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For optimal control problems constrained by a initial-valued parabolic PDE, we have to solve a large scale saddle point algebraic system consisting of considering the discrete space and time points all together. A popular strategy to handle such a system is the Krylov subspace method, for which an efficient preconditioner plays a crucial role. The matching-Schur-complement preconditioner has been extensively studied in literature and the implementation of this preconditioner lies in solving the underlying PDEs twice, sequentially in time. In this paper, we propose a new preconditioner for the Schur complement, which can be used parallel-in-time (PinT) via the so called diagonalization technique. We show that the eigenvalues of the preconditioned matrix are low and upper bounded by positive constants independent of matrix size and the regularization parameter. The uniform boundedness of the eigenvalues leads to an optimal linear convergence rate of conjugate gradient solver for the preconditioned Schur complement system. To the best of our knowledge, it is the first time to have an optimal convergence analysis for a PinT preconditioning technique of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner is robust with respect to the discretization step-sizes and the regularization parameter.

We study the mixing time of the Metropolis-adjusted Langevin algorithm (MALA) for sampling from a log-smooth and strongly log-concave distribution. We establish its optimal minimax mixing time under a warm start. Our main contribution is two-fold. First, for a $d$-dimensional log-concave density with condition number $\kappa$, we show that MALA with a warm start mixes in $\tilde O(\kappa \sqrt{d})$ iterations up to logarithmic factors. This improves upon the previous work on the dependency of either the condition number $\kappa$ or the dimension $d$. Our proof relies on comparing the leapfrog integrator with the continuous Hamiltonian dynamics, where we establish a new concentration bound for the acceptance rate. Second, we prove a spectral gap based mixing time lower bound for reversible MCMC algorithms on general state spaces. We apply this lower bound result to construct a hard distribution for which MALA requires at least $\tilde \Omega (\kappa \sqrt{d})$ steps to mix. The lower bound for MALA matches our upper bound in terms of condition number and dimension. Finally, numerical experiments are included to validate our theoretical results.

We revisit the $k$-Hessian eigenvalue problem on a smooth, bounded, $(k-1)$-convex domain in $\mathbb R^n$. First, we obtain a spectral characterization of the $k$-Hessian eigenvalue as the infimum of the first eigenvalues of linear second-order elliptic operators whose coefficients belong to the dual of the corresponding G\r{a}rding cone. Second, we introduce a non-degenerate inverse iterative scheme to solve the eigenvalue problem for the $k$-Hessian operator. We show that the scheme converges, with a rate, to the $k$-Hessian eigenvalue for all $k$. When $2\leq k\leq n$, we also prove a local $L^1$ convergence of the Hessian of solutions of the scheme. Hyperbolic polynomials play an important role in our analysis.

We obtain explicit $p$-Wasserstein distance error bounds between the distribution of the multi-parameter MLE and the multivariate normal distribution. Our general bounds are given for possibly high-dimensional, independent and identically distributed random vectors. Our general bounds are of the optimal $\mathcal{O}(n^{-1/2})$ order. Explicit numerical constants are given when $p\in(1,2]$, and in the case $p>2$ the bounds are explicit up to a constant factor that only depends on $p$. We apply our general bounds to derive Wasserstein distance error bounds for the multivariate normal approximation of the MLE in several settings; these being single-parameter exponential families, the normal distribution under canonical parametrisation, and the multivariate normal distribution under non-canonical parametrisation. In addition, we provide upper bounds with respect to the bounded Wasserstein distance when the MLE is implicitly defined.

We develop a computationally tractable method for estimating the optimal map between two distributions over $\mathbb{R}^d$ with rigorous finite-sample guarantees. Leveraging an entropic version of Brenier's theorem, we show that our estimator -- the barycentric projection of the optimal entropic plan -- is easy to compute using Sinkhorn's algorithm. As a result, unlike current approaches for map estimation, which are slow to evaluate when the dimension or number of samples is large, our approach is parallelizable and extremely efficient even for massive data sets. Under smoothness assumptions on the optimal map, we show that our estimator enjoys comparable statistical performance to other estimators in the literature, but with much lower computational cost. We showcase the efficacy of our proposed estimator through numerical examples. Our proofs are based on a modified duality principle for entropic optimal transport and on a method for approximating optimal entropic plans due to Pal (2019).

We consider n robots with limited visibility: each robot can observe other robots only up to a constant distance denoted as the viewing range. The robots operate in discrete rounds that are either fully synchronous (FSync) or semi-synchronized (SSync). Most previously studied formation problems in this setting seek to bring the robots closer together (e.g., Gathering or Chain-Formation). In this work, we introduce the Max-Line-Formation problem, which has a contrary goal: to arrange the robots on a straight line of maximal length. First, we prove that the problem is impossible to solve by robots with a constant sized circular viewing range. The impossibility holds under comparably strong assumptions: robots that agree on both axes of their local coordinate systems in FSync. On the positive side, we show that the problem is solvable by robots with a constant square viewing range, i.e., the robots can observe other robots that lie within a constant-sized square centered at their position. In this case, the robots need to agree on only one axis of their local coordinate systems. We derive two algorithms: the first algorithm considers oblivious robots and converges to the optimal configuration in time $\mathcal{O}(n^2 \cdot \log (n/\varepsilon))$ under the SSync scheduler. The other algorithm makes use of locally visible lights (LUMI). It is designed for the FSync scheduler and can solve the problem exactly in optimal time $\Theta(n)$. Afterward, we show that both the algorithmic and the analysis techniques can also be applied to the Gathering and Chain-Formation problem: we introduce an algorithm with a reduced viewing range for Gathering and give new and improved runtime bounds for the Chain-Formation problem.

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.

This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.