Brando Miranda

Rylan Schaeffer, Brando Miranda, Sanmi Koyejo

Recent work claims that large language models display emergent abilities, abilities not present in smaller-scale models that are present in larger-scale models. What makes emergent abilities intriguing is two-fold: their sharpness, transitioning seemingly instantaneously from not present to present, and their unpredictability, appearing at seemingly unforeseeable model scales. Here, we present an alternative explanation for emergent abilities: that for a particular task and model family, when analyzing fixed model outputs, emergent abilities appear due to the researcher's choice of metric rather than due to fundamental changes in model behavior with scale. Specifically, nonlinear or discontinuous metrics produce apparent emergent abilities, whereas linear or continuous metrics produce smooth, continuous predictable changes in model performance. We present our alternative explanation in a simple mathematical model, then test it in three complementary ways: we (1) make, test and confirm three predictions on the effect of metric choice using the InstructGPT/GPT-3 family on tasks with claimed emergent abilities; (2) make, test and confirm two predictions about metric choices in a meta-analysis of emergent abilities on BIG-Bench; and (3) show to choose metrics to produce never-before-seen seemingly emergent abilities in multiple vision tasks across diverse deep networks. Via all three analyses, we provide evidence that alleged emergent abilities evaporate with different metrics or with better statistics, and may not be a fundamental property of scaling AI models.

Brando Miranda, Patrick Yu, Yu-Xiong Wang et al.

Recently, it has been observed that a transfer learning solution might be all we need to solve many few-shot learning benchmarks -- thus raising important questions about when and how meta-learning algorithms should be deployed. In this paper, we seek to clarify these questions by 1. proposing a novel metric -- the diversity coefficient -- to measure the diversity of tasks in a few-shot learning benchmark and 2. by comparing Model-Agnostic Meta-Learning (MAML) and transfer learning under fair conditions (same architecture, same optimizer, and all models trained to convergence). Using the diversity coefficient, we show that the popular MiniImageNet and CIFAR-FS few-shot learning benchmarks have low diversity. This novel insight contextualizes claims that transfer learning solutions are better than meta-learned solutions in the regime of low diversity under a fair comparison. Specifically, we empirically find that a low diversity coefficient correlates with a high similarity between transfer learning and MAML learned solutions in terms of accuracy at meta-test time and classification layer similarity (using feature based distance metrics like SVCCA, PWCCA, CKA, and OPD). To further support our claim, we find this meta-test accuracy holds even as the model size changes. Therefore, we conclude that in the low diversity regime, MAML and transfer learning have equivalent meta-test performance when both are compared fairly. We also hope our work inspires more thoughtful constructions and quantitative evaluations of meta-learning benchmarks in the future.

Brando Miranda, Patrick Yu, Saumya Goyal et al.

In the context of few-shot learning, it is currently believed that a fixed pre-trained (PT) model, along with fine-tuning the final layer during evaluation, outperforms standard meta-learning algorithms. We re-evaluate these claims under an in-depth empirical examination of an extensive set of formally diverse datasets and compare PT to Model Agnostic Meta-Learning (MAML). Unlike previous work, we emphasize a fair comparison by using: the same architecture, the same optimizer, and all models trained to convergence. Crucially, we use a more rigorous statistical tool -- the effect size (Cohen's d) -- to determine the practical significance of the difference between a model trained with PT vs. a MAML. We then use a previously proposed metric -- the diversity coefficient -- to compute the average formal diversity of a dataset. Using this analysis, we demonstrate the following: 1. when the formal diversity of a data set is low, PT beats MAML on average and 2. when the formal diversity is high, MAML beats PT on average. The caveat is that the magnitude of the average difference between a PT vs. MAML using the effect size is low (according to classical statistical thresholds) -- less than 0.2. Nevertheless, this observation is contrary to the currently held belief that a pre-trained model is always better than a meta-learning model. Our extensive experiments consider 21 few-shot learning benchmarks, including the large-scale few-shot learning dataset Meta-Data set. We also show no significant difference between a MAML model vs. a PT model with GPT-2 on Openwebtext. We, therefore, conclude that a pre-trained model does not always beat a meta-learned model and that the formal diversity of a dataset is a driving factor.

