Daily Papers

We propose Deep Patch Visual Odometry (DPVO), a new deep learning system for monocular Visual Odometry (VO). DPVO uses a novel recurrent network architecture designed for tracking image patches across time. Recent approaches to VO have significantly improved the state-of-the-art accuracy by using deep networks to predict dense flow between video frames. However, using dense flow incurs a large computational cost, making these previous methods impractical for many use cases. Despite this, it has been assumed that dense flow is important as it provides additional redundancy against incorrect matches. DPVO disproves this assumption, showing that it is possible to get the best accuracy and efficiency by exploiting the advantages of sparse patch-based matching over dense flow. DPVO introduces a novel recurrent update operator for patch based correspondence coupled with differentiable bundle adjustment. On Standard benchmarks, DPVO outperforms all prior work, including the learning-based state-of-the-art VO-system (DROID) using a third of the memory while running 3x faster on average. Code is available at https://github.com/princeton-vl/DPVO

Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural representations. This mismatch leaves no suitable method to solve surface PDEs directly within the neural domain, forcing explicit mesh extraction or per-instance residual training, preventing end-to-end workflows. We present a novel, mesh-free formulation that learns a local update operator conditioned on neural (local) shape attributes, enabling surface PDEs to be solved directly where the (neural) data lives. The operator integrates naturally with prevalent neural surface representations, is trained once on a single representative shape, and generalizes across shape and topology variations, enabling accurate, fast inference without explicit meshing or per-instance optimization while preserving differentiability. Across analytic benchmarks (heat equation and Poisson solve on sphere) and real neural assets across different representations, our method slightly outperforms CPM while remaining reasonably close to FEM, and, to our knowledge, delivers the first end-to-end pipeline that solves surface PDEs on both neural and classical surface representations. Code will be released on acceptance.

Self-improving autonomous agents are moving from research prototypes to deployed systems. The primary goal is controllable evolution, or adaptation, from experience with minimal or even no human input. This survey frames modern self-improving agents as adaptive systems that convert experience into accumulated capability gains. We offer a system-level framework that represents a modern agent as a configuration coupling a foundation model with an operational scaffold of prompts, memory, tools, and control logic. Within this framework, self-improvement is formalized as a self-induced update operator that obtains and commits updates to model parameters or scaffold components. We organize prior work by update target and by the signals that drive change, then review applications and discuss evaluation, before closing with open problems and future directions. For convenience, we track technical updates on https://github.com/selfimproving-agent/awesome-Self-Improving-Agents.

Offline reinforcement learning (RL) can fail spectacularly when bootstrapped temporal-difference (TD) updates amplify their own errors, driving the critic toward extreme and unusable Q-values. A key counterintuitive insight of this work is that collapse is not only a property of the backup rule or network architecture: optimizer dynamics themselves can directly trigger or suppress instability. From a control-theoretic viewpoint, we model offline TD learning as a feedback system and analyze Adam-based critic updates. This yields a necessary and sufficient condition for stability of the induced local update dynamics: within the regime we analyze, these dynamics are stable if and only if the spectral radius of the corresponding update operator is strictly below one. Further analysis suggests that standard Adam updates can inadvertently distort the parameter geometry, motivating explicit orthogonality constraints to prevent TD error amplification. To this end, we propose AdamO, an Adam-based optimizer with a decoupled orthogonality correction regulated by a strict task-alignment budget. We prove that this design theoretically guarantees worst-case task safety and preserves Adam's continuous-time dissipative dynamics. Empirically, AdamO is broadly compatible with diverse offline RL baselines, improving stability and returns across a broad suite of benchmarks.

