We propose a novel algorithm for multi-player multi-armed bandits without collision sensing information. Our algorithm circumvents two problems shared by all state-of-the-art algorithms: it does not need as an input a lower bound on the minimal expected reward of an arm, and its performance does not scale inversely proportionally to the minimal expected reward. We prove a theoretical regret upper bound to justify these claims. We complement our theoretical results with numerical experiments, showing that the proposed algorithm outperforms state-of-the-art in practice as well.
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Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the finite element method. By decomposing the error estimates into contributions from individual elements, it is possible to formulate adaptation methods, which modify the mesh with the objective of minimising the resulting QoI error. However, the standard error estimate formulation involves the true adjoint solution, which is unknown in practice. As such, it is common practice to approximate it with an 'enriched' approximation (e.g. in a higher order space or on a refined mesh). Doing so generally results in a significant increase in computational cost, which can be a bottleneck compromising the competitiveness of (goal-oriented) adaptive simulations. The central idea of this paper is to develop a "data-driven" goal-oriented mesh adaptation approach through the selective replacement of the expensive error estimation step with an appropriately configured and trained neural network. In doing so, the error estimator may be obtained without even constructing the enriched spaces. An element-by-element construction is employed here, whereby local values of various parameters related to the mesh geometry and underlying problem physics are taken as inputs, and the corresponding contribution to the error estimator is taken as output. We demonstrate that this approach is able to obtain the same accuracy with a reduced computational cost, for adaptive mesh test cases related to flow around tidal turbines, which interact via their downstream wakes, and where the overall power output of the farm is taken as the QoI. Moreover, we demonstrate that the element-by-element approach implies reasonably low training costs.
We present regret minimization algorithms for stochastic contextual MDPs under minimum reachability assumption, using an access to an offline least square regression oracle. We analyze three different settings: where the dynamics is known, where the dynamics is unknown but independent of the context and the most challenging setting where the dynamics is unknown and context-dependent. For the latter, our algorithm obtains $ \tilde{O}\left( \max\{H,{1}/{p_{min}}\}H|S|^{3/2}\sqrt{|A|T\log(\max\{|\mathcal{F}|,|\mathcal{P}|\}/\delta)} \right)$ regret bound, with probability $1-\delta$, where $\mathcal{P}$ and $\mathcal{F}$ are finite and realizable function classes used to approximate the dynamics and rewards respectively, $p_{min}$ is the minimum reachability parameter, $S$ is the set of states, $A$ the set of actions, $H$ the horizon, and $T$ the number of episodes. To our knowledge, our approach is the first optimistic approach applied to contextual MDPs with general function approximation (i.e., without additional knowledge regarding the function class, such as it being linear and etc.). In addition, we present a lower bound of $\Omega(\sqrt{T H |S| |A| \ln(|\mathcal{F}|/|S|)/\ln(|A|)})$, on the expected regret which holds even in the case of known dynamics.
We present a generic framework for scale-aware direct monocular odometry based on depth prediction from a deep neural network. In contrast with previous methods where depth information is only partially exploited, we formulate a novel depth prediction residual which allows us to incorporate multi-view depth information. In addition, we propose to use a truncated robust cost function which prevents considering inconsistent depth estimations. The photometric and depth-prediction measurements are integrated into a tightly-coupled optimization leading to a scale-aware monocular system which does not accumulate scale drift. Our proposal does not particularize for a concrete neural network, being able to work along with the vast majority of the existing depth prediction solutions. We demonstrate the validity and generality of our proposal evaluating it on the KITTI odometry dataset, using two publicly available neural networks and comparing it with similar approaches and the state-of-the-art for monocular and stereo SLAM. Experiments show that our proposal largely outperforms classic monocular SLAM, being 5 to 9 times more precise, beating similar approaches and having an accuracy which is closer to that of stereo systems.
In this paper, we study a sequential decision making problem faced by e-commerce carriers related to when to send out a vehicle from the central depot to serve customer requests, and in which order to provide the service, under the assumption that the time at which parcels arrive at the depot is stochastic and dynamic. The objective is to maximize the number of parcels that can be delivered during the service hours. We propose two reinforcement learning approaches for solving this problem, one based on a policy function approximation (PFA) and the second on a value function approximation (VFA). Both methods are combined with a look-ahead strategy, in which future release dates are sampled in a Monte-Carlo fashion and a tailored batch approach is used to approximate the value of future states. Our PFA and VFA make a good use of branch-and-cut-based exact methods to improve the quality of decisions. We also establish sufficient conditions for partial characterization of optimal policy and integrate them into PFA/VFA. In an empirical study based on 720 benchmark instances, we conduct a competitive analysis using upper bounds with perfect information and we show that PFA and VFA greatly outperform two alternative myopic approaches. Overall, PFA provides best solutions, while VFA (which benefits from a two-stage stochastic optimization model) achieves a better tradeoff between solution quality and computing time.
