Continuous-Time Fitted Value Iteration for Robust Policies

Solving the Hamilton-Jacobi-Bellman equation is important in many domains including control, robotics and economics. Especially for continuous control, solving this differential equation and its extension the Hamilton-Jacobi-Isaacs equation, is important as it yields the optimal policy that achieves the maximum reward on a give task. In the case of the Hamilton-Jacobi-Isaacs equation, which includes an adversary controlling the environment and minimizing the reward, the obtained policy is also robust to perturbations of the dynamics. In this paper we propose continuous fitted value iteration (cFVI) and robust fitted value iteration (rFVI). These algorithms leverage the non-linear control-affine dynamics and separable state and action reward of many continuous control problems to derive the optimal policy and optimal adversary in closed form. This analytic expression simplifies the differential equations and enables us to solve for the optimal value function using value iteration for continuous actions and states as well as the adversarial case. Notably, the resulting algorithms do not require discretization of states or actions. We apply the resulting algorithms to the Furuta pendulum and cartpole. We show that both algorithms obtain the optimal policy. The robustness Sim2Real experiments on the physical systems show that the policies successfully achieve the task in the real-world. When changing the masses of the pendulum, we observe that robust value iteration is more robust compared to deep reinforcement learning algorithm and the non-robust version of the algorithm. Videos of the experiments are shown at https://sites.google.com/view/rfvi

Learning Green's functions associated with parabolic partial differential equations

Given input-output pairs from a parabolic partial differential equation (PDE) in any spatial dimension $n\geq 1$, we derive the first theoretically rigorous scheme for learning the associated Green's function $G$. Until now, rigorously learning Green's functions associated with parabolic operators has been a major challenge in the field of scientific machine learning because $G$ may not be square-integrable when $n>1$, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of $G$ together with the randomized singular value decomposition, we construct an approximant to $G$ that achieves a relative error of $\smash{\mathcal{O}(\Gamma_\epsilon^{-1/2}\epsilon)}$ in the $L^1$-norm with high probability by using at most $\smash{\mathcal{O}(\epsilon^{-\frac{n+2}{2}}\log(1/\epsilon))}$ input-output training pairs, where $\Gamma_\epsilon$ is a measure of the quality of the training dataset for learning $G$, and $\epsilon>0$ is sufficiently small. Along the way, we extend the low-rank theory of Bebendorf and Hackbusch from elliptic PDEs in dimension $1\leq n\leq 3$ to parabolic PDEs in any dimensions, which shows that Green's functions associated with parabolic PDEs admit a low-rank structure on well-separated domains.

Exact methods and lower bounds for the Oven Scheduling Problem

The Oven Scheduling Problem (OSP) is a new parallel batch scheduling problem that arises in the area of electronic component manufacturing. Jobs need to be scheduled to one of several ovens and may be processed simultaneously in one batch if they have compatible requirements. The scheduling of jobs must respect several constraints concerning eligibility and availability of ovens, release dates of jobs, setup times between batches as well as oven capacities. Running the ovens is highly energy-intensive and thus the main objective, besides finishing jobs on time, is to minimize the cumulative batch processing time across all ovens. This objective distinguishes the OSP from other batch processing problems which typically minimize objectives related to makespan, tardiness or lateness. We propose to solve this NP-hard scheduling problem via constraint programming (CP) and integer linear programming (ILP) and present corresponding models. For an experimental evaluation, we introduce a multi-parameter random instance generator to provide a diverse set of problem instances. Using state-of-the-art solvers, we evaluate the quality and compare the performance of our CP- and ILP-models. We show that our models can find feasible solutions for instances of realistic size, many of those being provably optimal or nearly optimal solutions. Finally, we derive theoretical lower bounds on the solution cost of feasible solutions to the OSP; these can be computed within a few seconds. We show that these lower bounds are competitive with those derived by state-of-the-art solvers.

Condition-Invariant and Compact Visual Place Description by Convolutional Autoencoder

Visual place recognition (VPR) in condition-varying environments is still an open problem. Popular solutions are CNN-based image descriptors, which have been shown to outperform traditional image descriptors based on hand-crafted visual features. However, there are two drawbacks of current CNN-based descriptors: a) their high dimension and b) lack of generalization, leading to low efficiency and poor performance in applications. In this paper, we propose to use a convolutional autoencoder (CAE) to tackle this problem. We employ a high-level layer of a pre-trained CNN to generate features, and train a CAE to map the features to a low-dimensional space to improve the condition invariance property of the descriptor and reduce its dimension at the same time. We verify our method in three challenging datasets involving significant illumination changes, and our method is shown to be superior to the state-of-the-art. For the benefit of the community, we make public the source code.