Many LLM-based open-ended search systems freeze the foundation model that proposes improvements to existing solutions, which may bottleneck long-run progress. Recent work has explored updating the proposal model at test time [arXiv:2511.23473], but the update strategy is still typically hand-specified. Therefore, this study investigated whether an LLM can use task feedback to decide how it should update its weights. For tractability, we focused on the simpler case where there is only one round of self-improvement, and restricted the update operator to self-supervised next token prediction (NTP), leaving the model freedom in choosing its training data and key NTP hyperparameters. Using the Self-Adapting Language Models (SEAL) [arXiv:2506.10943] framework as a testbed, we relaxed its fixed human template constraint and allowed the model to generate its own self-edit templates, thereby giving it more control over its training data and hyperparameters. Two variants were studied, differing in whether template generation was conditioned on a lightweight archive of past templates. In SEAL's Single-Passage Knowledge Incorporation setting with Qwen3-8B on SQuAD [arXiv:1606.05250], the no-archive variant performed comparably to the weaker "Implications" baseline, while the archive variant outperformed "Implications" and approached the strongest human-designed "Rewrite" baseline without surpassing it. Further analysis of collapse in the model's exploration revealed that a naive archive can confer some short-term robustness but can also accelerate homogenization, suggesting that explicit novelty pressure may be required to consistently advance beyond carefully optimized human strategies. Our code is available at https://github.com/cheongalc/search-self-edit-strategies .

Koopman theory turns nonlinear dynamics into a linear spectral problem. In computation, however, everything depends on a hard finite-dimensional choice: the observables must be expressive, nearly invariant under the dynamics, and, ideally, compatible with composition. Deep Koopman methods learn flexible coordinates, whereas structure-preserving methods enforce operator identities on fixed dictionaries. We combine these ideas by introducing Deep Embedded Multiplicative Dynamic Mode Decomposition (DeepMDMD), a method that learns a latent space and a partition of it, while enforcing the Koopman product rule as an exact algebraic constraint. Training alternates between an exact multiplicative operator update and a differentiable latent-clustering step that promotes Koopman closure. The result is a finite transition map on learned latent cells. Its nonzero spectrum lies on the unit circle, its dictionary is shaped by the dynamics rather than by ambient geometry, and forecasts are made in latent coordinates before being decoded to physical space. Across Hamiltonian, chaotic, and fluid examples, DeepMDMD learns dictionaries that are far more compact and dynamically coherent than those produced by geometric MDMD partitions. It reduces spectral pollution, reveals richer continuous-spectrum structure, and gives stable forecasts under severe noise. In high-dimensional flows, including a 158,624-dimensional cylinder wake and a noisy Re=20,000 lid-driven cavity, it preserves coherent structures and long-time spectral statistics where state-space MDMD fails. These results suggest a practical rule for Koopman learning: learn the coordinates, constrain the algebra.

Geometric Attention (GA) specifies an attention layer by four independent inputs: a finite carrier (what indices are addressable), an evidence-kernel rule (how masked proto-scores and a link induce nonnegative weights), a probe family (which observables are treated as admissible), and an anchor/update rule (which representative kernel is selected and how it is applied). Probe families induce an operational equivalence relation on kernels and therefore a gauge; anchors select representatives relative to that probe. Under a scalar relational-work representation and a multiplicative compositionality law for evidence, the admissible link family is exponential, yielding Gibbs weights; with row anchoring this includes the softmax kernel family as a subregime. After quotienting unary row/column score fields, the remaining interaction component admits a canonical rank-r normal form (Eckart-Young/SVD); dot-product score charts implement the corresponding low-rank interaction regime. Fixing the carrier and extensionalizing the update yields the standard fixed-token Transformer attention operator; allowing carrier updates yields adaptive-carrier and staged-depth regimes. The operator language also supports multihead/mixed kernels, plan-based anchors (e.g., entropic OT/Sinkhorn), and unary operators (e.g., FFN-style fields) as explicit regime choices. This separates invariant structure from modeling choice, enabling principled comparison and extension of attention mechanisms, and attention-based architectures.

