  First-order optimization (FOO) algorithms are pivotal in numerous
computational domains such as machine learning and signal denoising. However,
their application to complex tasks like neural network training often entails
significant inefficiencies due to the need for many sequential iterations for
convergence. In response, we introduce first-order optimization expedited with
approximately parallelized iterations (OptEx), the first framework that
enhances the efficiency of FOO by leveraging parallel computing to mitigate its
iterative bottleneck. OptEx employs kernelized gradient estimation to make use
of gradient history for future gradient prediction, enabling parallelization of
iterations -- a strategy once considered impractical because of the inherent
iterative dependency in FOO. We provide theoretical guarantees for the
reliability of our kernelized gradient estimation and the iteration complexity
of SGD-based OptEx, confirming that estimation errors diminish to zero as
historical gradients accumulate and that SGD-based OptEx enjoys an effective
acceleration rate of $\Omega(\sqrt{N})$ over standard SGD given parallelism of
N. We also use extensive empirical studies, including synthetic functions,
reinforcement learning tasks, and neural network training across various
datasets, to underscore the substantial efficiency improvements achieved by
OptEx.
