Previous work has focused mainly on bounding either the expected loss of a predictor or the probability that an individual prediction will incur a loss value in a specified range.

The differential equation (DE) approach for convex optimization, which relates optimization methods to specific continuous DEs with rate-revealing Lyapunov functionals, has gained increasing interest since the seminal paper by Su-Boyd-Candès (2014).

Müller et al. [2021], Nguyen et al. [2022] builds a GP model for both the function value and the

Bayesian Optimization with multi-objective acquisition functions such as q-Expected Hypervolume Improvement (qEHVI) requires efficient candidate optimization to maximize acquisition function values. Traditional approaches rely on continuous optimization methods like Sequential Least Squares Programming (SLSQP) for candidate selection. However, these gradient-based methods can become trapped in local optima, particularly in complex or high-dimensional objective landscapes. This paper presents a simulated annealing-based approach for candidate optimization in batch acquisition functions as an alternative to conventional continuous optimization methods. We evaluate our simulated annealing approach against SLSQP across four benchmark multi-objective optimization problems: ZDT1 (30D, 2 objectives), DTLZ2 (7D, 3 objectives), Kursawe (3D, 2 objectives), and Latent-Aware (4D, 2 objectives). Our results demonstrate that simulated annealing consistently achieves superior hypervolume performance compared to SLSQP in most test functions. The improvement is particularly pronounced for DTLZ2 and Latent-Aware problems, where simulated annealing reaches significantly higher hypervolume values and maintains better convergence characteristics. The histogram analysis of objective space coverage further reveals that simulated annealing explores more diverse and optimal regions of the Pareto front. These findings suggest that metaheuristic optimization approaches like simulated annealing can provide more robust and effective candidate optimization for multi-objective Bayesian optimization, offering a promising alternative to traditional gradient-based methods for batch acquisition function optimization.

Artificial intelligence has advanced rapidly through large neural networks trained on massive datasets using thousands of GPUs or TPUs. Such training can occupy entire data centers for weeks and requires enormous computational and energy resources. Yet the optimization algorithms behind these runs have not kept pace. Most large scale training still relies on synchronous methods, where workers must wait for the slowest device, wasting compute and amplifying the effects of hardware and network variability. Removing synchronization seems like a simple fix, but asynchrony introduces staleness, meaning updates computed on outdated models. This makes analysis difficult, especially when delays arise from system level randomness rather than algorithmic choices. As a result, the time complexity of asynchronous methods remains poorly understood. This dissertation develops a rigorous framework for asynchronous first order stochastic optimization, focusing on the core challenge of heterogeneous worker speeds. Within this framework, we show that with proper design, asynchronous SGD can achieve optimal time complexity, matching guarantees previously known only for synchronous methods. Our first contribution, Ringmaster ASGD, attains optimal time complexity in the homogeneous data setting by selectively discarding stale updates. The second, Ringleader ASGD, extends optimality to heterogeneous data, common in federated learning, using a structured gradient table mechanism. Finally, ATA improves resource efficiency by learning worker compute time distributions and allocating tasks adaptively, achieving near optimal wall clock time with less computation. Together, these results establish asynchronous optimization as a theoretically sound and practically efficient foundation for distributed learning, showing that coordination without synchronization can be both feasible and optimal.