Pierfrancesco Beneventano

Mohua Das, Pierfrancesco Beneventano, Shibshankar Dey et al.

Randomly initialized neural networks induce a prior over functions, but the predictor used in practice is produced only after training. We ask how much of this initial bias survives the training pipeline. To make the question measurable, we introduce initialization memory: the dependence of the validation-selected predictor on the scale of the random initialization. We perform controlled CIFAR-10 experiments on ResNets where initialization memory already sharply separates training regimes. Low-learning-rate SGD can interpolate while still remembering its initialization: on ResNet-9 with batch size $b=128$, test accuracy varies by $26.5$ percentage points across initialization scales despite $\ge99.5\%$ training accuracy. This is not undertraining: extending the same low-learning-rate regime to $5{,}000$ epochs leaves the spread essentially unchanged. In contrast, Adam-family methods largely erase the dependence. SGD can also be made to forget when larger learning rates are paired with explicit $L_2$ norm control. We interpret these findings in terms of the time scale of forgetting: gradient-flow-like dynamics can preserve initialization memory, whereas stochastic finite-step effects, explicit norm decay, and adaptive preconditioning erase it on scales governed by the size of explicit or implicit regularization. The practical inductive bias of a trained network is therefore not the architectural prior alone, but the architectural prior after being filtered by the forgetting dynamics of the training pipeline; and the same regularizers that improve generalization are precisely those that erase memory of initialization.

Shauna Kwag, Anakha Ganesh, Tomaso Poggio et al.

Existing analyses of the edge of stability (EoS) treat it as a global property of optimization. We show that it is also selective: the stability constraint redistributes learning across subsets of the training distribution, amplifying progress on some groups while suppressing progress on others. Using a branching intervention that enters or exits the EoS regime from the same training state, we causally demonstrate this trade-off and identify two necessary conditions for a group to benefit. First, its aggregate gradient must align with the top Hessian eigenvector. We isolate this mechanism with a controlled perturbation that preserves distance but randomizes direction, destroying alignment and eliminating the advantage. Second, the group must sustain non-vanishing gradient magnitude over time. Under cross-entropy loss, gradient saturation decouples confidently classified groups, shifting the advantage to output-outliers, whose gradients persist. Together, these results show that EoS functions not only as a stability boundary, but as a mechanism governing the allocation of learning across the data distribution.

Pierfrancesco Beneventano, Patrick Cheridito, Robin Graeber et al.

The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of dimensionality in the approximation of a large class of functions. More precisely, we prove that these functions can be approximated by DNNs on compact sets such that the number of parameters necessary to represent the approximating DNNs grows at most polynomially in the reciprocal $1/\varepsilon$ of the prescribed approximation error $\varepsilon>0$ and in the input dimension $d\in\mathbb N$. To this end, we introduce certain approximation spaces, consisting of sequences of functions that can be efficiently approximated by DNNs. We then establish closure properties which we combine with known and new bounds on the number of parameters necessary to approximate locally Lipschitz continuous functions, maximum functions, and product functions by DNNs. The main result of this article demonstrates that DNNs have sufficient expressive power to approximate, without the curse of dimensionality, certain sequences of functions which can be constructed by means of a finite number of compositions using locally Lipschitz continuous functions, maxima, and products.

Mahmoud Abdelmoneum, Pierfrancesco Beneventano, Tomaso Poggio

We present pAI/MSc, an open-source, customizable, modular multi-agent system for academic research workflows. Our goal is not autonomous scientific ideation, nor fully automated research. It is narrower and more practical: to reduce by orders of magnitude the human steering required to turn a specified hypothesis into a literature-grounded, mathematically established, experimentally supported, submission-oriented manuscript draft. pAI/MSc is built with a current emphasis on machine learning theory and adjacent quantitative fields.

Arseniy Andreyev, Advikar Ananthkumar, Marc Walden et al.

