Probabilistic machine learning provides a framework in which it is possible to reason about uncertainty for both models and predictions (Ghahramani, 2015). It is often argued that especially in high-stakes applications (healthcare, robotics, etc.), uncertainty estimates for decisions/predictions should be a central component and that they should be well-calibrated (Kuleshov and Deshpande, 2022). The intuition behind calibration is that the uncertainty estimates should accurately reflect the reality; for example if a classification model predicts 80% probability of belonging to class A on 10 datapoints, then (on average) we would expect 8 of those 10 samples actually belong to class A. Likewise - but less intuitively - in regression, if a calibrated model generates a prediction µ and uncertainty estimate σ, we would see p percent of the data lying inside a p percentile confidence interval of µ (Busk et al., 2021). In general, uncertainty can be divided into aleatoric (irreducible inherent randomness in the data-generating process) and epistemic (lack of knowledge, i.e. it can be reduced if more data is collected) (Hüllermeier and Waegeman, 2021). However, this distinction is rarely used when evaluating uncertainty estimates for regression tasks and although this has been critiqued (Sluijterman et al., 2021), it is highly non-trivial to achieve for real-world applications because it normally requires access to the underlying true function. Uncertainty also plays a central role Bayesian Optimization (BO) (Snoek et al., 2012), which will be the focus of this paper. As a sequential design strategy for global optimization, BO has several applications with perhaps the most popular ones being general experimental design (Shahriari et al., 2015) and model selection for machine learning

In this paper, we revisit the class of iterative shrinkage-thresholding algorithms (ISTA) for solving the linear inverse problem with sparse representation, which arises in signal and image processing. It is shown in the numerical experiment to deblur an image that the convergence behavior in the logarithmic-scale ordinate tends to be linear instead of logarithmic, approximating to be flat. Making meticulous observations, we find that the previous assumption for the smooth part to be convex weakens the least-square model. Specifically, assuming the smooth part to be strongly convex is more reasonable for the least-square model, even though the image matrix is probably ill-conditioned. Furthermore, we improve the pivotal inequality tighter for composite optimization with the smooth part to be strongly convex instead of general convex, which is first found in [Li et al., 2022]. Based on this pivotal inequality, we generalize the linear convergence to composite optimization in both the objective value and the squared proximal subgradient norm. Meanwhile, we set a simple ill-conditioned matrix which is easy to compute the singular values instead of the original blur matrix. The new numerical experiment shows the proximal generalization of Nesterov's accelerated gradient descent (NAG) for the strongly convex function has a faster linear convergence rate than ISTA. Based on the tighter pivotal inequality, we also generalize the faster linear convergence rate to composite optimization, in both the objective value and the squared proximal subgradient norm, by taking advantage of the well-constructed Lyapunov function with a slight modification and the phase-space representation based on the high-resolution differential equation framework from the implicit-velocity scheme.

We propose K-Deep Simplex (KDS) which, given a set of data points, learns a dictionary comprising synthetic landmarks, along with representation coefficients supported on a simplex. KDS integrates manifold learning and sparse coding/dictionary learning: reconstruction term, as in classical dictionary learning, and a novel local weighted $\ell_1$ penalty that encourages each data point to represent itself as a convex combination of nearby landmarks. We solve the proposed optimization program using alternating minimization and design an efficient, interpretable autoencoder using algorithm enrolling. We theoretically analyze the proposed program by relating the weighted $\ell_1$ penalty in KDS to a weighted $\ell_0$ program. Assuming that the data are generated from a Delaunay triangulation, we prove the equivalence of the weighted $\ell_1$ and weighted $\ell_0$ programs. If the representation coefficients are given, we prove that the resulting dictionary is unique. Further, we show that low-dimensional representations can be efficiently obtained from the covariance of the coefficient matrix. We apply KDS to the unsupervised clustering problem and prove theoretical performance guarantees. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.

Much recent research effort has been directed to the development of efficient algorithms for solving minimax problems with theoretical convergence guarantees due to the relevance of these problems to a few emergent applications. In this paper, we propose a unified single-loop alternating gradient projection (AGP) algorithm for solving smooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. AGP employs simple gradient projection steps for updating the primal and dual variables alternatively at each iteration. We show that it can find an $\varepsilon$-stationary point of the objective function in $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp. $\mathcal{O}\left( \varepsilon ^{-4} \right)$) iterations under nonconvex-strongly concave (resp. nonconvex-concave) setting. Moreover, its gradient complexity to obtain an $\varepsilon$-stationary point of the objective function is bounded by $\mathcal{O}\left( \varepsilon ^{-2} \right)$ (resp., $\mathcal{O}\left( \varepsilon ^{-4} \right)$) under the strongly convex-nonconcave (resp., convex-nonconcave) setting. To the best of our knowledge, this is the first time that a simple and unified single-loop algorithm is developed for solving both nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. Moreover, the complexity results for solving the latter (strongly) convex-nonconcave minimax problems have never been obtained before in the literature. Numerical results show the efficiency of the proposed AGP algorithm. Furthermore, we extend the AGP algorithm by presenting a block alternating proximal gradient (BAPG) algorithm for solving more general multi-block nonsmooth nonconvex-(strongly) concave and (strongly) convex-nonconcave minimax problems. We can similarly establish the gradient complexity of the proposed algorithm under these four different settings.

