The issue of fairness in AI has received an increasing amount of attention in
recent years. The problem can be approached by looking at different protected
attributes (e.g., ethnicity, gender, etc) independently, but fairness for
individual protected attributes does not imply intersectional fairness. In this
work, we frame the problem of intersectional fairness within a geometrical
setting. We project our data onto a hypercube, and split the analysis of
fairness by levels, where each level encodes the number of protected attributes
we are intersecting over. We prove mathematically that, while fairness does not
propagate "down" the levels, it does propagate "up" the levels. This means that
ensuring fairness for all subgroups at the lowest intersectional level (e.g.,
black women, white women, black men and white men), will necessarily result in
fairness for all the above levels, including each of the protected attributes
(e.g., ethnicity and gender) taken independently. We also derive a formula
describing the variance of the set of estimated success rates on each level,
under the assumption of perfect fairness. Using this theoretical finding as a
benchmark, we define a family of metrics which capture overall intersectional
bias. Finally, we propose that fairness can be metaphorically thought of as a
"fractal" problem. In fractals, patterns at the smallest scale repeat at a
larger scale. We see from this example that tackling the problem at the lowest
possible level, in a bottom-up manner, leads to the natural emergence of fair
AI. We suggest that trustworthiness is necessarily an emergent, fractal and
relational property of the AI system.

通过在超立方体的几何设置下，将交叉公平问题分层分析，证明了公平并不向下传递，但会向上传递，探讨了公平作为自相似、分形、关联属性在人工智能中的自然涌现。

交叉公平性：分形方法

Intersectional Fairness: A Fractal Approach

One way to resolve the actuation problem of metaphorical language change is
to provide a statistical profile of metaphorical constructions and generative
rules with antecedent conditions. Based on arguments from the view of language
as complex systems and the dynamic view of metaphor, this paper argues that
metaphorical language change qualifies as a self-organized criticality state
and the linguistic expressions of a metaphor can be profiled as a fractal with
spatio-temporal correlations. Synchronously, these metaphorical expressions
self-organize into a self-similar, scale-invariant fractal that follows a
power-law distribution; temporally, long range inter-dependence constrains the
self-organization process by the way of transformation rules that are intrinsic
of a language system. This argument is verified in the paper with statistical
analyses of twelve randomly selected Chinese verb metaphors in a large-scale
diachronic corpus.

通过大规模的历时语料统计分析，本文论证了随机选择的 12 个中文动词隐喻在时间和空间上符合自组织临界状态和分形自我相似的规律。

隐喻语言变化是自组织临界性

Metaphorical Language Change Is Self-Organized Criticality

We analyze the Gambler's problem, a simple reinforcement learning problem
where the gambler has the chance to double or lose the bets until the target is
reached. This is an early example introduced in the reinforcement learning
textbook by Sutton and Barto (2018), where they mention an interesting pattern
of the optimal value function with high-frequency components and repeating
non-smooth points. It is however without further investigation. We provide the
exact formula for the optimal value function for both the discrete and the
continuous cases. Though simple as it might seem, the value function is
pathological: fractal, self-similar, derivative taking either zero or infinity,
and not written as elementary functions. It is in fact one of the generalized
Cantor functions, where it holds a complexity that has been uncharted thus far.
Our analyses could provide insights into improving value function
approximation, gradient-based algorithms, and Q-learning, in real applications
and implementations.

分析了 Sutton 和 Barto（2018）中介绍的简单强化学习问题中赌徒问题的最优价值函数的确切公式，并发现它是分形的，并且是广义 Cantor 函数之一。