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Approximate Bayesian inference methods provide a powerful suite of tools for finding approximations to intractable posterior distributions. However, machine learning applications typically involve selecting actions, which -- in a Bayesian setting -- depend on the posterior distribution only via its contribution to expected utility. A growing body of work on loss-calibrated approximate inference methods has therefore sought to develop posterior approximations sensitive to the influence of the utility function. Here we introduce loss-calibrated expectation propagation (Loss-EP), a loss-calibrated variant of expectation propagation. This method resembles standard EP with an additional factor that "tilts" the posterior towards higher-utility decisions. We show applications to Gaussian process classification under binary utility functions with asymmetric penalties on False Negative and False Positive errors, and show how this asymmetry can have dramatic consequences on what information is "useful" to capture in an approximation.

We derive novel error estimates for Hybrid High-Order (HHO) discretizations of Leray-Lions problems set in W^(1,p) with p in (1,2]. Specifically, we prove that, depending on the degeneracy of the problem, the convergence rate may vary between (k+1)(p-1) and (k+1), with k denoting the degree of the HHO approximation. These regime-dependent error estimates are illustrated by a complete panel of numerical experiments.

Nash equilibrium is a central concept in game theory. Several Nash solvers exist, yet none scale to normal-form games with many actions and many players, especially those with payoff tensors too big to be stored in memory. In this work, we propose an approach that iteratively improves an approximation to a Nash equilibrium through joint play. It accomplishes this by tracing a previously established homotopy that defines a continuum of equilibria for the game regularized with decaying levels of entropy. This continuum asymptotically approaches the limiting logit equilibrium, proven by McKelvey and Palfrey (1995) to be unique in almost all games, thereby partially circumventing the well-known equilibrium selection problem of many-player games. To encourage iterates to remain near this path, we efficiently minimize average deviation incentive via stochastic gradient descent, intelligently sampling entries in the payoff tensor as needed. Monte Carlo estimates of the stochastic gradient from joint play are biased due to the appearance of a nonlinear max operator in the objective, so we introduce additional innovations to the algorithm to alleviate gradient bias. The descent process can also be viewed as repeatedly constructing and reacting to a polymatrix approximation to the game. In these ways, our proposed approach, average deviation incentive descent with adaptive sampling (ADIDAS), is most similar to three classical approaches, namely homotopy-type, Lyapunov, and iterative polymatrix solvers. The lack of local convergence guarantees for biased gradient descent prevents guaranteed convergence to Nash, however, we demonstrate through extensive experiments the ability of this approach to approximate a unique Nash in normal-form games with as many as seven players and twenty one actions (several billion outcomes) that are orders of magnitude larger than those possible with prior algorithms.

In this article, we study the pointwise asymptotic behavior of iterated convolutions on Z. We generalize the so-called local limit theorem in probability theory to complex valued sequences. A sharp rate of convergence is proved together with a generalized Gaussian bound for the first corrector.

The goal of this paper is to reduce the total complexity of gradient-based methods for two classes of problems: affine-constrained composite convex optimization and bilinear saddle-point structured non-smooth convex optimization. Our technique is based on a double-loop inexact accelerated proximal gradient (APG) method for minimizing the summation of a non-smooth but proximable convex function and two smooth convex functions with different smoothness constants and computational costs. Compared to the standard APG method, the inexact APG method can reduce the total computation cost if one smooth component has higher computational cost but a smaller smoothness constant than the other. With this property, the inexact APG method can be applied to approximately solve the subproblems of a proximal augmented Lagrangian method for affine-constrained composite convex optimization and the smooth approximation for bilinear saddle-point structured non-smooth convex optimization, where the smooth function with a smaller smoothness constant has significantly higher computational cost. Thus it can reduce total complexity for finding an approximately optimal/stationary solution. This technique is similar to the gradient sliding technique in the literature. The difference is that our inexact APG method can efficiently stop the inner loop by using a computable condition based on a measure of stationarity violation, while the gradient sliding methods need to pre-specify the number of iterations for the inner loop. Numerical experiments demonstrate significantly higher efficiency of our methods over an optimal primal-dual first-order method and the gradient sliding methods.

Analyzing numerous or long time series is difficult in practice due to the high storage costs and computational requirements. Therefore, techniques have been proposed to generate compact similarity-preserving representations of time series, enabling real-time similarity search on large in-memory data collections. However, the existing techniques are not ideally suited for assessing similarity when sequences are locally out of phase. In this paper, we propose the use of product quantization for efficient similarity-based comparison of time series under time warping. The idea is to first compress the data by partitioning the time series into equal length sub-sequences which are represented by a short code. The distance between two time series can then be efficiently approximated by pre-computed elastic distances between their codes. The partitioning into sub-sequences forces unwanted alignments, which we address with a pre-alignment step using the maximal overlap discrete wavelet transform (MODWT). To demonstrate the efficiency and accuracy of our method, we perform an extensive experimental evaluation on benchmark datasets in nearest neighbors classification and clustering applications. Overall, the proposed solution emerges as a highly efficient (both in terms of memory usage and computation time) replacement for elastic measures in time series applications.

Safe operation of systems such as robots requires them to plan and execute trajectories subject to safety constraints. When those systems are subject to uncertainties in their dynamics, it is challenging to ensure that the constraints are not violated. In this paper, we propose Safe-CDDP, a safe trajectory optimization and control approach for systems under additive uncertainties and non-linear safety constraints based on constrained differential dynamic programming (DDP). The safety of the robot during its motion is formulated as chance constraints with user-chosen probabilities of constraint satisfaction. The chance constraints are transformed into deterministic ones in DDP formulation by constraint tightening. To avoid over-conservatism during constraint tightening, linear control gains of the feedback policy derived from the constrained DDP are used in the approximation of closed-loop uncertainty propagation in prediction. The proposed algorithm is empirically evaluated on three different robot dynamics with up to 12 degrees of freedom in simulation. The computational feasibility and applicability of the approach are demonstrated with a physical hardware implementation.

By executing offloaded tasks from mobile users, edge computing augments mobile user equipments (UEs) with computing/communications resources from edge nodes (ENs), enabling new services (e.g., real-time gaming). However, despite being more resourceful than UEs, allocating ENs' resources to a given favorable set of users (e.g., closer to ENs) may block other UEs from their services. This is often the case for most existing approaches that only aim to maximize the network social welfare or minimize the total energy consumption but do not consider the computing/battery status of each UE. This work develops an energy-based proportional-fair framework to serve all users with multiple tasks while considering both their service requirements and energy/battery levels in a multi-layer edge network. The resulting problem for offloading tasks and allocating resources toward the tasks is a Mixed-Integer Nonlinear Programming, which is NP-hard. To tackle it, we leverage the fact that the relaxed problem is convex and propose a distributed algorithm, namely the dynamic branch-and-bound Benders decomposition (DBBD). DBBD decomposes the original problem into a master problem (MP) for the offloading decisions and multiple subproblems (SPs) for resource allocation. To quickly eliminate inefficient offloading solutions, MP is integrated with powerful Benders cuts exploiting the ENs' resource constraints. We then develop a dynamic branch-and-bound algorithm (DBB) to efficiently solve MP considering the load balance among ENs. SPs can either be solved for their closed-form solutions or be solved in parallel at ENs, thus reducing the complexity. The numerical results show that DBBD returns the optimal solution in maximizing the proportional fairness among UEs. DBBD has higher fairness indexes, i.e., Jain's index and min-max ratio, in comparison with the existing ones that minimize the total consumed energy.

Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.