(Abridged) Designing computationally efficient algorithms in the agnostic learning model (Haussler, 1992; Kearns et al., 1994) is notoriously difficult. In this work, we consider agnostic learning with membership queries for touchstone classes at the frontier of agnostic learning, with a focus on how much computation can be saved over the trivial runtime of 2^n$. This approach is inspired by and continues the study of ``learning with nontrivial savings'' (Servedio and Tan, 2017). To this end, we establish multiple agnostic learning algorithms, highlighted by: 1. An agnostic learning algorithm for circuits consisting of a sublinear number of gates, which can each be any function computable by a sublogarithmic degree k polynomial threshold function (the depth of the circuit is bounded only by size). This algorithm runs in time 2^{n -s(n)} for s(n) \approx n/(k+1), and learns over the uniform distribution over unlabelled examples on \{0,1\}^n. 2. An agnostic learning algorithm for circuits consisting of a sublinear number of gates, where each can be any function computable by a \sym^+ circuit of subexponential size and sublogarithmic degree k. This algorithm runs in time 2^{n-s(n)} for s(n) \approx n/(k+1), and learns over distributions of unlabelled examples that are products of k+1 arbitrary and unknown distributions, each over \{0,1\}^{n/(k+1)} (assume without loss of generality that k+1 divides n).