Rylan Schaeffer, Dan Valentine, Luke Bailey et al.

The integration of new modalities into frontier AI systems offers exciting capabilities, but also increases the possibility such systems can be adversarially manipulated in undesirable ways. In this work, we focus on a popular class of vision-language models (VLMs) that generate text outputs conditioned on visual and textual inputs. We conducted a large-scale empirical study to assess the transferability of gradient-based universal image ``jailbreaks" using a diverse set of over 40 open-parameter VLMs, including 18 new VLMs that we publicly release. Overall, we find that transferable gradient-based image jailbreaks are extremely difficult to obtain. When an image jailbreak is optimized against a single VLM or against an ensemble of VLMs, the jailbreak successfully jailbreaks the attacked VLM(s), but exhibits little-to-no transfer to any other VLMs; transfer is not affected by whether the attacked and target VLMs possess matching vision backbones or language models, whether the language model underwent instruction-following and/or safety-alignment training, or many other factors. Only two settings display partially successful transfer: between identically-pretrained and identically-initialized VLMs with slightly different VLM training data, and between different training checkpoints of a single VLM. Leveraging these results, we then demonstrate that transfer can be significantly improved against a specific target VLM by attacking larger ensembles of ``highly-similar" VLMs. These results stand in stark contrast to existing evidence of universal and transferable text jailbreaks against language models and transferable adversarial attacks against image classifiers, suggesting that VLMs may be more robust to gradient-based transfer attacks.

Brando Miranda, Alycia Lee, Sudharsan Sundar et al.

Current trends in pre-training Large Language Models (LLMs) primarily focus on the scaling of model and dataset size. While the quality of pre-training data is considered an important factor for training powerful LLMs, it remains a nebulous concept that has not been rigorously characterized. To this end, we propose a formalization of one key aspect of data quality -- measuring the variability of natural language data -- specifically via a measure we call the diversity coefficient. Our empirical analysis shows that the proposed diversity coefficient aligns with the intuitive properties of diversity and variability, e.g., it increases as the number of latent concepts increases. Then, we measure the diversity coefficient of publicly available pre-training datasets and demonstrate that their formal diversity is high compared to theoretical lower and upper bounds. Finally, we conduct a comprehensive set of controlled interventional experiments with GPT-2 and LLaMAv2 that demonstrate the diversity coefficient of pre-training data characterizes useful aspects of downstream model evaluation performance -- totaling 44 models of various sizes (51M to 7B parameters). We conclude that our formal notion of diversity is an important aspect of data quality that captures variability and causally leads to improved evaluation performance.

Brando Miranda, Avi Shinnar, Vasily Pestun et al.

Despite a growing body of work at the intersection of deep learning and formal languages, there has been relatively little systematic exploration of transformer models for reasoning about typed lambda calculi. This is an interesting area of inquiry for two reasons. First, typed lambda calculi are the lingua franc of programming languages. A set of heuristics that relate various typed lambda calculi to effective neural architectures would provide a systematic method for mapping language features (e.g., polymorphism, subtyping, inheritance, etc.) to architecture choices. Second, transformer models are widely used in deep learning architectures applied to code, but the design and hyperparameter space for them is large and relatively unexplored in programming language applications. Therefore, we suggest a benchmark that allows us to explore exactly this through perhaps the simplest and most fundamental property of a programming language: the relationship between terms and types. Consequently, we begin this inquiry of transformer architectures for typed lambda calculi by exploring the effect of transformer warm-up and optimizer selection in the task of type inference: i.e., predicting the types of lambda calculus terms using only transformers. We find that the optimization landscape is difficult even in this simple setting. One particular experimental finding is that optimization by Adafactor converges much faster compared to the optimization by Adam and RAdam. We conjecture that such different performance of optimizers might be related to the difficulties of generalization over formally generated dataset.

