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Always-valid concentration inequalities are increasingly used as performancemeasures for online statistical learning, notably in the learning of generativemodels and supervised learning. Such inequality advances the online learningalgorithms design by allowing random, adaptively chosen sample sizes instead ofa fixed pre-specified size in offline statistical learning. However,establishing such an always-valid type result for the task of matrix completionis challenging and far from understood in the literature. Due to the importanceof such type of result, this work establishes and devises the always-valid riskbound process for online matrix completion problems. Such theoretical advancesare made possible by a novel combination of non-asymptotic martingaleconcentration and regularized low-rank matrix regression. Our result enables amore sample-efficient online algorithm design and serves as a foundation toevaluate online experiment policies on the task of online matrix completion.

Always Valid Risk Monitoring for Online Matrix Completion