arXiv subjects

Jianjun Yuan

Publications and source records attributed to Jianjun Yuan.

11 recordsLinked to original sources

arXiv2026

A Globally Convergent Variational Framework for Mode Number Detection via Spectral Cutting Curves

Automatically determining the number of intrinsic mode functions (IMFs) and their center frequencies in Variational Mode Decomposition (VMD) remains an open mathematical challenge. Existing methods rely on heuristic settings, trial-and-error, or recursive extraction lacking theoretical convergence guarantees. We propose a variational framework that endogenously determines the number of modes. Any curve below the spectral amplitude divides the area under the spectrum into 2 parts and generate the connected intervals where spectrum locates above it, whose count defines the modal number K[g] -- a topological functional induced by the cutting curve. Since K[g] is discontinuous and intractable for direct optimization, we seek the optimal cutting curve as a continuous variational surrogate: it separates distinct spectral peaks into individual regions above it while merging noise-induced fragments below. This surrogate adversarially maximizes the integral of g while penalizing its curvature, transforming the problem into iteratively solving a fourth-order boundary value problem via Lagrangian duality. We establish a rigorous proof of global convergence for the dual ascent algorithm in function space. Comprehensive numerical experiments on artificial and real-world signals including ECG data show accurate estimates of IMFs and center frequencies, avoiding redundant modes while ensuring recovery of necessary components, providing a robust, theoretically grounded initialization routine for VMD.

math-ph↗

arXiv2023

Adversarial Sleeping Bandit Problems with Multiple Plays: Algorithm and Ranking Application

This paper presents an efficient algorithm to solve the sleeping bandit with multiple plays problem in the context of an online recommendation system. The problem involves bounded, adversarial loss and unknown i.i.d. distributions for arm availability. The proposed algorithm extends the sleeping bandit algorithm for single arm selection and is guaranteed to achieve theoretical performance with regret upper bounded by $\bigO(kN^2\sqrt{T\log T})$, where $k$ is the number of arms selected per time step, $N$ is the total number of arms, and $T$ is the time horizon.

cs.LG↗

arXiv2022

Balancing Exploration for Online Receding Horizon Learning Control with Provable Regret Guarantees

We address the problem of simultaneously learning and control in an online receding horizon control setting. We consider the control of an unknown linear dynamical system with general cost functions and affine constraints on the control input. Our goal is to develop an online learning algorithm that minimizes the dynamic regret, which is defined as the difference between the cumulative cost incurred by the algorithm and that of the best policy with full knowledge of the system, cost functions and state and that satisfies the control input constraints. We propose a novel approach to explore in an online receding horizon setting. The key challenge is to ensure that the control generated by the receding horizon controller is persistently exciting. Our approach is to apply a perturbation to the control input generated by the receding horizon controller that balances the exploration-exploitation trade-off. By exploring the conditions under which sub-linear regret is guaranteed, We show that the proposed controller's performance is upper bounded by $\tilde{\mathcal{O}}(T^{3/4})$ for both regret and cumulative constraint violation when the controller has preview of the cost functions for the interval that doubles in size from one interval to the next.

math.OC↗

arXiv2022

Online Learning for Predictive Control with Provable Regret Guarantees

We study the problem of online learning in predictive control of an unknown linear dynamical system with time varying cost functions which are unknown apriori. Specifically, we study the online learning problem where the control algorithm does not know the true system model and has only access to a fixed-length (that does not grow with the control horizon) preview of the future cost functions. The goal of the online algorithm is to minimize the dynamic regret, defined as the difference between the cumulative cost incurred by the algorithm and that of the best sequence of actions in hindsight. We propose two different online Model Predictive Control (MPC) algorithms to address this problem, namely Certainty Equivalence MPC (CE-MPC) algorithm and Optimistic MPC (O-MPC) algorithm. We show that under the standard stability assumption for the model estimate, the CE-MPC algorithm achieves $\mathcal{O}(T^{2/3})$ dynamic regret. We then extend this result to the setting where the stability assumption holds only for the true system model by proposing the O-MPC algorithm. We show that the O-MPC algorithm also achieves $\mathcal{O}(T^{2/3})$ dynamic regret, at the cost of some additional computation. We also present numerical studies to demonstrate the performance of our algorithm.

cs.LG↗

arXiv2022

Online Convex Optimization with Long Term Constraints for Predictable Sequences

In this paper, we investigate the framework of Online Convex Optimization (OCO) for online learning. OCO offers a very powerful online learning framework for many applications. In this context, we study a specific framework of OCO called {\it OCO with long term constraints}. Long term constraints are introduced typically as an alternative to reduce the complexity of the projection at every update step in online optimization. While many algorithmic advances have been made towards online optimization with long term constraints, these algorithms typically assume that the sequence of cost functions over a certain $T$ finite steps that determine the cost to the online learner are adversarially generated. In many circumstances, the sequence of cost functions may not be unrelated, and thus predictable from those observed till a point of time. In this paper, we study the setting where the sequences are predictable. We present a novel online optimization algorithm for online optimization with long term constraints that can leverage such predictability. We show that, with a predictor that can supply the gradient information of the next function in the sequence, our algorithm can achieve an overall regret and constraint violation rate that is strictly less than the rate that is achievable without prediction.

