Within distributed learning, workers typically compute gradients on their assigned dataset chunks and send them to the parameter server (PS), which aggregates them to compute either an exact or approximate version of $\nabla L$ (gradient of the loss function $L$). However, in large-scale clusters, many workers are slower than their promised speed or even failure-prone. A gradient coding solution introduces redundancy within the assignment of chunks to the workers and uses coding theoretic ideas to allow the PS to recover $\nabla L$ (exactly or approximately), even in the presence of stragglers. Unfortunately, most existing gradient coding protocols are inefficient from a computation perspective as they coarsely classify workers as operational or failed; the potentially valuable work performed by slow workers (partial stragglers) is ignored. In this work, we present novel gradient coding protocols that judiciously leverage the work performed by partial stragglers. Our protocols are efficient from a computation and communication perspective and numerically stable. For an important class of chunk assignments, we present efficient algorithms for optimizing the relative ordering of chunks within the workers; this ordering affects the overall execution time. For exact gradient reconstruction, our protocol is around $2\times$ faster than the original class of protocols and for approximate gradient reconstruction, the mean-squared-error of our reconstructed gradient is several orders of magnitude better.

Abstract Bicycle motion simulation is fundamental to urban transportation planning, virtual reality and other areas. This article proposes a data‐driven based double‐layer bicycle simulation model to consider the cyclist's decision‐making process and the bicycle's kinematic structure. This proposed model consists of two layers, the decision‐making layer and the motion layer. First, the decision‐making layer using machine learning algorithms models the decision‐making process as a regression problem to output the cyclist's decision. Then, the motion layer applies a bicycle kinematics model to output bicycle motion under physical constraints. In addition, a solution to calculate bicycles' dynamic information is proposed for the data‐driven method. Quantitative and qualitative experiments have been conducted, and results show that the double‐layer model and the parameter calculation solution can generate realistic bicycle motion simulations.

Abstract This paper proposes a novel crowd simulation method which integrates not only modelling ideas but also advantages from both data‐driven methods and crowd dynamics methods. To seamlessly integrate these two different modelling ideas, first, a fusion crowd motion model is developed. In this model the motion of crowd are driven dynamically by different forces. Part of the forces are modeled under a universal interaction mechanism, which describe the common parts of crowd dynamics. Others are modeled by examples from real data, which describe the personality parts of the agent motion. Second, a construction method for example dataset is proposed to support the fusion model. In the dataset, crowd trajectories captured in the real world are decomposed and re‐described under the structure of the fusion model. Thus, personality parts hidden in the real data could be locked and extracted, making the data understandable and migratable for our fusion model. A comprehensive crowd motion generation workflow using the fusion model and example dataset is also proposed. Quantitative and qualitative experiments and user studies are conducted. Results show that the proposed fusion crowd simulation method can generate crowd motion with the great motion fidelity, which not only match the macro characteristics of real data, but also has lots of micro personality showing the diversity of crowd motion.

Abstract Crowd simulation is a challenging problem, aiming to generate realistic pedestrians motions in virtual environment. Nowadays, ORCA is a widely used simulation algorithm in practice because of its stable and efficient performance. However, this algorithm cannot regenerate continuity and diversity of pedestrian motions in real data, leading to defects in motion fidelity. Otherwise, trajectory prediction methods based on deep learning have progressed in real pedestrians movement patterns mining. However, they are rarely applied in simulation due to the lack of ability to avoid collision and adapt to manufactured scenarios. Our work proposes a simulation method DeepORCA that integrates ORCA with a CVAE‐based velocity probability generator, which can model motion continuity, variable intentions, and scene semantics. Moreover, DeepORCA converts the velocity optimization into quadratic programming, which accelerates the calculation while maintaining the collision‐avoidance ability of ORCA. In the experiments of real and artificial scenes, our method produces more realistic crowd simulation results than ORCA quantitatively and qualitatively, while keeps the computational efficiency at the same order of magnitude.

Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f:[0,1] \to [0,1]$ is $r$-regular if there is a B\"{u}chi automaton that accepts precisely the set of base $r \in \mathbb{N}$ representations of elements of the graph of $f$. We show that a continuous $r$-regular function $f$ is locally affine away from a nowhere dense, Lebesgue null, subset of $[0,1]$. As a corollary we establish that every differentiable $r$-regular function is affine. It follows that checking whether an $r$-regular function is differentiable is in $\operatorname{PSPACE}$. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.