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The nonlinear Schr{\"o}dinger and the Schr{\"o}dinger-Newton equations model many phenomena in various fields. Here, we perform an extensive numerical comparison between splitting methods (often employed to numerically solve these equations) and the integrating factor technique, also called Lawson method. Indeed, the latter is known to perform very well for the nonlinear Schr{\"o}dinger equation, but has not been thoroughly investigated for the Schr{\"o}dinger-Newton equation. Comparisons are made in one and two spatial dimensions, exploring different boundary conditions and parameters values. We show that for the short range potential of the nonlinear Schr{\"o}dinger equation, the integrating factor technique performs better than splitting algorithms, while, for the long range potential of the Schr{\"o}dinger-Newton equation, it depends on the particular system considered.

This paper analyzes comparatively the performance of Random Forests and Gradient Boosting algorithms in the field of forecasting the energy consumption based on historical data. The two algorithms are applied in order to forecast the energy consumption individually, and then combined together by using a Weighted Average Ensemble Method. The comparison among the achieved experimental results proves that the Weighted Average Ensemble Method provides more accurate results than each of the two algorithms applied alone.

A dialogue policy module is an essential part of task-completion dialogue systems. Recently, increasing interest has focused on reinforcement learning (RL)-based dialogue policy. Its favorable performance and wise action decisions rely on an accurate estimation of action values. The overestimation problem is a widely known issue of RL since its estimate of the maximum action value is larger than the ground truth, which results in an unstable learning process and suboptimal policy. This problem is detrimental to RL-based dialogue policy learning. To mitigate this problem, this paper proposes a dynamic partial average estimator (DPAV) of the ground truth maximum action value. DPAV calculates the partial average between the predicted maximum action value and minimum action value, where the weights are dynamically adaptive and problem-dependent. We incorporate DPAV into a deep Q-network as the dialogue policy and show that our method can achieve better or comparable results compared to top baselines on three dialogue datasets of different domains with a lower computational load. In addition, we also theoretically prove the convergence and derive the upper and lower bounds of the bias compared with those of other methods.

We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the $L_1$-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our $L_1$-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for {\it randomized} algorithms robust to a white-box adversary. In particular, our results show that for all $p\ge 0$, there exists a constant $C_p>1$ such that any $C_p$-approximation algorithm for $F_p$ moment estimation in insertion-only streams with a white-box adversary requires $\Omega(n)$ space for a universe of size $n$. Similarly, there is a constant $C>1$ such that any $C$-approximation algorithm in an insertion-only stream for matrix rank requires $\Omega(n)$ space with a white-box adversary. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. (Abstract shortened to meet arXiv limits.)

It is a well-known fact that there is no complete and discrete invariant on the collection of all multiparameter persistence modules. Nonetheless, many invariants have been proposed in the literature to study multiparameter persistence modules, though each invariant will lose some amount of information. One such invariant is the generalized rank invariant. This invariant is known to be complete on the class of interval decomposable persistence modules in general, under mild assumptions on the indexing poset $P$. There is often a trade-off, where the stronger an invariant is, the more expensive it is to compute in practice. The generalized rank invariant on its own is difficult to compute, whereas the standard rank invariant is readily computable through software implementations such as RIVET. We can interpolate between these two to induce new invariants via restricting the domain of the generalized rank invariant, and this family exhibits the aforementioned trade-off. This work studies the tension which exists between computational efficiency and retaining strength when restricting the domain of the generalized rank invariant. We provide a characterization result on where such restrictions are complete invariants in the setting where $P$ is finite, and furthermore show that such restricted generalized rank invariants are stable.

