It is shown that the contradiction in experience with neural surrogates in derivative-free optimisation dissolves once three factors are stated, and that these, rather than the fit accuracy a training curve reports, are what delimit when a learned local model pays.
Abstract
Published experience with neural surrogates in derivative-free optimisation is contradictory: the same family of models that cuts the evaluation count of one solver leaves another unchanged, or makes it worse. We show that the contradiction dissolves once three factors are stated, and that these, rather than the fit accuracy a training curve reports, are what delimit when a learned local model pays. Role: a surrogate that proposes candidates the true objective must still approve helps, while one that replaces a gradient the solver depends on hurts. Radius: a model fitted to an optimisation path is reliable only inside a bounded neighbourhood, and its error neither vanishes as that neighbourhood shrinks nor survives its growth. Room: a surrogate can only accelerate progress the base method is still able to make. We formalise radius-aware local generalisation, relate it to the classical fully linear condition, and test each factor with the surrogate class, training pipeline and base method held fixed. Over 117 benchmark instances safeguarded assistance raises the instances solved to high accuracy from 67 to 84 while gradient replacement lowers them to 65; removing the gradient term from the training loss cuts surrogate acceptance from 0.703 to 0.148; and 1000 paired comparisons over ten noise levels show no noise threshold, only a base method that stops early. The same factors bound the gain: a model-based trust-region solver, which leaves little room, drops from 88 to 86 when the identical surrogate is attached, and released interpolation software stays ahead at 103, and on a Monte-Carlo inventory model repairing the acceptance interface is worth 10.40 cost units against 0.00 for the surrogate.
Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation. This thesis develops a principled grounding for Adam and motivates new designs. First, we revisit Adam's divergence--convergence debate and show the existence of a problem-dependent phase transition: with properly chosen, batch-size-dependent hyperparameters, Adam converges, whereas under small-$\beta_2$ regimes it can diverge. Second, we investigate why Adam substantially outperforms SGD on Transformers through Hessian structure. We find that the Hessian evolves toward a near-block-diagonal form along training, accompanied by strong block heterogeneity. We prove that this structure makes Adam's diagonal preconditioner effective. We further show that this special Hessian structure originates from consecutive multiplications of large matrix variables, and we provide a rigorous analysis based on random matrix theory. Finally, these insights motivate Adam-mini, a new optimizer that reduces Adam's memory footprint by 50\% while preserving its performance. Our results also have broader implications beyond Adam: they reveal new local structures in matrix-based nonconvex problems, and also help understand and improve recent NN optimizers, such as Muon.
The approach proposed in this paper enables a more capable DDDAS paradigm by improving the efficiency of the data-model-optimization loop by extracting a generalization bound based on Rademacher complexity that reveals the role of the $k-neighborhoods and related parameters.
Reinforcement learning with verifiable rewards (RLVR) can improve one-sample accuracy while making a model worse under repeated sampling. We study this pass@k inversion: after training, the policy may solve fewer distinct problems than its base model at large $k$. The failure concentrates on boundary prompts, where the base model contains rare correct trajectories that are recoverable by sampling but too sparse to reliably appear in finite RLVR rollout groups. We argue that a two-mode account explains this as an absence-of-evidence failure: rare correct trajectories may disappear before RLVR samples and reinforces them often enough. The main contribution is this diagnostic and mechanistic framing. Per-Problem Base Anchoring (PBA) is a deliberately simple proof-of-concept: sharpen prompts with sufficient frozen-base correct evidence, and anchor risky prompts to the base distribution. Across three training seeds on Omni-MATH-Test, with MATH500 as a secondary high-coverage validation benchmark, PBA improves both \PassK{1} and high-budget coverage over matched GRPO. A 3000-prompt regime-controlled diagnostic study is consistent across seeds with the expected signature: ordinary GRPO loses base-solvable boundary prompts, while PBA preserves rare verifier-positive trajectories. We use mathematical verifiers as a controlled testbed for verifier-guided optimization; the same pass@k inversion risk applies to ECCV-relevant vision-language agents when repeated visual, spatial, or chart-reasoning attempts are checked by external tools or verifiers. Reasoning post-training should decide not only how strongly to optimize, but which prompts are safe to optimize.
Deep neural networks generalize well despite their highly nonconvex, overparameterized loss landscapes, a phenomenon often associated with the geometry of the minima found by stochastic optimization. We study how incremental grow-and-optimize strategies bias training toward flatter regions by viewing growth as progressive constraint relaxation. Starting from a low-dimensional submodel, we iteratively expand the trainable parameters by unlocking nested random subspaces while freezing the orthogonal complement at the network initialization, re-optimizing after each expansion until the full architecture is reached. Under standard local regularity conditions around non-degenerate minima, we prove that local sublevel sets are well approximated by ellipsoids and that basin accessibility under frozen constraints can be characterized by an explicit effective curvature in the frozen directions. This leads to an explanation of the bias: progressive growth increases the relative weight of wide basins and suppresses sharp ones through a volume effect induced by the frozen constraints. We empirically validate these predictions in controlled toy landscapes and in a realistic ResNet/CIFAR-100 setting and confirm that although progressive subspace growth reliably produces flatter solutions, curvature reductions do not universally translate into improved test performance, highlighting subtleties in the flatness-generalization connection. The code is available at https://github.com/p0lcAi/Across-the-Loss-Landscape.
Paul Caillon, Christophe Cerisara, Alexandre Allauzen· 0 citations
As constrained learning becomes increasingly common, models are trained under explicit feasibility requirements to enforce fairness, safety, robustness, regulariza- tion, and physics or logic constraints. Understanding how training samples in- fluence the model solution (e.g., learned parameters) is crucial for interpretability and robustness. The classical influence function (IF) estimates sample contribu- tions via local sensitivity analysis, measuring how the solution changes when a specific training sample is perturbed or removed. However, IF becomes unreli- able in constrained settings: data perturbations can reshape both the objective and the feasible region, leading to estimates that violate feasibility. In response, we propose the Directional Influence Function (DIF), a novel estimator that explicitly incorporates these constraints into influence estimation. DIF formulates the opti- mality conditions of constrained learning as a variational inequality (VI) and ana- lyzes how perturbing training data affects this VI. We validate DIF on constrained linear regression and demonstrate that it recovers leave-one-out retraining results, whereas IF and penalty-based IF exhibit significant bias. We further apply DIF to fairness-constrained CNNs, where DIF accurately predicts test loss changes under data removal and aligns closely with actual retraining. Our results establish DIF as an efficient and reliable tool for data attribution in constrained learning.
ReBRAC-v2 is introduced, which directly trains an exact-likelihood normalizing flow as the RL actor, combines likelihood, MSE, and MAE behavior regularization, and integrates a classification-based residual critic, staged optimization, and multi-sample test-time action selection, and ranks first in eight categories.
Denis Tarasov, Robert K. Katzschmann· 0 citations
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