When the expected rewards of all arms are the same, we know that the arm with the lowest index will be chosen and thus the first K pulls will be π1 = 1,...,πK = K. We will complete the proof through induction. Suppose that the greedy pull sequence is periodic with π1 = 1,...,πK = K and πt+K = πt until time h>K. We will show that πh+1 = 1 if πh = K and πh+1 = πh + 1 otherwise. When k0 = 0 (i.e., πh = K), all arms have been pulled exactly ntimes as of time h. Therefore, by (3), at time h+ 1, arm 1 has the highest expected reward and will be chosen.

A fundamental structural model for data is that the data points lie close to an unknown subspace, meaning that the matrix created by concatenating the data vectors has low rank. We address a particular low-rank matrix recovery problem where we wish to recover a set of vectors from a low-dimensional subspace after they have been individually compressed (or "sketched").

Hence, the function over this constraint set isG-Lipschitz. Finally, in Lemma6, we provide bounds on excess empirical risk and average regret of gradient descent. Let ℓ be a non-negative eH smooth convex loss function. Let bw:= A(S), S(i) be the dataset where thei-th data point is replaced by an i.i.d. A.4 HighDimensionProofofTheorem 2. Let α 1 be a parameter to be set later.

We study the problem of(ǫ,δ)-differentially private learning of linear predictors with convexlosses.

These assumptions imply that an optimal solutionx?