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The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz (2017). In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.

Missing values challenge the probabilistic wind power forecasting at both parameter estimation and operational forecasting stages. In this paper, we illustrate that we are allowed to estimate forecasting functions for each missing patterns conveniently, and propose an adaptive quantile regression model whose parameters can adapt to missing patterns. For that, we particularly design a feature extraction block within the quantile regression model, where parameters are set as a function of missingness pattern and only account for observed values. To avoidthe quantile-crossing phenomena, we design a multi-task model to ensure the monotonicity of quantiles, where higher quantiles are derived by the addition between lower quantiles and non-negative increments modeled by neural networks. The proposed approach is distribution-free and applicable to both missing-at-random and missing-not-at-random cases. Case studies demonstrate that the proposed approach achieves the state-of-the-art in terms of the continuous ranked probability score.

The efficient representation of random fields on geometrically complex domains is crucial for Bayesian modelling in engineering and machine learning. Today's prevalent random field representations are restricted to unbounded domains or are too restrictive in terms of possible field properties. As a result, new techniques leveraging the historically established link between stochastic PDEs (SPDEs) and random fields are especially appealing for engineering applications with complex geometries which already have a finite element discretisation for solving the physical conservation equations. Unlike the dense covariance matrix of a random field, its inverse, the precision matrix, is usually sparse and equal to the stiffness matrix of a Helmholtz-like SPDE. In this paper, we use the SPDE representation to develop a scalable framework for large-scale statistical finite element analysis (statFEM) and Gaussian process (GP) regression on geometrically complex domains. We use the SPDE formulation to obtain the relevant prior probability densities with a sparse precision matrix. The properties of the priors are governed by the parameters and possibly fractional order of the Helmholtz-like SPDE so that we can model on bounded domains and manifolds anisotropic, non-homogeneous random fields with arbitrary smoothness. We use for assembling the sparse precision matrix the same finite element mesh used for solving the physical conservation equations. The observation models for statFEM and GP regression are such that the posterior probability densities are Gaussians with a closed-form mean and precision. The expressions for the mean vector and the precision matrix can be evaluated using only sparse matrix operations. We demonstrate the versatility of the proposed framework and its convergence properties with one and two-dimensional Poisson and thin-shell examples.

We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our theta-scheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order $\mathcal{O}(h^r)$ for the theta-scheme, where $h$ is the step length of the space variable and $r \in (0,1)$ is related to the H\"older continuity of the solution of the continuous problem and some of its derivatives.

We explore the performance of polynomial-time incentive-compatible mechanisms in single-crossing domains. Single-crossing domains were extensively studied in the economics literature. Roughly speaking, a domain is single crossing if monotonicity characterizes incentive compatibility. That is, single-crossing domains are the standard mathematical formulation of domains that are informally known as ``single parameter''. In all major single-crossing domains studied so far (e.g., welfare maximization in various auctions with single-minded bidders, makespan minimization on related machines), the performance of the best polynomial-time incentive-compatible mechanisms matches the performance of the best polynomial-time non-incentive-compatible algorithms. Our two main results make progress in understanding the power of incentive-compatible polynomial-time mechanisms in single-crossing domains: We provide the first proof of a gap in the power of polynomial-time incentive-compatible mechanisms and polynomial-time non-incentive-compatible algorithms: we present an objective function in a single-crossing multi-unit auction for which there is a polynomial-time algorithm that provides an approximation ratio of $\frac{1}{2}$, yet no polynomial-time incentive-compatible mechanism provides a finite approximation (under standard computational complexity assumptions). The objective function used above is not natural. We show that to some extent this is unavoidable by providing a sweeping positive result for the most natural objective function in multi-unit auctions, that of welfare maximization. We present an incentive-compatible FPTAS mechanism for every multi-unit auction with single-crossing domains. This improves over the mechanism of Briest et al. [STOC'05] that only applies to the much simpler case of single-minded bidders.

