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Algorithmic decisions made by machine learning models in high-stakes domains may have lasting impacts over time. However, naive applications of standard fairness criterion in static settings over temporal domains may lead to delayed and adverse effects. To understand the dynamics of performance disparity, we study a fairness problem in Markov decision processes (MDPs). Specifically, we propose return parity, a fairness notion that requires MDPs from different demographic groups that share the same state and action spaces to achieve approximately the same expected time-discounted rewards. We first provide a decomposition theorem for return disparity, which decomposes the return disparity of any two MDPs sharing the same state and action spaces into the distance between group-wise reward functions, the discrepancy of group policies, and the discrepancy between state visitation distributions induced by the group policies. Motivated by our decomposition theorem, we propose algorithms to mitigate return disparity via learning a shared group policy with state visitation distributional alignment using integral probability metrics. We conduct experiments to corroborate our results, showing that the proposed algorithm can successfully close the disparity gap while maintaining the performance of policies on two real-world recommender system benchmark datasets. 

We introduce a new class of photonic resonators with resonant modes that feature hybrid standing-travelling waves. 

Deep penetration of energetic electrons (10s–100s of keV) to lowL‐shells (L < 4), as an important source of inner belt electrons, is commonly observed during geomagnetically active times. However, such deep penetration is not observed as frequently for similar energy protons, for which underlying mechanisms are not fully understood. To study their differential deep penetration, we conducted a statistical analysis using phase space densities (PSDs) ofµ = 10–50 MeV/G,K = 0.14 G^1/2Re electrons and protons from multiyear Van Allen Probes observations. The results suggest systematic differences in electron and proton deep penetration: electron PSD enhancements at lowL‐shells occur more frequently, deeply, and faster than protons. Forµ = 10–50 MeV/G electrons, the occurrence rate of deep penetration events (defined as daily‐averaged PSD enhanced by at least a factor of 2 within a day atL < 4) is ∼2–3 events/month. For protons, only ∼1 event/month was observed forµ = 10 MeV/G, and much fewer events were identified forµ > 20 MeV/G. Leveraging dual‐Probe configurations, fast electron deep penetrations atL < 4 are revealed: ∼70% of electron deep penetration events occurred within ∼9 hr; ∼8%–13% occurred even within 3 hr, with lower‐µelectrons penetrating faster than higher‐µelectrons. These results suggest nondiffusive radial transport as the main mechanism of electron deep penetrations. In comparison, proton deep penetration happens at a slower pace. Statistics also show that the electron PSD radial gradient is much steeper than protons prior to deep penetration events, which can be responsible for these differential behaviors of electron and proton deep penetrations.