arXiv subjects

Hanxiao Wang

Publications and source records attributed to Hanxiao Wang.

At least 37 records · Page 2Linked to original sources

arXiv2022

Optimal Controls for Forward-Backward Stochastic Differential Equations: Time-Inconsistency and Time-Consistent Solutions

This paper is concerned with an optimal control problem for a forward-backward stochastic differential equation (FBSDE, for short) with a recursive cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). It is found that such an optimal control problem is time-inconsistent in general, even if the cost functional is reduced to a classical Bolza type one as in Peng [50], Lim-Zhou [41], and Yong [74]. Therefore, instead of finding a global optimal control (which is time-inconsistent), we will look for a time-consistent and locally optimal equilibrium strategy, which can be constructed via the solution of an associated equilibrium Hamilton-Jacobi-Bellman (HJB, for short) equation. A verification theorem for the local optimality of the equilibrium strategy is proved by means of the generalized Feynman-Kac formula for BSVIEs and some stability estimates of the representation for parabolic partial differential equations (PDEs, for short). Under certain conditions, it is proved that the equilibrium HJB equation, which is a nonlocal PDE, admits a unique classical solution. As special cases and applications, the linear-quadratic problems, a mean-variance model, a social planner problem with heterogeneous Epstein-Zin utilities, and a Stackelberg game are briefly investigated. It turns out that our framework can cover not only the optimal control problems for FBSDEs studied in [50,41,74], and so on, but also the problems of the general discounting and some nonlinear appearance of conditional expectations for the terminal state, studied in Yong [75,77] and Björk-Khapko-Murgoci [7].

math.OC↗

arXiv2022

Backward Stochastic Differential Equations and Backward Stochastic Volterra Integral Equations with Anticipating Generators

For a backward stochastic differential equation (BSDE, for short), when the generator is not progressively measurable, it might not admit adapted solutions, shown by an example. However, for backward stochastic Volterra integral equations (BSVIEs, for short), the generators are allowed to be anticipating. This gives, among other things, an essential difference between BSDEs and BSVIEs. Under some proper conditions, the well-posedness of such kinds of BSVIEs is established. Further, the results are extended to path-dependent BSVIEs, in which the generators can depend on the future paths of unknown processes. An additional finding is that for path-dependent BSVIEs, in general, the situation of anticipating generators is not avoidable and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng--Yang [22] is not necessary.

math.PR↗

arXiv2022

Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations

A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE, for short) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a Type-II backward SVIE (BSVIE, for short). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a Type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding Type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback.

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arXiv2022

Turnpike Properties for Stochastic Linear-Quadratic Optimal Control Problems

This paper analyzes the limiting behavior of stochastic linear-quadratic optimal control problems in finite time horizon $[0,T]$ as $T\rightarrow\infty$. The so-called turnpike properties are established for such problems, under stabilizability condition which is weaker than the controllability, normally imposed in the similar problem for ordinary differential systems. In dealing with the turnpike problem, a crucial issue is to determine the corresponding static optimization problem. Intuitively mimicking deterministic situations, it seems to be natural to include both the drift and the diffusion as constraints in the static optimization problem. However, this would lead us to a wrong direction. It is found that the correct static problem should contain the diffusion as a part of the objective function, which reveals a deep feature of the stochastic turnpike problem.

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arXiv2021

Zero-Sum Stackelberg Stochastic Linear-Quadratic Differential Games

The paper is concerned with a zero-sum Stackelberg stochastic linear-quadratic (LQ, for short) differential game over finite horizons. Under a fairly weak condition, the Stackelberg equilibrium is explicitly obtained by first solving a forward stochastic LQ optimal control problem (SLQ problem, for short) and then a backward SLQ problem. Two Riccati equations are derived in constructing the Stackelberg equilibrium. An interesting finding is that the difference of these two Riccati equations coincides with the Riccati equation associated with the zero-sum Nash stochastic LQ differential game, which implies that the Stackelberg equilibrium and the Nash equilibrium are actually identical. Consequently, the Stackelberg equilibrium admits a linear state feedback representation, and the Nash game can be solved in a leader-follower manner.

