Energy Grid Optimization

Large language models have achieved major advances across domains, yet training them remains extremely resource-intensive. We revisit Sign-SGD, which serves both as a memory-efficient optimizer for single-node training and as a gradient compression mechanism for distributed learning. This paper addresses a central limitation: the effective stepsize cannot be determined a priori because it relies on unknown, problem-specific quantities. We present a parameter-free Sign-SGD that removes manual stepsize selection. We analyze the deterministic single-node case, and extend the method to stochastic single-node training and multi-node settings. We also incorporate the momentum technique into our algorithms and propose a memory-efficient variant that stores only gradient signs instead of full gradients. We evaluate our methods on pre-training LLaMA models (130M and 350M) and fine-tuning a Swin Transformer (28M). Across considered tasks, the proposed methods match the performance of tuned Sign-SGD and AdamW (grid-searched stepsizes with a cosine schedule), while avoiding tuning overhead. Employing parameter-free training yields approximately $1.5\times$ end-to-end speedup compared to runs with grid-searched stepsizes.

Standard attention stores keys/values losslessly but reads them via a per-head convex average, blocking channel-wise selection. We propose the Free Energy Mixer (FEM): a free-energy (log-sum-exp) read that applies a value-driven, per-channel log-linear tilt to a fast prior (e.g., from queries/keys in standard attention) over indices. Unlike methods that attempt to improve and enrich the $(q,k)$ scoring distribution, FEM treats it as a prior and yields a value-aware posterior read at unchanged complexity, smoothly moving from averaging to per-channel selection as the learnable inverse temperature increases, while still preserving parallelism and the original asymptotic complexity ($O(T^2)$ for softmax; $O(T)$ for linearizable variants). We instantiate a two-level gated FEM that is plug-and-play with standard and linear attention, linear RNNs and SSMs. It consistently outperforms strong baselines on NLP, vision, and time-series at matched parameter budgets.

A key step in protein optimization is modeling the fitness landscape, which maps proteins to functional assay readouts. Existing methods typically either use property-agnostic likelihoods/embeddings from pretrained protein language models (PLMs) for fitness prediction, or assume independent mutational effects, limiting their ability to capture higher-order interactions. In this work, we introduce RankFlow, a conditional flow framework that refines PLM representations to be a property-aligned distribution via a tailored energy function and captures multi-mutation interactions through learnable embeddings. To align optimization with evaluation protocols, we propose the Rank-Consistent Conditional Flow Loss (RC$^2$), a differentiable ranking objective that enforces the correct order of mutants rather than absolute values, which improves out-of-distribution generalization. Finally, we introduce a Property-guided Steering Gate (PSG) that concentrates learning on positions carrying signals for the target property while suppressing unrelated evolutionary biases. Across ProteinGym, PEER, and FLIP benchmarks, RankFlow obtains state-of-the-art ranking accuracy and stronger generalization.

Learning population dynamics involves recovering the underlying process that governs particle evolution, given evolutionary snapshots of samples at discrete time points. Recent methods frame this as an energy minimization problem in probability space and leverage the celebrated JKO scheme for efficient time discretization. In this work, we introduce ``iJKOnet``, an approach that combines the JKO framework with inverse optimization techniques to learn population dynamics. Our method relies on a conventional *end-to-end* adversarial training procedure and does not require restrictive architectural choices, e.g., input-convex neural networks. We establish theoretical guarantees for our methodology and demonstrate improved performance over prior JKO-based methods.

Inspired by behavioral science, we propose Behavior Learning (BL), a novel general-purpose machine learning framework that learns interpretable and identifiable optimization structures from data, ranging from single optimization problems to hierarchical compositions. It unifies predictive performance, intrinsic interpretability, and identifiability, with broad applicability to scientific domains involving optimization. BL parameterizes a compositional utility function built from intrinsically interpretable modular blocks, which induces a data distribution for prediction and generation. Each block represents and can be written in symbolic form as a utility maximization problem (UMP), a foundational paradigm in behavioral science and a universal framework of optimization. BL supports architectures ranging from a single UMP to hierarchical compositions, the latter modeling hierarchical optimization structures that offer both expressiveness and structural transparency. Its smooth and monotone variant (IBL) guarantees identifiability under mild conditions. Theoretically, we establish the universal approximation property of both BL and IBL, and analyze the M-estimation properties of IBL. Empirically, BL demonstrates strong predictive performance, intrinsic interpretability and scalability to high-dimensional data. Code: https://github.com/MoonYLiang/Behavior-Learning; installable via pip install blnetwork.

The Schrödinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schrödinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method.

