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Deep Learning · Large Language Models

Mayee Chen, Tyler Murray, David Heineman, Matt Jordan, Hannaneh Hajishirzi, Christopher Re, Luca Soldaini, Kyle Lo

Data mixing---determining the ratios of data from different domains---is a first-order concern for training language models (LMs), but existing mixing methods have poorly understood design choices and assume that the set of domains remain fixed throughout development. We present Olmix, a framework that addresses two challenges encountered during LM development. First, the configuration space for developing a mixing method is not well understood---design choices across existing methods lack justification or consensus and overlook practical issues like data constraints. We conduct a comprehensive empirical study of this space, identifying which design choices lead to a strong mixing method. Second, the domain set evolves throughout LM development as datasets are revised and expanded---a problem setting largely unaddressed by existing works. We study how to efficiently recompute the mixture after the domain set is updated, given an existing mix from before the update. We introduce mixture reuse, a mechanism that reuses existing relative ratios and recomputes ratios only for domains affected by an update. Over a sequence of five domain-set updates mirroring real-world LM development, mixture reuse matches the performance of fully recomputing the mix after each update with 74% less compute and improves over training without mixing by 11.6% on downstream tasks.

Deep Learning · Large Language Models

Fahim Tajwar, Guanning Zeng, Yueer Zhou, Yuda Song, Daman Arora, Yiding Jiang, Jeff Schneider, Russ Salakhutdinov, Haiwen Feng, Andrea Zanette

Maximum likelihood is fundamental to supervised learning but it cannot be directly applied in correctness-based problems with non-differentiable sampling. In these settings, reinforcement learning (RL) is typically used to maximize expected reward. We show that for binary correctness tasks, expected-reward RL is a first-order approximation of the maximum likelihood objective, yielding vanishing learning signal on low-success inputs. We introduce **Maximum Likelihood Reinforcement Learning (MaxRL)**, a compute-indexed family of sampling-based objectives derived from a pass@k expansion of the likelihood, which interpolates between standard RL and exact maximum likelihood as compute increases. MaxRL admits a simple unbiased policy-gradient estimator whose optimized objective improves with additional compute. Across multiple domains, MaxRL consistently outperforms standard RL and GRPO, achieving higher $pass@1$ and substantially improved $pass@k$.

Deep Learning · Large Language Models

Zixi Chen, Yinyu Ye, Zijie Zhou

We study the online routing problem in large language model serving, where requests arrive sequentially and must be dispatched to parallel decode workers under tight batch-size and KV-cache constraints. Unlike widely used routing heuristics that are not tied to explicit service-level objectives (SLOs) and offer limited control over latency–throughput trade-offs, we introduce an multi-objective optimization framework that formulates routing as an online linear programming with interpretable decision rewards. We apply an efficient bid-price control policy based on the online linear programming that admits requests when their SLO-weighted benefit exceeds their shadow prices. To meet millisecond decision requirements, we develop a warm-started, projected first-order updates that track the evolving dual shadow prices online with predictable runtime. We integrate our router into the Vidur simulator and demonstrate substantial improvements over standard baselines across multiple SLO regimes, including end-to-end latency, time-to-first-token, throughput, and tail performance. A big picture from our result: a science-based approach outperforms others based on heuristics.

Theory · Optimization

Zijian Liu

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular $\mathtt{Adam}$ and $\mathtt{AdamW}$, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case, $\mathtt{AdaGrad}$, the origin of adaptive gradient methods. We provide the first provable convergence rate for $\mathtt{AdaGrad}$ in non-convex optimization when the tail index $p$ satisfies $\frac{4}{3}<p\leq2$. Notably, this result is achieved without requiring any prior knowledge of $p$ and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by $\mathtt{AdaGrad}$. Lastly, we consider $\mathtt{AdaGrad}\text{-}\mathtt{Norm}$, a popular variant of $\mathtt{AdaGrad}$ in theoretical studies, and show an improved rate that holds for any $1<p\leq2$ under an extra mild assumption.

