This paper develops and analyzes a stochastic derivative-free optimization strategy. A key feature is the state-dependent adaptive variance. We prove global convergence in probability with algebraic rate and give the quantitative results in numerical examples. A striking fact is that convergence is achieved without explicit information of the gradient and even without comparing different objective function values as in established methods such as the simplex method and simulated annealing. It can otherwise be compared to annealing with state-dependent temperature.

While most methods for solving mixed-integer optimization problems compute a single optimal solution, a diverse set of near-optimal solutions can often lead to improved outcomes. We present a new method for finding a set of diverse solutions by emphasizing diversity within the search for near-optimal solutions. Specifically, within a branch-and-bound framework, we investigated parameterized node selection rules that explicitly consider diversity. Our results indicate that our approach significantly increases the diversity of the final solution set. When compared with two existing methods, our method runs with similar runtime as regular node selection methods and gives a diversity improvement between 12% and 190%. In contrast, popular node selection rules, such as best-first search, in some instances performed worse than state-of-the-art methods by more than 35% and gave an improvement of no more than 130%. Further, we find that our method is most effective when diversity in node selection is continuously emphasized after reaching a minimal depth in the tree and when the solution set has grown sufficiently large. Our method can be easily incorporated into integer programming solvers and has the potential to significantly increase the diversity of solution sets.

Offline reinforcement learning (RL) aims to find an optimal policy for sequential decision-making using a pre-collected dataset, without further interaction with the environment. Recent theoretical progress has focused on developing sample-efficient offline RL algorithms with various relaxed assumptions on data coverage and function approximators, especially to handle the case with excessively large state-action spaces. Among them, the framework based on the linear-programming (LP) reformulation of Markov decision processes has shown promise: it enables sample-efficient offline RL with function approximation, under only partial data coverage and realizability assumptions on the function classes, with favorable computational tractability. In this work, we revisit the LP framework for offline RL, and provide a new reformulation that advances the existing results in several aspects, relaxing certain assumptions and achieving optimal statistical rates in terms of sample size. Our key enabler is to introduce proper constraints in the reformulation, instead of using any regularization as in the literature, also with careful choices of the function classes and initial state distributions. We hope our insights bring into light the use of LP formulations and the induced primal-dual minimax optimization, in offline RL.

The Kullback-Leibler (KL) divergence is widely used in state-of-the-art Bayesian Neural Networks (BNNs) to approximate the posterior distribution of weights. However, the KL divergence is unbounded and asymmetric, which may lead to instabilities during optimization or may yield poor generalizations. To overcome these limitations, we examine the Jensen-Shannon (JS) divergence that is bounded, symmetric, and more general. Towards this, we propose two novel loss functions for BNNs. The first loss function uses the geometric JS divergence (JS-G) that is symmetric, unbounded, and offers an analytical expression for Gaussian priors. The second loss function uses the generalized JS divergence (JS-A) that is symmetric and bounded. We show that the conventional KL divergence-based loss function is a special case of the two loss functions presented in this work. To evaluate the divergence part of the loss we use analytical expressions for JS-G and use Monte Carlo methods for JS-A. We provide algorithms to optimize the loss function using both these methods. The proposed loss functions offer additional parameters that can be tuned to control the degree of regularisation. The regularization performance of the JS divergences is analyzed to demonstrate their superiority over the state-of-the-art. Further, we derive the conditions for better regularization by the proposed JS-G divergence-based loss function than the KL divergence-based loss function. Bayesian convolutional neural networks (BCNN) based on the proposed JS divergences perform better than the state-of-the-art BCNN, which is shown for the classification of the CIFAR data set having various degrees of noise and a histopathology data set having a high bias.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Selecting period values for tasks is a very important step in the design process of a real-time system, especially due to the significance of its impact on system schedulability. It is well known that, under RMS, the utilization bound for a harmonic task set is 100%. Also, polynomial-time algorithms have been developed for response-time analysis of harmonic task sets. In practice, the largest acceptable value for the period of a task is determined by the performance and safety requirements of the application. In this paper, we address the problem of assigning harmonic periods to a task set such that every task gets assigned an integer period less than or equal to its application specified upper bound and the task utilization of every task is less than 1. We focus on integer solutions given the discrete nature of time in real-time computer systems. We first express this problem of assigning harmonic periods to a task set as a discrete piecewise optimization problem. We then present the 'Discrete Piecewise Harmonic Search' (DPHS) algorithm that outputs an optimal harmonic task assignment. We then define conditions for a metric to be rational for harmonization. We show that commonly used metrics like, the total percentage error (TPE), total system utilization (TSU), first order error (FOE), and maximum percentage error (MPE), are rational. We next prove that the DPHS algorithm finds the optimal feasible assignment, if one exists, for these rational metrics. We apply the DPHS algorithm to harmonize task sets used in real-world applications to highlight its benefits. We compare the performance of the DPHS algorithm against a brute-force search and find that the DPHS searches up to 94\% fewer task sets than the brute-force search that obtains the optimal solution.

