Long-term non-uniform motion poses a significant challenge for feed-forward Novel View Synthesis (\textbf{NVS}), as it requires modeling higher-order motion, such as acceleration. Existing methods primarily rely on deformation fields or scene flow, which are limited to first-order approximations. Due to neglecting higher-order motion representations and supervision, these approaches suffer from long-term non-uniform motion scenarios. Inspired by Taylor’s theorem, we propose Taylor-Gaussians-Flow (\textbf{TGsF}) to represent and supervise non-uniform motion through first-order and second-order motion components. TGsF comprises two key modules: Taylor-Gaussians (\textbf{TGs}) and Taylor-Gaussians-Flow (\textbf{TGs-Flow}). TGs represent motion using Gaussian means with a quadratic temporal term and time-dependent opacity. Unlike previous methods, TGs-Flow decouples scene-flow supervision into separate depth and 2D optical-flow constraints. This approach effectively mitigates error propagation from either depth or motion estimation while circumventing the scarcity of labeled scene flow data. Guided by the above analysis, we develop the Feed-Forward Taylor-Gaussians-Flow (\textbf{FF-TGsF}) framework, which sets a new state-of-the-art on four dynamic benchmarks.
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Reinforcement Learning · Deep RL
Safe reinforcement learning (Safe RL) aims to maximize expected return while satisfying safety constraints, typically modeled as constrained Markov decision processes. While primal-dual methods scale well to deep RL, they often suffer from delayed constraint correction, leading to oscillatory behavior and prolonged safety violations. In this paper, we propose *Constraint-Sensitive Policy Optimization (CSPO)*, a first-order primal-dual method that incorporates local constraint sensitivity into policy updates. CSPO augments the primal objective with a constraint-sensitive correction derived from the shortest signed distance to the safety boundary, enabling smarter recovery steps back to safety, compensating for delayed Lagrange multiplier updates, and reducing oscillations near the boundary, while preserving the KKT solutions of the original constrained problem. Extensive experiments on navigation and locomotion benchmarks demonstrate that CSPO achieves faster safety recovery and high reward preservation, resulting in higher constrained returns (+15.6\% average improvement) compared to state-of-the-art primal-dual and penalty-based methods.
Optimization · Stochastic
Flatness of the loss landscape has been widely studied as an important perspective for understanding the behavior and generalization of deep learning algorithms. Motivated by this view, we propose Flatness-Aware Stochastic Gradient Langevin Dynamics (fSGLD), a first-order optimization method that biases learning its dynamics toward flat basins while retaining the computational and memory efficiency of SGD and SGLD. We provide a non-asymptotic theoretical analysis showing that fSGLD converges to a flatness-biased Gibbs distribution under a theoretically prescribed coupling between the noise scale $\sigma$ and the inverse temperature $\beta$, together with explicit excess risk guarantees. We empirically evaluate fSGLD across standard optimizer benchmarks, Bayesian image classification, uncertainty quantification, and out-of-distribution detection, demonstrating consistently strong performance and reliable uncertainty estimates. Additional experiments confirm the effectiveness of the theoretically prescribed $\beta$–$\sigma$ coupling compared to decoupled choices.
Deep Learning · Algorithms
Classical dynamic programming algorithms such as Smith-Waterman, edit distance, and constituency parsing solve structured combinatorial problems using hard constraints. Soft relaxations replace hard or operators with differentiable probabilistic models whose gradients correspond to alignment or parsing marginals, but existing approaches typically treat the DP algorithm itself as fixed, relying on hand-tuned gap penalties, edit costs, span penalties, and temperatures. We show how to learn these parameters directly from data: the marginals these algorithms produce are themselves first-order derivatives, so learning them is inherently a second-order problem. We derive efficient Hessian-vector products and cross-Jacobians for twelve dynamic programming algorithms spanning alignment, edit distance, and parsing; both derivative families admit closed-form covariance expressions under the induced Gibbs distribution. We implement these operators as fused CUDA kernels in dp, achieving -- speedups over standard PyTorch and making end-to-end parameter learning practical at modern scales. We then demonstrate that learning these parameters is critical in practice. In protein structure alignment, freezing gap penalties collapses performance from to (and to at lower encoder capacity), while jointly learning them recovers biologically meaningful gap regimes and reaches and lDDT, % of the TM-align lDDT ceiling. The same machinery transfers to constituency parsing, where a structured CKY CRF matches dense per-span supervision to within on English with no direct parse-tree structural supervision.
