Constraint Optimization Problems (COP) are often considered without sufficient knowledge on the boundaries of the objective variable to optimize. When available, tight boundaries are helpful to prune the search space or estimate problem characteristics. Finding close boundaries, that correctly under- and overestimate the optimum, is almost impossible without actually solving the COP. This paper introduces Bion, a novel approach for boundary estimation by learning from previously solved instances of the COP. Based on supervised machine learning, Bion is problem-specific and solver-independent and can be applied to any COP which is repeatedly solved with different data inputs. An experimental evaluation over seven realistic COPs shows that an estimation model can be trained to prune the objective variables' domains by over 80%. By evaluating the estimated boundaries with various COP solvers, we find that Bion improves the solving process for some problems, although the effect of closer bounds is generally problem-dependent.

We consider an inverse reinforcement learning problem involving us versus an enemy radar equipped with a Bayesian tracker. By observing the emissions of the enemy radar,how can we identify if the radar is cognitive (constrained utility maximizer)? Given the observed sequence of actions taken by the enemy's radar, we consider three problems: (i) Are the enemy radar's actions (waveform choice, beam scheduling) consistent with constrained utility maximization? If so how can we estimate the cognitive radar's utility function that is consistent with its actions. We formulate and solve the problem in terms of the spectra (eigenvalues) of the state and observation noise covariance matrices, and the algebraic Riccati equation. (ii) How to construct a statistical test for detecting a cognitive radar (constrained utility maximization) when we observe the radar's actions in noise or the radar observes our probe signal in noise? We propose a statistical detector with a tight Type-II error bound. (iii) How can we optimally probe (interrogate) the enemy's radar by choosing our state to minimize the Type-II error of detecting if the radar is deploying an economic rational strategy, subject to a constraint on the Type-I detection error? We present a stochastic optimization algorithm to optimize our probe signal. The main analysis framework used in this paper is that of revealed preferences from microeconomics.

This paper proposes near-optimal algorithms for the pure-exploration linear bandit problem in the fixed confidence and fixed budget settings. Leveraging ideas from the theory of suprema of empirical processes, we provide an algorithm whose sample complexity scales with the geometry of the instance and avoids an explicit union bound over the number of arms. Unlike previous approaches which sample based on minimizing a worst-case variance (e.g. G-optimal design), we define an experimental design objective based on the Gaussian-width of the underlying arm set. We provide a novel lower bound in terms of this objective that highlights its fundamental role in the sample complexity. The sample complexity of our fixed confidence algorithm matches this lower bound, and in addition is computationally efficient for combinatorial classes, e.g. shortest-path, matchings and matroids, where the arm sets can be exponentially large in the dimension. Finally, we propose the first algorithm for linear bandits in the the fixed budget setting. Its guarantee matches our lower bound up to logarithmic factors.

Having a perfect model to compute the optimal policy is often infeasible in reinforcement learning. It is important in high-stakes domains to quantify and manage risk induced by model uncertainties. Entropic risk measure is an exponential utility-based convex risk measure that satisfies many reasonable properties. In this paper, we propose an entropic risk constrained policy gradient and actor-critic algorithms that are risk-averse to the model uncertainty. We demonstrate the usefulness of our algorithms on several problem domains.

We study the problem of efficient exact partitioning of the hypergraphs generated by high-order planted models. A high-order planted model assumes some underlying cluster structures, and simulates high-order interactions by placing hyperedges among nodes. Example models include the disjoint hypercliques, the densest subhypergraphs, and the hypergraph stochastic block models. We show that exact partitioning of high-order planted models (a NP-hard problem in general) is achievable through solving a computationally efficient convex optimization problem with a tensor nuclear norm constraint. Our analysis provides the conditions for our approach to succeed on recovering the true underlying cluster structures, with high probability.

