REVIEW 4 major objections 5 minor 56 references
CASTLE: Regularization via Auxiliary Causal Graph Discovery
T0 review · 4 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read CASTLE regularizes neural networks by jointly learning the causal DAG among features and target, claiming better out-of-sample generalization than standard regularizers.
desk verdict CASTLE is a promising and pragmatic regularizer for causal tabular data, but the core DAG mechanism is mis-specified in the text and the 'never degrades' claim is overreach. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the adjacency matrix $M$ embedded in the input-layer weights of a feed-forward network with $d+1$ sub-networks: $[M]_{k,j}$ is the $\ell^2$-norm of the $k$-th row of sub-network $j$'s input weight matrix, so it records the strength of edge $k\to j$. The mechanism that carries the argument is the continuous acyclicity penalty $R_{\Theta_1}=(\mathrm{Tr}(e^{M\odot M})-d-1)^2$, which is zero exactly when the learned graph is a DAG, combined with the reconstruction loss $L_N(f_\Theta)=\frac{1}{N}\|\tilde{X}-f_\Theta(\tilde{X})\|_F^2$ that reconstructs each variable only from the others. The PAC-Bayes bound in Theorem 1 reinterprets this combination as a prior on the input weights and controls the expected reconstruction loss by the empirical loss plus $(R_{\Theta_1}+C_1(V_{\Theta_1}+V_{\Theta_2})+\log(8/\delta))/N$, showing the acyclicity term acts as a capacity term with its largest effect at small sample sizes.
What would settle it
Simulate data from a known DAG in which the target is an exact invertible function of one of its own descendants, so an anti-causal predictor beats the true parents; if CASTLE still improves out-of-sample error without assigning weight to the true parents, the causal-DAG mechanism is not what produces the gain. Conversely, on data with no causal structure at all, independent features unrelated to the target, CASTLE should not beat early stopping; if it does, the benefit is not tied to the DAG assumption.
Extended reading notes
Core claim
The central claim is that a supervised network generalizes better when the training objective itself discovers the causal DAG among $[Y,X]$ rather than using a causality-agnostic penalty. CASTLE's objective is $\min_\Theta \frac{1}{N}\|Y-[f_\Theta(\tilde{X})]_{:,1}\|^2 + \lambda R_{DAG}(\tilde{X},f_\Theta)$, where the regularizer is the sum of a neighbor-only reconstruction loss, an $\ell^1$ sparsity term, and the acyclicity penalty $R_{\Theta_1}=(\mathrm{Tr}(e^{M\odot M})-d-1)^2$, with $M$ encoding the $\ell^2$ norms of the input-layer weight rows. The paper asserts that minimizing this joint objective achieves both causal prediction and target selection, and reports lower test error than early stopping, L1, L2, dropout, batch normalization, input noise, MixUp, and supervised autoencoders across its synthetic and real benchmark datasets.
Load-bearing premise
The regularizer only helps when the observed features and target really are generated by an underlying causal DAG and the continuous acyclicity penalty recovers that DAG from finite data; where that premise fails, as the paper itself concedes for image data, CASTLE has no grounding and can degrade predictions.
Editorial extensions
If this is right
- On tabular data with an underlying causal structure, CASTLE should give lower out-of-sample error than capacity-based regularizers and supervised autoencoders at matched architectures.
- Reconstruction-based regularization should reconstruct only variables with causal neighbors, because reconstructing noise variables, as standard autoencoders do, can actively hurt generalization.
- The PAC-Bayes bound predicts the benefit of the acyclicity term shrinks as sample size grows, so CASTLE's advantage should be largest in small-$N$ regimes.
- Causal parents are the preferred predictors when they exist; when they do not, the method falls back to children of the target rather than unrelated correlated features.
- CASTLE can be combined with conventional capacity control ($V_{\Theta_2}$) since the bound includes both terms, potentially giving further gains.
