One of the challenges in analyzing a learning algorithm is the circular entanglement between the objective value and the stochastic noise. This is also known as the "chicken and egg" phenomenon. Traditionally, people tackle this issue with the special structure of the problem and hence the analysis is difficult to generalize. In this paper, we present a general framework for analyzing high-probability bounds for stochastic dynamics in learning algorithms. Our framework composes standard techniques from probability theory to give a streamlined three-step recipe with a general and flexible principle to tackle the "chicken and egg" problem. We demonstrate the power and the flexibility of our framework by giving unifying analysis for three very different learning problems with both the last iterate and the strong uniform high probability convergence guarantee. The problems are stochastic gradient descent for strongly convex functions, streaming principal component analysis and linear bandit with stochastic gradient descent updates. We either improve or match the state-of-the-art bounds on all three dynamics.
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