Recent work suggests that (stochastic) gradient descent self-organizes near an instability boundary, shaping both optimization and the solutions found. Momentum and mini-batch gradients are widely used in practical deep learning optimization, but it remains unclear whether they operate in a comparable regime of instability. We demonstrate that SGD with momentum exhibits an Edge of Stochastic Stability (EoSS)-like regime with batch-size-dependent behavior that cannot be explained by a single momentum-adjusted stability threshold. Batch Sharpness (the expected directional mini-batch curvature) stabilizes in two distinct regimes: at small batch sizes it converges to a lower plateau $2(1-β)/η$, reflecting amplification of stochastic fluctuations by momentum and favoring flatter regions than vanilla SGD; at large batch sizes it converges to a higher plateau $2(1+β)/η$, where momentum recovers its classical stabilizing effect and favors sharper regions consistent with full-batch dynamics. We further show that this aligns with linear stability thresholds and discuss the implications for hyperparameter tuning and coupling.

Marius Saether, Amir Kolic, Tomaso Poggio et al.

In modern deep learning, weight decay is often credited with "stabilizing" training dynamics, diverging from its classical role as a static regularization penalty. We investigate a fundamental question: *does weight decay stabilize training dynamics, and if so, through which mechanism?* Indeed, training stability is understood through different but related notions in the literature. We consider how weight decay affects the parameter-space dynamics and loss sharpness by analyzing its effects at the \emph{Edge of Stability} (EoS). We show that weight decay robustly slows *progressive sharpening}. Furthermore, we uncover a striking architecture-dependent phase transition. In CNNs, weight decay dampens the oscillations at the EoS, while in MLPs, increasing weight decay causes a phase transition in which the sharpness stabilizes at a threshold significantly below the theoretical $\frac{2}η$ boundary. We develop a mathematical framework that accurately models these phenomena and identify the global alignment of the parameter vector and the sharpness gradient as the mechanistic driver of the phase transition. Importantly, we show that these phenomena translate into stability in terms of search in function-space (NTK). Last, this shows that curvature thresholds obtained from convex/quadratic heuristics may not be reliable stability diagnostics under regularization.

Federico Bottino, Carlo Ferrero, Nicholas Dosio et al.

Organizational knowledge used by AI agents typically lacks epistemic structure: retrieval systems surface semantically relevant content without distinguishing binding decisions from abandoned hypotheses, contested claims from settled ones, or known facts from unresolved questions. We argue that the ceiling on organizational AI is not retrieval fidelity but \emph{epistemic} fidelity--the system's ability to represent commitment strength, contradiction status, and organizational ignorance as computable properties. We present OIDA, a framework that structures organizational knowledge as typed Knowledge Objects carrying epistemic class, importance scores with class-specific decay, and signed contradiction edges. The Knowledge Gravity Engine maintains scores deterministically with proved convergence guarantees (sufficient condition: max degree $< 7$; empirically robust to degree 43). OIDA introduces QUESTION-as-modeled-ignorance: a primitive with inverse decay that surfaces what an organization does \emph{not} know with increasing urgency--a mechanism absent from all surveyed systems. We describe the Epistemic Quality Score (EQS), a five-component evaluation methodology with explicit circularity analysis. In a controlled comparison ($n{=}10$ response pairs), OIDA's RAG condition (3,868 tokens) achieves EQS 0.530 vs.\ 0.848 for a full-context baseline (108,687 tokens); the $28.1\times$ token budget difference is the primary confound. The QUESTION mechanism is statistically validated (Fisher $p{=}0.0325$, OR$=21.0$). The formal properties are established; the decisive ablation at equal token budget (E4) is pre-registered and not yet run.

Saharsh Koganti, Priyadarsi Mishra, Pierfrancesco Beneventano et al.

We study learning when the learned object is executable solver code rather than a predictor. In this setting, correctness is not enough: two solvers may both return valid solutions on the deployment distribution while differing substantially in runtime. Given samples from an unknown task distribution, the learner returns code evaluated on fresh instances by both solution quality and execution time. Our central abstraction is a \emph{solver hint}: reusable structure inferred from samples and compiled into specialized solver code. We prove that the empirically fastest sample-consistent solver from a fixed library generalizes in both correctness and runtime, and that statistically identifiable hints can be recovered and compiled from polynomially many samples. Empirically, we instantiate the framework with LLM code agents on \(21\) structured combinatorial-optimization target distributions across seven problem classes. The synthesized solvers reach mean normalized quality \(0.971\), improve by \(+0.224\) over the average heuristic pool and by \(+0.098\) over the highest-quality heuristic, and are \(336.9\times\), \(342.8\times\), and \(16.1\times\) faster than the quality-best heuristic, Gurobi, and the selected time-limited exact backend, respectively. On released PACE 2025 Dominating Set private instances, the synthesized solver is valid on all \(100\) graphs and runs about two orders of magnitude faster than top competition solvers, with a moderate quality gap. Inspection shows that many gains come from changing the computational scale: replacing ambient exponential search or general-purpose optimization with compiled distribution-specific computation.

