Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences.

A.1 Proof of Proposition 1 We note that 1 is restated and was proved in [25, Appendix A.1] Proof of 2: Non-negativity directly follows by non-negativity of mutual information. Proof of 5: The proof relies on the independence of functions of independent random variables. This concludes the proof. 1 A.2 Proof of Proposition 2 By translation invariance of mutual information, we may assume w.l.o.g. that the means are Next, we show that we may equivalently optimize with the added unit variance constraint. Example 3.4]), we have I (A B) null, where the last equality uses the unit variance property and Schur's determinant formula. Armed with Lemma 1, we are in place to prove Proposition 2. Since the CCA solutions Theorem 2.2], which is restated next for completeness.

We list detailed hyper-parameter settings here for reproducibility.

Neural Information Processing Systems http://nips.cc/

This paper studies the use of a machine learning-based estimator as a control variate for mitigating the variance of Monte Carlo sampling. Specifically, we seek to uncover the key factors that influence the efficiency of control variates in reducing variance.

For an overview of its proof, see Appendix B. Lemma A.1. In the following lemma, we use Lemma A.1 in order to show RSA T -hardness of By Assumption 2.1, there is K such that CSP K literals in the clause are satisfied by ψ, and otherwise null z, w null 1 . A.3 Hardness of learning random fully-connected neural networks Let n = ( n Let M be a diagonal-blocks matrix. By Lemma A.3, we have s By Lemma A.4, we have with probability 1 o Finally, Theorem 3.1 follows immediately from Theorem A.1 and the following lemma. By Lemma A.6, we have that By Theorem A.1, we need to show that SCAT We say that a distribution is isotropic if it has mean zero and its covariance matrix is the identity.

Neural networks are nowadays highly successful despite strong hardness results.

It is becoming increasingly important to understand the vulnerability of machine learning models to adversarial attacks. In this paper we study the feasibility of robust learning from the perspective of computational learning theory, considering both sample and computational complexity. In particular, our definition of robust learnability requires polynomial sample complexity. We start with two negative results. We show that no non-trivial concept class can be robustly learned in the distribution-free setting against an adversary who can perturb just a single input bit. We show moreover that the class of monotone conjunctions cannot be robustly learned under the uniform distribution against an adversary who can perturb $\omega(\log n)$ input bits. However if the adversary is restricted to perturbing $O(\log n)$ bits, then the class of monotone conjunctions can be robustly learned with respect to a general class of distributions (that includes the uniform distribution). Finally, we provide a simple proof of the computational hardness of robust learning on the boolean hypercube. Unlike previous results of this nature, our result does not rely on another computational model (e.g. the statistical query model) nor on any hardness assumption other than the existence of a hard learning problem in the PAC framework.