involved methods and principles and creates room for their cross-fertilisation, ripening and further development. Furthermore, from a more pragmatic viewpoint, such a unification is a necessary step in addressing updates of hybrid knowledge bases consisting of both a classical and a rule component. This paper bridges the seemingly irreconcilable approaches to updates. It introduces a novel monotonic characterisation of rules, dubbed RE-models, and shows it to be a more suitable semantic foundation for rule updates than SE-models. Then it proposes a generic scheme for specifying semantic rule update operators, based on the idea of viewing a program as the set of sets of RE-models of its rules; updates are performed by introducing additional interpretations – exceptions – to the sets of RE-models of rules in the original program. The introduced scheme is then used to define particular rule update operators that are closely related to both classical update principles and traditional approaches to rules updates, enjoying a range of plausible syntactic as well as semantic properties. In addition, these operators serve as a basis for a solution to the long-standing problem of state condensing for two of the foundational rule update semantics, showing how they can be equivalently defined as binary operators on some class of logic programs. Finally, the essence of these ideas is extracted to define an abstract framework for exception-based update operators, viewing a knowledge base as the set of sets of models of its elements. It is shown that the framework can capture a wide range of both modeland formula-based classical update operators, and thus serves as the first firm formal ground connecting classical and rule updates.
[EX A]: Exception-Based Knowledge Updates

[EX Q]: We propose an online convex optimization algorithm (RESCALEDEXP) that achieves<lb>optimal regret in the unconstrained setting without prior knowledge of any bounds<lb>on the loss functions. We prove a lower bound showing an exponential sep-<lb>aration between the regret of existing algorithms that require a known bound<lb>on the loss functions and any algorithm that does not require such knowledge.<lb>RESCALEDEXP matches this lower bound asymptotically in the number of itera-<lb>tions. RESCALEDEXP is naturally hyperparameter-free and we demonstrate empir-<lb>ically that it matches prior optimization algorithms that require hyperparameter<lb>optimization. 1 Online Convex Optimization Online Convex Optimization (OCO) [1, 2] provides an elegant framework for modeling noisy,<lb>antagonistic or changing environments. The problem can be stated formally with the help of the<lb>following definitions:<lb>Convex Set: A setW is convex ifW is contained in some real vector space and tw+(1− t)w′ ∈W<lb>for all w,w′ ∈W and t ∈ [0, 1].<lb>Convex Function: f :W → R is a convex function if f(tw + (1− t)w′) ≤ tf(w) + (1− t)f(w′)<lb>for all w,w′ ∈W and t ∈ [0, 1]. An OCO problem is a game of repeated rounds in which on round t a learner first chooses an element<lb>wt in some convex space W , then receives a convex loss function `t, and suffers loss `t(wt). The<lb>regret of the learner with respect to some other u ∈W is defined by
[EX A]:
Online Convex Optimization with Unconstrained Domains and Losses