Reinforcement Learning · Batch/Offline
Off-policy reinforcement learning suffers from extrapolation errors when a learned policy selects actions that are weakly supported in the replay buffer. In this study, we address this issue by drawing an analogy to static friction in classical mechanics. From this perspective, the replay buffer is represented as a smooth, low-dimensional action manifold, where the support directions correspond to the tangential component, while the normal component captures the dominant first-order extrapolation error. This decomposition reveals an intrinsic anisotropy in value sensitivity that naturally induces a stability condition analogous to a friction threshold. To mitigate deviations toward unsupported actions, we propose Frictional Q-Learning, an off-policy algorithm that encodes supported actions as tangent directions using a contrastive variational autoencoder. We further show that an orthonormal basis of the orthogonal complement corresponds to normal components under mild local isometry assumptions. Empirical results on standard continuous-control benchmarks demonstrate robust, stable performance compared with existing baselines.
General Machine Learning · Representation Learning
The explosive growth of open-source model repositories has created a Model Jungle, where checkpoints are frequently shared without adequate documentation or metadata. While weight-space learning offers a pathway to identify and analyze these models directly from their parameters, processing full-scale weights is computationally prohibitive. Probing-based methods have emerged as a lightweight alternative, extracting permutation-equivariant representations via learnable probe vectors. However, existing probing methods are limited by a single-view design: they capture first-order structures but fail to encode the rich, higher-order correlation patterns inherent in row--column interactions. To bridge this gap, we introduce MVProbe, a multi-perspective probing framework that synthesizes first-order signals with interaction-aware (Gram-based) views. Our approach is theoretically grounded; we analyze the scaling laws of different probing orders to derive a principled standardization and fusion strategy that ensures balanced contributions from all branches. On the Model Jungle benchmark, MVProbe consistently outperforms the state-of-the-art ProbeX across diverse architectures, including ResNet, SupViT, MAE, and DINO.
Social Aspects · Privacy
We provide the first study of the problem of finding differentially private (DP) second-order stationary points (SOSP) in stochastic (non-convex) minimax optimization. Existing literature either focuses only on first-order stationary points for minimax problems or on SOSP for classical stochastic minimization problems. This work provides, for the first time, a unified and detailed treatment of both empirical and population risks. Specifically, we propose a purely first-order method that combines a nested gradient descent--ascent scheme with SPIDER-style variance reduction and Gaussian perturbations to ensure privacy. A key technical device is a block-wise ($q$-period) analysis that controls the accumulation of stochastic variance and privacy noise without summing over the full iteration horizon, yielding a unified treatment of both empirical-risk and population formulations. Under standard smoothness, Hessian-Lipschitzness, and strong concavity assumptions, we establish high-probability guarantees for reaching an $(\alpha,\sqrt{\rho_\Phi \alpha})$-approximate second-order stationary point with $\alpha = \mathcal{O}( (\frac{\sqrt{d}}{n\varepsilon})^{2/3})$ for empirical risk objectives and $\mathcal{O}(\frac{1}{n^{1/3}} + (\frac{\sqrt{d}}{n\varepsilon})^{1/2})$ for population objectives, matching the best known rates for private first-order stationarity.
Optimization · Convex
Recent work has analyzed the convergence of first-order methods under non-uniform smoothness assumptions that better model the loss landscape in machine learning tasks. We generalize this assumption to objectives whose curvature is an affine function of the objective value. This property is satisfied by a broad class of problems, including logistic regression, generalized linear models with a logistic link function, softmax policy gradient in reinforcement learning, and a class of neural networks. Under this assumption and gradient domination conditions, we establish a general convergence rate for the steepest descent method, and deterministic, diagonal variants of RMSProp and Adam. Our results imply that for logistic regression on separable data and the softmax policy gradient objective, sign GD converges linearly and is provably faster than GD. Furthermore, we show that for a class of two-layer neural networks on separable data, RMSProp and Adam can converge at a linear rate with a constant step-size and momentum parameter. Finally, we present a lower bound demonstrating that, under our assumption, RMSProp and Adam are provably faster than AdaGrad, AMSGrad, gradient descent, and heavy-ball momentum.
