We lay the theoretical foundation for automating optimizer design in gradient-based learning. Based on the greedy principle, we formulate the problem of designing optimizers as maximizing the instantaneous decrease in loss. By treating an optimizer as a function that translates loss gradient signals into parameter motions, the problem reduces to a family of convex optimization problems over the space of optimizers. Solving these problems under various constraints not only recovers a wide range of popular optimizers as closed-form solutions, but also produces the optimal hyperparameters of these optimizers with respect to the problems at hand. This enables a systematic approach to design optimizers and tune their hyperparameters according to the gradient statistics that are collected during the training process. Furthermore, this optimization of optimization can be performed dynamically during training. Just as optimizers train their models by feeding them parameter velocities θ, models can also fit the optimizers to the underlying tasks by feeding gradients g. We are interested in the problem of designing optimiz-ers that maximize the utility of gradient-based learning for a given task. The process of learning manifests as the parameter motion θ driven by the gradient force g applied at each step t. Physics requires a constitutive law that relates kinematic motion to its motive force. In gradient-based learning, optimizers take that role. We can represent an optimizer as a positive semidefinite operator Q 0 that linearly translates the gradients into the parameter updates, θ = Q g. (1) Later sections will reveal that many existing optimizers fall into this category. Q g. (2) Adhering to the greedy paradigm, we turn our original problem of maximizing the utility of learning into a different optimization problem that maximizes this loss drop with respect to the optimizer Q: maximize Problem P1 reveals two design options that bound this maximum: (1) the trust region implied by the feasible set Q Q, and (2) the gradient distribution under the expectation E. Our main focus is on how these two factors determine the optimal optimizer Q Optimizers and their hyperparameters can be dynamically tuned or even be replaced by better ones according to the intermediate probes from the gradients in the middle of training. By reverse engineering commonly used optimizers, we draw the landscape of optimizers that have driven the success of machine learning (Robbins & Monro, 1951; Kingma & Ba, 2015; Loshchilov & Hutter, 2019; Gupta et al., 2018; Martens & Grosse, 2015) into a single picture. This lets us better use the well-studied optimizers in practice and also suggest extensions to them. Note that Σ is a symmetric and positive semidefinite (PSD) matrix of shape d d.

In this work, we study contextual strongly convex simulation optimization and adopt an "optimize then predict" (OTP) approach for real-time decision making. In the offline stage, simulation optimization is conducted across a set of covariates to approximate the optimal-solution function; in the online stage, decisions are obtained by evaluating this approximation at the observed covariate. The central theoretical challenge is to understand how the inexactness of solutions generated by simulation-optimization algorithms affects the optimality gap, which is overlooked in existing studies. To address this, we develop a unified analysis framework that explicitly accounts for both solution bias and variance. Using Polyak-Ruppert averaging SGD as an illustrative simulation-optimization algorithm, we analyze the optimality gap of OTP under four representative smoothing techniques: $k$ nearest neighbor, kernel smoothing, linear regression, and kernel ridge regression. We establish convergence rates, derive the optimal allocation of the computational budget $Γ$ between the number of design covariates and the per-covariate simulation effort, and demonstrate the convergence rate can approximately achieve $Γ^{-1}$ under appropriate smoothing technique and sample-allocation rule. Finally, through a numerical study, we validate the theoretical findings and demonstrate the effectiveness and practical value of the proposed approach.

