M+Adam

We introduce AdamX, a first-order optimizer that incorporates cosine similarity as an adaptive mechanism for controlling update magnitudes. The proposed method is scalable, model-agnostic, and straightforward to integrate into existing training pipelines. We further introduce a variance rectification scheme that promotes smoother optimization during the early stages of training. Overall, we provide empirical evidence that AdamX achieves competitive convergence rates across a range of benchmark datasets and architectures. Performance is evaluated in terms of the number of epochs required to reach predefined performance thresholds under a fixed hyperparameter budget. Code and Experiments available at: https://github.com/FranciscoCaldas/adamX.

Multi-Token Prediction (MTP) has emerged as an effective paradigm that augments a shared Large Language Model backbone with auxiliary heads, training the model to predict several future tokens in parallel to enrich its supervision signal and accelerate inference. However, existing training frameworks adopt a rigid, fixed-length prediction horizon, disregarding the highly non-uniform information density of natural language and code. Forcing the auxiliary heads to predict across high-entropy semantic boundaries injects noisy, conflicting training signals; because these heads share the backbone's latent representations, the resulting gradients backpropagate and interfere with the model's core capabilities. We propose AdaMTP, an adaptive training paradigm that dynamically aligns the prediction horizon with the intrinsic predictability of the sequence. At its core, an entropy-based segmentation algorithm leverages the base model to detect sudden surges in uncertainty as semantic boundaries, partitioning sequences into variable-length groups. Each token is assigned an adaptive prediction depth, and a dynamically masked MTP objective suppresses the loss for predictions that cross these boundaries, attenuating the noisy gradients that degrade the backbone. Across mathematical reasoning, code generation, and general benchmarks on three backbones (Llama-3.1-8B, Qwen-2.5-7B, Gemma-3-12B), AdaMTP consistently outperforms standard MTP in both task performance and inference speedup.

Long-term multimodal memory must support not only retrieving relevant information but also computing over observations accumulated across interactions. Existing systems largely emphasize \emph{retrieval memory}, organizing interaction histories through summaries and indexes to return query-relevant information at multiple granularities, from high-level abstractions to underlying records. In this paper, we formulate \emph{analytic memory} as a complementary abstraction that organizes recurring multimodal observations into queryable structures supporting filtering, aggregation, ranking, and temporal comparison. We present AdaMM, a framework that jointly supports retrieval and analytic memory. Rather than relying on application-defined schemas, AdaMM extracts provenance-linked attribute-value observations from dialogue, images, and contextual metadata, discovers recurring field structures, and materializes them for analytical access. At inference time, a memory-aware planner decomposes queries into retrieval and analytic operations and routes each operation to the appropriate tools. Experiments on two long-term multimodal memory benchmarks, MemEye and MemGallery, show that AdaMM improves performance by up to 11.3% and 6.9%, respectively.

Interpreting optimizers as gradient-flow discretizations has motivated applying higher-order Runge-Kutta (RK) integrators to neural networks. We build a representative Adam variant (Bogacki-Shampine 3(2) RK pair, FSAL reuse, local-error step control) and evaluate it under a strict compute-matched protocol giving every method the same gradient-evaluation budget - an accounting this literature rarely enforces. Under it the RK variant loses to plain Adam on training loss in both minibatch and full-batch (RK's best-case) training. Instrumenting it shows the "adaptivity" is illusory: normalized error stays far below tolerance, the step size pins at its growth cap from step one (98-100 percent of steps), and no rtol x hmax x h0 setting makes it act; tolerances spanning 100x give bit-identical trajectories. The method is exactly fixed-step Adam with an averaged gradient at 3-4x cost. Repairing it (true reject branch; error on the applied map) reverses the full-batch result - about 40x lower training loss than tuned Adam - and a fixed-step control isolates adaptivity (an emergent warmup-and-growth schedule) as the mechanism. But the gain is fragile to the initial step size and does not reach test accuracy. A pre-registered follow-up rules out the obvious explanations: deeper minimization does not overfit, and an explicit temperature knob only hurts - leaving a trajectory effect, the controller selecting a minimum generalizing 1.3-3.4 points below first-order descent at equal depth. An n=10 study confirms one secondary effect: gradient averaging is a genuine implicit regularizer, beating lr-matched Adam and AdamW on 10/10 seeds - yet RMSprop and NAdam match or beat it at a third the per-step cost. Higher-order adaptive integration buys deeper deterministic minimization and a small regularization effect, but nothing a cheaper, well-tuned first-order baseline does not already provide.

