Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.38540

When a Flatness Proxy Is Not a Function: Robustness Certificates and Training Interventions

Vicente Opazo·Jose Calatayud-Mateu·Cristobal Rojas·Cristian Buc Calderon
View original source ↗Read source document ↗Export metadata ↓

Abstract

A valid curvature upper bound need not justify either a robustness certificate or an intervention on an intrinsic predictor property. We demonstrate this distinction for a last-layer relative-flatness proxy used in both settings. First, empirical-risk stationarity does not eliminate pointwise first-order loss terms: at a finite global empirical-risk minimum, the retained certificate expression underestimates a loss increase by over $210\times$. We derive a globally valid, gauge-invariant feature-space repair. Second, common-row softmax shifts preserve predictions and the exact contraction while making the proxy unbounded. Even standard reference-class choices double it on average relative to the centered representation. For a single fixed-feature example with at least three classes, scalar retuning generically cannot align the induced probability updates. Row centering gives the orbit-minimized bound and restores value and full-model gradient invariance under this symmetry. Across 45 paired one-step tests on algorithmic and image models, amplified shifts separate raw-regularized predictors while quotient-regularized predictors remain aligned. Long-horizon CIFAR-10 experiments show substantial, reversible suppression of generalization, while evidence for selective delay after memorization is less consistent. Together, these results show that validity as a curvature upper bound does not by itself justify either inversion into a robustness certificate or differentiation into an intrinsic training intervention.

Explore related subjects

cs.LG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vicente Opazo, Jose Calatayud-Mateu, Cristobal Rojas, Cristian Buc Calderon. 2026-09-29. When a Flatness Proxy Is Not a Function: Robustness Certificates and Training Interventions. https://arxiv.org/abs/2609.38540

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Is the Policy Gradient a Gradient?

The policy gradient theorem describes the gradient of the expected discounted return with respect to an agent's policy parameters. However, many policy gradient methods omit the outer discount factor, yielding an update that need not equal the gradient of the discounted objective. Is the resulting update the gradient of another objective? This has been an open question for several years, and this lack of theoretical clarity has led to an abundance of misstatements in the literature. We answer this question for the resulting episode-summed, on-policy update by proving that it need not equal the gradient of any function. Further, we construct a counterexample in which exact expected updates drive the policy toward a deterministic limit that is globally pessimal with respect to both the discounted and undiscounted objectives, while the parameter diverges. We motivate this work by surveying the literature and showing that there remains a widespread misunderstanding regarding discounted policy gradient methods, with errors present even in highly-cited papers published at top conferences.

cs.LG↗
arXiv2026

Harnessing Trust in Directed Graphs: Redefining Robustness of Graph Learning

Existing research on robust Graph Neural Networks (GNNs) focuses predominantly on undirected graphs, overlooking the trustworthiness inherent in the link directions of directed graphs. In this work, we analyze the limitations of current approaches from both the attack and defense perspectives and redefine the robustness of GNNs in directed graphs. From the attack side, we introduce a more realistic directed graph attack setting that addresses the shortcomings of existing attacks. From the defense side, we propose a simple yet effective message-passing framework that serves as a plug-in layer to strengthen GNN robustness while avoiding a false sense of robustness. Our findings show that the trust signals encoded in directed graphs can be harnessed to substantially improve the robustness of GNNs. When combined with existing defense strategies, the framework achieves state-of-the-art robust performance alongside strong clean accuracy against both transfer and adaptive attacks. These results point to a novel and promising direction for strengthening the robustness of graph deep learning.

cs.LG↗
arXiv2026

Learning Agentic Time Series Forecasting via Trajectory Distillation and Memory-Based Reinforcement Learning

Recent agentic time series forecasting (TSF) approaches use large language models (LLMs) to autonomously orchestrate analytical and forecasting tools and reason about temporal patterns using the resulting evidence. However, many such approaches rely on powerful LLMs in training-free, prompt-driven workflows, leaving open whether smaller models can learn these interaction and reasoning capabilities. In this paper, we introduce Cast-R1, an 8Bparameter agentic forecasting model trained through trajectory distillation and multi-turn reinforcement learning with structured memory. First, we develop a forecasting agent harness with specialized analysis and prediction tools to generate multi-turn interaction trajectories using a teacher LLM, which are subsequently evaluated and selected for supervised fine-tuning. Second, we apply group relative policy optimization (GRPO) across multi-turn forecasting episodes, with structured memory preserving diagnostic evidence and intermediate forecasts for state construction. Rewards based on prediction accuracy and temporal structure guide tool selection, model routing, and forecast refinement. A difficulty-aware curriculum combines sample filtering and progressive sample ordering during training. Extensive experiments on multiple datasets demonstrate the effectiveness of Cast-R1. These findings highlight the potential of forecasting-specific post-training to equip smaller LLMs with multi-turn, tool-augmented forecasting capabilities within structured workflows.

cs.LG↗