We develop a theory of finite-dimensional polyhedral subsets over the Wasserstein space and optimization of functionals over them via first-order methods. Our main application is to the problem of mean-field variational inference, which seeks to approximate a distribution $\pi$ over $\mathbb{R}^d$ by a product measure $\pi^\star$. When $\pi$ is strongly log-concave and log-smooth, we provide (1) approximation rates certifying that $\pi^\star$ is close to the minimizer $\pi^\star_\diamond$ of the KL divergence over a \emph{polyhedral} set $\mathcal{P}_\diamond$, and (2) an algorithm for minimizing $\text{KL}(\cdot\|\pi)$ over $\mathcal{P}_\diamond$ with accelerated complexity $O(\sqrt \kappa \log(\kappa d/\varepsilon^2))$, where $\kappa$ is the condition number of $\pi$.

In this paper, we present a MLP-like architecture for sequential recommendation, namely TriMLP, with a novel Triangular Mixer for cross-token communications. In designing Triangular Mixer, we simplify the cross-token operation in MLP as the basic matrix multiplication, and drop the lower-triangle neurons of the weight matrix to block the anti-chronological order connections from future tokens. Accordingly, the information leakage issue can be remedied and the prediction capability of MLP can be fully excavated under the standard auto-regressive mode. Take a step further, the mixer serially alternates two delicate MLPs with triangular shape, tagged as global and local mixing, to separately capture the long range dependencies and local patterns on fine-grained level, i.e., long and short-term preferences. Empirical study on 12 datasets of different scales (50K\textasciitilde 10M user-item interactions) from 4 benchmarks (Amazon, MovieLens, Tenrec and LBSN) show that TriMLP consistently attains promising accuracy/efficiency trade-off, where the average performance boost against several state-of-the-art baselines achieves up to 14.88% with 8.65% less inference cost.