Deep learning has achieved impressive successes on a number of domain applications, thanks largely to innovations on algorithmic and architectural design, and equally importantly to the tremendous amount of computational power one can harness through GPUs, computer clusters and dedicated software and hardware. Edge devices, such as smart phones, tablets, routers, car devices, home sensors, etc., due to their ubiquity and moderate computational power, impose new opportunities and challenges for deep learning. On the one hand, edge devices have direct access to privacy sensitive data that users may be reluctant to share (with say data centers), and they are much more powerful than their predecessors, capable of conducting a significant amount of on-device computations. On the other hand, edge devices are largely heterogeneous in terms of capacity, power, data, availability, communication, memory, etc., posing new challenges beyond conventional in-house training of machine learning models. Thus, a new paradigm, known as federated learning (FL) [1] that aims at harvesting the prospects of edge devices, has recently emerged. Developing new FL algorithms and systems on edge devices has since become a hot research topic in machine learning. From the beginning of its birth, FL has close ties to conventional distributed optimization. However, FL emerged from the pressing need to address news challenges in the mobile era that existing distributed optimization algorithms were not designed for per se. We mention the following characteristics ofFL that are most relevant to our work, and refer to the excellent surveys [2, 3, 4] and the references therein for more challenges and applications inFL.

In this paper, we propose a new end-to-end methodology to optimize the energy performance and the comfort, air quality and hygiene of large buildings. A metamodel based on a Transformer network is introduced and trained using a dataset sampled with a simulation program. Then, a few physical parameters and the building management system settings of this metamodel are calibrated using the CMA-ES optimization algorithm and real data obtained from sensors. Finally, the optimal settings to minimize the energy loads while maintaining a target thermal comfort and air quality are obtained using a multi-objective optimization procedure. The numerical experiments illustrate how this metamodel ensures a significant gain in energy efficiency while being computationally much more appealing than models requiring a huge number of physical parameters to be estimated.

Modern large-scale machine learning applications require stochastic optimization algorithms to be implemented on distributed compute systems. A key bottleneck of such systems is the communication overhead for exchanging information across the workers, such as stochastic gradients. Among the many techniques proposed to remedy this issue, one of the most successful is the framework of compressed communication with error feedback (EF). EF remains the only known technique that can deal with the error induced by contractive compressors which are not unbiased, such as Top-$K$. In this paper, we propose a new and theoretically and practically better alternative to EF for dealing with contractive compressors. In particular, we propose a construction which can transform any contractive compressor into an induced unbiased compressor. Following this transformation, existing methods able to work with unbiased compressors can be applied. We show that our approach leads to vast improvements over EF, including reduced memory requirements, better communication complexity guarantees and fewer assumptions. We further extend our results to federated learning with partial participation following an arbitrary distribution over the nodes, and demonstrate the benefits thereof. We perform several numerical experiments which validate our theoretical findings.

In this paper, we propose a framework to train a generative model for texture image synthesis from a single example. To do so, we exploit the local representation of images via the space of patches, that is, square sub-images of fixed size (e.g. 4 4). Our main contribution is to consider optimal transport to enforce the multiscale patch distribution of generated images, which leads to two different formulations. First, a pixel-based optimization method is proposed, relying on discrete optimal transport. We show that it is related to a well-known texture optimization framework based on iterated patch nearest-neighbor projections, while avoiding some of its shortcomings. Second, in a semi-discrete setting, we exploit the differential properties of Wasserstein distances to learn a fully convolutional network for texture generation. Once estimated, this network produces realistic and arbitrarily large texture samples in real time. The two formulations result in non-convex concave problems that can be optimized efficiently with convergence properties and improved stability compared to adversarial approaches, without relying on any regularization. By directly dealing with the patch distribution of synthesized images, we also overcome limitations of state-of-the art techniques, such as patch aggregation issues that usually lead to low frequency artifacts (e.g.

Sum-of-norms clustering is a clustering formulation based on convex optimization that automatically induces hierarchy. Multiple algorithms have been proposed to solve the optimization problem: subgradient descent by Hocking et al.\ \cite{hocking}, ADMM and ADA by Chi and Lange\ \cite{Chi}, stochastic incremental algorithm by Panahi et al.\ \cite{Panahi} and semismooth Newton-CG augmented Lagrangian method by Yuan et al.\ \cite{dsun1}. All algorithms yield approximate solutions, even though an exact solution is demanded to determine the correct cluster assignment. The purpose of this paper is to close the gap between the output from existing algorithms and the exact solution to the optimization problem. We present a clustering test which identifies and certifies the correct cluster assignment from an approximate solution yielded by any primal-dual algorithm. The test may not succeed if the approximation is inaccurate. However, we show the correct cluster assignment is guaranteed to be found by a symmetric primal-dual path following algorithm after sufficiently many iterations, provided that the model parameter $\lambda$ avoids a finite number of bad values. Numerical experiments are implemented to support our results.