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arXiv2021

Mean-Field Linear-Quadratic Stochastic Differential Games

The paper is concerned with two-person zero-sum mean-field linear-quadratic stochastic differential games over finite horizons. By a Hilbert space method, a necessary condition and a sufficient condition are derived for the existence of an open-loop saddle point. It is shown that under the sufficient condition, the associated two Riccati equations admit unique strongly regular solutions, in terms of which the open-loop saddle point can be represented as a linear feedback of the current state. When the game only satisfies the necessary condition, an approximate sequence is constructed by solving a family of Riccati equations and closed-loop systems.The convergence of the approximate sequence turns out to be equivalent to the open-loop solvability of the game, and the limit is exactly an open-loop saddle point, provided that the game is open-loop solvable.

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arXiv2021

Linear-Quadratic Optimal Control for Backward Stochastic Differential Equations with Random Coefficients

This paper is concerned with a linear-quadratic (LQ, for short) optimal control problem for backward stochastic differential equations (BSDEs, for short), where the coefficients of the backward control system and the weighting matrices in the cost functional are allowed to be random. By a variational method, the optimality system, which is a coupled linear forward-backward stochastic differential equation (FBSDE, for short), is derived, and by a Hilbert space method, the unique solvability of the optimality system is obtained. In order to construct the optimal control, a new stochastic Riccati-type equation is introduced. It is proved that an adapted solution (possibly non-unique) to the Riccati equation exists and decouples the optimality system. With this solution, the optimal control is obtained in an explicit way.

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arXiv2021

Path Dependent Feynman-Kac Formula for Forward Backward Stochastic Volterra Integral Equations

This paper is concerned with the relationship between forward-backward stochastic Volterra integral equations (FBSVIEs, for short) and a system of (non-local in time) path dependent partial differential equations (PPDEs, for short). Due to the nature of Volterra type equations, the usual flow property (or semigroup property) does not hold. Inspired by Viens-Zhang \cite{Viens-Zhang-2019} and Wang-Yong \cite{Wang-Yong-2019}, auxiliary processes are introduced so that the flow property of adapted solutions to the FBSVIEs is recovered in a suitable sense, and thus the functional Itô's formula is applicable. Having achieved this stage, a natural PPDE is found so that the adapted solution of the backward SVIEs admits a representation in terms of the solution to the forward SVIE via the solution to a PPDE. On the other hand, the solution of the PPDE admits a representation in terms of adapted solution to the (path dependent) FBSVIE, which is referred to as a Feynman-Kac formula. This leads to the existence and uniqueness of a classical solution to the PPDE, under smoothness conditions on the coefficients of the FBSVIEs. Further, when the smoothness conditions are relaxed with the backward component of FBSVIE being one-dimensional, a new (and suitable) notion of viscosity solution is introduced for the PPDE, for which a comparison principle of the viscosity solutions is established, leading to the uniqueness of the viscosity solution. Finally, some results have been extended to coupled FBSVIEs and type-II BSVIEs, and a representation formula for the path derivatives of PPDE solution is obtained by a closer investigation of linear FBSVIEs.

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arXiv2020

Dont Even Look Once: Synthesizing Features for Zero-Shot Detection

Zero-shot detection, namely, localizing both seen and unseen objects, increasingly gains importance for large-scale applications, with large number of object classes, since, collecting sufficient annotated data with ground truth bounding boxes is simply not scalable. While vanilla deep neural networks deliver high performance for objects available during training, unseen object detection degrades significantly. At a fundamental level, while vanilla detectors are capable of proposing bounding boxes, which include unseen objects, they are often incapable of assigning high-confidence to unseen objects, due to the inherent precision/recall tradeoffs that requires rejecting background objects. We propose a novel detection algorithm Dont Even Look Once (DELO), that synthesizes visual features for unseen objects and augments existing training algorithms to incorporate unseen object detection. Our proposed scheme is evaluated on Pascal VOC and MSCOCO, and we demonstrate significant improvements in test accuracy over vanilla and other state-of-art zero-shot detectors

cs.CV↗

arXiv2019

Recursive Utility Processes, Dynamic Risk Measures and Quadratic Backward Stochastic Volterra Integral Equations