Reading between the lines
- The paper's generalization bound is stated for the reconstruction loss of the whole vector $\tilde{X}$, not directly for the prediction loss of $Y$; interpreting the bound as a guarantee for the supervised task requires an extra step the paper leaves implicit.
- A natural stress test the authors do not run: vary the proportion of noise (neighborless) variables while keeping the true parents of $Y$ weak; CASTLE should separate from SAE exactly where target selection matters most.
- The computational cost of training the joint network grows sharply with the number of features, so a practical successor would need a theory for the feature-subsampling trick the paper mentions but does not analyze.
- Because the method assumes no hidden confounders, an immediate extension would be to incorporate latent-variable causal discovery; without it, CASTLE may learn a DAG over proxies rather than the true causal graph.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes CASTLE, a regularization method that jointly learns a causal DAG over the input features and target variable while training a neural network for supervised prediction. The DAG is embedded in the input-layer weights of a multi-output network, with a continuous acyclicity penalty and a sparsity penalty used as the regularizer, and the auxiliary task is reconstruction of only the variables that have causal neighbors. The authors provide a PAC-Bayes generalization bound for the reconstruction loss, an ablation study, and large-scale experiments on synthetic and real tabular datasets comparing CASTLE with capacity-based and reconstruction-based regularizers. The central claim is that this causal inductive bias improves out-of-sample prediction and target selection relative to existing regularizers.
Significance. The core idea—using causal structure learning as an inductive bias for regularization and for selecting which variables to reconstruct—is original and potentially valuable. If the formal mechanism is made precise, the paper could make a useful contribution to the intersection of causality and deep learning regularization. The authors provide code, extensive experiments across many datasets, and an explicit attempt at a theoretical bound. However, the current manuscript has a load-bearing mathematical inconsistency in the definition of the DAG regularizer, and the theoretical bound addresses reconstruction loss rather than the supervised prediction loss that is the paper's headline. These issues condition the significance on corrections rather than establishing the claims as written.
major comments (4)
- [Section 3.3, Eq. (3), Eq. (7)] The dimensions and indexing of the input weight matrices W_k^1 are internally inconsistent. The text declares W_k^1 to be an h x (d+1) matrix, but the definition f_k(\tilde X) = φ(...φ(\tilde X W_k^1)...) requires W_k^1 to have d+1 rows so that the product with the N x (d+1) input matrix is defined. With the declared h x (d+1) shape, the row dimension of W_j^1 runs over hidden units, so [M]_{k,j} = ||row_k(W_j^1)||_2 in Eq. (7) does not define a (d+1)-variable adjacency matrix; it mixes hidden-unit indices with variable indices. The masking instruction "set the k-th column of W_k^1 to zero" is also incompatible with the only shape that makes Eq. (3) well-defined, where the k-th row must be zeroed. As written, the manuscript does not formally define the causal DAG adjacency matrix that is claimed to be embedded in the network, so the central mechanism of CASTLE is not checkable from the text alone. This should be corrected and the derivation of Eq. (7) restated consistently; the released code may resolve the intended convention, but the paper must be self-contained.
- [Section 3.4, Theorem 1, Eq. (8)] The theoretical result bounds the expected reconstruction loss L_P(f_Θ) = E||f_Θ(\tilde X) - \tilde X||^2, not the supervised prediction loss for Y. The abstract and introduction claim that CASTLE improves out-of-sample prediction of Y, but the bound contains no term for the prediction error of the first output [f_Θ]_{:,1}. The paragraph following Eq. (8) asserts that the reconstruction-loss result "explains the benefit" for predicting Y, but no Lipschitz or other transfer argument is given to connect reconstruction accuracy of the joint vector to prediction accuracy of its first component. If the theorem is meant to support the generalization of the prediction task, an additional argument or a modified bound is needed; otherwise the claim should be narrowed explicitly.
