Theorem 1.1 (Informal; see Theorem 3.5) .

While hyperparameter tuning shows theoretical promise, its practical efficacy is not universally superior, as evidenced in Figure 4. The results thus offer a balanced perspective that marries theoretical rigor with empirical validation, fulfilling both academic and practical requirements. Implications for the Field of Machine Learning and Predictive Modeling Robustness and Generalization: The ensemble and stacking methods offer a mathematically substantiated pathway to improve the generalization capabilities of predictive models. Interpretability: The feature integration techniques not only improve model performance but also offer better interpretability by highlighting important features through mathematical formulations. Optimization: The proven convergence of Bayesian Optimization to the global optimum has far-reaching implications for hyperparameter tuning in models, as formalized by the EI equation. Unified Framework: This research provides a unified, mathematically rigorous framework for integrating Bayesian and non-Bayesian approaches, thereby setting a new benchmark for hybrid predictive systems. Future Research Directions Scalability: Investigating the scalability of the proposed methods, particularly in the context of the ensemble and Bayesian optimization equations, for larger datasets and more models. Real-world Applications: Extending this research to specific domains like healthcare, finance, and natural language processing to assess the practical utility of the proposed methods. Advanced Optimization Techniques: Exploring other optimization techniques that could further improve the efficiency and effectiveness of the proposed hybrid models, perhaps by introducing new mathematical formulations.

We study the piecewise stationary combinatorial semi-bandit problem with causally related rewards. In our nonstationary environment, variations in the base arms' distributions, causal relationships between rewards, or both, change the reward generation process. In such an environment, an optimal decision-maker must follow both sources of change and adapt accordingly. The problem becomes aggravated in the combinatorial semi-bandit setting, where the decision-maker only observes the outcome of the selected bundle of arms. The core of our proposed policy is the Upper Confidence Bound (UCB) algorithm. We assume the agent relies on an adaptive approach to overcome the challenge. More specifically, it employs a change-point detector based on the Generalized Likelihood Ratio (GLR) test. Besides, we introduce the notion of group restart as a new alternative restarting strategy in the decision making process in structured environments. Finally, our algorithm integrates a mechanism to trace the variations of the underlying graph structure, which captures the causal relationships between the rewards in the bandit setting. Theoretically, we establish a regret upper bound that reflects the effects of the number of structural- and distribution changes on the performance. The outcome of our numerical experiments in real-world scenarios exhibits applicability and superior performance of our proposal compared to the state-of-the-art benchmarks.

Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator learning neural network. It is designed to seed numerical solutions for a class of partial differential equations instead of a single partial differential equation, such as partial differential equations with different parameters or different initial conditions. The network consists of two sub-networks, the Branch net and the Trunk net. For an objective operator G, the Branch net encodes different input functions u at the same number of sensors, and the Trunk net evaluates the output function at any location. By minimizing the error between the evaluated output q and the expected output G(u)(y), DeepONet generates a good approximation of the operator G. In order to preserve essential physical properties of PDEs, such as the Energy Dissipation Law, we adopt a scalar auxiliary variable approach to generate the minimization problem. It introduces a modified energy and enables unconditional energy dissipation law at the discrete level. By taking the parameter as a function of time t, this network can predict the accurate solution at any further time with feeding data only at the initial state. The data needed can be generated by the initial conditions, which are readily available. In order to validate the accuracy and efficiency of our neural networks, we provide numerical simulations of several partial differential equations, including heat equations, parametric heat equations and Allen-Cahn equations.

Time-constrained decision processes have been ubiquitous in many fundamental applications in physics, biology and computer science. Recently, restart strategies have gained significant attention for boosting the efficiency of time-constrained processes by expediting the completion times. In this work, we investigate the bandit problem with controlled restarts for time-constrained decision processes, and develop provably good learning algorithms. In particular, we consider a bandit setting where each decision takes a random completion time, and yields a random and correlated reward at the end, with unknown values at the time of decision. The goal of the decision-maker is to maximize the expected total reward subject to a time constraint $\tau$. As an additional control, we allow the decision-maker to interrupt an ongoing task and forgo its reward for a potentially more rewarding alternative. For this problem, we develop efficient online learning algorithms with $O(\log(\tau))$ and $O(\sqrt{\tau\log(\tau)})$ regret in a finite and continuous action space of restart strategies, respectively. We demonstrate an applicability of our algorithm by using it to boost the performance of SAT solvers.

Search methods, such as Genetic Algorithms and Simulated Annealing are typically used to achieve the required scalability in challenging problems for which it is hard to find optimal, or even just "good enough' solutions. The majority of these methods involve steps where the state of the algorithm is modified in some way to escape a local optimum. The aim is to avoid premature convergence, which is when the search method converges (usually very early in the search) to a local optimum of poor quality [1, 2]. Previous research has shown that the performance of search strategies is affected by the structure of the fitness landscape [3, 4]. A fitness landscape is defined by three components: i) the search space, which is the set of all candidate solutions, ii) the fitness function, which assigns a fitness value to each solution, and the neighbourhood operator, which defines how solutions are connected, and as a results how the search strategy can traverse the landscape.

This work analyses the potential of restarts for probSAT, a quite successful algorithm for k-SAT, by estimating its runtime distributions on random 3-SAT instances that are close to the phase transition. We estimate an optimal restart time from empirical data, reaching a potential speedup factor of 1.39. Calculating restart times from fitted probability distributions reduces this factor to a maximum of 1.30. A spin-off result is that the Weibull distribution approximates the runtime distribution for over 93% of the used instances well. A machine learning pipeline is presented to compute a restart time for a fixed-cutoff strategy to exploit this potential. The main components of the pipeline are a random forest for determining the distribution type and a neural network for the distribution's parameters. ProbSAT performs statistically significantly better than Luby's restart strategy and the policy without restarts when using the presented approach. The structure is particularly advantageous on hard problems.

The tabu and restart are two fundamental strategies for local search. In this paper, we improve the local search algorithms for solving the Maximum Weight Clique (MWC) problem by introducing new tabu and restart strategies. Both the tabu and restart strategies proposed are based on the notion of a local search scenario, which involves not only a candidate solution but also the tabu status and unlocking relationship. Compared to the strategy of configuration checking, our tabu mechanism discourages forming a cycle of unlocking operations. Our new restart strategy is based on the re-occurrence of a local search scenario instead of that of a candidate solution. Experimental results show that the resulting MWC solver outperforms several state-of-the-art solvers on the DIMACS, BHOSLIB, and two benchmarks from practical applications.

A common strategy for improving optimization algorithms is to restart the algorithm when it is believed to be trapped in an inferior part of the search space. However, while specific restart strategies have been developed for specific problems (and specific algorithms), restarts are typically not regarded as a general tool to speed up an optimization algorithm. In fact, many optimization algorithms do not employ restarts at all. Recently, "bet-and-run" was introduced in the context of mixed-integer programming, where first a number of short runs with randomized initial conditions is made, and then the most promising run of these is continued. In this article, we consider two classical NP-complete combinatorial optimization problems, traveling salesperson and minimum vertex cover, and study the effectiveness of different bet-and-run strategies. In particular, our restart strategies do not take any problem knowledge into account, nor are tailored to the optimization algorithm. Therefore, they can be used off-the-shelf. We observe that state-of-the-art solvers for these problems can benefit significantly from restarts on standard benchmark instances.