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Order execution is a fundamental task in quantitative finance, aiming at finishing acquisition or liquidation for a number of trading orders of the specific assets. Recent advance in model-free reinforcement learning (RL) provides a data-driven solution to the order execution problem. However, the existing works always optimize execution for an individual order, overlooking the practice that multiple orders are specified to execute simultaneously, resulting in suboptimality and bias. In this paper, we first present a multi-agent RL (MARL) method for multi-order execution considering practical constraints. Specifically, we treat every agent as an individual operator to trade one specific order, while keeping communicating with each other and collaborating for maximizing the overall profits. Nevertheless, the existing MARL algorithms often incorporate communication among agents by exchanging only the information of their partial observations, which is inefficient in complicated financial market. To improve collaboration, we then propose a learnable multi-round communication protocol, for the agents communicating the intended actions with each other and refining accordingly. It is optimized through a novel action value attribution method which is provably consistent with the original learning objective yet more efficient. The experiments on the data from two real-world markets have illustrated superior performance with significantly better collaboration effectiveness achieved by our method.

We consider the problem of mixed sparse linear regression with two components, where two real $k$-sparse signals $\beta_1, \beta_2$ are to be recovered from $n$ unlabelled noisy linear measurements. The sparsity is allowed to be sublinear in the dimension, and additive noise is assumed to be independent Gaussian with variance $\sigma^2$. Prior work has shown that the problem suffers from a $\frac{k}{SNR^2}$-to-$\frac{k^2}{SNR^2}$ statistical-to-computational gap, resembling other computationally challenging high-dimensional inference problems such as Sparse PCA and Robust Sparse Mean Estimation; here $SNR$ is the signal-to-noise ratio. We establish the existence of a more extensive computational barrier for this problem through the method of low-degree polynomials, but show that the problem is computationally hard only in a very narrow symmetric parameter regime. We identify a smooth information-computation tradeoff between the sample complexity $n$ and runtime for any randomized algorithm in this hard regime. Via a simple reduction, this provides novel rigorous evidence for the existence of a computational barrier to solving exact support recovery in sparse phase retrieval with sample complexity $n = \tilde{o}(k^2)$. Our second contribution is to analyze a simple thresholding algorithm which, outside of the narrow regime where the problem is hard, solves the associated mixed regression detection problem in $O(np)$ time with square-root the number of samples and matches the sample complexity required for (non-mixed) sparse linear regression; this allows the recovery problem to be subsequently solved by state-of-the-art techniques from the dense case. As a special case of our results, we show that this simple algorithm is order-optimal among a large family of algorithms in solving exact signed support recovery in sparse linear regression.

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\text{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, the Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets. We overcome this bottleneck and present the first $(1+\epsilon)$-approximate algorithm for estimating the Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\varepsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to \emph{report} a $(1+\varepsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

As Internet of Things (IoT) devices proliferate, sustainable methods for powering them are becoming indispensable. The wireless provision of power enables battery-free operation and is crucial for complying with weight and size restrictions. For the energy harvesting components of these devices to be small, a high operating frequency is necessary. In conjunction with an electrically large antenna, the receivers may be located in the radiating near-field (Fresnel) region, e.g., in indoor scenarios. In this paper, we propose a wireless power transfer system to ensure a reliable supply of power to an arbitrary number of mobile, low-power, and single-antenna receivers, which are located in a three-dimensional cuboid room. To this end, we formulate a max-min optimisation problem to determine the optimal allocation of transmit power among an infinite number of radiating elements of the system's transmit antenna array. Thereby, the optimal deployment, i.e, the set of transmit antenna positions that are allocated non-zero transmit power according to the optimal allocation, is obtained implicitly. Generally, the set of transmit antenna positions corresponding to the optimal deployment has Lebesgue measure zero and the closure of the set has empty interior. Moreover, for a one-dimensional transmit antenna array, the set of transmit antenna positions is proven to be finite. The proposed optimal solution is validated through simulation. Simulation results indicate that the optimal deployment requires a finite number of transmit antennas and depends on the geometry of the environment and the dimensionality of the transmit antenna array. The robustness of the solution, which is obtained under a line-of-sight (LoS) assumption between the transmitter and receiver, is assessed in an isotropic scattering environment containing a strong LoS component.

Robot swarm is a hot spot in robotic research community. In this paper, we propose a decentralized framework for car-like robotic swarm which is capable of real-time planning in cluttered environments. In this system, path finding is guided by environmental topology information to avoid frequent topological change, and search-based speed planning is leveraged to escape from infeasible initial value's local minima. Then spatial-temporal optimization is employed to generate a safe, smooth and dynamically feasible trajectory. During optimization, the trajectory is discretized by fixed time steps. Penalty is imposed on the signed distance between agents to realize collision avoidance, and differential flatness cooperated with limitation on front steer angle satisfies the non-holonomic constraints. With trajectories broadcast to the wireless network, agents are able to check and prevent potential collisions. We validate the robustness of our system in simulation and real-world experiments. Code will be released as open-source packages.

