Combined with Theorem A.1, this yields a generalization bound for the SPO+ loss which, when Recall Theorem 3.1, the biconjugate of Lemma B.1 provides the relationship between excess SPO risk and the optimal solution of (d 1) Lemma B.4 provide a lower bound of the conditional SPO+ risk condition on the first (d 1) Lemma B.5 provide a lower bound of the conditional SPO+ risk when the distribution of By result in Lemma B.4, it holds that E By Lemma B.3, it holds that Now we present a general version of Theorem 3.1. By Lemma B.5, we know that R (d 1) Herein we provide an example to show the tightness of the lower bound in Theorem B.1. First we provide some useful properties in the following lemma. Now we provide the proofs of Lemma C.1 and C.2. Proof of Lemma C.1. In Lemma C.1 we show that c From Theorem C.2 and Lemma C.5, we know that Proposition C.2. Suppose P P Let ω = c c. Since p (c) = p (2 c c), we have E Also, α ζ (α) is a non-decreasing function.

Visualizations of the real and learned state-values of IA VI, IQL and DIQL can be found in Figure 7.Figure 7: Visualization of state-values for different numbers of trajectories in Objectworld. Table 2: Comparison between online and offline estimation of state-action visitations for the Ob-jectworld environment, given a data set with an action distribution equivalent to the true optimal Boltzmann distribution. The pseudocode of the tabular variant of Constrained Inverse Q-learning can be found in Algorithm 4. See [4] for further details of Constrained Q-learning.Algorithm 4: Tabular Model-free Constrained Inverse Q-learning The pseudocode of Deep Constrained Inverse Q-learning can be found in Algorithm 5. The lower row shows the EVD. 3 For DIQL, the parameters were optimized in the range of Hence, it can only increase.

Estimation of the average treatment effect (ATE) is a central problem in causal inference. In recent times, inference for the ATE in the presence of high-dimensional covariates has been extensively studied. Among the diverse approaches that have been proposed, augmented inverse probability weighting (AIPW) with cross-fitting has emerged a popular choice in practice. In this work, we study this cross-fit AIPW estimator under well-specified outcome regression and propensity score models in a high-dimensional regime where the number of features and samples are both large and comparable. Under assumptions on the covariate distribution, we establish a new central limit theorem for the suitably scaled cross-fit AIPW that applies without any sparsity assumptions on the underlying high-dimensional parameters. Our CLT uncovers two crucial phenomena among others: (i) the AIPW exhibits a substantial variance inflation that can be precisely quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the pre-cross-fit estimators is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts. On the technical front, our work utilizes a novel interplay between three distinct tools--approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach. We believe our proof techniques should be useful for analyzing other two-stage estimators in this high-dimensional regime. Finally, we complement our theoretical results with simulations that demonstrate both the finite sample efficacy of our CLT and its robustness to our assumptions.

Most modern formalisms used in Databases and Artificial Intelligence for describing an application domain are based on the notions of class (or concept) and relationship among classes. One interesting feature of such formalisms is the possibility of defining a class, i.e., providing a set of properties that precisely characterize the instances of the class. Many recent articles point out that there are several ways of assigning a meaning to a class definition containing some sort of recursion. In this paper, we argue that, instead of choosing a single style of semantics, we achieve better results by adopting a formalism that allows for different semantics to coexist. We demonstrate the feasibility of our argument, by presenting a knowledge representation formalism, the description logic muALCQ, with the above characteristics. In addition to the constructs for conjunction, disjunction, negation, quantifiers, and qualified number restrictions, muALCQ includes special fixpoint constructs to express (suitably interpreted) recursive definitions. These constructs enable the usual frame-based descriptions to be combined with definitions of recursive data structures such as directed acyclic graphs, lists, streams, etc. We establish several properties of muALCQ, including the decidability and the computational complexity of reasoning, by formulating a correspondence with a particular modal logic of programs called the modal mu-calculus.

Decomposable dependency models possess a number of interesting and useful properties. This paper presents new characterizations of decomposable models in terms of independence relationships, which are obtained by adding a single axiom to the well-known set characterizing dependency models that are isomorphic to undirected graphs. We also briefly discuss a potential application of our results to the problem of learning graphical models from data.

Terminological knowledge representation systems (TKRSs) are tools for designing and using knowledge bases that make use of terminological languages (or concept languages). We analyze from a theoretical point of view a TKRS whose capabilities go beyond the ones of presently available TKRSs. The new features studied, often required in practical applications, can be summarized in three main points. First, we consider a highly expressive terminological language, called ALCNR, including general complements of concepts, number restrictions and role conjunction. Second, we allow to express inclusion statements between general concepts, and terminological cycles as a particular case. Third, we prove the decidability of a number of desirable TKRS-deduction services (like satisfiability, subsumption and instance checking) through a sound, complete and terminating calculus for reasoning in ALCNR-knowledge bases. Our calculus extends the general technique of constraint systems. As a byproduct of the proof, we get also the result that inclusion statements in ALCNR can be simulated by terminological cycles, if descriptive semantics is adopted.