- [Appendix A, Eqs. (14)-(15)] The KL-divergence bound appears to have an algebraic inconsistency with the final theorem statement. Eq. (13) gives L_P ≤ 4L_N + (2/N)[2 KL + log(8/δ)] + C_2, which has coefficient 4/N on the KL term. Substituting the bound KL ≤ (1/2)R_θ1 + (1/(2σ^2))(V_Θ1+V_Θ2) + ... would yield a coefficient of 2/N for R_θ1 in the final display, not 1/N as printed in Eq. (15) and Theorem 1. Please recheck the constants and ensure that the displayed bound follows from the preceding display; the current proof does not verify the theorem as stated.
- [Section 5 and Tables 3, 5, 6] The conclusion states that "in the worst case, never degrades performance," but the empirical evidence is based on means and standard deviations without any paired significance tests across folds or datasets. In several cases the CASTLE improvement is within one standard deviation of the baseline (e.g., BC regression 0.318±0.036 vs. Baseline 0.359±0.061; LV classification 0.595±0.032 vs. Baseline 0.559±0.061). The "never degrades" claim is stronger than what is demonstrated; either provide appropriate statistical tests (e.g., paired tests over folds) or soften the claim to the observed datasets and conditions.
minor comments (5)
- [Section 3.3, "Target Selection"] The statement that CASTLE "reconstructs only the features in the causal DAG that have a causal neighbor" is not a formal property of the objective: the reconstruction loss L_N(f_Θ) attempts to reconstruct every variable except the self-masked one, and the near-zero weights for noise variables arise from the L1 penalty and the optimization, not from an explicit constraint. Consider presenting this as an empirical property rather than a structural guarantee.
- [Section 4.1, Dissecting CASTLE] The text says "the best and worst possible rank is one and r, respectively (i.e., the higher the rank the better)"; if rank 1 is best, a higher numeric rank is worse. Please clarify the direction of the ranking or correct the wording, as the figure caption and accompanying discussion are ambiguous.
- [Theorem 1 in Section 3.4] The displayed bound in Eq. (8) uses a constant C_3, while the surrounding text refers to C_1 and C_2, and the appendix version of the theorem uses C_2. Unify the notation to avoid confusion.
- [Section 4, Experimental setup] The preprocessing description says all continuous variables are standardized, but the classification datasets surely contain categorical variables; please specify the encoding (e.g., one-hot) and whether those were standardized, since the DAG and the weight-matrix mechanism depend on the input representation.
- [Appendix C.1, Sensitivity analysis] The sensitivity analysis for λ is performed on synthetic DAGs only, yet λ = 1 is then used for all real datasets; since the scale of the reconstruction loss and the acyclicity penalty may vary across datasets even after standardization, a brief justification or a real-data sensitivity check would strengthen the hyperparameter choice.
Circularity Check
No significant circularity: the DAG regularizer is an external, independently grounded inductive bias and the generalization bound is a standard complexity-style bound, not a fitted quantity.
full rationale
CASTLE is a regularization method that adds a DAG-structure penalty to a supervised objective. The central elements are (a) the continuous acyclicity penalty from the external works [37, 38], (b) a reconstruction loss over all variables, and (c) a PAC-Bayes generalization bound in which the DAG penalty appears as a complexity term. None of these reduces to the paper's own empirical claim by construction. The acyclicity penalty R_Theta1 is not defined in terms of the target prediction or generalization performance; it is a function of the learned weight matrices and is imported from prior work by Zheng et al., not from the present authors. The objective in Eq. (5) genuinely combines a prediction loss with a regularizer, and the experiments compare against independent benchmarks, so there is no fitted parameter that is then renamed as a prediction. The PAC-Bayes bound in Theorem 1 is valid for all parameters simultaneously and treats R_Theta1 as part of the complexity term; minimizing it lowers an upper bound, but this is standard regularization theory rather than a circular equation. The 'target selection' property, i.e., reconstructing only variables with neighbors in the learned DAG, follows from the sparsity and acyclicity penalties rather than being assumed as the conclusion. The citations to the authors' own earlier work are not load-bearing for the main derivation. There is a separate formal clarity concern about the stated dimensions of W_k^1 and the indexing in Eq. (7), but that is an internal consistency issue, not a circular reduction of the claimed result to its inputs. Overall, no circular step meeting the required standard was found.