Autoregressive learning of time-stepping operators provides an effective approach to data-driven partial differential equation (PDE) simulation, yet for conservation laws, they face a fundamental challenge: learned updates may violate global conservation over long rollouts. For the important subclass of mass-conservation-type equations, the problem is compounded by inherent physical bounds (e.g., nonnegativity or concentrations in [0,1]) whose violation further destabilizes predictions. We introduce FluxNet, which learns cumulative transport amounts representing the total conserved quantity redistributed between each cell and a configurable neighborhood over the full surrogate interval. A conservative update guarantees exact discrete conservation by construction; modular capacity-constrained transport heads (L, U, and D) enforce lower bounds, upper bounds, or near-zero dual-bound violations through architectural design. Unlike flux-rate surrogates that require temporal integration and thus inherit CFL constraints, FluxNet involves no such integration; configurable transport neighborhoods enable large-timestep prediction at full spatial resolution. Ghost cells extend the framework to non-periodic boundaries. Experiments on four benchmarks (1D convection--diffusion, 2D shallow water, 1D traffic flow, 2D Cahn--Hilliard) demonstrate exact conservation, structural bound preservation, architecture modularity, and superior stability over flux-rate surrogates at large temporal strides. The code is publicly available at: https://github.com/Lan-zs/FluxNet.

We propose a delay-agnostic asynchronous coordinate update algorithm (DEGAS) for computing operator fixed points, with applications to asynchronous optimization. DEGAS includes novel asynchronous variants of ADMM and block-coordinate descent as special cases. We prove that DEGAS converges under both bounded and unbounded delays under delay-free parameter conditions. We also validate by theory and experiments that DEGAS adapts well to the actual delays. The effectiveness of DEGAS is demonstrated by numerical experiments on classification problems.

We propose Super-resolution Neural Operator (SRNO), a deep operator learning framework that can resolve high-resolution (HR) images at arbitrary scales from the low-resolution (LR) counterparts. Treating the LR-HR image pairs as continuous functions approximated with different grid sizes, SRNO learns the mapping between the corresponding function spaces. From the perspective of approximation theory, SRNO first embeds the LR input into a higher-dimensional latent representation space, trying to capture sufficient basis functions, and then iteratively approximates the implicit image function with a kernel integral mechanism, followed by a final dimensionality reduction step to generate the RGB representation at the target coordinates. The key characteristics distinguishing SRNO from prior continuous SR works are: 1) the kernel integral in each layer is efficiently implemented via the Galerkin-type attention, which possesses non-local properties in the spatial domain and therefore benefits the grid-free continuum; and 2) the multilayer attention architecture allows for the dynamic latent basis update, which is crucial for SR problems to "hallucinate" high-frequency information from the LR image. Experiments show that SRNO outperforms existing continuous SR methods in terms of both accuracy and running time. Our code is at https://github.com/2y7c3/Super-Resolution-Neural-Operator

We present a data-adaptive method for parameter-efficient fine-tuning of large neural networks. Standard low-rank adaptation methods improve efficiency by restricting each layer update to a fixed low-rank form, but this static parameterization can be too rigid when the appropriate correction depends on the input and on the evolving depth-wise computation of the network. Our approach replaces a purely layer-local adapter with a shared queryable memory of low-rank update atoms. For each block of layers, the model forms a query from the current low-rank state and a running summary of previous blocks, uses this query to retrieve a content-dependent combination of shared update components via attention, and applies the resulting routed operator within the low-rank bottleneck. In this way, the method retains the efficiency and scalability of low-rank adaptation while allowing the effective update to vary across inputs and to share reusable structure across layers. The resulting architecture provides a principled middle ground between static LoRA-style updates and fully generated parameter updates: it remains compact and parameter-efficient while supporting dynamic, context-sensitive adaptation. Further, we incorporate instruction-regularization by augmenting routing logits with a language-induced prior over update atoms, thereby biasing the selection of low-rank transformations toward semantically relevant directions without generating unconstrained parameter updates. Experiments on noisy non-linear regression tasks and LLM fine-tuning suggest that this queryable update-memory formulation can improve final test performance and training stability compared to standard low-rank adaptation, while using a comparable number of trainable parameters.