For an $\cF_T$-measurable payoff of a European type contingent claim, the recursive utility process/dynamic risk measure can be described by the adapted solution to a backward stochastic differential equation (BSDE). However, for an $\cF_T$-measurable stochastic process (called a position process, not necessarily $\dbF$-adapted), mimicking BSDE's approach will lead to a time-inconsistent recursive utility/dynamic risk measure. It is found that a more proper approach is to use the adapted solution to a backward stochastic Volterra integral equation (BSVIE). The corresponding notions are called equilibrium recursive utility and equilibrium dynamic risk measure, respectively. Motivated by this, the current paper is concerned with BSVIEs whose generators are allowed to have quadratic growth (in $Z(t,s)$). The existence and uniqueness for both the so-called adapted solutions and adapted M-solutions are established. A comparison theorem for adapted solutions to the so-called Type-I BSVIEs is established as well. As consequences of these results, some general continuous-time equilibrium dynamic risk measures and equilibrium recursive utility processes are constructed.

math.PR↗

arXiv2019

Time-Inconsistent Stochastic Optimal Control Problems and Backward Stochastic Volterra Integral Equations

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situation with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB, for short) equation is derived, through which the equilibrium valued function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman-Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value $Z(r,r)$ of $Z(\cd,\cd)$) is naturally introduced and the well-posedness of such kind of equations is briefly presented.

math.OC↗

arXiv2019

Mean-Field Stochastic Linear-Quadratic Optimal Control Problems: Weak Closed-Loop Solvability

This paper is concerned with mean-field stochastic linear-quadratic (MF-SLQ, for short) optimal control problems with deterministic coefficients. The notion of weak closed-loop optimal strategy is introduced. It is shown that the open-loop solvability is equivalent to the existence of a weak closed-loop optimal strategy. Moreover, when open-loop optimal controls exist, there is at least one of them admitting a state feedback representation, which is the outcome of a weak closed-loop optimal strategy. Finally, an example is presented to illustrate the procedure for finding weak closed-loop optimal strategies.

math.OC↗

arXiv2019

Extended Backward Stochastic Volterra Integral Equations, Quasilinear Parabolic Equations, and Feynman-Kac Formula

In this paper, we establish the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. We first introduce the extended backward stochastic Volterra integral equations (EBSVIEs, for short). Under some mild conditions, we establish the well-posedness of EBSVIEs and obtain some regularity results of the adapted solution to the EBSVIEs via Malliavin calculus. We show that a given function expressed in terms of the solution to the EBSVIEs solves a certain system of non-local parabolic partial differential equations (PDEs, for short), which generalizes the famous nonlinear Feynman-Kac formula in Pardoux{Peng [21].

math.PR↗

arXiv2019

Learning Classifiers for Domain Adaptation, Zero and Few-Shot Recognition Based on Learning Latent Semantic Parts

In computer vision applications, such as domain adaptation (DA), few shot learning (FSL) and zero-shot learning (ZSL), we encounter new objects and environments, for which insufficient examples exist to allow for training "models from scratch," and methods that adapt existing models, trained on the presented training environment, to the new scenario are required. We propose a novel visual attribute encoding method that encodes each image as a low-dimensional probability vector composed of prototypical part-type probabilities. The prototypes are learnt to be representative of all training data. At test-time we utilize this encoding as an input to a classifier. At test-time we freeze the encoder and only learn/adapt the classifier component to limited annotated labels in FSL; new semantic attributes in ZSL. We conduct extensive experiments on benchmark datasets. Our method outperforms state-of-art methods trained for the specific contexts (ZSL, FSL, DA).

