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For any Boolean functions $f$ and $g$, the question whether $R(f\circ g) =\tilde{\Theta}(R(f)R(g))$, is known as the composition question for therandomized query complexity. Similarly, the composition question for theapproximate degree asks whether $\widetilde{deg}(f\circ g) =\tilde{\Theta}(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$. These questions aretwo of the most important and well-studied problems, and yet we are far fromanswering them satisfactorily.  It is known that the measures compose if one assumes various properties ofthe outer function $f$ (or inner function $g$). This paper extends the class ofouter functions for which $\text{R}$ and $\widetilde{\text{deg}}$ compose.  A recent landmark result (Ben-David and Blais, 2020) showed that $R(f \circg) = \Omega(noisyR(f)\cdot R(g))$. This implies that composition holds whenever$noisyR(f) = \Tilde{\Theta}(R(f))$. We show two results:  (1)When $R(f) = \Theta(n)$, then $noisyR(f) = \Theta(R(f))$.  (2) If $\text{R}$ composes with respect to an outer function, then$\text{noisyR}$ also composes with respect to the same outer function. On theother hand, no result of the type $\widetilde{deg}(f \circ g) = \Omega(M(f)\cdot \widetilde{deg}(g))$ (for some non-trivial complexity measure $M(\cdot)$)was known to the best of our knowledge. We prove that  $\widetilde{deg}(f\circ g) = \widetilde{\Omega}(\sqrt{bs(f)} \cdot\widetilde{deg}(g)),$  where $bs(f)$ is the block sensitivity of $f$. This implies that$\widetilde{\text{deg}}$ composes when $\widetilde{\text{deg}}(f)$ isasymptotically equal to $\sqrt{\text{bs}(f)}$.  It is already known that both $\text{R}$ and $\widetilde{\text{deg}}$ composewhen the outer function is symmetric. We also extend these results to weakernotions of symmetry with respect to the outer function.

On the Composition of Randomized Query Complexity and Approximate Degree