Variance reduction techniques have been successfully applied to temporal-difference (TD) learning and help to improve the sample complexity in policy evaluation. However, the existing work applied variance reduction to either the less popular one time-scale TD algorithm or the two time-scale GTD algorithm but with a finite number of i.i.d.\ samples, and both algorithms apply to only the on-policy setting. In this work, we develop a variance reduction scheme for the two time-scale TDC algorithm in the off-policy setting and analyze its non-asymptotic convergence rate over both i.i.d.\ and Markovian samples. In the i.i.d.\ setting, our algorithm {matches the best-known lower bound $\tilde{O}(\epsilon^{-1}$).} In the Markovian setting, our algorithm achieves the state-of-the-art sample complexity $O(\epsilon^{-1} \log {\epsilon}^{-1})$ that is near-optimal. Experiments demonstrate that the proposed variance-reduced TDC achieves a smaller asymptotic convergence error than both the conventional TDC and the variance-reduced TD.

Adversarial team games model multiplayer strategic interactions in which a team of identically-interested players is competing against an adversarial player in a zero-sum game. Such games capture many well-studied settings in game theory, such as congestion games, but go well-beyond to environments wherein the cooperation of one team -- in the absence of explicit communication -- is obstructed by competing entities; the latter setting remains poorly understood despite its numerous applications. Since the seminal work of Von Stengel and Koller (GEB `97), different solution concepts have received attention from an algorithmic standpoint. Yet, the complexity of the standard Nash equilibrium has remained open. In this paper, we settle this question by showing that computing a Nash equilibrium in adversarial team games belongs to the class continuous local search (CLS), thereby establishing CLS-completeness by virtue of the recent CLS-hardness result of Rubinstein and Babichenko (STOC `21) in potential games. To do so, we leverage linear programming duality to prove that any $\epsilon$-approximate stationary strategy for the team can be extended in polynomial time to an $O(\epsilon)$-approximate Nash equilibrium, where the $O(\cdot)$ notation suppresses polynomial factors in the description of the game. As a consequence, we show that the Moreau envelop of a suitable best response function acts as a potential under certain natural gradient-based dynamics.

A nonlinear sea-ice problem is considered in a least-squares finite element setting. The corresponding variational formulation approximating simultaneously the stress tensor and the velocity is analysed. In particular, the least-squares functional is coercive and continuous in an appropriate solution space and this proves the well-posedness of the problem. As the method does not require a compatibility condition between the finite element space, the formulation allows the use of piecewise polynomial spaces of the same approximation order for both the stress and the velocity approximations. A Newton-type iterative method is used to linearize the problem and numerical tests are provided to illustrate the theory.

Matrix decomposition is a very important mathematical tool in numerical linear algebra for data processing. In this paper, we introduce a new randomized matrix decomposition algorithm, which is called randomized approximate SVD based on Qatar Riyal decomposition (RCSVD-QR). Our method utilize random sampling and the OR decomposition to address a serious bottlenck associated with classical SVD. RCSVD-QR gives satisfactory convergence speed as well as accuracy as compared to those state-of-the-art algorithms. In addition, we provides an estimate for the expected approximation error in Frobenius norm. Numerical experiments verify these claims.

We give query complexity lower bounds for convex optimization and the related feasibility problem. We show that quadratic memory is necessary to achieve the optimal oracle complexity for first-order convex optimization. In particular, this shows that center-of-mass cutting-planes algorithms in dimension $d$ which use $\tilde O(d^2)$ memory and $\tilde O(d)$ queries are Pareto-optimal for both convex optimization and the feasibility problem, up to logarithmic factors. Precisely, we prove that to minimize $1$-Lipschitz convex functions over the unit ball to $1/d^4$ accuracy, any deterministic first-order algorithms using at most $d^{2-\delta}$ bits of memory must make $\tilde\Omega(d^{1+\delta/3})$ queries, for any $\delta\in[0,1]$. For the feasibility problem, in which an algorithm only has access to a separation oracle, we show a stronger trade-off: for at most $d^{2-\delta}$ memory, the number of queries required is $\tilde\Omega(d^{1+\delta})$. This resolves a COLT 2019 open problem of Woodworth and Srebro.