Stereo depth estimation is fundamental to underwater robotic perception, yet suffers from severe domain shifts caused by wavelength-dependent light attenuation, scattering, and refraction. Recent approaches leverage monocular foundation models with GRU-based iterative refinement for underwater adaptation; however, the sequential gating and local convolutional kernels in GRUs necessitate multiple iterations for long-range disparity propagation, limiting performance in large-disparity and textureless underwater regions. In this paper, we propose StereoAdapter-2, which replaces the conventional ConvGRU updater with a novel ConvSS2D operator based on selective state space models. The proposed operator employs a four-directional scanning strategy that naturally aligns with epipolar geometry while capturing vertical structural consistency, enabling efficient long-range spatial propagation within a single update step at linear computational complexity. Furthermore, we construct UW-StereoDepth-80K, a large-scale synthetic underwater stereo dataset featuring diverse baselines, attenuation coefficients, and scattering parameters through a two-stage generative pipeline combining semantic-aware style transfer and geometry-consistent novel view synthesis. Combined with dynamic LoRA adaptation inherited from StereoAdapter, our framework achieves state-of-the-art zero-shot performance on underwater benchmarks with 17% improvement on TartanAir-UW and 7.2% improvment on SQUID, with real-world validation on the BlueROV2 platform demonstrates the robustness of our approach. Code: https://github.com/AIGeeksGroup/StereoAdapter-2. Website: https://aigeeksgroup.github.io/StereoAdapter-2.

Recently, evolutionary reinforcement learning has obtained much attention in various domains. Maintaining a population of actors, evolutionary reinforcement learning utilises the collected experiences to improve the behaviour policy through efficient exploration. However, the poor scalability of genetic operators limits the efficiency of optimising high-dimensional neural networks. To address this issue, this paper proposes a novel cooperative coevolutionary reinforcement learning (CoERL) algorithm. Inspired by cooperative coevolution, CoERL periodically and adaptively decomposes the policy optimisation problem into multiple subproblems and evolves a population of neural networks for each of the subproblems. Instead of using genetic operators, CoERL directly searches for partial gradients to update the policy. Updating policy with partial gradients maintains consistency between the behaviour spaces of parents and offspring across generations. The experiences collected by the population are then used to improve the entire policy, which enhances the sampling efficiency. Experiments on six benchmark locomotion tasks demonstrate that CoERL outperforms seven state-of-the-art algorithms and baselines. Ablation study verifies the unique contribution of CoERL's core ingredients.

Optimizers are crucial for the efficient training of Large Language Models (LLMs). While AdamW is the de facto standard, recent structure-aware optimizers like Muon have emerged, which regularize gradient updates by operating on entire weight matrices. The Muon optimizer balances the gradient updates along all the directions. However, Muon's reliance on the matrix sign function can lead to training instability, exhibits incompatibility when fine-tuning models pre-trained with AdamW. To address these limitations, we propose REG, a novel optimizer that replaces Muon's aggressive matrix sign operator with the Row-and-Column-Scaling (RACS) operator. Theoretically grounded in balancing a matrix, the RACS operator regularizes the update steps in a less drastic manner, making it simpler to implement and more compatible with established training dynamics. Through extensive empirical experiments on LLM training, we demonstrate that our REG optimizer not only achieves superior performance and stability over AdamW, but also maintains consistency with the AdamW training paradigm. This consistency is particularly evident during the fine-tuning stage, where REG optimizer avoids the performance degradation observed with Muon.

We propose a novel theoretical framework to understand contrastive self-supervised learning (SSL) methods that employ dual pairs of deep ReLU networks (e.g., SimCLR). First, we prove that in each SGD update of SimCLR with various loss functions, including simple contrastive loss, soft Triplet loss and InfoNCE loss, the weights at each layer are updated by a covariance operator that specifically amplifies initial random selectivities that vary across data samples but survive averages over data augmentations. To further study what role the covariance operator plays and which features are learned in such a process, we model data generation and augmentation processes through a hierarchical latent tree model (HLTM) and prove that the hidden neurons of deep ReLU networks can learn the latent variables in HLTM, despite the fact that the network receives no direct supervision from these unobserved latent variables. This leads to a provable emergence of hierarchical features through the amplification of initially random selectivities through contrastive SSL. Extensive numerical studies justify our theoretical findings. Code is released in https://github.com/facebookresearch/luckmatters/tree/master/ssl.

