Abstract
A correction regarding [Latz 2021, Stat. Comput. 31, 39].
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In Latz (2021), the space of continuous paths \(C^0([0, \infty ); X)\) needs to be equipped with the metric given by
rather than the supremum norm \(\Vert x\Vert _\infty = \sup _{t \in [0,\infty )}\Vert x(t)\Vert \), see Kushner (1984). This has the following implications:
(1) The convergence results in Theorem 1 and Lemma 2 are now weaker, as the metric \(\rho \) is weaker than the metric induced by \(\Vert \cdot \Vert _\infty \).
(2) Proposition 4(i) does not hold uniformly in time. The function F in the proof of this proposition can be chosen as \(F((\xi (t), \tau (t), {\varvec{j}}(t))_{t \ge 0},t):= \min \{1, \Vert \xi (t)\Vert \}2^{-(t+1)}\) – it now depends on t. This F is bounded and Lipschitz continuous in \((\xi (t))_{t \ge 0}\) with respect to \(\rho \), continuous in \(t>0\) and constant with respect to the other inputs. To show Lipschitz continuity, we observe
for \(t > 0\), \((\xi (t))_{t \ge 0}, (\xi '(t))_{t \ge 0}, (\tau (t))_{t \ge 0}, (\tau '(t))_{t \ge 0} \in C^0([0, \infty ); X)\) and \(({\varvec{j}}(t))_{t \ge 0}, ({\varvec{j}}'(t))_{t \ge 0}\) being Markov jump processes on I. Thus, the weak convergence shown in Lemma 2 now implies that
and the inequality in Proposition 4(i) now reads
with \(\alpha (0,t) = 0\) and \(\varepsilon \mapsto \alpha (\varepsilon , t)\) is continuous at 0 for any \(t>0\).
We also note that this \(\alpha '\) can depend on the initial values \(\xi _0\) and \(j_0\) and that the definition of \(t_0\) in this proposition is not necessary.
(3) Theorem 4 is still correct as stated with a refined argument in the proof. In the last inequality of the proof of Theorem 4, we can choose \(\varepsilon _t\) to depend on t. The continuity of \(\alpha '\) and \(\alpha ''\) implies that for any \(\delta > 0\), we can find \(\varepsilon _t > 0\) small enough such that \(\alpha '(\varepsilon _t,t) + \alpha ''(\varepsilon _t) \le \delta \) for all \(t > 0\), which is sufficient to prove convergence as stated in Theorem 4.
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References
Kushner, H.J.: Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory. MIT Press (1984)
Latz, J.: Analysis of stochastic gradient descent in continuous time. Stat. Comput. 31, 39 (2021)
Acknowledgements
JL thanks Chenguang Liu for helpful discussions supporting these corrections.
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Latz, J. Correction to: analysis of stochastic gradient descent in continuous time. Stat Comput 34, 146 (2024). https://doi.org/10.1007/s11222-024-10450-4
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DOI: https://doi.org/10.1007/s11222-024-10450-4