arXiv subjects

Tomaso Poggio

Publications and source records attributed to Tomaso Poggio.

At least 19 recordsLinked to original sources

arXiv2026

Neural Quadratic Forms: A Unified Minimal Model for Sudden Learning and Scaling Laws

Neural networks trained by gradient descent on a smooth cost function can nevertheless learn in steps: the cost holds on long plateaus and then drops abruptly. Meanwhile, training losses instead follow smooth power laws. Variants of both behaviors occur in architectures with very different microscopic structures, which is the signature of a few relevant collective variables. We show that a symmetry fixes what those variables are: a network layer is a sum over interchangeable units, so relabeling the units leaves it unchanged; given smoothness and the condition that a unit's gradient vanish at the origin, symmetry then enforces a universal leading form for the expansion about the near-zero weights present at the start of training, the quadratic $\Tr[WW^{\top}A(x)]$, in which every architectural detail is confined to a single ``structure matrix" $A(x)$ that we compute for each architecture. Perceptrons, attention layers, mixtures of experts, and convolutions become one model at different $A$. Its training dynamics then close on the ``order parameter" $M=WW^{\top}$ and, whenever the data matrices share an eigenbasis, reduce to a Lotka--Volterra equation whose modes switch on one after another. The smaller the initial weights, the further apart the switch-on times, and the plateaus appear as a singular limit of a smooth flow; when many modes are unresolved the same events merge into a power law in training time whose exponent the theory predicts. We confirm both numerically across training methods and architectures.

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arXiv2026

The Spectral Dynamics and Noise Geometry of Muon

Muon replaces a matrix gradient $G=U\Sigma V^\top$ by its polar factor $UV^\top$. This keeps the singular directions selected by the gradient, but makes the update spectrum flat. We study the optimization bias created by this operation. Under explicit alignment assumptions, we prove that the polar update is the one-step entropy-maximizing choice among bounded updates that use the gradient singular directions and do not adapt to the current weight spectrum. In an underdetermined regression model, we derive exact singular-value dynamics for continuous-time Muon and identify a measurement-dependent condition under which the normalized spectrum moves toward equal nonzero singular values. This geometry also rules out a common low-rank interpretation: at fixed Frobenius norm, Muon's distinguished state has a flat spectrum, whereas nuclear-norm minimization favors spectral concentration. Controlled matrix-sensing experiments separate the effect from simple gradient rescaling, show that norm-matched gradient descent does not reproduce Muon, and recover the predicted flattening trend across broad ablations. In small NanoGPT pretraining, Muon preserves stable rank, has a broad learning-rate plateau, and improves validation loss relative to AdamW; in a matched small-ViT control, the ranking reverses. The resulting picture is regime-dependent: Muon is not universally superior, but its flat-spectrum bias can help when many spectral directions need to remain active.

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arXiv2026

Edge of Stability Selectively Shapes Learning Across the Data Distribution

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.

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arXiv2026

Do Deep Networks Forget Initialization? A Forgetting-Time View of Practical Inductive Bias

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.

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arXiv2026

Learning Sparse Compositional Functions with Norm-Constrained Neural Networks

The ability of deep neural networks to learn hierarchical features is widely regarded as a key mechanism underlying their success in high-dimensional learning. Existing theory partially supports this view by establishing approximation rates based on parameter counts and sample complexity guarantees for compositional models without incurring the curse of dimensionality (CoD). To study overparameterized regimes, where the number of parameters exceeds the sample size, we develop a framework that measures complexity via the parameter norm. Within this approach, we establish approximation rates and excess risk bounds for learning sparse compositional functions whose compositional structure is represented by directed acyclic graphs (DAGs), using Frobenius norm-constrained deep neural networks. Our results have broad applicability since every function that is efficiently Turing computable admits sparse compositional representations. In particular, we cover a range of representative models, including multi-index models, binary tree structures, and general compositional architectures. The rates we derive show that deep networks can exploit the compositional structure of the target functions, effectively avoiding the CoD through hierarchical representations.

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arXiv2026

Does Weight Decay Enhance Training Stability?

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 $\frac{2}{\eta}$ 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.

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arXiv2026

Agentic Systems as Boosting Weak Reasoning Models

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.

cs.AI↗

arXiv2026

The Generalized Turing Test: A Foundation for Comparing Intelligence

We introduce the Generalized Turing Test (GTT), a formal framework for comparing the capabilities of arbitrary agents via indistinguishability. For agents A and B, we define the Turing comparator A $\geq$ B to hold if B, acting as a distinguisher, cannot reliably distinguish between interactions with A (instructed to imitate B) and another instance of B. This yields a dataset- and task-agnostic notion of relative intelligence. We study the comparator's structure, including conditions under which it is transitive and therefore induces an ordering over equivalence classes, and we define and analyze variants with querying, bounded interaction, and fixed distinguishers. To complement the theory, we instantiate the framework on a collection of modern models, empirically evaluating pairwise indistinguishability across thousands of trials. The resulting comparisons exhibit a stratified structure consistent with existing rankings, hinting that the proposed framework yields meaningful empirical orderings. Our results position indistinguishability as a unifying lens for reasoning about intelligence, suggesting a foundation for evaluation and, potentially, training objectives that are inherently independent of fixed datasets or benchmarks.

cs.AI↗

arXiv2026

Too Sharp, Too Sure: When Calibration Follows Curvature

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.

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arXiv2026

pAI/MSc: ML Theory Research with Humans on the Loop

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.

cs.AI↗

arXiv2026

Momentum Further Constrains Sharpness at the Edge of Stochastic Stability

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 $2(1-\beta)/\eta$, reflecting amplification of stochastic fluctuations by momentum and favoring flatter regions than vanilla SGD; at large batch sizes it converges to a higher plateau $2(1+\beta)/\eta$, 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.