Abstract--Autonomous robots rely on geometric maps to inform a diverse set of perception and decision-making algorithms. As autonomy requires reasoning and planning on multiple scales of the environment, each algorithm may require a different map for optimal performance. Light Detection And Ranging (LiDAR) sensors generate an abundance of geometric data to satisfy these diverse requirements, but selecting informative, size-constrained maps is computationally challenging as it requires solving an NP-hard combinatorial optimization. In this work we present OptMap: a geometric map distillation algorithm which achieves real-time, application-specific map generation via multiple theoretical and algorithmic innovations. A central feature is the maximization of set functions that exhibit diminishing returns, i.e., submodularity, using polynomial-time algorithms with provably near-optimal solutions. We formulate a novel submodular reward function which quantifies informativeness, reduces input set sizes, and minimizes bias in sequentially collected datasets. Further, we propose a dynamically reordered streaming submod-ular algorithm which improves empirical solution quality and addresses input order bias via an online approximation of the value of all scans. T esting was conducted on open-source and custom datasets with an emphasis on long-duration mapping sessions, highlighting OptMap's minimal computation requirements. Open-source ROS1 and ROS2 packages are available and can be used alongside any LiDAR SLAM algorithm. ODERN autonomous systems use a modular software architecture with separate algorithms for perceiving the environment, planning collision-free paths, estimating vehicle motion, and making higher-level decisions to complete their tasks. Many of these algorithms depend on geometric information about the environment to function properly. As a result, their performance and processing time can vary greatly depending on the quality of the geometric data. For example, trajectory planners use geometric maps to plan collision-free paths, but the density of geometric data is critical for balancing real-time replanning requirements against reliable collision detection. This trade-off is best served by dense geometric maps that specifically capture the intended trajectory corridor (Figure 1a). In contrast, localization entails aligning a source and reference point cloud, a process best served by using a sparse and global reference point could to minimize computation time while maximizing alignment accuracy (Figure 1b). Distribution Statement A: Approved for public release; distribution is unlimited. Map is dense while remaining efficient as only points near the intended trajectory are returned.

We present CompassMax-V3-Thinking, a hundred-billion-scale MoE reasoning model trained with a new RL framework built on one principle: each prompt must matter. Scaling RL to this size exposes critical inefficiencies-zero-variance prompts that waste rollouts, unstable importance sampling over long horizons, advantage inversion from standard reward models, and systemic bottlenecks in rollout processing. To overcome these challenges, we introduce several unified innovations: (1) Multi-Stage Zero-Variance Elimination, which filters out non-informative prompts and stabilizes group-based policy optimization (e.g. GRPO) by removing wasted rollouts; (2) ESPO, an entropy-adaptive optimization method that balances token-level and sequence-level importance sampling to maintain stable learning dynamics; (3) a Router Replay strategy that aligns training-time MoE router decisions with inference-time behavior to mitigate train-infer discrepancies, coupled with a reward model adjustment to prevent advantage inversion; (4) a high-throughput RL system with FP8-precision rollouts, overlapped reward computation, and length-aware scheduling to eliminate performance bottlenecks. Together, these contributions form a cohesive pipeline that makes RL on hundred-billion-scale MoE models stable and efficient. The resulting model delivers strong performance across both internal and public evaluations.

This work aims to learn the directed acyclic graph (DAG) that captures the instantaneous dependencies underlying a multivariate time series. The observed data follow a linear structural vector autoregressive model (SVARM) with both instantaneous and time-lagged dependencies, where the instantaneous structure is modeled by a DAG to reflect potential causal relationships. While recent continuous relaxation approaches impose acyclicity through smooth constraint functions involving powers of the adjacency matrix, they lead to non-convex optimization problems that are challenging to solve. In contrast, we assume that the underlying DAG has only non-negative edge weights, and leverage this additional structure to impose acyclicity via a convex constraint. This enables us to cast the problem of non-negative DAG recovery from multivariate time-series data as a convex optimization problem in abstract form, which we solve using the method of multipliers. Crucially, the convex formulation guarantees global optimality of the solution. Finally, we assess the performance of the proposed method on synthetic time-series data, where it outperforms existing alternatives.

The Lottery Ticket Hypothesis asserts the existence of highly sparse, trainable subnetworks ('winning tickets') within dense, randomly initialized neural networks. However, state-of-the-art methods of drawing these tickets, like Lottery Ticket Rewinding (LTR), are computationally prohibitive, while more efficient saliency-based Pruning-at-Initialization (PaI) techniques suffer from a significant accuracy-sparsity trade-off and fail basic sanity checks. In this work, we argue that PaI's reliance on first-order saliency metrics, which ignore inter-weight dependencies, contributes substantially to this performance gap, especially in the sparse regime. To address this, we introduce Concrete Ticket Search (CTS), an algorithm that frames subnetwork discovery as a holistic combinatorial optimization problem. By leveraging a Concrete relaxation of the discrete search space and a novel gradient balancing scheme (GRADBALANCE) to control sparsity, CTS efficiently identifies high-performing subnetworks near initialization without requiring sensitive hyperparameter tuning. Motivated by recent works on lottery ticket training dynamics, we further propose a knowledge distillation-inspired family of pruning objectives, finding that minimizing the reverse Kullback-Leibler divergence between sparse and dense network outputs (CTS-KL) is particularly effective. Experiments on varying image classification tasks show that CTS produces subnetworks that robustly pass sanity checks and achieve accuracy comparable to or exceeding LTR, while requiring only a small fraction of the computation. For example, on ResNet-20 on CIFAR10, it reaches 99.3% sparsity with 74.0% accuracy in 7.9 minutes, while LTR attains the same sparsity with 68.3% accuracy in 95.2 minutes. CTS's subnetworks outperform saliency-based methods across all sparsities, but its advantage over LTR is most pronounced in the highly sparse regime.