Aryan Gulati, Brando Miranda, Eric Chen et al.

Current mathematical reasoning benchmarks for large language models (LLMs) are approaching saturation, with some achieving > 90% accuracy, and are increasingly compromised by training-set contamination. We introduce Putnam-AXIOM, a benchmark of 522 university-level competition problems drawn from the prestigious William Lowell Putnam Mathematical Competition, and Putnam-AXIOM Variation, an unseen companion set of 100 functional variants generated by programmatically perturbing variables and constants. The variation protocol produces an unlimited stream of equally difficult, unseen instances -- yielding a contamination-resilient test bed. On the Original set, OpenAI's o1-preview -- the strongest evaluated model -- scores 41.9%, but its accuracy drops by 19.6% (46.8% relative decrease) on the paired Variations. The remaining eighteen models show the same downward trend, ten of them with non-overlapping 95% confidence intervals. These gaps suggest memorization and highlight the necessity of dynamic benchmarks. We complement "boxed" accuracy with Teacher-Forced Accuracy (TFA), a lightweight metric that directly scores reasoning traces and automates natural language proof evaluations. Putnam-AXIOM therefore provides a rigorous, contamination-resilient evaluation framework for assessing advanced mathematical reasoning of LLMs. Data and evaluation code are publicly available at https://github.com/brando90/putnam-axiom.

Zhanke Zhou, Chentao Cao, Xiao Feng et al.

We present AlphaApollo, a self-evolving agentic reasoning system that aims to address two bottlenecks in foundation model (FM) reasoning-limited model-intrinsic capacity and unreliable test-time iteration. AlphaApollo orchestrates multiple models with professional tools to enable deliberate, verifiable reasoning. It couples (i) a computation tool (Python with numerical and symbolic libraries) and (ii) a retrieval tool (task-relevant external information) to execute exact calculations and ground decisions. The system further supports multi-round, multi-model solution evolution via a shared state map that records candidates, executable checks, and feedback for iterative refinement. In evaluations on AIME 2024/2025 across multiple models, AlphaApollo delivers consistent gains: +5.15% Average@32 and +23.34% Pass@32 for Qwen2.5-14B-Instruct, and +8.91% Average@32 with +26.67% Pass@32 for Llama-3.3-70B-Instruct. Tool-use analysis shows that more than 80% of tool calls are successfully executed, with consistent outperformance of non-tool baselines, thereby lifting the capability ceiling of FMs. More empirical results and implementation details will be updated at https://github.com/tmlr-group/AlphaApollo.

Leni Aniva, Chuyue Sun, Brando Miranda et al.

Machine-assisted theorem proving refers to the process of conducting structured reasoning to automatically generate proofs for mathematical theorems. Recently, there has been a surge of interest in using machine learning models in conjunction with proof assistants to perform this task. In this paper, we introduce Pantograph, a tool that provides a versatile interface to the Lean 4 proof assistant and enables efficient proof search via powerful search algorithms such as Monte Carlo Tree Search. In addition, Pantograph enables high-level reasoning by enabling a more robust handling of Lean 4's inference steps. We provide an overview of Pantograph's architecture and features. We also report on an illustrative use case: using machine learning models and proof sketches to prove Lean 4 theorems. Pantograph's innovative features pave the way for more advanced machine learning models to perform complex proof searches and high-level reasoning, equipping future researchers to design more versatile and powerful theorem provers.

Zeyu Tang, Sang T. Truong, Deonna Owens et al.

