We consider the reinforcement learning problem for the constrained Markov decision process (CMDP), which plays a central role in satisfying safety or resource constraints in sequential learning and decision-making.

Section A.1 presents the lemmas used to prove the main results. Section A.2 presents the main results The first two inequalities are owing to the triangle inequality, and the third inequality is due to the definition of L-divergence Eq.(5). We complete the proof by applying Lemma A.1 to bound F ollowing the conditions of Theorem 4.1, the upper bound of null V arnull null D Based on the conditions of Theorem 4.1, we assume We complete the proof by applying Lemma A.3 and Lemma A.4 to bound the Rademacher Following the proof of Theorem 4.1, we have |D F ollowing the conditions of Proposition 4.3, as N, we have, null D Based on the result on Proposition 4.3, for any δ (0, 1), we know that 4LB ( 2 D ln 2 + 1)null We complete the proof by applying the triangle inequality. III: Samples from p and q are labeled with 0 and 1, respectively. All values are averaged over five trials.

Many applications seek to recover low-rank approximations of noisy tensor data.

We use BLIP-2 models built on the FLAN-T5 language model family. We use the same padding side as the FLAN-T5 models. We use a batch size of 8 for all datasets and models. The Q-former is kept in full precision. To produce decompositions, we use multinomial beam search sampling with 5 beams and a top-p of 0.95.

Our tight mechanism-specific bounds outperform tight mechanism-agnostic bounds and classic group privacy results.