arXiv subjects

Emmanuel Abbe

Publications and source records attributed to Emmanuel Abbe.

At least 37 records · Page 2Linked to original sources

arXiv2024

Transformation-Invariant Learning and Theoretical Guarantees for OOD Generalization

Learning with identical train and test distributions has been extensively investigated both practically and theoretically. Much remains to be understood, however, in statistical learning under distribution shifts. This paper focuses on a distribution shift setting where train and test distributions can be related by classes of (data) transformation maps. We initiate a theoretical study for this framework, investigating learning scenarios where the target class of transformations is either known or unknown. We establish learning rules and algorithmic reductions to Empirical Risk Minimization (ERM), accompanied with learning guarantees. We obtain upper bounds on the sample complexity in terms of the VC dimension of the class composing predictors with transformations, which we show in many cases is not much larger than the VC dimension of the class of predictors. We highlight that the learning rules we derive offer a game-theoretic viewpoint on distribution shift: a learner searching for predictors and an adversary searching for transformation maps to respectively minimize and maximize the worst-case loss.

cs.LG↗

arXiv2024

The merged-staircase property: a necessary and nearly sufficient condition for SGD learning of sparse functions on two-layer neural networks

It is currently known how to characterize functions that neural networks can learn with SGD for two extremal parameterizations: neural networks in the linear regime, and neural networks with no structural constraints. However, for the main parametrization of interest (non-linear but regular networks) no tight characterization has yet been achieved, despite significant developments. We take a step in this direction by considering depth-2 neural networks trained by SGD in the mean-field regime. We consider functions on binary inputs that depend on a latent low-dimensional subspace (i.e., small number of coordinates). This regime is of interest since it is poorly understood how neural networks routinely tackle high-dimensional datasets and adapt to latent low-dimensional structure without suffering from the curse of dimensionality. Accordingly, we study SGD-learnability with $O(d)$ sample complexity in a large ambient dimension $d$. Our main results characterize a hierarchical property, the "merged-staircase property", that is both necessary and nearly sufficient for learning in this setting. We further show that non-linear training is necessary: for this class of functions, linear methods on any feature map (e.g., the NTK) are not capable of learning efficiently. The key tools are a new "dimension-free" dynamics approximation result that applies to functions defined on a latent space of low-dimension, a proof of global convergence based on polynomial identity testing, and an improvement of lower bounds against linear methods for non-almost orthogonal functions.

cs.LG↗

arXiv2024

On the Minimal Degree Bias in Generalization on the Unseen for non-Boolean Functions

We investigate the out-of-domain generalization of random feature (RF) models and Transformers. We first prove that in the `generalization on the unseen (GOTU)' setting, where training data is fully seen in some part of the domain but testing is made on another part, and for RF models in the small feature regime, the convergence takes place to interpolators of minimal degree as in the Boolean case (Abbe et al., 2023). We then consider the sparse target regime and explain how this regime relates to the small feature regime, but with a different regularization term that can alter the picture in the non-Boolean case. We show two different outcomes for the sparse regime with q-ary data tokens: (1) if the data is embedded with roots of unities, then a min-degree interpolator is learned like in the Boolean case for RF models, (2) if the data is not embedded as such, e.g., simply as integers, then RF models and Transformers may not learn minimal degree interpolators. This shows that the Boolean setting and its roots of unities generalization are special cases where the minimal degree interpolator offers a rare characterization of how learning takes place. For more general integer and real-valued settings, a more nuanced picture remains to be fully characterized.

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arXiv2024

When can transformers reason with abstract symbols?

We investigate the capabilities of transformer models on relational reasoning tasks. In these tasks, models are trained on a set of strings encoding abstract relations, and are then tested out-of-distribution on data that contains symbols that did not appear in the training dataset. We prove that for any relational reasoning task in a large family of tasks, transformers learn the abstract relations and generalize to the test set when trained by gradient descent on sufficiently large quantities of training data. This is in contrast to classical fully-connected networks, which we prove fail to learn to reason. Our results inspire modifications of the transformer architecture that add only two trainable parameters per head, and that we empirically demonstrate improve data efficiency for learning to reason.

cs.CL↗

arXiv2023

Transformers learn through gradual rank increase

We identify incremental learning dynamics in transformers, where the difference between trained and initial weights progressively increases in rank. We rigorously prove this occurs under the simplifying assumptions of diagonal weight matrices and small initialization. Our experiments support the theory and also show that phenomenon can occur in practice without the simplifying assumptions.

cs.LG↗

arXiv2023

Reed-Muller codes have vanishing bit-error probability below capacity: a simple tighter proof via camellia boosting

This paper shows that a class of codes such as Reed-Muller (RM) codes have vanishing bit-error probability below capacity on symmetric channels. The proof relies on the notion of `camellia codes': a class of symmetric codes decomposable into `camellias', i.e., set systems that differ from sunflowers by allowing for scattered petal overlaps. The proof then follows from a boosting argument on the camellia petals with second moment Fourier analysis. For erasure channels, this gives a self-contained proof of the bit-error result in Kudekar et al.'17, without relying on sharp thresholds for monotone properties Friedgut-Kalai'96. For error channels, this gives a shortened proof of Reeves-Pfister'23 with an exponentially tighter bound, and a proof variant of the bit-error result in Abbe-Sandon'23. The control of the full (block) error probability still requires Abbe-Sandon'23 for RM codes.