Decoding-based regression, which reformulates regression as a sequence generation task, has emerged as a promising paradigm of applying large language models for numerical prediction. However, its progress is hindered by the misalignment between discrete token-level objectives (e.g., cross-entropy) and continuous numerical values. Existing approaches relying on token-level constraints often fail to capture the global magnitude of the target value, limiting their precision and generalization. In this paper, we propose to unlock the potential of decoding-based regression via Reinforcement Learning (RL). We formulate the generation process as a Markov Decision Process, utilizing sequence-level rewards to enforce global numerical coherence. Extensive experiments on tabular regression and code metric regression demonstrate that our method (specifically with ReMax and GRPO) consistently outperforms both state-of-the-art token-level baselines and traditional regression heads, showing the superiority of introducing sequence-level signals. Our analysis further reveals that RL significantly enhances sampling efficiency and predictive precision, establishing decoding-based regression as a robust and accurate paradigm for general-purpose numerical prediction.

Merging neural networks without retraining is central to federated and distributed learning. Common methods such as weight averaging or Fisher merging often lose accuracy and are unstable across seeds. CoGraM (Contextual Granular Merging) is a multi-stage, context-sensitive, loss-based, and iterative optimization method across layers, neurons, and weight levels that aligns decisions with loss differences and thresholds and prevents harmful updates through rollback. CoGraM is an optimization method that addresses the weaknesses of methods such as Fisher and can significantly improve the merged network.

The emergence of Multimodal Large Language Models (MLLMs) has propelled the development of autonomous agents that operate on Graphical User Interfaces (GUIs) using pure visual input. A fundamental challenge is robustly grounding natural language instructions. This requires a precise spatial alignment, which accurately locates the coordinates of each element, and, more critically, a correct semantic alignment, which matches the instructions to the functionally appropriate UI element. Although Reinforcement Learning with V erifiable Rewards (RL VR) has proven to be effective at improving spatial alignment for these MLLMs, we find that inefficient exploration bottlenecks semantic alignment, which prevent models from learning difficult semantic associations. To address this exploration problem, we present Adaptive Exploration Policy Optimization (AEPO), a new policy optimization framework. AEPO employs a multi-answer generation strategy to enforce broader exploration, which is then guided by a theoretically grounded Adaptive Exploration Reward (AER) function derived from first principles of efficiency η = U/C . Our AEPO-trained models, InfiGUI-G1-3B and InfiGUI-G1-7B, establish new state-of-the-art results across multiple challenging GUI grounding benchmarks, achieving significant relative improvements of up to 9.0% against the naive RL VR baseline on benchmarks designed to test generalization and semantic understanding. Resources are available at https://github.

Designing effective algorithmic components remains a fundamental obstacle in tackling NP-hard combinatorial optimization problems (COPs), where solvers often rely on carefully hand-crafted strategies. Despite recent advances in using large language models (LLMs) to synthesize high-quality components, most approaches restrict the search to a single element - commonly a heuristic scoring function - thus missing broader opportunities for innovation. In this paper, we introduce a broader formulation of solver design as a multi-strategy optimization problem, which seeks to jointly improve a set of interdependent components under a unified objective. To address this, we propose Multi-strategy Optimization via Turn-based Interactive Framework (MOTIF) - a novel framework based on Monte Carlo Tree Search that facilitates turn-based optimization between two LLM agents. At each turn, an agent improves one component by leveraging the history of both its own and its opponent's prior updates, promoting both competitive pressure and emergent cooperation. This structured interaction broadens the search landscape and encourages the discovery of diverse, high-performing solutions. Experiments across multiple COP domains show that MOTIF consistently outperforms state-of-the-art methods, highlighting the promise of turn-based, multi-agent prompting for fully automated solver design.