Tomaso Poggio

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.

Daniel Mitropolsky, Susan S. Hong, Riccardo Neumarker et al.

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.

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.

Andrea Pinto, Akshay Rangamani, Tomaso Poggio

While previous optimization results have suggested that deep neural networks tend to favour low-rank weight matrices, the implications of this inductive bias on generalization bounds remain underexplored. In this paper, we apply Maurer's chain rule for Gaussian complexity to analyze how low-rank layers in deep networks can prevent the accumulation of rank and dimensionality factors that typically multiply across layers. This approach yields generalization bounds for rank and spectral norm constrained networks. We compare our results to prior generalization bounds for deep networks, highlighting how deep networks with low-rank layers can achieve better generalization than those with full-rank layers. Additionally, we discuss how this framework provides new perspectives on the generalization capabilities of deep networks exhibiting neural collapse.

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.