Varun Sunkaraneni, Pierfrancesco Beneventano, Riccardo Neumarker et al.

Can a committee of weak reasoning-model calls reach the performance of much stronger models? We study verifier-backed committee search as inference-time boosting for reasoning language models. The mechanism is not simply that ``more agents help'': samples expose latent correct solutions, while critics and comparators must recover them without access to the hidden verifier. We formalize this view by separating proposal coverage, local identifiability, progress, and diversity. We prove that coverage can be amplified by repeated sampling, but cannot by itself create useful critics or comparators; reliable amplification requires an additional local soundness signal, such as execution, proof checking, type checking, tests, or constraint solving. We give rank-based bounds showing when local selection errors compose into reliable trajectories, and characterize the proposer-side ceiling: oracle best-of-\(k\) converges only to the mass of task slices on which the proposal system assigns nonzero useful probability. Empirically, on SWE-bench Verified, a single \texttt{GPT-5.4 nano} proposal solves \(67.0\%\) of tasks. Using the same nano model, our critic--comparator orchestration reaches \(76.4\%\) with \(k=8\) proposals, matching the standalone performance of \texttt{Gemini 3 Pro} and \texttt{Claude Opus 4.5} Thinking and approaching the \(79.0\%\) oracle best-of-\(8\) upper bound. Thus, many correct patches are already present in weak-model proposal pools; the main challenge is selecting them. The remaining failures are mostly proposal-coverage failures, indicating shared blind spots that stronger selection alone cannot close.

Federico Vittorio Cortesi, Giuseppe Iannone, Giulia Crippa et al.

Neural networks applied to financial time series operate in a regime of underspecification, where model predictors achieve indistinguishable out-of-sample error. Using large-scale volatility forecasting for S$\&$P 500 stocks, we show that different model-training-pipeline pairs with identical test loss learn qualitatively different functions. Across architectures, predictive accuracy remains unchanged, yet optimizer choice reshapes non-linear response profiles and temporal dependence differently. These divergences have material consequences for decisions: volatility-ranked portfolios trace a near-vertical Sharpe-turnover frontier, with nearly $3\times$ turnover dispersion at comparable Sharpe ratios. We conclude that in underspecified settings, optimization acts as a consequential source of inductive bias, thus model evaluation should extend beyond scalar loss to encompass functional and decision-level implications.

Pierfrancesco Beneventano

This article examines the implicit regularization effect of Stochastic Gradient Descent (SGD). We consider the case of SGD without replacement, the variant typically used to optimize large-scale neural networks. We analyze this algorithm in a more realistic regime than typically considered in theoretical works on SGD, as, e.g., we allow the product of the learning rate and Hessian to be $O(1)$ and we do not specify any model architecture, learning task, or loss (objective) function. Our core theoretical result is that optimizing with SGD without replacement is locally equivalent to making an additional step on a novel regularizer. This implies that the expected trajectories of SGD without replacement can be decoupled in (i) following SGD with replacement (in which batches are sampled i.i.d.) along the directions of high curvature, and (ii) regularizing the trace of the noise covariance along the flat ones. As a consequence, SGD without replacement travels flat areas and may escape saddles significantly faster than SGD with replacement. On several vision tasks, the novel regularizer penalizes a weighted trace of the Fisher Matrix, thus encouraging sparsity in the spectrum of the Hessian of the loss in line with empirical observations from prior work. We also propose an explanation for why SGD does not train at the edge of stability (as opposed to GD).

Alessandro Morosini, Matea Gjika, Tomaso Poggio et al.