LLM fairness should be evaluated through in-situ conversational behavior rather than standardized-test Q&A benchmarks. We show that the standardized-test paradigm can be structurally unreliable: surface-level prompt construction choices, although entirely orthogonal to the fairness question being tested, account for the majority of score variance, shift fairness conclusions in both the direction and the magnitude, and result in severe discordance in model rankings. We develop MAC-Fairness, a multi-agent conversational framework that embeds controlled variation factors into multi-round dialogue for in-situ behavior evaluation, examining how models' conversational behavior shifts when identity is varied as part of natural multi-agent interaction. Repurposing standardized-test questions as conversation seeds rather than as the evaluation instrument, we evaluate position persistence (how they hold positions, from the self-perspective) and peer receptiveness (how receptive they are to peers, from the other-perspective) across 8 million conversation transcripts spanning multiple models and identity presence configurations. In-situ behavioral evaluation reveals stable, model-specific behavioral signatures that could generalize across benchmarks differing in fairness targets and evaluation methodologies, a form of evidence the standardized-test paradigm does not offer.

Willy Chan, Michael Souliman, Jakob Nordhagen et al.

Autoformalization, the process of transforming informal mathematical language into formal specifications and proofs remains a difficult task for state-of-the-art (large) language models. Existing works point to competing explanations for the performance gap. To this end, we introduce a novel methodology that leverages back-translation with hand-curated prompts to enhance the mathematical capabilities of language models, particularly addressing the challenge posed by the scarcity of labeled data. Specifically, we evaluate three primary variations of this strategy: (1) on-the-fly (online) backtranslation, (2) distilled (offline) backtranslation with few-shot amplification, and (3) line-by-line proof analysis integrated with proof state information. Each variant is designed to optimize data quality over quantity, focusing on the high fidelity of generated proofs rather than sheer data scale. Our findings provide evidence that employing our proposed approaches to generate synthetic data, which prioritizes quality over volume, improves the Autoformalization performance of LLMs as measured by standard benchmarks such as ProofNet. Crucially, our approach outperforms pretrained models using a minimal number of tokens. We also show, through strategic prompting and backtranslation, that our approaches surpass the performance of fine-tuning with extensive multilingual datasets such as MMA on ProofNet with only 1/150th of the tokens. Taken together, our methods show a promising new approach to significantly reduce the resources required to formalize proofs, thereby accelerating AI for math.

Rylan Schaeffer, Joshua Kazdan, Yegor Denisov-Blanch et al.

Science progresses by iteratively advancing and correcting humanity's understanding of the world. In machine learning (ML) research, rapid advancements have led to an explosion of publications, but have also led to misleading, incorrect, flawed or perhaps even fraudulent studies being accepted and sometimes highlighted at ML conferences due to the fallibility of peer review. While such mistakes are understandable, ML conferences do not offer robust processes to help the field systematically correct when such errors are made. This position paper argues that ML conferences should establish a dedicated "Refutations and Critiques" (R&C) Track. This R&C Track would provide a high-profile, reputable platform to support vital research that critically challenges prior research, thereby fostering a dynamic self-correcting research ecosystem. We discuss key considerations including track design, review principles, potential pitfalls, and provide an illustrative example submission concerning a recent ICLR 2025 Oral. We conclude that ML conferences should create official, reputable mechanisms to help ML research self-correct.

Rylan Schaeffer, Joshua Kazdan, Baber Abbasi et al.

As frontier AI systems are pretrained on web-scale data, test set contamination has become a critical concern for accurately assessing their capabilities. While research has thoroughly investigated the impact of test set contamination on discriminative evaluations like multiple-choice question-answering, comparatively little research has studied the impact of test set contamination on generative evaluations. In this work, we quantitatively assess the effect of test set contamination on generative evaluations through the language model lifecycle. We pretrain language models on mixtures of web data and the MATH benchmark, sweeping model sizes and number of test set replicas contaminating the pretraining corpus; performance improves with contamination and model size. Using scaling laws, we make a surprising discovery: including even a single test set replica enables models to achieve lower loss than the irreducible error of training on the uncontaminated corpus. We then study further training: overtraining with fresh data reduces the effects of contamination, whereas supervised finetuning on the training set can either increase or decrease performance on test data, depending on the amount of pretraining contamination. Finally, at inference, we identify factors that modulate memorization: high sampling temperatures mitigate contamination effects, and longer solutions are exponentially more difficult to memorize than shorter ones, presenting a contrast with discriminative evaluations, where solutions are only a few tokens in length. By characterizing how generation and memorization interact, we highlight a new layer of complexity for trustworthy evaluation of AI systems.

