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Information-theoretic Bayesian optimization techniques have become popular for optimizing expensive-to-evaluate black-box functions due to their non-myopic qualities. Entropy Search and Predictive Entropy Search both consider the entropy over the optimum in the input space, while the recent Max-value Entropy Search considers the entropy over the optimal value in the output space. We propose Joint Entropy Search (JES), a novel information-theoretic acquisition function that considers an entirely new quantity, namely the entropy over the joint optimal probability density over both input and output space. To incorporate this information, we consider the reduction in entropy from conditioning on fantasized optimal input/output pairs. The resulting approach primarily relies on standard GP machinery and removes complex approximations typically associated with information-theoretic methods. With minimal computational overhead, JES shows superior decision-making, and yields state-of-the-art performance for information-theoretic approaches across a wide suite of tasks. As a light-weight approach with superior results, JES provides a new go-to acquisition function for Bayesian optimization.

Conditional normalizing flows can generate diverse image samples for solving inverse problems. Most normalizing flows for inverse problems in imaging employ the conditional affine coupling layer that can generate diverse images quickly. However, unintended severe artifacts are occasionally observed in the output of them. In this work, we address this critical issue by investigating the origins of these artifacts and proposing the conditions to avoid them. First of all, we empirically and theoretically reveal that these problems are caused by ``exploding variance'' in the conditional affine coupling layer for certain out-of-distribution (OOD) conditional inputs. Then, we further validated that the probability of causing erroneous artifacts in pixels is highly correlated with a Mahalanobis distance-based OOD score for inverse problems in imaging. Lastly, based on our investigations, we propose a remark to avoid exploding variance and then based on it, we suggest a simple remedy that substitutes the affine coupling layers with the modified rational quadratic spline coupling layers in normalizing flows, to encourage the robustness of generated image samples. Our experimental results demonstrated that our suggested methods effectively suppressed critical artifacts occurring in normalizing flows for super-resolution space generation and low-light image enhancement without compromising performance.

The problem of robust mean estimation in high dimensions is studied, in which a certain fraction (less than half) of the datapoints can be arbitrarily corrupted. Motivated by compressive sensing, the robust mean estimation problem is formulated as the minimization of the $\ell_0$-`norm' of an \emph{outlier indicator vector}, under a second moment constraint on the datapoints. The $\ell_0$-`norm' is then relaxed to the $\ell_p$-norm ($0<p\leq 1$) in the objective, and it is shown that the global minima for each of these objectives are order-optimal and have optimal breakdown point for the robust mean estimation problem. Furthermore, a computationally tractable iterative $\ell_p$-minimization and hard thresholding algorithm is proposed that outputs an order-optimal robust estimate of the population mean. The proposed algorithm (with breakdown point $\approx 0.3$) does not require prior knowledge of the fraction of outliers, in contrast with most existing algorithms, and for $p=1$ it has near-linear time complexity. Both synthetic and real data experiments demonstrate that the proposed algorithm outperforms state-of-the-art robust mean estimation methods.

Gradient-based first-order convex optimization algorithms find widespread applicability in a variety of domains, including machine learning tasks. Motivated by the recent advances in fixed-time stability theory of continuous-time dynamical systems, we introduce a generalized framework for designing accelerated optimization algorithms with strongest convergence guarantees that further extend to a subclass of non-convex functions. In particular, we introduce the \emph{GenFlow} algorithm and its momentum variant that provably converge to the optimal solution of objective functions satisfying the Polyak-{\L}ojasiewicz (PL) inequality, in a fixed-time. Moreover for functions that admit non-degenerate saddle-points, we show that for the proposed GenFlow algorithm, the time required to evade these saddle-points is bounded uniformly for all initial conditions. Finally, for strongly convex-strongly concave minimax problems whose optimal solution is a saddle point, a similar scheme is shown to arrive at the optimal solution again in a fixed-time. The superior convergence properties of our algorithm are validated experimentally on a variety of benchmark datasets.

We consider the problem of estimating the optimal transport map between a (fixed) source distribution $P$ and an unknown target distribution $Q$, based on samples from $Q$. The estimation of such optimal transport maps has become increasingly relevant in modern statistical applications, such as generative modeling. At present, estimation rates are only known in a few settings (e.g. when $P$ and $Q$ have densities bounded above and below and when the transport map lies in a H\"older class), which are often not reflected in practice. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure $P$ satisfies a Poincar\'e inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for bounded densities and H\"older transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when $P$ is the normal distribution and the transport map is given by an infinite-width shallow neural network.

