We study the problem of learning a mixture of multiple linear dynamical systems (LDSs) from unlabeled short sample trajectories, each generated by one of the LDS models. Despite the wide applicability of mixture models for time-series data, learning algorithms that come with end-to-end performance guarantees are largely absent from existing literature. There are multiple sources of technical challenges, including but not limited to (1) the presence of latent variables (i.e. the unknown labels of trajectories); (2) the possibility that the sample trajectories might have lengths much smaller than the dimension $d$ of the LDS models; and (3) the complicated temporal dependence inherent to time-series data. To tackle these challenges, we develop a two-stage meta-algorithm, which is guaranteed to efficiently recover each ground-truth LDS model up to error $\tilde{O}(\sqrt{d/T})$, where $T$ is the total sample size. We validate our theoretical studies with numerical experiments, confirming the efficacy of the proposed algorithm.
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Anomaly detection among a large number of processes arises in many applications ranging from dynamic spectrum access to cybersecurity. In such problems one can often obtain noisy observations aggregated from a chosen subset of processes that conforms to a tree structure. The distribution of these observations, based on which the presence of anomalies is detected, may be only partially known. This gives rise to the need for a search strategy designed to account for both the sample complexity and the detection accuracy, as well as cope with statistical models that are known only up to some missing parameters. In this work we propose a sequential search strategy using two variations of the Generalized Local Likelihood Ratio statistic. Our proposed Hierarchical Dynamic Search (HDS) strategy is shown to be order-optimal with respect to the size of the search space and asymptotically optimal with respect to the detection accuracy. An explicit upper bound on the error probability of HDS is established for the finite sample regime. Extensive experiments are conducted, demonstrating the performance gains of HDS over existing methods.
Learning Markov decision processes (MDPs) in the presence of the adversary is a challenging problem in reinforcement learning (RL). In this paper, we study RL in episodic MDPs with adversarial reward and full information feedback, where the unknown transition probability function is a linear function of a given feature mapping, and the reward function can change arbitrarily episode by episode. We propose an optimistic policy optimization algorithm POWERS and show that it can achieve $\tilde{O}(dH\sqrt{T})$ regret, where $H$ is the length of the episode, $T$ is the number of interactions with the MDP, and $d$ is the dimension of the feature mapping. Furthermore, we also prove a matching lower bound of $\tilde{\Omega}(dH\sqrt{T})$ up to logarithmic factors. Our key technical contributions are two-fold: (1) a new value function estimator based on importance weighting; and (2) a tighter confidence set for the transition kernel. They together lead to the nearly minimax optimal regret.
We study reinforcement learning for two-player zero-sum Markov games with simultaneous moves in the finite-horizon setting, where the transition kernel of the underlying Markov games can be parameterized by a linear function over the current state, both players' actions and the next state. In particular, we assume that we can control both players and aim to find the Nash Equilibrium by minimizing the duality gap. We propose an algorithm Nash-UCRL based on the principle "Optimism-in-Face-of-Uncertainty". Our algorithm only needs to find a Coarse Correlated Equilibrium (CCE), which is computationally efficient. Specifically, we show that Nash-UCRL can provably achieve an $\tilde{O}(dH\sqrt{T})$ regret, where $d$ is the linear function dimension, $H$ is the length of the game and $T$ is the total number of steps in the game. To assess the optimality of our algorithm, we also prove an $\tilde{\Omega}( dH\sqrt{T})$ lower bound on the regret. Our upper bound matches the lower bound up to logarithmic factors, which suggests the optimality of our algorithm.
The success of large-scale models in recent years has increased the importance of statistical models with numerous parameters. Several studies have analyzed over-parameterized linear models with high-dimensional data that may not be sparse; however, existing results depend on the independent setting of samples. In this study, we analyze a linear regression model with dependent time series data under over-parameterization settings. We consider an estimator via interpolation and developed a theory for excess risk of the estimator under multiple dependence types. This theory can treat infinite-dimensional data without sparsity and handle long-memory processes in a unified manner. Moreover, we bound the risk in our theory via the integrated covariance and nondegeneracy of autocorrelation matrices. The results show that the convergence rate of risks with short-memory processes is identical to that of cases with independent data, while long-memory processes slow the convergence rate. We also present several examples of specific dependent processes that can be applied to our setting.
