In his "ship of state" allegory (\textit{Republic}, Book VI, 488) Plato poses
a question -- how can a crew of sailors presumed to know little about the art
of navigation recognize the true pilot among them? The allegory argues that a
simple majority voting procedure cannot safely determine who is most qualified
to pilot a ship when the voting members are ignorant or biased. We formalize
Plato's concerns by considering the problem in AI safety of monitoring noisy AI
agents in unsupervised settings. An algorithm evaluating AI agents using
unlabeled data would be subject to the evaluation dilemma - how would we know
the evaluation algorithm was correct itself? This endless validation chain can
be avoided by considering purely algebraic functions of the observed responses.
We can construct complete postulates than can prove or disprove the logical
consistency of any grading algorithm. A complete set of postulates exists
whenever we are evaluating $N$ experts that took $T$ tests with $Q$ questions
with $R$ responses each. We discuss evaluating binary classifiers that have
taken a single test - the $(N,T=1,Q,R=2)$ tests. We show how some of the
postulates have been previously identified in the ML literature but not
recognized as such - the \textbf{agreement equations} of Platanios. The
complete postulates for pair correlated binary classifiers are considered and
we show how it allows for error correlations to be quickly calculated. An
algebraic evaluator based on the assumption that the ensemble is error
independent is compared with grading by majority voting on evaluations using
the \uciadult and and \texttt{two-norm} datasets. Throughout, we demonstrate
how the formalism of logical consistency via algebraic postulates of evaluation
can help increase the safety of machines using AI algorithms.

通过代数学公设，本研究构建了一种全面的评估算法来解决无监督环境下监控 AI 代理的问题，在评估二分类器的时候，考虑了错误相关性，与通过多数投票评分相比，该代数评估器能够提高机器使用 AI 算法的安全性。

噪声 AI 代理的 NTQR 评估逻辑：完备公设与逻辑一致的误差相关性

The logic of NTQR evaluations of noisy AI agents: Complete postulates  and logically consistent error correlations

ALAMO is a computational methodology for leaning algebraic functions from
data. Given a data set, the approach begins by building a low-complexity,
linear model composed of explicit non-linear transformations of the independent
variables. Linear combinations of these non-linear transformations allow a
linear model to better approximate complex behavior observed in real processes.
The model is refined, as additional data are obtained in an adaptive fashion
through error maximization sampling using derivative-free optimization. Models
built using ALAMO can enforce constraints on the response variables to
incorporate first-principles knowledge. The ability of ALAMO to generate simple
and accurate models for a number of reaction problems is demonstrated. The
error maximization sampling is compared with Latin hypercube designs to
demonstrate its sampling efficiency. ALAMO's constrained regression methodology
is used to further refine concentration models, resulting in models that
perform better on validation data and satisfy upper and lower bounds placed on
model outputs.

ALAMO 是一种用于从数据中学习代数函数的计算方法，通过从数据中构建低复杂度的线性模型，并使用非线性变换，将线性模型精确逼近实际过程中的复杂行为，并使用无导数优化的误差最大化采样逐步细化模型。同时，该方法可以强制约束反应变量，以纳入基础知识，并用于进一步完善浓度模型，生成表现更好且满足模型输出上下限约束的模型。

机器学习的 ALAMO 方法

The ALAMO approach to machine learning

Consider the polynomial optimization problem whose objective and constraints
are all described by multivariate polynomials. Under some genericity
assumptions, %% on these polynomials, we prove that the optimality conditions
always hold on optimizers, and the coordinates of optimizers are algebraic
functions of the coefficients of the input polynomials. We also give a general
formula for the algebraic degree of the optimal coordinates. The derivation of
the algebraic degree is equivalent to counting the number of all complex
critical points. As special cases, we obtain the algebraic degrees of
quadratically constrained quadratic programming (QCQP), second order cone
programming (SOCP) and $p$-th order cone programming (pOCP), in analogy to the
algebraic degree of semidefinite programming.

针对多项式优化问题，本文证明了在一些范畴的多项式的约束下，优化条件总是满足优化器，优化器坐标是输入多项式系数的代数函数，并且给出了最优坐标的代数度数的一般公式，该公式的推导类似于计算所有复的临界点的数量。作为特例，我们还获得了类比于半定规划的代数度量的 QCQP、SOCP 和 pOCP 的代数度量。