State space models (SSMs) often sacrifice capacity, search space, or stability to offset the memory and compute costs of large state dimensions. We introduce a structured post-training pruning method for SSMs — AIRE-Prune (Asymptotic Impulse- Response Energy for State PRUN(E)ing ) — that reduces each layer’s state dimension by directly minimizing long-run output-energy distortion. AIRE-Prune assigns every state a closed-form asymptotic impulse-response energy based score, i.e., the total impulse-response energy it contributes over an infinite horizon (time), and normalizes these scores layer-wise to enable global cross-layer comparison and selection. This extends modal truncation from single systems to deep stacks and aligns pruning with asymptotic response energy rather than worst-case gain. Across diverse sequence benchmarks, AIRE-Prune reveals substantial redundancy in SISO and MIMO SSMs with average pruning of 60.8%, with average accuracy drop of 0.29% without retraining while significantly lowering compute.

This paper introduces a new constraint-free concave dual formulation for the Wasserstein barycenter. Tailoring the vanilla dual gradient ascent algorithm to the Sobolev geometry, we derive a scalable Sobolev gradient ascent (SGA) algorithm to compute the barycenter for input distributions supported on a regular grid. Despite the algorithmic simplicity, we provide a global convergence analysis that achieves the same rate as the classical subgradient descent methods for minimizing nonsmooth convex functions in the Euclidean space. A central feature of our SGA algorithm is that the computationally expensive $c$-concavity projection operator enforced on the Kantorovich dual potentials is unnecessary to guarantee convergence, leading to significant algorithmic and theoretical simplifications over all existing primal and dual methods for computing the exact barycenter. Our numerical experiments demonstrate the superior empirical performance of SGA over the existing optimal transport barycenter solvers.

Transformer-based models have achieved remarkable success, but their core components, Transformer layers, are largely heuristics-driven and engineered from the bottom up, calling for a prototypical model with high interpretability and practical competence. To this end, we conceptualize a principled, top-down approach grounded in energy-based interpretation. Specifically, we formalize token dynamics as a joint maximum likelihood estimation on the hypersphere, featuring two properties: semantic alignment in the high-dimensional space and distributional uniformity in the low-dimensional space. By quantifying them with extended Hopfield energy functions, we instantiate this idea as a constrained energy minimization problem, which enables designs of symmetric attention and feedforward modules with RMS normalization. We further present *Hyper-Spherical Energy Transformer* (Hyper-SET), a recurrent-depth alternative to vanilla Transformers naturally emerging from iterative energy optimization on the hypersphere. With shared parameters across layers, Hyper-SET can scale to arbitrary depth with fewer parameters. Theoretically grounded and compact, it achieves competitive or superior performance across diverse tasks, including Sudoku solving, image classification, and masked image modeling. We also design novel variations under the proposed general principle, such as linear attention and gated feedforward layer, and showcase its scalability with depth-wise LoRA. Our results highlight Hyper-SET as a step toward interpretable and principled Transformer design.

Large language models (LLMs) require constant updates to remain aligned with evolving real-world knowledge. Model editing offers a lightweight alternative to retraining, but sequential editing that updates the LLM knowledge through multiple successive edits often destabilizes representations and induces catastrophic forgetting. In this work, we seek to better understand and mitigate performance degradation caused by sequential editing. We hypothesize that hyperspherical uniformity, a property that maintains uniform distribution of neuron weights on a hypersphere, helps the model remain stable, retain prior knowledge, while still accommodate new updates. We use Hyperspherical Energy (HE) to quantify neuron uniformity during editing, and examine its correlation with editing performance. Empirical studies across widely used editing methods reveals a strong correlation between HE dynamics and editing performance, with editing failures consistently coinciding with uncontrolled HE fluctuations. We further theoretically prove that HE dynamics impose a lower bound on the degradation of pretrained knowledge, highlighting why HE stability is crucial for knowledge retention. Motivated by these insights, we propose SPHERE (Sparse Projection for Hyperspherical Energy-Regularized Editing), an HE-driven regularization strategy that stabilizes neuron weight distributions, ultimately preserving prior knowledge while enabling reliable sequential updates. Specifically, SPHERE identifies a sparse space complementary to the principal hyperspherical directions of the pretrained weight matrices and projects new knowledge onto it, attenuating perturbations on the principal directions. Extensive experiments on LLaMA3 (8B) and Qwen2.5 (7B) show that SPHERE outperforms the best baseline in editing capability by an average of 16.41%, while most faithfully preserving general model performance, thereby offering a principled path toward reliable large-scale knowledge editing.

Generative Pre-trained Transformer (GPT) architectures are the most popular design for language modeling. Energy-based modeling is a different paradigm that views inference as a dynamical process operating on an energy landscape. We propose a minimal modification of the GPT setting to unify it with the EBM framework. The inference step of our model, which we call eNeRgy-GPT (NRGPT), is conceptualized as an exploration of the tokens on the energy landscape. We prove, and verify empirically, that under certain circumstances this exploration becomes gradient descent, although they don’t necessarily lead to the best performing models. We demonstrate that our model performs well for simple language (Shakespeare dataset), algebraic ListOPS tasks, and richer settings such as OpenWebText language modeling. We also observe that our models may be more resistant to overfitting, doing so only during very long training.