Stability certification and identification of the stabilizable operating region of a dynamical system are two important concerns to ensure its operational safety/security and robustness. With the advent of machine-learning tools, these issues are especially important for systems with machine-learned components in the feedback loop. Here, in presence of unknown discrete variation (DV) of its parameters within a bounded range, a system controlled by a static feedback controller in which the closed-loop (CL) equilibria are subject to variation-induced drift is equivalently represented using a class of time-invariant systems, each with the same control policy. To develop a general theory for stability and stabilizability of such a class of neural-network (NN) controlled nonlinear systems, a Lyapunov-based convex stability certificate is proposed and is further used to devise an estimate of a local Lipschitz upper bound for the NN and a corresponding operating domain in the state space containing an initialization set, starting from where the CL local asymptotic stability of each system in the class is guaranteed, while the trajectory of the original system remains confined to the domain if the DV of the parameters satisfies a certain quasi-stationarity condition. To compute such a robustly stabilizing NN controller, a stability-guaranteed training (SGT) algorithm is also proposed. The effectiveness of the proposed framework is demonstrated using illustrative examples.
In this work, we study the transfer learning problem under high-dimensional generalized linear models (GLMs), which aim to improve the fit on target data by borrowing information from useful source data. Given which sources to transfer, we propose a transfer learning algorithm on GLM, and derive its $\ell_1/\ell_2$-estimation error bounds as well as a bound for a prediction error measure. The theoretical analysis shows that when the target and source are sufficiently close to each other, these bounds could be improved over those of the classical penalized estimator using only target data under mild conditions. When we don't know which sources to transfer, an algorithm-free transferable source detection approach is introduced to detect informative sources. The detection consistency is proved under the high-dimensional GLM transfer learning setting. We also propose an algorithm to construct confidence intervals of each coefficient component, and the corresponding theories are provided. Extensive simulations and a real-data experiment verify the effectiveness of our algorithms. We implement the proposed GLM transfer learning algorithms in a new R package glmtrans, which is available on CRAN.
There are many important high dimensional function classes that have fast agnostic learning algorithms when strong assumptions on the distribution of examples can be made, such as Gaussianity or uniformity over the domain. But how can one be sufficiently confident that the data indeed satisfies the distributional assumption, so that one can trust in the output quality of the agnostic learning algorithm? We propose a model by which to systematically study the design of tester-learner pairs $(\mathcal{A},\mathcal{T})$, such that if the distribution on examples in the data passes the tester $\mathcal{T}$ then one can safely trust the output of the agnostic learner $\mathcal{A}$ on the data. To demonstrate the power of the model, we apply it to the classical problem of agnostically learning halfspaces under the standard Gaussian distribution and present a tester-learner pair with a combined run-time of $n^{\tilde{O}(1/\epsilon^4)}$. This qualitatively matches that of the best known ordinary agnostic learning algorithms for this task. In contrast, finite sample Gaussian distribution testers do not exist for the $L_1$ and EMD distance measures. A key step in the analysis is a novel characterization of concentration and anti-concentration properties of a distribution whose low-degree moments approximately match those of a Gaussian. We also use tools from polynomial approximation theory. In contrast, we show strong lower bounds on the combined run-times of tester-learner pairs for the problems of agnostically learning convex sets under the Gaussian distribution and for monotone Boolean functions under the uniform distribution over $\{0,1\}^n$. Through these lower bounds we exhibit natural problems where there is a dramatic gap between standard agnostic learning run-time and the run-time of the best tester-learner pair.
Dynamic neural network is an emerging research topic in deep learning. Compared to static models which have fixed computational graphs and parameters at the inference stage, dynamic networks can adapt their structures or parameters to different inputs, leading to notable advantages in terms of accuracy, computational efficiency, adaptiveness, etc. In this survey, we comprehensively review this rapidly developing area by dividing dynamic networks into three main categories: 1) instance-wise dynamic models that process each instance with data-dependent architectures or parameters; 2) spatial-wise dynamic networks that conduct adaptive computation with respect to different spatial locations of image data and 3) temporal-wise dynamic models that perform adaptive inference along the temporal dimension for sequential data such as videos and texts. The important research problems of dynamic networks, e.g., architecture design, decision making scheme, optimization technique and applications, are reviewed systematically. Finally, we discuss the open problems in this field together with interesting future research directions.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
In this paper, we propose a deep reinforcement learning framework called GCOMB to learn algorithms that can solve combinatorial problems over large graphs. GCOMB mimics the greedy algorithm in the original problem and incrementally constructs a solution. The proposed framework utilizes Graph Convolutional Network (GCN) to generate node embeddings that predicts the potential nodes in the solution set from the entire node set. These embeddings enable an efficient training process to learn the greedy policy via Q-learning. Through extensive evaluation on several real and synthetic datasets containing up to a million nodes, we establish that GCOMB is up to 41% better than the state of the art, up to seven times faster than the greedy algorithm, robust and scalable to large dynamic networks.
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