Finding the optimal design of a hydrodynamic or aerodynamic surface is often impossible due to the expense of evaluating the cost functions (say, with computational fluid dynamics) needed to determine the performances of the flows that the surface controls. In addition, inherent limitations of the design space itself due to imposed geometric constraints, conventional parameterization methods, and user bias can restrict {\it all} of the designs within a chosen design space regardless of whether traditional optimization methods or newer, data-driven design algorithms with machine learning are used to search the design space. We present a 2-pronged attack to address these difficulties: we propose (1) a methodology to create the design space using morphing that we call {\it Design-by-Morphing} (DbM); and (2) an optimization algorithm to search that space that uses a novel Bayesian Optimization (BO) strategy that we call {\it Mixed variable, Multi-Objective Bayesian Optimization} (MixMOBO). We apply this shape optimization strategy to maximize the power output of a hydrokinetic turbine. Applying these two strategies in tandem, we demonstrate that we can create a novel, geometrically-unconstrained, design space of a draft tube and hub shape and then optimize them simultaneously with a {\it minimum} number of cost function calls. Our framework is versatile and can be applied to the shape optimization of a variety of fluid problems.

Optimal execution is a sequential decision-making problem for cost-saving in algorithmic trading. Studies have found that reinforcement learning (RL) can help decide the order-splitting sizes. However, a problem remains unsolved: how to place limit orders at appropriate limit prices? The key challenge lies in the "continuous-discrete duality" of the action space. On the one hand, the continuous action space using percentage changes in prices is preferred for generalization. On the other hand, the trader eventually needs to choose limit prices discretely due to the existence of the tick size, which requires specialization for every single stock with different characteristics (e.g., the liquidity and the price range). So we need continuous control for generalization and discrete control for specialization. To this end, we propose a hybrid RL method to combine the advantages of both of them. We first use a continuous control agent to scope an action subset, then deploy a fine-grained agent to choose a specific limit price. Extensive experiments show that our method has higher sample efficiency and better training stability than existing RL algorithms and significantly outperforms previous learning-based methods for order execution.

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.

Predicting the future states of surrounding traffic participants and planning a safe, smooth, and socially compliant trajectory accordingly is crucial for autonomous vehicles. There are two major issues with the current autonomous driving system: the prediction module is often decoupled from the planning module and the cost function for planning is hard to specify and tune. To tackle these issues, we propose an end-to-end differentiable framework that integrates prediction and planning modules and is able to learn the cost function from data. Specifically, we employ a differentiable nonlinear optimizer as the motion planner, which takes the predicted trajectories of surrounding agents given by the neural network as input and optimizes the trajectory for the autonomous vehicle, thus enabling all operations in the framework to be differentiable including the cost function weights. The proposed framework is trained on a large-scale real-world driving dataset to imitate human driving trajectories in the entire driving scene and validated in both open-loop and closed-loop manners. The open-loop testing results reveal that the proposed method outperforms the baseline methods across a variety of metrics and delivers planning-centric prediction results, allowing the planning module to output close-to-human trajectories. In closed-loop testing, the proposed method shows the ability to handle complex urban driving scenarios and robustness against the distributional shift that imitation learning methods suffer from. Importantly, we find that joint training of planning and prediction modules achieves better performance than planning with a separate trained prediction module in both open-loop and closed-loop tests. Moreover, the ablation study indicates that the learnable components in the framework are essential to ensure planning stability and performance.

This paper studies \emph{linear} and \emph{affine} error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to have information rate at most $1/2$ (achieved by the trivial 2-fold repetition code). Previously, it was (erroneously) reported that more generally no non-trivial linear codes correcting $k$ deletions exist, i.e., that the $(k+1)$-fold repetition codes and its rate of $1/(k+1)$ are basically optimal for any $k$. We disprove this and show the existence of binary linear codes of length $n$ and rate just below $1/2$ capable of correcting $\Omega(n)$ insertions and deletions. This identifies rate $1/2$ as a sharp threshold for recovery from deletions for linear codes, and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically-good linear code for Hamming errors into an asymptotically-good linear code for insdel errors. Lastly, we show that the $\frac{1}{2}$-rate limitation does not hold for affine codes by giving an explicit affine code of rate $1-\epsilon$ which can efficiently correct a constant fraction of insdel errors.