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We consider the question of adaptive data analysis within the framework of convex optimization. We ask how many samples are needed in order to compute $\epsilon$-accurate estimates of $O(1/\epsilon^2)$ gradients queried by gradient descent, and we provide two intermediate answers to this question. First, we show that for a general analyst (not necessarily gradient descent) $\Omega(1/\epsilon^3)$ samples are required. This rules out the possibility of a foolproof mechanism. Our construction builds upon a new lower bound (that may be of interest of its own right) for an analyst that may ask several non adaptive questions in a batch of fixed and known $T$ rounds of adaptivity and requires a fraction of true discoveries. We show that for such an analyst $\Omega (\sqrt{T}/\epsilon^2)$ samples are necessary. Second, we show that, under certain assumptions on the oracle, in an interaction with gradient descent $\tilde \Omega(1/\epsilon^{2.5})$ samples are necessary. Our assumptions are that the oracle has only \emph{first order access} and is \emph{post-hoc generalizing}. First order access means that it can only compute the gradients of the sampled function at points queried by the algorithm. Our assumption of \emph{post-hoc generalization} follows from existing lower bounds for statistical queries. More generally then, we provide a generic reduction from the standard setting of statistical queries to the problem of estimating gradients queried by gradient descent. These results are in contrast with classical bounds that show that with $O(1/\epsilon^2)$ samples one can optimize the population risk to accuracy of $O(\epsilon)$ but, as it turns out, with spurious gradients.

Reinforcement Learning (RL) approaches are lately deployed for orchestrating wireless communications empowered by Reconfigurable Intelligent Surfaces (RISs), leveraging their online optimization capabilities. Most commonly, in RL-based formulations for realistic RISs with low resolution phase-tunable elements, each configuration is modeled as a distinct reflection action, resulting to inefficient exploration due to the exponential nature of the search space. In this paper, we consider RISs with 1-bit phase resolution elements, and model the action of each of them as a binary vector including the feasible reflection coefficients. We then introduce two variations of the well-established Deep Q-Network (DQN) and Deep Deterministic Policy Gradient (DDPG) agents, aiming for effective exploration of the binary action spaces. For the case of DQN, we make use of an efficient approximation of the Q-function, whereas a discretization post-processing step is applied to the output of DDPG. Our simulation results showcase that the proposed techniques greatly outperform the baseline in terms of the rate maximization objective, when large-scale RISs are considered. In addition, when dealing with moderate scale RIS sizes, where the conventional DQN based on configuration-based action spaces is feasible, the performance of the latter technique is similar to the proposed learning approach.

We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.

In this paper, we propose a novel design, called MixNN, for protecting deep learning model structure and parameters. The layers in a deep learning model of MixNN are fully decentralized. It hides communication address, layer parameters and operations, and forward as well as backward message flows among non-adjacent layers using the ideas from mix networks. MixNN has following advantages: 1) an adversary cannot fully control all layers of a model including the structure and parameters, 2) even some layers may collude but they cannot tamper with other honest layers, 3) model privacy is preserved in the training phase. We provide detailed descriptions for deployment. In one classification experiment, we compared a neural network deployed in a virtual machine with the same one using the MixNN design on the AWS EC2. The result shows that our MixNN retains less than 0.001 difference in terms of classification accuracy, while the whole running time of MixNN is about 7.5 times slower than the one running on a single virtual machine.

Music Structure Analysis (MSA) consists in segmenting a music piece in several distinct sections. We approach MSA within a compression framework, under the hypothesis that the structure is more easily revealed by a simplified representation of the original content of the song. More specifically, under the hypothesis that MSA is correlated with similarities occurring at the bar scale, this article introduces the use of linear and non-linear compression schemes on barwise audio signals. Compressed representations capture the most salient components of the different bars in the song and are then used to infer the song structure using a dynamic programming algorithm. This work explores both low-rank approximation models such as Principal Component Analysis or Nonnegative Matrix Factorization and "piece-specific" Auto-Encoding Neural Networks, with the objective to learn latent representations specific to a given song. Such approaches do not rely on supervision nor annotations, which are well-known to be tedious to collect and possibly ambiguous in MSA description. In our experiments, several unsupervised compression schemes achieve a level of performance comparable to that of state-of-the-art supervised methods (for 3s tolerance) on the RWC-Pop dataset, showcasing the importance of the barwise compression processing for MSA.

Dynamic Linear Models (DLMs) are commonly employed for time series analysis due to their versatile structure, simple recursive updating, ability to handle missing data, and probabilistic forecasting. However, the options for count time series are limited: Gaussian DLMs require continuous data, while Poisson-based alternatives often lack sufficient modeling flexibility. We introduce a novel semiparametric methodology for count time series by warping a Gaussian DLM. The warping function has two components: a (nonparametric) transformation operator that provides distributional flexibility and a rounding operator that ensures the correct support for the discrete data-generating process. We develop conjugate inference for the warped DLM, which enables analytic and recursive updates for the state space filtering and smoothing distributions. We leverage these results to produce customized and efficient algorithms for inference and forecasting, including Monte Carlo simulation for offline analysis and an optimal particle filter for online inference. This framework unifies and extends a variety of discrete time series models and is valid for natural counts, rounded values, and multivariate observations. Simulation studies illustrate the excellent forecasting capabilities of the warped DLM. The proposed approach is applied to a multivariate time series of daily overdose counts and demonstrates both modeling and computational successes.

Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving l_1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.

One of the most important problems in system identification and statistics is how to estimate the unknown parameters of a given model. Optimization methods and specialized procedures, such as Empirical Minimization (EM) can be used in case the likelihood function can be computed. For situations where one can only simulate from a parametric model, but the likelihood is difficult or impossible to evaluate, a technique known as the Two-Stage (TS) Approach can be applied to obtain reliable parametric estimates. Unfortunately, there is currently a lack of theoretical justification for TS. In this paper, we propose a statistical decision-theoretical derivation of TS, which leads to Bayesian and Minimax estimators. We also show how to apply the TS approach on models for independent and identically distributed samples, by computing quantiles of the data as a first step, and using a linear function as the second stage. The proposed method is illustrated via numerical simulations.

Multi-fidelity models are of great importance due to their capability of fusing information coming from different simulations and sensors. In the context of Gaussian process regression we can exploit low-fidelity models to better capture the latent manifold thus improving the accuracy of the model. We focus on the approximation of high-dimensional scalar functions with low intrinsic dimensionality. By introducing a low dimensional bias in a chain of Gaussian processes with different fidelities we can fight the curse of dimensionality affecting these kind of quantities of interest, especially for many-query applications. In particular we seek a gradient-based reduction of the parameter space through linear active subspaces or a nonlinear transformation of the input space. Then we build a low-fidelity response surface based on such reduction, thus enabling multi-fidelity Gaussian process regression without the need of running new simulations with simplified physical models. This has a great potential in the data scarcity regime affecting many engineering applications. In this work we present a new multi-fidelity approach -- starting from the preliminary analysis conducted in Romor et al. 2020 -- involving active subspaces and nonlinear level-set learning method. The proposed numerical method is tested on two high-dimensional benchmark functions, and on a more complex car aerodynamics problem. We show how a low intrinsic dimensionality bias can increase the accuracy of Gaussian process response surfaces.