In the framework of tensor spaces, we consider orthogonalization kernels to generate an orthogonal basis of a tensor subspace from a set of linearly independent tensors. In particular, we investigate numerically the loss of orthogonality of six orthogonalization methods, namely Classical and Modified Gram-Schmidt with (CGS2, MGS2) and without (CGS, MGS) re-orthogonalization, the Gram approach, and the Householder transformation. To tackle the curse of dimensionality, we represent tensor with low rank approximation using the Tensor Train (TT) formalism, and we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed accuracy. After describing the algorithm structure and properties, we illustrate numerically that the theoretical bounds for the loss of orthogonality in the classical matrix computation round-off analysis results are maintained, with the unit round-off replaced by the TT-rounding accuracy. The computational analysis for each orthogonalization kernel in terms of the memory requirement and the computational complexity measured as a function of the number of TT-rounding, which happens to be the computational most expensive operation, completes the study.

In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that $F(a_1,\ldots,a_k) = 0$. We consider a generalization of this problem in which $F$ is an arbitrary constant-degree polynomial, we are given $k$ sets of $n$ numbers, and have to determine whether there exist a $k$-tuple of numbers, one in each set, on which $F$ vanishes. We give the first improvement over the na\"ive $O^*(n^{k-1})$ algorithm for this problem (where the $O^*(\cdot)$ notation omits subpolynomial factors). We show that the problem can be solved in time $O^*\left( n^{k - 2 + \frac 4{k+2}}\right)$ for even $k$ and in time $O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right)$ for odd $k$ in the real RAM model of computation. We also prove that for $k=4$, the problem can be solved in time $O^*(n^{2.625})$ in the algebraic decision tree model, and for $k=5$ it can be solved in time $O^*(n^{3.56})$ in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for $k$-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft's point-line incidence detection problem in any dimension.

This paper considers the problem of kernel regression and classification with possibly unobservable response variables in the data, where the mechanism that causes the absence of information is unknown and can depend on both predictors and the response variables. Our proposed approach involves two steps: In the first step, we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. The main focus of the paper is to look into the theoretical properties of these estimators. The issue of identifiability is also addressed. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general Lp norms, where we also allow the size of the cover or subclass to diverge as the sample size n increases. These bounds immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of statistical classification based on the proposed regression estimators and also look into their rates of convergence under different settings. Although this work is mainly stated for kernel-type estimators, they can also be extended to other popular local-averaging methods such as nearest-neighbor estimators, and histogram estimators.

Game theory has by now found numerous applications in various fields, including economics, industry, jurisprudence, and artificial intelligence, where each player only cares about its own interest in a noncooperative or cooperative manner, but without obvious malice to other players. However, in many practical applications, such as poker, chess, evader pursuing, drug interdiction, coast guard, cyber-security, and national defense, players often have apparently adversarial stances, that is, selfish actions of each player inevitably or intentionally inflict loss or wreak havoc on other players. Along this line, this paper provides a systematic survey on three main game models widely employed in adversarial games, i.e., zero-sum normal-form and extensive-form games, Stackelberg (security) games, zero-sum differential games, from an array of perspectives, including basic knowledge of game models, (approximate) equilibrium concepts, problem classifications, research frontiers, (approximate) optimal strategy seeking techniques, prevailing algorithms, and practical applications. Finally, promising future research directions are also discussed for relevant adversarial games.

Games and simulators can be a valuable platform to execute complex multi-agent, multiplayer, imperfect information scenarios with significant parallels to military applications: multiple participants manage resources and make decisions that command assets to secure specific areas of a map or neutralize opposing forces. These characteristics have attracted the artificial intelligence (AI) community by supporting development of algorithms with complex benchmarks and the capability to rapidly iterate over new ideas. The success of artificial intelligence algorithms in real-time strategy games such as StarCraft II have also attracted the attention of the military research community aiming to explore similar techniques in military counterpart scenarios. Aiming to bridge the connection between games and military applications, this work discusses past and current efforts on how games and simulators, together with the artificial intelligence algorithms, have been adapted to simulate certain aspects of military missions and how they might impact the future battlefield. This paper also investigates how advances in virtual reality and visual augmentation systems open new possibilities in human interfaces with gaming platforms and their military parallels.