Rylan Schaeffer, Noam Levi, Brando Miranda et al.

Neural scaling laws have played a central role in modern machine learning, driving the field's ever-expanding scaling of parameters, data and compute. While much research has gone into fitting scaling laws and predicting performance on pretraining losses and on discriminative evaluations such as multiple-choice question-answering, comparatively little research has been done on fitting scaling laws and predicting performance on generative evaluations such as mathematical problem-solving or software engineering. We propose and evaluate three different pretraining scaling laws for fitting pass-at-$k$ on generative evaluations and for predicting pass-at-$k$ of the most expensive model using the performance of cheaper models. Our three scaling laws differ in the covariates used: (1) compute, (2) model parameters and tokens, (3) log likelihoods of gold reference solutions. We make four main contributions: (1) We show how generative evaluations offer new hyperparameters (in our setting, $k$) that researchers can use to control the scaling laws parameters and the predictability of performance. (2) In terms of scaling law parameters, we find that the compute scaling law and parameters\,+\,tokens scaling law stabilize for the last ~$1.5{-}2.5$ orders of magnitude, whereas the gold reference likelihood scaling law stabilizes for the last ~$5$ orders of magnitude. (3) In terms of predictive performance, we find all three scaling laws perform comparably, although the compute scaling law predicts slightly worse for small $k$ and the log likelihoods of gold reference solutions predicts slightly worse for large $k$. (4) We establish a theoretical connection that the compute scaling law emerges as the compute-optimal envelope of the parameters-and-tokens scaling law. Our framework provides researchers and practitioners with insights and methodologies to forecast generative performance.

Rylan Schaeffer, Noam Levi, Andreas Kirsch et al.

Hoffman et al (2022)'s Chinchilla paper introduced the principle of compute-optimal scaling, laying a foundation for future scaling of language models. In the years since, however, valid concerns about Chinchilla have been raised: wide confidence intervals, discrepancies between its three approaches, and incongruities with other scaling laws. This raises a critical question for the field: Can practitioners still rely on Chinchilla's prescriptions? Our work demonstrates the answer is yes. We begin by uncovering that the model parameters central to Chinchilla's analyses were ambiguous: three interpretations are possible, with relative differences between different interpretations of model parameters as high as 15.2%. We find that, perhaps surprisingly, which model parameters are used for the analyses do not meaningfully affect key results: the scaling law estimates and the compute-optimal tokens-to-parameter ratio. Indeed, under one interpretation, the tokens-to-parameter ratio becomes more constant with the target compute budget. We then ask how distorted the Chinchilla model parameters could have been without meaningfully affecting the key results. By deliberately perturbing model parameters in four structured ways, we find that key Chinchilla results are most sensitive to additive or systematic errors, which can alter the otherwise flat trend of the optimal tokens-to-parameter ratio, but overall, Chinchilla's key results withstand sizable perturbations. Altogether, our findings offer the field renewed confidence in Chinchilla as a durable guide for scaling language models.

Kavita Selva, Satita Vittayaareekul, Brando Miranda

Currently, data and model size dominate the narrative in the training of super-large, powerful models. However, there has been a lack of exploration on the effect of other attributes of the training dataset on model performance. We hypothesize that dataset diversity can impact the performance of vision models. Our study shows positive correlations between test set accuracy and data diversity, providing an argument for furthering the research of dataset attributes beyond size. We analyzed pre-training and model-agnostic meta-learning methods on twelve popular visual datasets (e.g., Omniglot, CIFAR-FS, Aircraft) and five model configurations, including MAML variants with different numbers of inner gradient steps and supervised learning. We show moderate to strong positive correlations (R-squared: 0.15-0.42) between accuracy and data diversity and weaker but significant correlations (R-squared: ~0.2) between loss and diversity. These findings support our hypothesis and demonstrate a promising way for a deeper exploration of how formal data diversity influences model performance. This initial study highlights the potential of (Task2Vec) data diversity as a valuable measure in the rapidly evolving field of large-scale learning and emphasizes that understanding the dataset is key to building more powerful and generalizable models.