Quantum Finite Automata and Quiver Algebras

We find an application in quantum finite automata for the ideas and results of [JL21] and [JL22]. We reformulate quantum finite automata with multiple-time measurements using the algebraic notion of near-ring. This gives a unified understanding towards quantum computing and deep learning. When the near-ring comes from a quiver, we have a nice moduli space of computing machines with metric that can be optimized by gradient descent.

Online Weak-form Sparse Identification of Partial Differential Equations

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the $\ell_0$-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.

PCP Theorems, SETH and More: Towards Proving Sub-linear Time Inapproximability

In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et. have devised the distributed PCP framework for sub-quadratic time inapproximability. We show that the distributed PCP theorem can be generalized for proving arbitrary polynomial time inapproximability, but fails in the linear case. We borrow similar proof techniques from \cite{Abboud2017} for proving the sub-linear time PCP, and show how to use it to prove both existing and new inapproximability results. These results exhibits the power of the sub-linear PCP theorem proposed in this paper. Considering the emerging research works on sub-linear time algorithms, the sub-linear PCP theorem is important in guiding the research in sub-linear time approximation algorithms.

Dynamic Deep Convolutional Candlestick Learner

Candlestick pattern is one of the most fundamental and valuable graphical tools in financial trading that supports traders observing the current market conditions to make the proper decision. This task has a long history and, most of the time, human experts. Recently, efforts have been made to automatically classify these patterns with the deep learning models. The GAF-CNN model is a well-suited way to imitate how human traders capture the candlestick pattern by integrating spatial features visually. However, with the great potential of the GAF encoding, this classification task can be extended to a more complicated object detection level. This work presents an innovative integration of modern object detection techniques and GAF time-series encoding on candlestick pattern tasks. We make crucial modifications to the representative yet straightforward YOLO version 1 model based on our time-series encoding method and the property of such data type. Powered by the deep neural networks and the unique architectural design, the proposed model performs pretty well in candlestick classification and location recognition. The results show tremendous potential in applying modern object detection techniques on time-series tasks in a real-time manner.

Interpretable Latent Variables in Deep State Space Models

We introduce a new version of deep state-space models (DSSMs) that combines a recurrent neural network with a state-space framework to forecast time series data. The model estimates the observed series as functions of latent variables that evolve non-linearly through time. Due to the complexity and non-linearity inherent in DSSMs, previous works on DSSMs typically produced latent variables that are very difficult to interpret. Our paper focus on producing interpretable latent parameters with two key modifications. First, we simplify the predictive decoder by restricting the response variables to be a linear transformation of the latent variables plus some noise. Second, we utilize shrinkage priors on the latent variables to reduce redundancy and improve robustness. These changes make the latent variables much easier to understand and allow us to interpret the resulting latent variables as random effects in a linear mixed model. We show through two public benchmark datasets the resulting model improves forecasting performances.

A De-raining semantic segmentation network for real-time foreground segmentation

Few researches have been proposed specifically for real-time semantic segmentation in rainy environments. However, the demand in this area is huge and it is challenging for lightweight networks. Therefore, this paper proposes a lightweight network which is specially designed for the foreground segmentation in rainy environments, named De-raining Semantic Segmentation Network (DRSNet). By analyzing the characteristics of raindrops, the MultiScaleSE Block is targetedly designed to encode the input image, it uses multi-scale dilated convolutions to increase the receptive field, and SE attention mechanism to learn the weights of each channels. In order to combine semantic information between different encoder and decoder layers, it is proposed to use Asymmetric Skip, that is, the higher semantic layer of encoder employs bilinear interpolation and the output passes through pointwise convolution, then added element-wise to the lower semantic layer of decoder. According to the control experiments, the performances of MultiScaleSE Block and Asymmetric Skip compared with SEResNet18 and Symmetric Skip respectively are improved to a certain degree on the Foreground Accuracy index. The parameters and the floating point of operations (FLOPs) of DRSNet is only 0.54M and 0.20GFLOPs separately. The state-of-the-art results and real-time performances are achieved on both the UESTC all-day Scenery add rain (UAS-add-rain) and the Baidu People Segmentation add rain (BPS-add-rain) benchmarks with the input sizes of 192*128, 384*256 and 768*512. The speed of DRSNet exceeds all the networks within 1GFLOPs, and Foreground Accuracy index is also the best among the similar magnitude networks on both benchmarks.