Scaling reinforcement learning for visual mathematical reasoning requires more than generating harder questions: as data volume grows, the reward labels themselves must remain reliable. Yet existing data pipelines scale supervision while trusting the labeller, and policy-side methods assume the underlying answers are already correct. We instead treat scaling as a verifiable data-construction problem and decouple two axes before any policy update: prompt difficulty, expanded by route-specific evolution operators, and answer reliability, enforced by offline hypothesis-test falsification. We instantiate this as VeriEvol, an iterative framework with two extensible components: a type-aware evolution module that rewrites low-difficulty image-question seeds into harder, image-grounded prompts; and HTV-Agent, a verifier that accepts an answer only after multi-source counter-evidence has failed to refute it. The resulting verified data scales in volume, extends by adding evolution routes or verifier channels, and plugs directly into existing GRPO-style RL recipes. On a five-benchmark visual-math suite, scaling evolved SFT data from 10K to 250K samples raises the mean accuracy from 35.42 to 54.73; then, with backbone, SFT initialization, and GRPO recipe held fixed, VeriEvol adds a cumulative +3.88 over an un-evolved RL baseline, of which +1.82 comes from evolved prompts and +2.06 from the HTV-Agent verifier. We release the prompts, data, models, code, and the full verifier trace of every sample, so that downstream work can scale and audit the pipeline rather than only inspect its outputs.

Sign-based optimization algorithms, such as SignSGD, have garnered significant attention for their remarkable performance in distributed learning and training large foundation models. Despite their empirical superiority, SignSGD is known to diverge on non-smooth objectives, which are ubiquitous in modern machine learning due to ReLUs, max-pools, and mixture-of-experts. To overcome this fundamental limitation, we propose StoSignSGD, an algorithm that injects structural stochasticity into the sign operator while maintaining an unbiased update step. In the regime of (online) convex optimization, our theoretical analysis shows that StoSignSGD rigorously resolves the non-convergence issues of SignSGD, achieving a sharp convergence rate matching the lower bound. For the more challenging non-convex non-smooth optimization, we introduce generalized stationary measures that encompass prior definitions, proving that StoSignSGD improves upon the best-known complexity bounds by dimensional factors. Empirically, StoSignSGD exhibits robust stability and superior efficiency across diverse large language model (LLM) training regimes. Notably, in low-precision FP8 pretraining -- a setting where AdamW fails catastrophically -- StoSignSGD remains highly stable and yields a remarkable 1.44times to 2.14times speedup relative to established baselines. Furthermore, when fine-tuning 7B LLMs on mathematical reasoning tasks, StoSignSGD delivers substantial performance gains over both AdamW and SignSGD. Finally, to dissect the mechanisms driving its success, we develop a sign conversion framework capable of transforming any general optimizer into its unbiased, sign-based counterpart. Utilizing this framework, we deconstruct the core components of StoSignSGD and present a comprehensive ablation study to empirically validate our algorithmic design choices.

Weight-space model merging is usually formulated as an algebraic operation on checkpoints, yet at LLM scale the limiting resource is often the set of expert weights that must be read. We introduce MergePipe, a budget-aware execution layer that casts LLM merging as an expert access-set problem: given a merge operator and a checkpoint family in a shared weight coordinate system, choose which expert delta blocks to access under an explicit I/O budget. MergePipe indexes parameter blocks, builds deterministic access plans, and executes the induced budgeted merge with replayable manifests. The plan is budget-sound by construction and recovers the full-read merge at full budget; for fixed-coefficient additive operators, the omitted-update error is bounded by the norm of omitted deltas. Across Qwen and Llama merging workloads, MergePipe reduces expert-read I/O by up to an order of magnitude and achieves up to 11times speedups. Representative budget sweeps show O(10^{-3}) parameter deviation from full-read merges and no monotonic degradation on downstream benchmarks.