Assumptions & free parameters
free parameters (2)
- lambda (DAG loss weight) =
1
- beta (sparsity penalty) =
Not reported per dataset; grid searched over {0.001, 0.01, 0.1, 1}
assumptions (4)
- domain assumption The variables [Y,X] satisfy a nonparametric structural equation model with an underlying DAG G (Definition 1).
- standard math A matrix W is a DAG if and only if Tr(e^{W⊙W}) - (d+1) = 0 (Theorem 1 in [37]).
- domain assumption The squared reconstruction loss is sub-Gaussian under the data distribution (Assumption 2).
- standard math PAC-Bayes generalization bounds and the derandomization of Neyshabur et al. [45] apply to deterministic ReLU networks.
Cite this review
Pith. "Pith review of CASTLE: Regularization via Auxiliary Causal Graph Discovery." pith.science (2026). https://pith.science/paper/MN2FMDTH
@misc{pith2026200913180,
author = {Pith},
title = {Pith review of: CASTLE: Regularization via Auxiliary Causal Graph Discovery},
year = {2026},
howpublished = {\url{https://pith.science/paper/MN2FMDTH}},
note = {Machine review of arXiv:2009.13180}
}
read the original abstract
Regularization improves generalization of supervised models to out-of-sample data. Prior works have shown that prediction in the causal direction (effect from cause) results in lower testing error than the anti-causal direction. However, existing regularization methods are agnostic of causality. We introduce Causal Structure Learning (CASTLE) regularization and propose to regularize a neural network by jointly learning the causal relationships between variables. CASTLE learns the causal directed acyclical graph (DAG) as an adjacency matrix embedded in the neural network's input layers, thereby facilitating the discovery of optimal predictors. Furthermore, CASTLE efficiently reconstructs only the features in the causal DAG that have a causal neighbor, whereas reconstruction-based regularizers suboptimally reconstruct all input features. We provide a theoretical generalization bound for our approach and conduct experiments on a plethora of synthetic and real publicly available datasets demonstrating that CASTLE consistently leads to better out-of-sample predictions as compared to other popular benchmark regularizers.
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Works this paper leans on
-
[1]
Larry S. Yaeger, Richard F. Lyon, and Brandyn J. Webb. Effective training of a neural network character classifier for word recognition. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems 9, pages 807–816. MIT Press, 1997
work page 1997
-
[2]
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep convolutional neural networks. In Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 1, NIPS’12, page 1097–1105, Red Hook, NY , USA,
-
[3]
Improving neural networks by preventing co-adaptation of feature detectors
Geoffrey E. Hinton, Nitish Srivastava, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhut- dinov. Improving neural networks by preventing co-adaptation of feature detectors. ArXiv, abs/1207.0580, 2012
work page Pith review arXiv 2012
-
[4]
Dropout training as adaptive regularization
Stefan Wager, Sida Wang, and Percy Liang. Dropout training as adaptive regularization. Advances in Neural Information Processing Systems, 07 2013
work page 2013
-
[5]
Dropout: A simple way to prevent neural networks from overfitting
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(56):1929–1958, 2014
work page 1929
-
[6]
Adversarial regularizers in inverse problems
Sebastian Lunz, Ozan Öktem, and Carola-Bibiane Schönlieb. Adversarial regularizers in inverse problems. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and 9 R. Garnett, editors, Advances in Neural Information Processing Systems 31, pages 8507–8516. Curran Associates, Inc., 2018
work page 2018
-
[7]
Deep label distribution learning with label ambiguity
Bin-Bin Gao, Chao Xing, Chen-Wei Xie, Jianxin Wu, and Xin Geng. Deep label distribution learning with label ambiguity. IEEE Transactions on Image Processing, 26:2825–2838, 04 2017
work page 2017
-
[8]
Greedy layer-wise training of deep networks
Yoshua Bengio, Pascal Lamblin, Dan Popovici, and Hugo Larochelle. Greedy layer-wise training of deep networks. In Proceedings of the 19th International Conference on Neural Information Processing Systems, NIPS’06, page 153–160, Cambridge, MA, USA, 2006. MIT Press
work page 2006
Show all 56 references
-
[9]
Semi-supervised learning of compact document representations with deep networks
Marc’Aurelio Ranzato and Martin Szummer. Semi-supervised learning of compact document representations with deep networks. In Proceedings of the 25th International Conference on Machine Learning, pages 792–799, 01 2008
2008
-
[10]
Layer-wise coordination between encoder and decoder for neural machine translation
Tianyu He, Xu Tan, Yingce Xia, Di He, Tao Qin, Zhibo Chen, and Tie-Yan Liu. Layer-wise coordination between encoder and decoder for neural machine translation. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Inf...