Univariate: When there is just one variable in the data and no mention of causes, effects, or causal links. For instance, the researcher may want to count the number of males and girls in a school when conducting a survey. The data, in this case, would just show a number, i.e., a variable and amount. The main goal of univariate data is to describe the data using mean, median, variance, mode, dispersion, range, standard deviation, etc. to identify patterns within the data. Bivariate: When the dataset contains two variables and researchers aim to undertake a comparison between the two datasets then the bivariate analysis is the right technique.

Stochastic Gradient Descent Langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numerical resolution of stochastic control problems through gradient descent, where the control is parametrized by a neural network. If the control is applied at many discretization times then solving the stochastic control problem reduces to minimizing the loss of a very deep neural network. We numerically show that Langevin algorithms improve the training on various stochastic control problems like hedging and resource management, and for different choices of gradient descent methods.

Polynomial chaos expansion (PCE) is a versatile tool widely used in uncertainty quantification and machine learning, but its successful application depends strongly on the accuracy and reliability of the resulting PCE-based response surface. High accuracy typically requires high polynomial degrees, demanding many training points especially in high-dimensional problems through the curse of dimensionality. So-called sparse PCE concepts work with a much smaller selection of basis polynomials compared to conventional PCE approaches and can overcome the curse of dimensionality very efficiently, but have to pay specific attention to their strategies of choosing training points. Furthermore, the approximation error resembles an uncertainty that most existing PCE-based methods do not estimate. In this study, we develop and evaluate a fully Bayesian approach to establish the PCE representation via joint shrinkage priors and Markov chain Monte Carlo. The suggested Bayesian PCE model directly aims to solve the two challenges named above: achieving a sparse PCE representation and estimating uncertainty of the PCE itself. The embedded Bayesian regularizing via the joint shrinkage prior allows using higher polynomial degrees for given training points due to its ability to handle underdetermined situations, where the number of considered PCE coefficients could be much larger than the number of available training points. We also explore multiple variable selection methods to construct sparse PCE expansions based on the established Bayesian representations, while globally selecting the most meaningful orthonormal polynomials given the available training data. We demonstrate the advantages of our Bayesian PCE and the corresponding sparsity-inducing methods on several benchmarks.

We introduce a class of variational actor-critic algorithms based on a variational formulation over both the value function and the policy. The objective function of the variational formulation consists of two parts: one for maximizing the value function and the other for minimizing the Bellman residual. Besides the vanilla gradient descent with both the value function and the policy updates, we propose two variants, the clipping method and the flipping method, in order to speed up the convergence. We also prove that, when the prefactor of the Bellman residual is sufficiently large, the fixed point of the algorithm is close to the optimal policy.

An individualized treatment regime (ITR) is a decision rule that assigns treatments based on patients' characteristics. The value function of an ITR is the expected outcome in a counterfactual world had this ITR been implemented. Recently, there has been increasing interest in combining heterogeneous data sources, such as leveraging the complementary features of randomized controlled trial (RCT) data and a large observational study (OS). Usually, a covariate shift exists between the source and target population, rendering the source-optimal ITR unnecessarily optimal for the target population. We present an efficient and robust transfer learning framework for estimating the optimal ITR with right-censored survival data that generalizes well to the target population. The value function accommodates a broad class of functionals of survival distributions, including survival probabilities and restrictive mean survival times (RMSTs). We propose a doubly robust estimator of the value function, and the optimal ITR is learned by maximizing the value function within a pre-specified class of ITRs. We establish the $N^{-1/3}$ rate of convergence for the estimated parameter indexing the optimal ITR, and show that the proposed optimal value estimator is consistent and asymptotically normal even with flexible machine learning methods for nuisance parameter estimation. We evaluate the empirical performance of the proposed method by simulation studies and a real data application of sodium bicarbonate therapy for patients with severe metabolic acidaemia in the intensive care unit (ICU), combining a RCT and an observational study with heterogeneity.

We present a novel scheme to boost detection power for kernel maximum mean discrepancy based sequential change-point detection procedures. Our proposed scheme features an optimal sub-sampling of the history data before the detection procedure, in order to tackle the power loss incurred by the random sub-sample from the enormous history data. We apply our proposed scheme to both Scan $B$ and Kernel Cumulative Sum (CUSUM) procedures, and improved performance is observed from extensive numerical experiments.