Retrieval-augmented generation (RAG) systems commonly improve robustness via query-time adaptations such as query expansion and iterative retrieval. While effective, these approaches are inherently stateless: adaptations are recomputed for each query and discarded thereafter, precluding cumulative learning and repeatedly incurring inference-time cost. Index-side approaches like key expansion introduce persistence but rely on offline preprocessing or heuristic updates that are weakly aligned with downstream task utility, leading to semantic drift and noise accumulation. We propose Evolving Retrieval Memory (ERM), a training-free framework that transforms transient query-time gains into persistent retrieval improvements. ERM updates the retrieval index through correctness-gated feedback, selectively attributes atomic expansion signals to the document keys they benefit, and progressively evolves keys via stable, norm-bounded updates. We show that query and key expansion are theoretically equivalent under standard similarity functions and prove convergence of ERM's selective updates, amortizing optimal query expansion into a stable index with zero inference-time overhead. Experiments on BEIR and BRIGHT across 13 domains demonstrate consistent gains in retrieval and generation, particularly on reasoning-intensive tasks, at native retrieval speed.

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

Residual connections are pivotal for deep neural networks, enabling greater depth by mitigating vanishing gradients. However, in standard residual updates, the module's output is directly added to the input stream. This can lead to updates that predominantly reinforce or modulate the existing stream direction, potentially underutilizing the module's capacity for learning entirely novel features. In this work, we introduce Orthogonal Residual Update: we decompose the module's output relative to the input stream and add only the component orthogonal to this stream. This design aims to guide modules to contribute primarily new representational directions, fostering richer feature learning while promoting more efficient training. We demonstrate that our orthogonal update strategy improves generalization accuracy and training stability across diverse architectures (ResNetV2, Vision Transformers) and datasets (CIFARs, TinyImageNet, ImageNet-1k), achieving, for instance, a +4.3\%p top-1 accuracy gain for ViT-B on ImageNet-1k.

In this work, we propose marginalized operators, a new class of off-policy evaluation operators for reinforcement learning. Marginalized operators strictly generalize generic multi-step operators, such as Retrace, as special cases. Marginalized operators also suggest a form of sample-based estimates with potential variance reduction, compared to sample-based estimates of the original multi-step operators. We show that the estimates for marginalized operators can be computed in a scalable way, which also generalizes prior results on marginalized importance sampling as special cases. Finally, we empirically demonstrate that marginalized operators provide performance gains to off-policy evaluation and downstream policy optimization algorithms.

Backward compatibility of model predictions is a desired property when updating a machine learning driven application. It allows to seamlessly improve the underlying model without introducing regression bugs. In classification tasks these bugs occur in the form of negative flips. This means an instance that was correctly classified by the old model is now classified incorrectly by the updated model. This has direct negative impact on the user experience of such systems e.g. a frequently used voice assistant query is suddenly misclassified. A common reason to update the model is when new training data becomes available and needs to be incorporated. Simply retraining the model with the updated data introduces the unwanted negative flips. We study the problem of regression during data updates and propose Backward Compatible Weight Interpolation (BCWI). This method interpolates between the weights of the old and new model and we show in extensive experiments that it reduces negative flips without sacrificing the improved accuracy of the new model. BCWI is straight forward to implement and does not increase inference cost. We also explore the use of importance weighting during interpolation and averaging the weights of multiple new models in order to further reduce negative flips.

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

We propose an abstract framework to establish the convergence of the iterates of stochastic versions of a broad range of monotone operator splitting methods in Hilbert spaces. This framework allows for the introduction of stochasticity at several levels: approximation of operators, selection of coordinates and operators in block-iterative implementations, and relaxation parameters. The proposed analysis involves a reduced inclusion model with two operators. At each iteration, stochastic approximations to points in the graphs of these two operators are used to form the update. The results are applied to derive the almost sure and L^2 convergence of stochastic versions of the proximal point algorithm, as well as of randomized block-iterative projective splitting methods for solving systems of coupled inclusions involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators combined via various monotonicity-preserving operations.