Training with quantized weights can reduce costs but often results in degraded accuracy, especially when optimization is carried out in low precision, without storing high-precision copies. We identify a key failure mode: under low precision, standard optimizers can get stuck and not make progress, especially at large weight magnitudes due to coarse mantissa resolution. To overcome this, multiplicative updates have been previously proposed, in place of additive updates in standard optimizers. While successful under extremely low precision, such as under the logarithmic number system, they suffer from failures near zero and across sign changes. The failure modes of additive and multiplicative updates are therefore complementary. To exploit this, we propose M+Adam, which combines both update types: additive steps handle sign changes and small magnitudes, while multiplicative steps ensure progress at large magnitudes when additive updates are zeroed out under rounding. We prove monotone descent for M+Adam under standard smoothness assumptions. Across LLaMA-style pretraining with 60M-1B models, 1x-8x Chinchilla budgets, and using only BF16, FP8, and FP4 master weights, M+Adam consistently improves low-precision training.

Long-term memory systems allow LLM agents to preserve information beyond a single context window, but most systems focus on storing and retrieving facts after extraction, leaving the write decision under-specified. What deserves memory can depend on the user's current task, topic, activity, or interaction partner, while uniform extraction applies one notion of importance across these different situations. We formulate this challenge as preference-conditioned write control and introduce AdaMem, which uses adaptive natural-language Memory Policies to personalize what an agent writes to memory. Each policy represents the user's memory preference for a particular interaction context, is updated from periodic feedback, and controls subsequent memory writing. We evaluate this loop in AdaMem-Bench, which assigns different memory preferences to six concurrent interaction personas across five ten-week stories. Across two extraction models and two feedback modes, AdaMem improves average QA accuracy over Mem0 from 80.0% to 84.35% while reducing persistent memory by 9.27%. Our analyses show that explicit feedback helps models learn better memory policies, but current models still struggle to translate those policies into reliably selective writing behavior. AdaMem thus demonstrates the promise of adaptive write control while exposing policy execution as a central limitation of current memory agents. Our code is publicly available: https://github.com/galaxyChen/AdaMem

Publicly documented accelerator architectures generally separate training computation from optimizer-state updates or rely on external memory and host orchestration. This paper presents NeuronFabric, a software reference architecture intended for future FPGA and ASIC implementations of transformer training with local Adam updates. A complete C# prototype implements forward pass, backpropagation, and Adam optimization without external machine-learning frameworks. The goal is to validate numerical correctness and memory requirements before hardware implementation. The evaluated model is a 334K-parameter autoregressive transformer (d=88, H=4, f=264, L=4, vocab=256) trained on the Shakespeare corpus. The BF16W configuration achieves evaluation loss 1.5426 after 80K samples, compared with 1.5224 for an FP32 GPU reference, while producing coherent character-level text. The paper introduces BF16W, which stores weights in BF16 while retaining Adam optimizer moments in FP32. This reduces memory requirements for on-chip training. A 334K-parameter FP32 model with Adam moments requires approximately 4.0 MB, matching the BRAM capacity of a Xilinx ZCU102 device. The BF16W variant requires approximately 3.34 MB, leaving memory available for activation storage. We describe the vocabulary-budget constraint observed during earlier experiments, quantify BF16W memory savings, and outline FPGA training as the next stage of development. No FPGA measurements are included in this paper. This publication serves as a public architectural disclosure and software reference implementation for future FPGA and ASIC exploration of the NeuronFabric architecture.