Despite the fast progress in neural operator learning, long-sequence modeling still is a standing challenge whereby latent states have been introduced with techniques well derived. Diverging from existing methods that treat latent states as transient variables or decoupled representations, CoEvol-NO introduces a {persistent state} to establish a {co-evolutionary framework}, where the latent state and mesh sequence are updated jointly and bidirectionally. Inspired by classical numerical methods, we model the layer-wise state evolution as a {Predictor-Corrector (PC)} process. Specifically, a ``Predictor'' generates a tentative target, followed by a ``Corrector'' that refines the persistent state via an {error-driven update mechanism}. Furthermore, our theoretical analysis reveals that the widely used \textit{direct substitution} and \textit{residual update} paradigms are essentially {first-order approximations} of this error-driven correction under different loss assumptions. We theoretically prove that CoEvol-NO achieves strict {linear time complexity}. Extensive experiments on five standard benchmarks and two large-scale industrial design tasks demonstrate that CoEvol-NO consistently achieves {state-of-the-art (SOTA)} performance.
Optimization · Non-Convex
We study optimization over non-convex constraint sets that are homeomorphic to a ball, encompassing important problem classes such as star-shaped sets that frequently arise in machine learning and engineering applications. We propose Hom-PGD$^+$, a learning-based and projection-efficient first-order method that efficiently solves such problems without requiring expensive projection or optimization oracles. Our approach leverages an invertible neural network (INN) to learn the homeomorphism between the non-convex constraint set and a unit ball, transforming the original problem into an equivalent ball-constrained optimization where projections admit closed-form solutions. We establish that Hom-PGD$^+$ achieves an $\mathcal{O}(\epsilon^{-2})$ convergence rate to an ($\epsilon + \mathcal{O}(\sqrt{\epsilon_{\rm inn}})$)-approximate stationary solution, where $\epsilon_{\rm inn}$ denotes the homeomorphism learning error. This rate significantly improves upon existing methods for optimization over non-convex sets, while maintaining a per-iteration complexity of only $\mathcal{O}(W)$ for $W$ INN parameters. Experiments on chance-constrained optimization problems in power systems demonstrate that Hom-PGD$^+$ achieves convergence rates comparable to state-of-the-art methods while delivering speedups of up to one order of magnitude.
Optimization · Convex
We present a batched first-order method for solving multiple linear programs in parallel on GPUs. Our approach extends the primal-dual hybrid gradient algorithm to efficiently solve batches of related linear programming problems that arise in mixed-integer programming techniques such as strong branching and bound tightening. By leveraging matrix-matrix operations instead of repeated matrix-vector operations, we obtain significant computational advantages on GPU architectures. We demonstrate the effectiveness of our approach on various case studies and identify the problem sizes where first-order methods outperform traditional simplex-based solvers depending on the computational environment one can use. This is a significant step for the design and development of integer programming algorithms tightly exploiting GPU capabilities where we argue that some specific operations should be allocated to GPUs and performed in full instead of using light-weight heuristic approaches on CPUs.
Deep Learning · Large Language Models
The collective intelligence of Large Language Model (LLM)-based Multi-Agent Systems (MAS) is fundamentally governed by the underlying communication graph. However, discovering task-adaptive structures within this combinatorial search space remains a significant challenge. Existing methods, ranging from heuristic pruning to autoregressive generation, often lack a unified theoretical framework to guide the self-organization of agents into efficient teams. In this paper, we bridge non-equilibrium thermodynamics and generative modeling to formalize multi-agent graph generation as an energy minimization process. Specifically, we frame the emergence of efficient collaboration as a thermodynamic "cooling" process, where initially stochastic interactions converge to a low-energy, structured equilibrium. To implement this, We propose MAGE (Multi-Agent Communication Graph Generation), a score-based diffusion framework that constructs communication graphs by navigating the energy landscape via iterative denoising and first-order gradient guidance. Extensive experiments on representative benchmarks demonstrate that MAGE achieves state-of-the-art performance. Furthermore, qualitative analysis reveals that the generated graphs mirroring the functional specialization of human organizations, validating our thermodynamic hypothesis.
