AUC (area under ROC curve) is an important evaluation criterion, which has been popularly used in many learning tasks such as class-imbalance learning, cost-sensitive learning, learning to rank, etc. Many learning approaches try to optimize AUC, while owing to the non-convexity and discontinuousness of AUC, almost all approaches work with surrogate loss functions. Thus, the consistency of AUC is crucial; however, it has been almost untouched before. In this paper, we provide a sufficient condition for the asymptotic consistency of learning approaches based on surrogate loss functions. Based on this result, we prove that exponential loss and logistic loss are consistent with AUC, but hinge loss is inconsistent. Then, we derive the $q$-norm hinge loss and general hinge loss that are consistent with AUC. We also derive the consistent bounds for exponential loss and logistic loss, and obtain the consistent bounds for many surrogate loss functions under the non-noise setting. Further, we disclose an equivalence between the exponential surrogate loss of AUC and exponential surrogate loss of accuracy, and one straightforward consequence of such finding is that AdaBoost and RankBoost are equivalent.

Great successes of deep neural networks have been witnessed in various real applications. Many algorithmic and implementation techniques have been developed, however, theoretical understanding of many aspects of deep neural networks is far from clear. A particular interesting issue is the usefulness of dropout, which was motivated from the intuition of preventing complex co-adaptation of feature detectors. In this paper, we study the Rademacher complexity of different types of dropout, and our theoretical results disclose that for shallow neural networks (with one or none hidden layer) dropout is able to reduce the Rademacher complexity in polynomial, whereas for deep neural networks it can amazingly lead to an exponential reduction of the Rademacher complexity.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.

Golog and ConGolog are languages defined in the situation calculus for cognitive robotics. Given a Golog program \delta, its semantics is defined by a macro Do(\delta,s,s') that expands to a logical sentence that captures the conditions under which performing \delta in s can terminate in s'. A similarmacro is defined for ConGolog programs. In general, the logical sentences that these macros expand to are second-order, and in the case of ConGolog, may involve quantification over programs. In this paper, we show that by making use of the foundational axioms in the situation calculus, in particular, the second-order closure axiom about the space of situations, these macro expressions can actually be defined using first-order sentences.