cs.CV↗

arXiv2019

Cost-Aware Fine-Grained Recognition for IoTs Based on Sequential Fixations

We consider the problem of fine-grained classification on an edge camera device that has limited power. The edge device must sparingly interact with the cloud to minimize communication bits to conserve power, and the cloud upon receiving the edge inputs returns a classification label. To deal with fine-grained classification, we adopt the perspective of sequential fixation with a foveated field-of-view to model cloud-edge interactions. We propose a novel deep reinforcement learning-based foveation model, DRIFT, that sequentially generates and recognizes mixed-acuity images.Training of DRIFT requires only image-level category labels and encourages fixations to contain task-relevant information, while maintaining data efficiency. Specifically, wetrain a foveation actor network with a novel Deep Deterministic Policy Gradient by Conditioned Critic and Coaching (DDPGC3) algorithm. In addition, we propose to shape the reward to provide informative feedback after each fixation to better guide RL training. We demonstrate the effectiveness of DRIFT on this task by evaluating on five fine-grained classification benchmark datasets, and show that the proposed approach achieves state-of-the-art performance with over 3X reduction in transmitted pixels.

cs.CV↗

arXiv2019

Generalized Zero-Shot Recognition based on Visually Semantic Embedding

We propose a novel Generalized Zero-Shot learning (GZSL) method that is agnostic to both unseen images and unseen semantic vectors during training. Prior works in this context propose to map high-dimensional visual features to the semantic domain, we believe contributes to the semantic gap. To bridge the gap, we propose a novel low-dimensional embedding of visual instances that is "visually semantic." Analogous to semantic data that quantifies the existence of an attribute in the presented instance, components of our visual embedding quantifies existence of a prototypical part-type in the presented instance. In parallel, as a thought experiment, we quantify the impact of noisy semantic data by utilizing a novel visual oracle to visually supervise a learner. These factors, namely semantic noise, visual-semantic gap and label noise lead us to propose a new graphical model for inference with pairwise interactions between label, semantic data, and inputs. We tabulate results on a number of benchmark datasets demonstrating significant improvement in accuracy over state-of-the-art under both semantic and visual supervision.

cs.CV↗

arXiv2019

Zero-Shot Detection

As we move towards large-scale object detection, it is unrealistic to expect annotated training data, in the form of bounding box annotations around objects, for all object classes at sufficient scale, and so methods capable of unseen object detection are required. We propose a novel zero-shot method based on training an end-to-end model that fuses semantic attribute prediction with visual features to propose object bounding boxes for seen and unseen classes. While we utilize semantic features during training, our method is agnostic to semantic information for unseen classes at test-time. Our method retains the efficiency and effectiveness of YOLOv2 for objects seen during training, while improving its performance for novel and unseen objects. The ability of state-of-art detection methods to learn discriminative object features to reject background proposals also limits their performance for unseen objects. We posit that, to detect unseen objects, we must incorporate semantic information into the visual domain so that the learned visual features reflect this information and leads to improved recall rates for unseen objects. We test our method on PASCAL VOC and MS COCO dataset and observed significant improvements on the average precision of unseen classes.

cs.CV↗

arXiv2018

Deep Reinforcement Learning Attention Selection for Person Re-Identification

Existing person re-identification (re-id) methods assume the provision of accurately cropped person bounding boxes with minimum background noise, mostly by manually cropping. This is significantly breached in practice when person bounding boxes must be detected automatically given a very large number of images and/or videos processed. Compared to carefully cropped manually, auto-detected bounding boxes are far less accurate with random amount of background clutter which can degrade notably person re-id matching accuracy. In this work, we develop a joint learning deep model that optimises person re-id attention selection within any auto-detected person bounding boxes by reinforcement learning of background clutter minimisation subject to re-id label pairwise constraints. Specifically, we formulate a novel unified re-id architecture called Identity DiscriminativE Attention reinforcement Learning (IDEAL) to accurately select re-id attention in auto-detected bounding boxes for optimising re-id performance. Our model can improve re-id accuracy comparable to that from exhaustive human manual cropping of bounding boxes with additional advantages from identity discriminative attention selection that specially benefits re-id tasks beyond human knowledge. Extensive comparative evaluations demonstrate the re-id advantages of the proposed IDEAL model over a wide range of state-of-the-art re-id methods on two auto-detected re-id benchmarks CUHK03 and Market-1501.

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