Xuanyuan Luo

Xuanyuan Luo, Luo Bei, Jian Li

Proving algorithm-dependent generalization error bounds for gradient-type optimization methods has attracted significant attention recently in learning theory. However, most existing trajectory-based analyses require either restrictive assumptions on the learning rate (e.g., fast decreasing learning rate), or continuous injected noise (such as the Gaussian noise in Langevin dynamics). In this paper, we introduce a new discrete data-dependent prior to the PAC-Bayesian framework, and prove a high probability generalization bound of order $O(\frac{1}{n}\cdot \sum_{t=1}^T(γ_t/\varepsilon_t)^2\left\|{\mathbf{g}_t}\right\|^2)$ for Floored GD (i.e. a version of gradient descent with precision level $\varepsilon_t$), where $n$ is the number of training samples, $γ_t$ is the learning rate at step $t$, $\mathbf{g}_t$ is roughly the difference of the gradient computed using all samples and that using only prior samples. $\left\|{\mathbf{g}_t}\right\|$ is upper bounded by and and typical much smaller than the gradient norm $\left\|{\nabla f(W_t)}\right\|$. We remark that our bound holds for nonconvex and nonsmooth scenarios. Moreover, our theoretical results provide numerically favorable upper bounds of testing errors (e.g., $0.037$ on MNIST). Using a similar technique, we can also obtain new generalization bounds for certain variants of SGD. Furthermore, we study the generalization bounds for gradient Langevin Dynamics (GLD). Using the same framework with a carefully constructed continuous prior, we show a new high probability generalization bound of order $O(\frac{1}{n} + \frac{L^2}{n^2}\sum_{t=1}^T(γ_t/σ_t)^2)$ for GLD. The new $1/n^2$ rate is due to the concentration of the difference between the gradient of training samples and that of the prior.

Lin Guan, Jia-Qi Yang, Zhishan Zhao et al.

Short-video recommenders such as Douyin must exploit extremely long user histories without breaking latency or cost budgets. We present an end-to-end system that scales long-sequence modeling to 10k-length histories in production. First, we introduce Stacked Target-to-History Cross Attention (STCA), which replaces history self-attention with stacked cross-attention from the target to the history, reducing complexity from quadratic to linear in sequence length and enabling efficient end-to-end training. Second, we propose Request Level Batching (RLB), a user-centric batching scheme that aggregates multiple targets for the same user/request to share the user-side encoding, substantially lowering sequence-related storage, communication, and compute without changing the learning objective. Third, we design a length-extrapolative training strategy -- train on shorter windows, infer on much longer ones -- so the model generalizes to 10k histories without additional training cost. Across offline and online experiments, we observe predictable, monotonic gains as we scale history length and model capacity, mirroring the scaling law behavior observed in large language models. Deployed at full traffic on Douyin, our system delivers significant improvements on key engagement metrics while meeting production latency, demonstrating a practical path to scaling end-to-end long-sequence recommendation to the 10k regime.

Jian Li, Xuanyuan Luo, Mingda Qiao

Generalization error (also known as the out-of-sample error) measures how well the hypothesis learned from training data generalizes to previously unseen data. Proving tight generalization error bounds is a central question in statistical learning theory. In this paper, we obtain generalization error bounds for learning general non-convex objectives, which has attracted significant attention in recent years. We develop a new framework, termed Bayes-Stability, for proving algorithm-dependent generalization error bounds. The new framework combines ideas from both the PAC-Bayesian theory and the notion of algorithmic stability. Applying the Bayes-Stability method, we obtain new data-dependent generalization bounds for stochastic gradient Langevin dynamics (SGLD) and several other noisy gradient methods (e.g., with momentum, mini-batch and acceleration, Entropy-SGD). Our result recovers (and is typically tighter than) a recent result in Mou et al. (2018) and improves upon the results in Pensia et al. (2018). Our experiments demonstrate that our data-dependent bounds can distinguish randomly labelled data from normal data, which provides an explanation to the intriguing phenomena observed in Zhang et al. (2017a). We also study the setting where the total loss is the sum of a bounded loss and an additional \ell_2 regularization term. We obtain new generalization bounds for the continuous Langevin dynamic in this setting by developing a new Log-Sobolev inequality for the parameter distribution at any time. Our new bounds are more desirable when the noisy level of the process is not small, and do not become vacuous even when T tends to infinity.