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In this paper, we study a sequential decision making problem faced by e-commerce carriers related to when to send out a vehicle from the central depot to serve customer requests, and in which order to provide the service, under the assumption that the time at which parcels arrive at the depot is stochastic and dynamic. The objective is to maximize the number of parcels that can be delivered during the service hours. We propose two reinforcement learning approaches for solving this problem, one based on a policy function approximation (PFA) and the second on a value function approximation (VFA). Both methods are combined with a look-ahead strategy, in which future release dates are sampled in a Monte-Carlo fashion and a tailored batch approach is used to approximate the value of future states. Our PFA and VFA make a good use of branch-and-cut-based exact methods to improve the quality of decisions. We also establish sufficient conditions for partial characterization of optimal policy and integrate them into PFA/VFA. In an empirical study based on 720 benchmark instances, we conduct a competitive analysis using upper bounds with perfect information and we show that PFA and VFA greatly outperform two alternative myopic approaches. Overall, PFA provides best solutions, while VFA (which benefits from a two-stage stochastic optimization model) achieves a better tradeoff between solution quality and computing time.

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, - ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories); - the invariant $\sigma$-algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much simpler than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

The monotone minimal perfect hash function (MMPHF) problem is the following indexing problem. Given a set $S= \{s_1,\ldots,s_n\}$ of $n$ distinct keys from a universe $U$ of size $u$, create a data structure $DS$ that answers the following query: \[ RankOp(q) = \text{rank of } q \text{ in } S \text{ for all } q\in S ~\text{ and arbitrary answer otherwise.} \] Solutions to the MMPHF problem are in widespread use in both theory and practice. The best upper bound known for the problem encodes $DS$ in $O(n\log\log\log u)$ bits and performs queries in $O(\log u)$ time. It has been an open problem to either improve the space upper bound or to show that this somewhat odd looking bound is tight. In this paper, we show the latter: specifically that any data structure (deterministic or randomized) for monotone minimal perfect hashing of any collection of $n$ elements from a universe of size $u$ requires $\Omega(n \cdot \log\log\log{u})$ expected bits to answer every query correctly. We achieve our lower bound by defining a graph $\mathbf{G}$ where the nodes are the possible ${u \choose n}$ inputs and where two nodes are adjacent if they cannot share the same $DS$. The size of $DS$ is then lower bounded by the log of the chromatic number of $\mathbf{G}$. Finally, we show that the fractional chromatic number (and hence the chromatic number) of $\mathbf{G}$ is lower bounded by $2^{\Omega(n \log\log\log u)}$.

With the aid of hardware and software developments, there has been a surge of interests in solving partial differential equations by deep learning techniques, and the integration with domain decomposition strategies has recently attracted considerable attention due to its enhanced representation and parallelization capacity of the network solution. While there are already several works that substitute the numerical solver of overlapping Schwarz methods with the deep learning approach, the non-overlapping counterpart has not been thoroughly studied yet because of the inevitable interface overfitting problem that would propagate the errors to neighbouring subdomains and eventually hamper the convergence of outer iteration. In this work, a novel learning approach, i.e., the compensated deep Ritz method, is proposed to enable the flux transmission across subregion interfaces with guaranteed accuracy, thereby allowing us to construct effective learning algorithms for realizing the more general non-overlapping domain decomposition methods in the presence of overfitted interface conditions. Numerical experiments on a series of elliptic boundary value problems including the regular and irregular interfaces, low and high dimensions, smooth and high-contrast coefficients on multidomains are carried out to validate the effectiveness of our proposed domain decomposition learning algorithms.

Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754 floating-point rounding modes. In some applications such as PDEs, ODEs and neural networks, SR empirically improves the numerical behavior and convergence to accurate solutions while no sound theoretical background has been provided. Recent works by Ipsen, Zhou, Higham, and Mary have computed SR probabilistic error bounds for basic linear algebra kernels. For example, the inner product SR probabilistic bound of the forward error is proportional to $\sqrt$ nu instead of nu for the default rounding mode. To compute the bounds, these works show that the errors accumulated in computation form a martingale. This paper proposes an alternative framework to characterize SR errors based on the computation of the variance. We pinpoint common error patterns in numerical algorithms and propose a lemma that bounds their variance. For each probability and through Bienaym{\'e}-Chebyshev inequality, this bound leads to better probabilistic error bound in several situations. Our method has the advantage of providing a tight probabilistic bound for all algorithms fitting our model. We show how the method can be applied to give SR error bounds for the inner product and Horner polynomial evaluation.

While the Vision Transformer (VT) architecture is becoming trendy in computer vision, pure VT models perform poorly on tiny datasets. To address this issue, this paper proposes the locality guidance for improving the performance of VTs on tiny datasets. We first analyze that the local information, which is of great importance for understanding images, is hard to be learned with limited data due to the high flexibility and intrinsic globality of the self-attention mechanism in VTs. To facilitate local information, we realize the locality guidance for VTs by imitating the features of an already trained convolutional neural network (CNN), inspired by the built-in local-to-global hierarchy of CNN. Under our dual-task learning paradigm, the locality guidance provided by a lightweight CNN trained on low-resolution images is adequate to accelerate the convergence and improve the performance of VTs to a large extent. Therefore, our locality guidance approach is very simple and efficient, and can serve as a basic performance enhancement method for VTs on tiny datasets. Extensive experiments demonstrate that our method can significantly improve VTs when training from scratch on tiny datasets and is compatible with different kinds of VTs and datasets. For example, our proposed method can boost the performance of various VTs on tiny datasets (e.g., 13.07% for DeiT, 8.98% for T2T and 7.85% for PVT), and enhance even stronger baseline PVTv2 by 1.86% to 79.30%, showing the potential of VTs on tiny datasets. The code is available at //github.com/lkhl/tiny-transformers.

Knowledge graphs have emerged as an effective tool for managing and standardizing semistructured domain knowledge in a human- and machine-interpretable way. In terms of graph-based domain applications, such as embeddings and graph neural networks, current research is increasingly taking into account the time-related evolution of the information encoded within a graph. Algorithms and models for stationary and static knowledge graphs are extended to make them accessible for time-aware domains, where time-awareness can be interpreted in different ways. In particular, a distinction needs to be made between the validity period and the traceability of facts as objectives of time-related knowledge graph extensions. In this context, terms and definitions such as dynamic and temporal are often used inconsistently or interchangeably in the literature. Therefore, with this paper we aim to provide a short but well-defined overview of time-aware knowledge graph extensions and thus faciliate future research in this field as well.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.