Competitive Gradient Descent

We introduce a new algorithm for the numerical computation of Nash equilibria of competitive two-player games. Our method is a natural generalization of gradient descent to the two-player setting where the update is given by the Nash equilibrium of a regularized bilinear local approximation of the underlying game. It avoids oscillatory and divergent behaviors seen in alternating gradient descent. Using numerical experiments and rigorous analysis, we provide a detailed comparison to methods based on \emph{optimism} and \emph{consensus} and show that our method avoids making any unnecessary changes to the gradient dynamics while achieving exponential (local) convergence for (locally) convex-concave zero sum games. Convergence and stability properties of our method are robust to strong interactions between the players, without adapting the stepsize, which is not the case with previous methods. In our numerical experiments on non-convex-concave problems, existing methods are prone to divergence and instability due to their sensitivity to interactions among the players, whereas we never observe divergence of our algorithm. The ability to choose larger stepsizes furthermore allows our algorithm to achieve faster convergence, as measured by the number of model evaluations.Comment: Appeared in NeurIPS 2019. This version corrects an error in theorem 2.2. Source code used for the numerical experiments can be found under http://github.com/f-t-s/CGD. A high-level overview of this work can be found under http://f-t-s.github.io/projects/cgd

Instrument subjects are agents or causers

It has often been noticed that one syntactic argument position can be realized by elements which seem to realize different thematic roles. This is notably the case with the external argument position of verbs of change of state which licenses volitional agents, instruments or natural forces/causers, showing the generality and abstractness of the external argument relation. (1) a. John broke the window (Agent) b. The hammer broke the window (Instrument) c. The storm broke the window (Causer) In order to capture this generality, Van Valin & Wilkins (1996) and Ramchand (2003) among others have proposed that the thematic role of the external argument position is in fact underspecified. The relevant notion is that of an effector (in Van Valin & Wilkins) or of an abstract causer/initiator (in Ramchand). In this paper we argue against a total underspecification of the external argument relation. While we agree that (1b) does not instantiate an instrument theta role in subject position, we argue that a complete underspecification of the external theta-position is not feasible, but that two types of external theta roles have to be distinguished, Agents and Causers. Our arguments are based on languages where Agents and Causers show morpho-syntactic independence (section 2.1) and the behavior of instrument subjects in English, Dutch, German and Greek (section 2.2 and 3). We show that instrument subjects are either Agent or Causer like. In section (4) we give an analysis how arguments realizing these thematic notions are introduced into syntax

PP licensing in nominalizations

In this paper we compare the distribution of PPs introducing external arguments in nominalizations with PPs introducing external arguments in the verbal domain. We show that several mismatches exist between the behavior of PPs in nominalizations and PPs in the verbal domain. This leads us to suggest that while PPs in the verbal domain are licensed by functional structure alone, within the nominal domain, PPs can also be licensed via an interplay of the encyclopaedic meaning of the root involved and the properties of the preposition itself. This second mechanism kicks in in the absence of functional structure

The properties of anticausatives crosslinguistically

The causative/anticausative alternation has been the topic of much typological and theoretical discussion in the linguistic literature. This alternation is characterized by verbs with transitive and intransitive uses, such that the transitive use of a verb V means roughly "cause to Vintransitive" (see Levin 1993). The discussion revolves around two issues: the first one concerns the similarities and differences between the anticausative and the passive, and the second one concerns the derivational relationship, if any, between the transitive and intransitive variant. With respect to the second issue, a number of approaches have been developed. Judging the approach conceptually unsatisfactory, according to which each variant is assigned an independent lexical entry, it was concluded that the two variants have to be derivationally related. The question then is which one of the two is basic and where this derivation takes place in the grammar. Our contribution to this discussion is to argue against derivational approaches to the causative / anticausative alternation. We focus on the distribution of PPs related to external arguments (agent, causer, instrument, causing event) in passives and anticausatives of English, German and Greek and the set of verbs undergoing the causative/anticausative alternation in these languages. We argue that the crosslinguistic differences in these two domains provide evidence against both causativization and detransitivization analyses of the causative / anticausative alternation. We offer an approach to this alternation which builds on a syntactic decomposition of change of state verbs into a Voice and a CAUS component. Crosslinguistic variation in passives and anticausatives depends on properties of Voice and its combinations with CAUS and various types of roots