2018
-
[11]
Extracting and composing robust features with denoising autoencoders
Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. InProceedings of the 25th international conference on Machine learning, pages 1096–1103. ACM, 2008
2008
-
[12]
Supervised autoencoders: Improving generaliza- tion performance with unsupervised regularizers
Lei Le, Andrew Patterson, and Martha White. Supervised autoencoders: Improving generaliza- tion performance with unsupervised regularizers. In Advances in Neural Information Processing Systems, pages 107–117, 2018
2018
-
[13]
Causal discovery with reinforcement learning
Shengyu Zhu and Zhitang Chen. Causal discovery with reinforcement learning. CoRR, abs/1906.04477, 2019
1906 arXiv
-
[14]
Triad constraints for learning causal structure of latent variables
Ruichu Cai, Feng Xie, Clark Glymour, Zhifeng Hao, and Kun Zhang. Triad constraints for learning causal structure of latent variables. In Advances in Neural Information Processing Systems 32, pages 12883–12892. Curran Associates, Inc., 2019
2019
-
[15]
Estimating individual treatment effect: generalization bounds and algorithms
Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect: generalization bounds and algorithms. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 3076–3085. JMLR. org, 2017
2017
-
[16]
Validating causal inference models via influence functions
Ahmed Alaa and Mihaela van der Schaar. Validating causal inference models via influence functions. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, volume 97 ofProceedings of Machine Learning Research...
2019
-
[17]
Cxplain: Causal explanations for model interpretation under uncertainty
Patrick Schwab and Walter Karlen. Cxplain: Causal explanations for model interpretation under uncertainty. In Advances in Neural Information Processing Systems 32, pages 10220–10230. Curran Associates, Inc., 2019
2019
-
[18]
Domain adaptation un- der target and conditional shift
Kun Zhang, Bernhard Schölkopf, Krikamol Muandet, and Zhikun Wang. Domain adaptation un- der target and conditional shift. In Sanjoy Dasgupta and David McAllester, editors,Proceedings of the 30th International Conference on Machine Learning (ICML), volume 28 of Proceedings of M...