We present a generalization of the proximal operator defined through a convex combination of convex objectives, where the coefficients are updated in a minimax fashion. We prove that this new operator is Bregman firmly nonexpansive with respect to a Bregman divergence that combines Euclidean and information geometries.

In several recently proposed stochastic optimization methods (e.g. RMSProp, Adam, Adadelta), parameter updates are scaled by the inverse square roots of exponential moving averages of squared past gradients. Maintaining these per-parameter second-moment estimators requires memory equal to the number of parameters. For the case of neural network weight matrices, we propose maintaining only the per-row and per-column sums of these moving averages, and estimating the per-parameter second moments based on these sums. We demonstrate empirically that this method produces similar results to the baseline. Secondly, we show that adaptive methods can produce larger-than-desired updates when the decay rate of the second moment accumulator is too slow. We propose update clipping and a gradually increasing decay rate scheme as remedies. Combining these methods and dropping momentum, we achieve comparable results to the published Adam regime in training the Transformer model on the WMT 2014 English-German machine translation task, while using very little auxiliary storage in the optimizer. Finally, we propose scaling the parameter updates based on the scale of the parameters themselves.

Retrieval-Augmented Generation (RAG) is a technique that enhances the capabilities of large language models (LLMs) by incorporating external knowledge sources. This method addresses common LLM limitations, including outdated information and the tendency to produce inaccurate "hallucinated" content. However, the evaluation of RAG systems is challenging, as existing benchmarks are limited in scope and diversity. Most of the current benchmarks predominantly assess question-answering applications, overlooking the broader spectrum of situations where RAG could prove advantageous. Moreover, they only evaluate the performance of the LLM component of the RAG pipeline in the experiments, and neglect the influence of the retrieval component and the external knowledge database. To address these issues, this paper constructs a large-scale and more comprehensive benchmark, and evaluates all the components of RAG systems in various RAG application scenarios. Specifically, we have categorized the range of RAG applications into four distinct types-Create, Read, Update, and Delete (CRUD), each representing a unique use case. "Create" refers to scenarios requiring the generation of original, varied content. "Read" involves responding to intricate questions in knowledge-intensive situations. "Update" focuses on revising and rectifying inaccuracies or inconsistencies in pre-existing texts. "Delete" pertains to the task of summarizing extensive texts into more concise forms. For each of these CRUD categories, we have developed comprehensive datasets to evaluate the performance of RAG systems. We also analyze the effects of various components of the RAG system, such as the retriever, the context length, the knowledge base construction, and the LLM. Finally, we provide useful insights for optimizing the RAG technology for different scenarios.

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon with three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). By rebalancing the momentum matrix before finite-step Newton--Schulz orthogonalization, {\method} improves the geometry seen by orthogonalization. We show that finite-step orthogonalization is governed by the input spectrum, especially stable rank and condition number, and that row/column normalization acts as a zeroth-order surrogate for whitening. For hidden matrix weights, R is the default variant. Theoretically, {\method} (R) retains the standard mathcal O(T^{-1/4}) Muon-type nonconvex stationarity guarantee with decoupled weight decay and a horizon-free diminishing learning-rate schedule, and extends it to finite-step NS5 up to an explicit inexactness constant. In LLaMA2 pretraining on C4, {\method} (R) consistently outperforms Muon on 130M, 350M, and 1B models, with faster convergence and lower validation perplexity. The code is available at the https://github.com/MaeChd/muon-eq{MuonEq codebase}.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.

In reinforcement learning, agents collect state information and rewards through environmental interactions, essential for policy refinement. This process is notably time-consuming, especially in complex robotic simulations and real-world applications. Traditional algorithms usually re-engage with the environment after processing a single batch of samples, thereby failing to fully capitalize on historical data. However, frequently observed states, with reliable value estimates, require minimal updates; in contrast, rare observed states necessitate more intensive updates for achieving accurate value estimations. To address uneven sample utilization, we propose Novelty-guided Sample Reuse (NSR). NSR provides extra updates for infrequent, novel states and skips additional updates for frequent states, maximizing sample use before interacting with the environment again. Our experiments show that NSR improves the convergence rate and success rate of algorithms without significantly increasing time consumption. Our code is publicly available at https://github.com/ppksigs/NSR-DDPG-HER.

