This is based on work from Knutti et al. The heteroscedastic loss function is prone to episodes of catastrophic forgetting. Synthetic experiment Ozone experimentSpatial coord scaling 2 2 Temporal coord scaling (month of year) 1 2 Temporal coord scaling (total months) 1 1 Number of physical models 4 15 Number of neural network ensemble members 50 65 Bias mean. Noise mean prior 0. 02 0 .015 In the following, we derive the anchored ensembling loss function for the heteroscedastic case.

Wasserstein distributionally robust optimization ( WDRO) is a popular model to enhance the robustness of machine learning with ambiguous data. However, the complexity of WDRO can be prohibitive in practice since solving its "min-imax" formulation requires a great amount of computation. Recently, several fast WDRO training algorithms for some specific machine learning tasks (e.g., logistic regression) have been developed. However, the research on designing efficient algorithms for general large-scale WDROs is still quite limited, to the best of our knowledge. Coreset is an important tool for compressing large dataset, and thus it has been widely applied to reduce the computational complexities for many optimization problems.

Causal inference, i.e., understanding the effect of an intervention in a stochastic system, is a key focus of research in statistics and machine learning [

(conclusion 1). (conclusion 2). Z contains and only contains exogenous noises w.r.t. " means source and " Based on Theorem 6, we can readily give proof to Theorem 2. Note that in our setting where " is equivalent to " Theorem 7 (Trek-separation for directed graphical models, Theorem 2.8 in [ We now show that Theorem 2 can also be proved by trek-separation theorem: Proof of Theorem 2 (another version). 's noise components that is not shared in Therefore, the direction between X and Y is unidentifiable. GIN( Z, Y) must hold, with solution ω .

Causal discovery aims to recover causal structures generating the observational data.

(Theorem 1). A.1 Graphical representation Theorem 1. V ertex X is removable in a MAG M over the variables V, if and only if 1. for any Y 2 Adj ( X) and Z 2 Ch ( X) [ N ( X) \{ Y }, Y and Z are adjacent, and 2. Let H denote the induced subgraph of M over V \{ X } . Since X is removable in M, by definition of removability, ( Y? M, Lemma 6 implies that u is not m-connecting relative to W in H . (: Lemma 6 implies that u is not m-connecting relative to W in M . This contradiction proves that X cannot have a descendant in { Y,Z }[ W, which implies that X blocks u in M .

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Causal inference, i.e., understanding the effect of an intervention in a stochastic system, is a key focus of research in statistics and machine learning [

(conclusion 1). (conclusion 2). Z contains and only contains exogenous noises w.r.t. " means source and " Based on Theorem 6, we can readily give proof to Theorem 2. Note that in our setting where " is equivalent to " Theorem 7 (Trek-separation for directed graphical models, Theorem 2.8 in [ We now show that Theorem 2 can also be proved by trek-separation theorem: Proof of Theorem 2 (another version). 's noise components that is not shared in Therefore, the direction between X and Y is unidentifiable. GIN( Z, Y) must hold, with solution ω .