Optimization · Stochastic

Haoxuan Wang, Xinchen Du, Sen Na

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring $O(d^2)$ time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with $O(d^3)$ complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a *sketch-and-project solver with Nesterov's acceleration*, matching $O(d^2)$ complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

Optimization · Zero-order and Black-box Optimization

Minhak Song, Liang Zhang, Bingcong Li, Niao He, Michael Muehlebach, Sewoong Oh

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across CNNs, ResNets, and Transformers on vision tasks. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

Deep Learning · Sequential Models, Time series

Thai Khanh Nguyen, Ngoc Bich Uyen Vo, Thieu Vo, Tan Nguyen, Cuong Pham

State-space models (SSMs) have emerged as efficient linear-time alternatives to attention for long-sequence modeling. However, existing SSMs often suffer from instability and memory degradation over extended horizons due to poorly conditioned first-order updates and uncontrolled spectral geometry. We introduce MuonSSM, a general framework that stabilizes SSM training by explicitly conditioning the geometry of memory updates rather than the recurrent transition matrix. MuonSSM augments standard SSMs with a momentum-based pathway and lightweight Newton–Schulz iterations on low-rank input injections, yielding approximately norm-preserving and spectrally balanced updates while preserving parallel scan complexity. Theoretical analysis demonstrates substantial improvements in gradient propagation and mitigation of vanishing gradients over long horizons. Extensive experiments across language, vision, and time-series benchmarks show consistent gains in accuracy, robustness, and long-context performance when integrated into diverse SSM backbones. These results establish geometric conditioning of updates as a principled pathway to stable, scalable sequence modeling.

Optimization · Everything Else

Gal Lifshitz, Shahar Zuler, Ori Fouks, Dan Raviv

End-to-end deep learning has achieved impressive results but often relies on large labeled datasets, exhibits limited generalization to unseen scenarios, and incurs substantial computational cost. Classical optimization methods, in contrast, are data-efficient and lightweight but frequently suffer from slow or unstable convergence. Learned optimizers aim to bridge this gap, yet existing approaches predominantly focus on first-order methods, leaving learned second-order optimization largely unexplored. We introduce L-SR1, a novel learned second-order optimizer that augments the classical Symmetric Rank-One (SR1) method with a lightweight, trainable preconditioning unit. This unit generates data-driven rank-one updates that are explicitly aligned with the secant condition via a learned projection mechanism, ensuring stable and consistent curvature estimation. Through controlled analytic benchmarks, we systematically analyze the stability, generalization across problem dimensions, and the quality of the resulting search directions. We further evaluate L-SR1 on the real-world task of Monocular Human Mesh Recovery (HMR), where it outperforms existing optimization-based and learned-optimization baselines. With a compact model and no reliance on task-specific fine-tuning or annotated data, L-SR1 demonstrates strong generalization and provides an effective drop-in optimizer for a wide range of iterative optimization problems.

Optimization · Zero-order and Black-box Optimization

Wei LIN, Yining Jiang, Qingyu Song, Qiao Xiang, Hong Xu

Zeroth-Order (ZO) optimization has emerged as a promising solution for fine-tuning LLMs under strict memory constraints, as it avoids the prohibitive memory cost of storing activations for backpropagation. However, existing ZO methods typically employ isotropic perturbations, neglecting the rich structural information available during the forward pass. In this paper, we identify a crucial link between gradient formation and activation structure: the gradient of a linear layer is confined to the subspace spanned by its input activations. Leveraging this insight, we propose Activation-Guided Zeroth-Order optimization (AGZO). Unlike prior methods, AGZO extracts a compact, activation-informed subspace on the fly during the forward pass and restricts perturbations to this low-rank subspace. We provide a theoretical framework showing that AGZO optimizes a subspace-smoothed objective and provably yields update directions with higher cosine similarity to the true gradient than isotropic baselines. Empirically, we evaluate AGZO on Qwen3 and Pangu models across various benchmarks. AGZO consistently outperforms state-of-the-art ZO baselines and significantly narrows the performance gap with first-order fine-tuning, while maintaining almost the same peak memory footprint as other ZO methods.

