Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.

We develop a unified mathematical framework for certified Top-$k$ attention truncation that quantifies approximation error at both the distribution and output levels. For a single attention distribution $P$ and its Top-$k$ truncation $\hat P$, we show that the total-variation distance coincides with the discarded softmax tail mass and satisfies $\mathrm{TV}(P,\hat P)=1-e^{-\mathrm{KL}(\hat P\Vert P)}$, yielding sharp Top-$k$-specific bounds in place of generic inequalities. From this we derive non-asymptotic deterministic bounds -- from a single boundary gap through multi-gap and blockwise variants -- that control $\mathrm{TV}(P,\hat P)$ using only the ordered logits. Using an exact head-tail decomposition, we prove that the output error factorizes as $\|\mathrm{Attn}(q,K,V)-\mathrm{Attn}_k(q,K,V)\|_2=τ\|μ_{\mathrm{tail}}-μ_{\mathrm{head}}\|_2$ with $τ=\mathrm{TV}(P,\hat P)$, yielding a new head-tail diameter bound $\|\mathrm{Attn}(q,K,V)-\mathrm{Attn}_k(q,K,V)\|_2\leτ\,\mathrm{diam}_{H,T}$ and refinements linking the error to $\mathrm{Var}_P(V)$. Under an i.i.d. Gaussian score model $s_i\sim\mathcal N(μ,σ^2)$ we derive closed-form tail masses and an asymptotic rule for the minimal $k_\varepsilon$ ensuring $\mathrm{TV}(P,\hat P)\le\varepsilon$, namely $k_\varepsilon/n\approxΦ_c(σ+Φ^{-1}(\varepsilon))$. Experiments on bert-base-uncased and synthetic logits confirm the predicted scaling of $k_\varepsilon/n$ and show that certified Top-$k$ can reduce scored keys by 2-4$\times$ on average while meeting the prescribed total-variation budget.

We use the PyTorch framework for our experiments. Similar to TD3, we implement our GRU-ODE in SAC. In this ablation study, we ask two questions in relation to numerical integration. Thus, simple numerical solvers are enough. We evaluate the time costs of different baselines on Walker-P environments.

The main contribution of the paper is that it shows that Runge-Kutta solutions of order 1, 2, 3 can be obtained from integrated Weiner processes.

We summarize the data statistics in our experiments in Table 1. For both fully and semi-supervised node classification tasks on the citation networks, Cora, Citeseer and Pubmed, we train our DGC following the hyper-parameters in SGC [5]. Specifically, we train DGC for 100 epochs using Adam [2] with learning rate 0.2. For weight decay, as in SGC, we tune this hyperparameter on each dataset using hyperopt [1] for 10,000 trails. For the large-scale inductive learning task on the Reddit network, we also follow the protocols of SGC [5], where we use L-BFGS [3] optimizer for 2 epochs with no weight decay.

Image samples are shown in Section I.