Algorithmic fairness has emerged as a central issue in ML, and it has become standard practice to adjust ML algorithms so that they will satisfy fairness requirements such as Equal Opportunity. In this paper we consider the effects of adopting such fair classifiers on the overall level of ecosystem fairness. Specifically, we introduce the study of fairness with competing firms, and demonstrate the failure of fair classifiers in yielding fair ecosystems. Our results quantify the loss of fairness in systems, under a variety of conditions, based on classifiers' correlation and the level of their data overlap. We show that even if competing classifiers are individually fair, the ecosystem's outcome may be unfair; and that adjusting biased algorithms to improve their individual fairness may lead to an overall decline in ecosystem fairness. In addition to these theoretical results, we also provide supporting experimental evidence. Together, our model and results provide a novel and essential call for action.

We study fair machine learning (ML) under predictive uncertainty to enable reliable and trustworthy decision-making. The seminal work of "equalized coverage" proposed an uncertainty-aware fairness notion. However, it does not guarantee equal coverage rates across more fine-grained groups (e.g., low-income females) conditioning on the true label and is biased in the assessment of uncertainty. To tackle these limitations, we propose a new uncertainty-aware fairness - Equal Opportunity of Coverage (EOC) - that aims to achieve two properties: (1) coverage rates for different groups with similar outcomes are close, and (2) the coverage rate for the entire population remains at a predetermined level. Further, the prediction intervals should be narrow to be informative. We propose Binned Fair Quantile Regression (BFQR), a distribution-free post-processing method to improve EOC with reasonable width for any trained ML models. It first calibrates a hold-out set to bound deviation from EOC, then leverages conformal prediction to maintain EOC on a test set, meanwhile optimizing prediction interval width. Experimental results demonstrate the effectiveness of our method in improving EOC.

The seminal work of "equalized coverage" proposed an uncertainty-aware fairness notion.

We study fair machine learning (ML) under predictive uncertainty to enable reliable and trustworthy decision-making. The seminal work of'equalized coverage' proposed an uncertainty-aware fairness notion. However, it does not guarantee equal coverage rates across more fine-grained groups (e.g., low-income females) conditioning on the true label and is biased in the assessment of uncertainty. To tackle these limitations, we propose a new uncertainty-aware fairness -- Equal Opportunity of Coverage (EOC) -- that aims to achieve two properties: (1) coverage rates for different groups with similar outcomes are close, and (2) the coverage rate for the entire population remains at a predetermined level. Further, the prediction intervals should be narrow to be informative. We propose Binned Fair Quantile Regression (BFQR), a distribution-free post-processing method to improve EOC with reasonable width for any trained ML models. It first calibrates a hold-out set to bound deviation from EOC, then leverages conformal prediction to maintain EOC on a test set, meanwhile optimizing prediction interval width. Experimental results demonstrate the effectiveness of our method in improving EOC.

Algorithmic fairness has emerged as a central issue in ML, and it has become standard practice to adjust ML algorithms so that they will satisfy fairness requirements such as Equal Opportunity. In this paper we consider the effects of adopting such fair classifiers on the overall level of ecosystem fairness. Specifically, we introduce the study of fairness with competing firms, and demonstrate the failure of fair classifiers in yielding fair ecosystems. Our results quantify the loss of fairness in systems, under a variety of conditions, based on classifiers' correlation and the level of their data overlap. We show that even if competing classifiers are individually fair, the ecosystem's outcome may be unfair; and that adjusting biased algorithms to improve their individual fairness may lead to an overall decline in ecosystem fairness. In addition to these theoretical results, we also provide supporting experimental evidence. Together, our model and results provide a novel and essential call for action.

The Extreme Operating Conditions Search (EOCS) problem is one of the key problems in relay setting calculation, which is used to ensure that the setting values of protection relays can adapt to the changing operating conditions of power systems over a period of time after deployment. The high penetration of renewable energy and the wide application of inverter-based resources make the operating conditions of renewable power systems more volatile, which urges the adoption of the online relay setting calculation strategy. However, the computation speed of existing EOCS methods based on local enumeration, heuristic algorithms, and mathematical programming cannot meet the efficiency requirement of online relay setting calculation. To reduce the time overhead, this paper, for the first time, proposes an efficient deep learning-based EOCS method suitable for online relay setting calculation. First, the power system information is formulated as four layers, i.e., a component parameter layer, a topological connection layer, an electrical distance layer, and a graph distance layer, which are fed into a parallel graph neural network (PGNN) model for feature extraction. Then, the four feature layers corresponding to each node are spliced and stretched, and then fed into the decision network to predict the extreme operating condition of the system. Finally, the proposed PGNN method is validated on the modified IEEE 39-bus and 118-bus test systems, where some of the synchronous generators are replaced by renewable generation units. The nonlinear fault characteristics of renewables are fully considered when computing fault currents. The experiment results show that the proposed PGNN method achieves higher accuracy than the existing methods in solving the EOCS problem. Meanwhile, it also provides greater improvements in online computation time.