Applications · Neuroscience, Cognitive Science

Kartikay Agrawal, Vaishnavi N, Abhijeet Vikram, Vedant Sharma, Ayon Borthakur

Spiking neural networks have attracted increasing attention for their energy efficiency, multiplication-free computation, and sparse event-based processing. In parallel, state space models have emerged as a scalable alternative to transformers for long-range sequence modelling by avoiding quadratic dependence on sequence length. We propose here a spiking heterogeneous harmonic resonate-and-fire state space model (S$H^2$RFSSM), a second-order spiking SSM for classification and regression on ultra-long sequences. S$H^2$RFSSM outperforms transformers and first-order SSMs on average while eliminating matrix multiplications, making it highly suitable for resource-constrained applications. Furthermore, we introduce a kernel-based spiking regressor that enables accurate modelling of dependencies in sequences of up to 50k steps. We also observe a reduction in spiking operations and improved performance with heterogeneity and discretisation in harmonic resonate-and-fire neuronal layers. Overall, we evaluate Harmonic Resonate-and Fire layers across 17 diverse datasets, spanning sensors, time series, and classification to long-term forecasting. Our results demonstrate that S$H^2$RFSSM achieves superior long-range modelling capability with energy efficiency, positioning it as a strong candidate for signal processing on resource-constrained devices for human activity recognition, time series classification, and regression.

Social Aspects · Accountability, Transparency, and Interpretability

Yilin Zhang, Cai Xu, You Wu, Ziyu Guan, Wei Zhao

Real-world deployments inevitably encounter distribution shifts, rendering the confidence estimates of deep neural networks highly unreliable, posing severe risks in safety-critical applications. Existing methods improve calibration via training-time regularization or post-hoc adjustment, but often rely on access to (or simulation of) target domains, limiting practicality. We propose Frequency-aware Gradient Rectification (FGR), a target-agnostic training framework for robust calibration. From a frequency perspective, FGR applies low-pass filtering to a subset of training images to diminish spurious high-frequency cues and encourage the learning of domain-invariant features. However, the associated information loss can degrade In-Distribution (ID) calibration. To resolve this trade-off, FGR treats ID calibration as a hard constraint and rectifies conflicting parameter updates via geometric projection. This ensures a first-order non-increase in the ID calibration objective without introducing additional weighting hyperparameters. Extensive experiments on synthetic, real-world, and semantic shift datasets demonstrate that FGR significantly improves calibration under diverse shifts while preserving ID performance, and it remains compatible with post-hoc calibration methods.

Theory · Online Learning and Bandits

Emmanuel Esposito, Andrew Jacobsen, Hao Qiu, Mengxiao Zhang

In this paper, we study dynamic regret in unconstrained online convex optimization (OCO) with movement costs. Specifically, we generalize the standard setting by allowing the movement cost coefficients $\lambda_t$ to vary arbitrarily over time. Our main contribution is a novel algorithm that establishes the first comparator-adaptive dynamic regret bound for this setting, guaranteeing $\widetilde{\mathcal{O}}(\sqrt{(1+P_T)(T+\sum_t \lambda_t)})$ regret, where $P_T$ is the path length of the comparator sequence over $T$ rounds. This recovers the optimal guarantees for both static and dynamic regret in standard OCO as a special case where $\lambda_t=0$ for all rounds. To demonstrate the versatility of our results, we consider two applications: *OCO with delayed feedback* and *OCO with time-varying memory*. We show that both problems can be translated into time-varying movement costs, establishing a *novel reduction* specifically for the delayed feedback setting that is of independent interest. A crucial observation is that the first-order dependence on movement costs in our regret bound plays a key role in enabling optimal comparator-adaptive dynamic regret guarantees in both settings.

Deep Learning · Robustness

Jinping Wang, Qinhan Liu, Zhiwu Xie, Zhiqiang Gao

Sharpness-Aware Minimization (SAM) improves generalization by minimizing the worst-case loss within a fixed parameter-space radius neighborhood. SAM and its variants mainly rely on a first-order linearized surrogate, while flat minima are inherently a second-order (curvature) notion. We revisit this mismatch and propose Loss-Equated SAM (LE-SAM), which inverts the traditional SAM mechanism that fixed perturbation radius with a fixed loss-space budget, effectively removing gradient-norm–dominated learning signals and shifting optimization toward curvature-dominated terms. Extensive experiments across diverse benchmarks and tasks demonstrate the strong generalization ability of LESAM that consistently outperforms SAM and even its variants, achieving the state-of-the-art performance.