Suppose we are given integer $k \leq n$ and $n$ boxes labeled $1,\ldots, n$ by an adversary, each containing a number chosen from an unknown distribution. We have to choose an order to sequentially open these boxes, and each time we open the next box in this order, we learn its number. If we reject a number in a box, the box cannot be recalled. Our goal is to accept the $k$ largest of these numbers, without necessarily opening all boxes. This is the free order multiple-choice secretary problem. Free order variants were studied extensively for the secretary and prophet problems. Kesselheim, Kleinberg, and Niazadeh KKN (STOC'15) initiated a study of randomness-efficient algorithms (with the cheapest order in terms of used random bits) for the free order secretary problems. We present an algorithm for free order multiple-choice secretary, which is simultaneously optimal for the competitive ratio and used amount of randomness. I.e., we construct a distribution on orders with optimal entropy $\Theta(\log\log n)$ such that a deterministic multiple-threshold algorithm is $1-O(\sqrt{\log k/k})$-competitive. This improves in three ways the previous best construction by KKN, whose competitive ratio is $1 - O(1/k^{1/3}) - o(1)$. Our competitive ratio is (near)optimal for the multiple-choice secretary problem; it works for exponentially larger parameter $k$; and our algorithm is a simple deterministic multiple-threshold algorithm, while that in KKN is randomized. We also prove a corresponding lower bound on the entropy of optimal solutions for the multiple-choice secretary problem, matching entropy of our algorithm, where no such previous lower bound was known. We obtain our algorithmic results with a host of new techniques, and with these techniques we also improve significantly the previous results of KKN about constructing entropy-optimal distributions for the classic free order secretary.
When learning disconnected distributions, Generative adversarial networks (GANs) are known to face model misspecification. Indeed, a continuous mapping from a unimodal latent distribution to a disconnected one is impossible, so GANs necessarily generate samples outside of the support of the target distribution. This raises a fundamental question: what is the latent space partition that minimizes the measure of these areas? Building on a recent result of geometric measure theory, we prove that an optimal GANs must structure its latent space as a 'simplicial cluster' - a Voronoi partition where cells are convex cones - when the dimension of the latent space is larger than the number of modes. In this configuration, each Voronoi cell maps to a distinct mode of the data. We derive both an upper and a lower bound on the optimal precision of GANs learning disconnected manifolds. Interestingly, these two bounds have the same order of decrease: $\sqrt{\log m}$, $m$ being the number of modes. Finally, we perform several experiments to exhibit the geometry of the latent space and experimentally show that GANs have a geometry with similar properties to the theoretical one.
We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ smooth functions with $s >0$ and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression.
Federated learning (FL) provides an effective paradigm to train machine learning models over distributed data with privacy protection. However, recent studies show that FL is subject to various security, privacy, and fairness threats due to the potentially malicious and heterogeneous local agents. For instance, it is vulnerable to local adversarial agents who only contribute low-quality data, with the goal of harming the performance of those with high-quality data. This kind of attack hence breaks existing definitions of fairness in FL that mainly focus on a certain notion of performance parity. In this work, we aim to address this limitation and propose a formal definition of fairness via agent-awareness for FL (FAA), which takes the heterogeneous data contributions of local agents into account. In addition, we propose a fair FL training algorithm based on agent clustering (FOCUS) to achieve FAA. Theoretically, we prove the convergence and optimality of FOCUS under mild conditions for linear models and general convex loss functions with bounded smoothness. We also prove that FOCUS always achieves higher fairness measured by FAA compared with standard FedAvg protocol under both linear models and general convex loss functions. Empirically, we evaluate FOCUS on four datasets, including synthetic data, images, and texts under different settings, and we show that FOCUS achieves significantly higher fairness based on FAA while maintaining similar or even higher prediction accuracy compared with FedAvg.
The Laplace approximation (LA) has been proposed as a method for approximating the marginal likelihood of statistical models with latent variables. However, the approximate maximum likelihood estimators (MLEs) based on the LA are often biased for binary or spatial data, and the corresponding Hessian matrix underestimates the standard errors of these approximate MLEs. A higher-order approximation has been proposed; however, it cannot be applied to complicated models such as correlated random effects models and does not provide consistent variance estimators. In this paper, we propose an enhanced LA (ELA) that provides the true MLE and its consistent variance estimator. We study its relationship to the variational Bayes method. We also introduce a new restricted maximum likelihood estimator (REMLE) for estimating dispersion parameters. The results of numerical studies show that the ELA provides a satisfactory MLE and REMLE, as well as their variance estimators for fixed parameters. The MLE and REMLE can be viewed as posterior mode and marginal posterior mode under flat priors, respectively. Some comparisons are also made with Bayesian procedures under different priors.
Deep operator learning has emerged as a promising tool for reduced-order modelling and PDE model discovery. Leveraging the expressive power of deep neural networks, especially in high dimensions, such methods learn the mapping between functional state variables. While proposed methods have assumed noise only in the dependent variables, experimental and numerical data for operator learning typically exhibit noise in the independent variables as well, since both variables represent signals that are subject to measurement error. In regression on scalar data, failure to account for noisy independent variables can lead to biased parameter estimates. With noisy independent variables, linear models fitted via ordinary least squares (OLS) will show attenuation bias, wherein the slope will be underestimated. In this work, we derive an analogue of attenuation bias for linear operator regression with white noise in both the independent and dependent variables. In the nonlinear setting, we computationally demonstrate underprediction of the action of the Burgers operator in the presence of noise in the independent variable. We propose error-in-variables (EiV) models for two operator regression methods, MOR-Physics and DeepONet, and demonstrate that these new models reduce bias in the presence of noisy independent variables for a variety of operator learning problems. Considering the Burgers operator in 1D and 2D, we demonstrate that EiV operator learning robustly recovers operators in high-noise regimes that defeat OLS operator learning. We also introduce an EiV model for time-evolving PDE discovery and show that OLS and EiV perform similarly in learning the Kuramoto-Sivashinsky evolution operator from corrupted data, suggesting that the effect of bias in OLS operator learning depends on the regularity of the target operator.
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