In real-world applications, it is important and desirable to learn a model that performs well on out-of-distribution (OOD) data. Recently, causality has become a powerful tool to tackle the OOD generalization problem, with the idea resting on the causal mechanism that is invariant across domains of interest. To leverage the generally unknown causal mechanism, existing works assume a linear form of causal feature or require sufficiently many and diverse training domains, which are usually restrictive in practice. In this work, we obviate these assumptions and tackle the OOD problem without explicitly recovering the causal feature. Our approach is based on transformations that modify the non-causal feature but leave the causal part unchanged, which can be either obtained from prior knowledge or learned from the training data in the multi-domain scenario. Under the setting of invariant causal mechanism, we theoretically show that if all such transformations are available, then we can learn a minimax optimal model across the domains using only single domain data. Noticing that knowing a complete set of these causal invariant transformations may be impractical, we further show that it suffices to know only a subset of these transformations. Based on the theoretical findings, a regularized training procedure is proposed to improve the OOD generalization capability. Extensive experimental results on both synthetic and real datasets verify the effectiveness of the proposed algorithm, even with only a few causal invariant transformations.

Cancer-associated fibroblasts (CAFs) mediate an immunosuppressive effect, but the underlying mechanism remains incompletely defined. Here we show that increasing prostatic stromal Foxf2 suppresses the growth and progression of both syngeneic and autochthonous mouse prostate cancer models in an immunocompetent context. Mechanistically, Foxf2 moderately attenuates the CAF phenotype and transcriptionally downregulates Cxcl5, which diminish the immunosuppressive myeloid cells and enhance T cell cytotoxicity. Increasing prostatic stromal Foxf2 sensitizes prostate cancer to the immune checkpoint blockade therapies. Augmenting lung stromal Foxf2 also mediates an immunosuppressive milieu and inhibits lung colonization of prostate cancer. FOXF2 is expressed higher in the stroma of human transition zone (TZ) than peripheral zone (PZ) prostate. The stromal FOXF2 expression level in primary prostate cancers inversely correlates with the Gleason grade. Our study establishes Foxf2 as a stromal transcription factor modulating the tumor immune microenvironment and potentially explains why cancers are relatively rare and indolent in the TZ prostate.