Probabilistic predictions can be evaluated through comparisons with observed label frequencies, that is, through the lens of calibration. Recent scholarship on algorithmic fairness has started to look at a growing variety of calibration-based objectives under the name of multi-calibration but has still remained fairly restricted. In this paper, we explore and analyse forms of evaluation through calibration by making explicit the choices involved in designing calibration scores. We organise these into three grouping choices and a choice concerning the agglomeration of group errors. This provides a framework for comparing previously proposed calibration scores and helps to formulate novel ones with desirable mathematical properties. In particular, we explore the possibility of grouping datapoints based on their input features rather than on predictions and formally demonstrate advantages of such approaches. We also characterise the space of suitable agglomeration functions for group errors, generalising previously proposed calibration scores. Complementary to such population-level scores, we explore calibration scores at the individual level and analyse their relationship to choices of grouping. We draw on these insights to introduce and axiomatise fairness deviation measures for population-level scores. We demonstrate that with appropriate choices of grouping, these novel global fairness scores can provide notions of (sub-)group or individual fairness.

We propose \emph{Choquet regularizers} to measure and manage the level of exploration for reinforcement learning (RL), and reformulate the continuous-time entropy-regularized RL problem of Wang et al. (2020, JMLR, 21(198)) in which we replace the differential entropy used for regularization with a Choquet regularizer. We derive the Hamilton--Jacobi--Bellman equation of the problem, and solve it explicitly in the linear--quadratic (LQ) case via maximizing statically a mean--variance constrained Choquet regularizer. Under the LQ setting, we derive explicit optimal distributions for several specific Choquet regularizers, and conversely identify the Choquet regularizers that generate a number of broadly used exploratory samplers such as $\epsilon$-greedy, exponential, uniform and Gaussian.

Designing reward functions for reinforcement learning is difficult: besides specifying which behavior is rewarded for a task, the reward also has to discourage undesired outcomes. Misspecified reward functions can lead to unintended negative side effects, and overall unsafe behavior. To overcome this problem, recent work proposed to augment the specified reward function with an impact regularizer that discourages behavior that has a big impact on the environment. Although initial results with impact regularizers seem promising in mitigating some types of side effects, important challenges remain. In this paper, we examine the main current challenges of impact regularizers and relate them to fundamental design decisions. We discuss in detail which challenges recent approaches address and which remain unsolved. Finally, we explore promising directions to overcome the unsolved challenges in preventing negative side effects with impact regularizers.

We present SafeLife, a publicly available reinforcement learning environment that tests the safety of reinforcement learning agents. It contains complex, dynamic, tunable, procedurally generated levels with many opportunities for unsafe behavior. Agents are graded both on their ability to maximize their explicit reward and on their ability to operate safely without unnecessary side effects. We train agents to maximize rewards using proximal policy optimization and score them on a suite of benchmark levels. The resulting agents are performant but not safe---they tend to cause large side effects in their environments---but they form a baseline against which future safety research can be measured.

Utilizing recently introduced concepts from statistics and quantitative risk management, we present a general variant of Batch Normalization (BN) that offers accelerated convergence of Neural Network training compared to conventional BN. In general, we show that mean and standard deviation are not always the most appropriate choice for the centering and scaling procedure within the BN transformation, particularly if ReLU follows the normalization step. We present a Generalized Batch Normalization (GBN) transformation, which can utilize a variety of alternative deviation measures for scaling and statistics for centering, choices which naturally arise from the theory of generalized deviation measures and risk theory in general. When used in conjunction with the ReLU non-linearity, the underlying risk theory suggests natural, arguably optimal choices for the deviation measure and statistic. Utilizing the suggested deviation measure and statistic, we show experimentally that training is accelerated more so than with conventional BN, often with improved error rate as well. Overall, we propose a more flexible BN transformation supported by a complimentary theoretical framework that can potentially guide design choices.

How can we design reinforcement learning agents that avoid causing unnecessary disruptions to their environment? We argue that current approaches to penalizing side effects can introduce bad incentives in tasks that require irreversible actions, and in environments that contain sources of change other than the agent. For example, some approaches give the agent an incentive to prevent any irreversible changes in the environment, including the actions of other agents. We introduce a general definition of side effects, based on relative reachability of states compared to a default state, that avoids these undesirable incentives. Using a set of gridworld experiments illustrating relevant scenarios, we empirically compare relative reachability to penalties based on existing definitions and show that it is the only penalty among those tested that produces the desired behavior in all the scenarios.