Elyas Obbad, Iddah Mlauzi, Brando Miranda et al.

Data selection is crucial for optimizing language model (LM) performance on specific tasks, yet most existing methods fail to effectively consider the target task distribution. Current approaches either ignore task-specific requirements entirely or rely on approximations that fail to capture the nuanced patterns needed for tasks like Autoformalization or code generation. Methods that do consider the target distribution often rely on simplistic, sometimes noisy, representations, like hashed n-gram features, which can lead to collisions and introduce noise. We introduce ZIP-FIT, a data selection framework that uses gzip compression to directly measure alignment between potential training data and the target task distribution. In extensive evaluations on Autoformalization and Python code generation, ZIP-FIT significantly outperforms leading baselines like DSIR and D4. Models trained on ZIP-FIT-selected data achieve their lowest cross-entropy loss up to 85.1\% faster than baselines, demonstrating that better task alignment leads to more efficient learning. In addition, ZIP-FIT performs selection up to 65.8\% faster than DSIR and two orders of magnitude faster than D4. Notably, ZIP-FIT shows that smaller, well-aligned datasets often outperform larger but less targeted ones, demonstrating that a small amount of higher quality data is superior to a large amount of lower quality data. Our results imply that task-aware data selection is crucial for efficient domain adaptation, and that compression offers a principled way to measure task alignment. By showing that targeted data selection can dramatically improve task-specific performance, our work provides new insights into the relationship between data quality, task alignment, and model learning efficiency.

Rylan Schaeffer, Hailey Schoelkopf, Brando Miranda et al.

Predicting changes from scaling advanced AI systems is a desirable property for engineers, economists, governments and industry alike, and, while a well-established literature exists on how pretraining performance scales, predictable scaling behavior on downstream capabilities remains elusive. While many factors are certainly responsible, this paper identifies a significant factor that makes predicting scaling behavior on widely used multiple-choice question answering benchmarks challenging and illuminates a path towards making such downstream evaluations predictable with scale. Using five model families and twelve well-established multiple-choice benchmarks, we demonstrate that downstream performance is computed from negative log likelihoods via a sequence of transformations that progressively degrades the statistical relationship between performance and scale. We then pinpoint the mechanism causing this degradation: downstream metrics require comparing the correct choice against a small number of specific incorrect choices, meaning accurately predicting downstream capabilities requires predicting not just how probability mass concentrates on the correct choice with scale, but also how probability mass fluctuates on the alternative incorrect choices with scale. We empirically study how probability mass on the correct choice co-varies with probability mass on incorrect choices with increasing compute, suggesting that scaling laws for \textit{incorrect} choices might be achievable. Our work also explains why pretraining scaling laws are commonly regarded as more predictable than downstream capabilities and contributes towards establishing scaling-predictable evaluations of frontier AI models.

Aryan Gulati, Devanshu Ladsaria, Shubhra Mishra et al.

Large Language Models (LLMs) hold the potential to revolutionize autoformalization. The introduction of Lean4, a mathematical programming language, presents an unprecedented opportunity to rigorously assess the autoformalization capabilities of LLMs. This paper introduces a novel evaluation benchmark designed for Lean4, applying it to test the abilities of state-of-the-art LLMs, including GPT-3.5, GPT-4, and Gemini Pro. Our comprehensive analysis reveals that, despite recent advancements, these LLMs still exhibit limitations in autoformalization, particularly in more complex areas of mathematics. These findings underscore the need for further development in LLMs to fully harness their potential in scientific research and development. This study not only benchmarks current LLM capabilities but also sets the stage for future enhancements in autoformalization.

