We develop a method for generating causal post-hoc explanations of black-box classifiers based on a learned low-dimensional representation of the data. The explanation is causal in the sense that changing learned latent factors produces a change in the classifier output statistics. To construct these explanations, we design a learning framework that leverages a generative model and information-theoretic measures of causal influence.

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

In Table 1, we compare our agnostic learning results. Our results in this setting come from Theorem 3.3. We note that the sample complexity for Diakonikolas et al. To prove Lemma 3.5, we use the following result of Y ehudai and Shamir [35]. We first consider the case when σ satisfies Assumption 3.1.

We assume we have access to a set of i.i.d.

Empirical risk minimization (ERM) is a fundamental learning rule for statistical learning problems where the data is generated according to some unknown distribution P and returns a hypothesis f chosen from a fixed class F with small loss null. In the parametric setting, depending upon (null, F, P) ERM can have slow (1 / n) or fast (1/n) rates of convergence of the excess risk as a function of the sample size n . There exist several results that give sufficient conditions for fast rates in terms of joint properties of null, F, and P, such as the margin condition and the Bernstein condition. In the non-statistical prediction with expert advice setting, there is an analogous slow and fast rate phenomenon, and it is entirely characterized in terms of the mixability of the loss null (there being no role there for F or P). The notion of stochastic mixability builds a bridge between these two models of learning, reducing to classical mixability in a special case. The present paper presents a direct proof of fast rates for ERM in terms of stochastic mixability of ( null, F, P), and in so doing provides new insight into the fast-rates phenomenon.

The use of momentum in stochastic gradient methods has become a widespread practice in machine learning. Different variants of momentum, including heavy-ball momentum, Nesterov's accelerated gradient (NAG), and quasi-hyperbolic momentum (QHM), have demonstrated success on various tasks.