Deep Learning · Generative Models and Autoencoders

Hyunmin Cho, Donghoon Ahn, Susung Hong, Jee Eun Kim, Seungryong Kim, Kyong Hwan Jin

Recent diffusion models achieve the state-of-the-art performance in image generation, but often suffer from semantic inconsistencies or *hallucinations*. While various inference-time guidance methods can enhance generation, they often operate *indirectly* by relying on external signals or architectural modifications, which introduces additional computational overhead. In this paper, we propose **T**angential **A**mplifying **G**uidance **(TAG)**, a theoretically grounded, training-free, computationally lightweight, and architecture-agnostic guidance method that operates solely on trajectory signals without modifying the underlying diffusion model. TAG leverages an intermediate sample as a projection basis and amplifies the tangential components of the estimated scores with respect to this basis to correct the sampling trajectory. We formalize this guidance process via a first-order Taylor analysis, showing that tangential amplification steers the state toward higher-probability regions of the data manifold, thereby reducing inconsistencies and improving sample fidelity. TAG is a plug-and-play module that integrates into existing diffusion samplers with minimal additional computation, offering a new perspective on diffusion guidance.

Optimization · Stochastic

Enea Monzio Compagnoni, Rustem Islamov, Frank Proske, Aurelien Lucchi, Antonio Orvieto, Eduard Gorbunov

Distributed stochastic optimization intertwines (i) stochastic gradient noise, (ii) communication compression, and (iii) adaptive/normalized updates. While each factor has been studied in isolation, their joint effect under realistic assumptions remains poorly understood. In this work, we develop a unified theoretical framework for Distributed Compressed SGD (DCSGD) and its sign variant Distributed SignSGD (DSignSGD) under the recently introduced $(L_0, L_1)$-smoothness condition. From a conceptual perspective, we show that the first- and second-order modified equations from the literature do not accurately model the discrete-time step-size/stability restrictions, especially under $(L_0,L_1)$-smoothness. From a technical perspective, we propose new first-order SDEs by carefully incorporating curvature-dependent terms into their drift: This helps capture the fine-grained relationship between learning rate restrictions, gradient noise, compression, and the geometry of the loss landscape. Importantly, we do so under general gradient noise assumptions, including heavy-tailed and affine-variance regimes, which extend beyond the classical bounded-variance setting. Our results suggest that normalizing the updates of DCSGD emerges as a natural condition for stability, with the degree of normalization precisely determined by the gradient noise structure, the landscape’s regularity, and the compression rate. In contrast, DSignSGD converges even under heavy-tailed noise with standard learning rate schedules. Together, these findings offer both new theoretical insights and perspectives, and practical guidance.

Deep Learning · Generative Models and Autoencoders

Wenbin Duan, Yan Shu, Zhuoyuan Fu, Fangmin Zhao, Yan Li, Yaru Zhao, Binyang Li

Rectified-flow-based diffusion transformers, particularly FLUX, have demonstrated outstanding performance in high-quality image generation. However, achieving fast and accurate inversion—transforming images back to latent noise for faithful reconstruction and editing—remains a challenging bottleneck due to the discretization errors of linear solvers. This paper introduces \textbf{SlerpFlow}, a straightforward yet highly effective zero-shot approach that unlocks the full potential of FLUX for high-fidelity inversion and editing. Unlike existing approaches (e.g., RF-Solver) that rely on complex numerical approximations such as high-order Taylor expansions to correct trajectory errors, we present a geometric view based on the Manifold Hypothesis: the empirically observed trajectory curvature is not a numerical artifact, but rather serves as a necessary ``centripetal force" that constrains the flow to remain on the data manifold. Guided by this insight, SlerpFlow integrates Spherical Linear Interpolation (Slerp) to rectify flow velocity directions on the hypersphere, strictly adhering to the intrinsic curvature of the latent space. Crucially, by caching the corrected velocity for subsequent steps, SlerpFlow achieves high-precision inversion while maintaining the computational efficiency of a first-order Euler solver. Extensive experiments on FLUX-based reconstruction and editing tasks demonstrate that our method delivers superior structural consistency and lower reconstruction errors compared to state-of-the-art baselines without requiring additional training.