cs.LG↗

arXiv2021

Adaptive Gradient Online Control

In this work we consider the online control of a known linear dynamic system with adversarial disturbance and adversarial controller cost. The goal in online control is to minimize the regret, defined as the difference between cumulative cost over a period $T$ and the cumulative cost for the best policy from a comparator class. For the setting we consider, we generalize the previously proposed online Disturbance Response Controller (DRC) to the adaptive gradient online Disturbance Response Controller. Using the modified controller, we present novel regret guarantees that improves the established regret guarantees for the same setting. We show that the proposed online learning controller is able to achieve intermediate intermediate regret rates between $\sqrt{T}$ and $\log{T}$ for intermediate convex conditions, while it recovers the previously established regret results for general convex controller cost and strongly convex controller cost.

math.OC↗

arXiv2020

Online Convex Optimization in Changing Environments and its Application to Resource Allocation

In the era of the big data, we create and collect lots of data from all different kinds of sources: the Internet, the sensors, the consumer market, and so on. Many of the data are coming sequentially, and would like to be processed and understood quickly. One classic way of analyzing data is based on batch processing, in which the data is stored and analyzed in an offline fashion. However, when the volume of the data is too large, it is much more difficult and time-consuming to do batch processing than sequential processing. What's more, sequential data is usually changing dynamically, and needs to be understood on-the-fly in order to capture the changes. Online Convex Optimization (OCO) is a popular framework that matches the above sequential data processing requirement. Applications using OCO include online routing, online auctions, online classification and regression, as well as online resource allocation. Due to the general applicability of OCO to the sequential data and the rigorous theoretical guarantee, it has attracted lots of researchers to develop useful algorithms to fulfill different needs. In this thesis, we show our contributions to OCO's development by designing algorithms to adapt to changing environments.

cs.LG↗

arXiv2019

Trading-Off Static and Dynamic Regret in Online Least-Squares and Beyond

Recursive least-squares algorithms often use forgetting factors as a heuristic to adapt to non-stationary data streams. The first contribution of this paper rigorously characterizes the effect of forgetting factors for a class of online Newton algorithms. For exp-concave and strongly convex objectives, the algorithms achieve the dynamic regret of $\max\{O(\log T),O(\sqrt{TV})\}$, where $V$ is a bound on the path length of the comparison sequence. In particular, we show how classic recursive least-squares with a forgetting factor achieves this dynamic regret bound. By varying $V$, we obtain a trade-off between static and dynamic regret. In order to obtain more computationally efficient algorithms, our second contribution is a novel gradient descent step size rule for strongly convex functions. Our gradient descent rule recovers the order optimal dynamic regret bounds described above. For smooth problems, we can also obtain static regret of $O(T^{1-β})$ and dynamic regret of $O(T^βV^*)$, where $β\in (0,1)$ and $V^*$ is the path length of the sequence of minimizers. By varying $β$, we obtain a trade-off between static and dynamic regret.

cs.LG↗

arXiv2019

Online Adaptive Principal Component Analysis and Its extensions

We propose algorithms for online principal component analysis (PCA) and variance minimization for adaptive settings. Previous literature has focused on upper bounding the static adversarial regret, whose comparator is the optimal fixed action in hindsight. However, static regret is not an appropriate metric when the underlying environment is changing. Instead, we adopt the adaptive regret metric from the previous literature and propose online adaptive algorithms for PCA and variance minimization, that have sub-linear adaptive regret guarantees. We demonstrate both theoretically and experimentally that the proposed algorithms can adapt to the changing environments.

cs.LG↗

arXiv2019

Online convex optimization for cumulative constraints

We propose the algorithms for online convex optimization which lead to cumulative squared constraint violations of the form $\sum\limits_{t=1}^T\big([g(x_t)]_+\big)^2=O(T^{1-β})$, where $β\in(0,1)$. Previous literature has focused on long-term constraints of the form $\sum\limits_{t=1}^Tg(x_t)$. There, strictly feasible solutions can cancel out the effects of violated constraints. In contrast, the new form heavily penalizes large constraint violations and cancellation effects cannot occur. Furthermore, useful bounds on the single step constraint violation $[g(x_t)]_+$ are derived. For convex objectives, our regret bounds generalize existing bounds, and for strongly convex objectives we give improved regret bounds. In numerical experiments, we show that our algorithm closely follows the constraint boundary leading to low cumulative violation.

cs.LG↗

arXiv2015

Global solutions of two coupled Maxwell systems in the temporal gauge

In this paper, we consider the Maxwell-Klein-Gordon and Maxwell-Chern-Simons-Higgs systems in the temporal gauge. By using the fact that when the spatial gauge potentials are in the Coulomb gauge, their $\dot{H}^1$ norms can be controlled by the energy of the corresponding system and their $L^2$ norms, and the gauge invariance of the systems, we show that finite energy solutions of these two systems exist globally in this gauge.

math.AP↗