The proof of infeasibility condition (Theorem 3.2) is provided in Section B. Explanations on conditions derived in Theorem 3.2 are included in Section C. The proof of properties of the proposed model (r)LogSpecT (Proposition 3.4 The truncated Hausdorff distance based proof details of Theorem 4.1 and Corollary 4.4 are Details of L-ADMM and its convergence analysis are in Section F. Additional experiments and discussions on synthetic data are included in Section G. ( m 1) Again, from Farkas' lemma, this implies that the following linear system does not have a solution: Example 3.1 we know δ = 2|h Since the constraint set S is a cone, it follows that for all γ > 0, γ S = S . Opt(C, α) = α Opt(C, 1), which completes the proof. The proof will be conducted by constructing a feasible solution for rLogSpecT. Since the LogSpecT is a convex problem and Slater's condition holds, the KKT conditions We show that it is feasible for rLogSpecT. R, its epigraph is defined as epi f: = {( x, y) | y f ( x) }. Before presenting the proof, we first introduce the following lemma.

First, note that we shall use the same MDPs defined in Appendix D.1 as follows

Let k (,) be the ν = 2 .5 Matérn kernel.

This paper aims to efficiently enable Large Language Models (LLMs) to use multi-modal tools. Advanced proprietary LLMs, such as ChatGPT and GPT -4, have shown great potential for tool usage through sophisticated prompt engineering.

The coefficients in this RF expansion are optimized to fit the given data of input-output pairs.

Further, we are interested in algorithms that enjoy sample efficiency while leveraging (value) function approximation.

The following lemma is standard in the literature, see e.g.