We introduce Evolution Guided General Optimization via Low-rank Learning (EGGROLL), an evolution strategies (ES) algorithm designed to scale backprop-free optimization to large population sizes for modern large neural network architectures with billions of parameters. ES is a set of powerful blackbox optimisation methods that can handle non-differentiable or noisy objectives with excellent scaling potential through parallelisation. Na{ï}ve ES becomes prohibitively expensive at scale due to the computational and memory costs associated with generating matrix perturbations EinR^{mtimes n} and the batched matrix multiplications needed to compute per-member forward passes. EGGROLL overcomes these bottlenecks by generating random matrices Ain R^{mtimes r}, Bin R^{ntimes r} with rll min(m,n) to form a low-rank matrix perturbation A B^top that are used in place of the full-rank perturbation E. As the overall update is an average across a population of N workers, this still results in a high-rank update but with significant memory and computation savings, reducing the auxiliary storage from mn to r(m+n) per layer and the cost of a forward pass from O(mn) to O(r(m+n)) when compared to full-rank ES. A theoretical analysis reveals our low-rank update converges to the full-rank update at a fast Oleft(1{r}right) rate. Our experiments show that (1) EGGROLL does not compromise the performance of ES in tabula-rasa RL settings, despite being faster, (2) it is competitive with GRPO as a technique for improving LLM reasoning, and (3) EGGROLL enables stable pre-training of nonlinear recurrent language models that operate purely in integer datatypes.

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

A sequence of operators T_n from a Hilbert space {mathfrak H} to Hilbert spaces {mathfrak K}_n which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator T from {mathfrak H} to a Hilbert space {mathfrak K}. Moreover, the closability or closedness of T_n is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to neural operators leads to significantly lower errors for long-term forecasts as well as longer time horizons without qualitative signs of divergence compared to the original models for these systems. We open-source our https://github.com/mikemccabe210/stabilizing_neural_operators{code} for reproducibility.

We introduce Qronos -- a new state-of-the-art post-training quantization algorithm that sequentially rounds and updates neural network weights. Qronos not only explicitly corrects errors due to both weight and activation quantization, but also errors resulting from quantizing previous layers. Our iterative algorithm is based on an interpretable and disciplined optimization framework that subsumes and surpasses existing data-driven approaches. At each step, Qronos alternates between error correction and diffusion via optimal update rules. Importantly, we prove that Qronos admits an efficient implementation that uses the Cholesky decomposition for solving least-squares problems. We also demonstrate that Qronos is compatible with existing transformation techniques such as Hadamard-based incoherence processing and weight-activation scaling equalization, among others. We evaluate Qronos using recent autoregressive language generation models in the Llama3 family; Qronos consistently outperforms previous state-of-the-art adaptive rounding methods when quantizing the weights, activations, and/or KV caches.

Class-Incremental Learning (CIL) requires a learning system to continually learn new classes without forgetting. Despite the strong performance of Pre-Trained Models (PTMs) in CIL, a critical issue persists: learning new classes often results in the overwriting of old ones. Excessive modification of the network causes forgetting, while minimal adjustments lead to an inadequate fit for new classes. As a result, it is desired to figure out a way of efficient model updating without harming former knowledge. In this paper, we propose ExpAndable Subspace Ensemble (EASE) for PTM-based CIL. To enable model updating without conflict, we train a distinct lightweight adapter module for each new task, aiming to create task-specific subspaces. These adapters span a high-dimensional feature space, enabling joint decision-making across multiple subspaces. As data evolves, the expanding subspaces render the old class classifiers incompatible with new-stage spaces. Correspondingly, we design a semantic-guided prototype complement strategy that synthesizes old classes' new features without using any old class instance. Extensive experiments on seven benchmark datasets verify EASE's state-of-the-art performance. Code is available at: https://github.com/sun-hailong/CVPR24-Ease