While Large Reasoning Models (LRMs) show strong performance in English, they often fail to reason in the language of the query, a phenomenon known as language collapse. Existing RL-based fixes typically add a binary language fidelity reward to the accuracy objective, yet still incur trade-off in accuracy, mid-trace code-switching, and excessive token usage. In this work, we propose AdaMame, a two-stage training recipe for multilingual mathematical reasoning that addresses these limitations by adaptively aligning the reasoning language to the query language without compromising accuracy. The first SFT stage fine-tunes on non-MT reasoning traces across five languages to establish multilingual reasoning capability. In the subsequent RL stage, we introduce AdaMame-GRPO, an adaptation of Group Relative Policy Optimization (GRPO) in which a query-conditioned alignment factor grows progressively during training, guiding the model to first explore diverse reasoning languages before exploiting reasoning in the query language. Evaluated across two benchmarks, two LRMs, and 12 languages, AdaMame-GRPO achieves Pareto-optimal performance across reasoning accuracy, language fidelity, and token efficiency over all baselines, with the strongest gains on out-of-domain, lower-resource languages.

This volume develops a knowledge theory of capital for economies in which productive capacity increasingly resides in software, data, models, routines, expertise, platforms, organizations, commons, and public epistemic infrastructure. Beginning from Adam Smith's theory of labour, stock, specialization, and market extent, it asks what changes when knowledge becomes stock-like, mobile across forms, scalable, governable, recombinable, and imperfectly visible in accounting. The book introduces knowledge-bearing stock as the central object and analyses how it is generated, converted into governable form, deployed, improved through feedback, enclosed or shared, measured, impaired, and used as input to future production. It distinguishes embodied, disembodied, institutionalized, commons, and public knowledge forms and develops concepts such as first conversion, cognitive enclosure, feedback capture, dark capital, and expected knowledge loss. The argument is conditional and testable: modern wealth depends not only on capital accumulation, but on how productive knowledge is governed.

Prior work has identified several factors that can contribute to the performance gap between Adam and SGD, spanning data aspects, architecture design, and optimization properties. Yet these explanations are often studied in isolation, leaving their relative importance unclear. In this work, we revisit these hypotheses through a controlled empirical study across vision, language, genomics, and graph tasks, spanning modern and classical architectures, and carefully designed training setups. Our results suggest that no single factor consistently explains the Adam--SGD gap. For instance, the Adam advantage can (1) persist under a uniform vocabulary distribution yet nearly disappear under a heavy-tailed one; (2) reverse in favor of SGD in softmax-attention models; and (3) become larger under soft architectural modifications, e.g., when ReLU is replaced by a GeLU nonlinearity. This suggests that the gap arises from nontrivial data and architecture interactions, rather than from a single common factor. Yet, we observe a pattern across our settings: a \emph{crossover batch size} at which the relative advantage shifts from SGD to Adam as the batch size scales. These empirical results are captured by our theoretical gap model, which predicts this batch-size-dependent crossover. Our perspective helps reconcile several existing hypotheses while offering practical insights across domains.

Large Language Models (LLMs) show promise as tool-using agents but remain limited in long-horizon tasks that require remembering, organizing, and reusing knowledge. Prior memory approaches aim to resolve the situation, but mainly focus on storing factual information. Recent work on procedural memory improves task reuse, yet often reduces to replaying past successes without addressing failure cases or online scalability. We introduce a unified and automatic memory framework that integrates semantic, episodic, and procedural memory in a bi-level design combining short-term and long-term stores. A multi-agent architecture with actor, memory, and critic agents enables automatic memory generation, reward annotation, and adaptive retrieval. Long-term memory is managed through reward-based evaluation, merging, and pruning, ensuring scalability and continual improvement. Experiments across various environments show that our approach improves robustness and success on long multi-turn tasks compared to existing baselines. This work highlights the importance of comprehensive, adaptive memory for advancing LLM-based agents.

A crucial component of machine learning algorithms is minimizing loss functions with less computational cost and less oscillations. While adaptive learning rate-based optimizers have been widely used for real-world tasks, they do not guarantee convergence, which is why AMSGrad was later introduced to investigate the non-convergence behaviour of Adam. In this paper, popular adaptive optimization methods like Adam and AMSGrad are critically reviewed with an emphasis on their fundamental design concepts. To address limitations of the above mentioned optimizers, a new optimizer variant, C-Adam, is proposed based on the line of sight approach. A theoretical proof for convergence is also provided and the optimizer is validated through a number of real-life based numerical experiments.