Modern neural networks can achieve high accuracy while remaining poorly calibrated, producing confidence estimates that do not match empirical correctness. Yet calibration is often treated as a post-hoc attribute. We take a different perspective: we study calibration as a training-time phenomenon on small vision tasks, and ask whether calibrated solutions can be obtained reliably by intervening on the training procedure. We identify a tight coupling between calibration, curvature, and margins during training of deep networks under multiple gradient-based methods. Empirically, Expected Calibration Error (ECE) closely tracks curvature-based sharpness throughout optimization. Mathematically, we show that both ECE and Gauss--Newton curvature are controlled, up to problem-specific constants, by the same margin-dependent exponential tail functional along the trajectory. Guided by this mechanism, we introduce a margin-aware training objective that explicitly targets robust-margin tails and local smoothness, yielding improved out-of-sample calibration across optimizers without sacrificing accuracy.

Arseniy Andreyev, Pierfrancesco Beneventano

Recent findings by Cohen et al., 2021, demonstrate that when training neural networks with full-batch gradient descent with a step size of $η$, the largest eigenvalue $λ_{\max}$ of the full-batch Hessian consistently stabilizes at $λ_{\max} = 2/η$. These results have significant implications for convergence and generalization. This, however, is not the case of mini-batch stochastic gradient descent (SGD), limiting the broader applicability of its consequences. We show that SGD trains in a different regime we term Edge of Stochastic Stability (EoSS). In this regime, what stabilizes at $2/η$ is *Batch Sharpness*: the expected directional curvature of mini-batch Hessians along their corresponding stochastic gradients. As a consequence $λ_{\max}$ -- which is generally smaller than Batch Sharpness -- is suppressed, aligning with the long-standing empirical observation that smaller batches and larger step sizes favor flatter minima. We further discuss implications for mathematical modeling of SGD trajectories.

Pierfrancesco Beneventano, Blake Woodworth

We study the gradient descent (GD) dynamics of a depth-2 linear neural network with a single input and output. We show that GD converges at an explicit linear rate to a global minimum of the training loss, even with a large stepsize -- about $2/\textrm{sharpness}$. It still converges for even larger stepsizes, but may do so very slowly. We also characterize the solution to which GD converges, which has lower norm and sharpness than the gradient flow solution. Our analysis reveals a trade off between the speed of convergence and the magnitude of implicit regularization. This sheds light on the benefits of training at the ``Edge of Stability'', which induces additional regularization by delaying convergence and may have implications for training more complex models.

Yizhou Xu, Pierfrancesco Beneventano, Isaac Chuang et al.

A large body of theory and empirical work hypothesizes a connection between the flatness of a neural network's loss landscape during training and its performance. However, there have been conceptually opposite pieces of evidence regarding when SGD prefers flatter or sharper solutions during training. In this work, we partially but causally clarify the flatness-seeking behavior of SGD by identifying and exactly solving an analytically solvable model that exhibits both flattening and sharpening behavior during training. In this model, the SGD training has no \textit{a priori} preference for flatness, but only a preference for minimal gradient fluctuations. This leads to the insight that, at least within this model, it is data distribution that uniquely determines the sharpness at convergence, and that a flat minimum is preferred if and only if the noise in the labels is isotropic across all output dimensions. When the noise in the labels is anisotropic, the model instead prefers sharpness and can converge to an arbitrarily sharp solution, depending on the imbalance in the noise in the labels spectrum. We reproduce this key insight in controlled settings with different model architectures such as MLP, RNN, and transformers.

Pierfrancesco Beneventano, Andrea Pinto, Tomaso Poggio

We investigate the ability of deep neural networks to identify the support of the target function. Our findings reveal that mini-batch SGD effectively learns the support in the first layer of the network by shrinking to zero the weights associated with irrelevant components of input. In contrast, we demonstrate that while vanilla GD also approximates the target function, it requires an explicit regularization term to learn the support in the first layer. We prove that this property of mini-batch SGD is due to a second-order implicit regularization effect which is proportional to $η/ b$ (step size / batch size). Our results are not only another proof that implicit regularization has a significant impact on training optimization dynamics but they also shed light on the structure of the features that are learned by the network. Additionally, they suggest that smaller batches enhance feature interpretability and reduce dependency on initialization.