Deep Learning · Algorithms

Tian Zhang, Yujia Tong, Junhao Dong, Ke Xu, Yuze Wang, Jingling Yuan

The deployment of quantized neural networks on edge devices, combined with privacy regulations like GDPR, creates an urgent need for machine unlearning in quantized models. However, existing methods face critical challenges: they induce forgetting by training models to memorize incorrect labels, conflating forgetting with misremembering, and employ scalar gradient reweighting that cannot resolve directional conflicts between gradients. We propose \textbf{OEU}, a novel Orthogonal Entropy Unlearning framework with two key innovations: 1) Entropy-guided unlearning maximizes prediction uncertainty on forgotten data, achieving genuine forgetting rather than confident misprediction, and 2) Gradient orthogonal projection eliminates interference by projecting forgetting gradients onto the orthogonal complement of retain gradients, providing theoretical guarantees for utility preservation under first-order approximation. Extensive experiments demonstrate that OEU outperforms existing methods in both forgetting effectiveness and retain accuracy.

Feiyang Wu, Ye Zhao, Anqi Wu

We propose a distributional framework for offline Inverse Reinforcement Learning (IRL) that jointly models uncertainty over reward functions and full distributions of returns. Unlike conventional IRL approaches that recover a deterministic reward estimate or match only expected returns, our method captures richer structure in expert behavior, particularly in learning the reward distribution, by minimizing first-order stochastic dominance (FSD) violations and thus integrating distortion risk measures (DRMs) into policy learning, enabling the recovery of both reward distributions and distribution-aware policies. This formulation is well-suited for behavior analysis and risk-aware imitation learning. Theoretical analysis show that the algorithm converge with $\mathcal{O}(\varepsilon^{-2})$ iteration complexity. Empirical results on synthetic benchmarks, real-world neurobehavioral data, and MuJoCo control tasks demonstrate that our method recovers expressive reward representations and achieves state-of-the-art imitation performance.

Reinforcement Learning · Inverse

Feiyang Wu, Ye Zhao, Anqi Wu

We propose a distributional framework for offline Inverse Reinforcement Learning (IRL) that jointly models uncertainty over reward functions and full distributions of returns. Unlike conventional IRL approaches that recover a deterministic reward estimate or match only expected returns, our method captures richer structure in expert behavior, particularly in learning the reward distribution, by minimizing first-order stochastic dominance (FSD) violations and thus integrating distortion risk measures (DRMs) into policy learning, enabling the recovery of both reward distributions and distribution-aware policies. This formulation is well-suited for behavior analysis and risk-aware imitation learning. Theoretical analysis show that the algorithm converge with $\mathcal{O}(\varepsilon^{-2})$ iteration complexity. Empirical results on synthetic benchmarks, real-world neurobehavioral data, and MuJoCo control tasks demonstrate that our method recovers expressive reward representations and achieves state-of-the-art imitation performance.

Theory · Online Learning and Bandits

Mingyi Li, Taira Tsuchiya, Kenji Yamanishi

This work studies online episodic tabular Markov decision processes (MDPs) with known transitions and develops best-of-both-worlds algorithms that achieve refined data-dependent regret bounds in the adversarial regime and variance-dependent regret bounds in the stochastic regime. We quantify MDP complexity using a first-order quantity and several new data-dependent measures for the adversarial regime, including a second-order quantity and a path-length measure, as well as variance-based measures for the stochastic regime. To adapt to these measures, we develop algorithms based on global optimization and policy optimization, both built on optimistic follow-the-regularized-leader with log-barrier regularization. For global optimization, our algorithms achieve first-order, second-order, and path-length regret bounds in the adversarial regime, and in the stochastic regime, they achieve a variance-aware gap-independent bound and a variance-aware gap-dependent bound that is polylogarithmic in the number of episodes. For policy optimization, our algorithms achieve the same data- and variance-dependent adaptivity, up to a factor of the episode horizon, by exploiting a new optimistic $Q$-function estimator. Finally, we establish regret lower bounds in terms of data-dependent complexity measures for the adversarial regime and a variance measure for the stochastic regime, implying that the regret upper bounds achieved by the global-optimization approach are nearly optimal.