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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 binary rank of a $0,1$ matrix is the smallest size of a partition of its ones into monochromatic combinatorial rectangles. A matrix $M$ is called $(k_1, \ldots, k_m ; n_1, \ldots, n_m)$ circulant block diagonal if it is a block matrix with $m$ diagonal blocks, such that for each $i \in [m]$, the $i$th diagonal block of $M$ is the circulant matrix whose first row has $k_i$ ones followed by $n_i-k_i$ zeros, and all of whose other entries are zeros. In this work, we study the binary rank of these matrices and of their complement. In particular, we compare the binary rank of these matrices to their rank over the reals, which forms a lower bound on the former. We present a general method for proving upper bounds on the binary rank of block matrices that have diagonal blocks of some specified structure and ones elsewhere. Using this method, we prove that the binary rank of the complement of a $(k_1, \ldots, k_m ; n_1, \ldots, n_m)$ circulant block diagonal matrix for integers satisfying $n_i>k_i>0$ for each $i \in [m]$ exceeds its real rank by no more than the maximum of $\gcd(n_i,k_i)-1$ over all $i \in [m]$. We further present several sufficient conditions for the binary rank of these matrices to strictly exceed their real rank. By combining the upper and lower bounds, we determine the exact binary rank of various families of matrices and, in addition, significantly generalize a result of Gregory. Motivated by a question of Pullman, we study the binary rank of $k$-regular $0,1$ matrices and of their complement. As an application of our results on circulant block diagonal matrices, we show that for every $k \geq 2$, there exist $k$-regular $0,1$ matrices whose binary rank is strictly larger than that of their complement. Furthermore, we exactly determine for every integer $r$, the smallest possible binary rank of the complement of a $2$-regular $0,1$ matrix with binary rank $r$.

Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.

In this paper we propose a methodology to accelerate the resolution of the so-called "Sorted L-One Penalized Estimation" (SLOPE) problem. Our method leverages the concept of "safe screening", well-studied in the literature for \textit{group-separable} sparsity-inducing norms, and aims at identifying the zeros in the solution of SLOPE. More specifically, we derive a set of \(\tfrac{n(n+1)}{2}\) inequalities for each element of the \(n\)-dimensional primal vector and prove that the latter can be safely screened if some subsets of these inequalities are verified. We propose moreover an efficient algorithm to jointly apply the proposed procedure to all the primal variables. Our procedure has a complexity \(\mathcal{O}(n\log n + LT)\) where \(T\leq n\) is a problem-dependent constant and \(L\) is the number of zeros identified by the tests. Numerical experiments confirm that, for a prescribed computational budget, the proposed methodology leads to significant improvements of the solving precision.

In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissner variational principle. We show the existence of suitable approximate fields leading to an inf-sup stable saddle-point problem. For both discretization approaches we prove convergence statements and in case of weak symmetry we illustrate the approximation behavior by means of several numerical experiments.

We study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, it may happen that a subset of the variables are uninformative in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. This g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with such property exists that only depends on a strict subset of the variables.

Bearing fault identification and analysis is an important research area in the field of machinery fault diagnosis. Aiming at the common faults of rolling bearings, we propose a data-driven diagnostic algorithm based on the characteristics of bearing vibrations called multi-size kernel based adaptive convolutional neural network (MSKACNN). Using raw bearing vibration signals as the inputs, MSKACNN provides vibration feature learning and signal classification capabilities to identify and analyze bearing faults. Ball mixing is a ball bearing production quality problem that is difficult to identify using traditional frequency domain analysis methods since it requires high frequency resolutions of the measurement signals and results in a long analyzing time. The proposed MSKACNN is shown to improve the efficiency and accuracy of ball mixing diagnosis. To further demonstrate the effectiveness of MSKACNN in bearing fault identification, a bearing vibration data acquisition system was developed, and vibration signal acquisition was performed on rolling bearings under five different fault conditions including ball mixing. The resulting datasets were used to analyze the performance of our proposed model. To validate the adaptive ability of MSKACNN, fault test data from the Case Western Reserve University Bearing Data Center were also used. Test results show that MSKACNN can identify the different bearing conditions with high accuracy with high generalization ability. We presented an implementation of the MSKACNN as a lightweight module for a real-time bearing fault diagnosis system that is suitable for production.

For a given nonnegative matrix $A=(A_{ij})$, the matrix scaling problem asks whether $A$ can be scaled to a doubly stochastic matrix $XAY$ for some positive diagonal matrices $X,Y$. The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization $A_{ij} \leftarrow A_{ij}/\sum_{j}A_{ij}$ and column-normalization $A_{ij} \leftarrow A_{ij}/\sum_{i}A_{ij}$ alternatively. By this algorithm, $A$ converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with $A$ has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph $G$, which is identified with the $0,1$-matrix $A_G$. Linial, Samorodnitsky, and Wigderson showed that a polynomial number of the Sinkhorn iterations for $A_G$ decides whether $G$ has a perfect matching. In this paper, we show an extension of this result: If $G$ has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker -- a certificate of the nonexistence of a perfect matching. Our analysis is based on an interpretation of the Sinkhorn algorithm as alternating KL-divergence minimization (Csisz\'{a}r and Tusn\'{a}dy 1984, Gietl and Reffel 2013) and its limiting behavior for a nonscalable matrix (Aas 2014). We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage-Mendelsohn decomposition of a bipartite graph.

We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The method we propose is capable of dealing with either variational and non-variational problems, and because of its meshless nature, it can also deal with problems posed in high-dimensional domains. We prove the $\Gamma$-convergence of the neural network approximation towards the solution of the continuous problem, and extend the convergence proof to some well-known related methods. Finally, we present several numerical examples illustrating the performance of our discretization.

Decomposition-based evolutionary algorithms have become fairly popular for many-objective optimization in recent years. However, the existing decomposition methods still are quite sensitive to the various shapes of frontiers of many-objective optimization problems (MaOPs). On the one hand, the cone decomposition methods such as the penalty-based boundary intersection (PBI) are incapable of acquiring uniform frontiers for MaOPs with very convex frontiers. On the other hand, the parallel reference lines of the parallel decomposition methods including the normal boundary intersection (NBI) might result in poor diversity because of under-sampling near the boundaries for MaOPs with concave frontiers. In this paper, a collaborative decomposition method is first proposed to integrate the advantages of parallel decomposition and cone decomposition to overcome their respective disadvantages. This method inherits the NBI-style Tchebycheff function as a convergence measure to heighten the convergence and uniformity of distribution of the PBI method. Moreover, this method also adaptively tunes the extent of rotating an NBI reference line towards a PBI reference line for every subproblem to enhance the diversity of distribution of the NBI method. Furthermore, a collaborative decomposition-based evolutionary algorithm (CoDEA) is presented for many-objective optimization. A collaborative decomposition-based environmental selection mechanism is primarily designed in CoDEA to rank all the individuals associated with the same PBI reference line in the boundary layer and pick out the best ranks. CoDEA is compared with several popular algorithms on 85 benchmark test instances. The experimental results show that CoDEA achieves high competitiveness benefiting from the collaborative decomposition maintaining a good balance among the convergence, uniformity, and diversity of distribution.