2013
-
[19]
Causal inference by using invariant prediction: identification and confidence intervals
Jonas Peters, Peter Bühlmann, and Nicolai Meinshausen. Causal inference by using invariant prediction: identification and confidence intervals. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(5):947–1012, 2016
2016
-
[20]
Domain adaptation by using causal inference to predict invariant conditional distributions
Sara Magliacane et al. Domain adaptation by using causal inference to predict invariant conditional distributions. In S. Bengio et al., editors,Advances in Neural Information Processing Systems 31, pages 10846–10856. Curran Associates, Inc., 2018
2018
-
[21]
Invariant models for causal transfer learning
Mateo Rojas-Carulla, Bernhard Schölkopf, Richard Turner, and Jonas Peters. Invariant models for causal transfer learning. Journal of Machine Learning Research, 19(36):1–34, 2018. 10
2018
-
[22]
On causal and anticausal learning
Bernhard Schoelkopf, Dominik Janzing, Jonas Peters, Eleni Sgouritsa, Kun Zhang, and Joris Mooij. On causal and anticausal learning. Proceedings of the 29th International Conference on Machine Learning, ICML 2012, 2, 06 2012
2012
-
[23]
Stewart, and Jimeng Sun
Mohammad Taha Bahadori, Krzysztof Chalupka, Edward Choi, Robert Chen, Walter F. Stewart, and Jimeng Sun. Causal regularization. CoRR, abs/1702.02604, 2017
2017 arXiv
-
[24]
Anchor regression: heterogeneous data meet causality
Dominik Rothenhausler, Nicolai Meinshausen, Peter Buhlmann, and Jonas Peters. Anchor regression: heterogeneous data meet causality. CoRR, abs/1801.06229, 2018
2018 arXiv
-
[25]
Causal regularization
Dominik Janzing. Causal regularization. In Advances in Neural Information Processing Systems 32, pages 12704–12714. Curran Associates, Inc., 2019
2019
-
[26]
On causal and anticausal learning
Bernhard Schölkopf, Dominik Janzing, Jonas Peters, Eleni Sgouritsa, Kun Zhang, and Joris Mooij. On causal and anticausal learning. In Proceedings of the 29th International Coference on International Conference on Machine Learning, ICML’12, page 459–466, Madison, WI, USA, 2012....
2012
-
[27]
Discovering causal signals in images
David Lopez-Paz, Robert Nishihara, Soumith Chintala, Bernhard Scholkopf, and Leon Bottou. Discovering causal signals in images. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017
2017
-
[28]
Semi-supervised interpolation in an anticausal learning scenario
Dominik Janzing and Bernhard Schölkopf. Semi-supervised interpolation in an anticausal learning scenario. J. Mach. Learn. Res., 16(1):1923–1948, January 2015
1923
-
[29]
Regression shrinkage and selection via the lasso
Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267–288, 1996
1996
-
[30]
Ridge regression: Biased estimation for nonorthogonal problems
Arthur Hoerl and Robert Kennard. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12:55–67, 04 2012
2012
-
[31]
Deep Learning
Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, 2016
2016
-
[32]
On the difficulty of training recurrent neural networks
Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. In ICML, 2012
2012
-
[33]
Batch normalization: Accelerating deep network training by reducing internal covariate shift
Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37 , ICML’15, page 448–456. JMLR.org, 2015
2015
-
[34]
Regularizing deep neural networks by noise: Its interpretation and optimization
Hyeonwoo Noh, Tackgeun You, Jonghwan Mun, and Bohyung Han. Regularizing deep neural networks by noise: Its interpretation and optimization. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page 5115–5124, Red Hook, NY , USA...