Brando Miranda, Yu-Xiong Wang, Sanmi Koyejo

Recent work has suggested that a good embedding is all we need to solve many few-shot learning benchmarks. Furthermore, other work has strongly suggested that Model Agnostic Meta-Learning (MAML) also works via this same method - by learning a good embedding. These observations highlight our lack of understanding of what meta-learning algorithms are doing and when they work. In this work, we provide empirical results that shed some light on how meta-learned MAML representations function. In particular, we identify three interesting properties: 1) In contrast to previous work, we show that it is possible to define a family of synthetic benchmarks that result in a low degree of feature re-use - suggesting that current few-shot learning benchmarks might not have the properties needed for the success of meta-learning algorithms; 2) meta-overfitting occurs when the number of classes (or concepts) are finite, and this issue disappears once the task has an unbounded number of concepts (e.g., online learning); 3) more adaptation at meta-test time with MAML does not necessarily result in a significant representation change or even an improvement in meta-test performance - even when training on our proposed synthetic benchmarks. Finally, we suggest that to understand meta-learning algorithms better, we must go beyond tracking only absolute performance and, in addition, formally quantify the degree of meta-learning and track both metrics together. Reporting results in future work this way will help us identify the sources of meta-overfitting more accurately and help us design more flexible meta-learning algorithms that learn beyond fixed feature re-use. Finally, we conjecture the core challenge of re-thinking meta-learning is in the design of few-shot learning data sets and benchmarks - rather than in the algorithms, as suggested by previous work.

Brando Miranda, Yu-Xiong Wang, Sanmi Koyejo

Recently, it has been observed that a transfer learning solution might be all we need to solve many few-shot learning benchmarks -- thus raising important questions about when and how meta-learning algorithms should be deployed. In this paper, we seek to clarify these questions by proposing a novel metric -- the diversity coefficient -- to measure the diversity of tasks in a few-shot learning benchmark. We hypothesize that the diversity coefficient of the few-shot learning benchmark is predictive of whether meta-learning solutions will succeed or not. Using the diversity coefficient, we show that the MiniImagenet benchmark has zero diversity. This novel insight contextualizes claims that transfer learning solutions are better than meta-learned solutions. Specifically, we empirically find that a diversity coefficient of zero correlates with a high similarity between transfer learning and Model-Agnostic Meta-Learning (MAML) learned solutions in terms of meta-accuracy (at meta-test time). Therefore, we conjecture meta-learned solutions have the same meta-test performance as transfer learning when the diversity coefficient is zero. Our work provides the first test of whether diversity correlates with meta-learning success.

Andrzej Banburski, Qianli Liao, Brando Miranda et al.

The key to generalization is controlling the complexity of the network. However, there is no obvious control of complexity -- such as an explicit regularization term -- in the training of deep networks for classification. We will show that a classical form of norm control -- but kind of hidden -- is present in deep networks trained with gradient descent techniques on exponential-type losses. In particular, gradient descent induces a dynamics of the normalized weights which converge for $t \to \infty$ to an equilibrium which corresponds to a minimum norm (or maximum margin) solution. For sufficiently large but finite $ρ$ -- and thus finite $t$ -- the dynamics converges to one of several margin maximizers, with the margin monotonically increasing towards a limit stationary point of the flow. In the usual case of stochastic gradient descent, most of the stationary points are likely to be convex minima corresponding to a constrained minimizer -- the network with normalized weights-- which corresponds to vanishing regularization. The solution has zero generalization gap, for fixed architecture, asymptotically for $N \to \infty$, where $N$ is the number of training examples. Our approach extends some of the original results of Srebro from linear networks to deep networks and provides a new perspective on the implicit bias of gradient descent. We believe that the elusive complexity control we describe is responsible for the puzzling empirical finding of good predictive performance by deep networks, despite overparametrization.

Qianli Liao, Brando Miranda, Andrzej Banburski et al.