Optimization · Non-Convex
Reinforcement learning (RL) is often hierarchical: an upper-level (UL) learner selects model parameters while a lower-level (LL) decision-making process responds, forming a nested two-level optimization structure captured by bilevel optimization. Most existing bilevel RL methods assume a single-policy LL Markov decision process (MDP), and thus miss the competitive structure in applications such as incentive design where multiple policies interact. We study a class of bilevel optimization problems whose LL is a regularized min--max zero-sum Markov game, and whose UL optimizes through the saddle-point equilibrium induced by the LL game. We propose a penalty-based first-order policy-gradient method built on the Nikaido–Isoda function, termed penalty-augmented Nikaido–Isoda descent–ascent (PANDA), which avoids UL hypergradients and requires no second-order information by exploiting the min--max game structure. We prove that PANDA converges to stationary points of this bilevel problem without restrictive assumptions such as convexity on either the UL or LL objectives. Moreover, PANDA reaches an $\epsilon$-stationary point in $\tilde{\mathcal{O}}(\epsilon^{-1})$ iterations with sample complexity $\tilde{\mathcal{O}}(\epsilon^{-3})$, matching the best-known rates for bilevel RL with single-policy LL MDPs. Experiments further demonstrate superior performance over closely related baselines.
Deep Learning · Other Representation Learning
While diffusion models have advanced anomaly detection, their reliance on multi-step noise schedules introduces significant computational complexity. In this paper, we demonstrate that the generative capability of diffusion is not required for tabular one-class anomaly detection. We revisit core principles of denoising without targeting data generation and present a deep-learning approach that streamlines these objectives into a fixed-noise formulation. Unlike standard denoising autoencoders that rely on reconstruction error, our method utilizes a preconditioning with an explicit linear reference channel. We train a denoising predictor to recover clean samples from perturbed observations and derive anomaly scores from the expected deviation under repeated perturbations. We theoretically motivate this with a stability proxy by a first-order approximation rather than just distance to the manifold. On the well-established ADBench benchmark, our approach achieves improvements over existing methods of 1.22\% in AUCROC and 1.13\% in AUCPR, the most informative and threshold-independent metrics. Our approach emphasizes structural simplicity and efficiency, indicating that a single-step, stability-based objective outperforms complex generative schedules for tabular data.
Optimization · Large Scale, Parallel and Distributed
Decentralized stochastic bi-level optimization has been actively studied in recent years. However, existing studies assume that the lower-level loss function is strongly convex, which limits their applicability to many machine learning models. To address this limitation, in this paper, we propose a novel decentralized stochastic first-order optimization algorithm, which does not require second-order Hessian or Jacobian matrices, for the setting where the lower-level loss function is nonconvex but satisfies the Polyak–Łojasiewicz (PL) condition. Additionally, unlike existing single-agent methods that introduce a regularization term to the lower-level loss function to artificially enforce strong convexity, our algorithm does not require such modification. Moreover, our algorithm employs a constant single-timescale learning rate for updating variables, which is different from the time-dependent and two-timescale learning rate schedules used in prior work. To establish the convergence rate, we develop a new convergence analysis framework for the pure PL condition, rather than relying on the artificial strong convexity introduced through regularization in existing single-agent methods. To the best of our knowledge, this is the first algorithm for nonconvex decentralized bi-level optimization that offers theoretical convergence guarantees under mild conditions. Finally, our extensive experimental results on hyperparameter optimization and model pruning applications validate the efficacy of the proposed algorithm.
Applications · Health / Medicine
Common deep learning approaches for antibody engineering focus on modeling the marginal distribution of sequences. By treating sequences as independent samples, however, these methods overlook affinity maturation as a rich and largely untapped source of information about the evolutionary process by which antibodies explore the underlying fitness landscape. In contrast, classical phylogenetic models explicitly represent evolutionary dynamics but lack the expressivity to capture complex epistatic interactions. We bridge this gap with **CoSiNE**, a continuous-time Markov chain parameterized by a deep neural network. Mathematically, we prove that CoSiNE provides a first-order approximation to the intractable sequential point mutation process, capturing epistatic effects with an error bound that is quadratic in branch length. Empirically, CoSiNE outperforms state-of-the-art language models in zero-shot variant effect prediction by explicitly disentangling selection from context-dependent somatic hypermutation. Finally, we introduce *Guided Gillespie*, a classifier-guided sampling scheme that steers CoSiNE at inference time, enabling efficient optimization of antibody binding affinity toward specific antigens.