cs.IT↗

arXiv2023

Community Detection and Stochastic Block Models

The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study the information-theoretic and computational tradeoffs that arise in combinatorial statistics and more generally data science. This monograph surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational tradeoffs, and for various recovery requirements such as exact, partial and weak recovery. The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal SNR-mutual information tradeoff for partial recovery, and the gap between information-theoretic and computational thresholds. The monograph gives a principled derivation of the main algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, (linearized) belief propagation, classical/nonbacktracking spectral methods and graph powering. Extensions to other block models, such as geometric block models, and a few open problems are also discussed.

math.PR↗

arXiv2023

SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics

We investigate the time complexity of SGD learning on fully-connected neural networks with isotropic data. We put forward a complexity measure -- the leap -- which measures how "hierarchical" target functions are. For $d$-dimensional uniform Boolean or isotropic Gaussian data, our main conjecture states that the time complexity to learn a function $f$ with low-dimensional support is $\tildeΘ(d^{\max(\mathrm{Leap}(f),2)})$. We prove a version of this conjecture for a class of functions on Gaussian isotropic data and 2-layer neural networks, under additional technical assumptions on how SGD is run. We show that the training sequentially learns the function support with a saddle-to-saddle dynamic. Our result departs from [Abbe et al. 2022] by going beyond leap 1 (merged-staircase functions), and by going beyond the mean-field and gradient flow approximations that prohibit the full complexity control obtained here. Finally, we note that this gives an SGD complexity for the full training trajectory that matches that of Correlational Statistical Query (CSQ) lower-bounds.

cs.LG↗

arXiv2023

Provable Advantage of Curriculum Learning on Parity Targets with Mixed Inputs

Experimental results have shown that curriculum learning, i.e., presenting simpler examples before more complex ones, can improve the efficiency of learning. Some recent theoretical results also showed that changing the sampling distribution can help neural networks learn parities, with formal results only for large learning rates and one-step arguments. Here we show a separation result in the number of training steps with standard (bounded) learning rates on a common sample distribution: if the data distribution is a mixture of sparse and dense inputs, there exists a regime in which a 2-layer ReLU neural network trained by a curriculum noisy-GD (or SGD) algorithm that uses sparse examples first, can learn parities of sufficiently large degree, while any fully connected neural network of possibly larger width or depth trained by noisy-GD on the unordered samples cannot learn without additional steps. We also provide experimental results supporting the qualitative separation beyond the specific regime of the theoretical results.

cs.LG↗

arXiv2023

A proof that Reed-Muller codes achieve Shannon capacity on symmetric channels

Reed-Muller codes were introduced in 1954, with a simple explicit construction based on polynomial evaluations, and have long been conjectured to achieve Shannon capacity on symmetric channels. Major progress was made towards a proof over the last decades; using combinatorial weight enumerator bounds, a breakthrough on the erasure channel from sharp thresholds, hypercontractivity arguments, and polarization theory. Another major progress recently established that the bit error probability vanishes slowly below capacity. However, when channels allow for errors, the results of Bourgain-Kalai do not apply for converting a vanishing bit to a vanishing block error probability, neither do the known weight enumerator bounds. The conjecture that RM codes achieve Shannon capacity on symmetric channels, with high probability of recovering the codewords, has thus remained open. This paper closes the conjecture's proof. It uses a new recursive boosting framework, which aggregates the decoding of codeword restrictions on `subspace-sunflowers', handling their dependencies via an $L_p$ Boolean Fourier analysis, and using a list-decoding argument with a weight enumerator bound from Sberlo-Shpilka. The proof does not require a vanishing bit error probability for the base case, but only a non-trivial probability, obtained here for general symmetric codes. This gives in particular a shortened and tightened argument for the vanishing bit error probability result of Reeves-Pfister, and with prior works, it implies the strong wire-tap secrecy of RM codes on pure-state classical-quantum channels.

cs.IT↗

arXiv2022

Learning to Reason with Neural Networks: Generalization, Unseen Data and Boolean Measures

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss.

cs.LG↗

arXiv2022

On the non-universality of deep learning: quantifying the cost of symmetry

We prove limitations on what neural networks trained by noisy gradient descent (GD) can efficiently learn. Our results apply whenever GD training is equivariant, which holds for many standard architectures and initializations. As applications, (i) we characterize the functions that fully-connected networks can weak-learn on the binary hypercube and unit sphere, demonstrating that depth-2 is as powerful as any other depth for this task; (ii) we extend the merged-staircase necessity result for learning with latent low-dimensional structure [ABM22] to beyond the mean-field regime. Under cryptographic assumptions, we also show hardness results for learning with fully-connected networks trained by stochastic gradient descent (SGD).