The remarkable success of the Adam in training neural networks has naturally led to the widespread use of its descent-ascent counterpart, Adam-DA, for solving zero-sum games. Despite its popularity in practice, a rigorous theoretical understanding of Adam-DA still lags behind. In this paper, we derive ordinary differential equations (ODEs) that serve as continuous-time limits of the Adam-DA. These ODEs closely approximate the discrete-time dynamics of Adam-DA, providing a tractable analytical framework for understanding its behavior in zero-sum games. Using this ODE approach, we investigate two fundamental aspects of Adam-DA: local convergence and implicit gradient regularization. Our analysis reveals that the roles of the first- and second-order momentum parameters in zero-sum games are exactly the opposite of their well-documented effects in minimization problems. We validate these predictions through GAN experiments across multiple architectures and datasets, demonstrating the practical implications of this reversed momentum effect.

We propose Adam-SHANG, a Lyapunov-guided Adam-type method that couples momentum, adaptive preconditioning, and a curvature-aware correction through a more stable lagged-preconditioner update. For stochastic smooth convex optimization, we prove convergence in expectation under an admissible stepsize condition that can always be satisfied by a conservative spectral bound, without imposing global monotonicity on the second-moment sequence. To obtain a less conservative practical rule, we introduce a computable trace-ratio stepsize, motivated by a local coordinatewise alignment condition. The same structural update is also tested beyond the convex setting with simplified parameters. Experiments validate the predicted stochastic decay and show competitive training performance against Adam and AdamW on deep learning tasks.

There are 7,407 languages in the world. But, what about the languages that are not there in the world? Are humans so narrow minded that we don't care about the languages aliens communicate in? Aliens are humans too! In the 2016 movie Arrival, Amy Adams plays a linguist, Dr. Louise Banks who, by learning to think in an alien language (Heptapod) formed of non-sequential sentences, gains the ability to transcend time and look into the future. In this work, I aim to explore the representation and reasoning of vision-language concepts in a neuro-symbolic language, and study improvement in analytical reasoning abilities and efficiency of "thinking systems". With Qwen3-VL-2B-Instruct as base model and 4 $\times$ Nvidia H200 GPU nodes, I achieve an accuracy improvement of 3.33% on a vision-language evaluation dataset consisting of math, science, and general knowledge questions, while reducing the reasoning tokens by 75% over SymPy. I've documented the compute challenges faced, scaling possibilities, and the future work to improve thinking in a neuro-symbolic language in vision-language models. The training and inference setup can be found here: https://github.com/i-like-bfs-and-dfs/wolfram-reasoning.

Beside the standard stochastic gradient descent (SGD) method, the Adam optimizer due to Kingma & Ba (2014) is currently probably the best-known optimization method for the training of deep neural networks in artificial intelligence (AI) systems. Despite the popularity and the success of Adam it remains an \emph{open research problem} to provide a rigorous convergence analysis for Adam even for the class of strongly convex SOPs. In one of the main results of this work we establish convergence rates for Adam in terms of the number of gradient steps (convergence rate \nicefrac{1}{2} w.r.t. the size of the learning rate), the size of the mini-batches (convergence rate 1 w.r.t. the size of the mini-batches), and the size of the second moment parameter of Adam (convergence rate 1 w.r.t. the distance of the second moment parameter to 1) for the class of strongly convex SOPs. In a further main result of this work, which we refer to as \emph{Adam symmetry theorem}, we illustrate the optimality of the established convergence rates by proving for a special class of simple quadratic strongly convex SOPs that Adam converges as the number of gradient steps increases to infinity to the solution of the SOP (the unique minimizer of the strongly convex objective function) if and \emph{only} if the random variables in the SOP (the data in the SOP) are \emph{symmetrically distributed}. In particular, in the standard case where the random variables in the SOP are not symmetrically distributed we \emph{disprove} that Adam converges to the minimizer of the SOP as the number of Adam steps increases to infinity. We also complement the conclusions of our convergence analysis and the Adam symmetry theorem by several numerical simulations that indicate the sharpness of the established convergence rates and that illustrate the practical appearance of the phenomena revealed in the \emph{Adam symmetry theorem}.