#01Jul 23, 2026
math.OC
Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension $n\geq 4$
Dawei Li, Xiaotian Jiang, Mingyi Hong
Barzilai--Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai--Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $ρ_{\min}=10^{-6},ρ_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.
#02Jul 23, 2026
quant-ph
Cautious optimism for deep parameterized quantum circuits
Marie Kempkes, Elies Gil-Fuster, Carlos Bravo-Prieto and 5 more
A central challenge in quantum machine learning is understanding the scaling behavior of parameterized quantum circuits (PQCs). In particular, it remains unclear how their performance on unseen data changes as the number of trainable parameters increases. Prior works have derived formal generalization guarantees for quantum models, but it is well-known that many such results do not fully characterize generalization behavior in practice. In this work, we show that gradient-based PQCs can exhibit improved performance on unseen data as model size increases, displaying the phenomenon of double descent. This contrasts with the traditional view that larger models lead to degraded generalization. We provide analytical results rigorously underpinning this behavior by leveraging add-one-in perturbation techniques and spectral properties of random matrices. We support these results with numerical experiments on re-uploading PQCs across several data sets and training set sizes, consistently observing the predicted double descent behavior. While other obstacles on the path toward practical quantum machine learning remain, our finding that deeper parameterized quantum circuits do not necessarily exhibit degraded performance provides reasons for cautious optimism.
#03Jul 23, 2026
cs.LG
Error Certificates for KV-Cache Eviction via Randomized Design
Peng Xie
Deterministic KV-cache eviction keeps the top-$k$ tokens under an importance score and deletes the rest. We prove that this design cannot know what it destroyed: evicted values can be altered so that everything the serving system retains is unchanged while the true attention-output error grows arbitrarily, so no serving-time estimator of that error is consistent. Randomized eviction restores identifiability. With a Poisson-sampled tail at known inclusion probabilities, one logit offset performs the Hájek correction inside the softmax, and a survey-sampling variance estimator over the retained set becomes a per-step error certificate with 0.97 empirical coverage at no accuracy cost. On real workloads we pre-registered seven claims and lost three: question-aware eviction at 25--50\% budgets is nearly free; output log-probability predicts failure better than the certificate; certificate-gated budget escalation adds nothing. What survives is attribution: the certificate separates cache-induced from inherent failures (AUC 0.73--0.75, against 0.47--0.54 for output confidence) and schedules recomputation better than random or confidence gating. Randomization buys attribution, not prediction.
#04Jul 23, 2026
cs.LG
A Diffusion-Model Subpopulation Digital Twin for Mobile Health Deployment: A Case Study on the HeartSteps Intervention
Ziping Xu, Yuyi Chang, Chenshun Ni and 5 more
Mobile-health interventions increasingly use online learning and decision making algorithms to personalize when to nudge users toward healthier behavior, but a poorly designed algorithm can burden and disengage participants. New algorithm design decisions should therefore be vetted against realistic simulated users before each real-life deployment. We propose a method to develop ``JITAI-Twins'': digital twins of a target subpopulation for comparing candidate online algorithms before a just-in-time adaptive intervention (JITAI) deployment. The method builds on a conditional time-series diffusion model that is temporally consistent (future actions do not affect the generated past), and it supports repeated updating from three sources of information, in three steps: pre-training on a large observational dataset, fine-tuning on small prior intervention deployments in related populations, and inference-time calibration to the next target population from domain-scientist expertise. We validate the twin at each pre-deployment stage of the long-running HeartSteps series (v2 through v4) of physical-activity suggestion intervention deployments, treating each successive deployment as an upcoming study. The proposed method reproduces the target subpopulation's temporal and between-participant structure better than simpler simulators. These results suggest that our twin can be used to simulate a target deployment before it runs, the prerequisite for testing and informing online algorithm design decisions.
#05Jul 23, 2026
cs.LG
Emergent Misalignment Recruits a Pre-existing Persona Subspace
Mohammed Suhail B Nadaf
Fine-tuning an aligned language model on a narrow stream of bad advice can make it broadly misaligned on questions unrelated to the training data, a phenomenon called emergent misalignment. We ask why the narrow lesson generalizes at all, and we find that narrow fine-tuning recruits a persona structure that is present in the model before the fine-tune exists. From a frozen instruction-tuned model (Qwen2.5-14B-Instruct) we extract per-domain persona subspaces by contrastive teacher forcing and find that 4 unrelated domains share one low-rank core at 657x a random-subspace null, with 82% of that core lying outside a style core built at matched diversity. The literal first optimizer step of fine-tuning on insecure code climbs a broad-misalignment margin harder than the same code framed as educational, and forecasts realized margin movement out to 375 steps. Projecting the subspace out of the residual stream throughout fine-tuning prevents broad misalignment (27.7% to 0.0% of judged generations) while a matched-rank random subspace changes nothing; injecting it into the never-fine-tuned model induces misalignment that grows with dose to 45.4%, past the fine-tuned model it is measured against. The same projection applied to the weight gradient is inert, and three post-hoc weight edits leave the disposition in place: the sharpest edit suppresses the behavior rather than removing it, and the ablated structure re-forms inside the subspace the edit cleared. Spreading a fixed budget of bad data across 4 domains produces more broad misalignment than mechanical weight superposition and matched diversity jointly account for. All measurements come from one model at 14B; the extraction is from an aligned instruction-tuned checkpoint, which leaves the structure's provenance open; and the intervention that prevents misalignment also abolishes the narrow trained behavior.