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arXiv2026

Same Error, Different Function: The Optimizer as an Implicit Prior in Financial Time Series

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.

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arXiv2026

Tool Building as a Path to "Superintelligence"

The Diligent Learner framework suggests LLMs can achieve superintelligence via test-time search, provided a sufficient step-success probability $\gamma$. In this work, we design a benchmark to measure $\gamma$ on logical out-of-distribution inference. We construct a class of tasks involving GF(2) circuit reconstruction that grow more difficult with each reasoning step, and that are, from an information-theoretic standpoint, impossible to reliably solve unless the LLM carefully integrates all of the information provided. Our analysis demonstrates that while the $\gamma$ value for small LLMs declines superlinearly as depth increases, frontier models exhibit partial robustness on this task. Furthermore, we find that successful reasoning at scale is contingent upon precise tool calls, identifying tool design as a critical capability for LLMs to achieve general superintelligence through the Diligent Learner framework.

cs.AI↗

arXiv2025

Training the Untrainable: Introducing Inductive Bias via Representational Alignment

We demonstrate that architectures which traditionally are considered to be ill-suited for a task can be trained using inductive biases from another architecture. We call a network untrainable when it overfits, underfits, or converges to poor results even when tuning their hyperparameters. For example, fully connected networks overfit on object recognition while deep convolutional networks without residual connections underfit. The traditional answer is to change the architecture to impose some inductive bias, although the nature of that bias is unknown. We introduce guidance, where a guide network steers a target network using a neural distance function. The target minimizes its task loss plus a layerwise representational similarity against the frozen guide. If the guide is trained, this transfers over the architectural prior and knowledge of the guide to the target. If the guide is untrained, this transfers over only part of the architectural prior of the guide. We show that guidance prevents FCN overfitting on ImageNet, narrows the vanilla RNN-Transformer gap, boosts plain CNNs toward ResNet accuracy, and aids Transformers on RNN-favored tasks. We further identify that guidance-driven initialization alone can mitigate FCN overfitting. Our method provides a mathematical tool to investigate priors and architectures, and in the long term, could automate architecture design.

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arXiv2025

Does Feedback Alignment Work at Biological Timescales?

Feedback alignment and related weight-transport-free algorithms are often proposed as biologically plausible alternatives to backpropagation, yet they are typically formulated in discrete phases with implicitly synchronized forward and error signals. We develop a continuous-time model of feedback-alignment-type learning in which neural activities and synaptic weights evolve together under coupled first-order dynamics with distinct propagation, plasticity, and decay time constants. We show that learning is governed by the temporal overlap between presynaptic drive and a locally projected error signal, providing an analytic explanation for robustness to moderate timing mismatch and for failure when mismatch eliminates overlap. Our results show that in order for feedback-alignment-type algorithms to work at biological timescales, they must obey the same temporal overlap principle that applies to other biological processes like eligibility traces.

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arXiv2025

On efficiently computable functions, deep networks and sparse compositionality

We show that \emph{efficient Turing computability} at any fixed input/output precision implies the existence of \emph{compositionally sparse} (bounded-fan-in, polynomial-size) DAG representations and of corresponding neural approximants achieving the target precision. Concretely: if $f:[0,1]^d\to\R^m$ is computable in time polynomial in the bit-depths, then for every pair of precisions $(n,m_{\mathrm{out}})$ there exists a bounded-fan-in Boolean circuit of size and depth $\poly(n+m_{\mathrm{out}})$ computing the discretized map; replacing each gate by a constant-size neural emulator yields a deep network of size/depth $\poly(n+m_{\mathrm{out}})$ that achieves accuracy $\varepsilon=2^{-m_{\mathrm{out}}}$. We also relate these constructions to compositional approximation rates \cite{MhaskarPoggio2016b,poggio_deep_shallow_2017,Poggio2017,Poggio2023HowDS} and to optimization viewed as hierarchical search over sparse structures.

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arXiv2025

Hierarchical Reasoning Models: Perspectives and Misconceptions

Transformers have demonstrated remarkable performance in natural language processing and related domains, as they largely focus on sequential, autoregressive next-token prediction tasks. Yet, they struggle in logical reasoning, not necessarily because of a fundamental limitation of these models, but possibly due to the lack of exploration of more creative uses, such as latent space and recurrent reasoning. An emerging exploration in this direction is the Hierarchical Reasoning Model (Wang et. al., 2025), which introduces a novel type of recurrent reasoning in the latent space of transformers, achieving remarkable performance on a wide range of 2D reasoning tasks. Despite the promising results, this line of models is still at an early stage and calls for in-depth investigation. In this work, we review this class of models, examine key design choices, test alternative variants and clarify common misconceptions.

cs.AI↗

arXiv2025

Topological Invariance and Breakdown in Learning

We prove that for a broad class of permutation-equivariant learning rules (including SGD, Adam, and others), the training process induces a bi-Lipschitz mapping between neurons and strongly constrains the topology of the neuron distribution during training. This result reveals a qualitative difference between small and large learning rates $η$. With a learning rate below a topological critical point $η^*$, the training is constrained to preserve all topological structure of the neurons. In contrast, above $η^*$, the learning process allows for topological simplification, making the neuron manifold progressively coarser and thereby reducing the model's expressivity. Viewed in combination with the recent discovery of the edge of stability phenomenon, the learning dynamics of neuron networks under gradient descent can be divided into two phases: first they undergo smooth optimization under topological constraints, and then enter a second phase where they learn through drastic topological simplifications. A key feature of our theory is that it is independent of specific architectures or loss functions, enabling the universal application of topological methods to the study of deep learning.

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