We study the concentration of the Neural Tangent Kernel (NTK) $K_\theta : \mathbb{R}^{m_0} \times \mathbb{R}^{m_0} \to \mathbb{R}^{m_l \times m_l}$ of $l$-layer Multilayer Perceptrons (MLPs) $N : \mathbb{R}^{m_0} \times \Theta \to \mathbb{R}^{m_l}$ equipped with activation functions $\phi(s) = a s + b \vert s \vert$ for some $a,b \in \mathbb{R}$ with the parameter $\theta \in \Theta$ being initialized at the Edge Of Chaos (EOC). Without relying on the gradient independence assumption that has only been shown to hold asymptotically in the infinitely wide limit, we prove that an approximate version of gradient independence holds at finite width. Showing that the NTK entries $K_\theta(x_{i_1},x_{i_2})$ for $i_1,i_2 \in [1:n]$ over a dataset $\{x_1,\cdots,x_n\} \subset \mathbb{R}^{m_0}$ concentrate simultaneously via maximal inequalities, we prove that the NTK matrix $K(\theta) = [\frac{1}{n} K_\theta(x_{i_1},x_{i_2}) : i_1,i_2 \in [1:n]] \in \mathbb{R}^{nm_l \times nm_l}$ concentrates around its infinitely wide limit $\overset{\scriptscriptstyle\infty}{K} \in \mathbb{R}^{nm_l \times nm_l}$ without the need for linear overparameterization. Our results imply that in order to accurately approximate the limit, hidden layer widths have to grow quadratically as $m_k = k^2 m$ for some $m \in \mathbb{N}+1$ for sufficient concentration. For such MLPs, we obtain the concentration bound $\mathbb{P}( \Vert K(\theta) - \overset{\scriptscriptstyle\infty}{K} \Vert \leq O((\Delta_\phi^{-2} + m_l^{\frac{1}{2}} l) \kappa_\phi^2 m^{-\frac{1}{2}})) \geq 1-O(m^{-1})$ modulo logarithmic terms, where we denoted $\Delta_\phi = \frac{b^2}{a^2+b^2}$ and $\kappa_\phi = \frac{\vert a \vert + \vert b \vert}{\sqrt{a^2 + b^2}}$. This reveals in particular that the absolute value ($\Delta_\phi=1$, $\kappa_\phi=1$) beats the ReLU ($\Delta_\phi=\frac{1}{2}$, $\kappa_\phi=\sqrt{2}$) in terms of the concentration of the NTK.

We study the properties of the Neural Tangent Kernel (NTK) $\overset{\scriptscriptstyle\infty}{K} : \mathbb{R}^{m_0} \times \mathbb{R}^{m_0} \to \mathbb{R}^{m_l \times m_l}$ corresponding to infinitely wide $l$-layer Multilayer Perceptrons (MLPs) taking inputs from $\mathbb{R}^{m_0}$ to outputs in $\mathbb{R}^{m_l}$ equipped with activation functions $\phi(s) = a s + b \vert s \vert$ for some $a,b \in \mathbb{R}$ and initialized at the Edge Of Chaos (EOC). We find that the entries $\overset{\scriptscriptstyle\infty}{K}(x_1,x_2)$ can be approximated by the inverses of the cosine distances of the activations corresponding to $x_1$ and $x_2$ increasingly better as the depth $l$ increases. By quantifying these inverse cosine distances and the spectrum of the matrix containing them, we obtain tight spectral bounds for the NTK matrix $\overset{\scriptscriptstyle\infty}{K} = [\frac{1}{n} \overset{\scriptscriptstyle\infty}{K}(x_{i_1},x_{i_2}) : i_1, i_2 \in [1:n]]$ over a dataset $\{x_1,\cdots,x_n\} \subset \mathbb{R}^{m_0}$, transferred from the inverse cosine distance matrix via our approximation result. Our results show that $\Delta_\phi = \frac{b^2}{a^2+b^2}$ determines the rate at which the condition number of the NTK matrix converges to its limit as depth increases, implying in particular that the absolute value ($\Delta_\phi=1$) is better than the ReLU ($\Delta_\phi=\frac{1}{2}$) in this regard.

Searching for the Extreme Operating Conditions (EOCs) is one of the core problems of power system relay protection setting calculation. The current methods based on brute-force search, heuristic algorithms, and mathematical programming can hardly meet the requirements of today's power systems in terms of computation speed due to the drastic changes in operating conditions induced by renewables and power electronics. This paper proposes an EOC fast search method, named Graph Dueling Double Deep Q Network (Graph D3QN), which combines graph neural network and deep reinforcement learning to address this challenge. First, the EOC search problem is modeled as a Markov decision process, where the information of the underlying power system is extracted using graph neural networks, so that the EOC of the system can be found via deep reinforcement learning. Then, a two-stage Guided Learning and Free Exploration (GLFE) training framework is constructed to accelerate the convergence speed of reinforcement learning. Finally, the proposed Graph D3QN method is validated through case studies of searching maximum fault current for relay protection setting calculation on the IEEE 39-bus and 118-bus systems. The experimental results demonstrate that Graph D3QN can reduce the computation time by 10 to 1000 times while guaranteeing the accuracy of the selected EOCs.

We study fair machine learning (ML) under predictive uncertainty to enable reliable and trustworthy decision-making. The seminal work of'equalized coverage' proposed an uncertainty-aware fairness notion. However, it does not guarantee equal coverage rates across more fine-grained groups (e.g., low-income females) conditioning on the true label and is biased in the assessment of uncertainty. To tackle these limitations, we propose a new uncertainty-aware fairness -- Equal Opportunity of Coverage (EOC) -- that aims to achieve two properties: (1) coverage rates for different groups with similar outcomes are close, and (2) the coverage rate for the entire population remains at a predetermined level. Further, the prediction intervals should be narrow to be informative.