Given two networks with the same training loss on a dataset, when would they have drastically different test losses and errors? Better understanding of this question of generalization may improve practical applications of deep networks. In this paper we show that with cross-entropy loss it is surprisingly simple to induce significantly different generalization performances for two networks that have the same architecture, the same meta parameters and the same training error: one can either pretrain the networks with different levels of "corrupted" data or simply initialize the networks with weights of different Gaussian standard deviations. A corollary of recent theoretical results on overfitting shows that these effects are due to an intrinsic problem of measuring test performance with a cross-entropy/exponential-type loss, which can be decomposed into two components both minimized by SGD -- one of which is not related to expected classification performance. However, if we factor out this component of the loss, a linear relationship emerges between training and test losses. Under this transformation, classical generalization bounds are surprisingly tight: the empirical/training loss is very close to the expected/test loss. Furthermore, the empirical relation between classification error and normalized cross-entropy loss seem to be approximately monotonic

Tomaso Poggio, Qianli Liao, Brando Miranda et al.

A main puzzle of deep neural networks (DNNs) revolves around the apparent absence of "overfitting", defined in this paper as follows: the expected error does not get worse when increasing the number of neurons or of iterations of gradient descent. This is surprising because of the large capacity demonstrated by DNNs to fit randomly labeled data and the absence of explicit regularization. Recent results by Srebro et al. provide a satisfying solution of the puzzle for linear networks used in binary classification. They prove that minimization of loss functions such as the logistic, the cross-entropy and the exp-loss yields asymptotic, "slow" convergence to the maximum margin solution for linearly separable datasets, independently of the initial conditions. Here we prove a similar result for nonlinear multilayer DNNs near zero minima of the empirical loss. The result holds for exponential-type losses but not for the square loss. In particular, we prove that the weight matrix at each layer of a deep network converges to a minimum norm solution up to a scale factor (in the separable case). Our analysis of the dynamical system corresponding to gradient descent of a multilayer network suggests a simple criterion for ranking the generalization performance of different zero minimizers of the empirical loss.

Chiyuan Zhang, Qianli Liao, Alexander Rakhlin et al.

In Theory IIb we characterize with a mix of theory and experiments the optimization of deep convolutional networks by Stochastic Gradient Descent. The main new result in this paper is theoretical and experimental evidence for the following conjecture about SGD: SGD concentrates in probability -- like the classical Langevin equation -- on large volume, "flat" minima, selecting flat minimizers which are with very high probability also global minimizers

Tomaso Poggio, Kenji Kawaguchi, Qianli Liao et al.

A main puzzle of deep networks revolves around the absence of overfitting despite large overparametrization and despite the large capacity demonstrated by zero training error on randomly labeled data. In this note, we show that the dynamics associated to gradient descent minimization of nonlinear networks is topologically equivalent, near the asymptotically stable minima of the empirical error, to linear gradient system in a quadratic potential with a degenerate (for square loss) or almost degenerate (for logistic or crossentropy loss) Hessian. The proposition depends on the qualitative theory of dynamical systems and is supported by numerical results. Our main propositions extend to deep nonlinear networks two properties of gradient descent for linear networks, that have been recently established (1) to be key to their generalization properties: 1. Gradient descent enforces a form of implicit regularization controlled by the number of iterations, and asymptotically converges to the minimum norm solution for appropriate initial conditions of gradient descent. This implies that there is usually an optimum early stopping that avoids overfitting of the loss. This property, valid for the square loss and many other loss functions, is relevant especially for regression. 2. For classification, the asymptotic convergence to the minimum norm solution implies convergence to the maximum margin solution which guarantees good classification error for "low noise" datasets. This property holds for loss functions such as the logistic and cross-entropy loss independently of the initial conditions. The robustness to overparametrization has suggestive implications for the robustness of the architecture of deep convolutional networks with respect to the curse of dimensionality.

Tomaso Poggio, Hrushikesh Mhaskar, Lorenzo Rosasco et al.

The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage.