Stochastic nonconvex optimization is central to training deep networks and LLMs in modern machine learning. We give a black-box reduction from stochastic nonconvex optimization to ordinary static regret minimization in online convex optimization (OCO), addressing the open problem posed by Chen and Hazan (2024). Our reduction maintains a predictable gradient tracker, while a black-box online learner $\mathcal A$ selects a preconditioner that transforms this tracker into the update direction. Given a $β$-smooth function with initial gap at most $M$ and an unbiased gradient oracle with variance bounded by $σ^2$, we bound the expected average squared gradient norm by $O(σ\sqrt{Mβ/T}+\sqrt{Mβ}\mathrm{Reg}_T(\mathcal A)/T+\frac{Mβ}{T})$, where $\mathrm{Reg}_T(\mathcal A)$ is the static regret of $\mathcal A$ against the identity preconditioner. Thus, any OCO oracle with $O(\sqrt T)$ regret recovers the classical $O(T^{-1/2})$ convergence rate. We also prove that the additive variance term cannot be removed: every conversion making a constant number of stochastic-oracle queries per round, whose guarantee is stated through static regret against a comparator that the online learner can play, must contain an additive term of order $σ\sqrt{Mβ/T}$ for sufficiently large $T$. Moreover, for every online learner, our conversion bound is tight up to universal constants. We also extend the framework to nonsmooth nonconvex optimization, still relying only on ordinary static regret, and attain the optimal convergence rate for Goldstein-type stationarity. Finally, we conduct numerical experiments on nonconvex objectives to illustrate how the reduction exploits online-selected preconditioners while using the same stochastic-oracle budget as stochastic gradient descent.
Multimodal Large Language Models (MLLMs) have made substantial progress in 3D scene understanding from multi-view RGB observations. Yet a fixed object and query can elicit inconsistent target-related responses across camera viewpoints, a phenomenon we characterize as viewpoint-induced semantic drift. We introduce LieVLM, a Lie routing framework for view-robust 3D understanding. Motivated by the rotational component of viewpoint change, Lie routing adapts visual representations by applying local $\mathrm{SO}(3)$ transformations to projected features within learned three-dimensional subspaces. Geometry tokens extracted by VGGT predict local rotation parameters and channel-wise routing coefficients. A closed-form Padé--Cayley map transforms projected visual features, whose identity-relative corrections are projected back and gated into the original visual and geometry features. This design retains the pretrained semantic and direct spatial context while learning view-dependent feature adjustments. Experiments on 3D captioning, grounding, detection, and spatial reasoning show consistent improvements over additive fusion across training scales and backbones. Controlled ablations examine the roles of geometric conditioning and transformation structure. Viewpoint analyses further reveal systematic variation in the predicted routing states alongside more localized target responses and stronger task performance across viewpoint changes.
Mixture-of-Experts (MoE) architecture has been widely adopted in large language models (LLMs) since it reduces the computational cost through sparse expert activation. However, the massive parameter scale of MoE models hinders their deployment on resource-constrained edge devices. Although existing mixed-precision quantization reduces the model size, they are inherently static, fixing the bit-width of each expert across decoding steps. Such static schemes yield suboptimal performance as they overlook the dynamic variation in expert importance during decoding. Furthermore, they fail to adapt to the dynamic latency constraint driven by the fluctuating system utilization. To address these challenges, we propose MatMoE, a dynamic mixed-precision quantization framework for efficient MoE inference. MatMoE features: (1) a Matryoshka MoE PTQ algorithm that enables experts to operate flexibly at multiple precisions within a stringent memory budget; (2) a floor-and-compensation quantization method that eliminates redundant memory accesses in the Matryoshka representation; and (3) a dynamic bit-width allocation strategy that determines the precision for each expert based on their routing scores to satisfy the dynamic latency requirement. We implement our MatMoE system atop llama.cpp for both CPU and GPU platforms. Extensive evaluations across three modern MoE models and challenging benchmarks demonstrate that MatMoE achieves $4.0\% \sim 18.3\%$ higher accuracy than state-of-the-art static mixed-precision method. Compared to llama.cpp, MatMoE delivers $1.3\times \sim 3.7\times$ speedup on Raspberry Pi 5, and $1.2\times \sim 1.3\times$ speedup on NVIDIA RTX 4090 and Jetson AGX Orin.
Large language model (LLM) distillation can propagate teacher bias under covariate shift when target reward feedback is unavailable. We propose Iterative Calibration and Learning (IterCaL), which alternates source-reward teacher calibration through token-level branching with student learning on target prompts. We prove polynomial convergence in expected average KL to the oracle student, the reference-regularized target-reward maximizer within a potentially misspecified student class; a quantified source-to-target calibration discrepancy gives an explicit additive true-return term. Our separation shows that regularized direct teacher matching retains positive oracle-KL error at its population optimum for every shared reference penalty, even when the teacher is stronger than every student. On the same instance, IterCaL converges to the oracle using source rewards alone.
Yannan Chen, Beichen Huang, Licheng Zhao, Chris Ding, Kaiming Shen
The normalized cut (NCut) problem is a fundamental one in the area of data clustering. Despite its fractional structure, the NCut problem is seldom considered from a fractional programming (FP) perspective in the existing literature. The main reason is that the conventional FP method is inadequate for this problem case: the classical Dinkelbach's transform can only handle a single ratio and hence is limited to the two-class clustering, whereas the multi-class clustering involves multiple ratio terms. This work proposes a novel so-called multidimensional quadratic transform that is capable of dealing with the multi-ratio case; it recasts the NCut problem into a bipartite matching problem---which can be immediately solved by linear search. Furthermore, we explore the connection between the proposed FP method and the minorization-maximization (MM) theory, and in turn characterize the convergence behavior of our algorithm. Finally, we conduct experiments to numerically demonstrate the advantages of the proposed FP-based algorithm in a variety of data clustering and image segmentation tasks.