On some deterministic properties of stochastic gradient descent and random projection sequences

This thesis investigates deterministic properties of random algorithms, focusing on boundedness and a property related to the convergence rate of sequences generated by stochastic gradient and random projection methods in both finite and infinite-dimensional Hilbert spaces. We begin by revisiting classical stochastic gradient descent methods and show that the sequence of iterates remains bounded under the mild assumption of coercivity, generalizing previous results that required strong convexity of the component functions. We then extend boundedness results for random projection algorithms, originally established by Meshulam for affine subspaces in ℝⁿ, to polyhedral sets in infinite-dimensional Hilbert spaces. We also note that certain assumptions are necessary to guarantee boundedness for affine subspaces in infinite-dimensional spaces. To address this problem, we introduce the so-called innate regularity assumption and prove that it ensures boundedness of the iterates. Finally, we generalize the Güntürk-Thao theorem, which characterizes summability properties of successive differences for sequences projected onto a finite collection of innately regular closed subspaces, to the case of polyhedral cones. We provide examples illustrating the sharpness of these results, demonstrating the importance of the innate regularity and the polyhedral cone assumption.