Deep Learning · Large Language Models
Activation steering promises to be an extremely parameter-efficient form of adaptation, but its effectiveness depends on critical design choices---such as intervention location and parameterization---that currently rely on empirical heuristics rather than a principled foundation. We establish a first-order equivalence between activation-space interventions and weight-space updates, deriving the conditions under which activation steering can replicate fine-tuning behavior. This equivalence yields a principled framework for steering design and identifies the post-block output as a theoretically-backed and highly expressive intervention site. We further explain why certain intervention locations outperform others and show that weight updates and activation updates play distinct, complementary functional roles. This analysis motivates a new approach---joint adaptation---that trains in both spaces simultaneously. Our post-block steering method achieves accuracy within 0.4%-1.5% of full-parameter tuning while training only $0.04% of model parameters, consistently outperforming prior activation steering methods such as ReFT and PEFT approaches including LoRA while using significantly fewer parameters. Finally, we show that joint adaptation often surpasses the performance ceilings of weight and activation updates in isolation, introducing a new paradigm for efficient model adaptation.
Optimization · Zero-order and Black-box Optimization
Zeroth-order optimizers have recently emerged as an attractive approach for fine-tuning large language models (LLMs), as they avoid backpropagation and can substantially reduce memory overhead relative to standard first-order training. However, existing zeroth-order methods rely on hand-crafted, static sampling strategies that are not adaptable to model-specific structures. To address this, we propose ZO Fine-tuner, a learning-based zeroth-order optimizer for LLMs that automatically learns efficient perturbation strategies through a compact and memory-efficient design. Motivated by the fact that a small set of base LLMs is repeatedly fine-tuned across tasks, ZO Fine-tuner supports one-time per-model training and reuse across downstream tasks with minimal overhead. Therefore, learning the optimizer once for a given LLM and reusing it across diverse downstream tasks is both feasible and highly desirable. Accordingly, ZO Fine-tuner is designed to scale learning to learn (L2L) to the foundation-model era by supporting one-time per-model training with minimal overhead. Experiments on 4 LLMs and 7 datasets show that ZO Fine-tuner outperforms prior zeroth-order baselines in 82.1\% of task-model combinations, thereby demonstrating strong performance and scalability for efficient LLM fine-tuning.
Deep Learning · Theory
Neural tangent kernels (NTKs) are a powerful tool for analyzing deep, non-linear neural networks. In the infinite-width limit, NTKs can easily be computed for most common architectures, yielding full analytic control over the training dynamics. However, at infinite width, important properties of training such as NTK evolution or feature learning are absent. Nevertheless, finite width effects can be included by computing corrections to the Gaussian statistics at infinite width. We introduce Feynman diagrams for computing finite-width corrections to NTK statistics. These dramatically simplify the necessary algebraic manipulations and enable the computation of layer-wise recursion relations for arbitrary statistics involving preactivations, NTKs and certain higher-derivative tensors (dNTK and ddNTK) required to predict the training dynamics at leading order. We demonstrate the feasibility of our framework by extending stability results for deep networks from preactivations to NTKs and proving the absence of finite-width corrections for scale-invariant nonlinearities such as ReLU on the diagonal of the Gram matrix of the NTK. We numerically implement the complete set of equations necessary to compute the first-order corrections for arbitrary inputs and demonstrate that the results follow the statistics of sampled neural networks for widths $n\gtrsim 20$.
Randomized experiments (or A/B tests) are widely used to evaluate interventions in dynamic systems such as recommendation platforms, marketplaces, and digital health. In these settings, interventions affect both current and future system states, so estimating the global average treatment effect (GATE) requires accounting for temporal dynamics, which is especially challenging in the presence of nonstationarity; existing approaches suffer from high bias, high variance, or both. In this paper, we address this challenge via the novel Truncated Policy Gradient (TPG) estimator, which replaces instantaneous outcomes with short-horizon outcome trajectories. The estimator admits a policy-gradient interpretation: it is a truncation of the first-order approximation to the GATE, yielding provable reductions in bias and variance in nonstationary Markovian settings. We further establish a central limit theorem for the TPG estimator and develop a consistent variance estimator that remains valid under nonstationarity with single-trajectory data. We validate our theory with two real-world case studies. The results show that a well-calibrated TPG estimator attains low bias and variance in practical nonstationary settings, and highlight the value of the policy gradient approach in the design of effective estimators despite complex dynamics.