To improve the convergence property of the randomized Kaczmarz (RK) method for solving linear systems, Bai and Wu (SIAM J. Sci. Comput., 40(1):A592--A606, 2018) originally introduced a greedy probability criterion for effectively selecting the working row from the coefficient matrix and constructed the greedy randomized Kaczmarz (GRK) method. Due to its simplicity and efficiency, this approach has inspired numerous subsequent works in recent years, such as the capped adaptive sampling rule, the greedy augmented randomized Kaczmarz method, and the greedy randomized coordinate descent method. Since the iterates of the GRK method are actually random variables, existing convergence analyses are all related to the expectation of the error. In this note, we prove that the linear convergence rate of the GRK method is deterministic, i.e. not in the sense of expectation. Moreover, the Polyak's heavy ball momentum technique is incorporated to improve the performance of the GRK method. We propose a refined convergence analysis, compared with the technique used in Loizou and Richt\'{a}rik (Comput. Optim. Appl., 77(3):653--710, 2020), of momentum variants of randomized iterative methods, which shows that the proposed GRK method with momentum (mGRK) also enjoys a deterministic linear convergence. Numerical experiments show that the mGRK method is more efficient than the GRK method.

This thesis captures the ongoing development of twisted cubes, which is a modification of cubes (in a topological sense) where its homotopy type theory does not require paths or higher paths to be invertible. My original motivation to develop the twisted cubes was to resolve the incompatibility between cubical type theory and directed type theory. The development of twisted cubes is still in the early stages and the intermediate goal, for now, is to define a twisted cube category and its twisted cubical sets that can be used to construct a potential definition of (infinity, n)-categories. The intermediate goal above leads me to discover a novel framework that uses graph theory to transform convex polytopes, such as simplices and (standard) cubes, into base categories. Intuitively, an n-dimensional polytope is transformed into a directed graph consists 0-faces (extreme points) of the polytope as its nodes and 1-faces of the polytope as its edges. Then, we define the base category as the full subcategory of the graph category induced by the family of these graphs from all n-dimensional cases. With this framework, the modification from cubes to twisted cubes can formally be done by reversing some edges of cube graphs. Equivalently, the twisted n-cube graph is the result of a certain endofunctor being applied n times to the singleton graph; this endofunctor (called twisted prism functor) duplicates the input, reverses all edges in the first copy, and then pairwisely links nodes from the first copy to the second copy. The core feature of a twisted graph is its unique Hamiltonian path, which is useful to prove many properties of twisted cubes. In particular, the reflexive transitive closure of a twisted graph is isomorphic to the simplex graph counterpart, which remarkably suggests that twisted cubes not only relate to (standard) cubes but also simplices.

In this paper, we devise a scheme for kernelizing, in sublinear space and polynomial time, various problems on planar graphs. The scheme exploits planarity to ensure that the resulting algorithms run in polynomial time and use O((sqrt(n) + k) log n) bits of space, where n is the number of vertices in the input instance and k is the intended solution size. As examples, we apply the scheme to Dominating Set and Vertex Cover. For Dominating Set, we also show that a well-known kernelization algorithm due to Alber et al. (JACM 2004) can be carried out in polynomial time and space O(k log n). Along the way, we devise restricted-memory procedures for computing region decompositions and approximating the aforementioned problems, which might be of independent interest.

This paper focuses on optimal beamforming to maximize the mean signal-to-noise ratio (SNR) for a reconfigurable intelligent surface (RIS)-aided MISO downlink system under correlated Rician fading. The beamforming problem becomes non-convex because of the unit modulus constraint of passive RIS elements. To tackle this, we propose a semidefinite relaxation-based iterative algorithm for obtaining statistically optimal transmit beamforming vector and RIS-phase shift matrix. Further, we analyze the outage probability (OP) and ergodic capacity (EC) to measure the performance of the proposed beamforming scheme. Just like the existing works, the OP and EC evaluations rely on the numerical computation of the iterative algorithm, which does not clearly reveal the functional dependence of system performance on key parameters. Therefore, we derive closed-form expressions for the optimal beamforming vector and phase shift matrix along with their OP performance for special cases of the general setup. Our analysis reveals that the i.i.d. fading is more beneficial than the correlated case in the presence of LoS components. This fact is analytically established for the setting in which the LoS is blocked. Furthermore, we demonstrate that the maximum mean SNR improves linearly/quadratically with the number of RIS elements in the absence/presence of LoS component under i.i.d. fading.

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.