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  • method This appendix supplies the predictive-variance machinery deferred from Section 2.5. Kriging conditional variance.Given an observation set Oj⊆{1,...,N}at sample j, the fixed-rank kriging conditional variance (Cressie and Johannesson, 2008) at a query locations∗is ˆv(s∗,j) :=ˆσ2 + ˆϕ(s∗)⊤ˆΛ cond(Oj)ˆϕ(s∗),(16) with plug-in conditional score covariance ˆΛ cond(Oj) := ( ˆΛ−1+ˆσ−2Φ⊤ OjΦOj )−1 .(17) WhenOj =∅(new-sample regime),ˆΛ cond =ˆΛ and ˆvreduces to the marginal variance; when Oj is the trainin
  • background Multi-Generation Regression Outputs.For a minibatch of inputs {x1, x2, . . . , xB}, GRPO samples K independent generation trajectories for each input. This yields a set of numeric predictions: q(xi) =  q1(xi), q2(xi), . . . , qK (xi) ⊤ ,(1) which naturally encode prediction variability. We summa- rize these outputs using their empirical mean: µ(xi) = 1 K KX k=1 qk(xi),(2) which provides a stable, low-variance estimate for each compared sample during reward computation. Batch-Level Relational C
  • background (28) In particular,Γ j,R is not a fold trace (e.g., it excludesz(y) =|y 1|along{y 1 = 0}), and the trace locally separatesU j,R into two nonempty strict-sign sides. (A3) Non-redundancy of intersecting traces (essentiality).For everyj∈ Kwith Zj ∩int(P)̸=∅, there exists a point y∈Z j ∩int(P)such thatz k(y)̸= 0∀k∈ K \ {j}.(29) Equivalently, Zj ∩int(P)̸⊆ [ k∈K\{j} Zk.(30) Lemma 3 (Counting cells intersectingPvs.int(P))LetP⊂R d be a convex closed polytope withint(P)̸=∅, and letPbe any collection of f
  • background In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016. Patrick Helber, Benjamin Bischke, Andreas Dengel, and Damian Borth. Eurosat: A novel dataset and deep learning benchmark for land use and land cover classification.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 12(7):2217-2226, 2019. JulianIbarz, JieTan, ChelseaFinn, MrinalKalakrishnan, PeterPastor, andSergeyLevine. Howtotrainyour robot with deep reinforcement l
  • background For FEDADAGRAD , we setβ1 =β2 = 0 (as typical versions of ADAGRAD do not use momentum). For FEDADAM and FEDYOGI we setβ1 = 0.9,β 2 = 0.99. While these parameters are generally 22 Published as a conference paper at ICLR 2021 Algorithm 5 FEDADAGRADFEDADAGRADFEDADAGRAD , FEDYOGIFEDYOGIFEDYOGI , and FEDADAMFEDADAMFEDADAM - Batched data Input:x0,v−1≥τ 2, optionalβ1,β 2∈ [0, 1) for FEDYOGI and FEDADAM fort = 0,··· ,T − 1 do Sample a subsetS of clients xt i =xt for each clienti∈S in parallel do fore =
  • background Jingkang Yang, Pengyun Wang, Dejian Zou, Zitang Zhou, Kunyuan Ding, Wenxuan Peng, Haoqi Wang, Guangyao Chen, Bo Li, Yiyou Sun, et al. Openood: Benchmarking generalized out-of- distribution detection.Advances in Neural Information Processing Systems, 35:32598-32611, 2022. Jingkang Yang, Kaiyang Zhou, Yixuan Li, and Ziwei Liu. Generalized out-of-distribution detection: A survey.International Journal of Computer Vision, 132(12):5635-5662, 2024. 11 A TRAININGCONFIGURATION All experiments share a com

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