The following proof is from Daniely and V ardi [15], and we give it here for completeness. By Lemma A.1, there exists a DNF formula We construct such an affine layer in Lemma A.2. At least one of the k size-n slices in z contains 0 more than once. We define the outputs of our affine layer as follows. Pr [z represents a hyperedge ] = n (n 1) ... (n k + 1) null 1 n null Pr null z Z null 1 2 log(n) .

Gaussian, and the weight matrix is non-degenerate.

The heads are then concatenated to form the LoRA adapter.

As a central example, we highlight the case one agent is no-swap and the other's regret is

Some related lower bounds include the work of Backurs et al. [2017] that solving kernel Support V ector Machines (SVM), ridge regression, or Principal Component Analysis (PCA) problems to high accuracy or approximating kernel density estimates up to a constant factor for kernels with