How should we leverage such an extra modality of feedback along with the typical reward signal in CBs? We study contextual bandit (CB) problems, While prior works have developed hybrid models such as where the user can sometimes respond with the learning with feedback graphs (e.g., (Mannor & Shamir, best action in a given context. Such an interaction 2011; Caron et al., 2012; Alon et al., 2017)) to capture a arises, for example, in text prediction or autocompletion continuum between supervised and CB learning, such settings settings, where a poor suggestion is simply are not a natural fit here. A key challenge in the ignored and the user enters the desired text feedback structure is that the extra supervised signal is only instead. Crucially, this extra feedback is usertriggered available on a subset of the contexts, which are chosen by on only a subset of the contexts. We develop the user as some unknown function of the algorithm's recommended a new framework to leverage such signals,

We consider a class of assortment optimization problems in an offline data-driven setting. A firm does not know the underlying customer choice model but has access to an offline dataset consisting of the historically offered assortment set, customer choice, and revenue. The objective is to use the offline dataset to find an optimal assortment. Due to the combinatorial nature of assortment optimization, the problem of insufficient data coverage is likely to occur in the offline dataset. Therefore, designing a provably efficient offline learning algorithm becomes a significant challenge. To this end, we propose an algorithm referred to as Pessimistic ASsortment opTimizAtion (PASTA for short) designed based on the principle of pessimism, that can correctly identify the optimal assortment by only requiring the offline data to cover the optimal assortment under general settings. In particular, we establish a regret bound for the offline assortment optimization problem under the celebrated multinomial logit model. We also propose an efficient computational procedure to solve our pessimistic assortment optimization problem. Numerical studies demonstrate the superiority of the proposed method over the existing baseline method.

Electric city bus gains popularity in recent years for its low greenhouse gas emission, low noise level, etc. Different from a passenger car, the weight of a city bus varies significantly with different amounts of onboard passengers. After analyzing the importance of battery aging and passenger load effects on an optimal energy management strategy, this study introduces the passenger load prediction into the hybrid-electric city buses energy management problem, which is not well studied in the existing literature. The average model, Decision Tree, Gradient Boost Decision Tree, and Neural Networks models are compared in the passenger load prediction. The Gradient Boost Decision Tree model is selected due to its best accuracy and high stability. Given the predicted passenger load, a dynamic programming algorithm determines the optimal power demand for supercapacitor and battery by optimizing the battery aging and energy usage leveraging cloud techniques. Then, rule extraction is conducted on dynamic programming results, and the rule is real-time loaded to the vehicle onboard controller to handle prediction errors and uncertainties. The proposed cloud-based Dynamic Programming and rule extraction framework with the passenger load prediction show 4% and 11% lower bus operating costs in off-peak and peak hours, respectively. The operating cost by the proposed framework is less than 1% of the dynamic programming with the true passenger load information.

There is a lack of consensus within the literature as to how `fairness' of algorithmic systems can be measured, and different metrics can often be at odds. In this paper, we approach this task by drawing on the ethical frameworks of utilitarianism and John Rawls. Informally, these two theories of distributive justice measure the `good' as either a population's sum of utility, or worst-off outcomes, respectively. We present a parameterized class of objective functions that interpolates between these two (possibly) conflicting notions of the `good'. This class is shown to represent a relaxation of the Rawlsian `veil of ignorance', and its sequence of optimal solutions converges to both a utilitarian and Rawlsian optimum. Several other properties of this class are studied, including: 1) a relationship to regularized optimization, 2) feasibility of consistent estimation, and 3) algorithmic cost. In several real-world datasets, we compute optimal solutions and construct the tradeoff between utilitarian and Rawlsian notions of the `good'. Empirically, we demonstrate that increasing model complexity can manifest strict improvements to both measures of the `good'. This work suggests that the proper degree of `fairness' can be informed by a designer's preferences over the space of induced utilitarian and Rawlsian `good'.