2017
-
[35]
mixup: Beyond empirical risk minimization
Hongyi Zhang, Moustapha Cisse, Yann Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. Proceedings of the 6th International Conference on Learning Representations (ICLR), 2018
2018
-
[36]
Estimating causal direction and confounding of two discrete variables
Krzysztof Chalupka, Frederick Eberhardt, and Pietro Perona. Estimating causal direction and confounding of two discrete variables. arXiv preprint arXiv:1611.01504, 2016
2016 arXiv
-
[37]
Xun Zheng, Bryon Aragam, Pradeep Ravikumar, and Eric P. Xing. DAGs with NO TEARS: Continuous Optimization for Structure Learning. In Advances in Neural Information Processing Systems, 2018
2018
-
[38]
Learning sparse nonparametric dags
Xun Zheng, Chen Dan, Bryon Aragam, Pradeep Ravikumar, and Eric P Xing. Learning sparse nonparametric dags. arXiv preprint arXiv:1909.13189, 2019
1909 arXiv
-
[39]
Gradient- based neural DAG learning
Sébastien Lachapelle, Philippe Brouillard, Tristan Deleu, and Simon Lacoste-Julien. Gradient- based neural DAG learning. In Proceedings of the 8th International Conference on Learning Representations (ICLR), 2020
2020
-
[40]
J. Pearl. Causality. Causality: Models, Reasoning, and Inference. Cambridge Univ. Press, 2009. 11
2009
-
[41]
The benefit of multitask representation learning
Andreas Maurer, Massimiliano Pontil, and Bernardino Romera-Paredes. The benefit of multitask representation learning. The Journal of Machine Learning Research, 17(1):2853–2884, 2016
2016
-
[42]
Pac-bayes & margins
John Langford and John Shawe-Taylor. Pac-bayes & margins. InAdvances in neural information processing systems, pages 439–446, 2003
2003
-
[43]
A pac analysis of a bayesian estimator
John Shawe-Taylor and Robert C Williamson. A pac analysis of a bayesian estimator. In Proceedings of the tenth annual conference on Computational learning theory, pages 2–9, 1997
1997
-
[44]
McAllester
David A. McAllester. Some pac-bayesian theorems. In Machine Learning, pages 230–234. ACM Press, 1998
1998
-
[45]
A pac-bayesian approach to spectrally-normalized margin bounds for neural networks
Behnam Neyshabur, Srinadh Bhojanapalli, and Nathan Srebro. A pac-bayesian approach to spectrally-normalized margin bounds for neural networks. arXiv preprint arXiv:1707.09564, 2017
2017 arXiv
-
[46]
Deterministic pac-bayesian generalization bounds for deep networks via generalizing noise-resilience
Vaishnavh Nagarajan and J Zico Kolter. Deterministic pac-bayesian generalization bounds for deep networks via generalizing noise-resilience. arXiv preprint arXiv:1905.13344, 2019
1905 arXiv
-
[47]
High-dimensional learning of linear causal networks via inverse covariance estimation
Po-Ling Loh and Peter Bühlmann. High-dimensional learning of linear causal networks via inverse covariance estimation. The Journal of Machine Learning Research, 15(1):3065–3105, 2014
2014
-
[48]
Learning directed acyclic graphs with penalized neighbourhood regression
Bryon Aragam, Arash A Amini, and Qing Zhou. Learning directed acyclic graphs with penalized neighbourhood regression. arXiv preprint arXiv:1511.08963, 2015
2015 arXiv
-
[49]
The Annals of Statistics, 41(2):536–567, 2013
Sara Van de Geer, Peter Bühlmann, et al.l0-penalized maximum likelihood for sparse directed acyclic graphs. The Annals of Statistics, 41(2):536–567, 2013
2013
-
[50]
UCI machine learning repository, 2020
Dheeru Dua and Casey Graff. UCI machine learning repository, 2020
2020
-
[51]
On large-batch training for deep learning: Generalization gap and sharp minima
Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. arXiv preprint arXiv:1609.04836, 2016
2016 arXiv
-
[52]
Pac-bayesian theory meets bayesian inference
Pascal Germain, Francis Bach, Alexandre Lacoste, and Simon Lacoste-Julien. Pac-bayesian theory meets bayesian inference. In Advances in Neural Information Processing Systems, pages 1884–1892, 2016
2016
-
[53]
User-friendly tail bounds for sums of random matrices
Joel A Tropp. User-friendly tail bounds for sums of random matrices. Foundations of computa- tional mathematics, 12(4):389–434, 2012. 12 A Proof of Theorem 1 In this paper, we consider learning a causal DAG as our regularizer. We use a squared loss in our objective function. W...
2012
-
[54]
Medical expenditure panel survey (meps), 2020
Agency for Healthcare Research and Quality. Medical expenditure panel survey (meps), 2020
2020
-
[55]
Wong et al
Chih M. Wong et al. Heart failure in younger patients: the Meta-analysis Global Group in Chronic Heart Failure (MAGGIC). European Heart Journal, 35(39):2714–2721, 06 2014. 20
2014
-
[2012]
Curran Associates Inc
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