cs.LG↗

arXiv2022

An initial alignment between neural network and target is needed for gradient descent to learn

This paper introduces the notion of ``Initial Alignment'' (INAL) between a neural network at initialization and a target function. It is proved that if a network and a Boolean target function do not have a noticeable INAL, then noisy gradient descent on a fully connected network with normalized i.i.d. initialization will not learn in polynomial time. Thus a certain amount of knowledge about the target (measured by the INAL) is needed in the architecture design. This also provides an answer to an open problem posed in [AS20]. The results are based on deriving lower-bounds for descent algorithms on symmetric neural networks without explicit knowledge of the target function beyond its INAL.

cs.LG↗

arXiv2022

An $\ell_p$ theory of PCA and spectral clustering

Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an $\ell_p$ perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel $\ell_p$ analysis of eigenvectors, we investigate entrywise behaviors of principal component score vectors and show that they can be approximated by linear functionals of the Gram matrix in $\ell_p$ norm, which includes $\ell_2$ and $\ell_\infty$ as special examples. For sub-Gaussian mixture models, the choice of $p$ giving optimal bounds depends on the signal-to-noise ratio, which further yields optimality guarantees for spectral clustering. For contextual community detection, the $\ell_p$ theory leads to a simple spectral algorithm that achieves the information threshold for exact recovery. These also provide optimal recovery results for Gaussian mixture and stochastic block models as special cases.

math.ST↗

arXiv2022

On the Power of Differentiable Learning versus PAC and SQ Learning

We study the power of learning via mini-batch stochastic gradient descent (SGD) on the population loss, and batch Gradient Descent (GD) on the empirical loss, of a differentiable model or neural network, and ask what learning problems can be learnt using these paradigms. We show that SGD and GD can always simulate learning with statistical queries (SQ), but their ability to go beyond that depends on the precision $ρ$ of the gradient calculations relative to the minibatch size $b$ (for SGD) and sample size $m$ (for GD). With fine enough precision relative to minibatch size, namely when $b ρ$ is small enough, SGD can go beyond SQ learning and simulate any sample-based learning algorithm and thus its learning power is equivalent to that of PAC learning; this extends prior work that achieved this result for $b=1$. Similarly, with fine enough precision relative to the sample size $m$, GD can also simulate any sample-based learning algorithm based on $m$ samples. In particular, with polynomially many bits of precision (i.e. when $ρ$ is exponentially small), SGD and GD can both simulate PAC learning regardless of the mini-batch size. On the other hand, when $b ρ^2$ is large enough, the power of SGD is equivalent to that of SQ learning.

cs.LG↗

arXiv2021

The staircase property: How hierarchical structure can guide deep learning

This paper identifies a structural property of data distributions that enables deep neural networks to learn hierarchically. We define the "staircase" property for functions over the Boolean hypercube, which posits that high-order Fourier coefficients are reachable from lower-order Fourier coefficients along increasing chains. We prove that functions satisfying this property can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks -- a class of network architectures and initializations that have homogeneity properties. Our analysis shows that for such staircase functions and neural networks, the gradient-based algorithm learns high-level features by greedily combining lower-level features along the depth of the network. We further back our theoretical results with experiments showing that staircase functions are also learnable by more standard ResNet architectures with stochastic gradient descent. Both the theoretical and experimental results support the fact that staircase properties have a role to play in understanding the capabilities of gradient-based learning on regular networks, in contrast to general polynomial-size networks that can emulate any SQ or PAC algorithms as recently shown.

cs.LG↗

arXiv2021

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory communities, with recent connections made to the performance of learning algorithms in Baldassi et al. '15. We establish that the partition function of this model, normalized by its expected value, converges to a lognormal distribution. As a consequence, this allows us to establish several conjectures for this model: (i) it proves the contiguity conjecture of Aubin et al. '19 between the planted and unplanted models in the satisfiable regime; (ii) it establishes the sharp threshold conjecture; (iii) it proves the frozen 1-RSB conjecture in the symmetric case, conjectured first by Krauth-Mézard '89 in the asymmetric case. In a recent work of Perkins-Xu '21, the last two conjectures were also established by proving that the partition function concentrates on an exponential scale, under an analytical assumption on a real-valued function. This left open the contiguity conjecture and the lognormal limit characterization, which are established here unconditionally, with the analytical assumption verified. In particular, our proof technique relies on a dense counter-part of the small graph conditioning method, which was developed for sparse models in the celebrated work of Robinson and Wormald.

math.PR↗

arXiv2021

Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster

It was recently shown that almost all solutions in the symmetric binary perceptron are isolated, even at low constraint densities, suggesting that finding typical solutions is hard. In contrast, some algorithms have been shown empirically to succeed in finding solutions at low density. This phenomenon has been justified numerically by the existence of subdominant and dense connected regions of solutions, which are accessible by simple learning algorithms. In this paper, we establish formally such a phenomenon for both the symmetric and asymmetric binary perceptrons. We show that at low constraint density (equivalently for overparametrized perceptrons), there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu '21. In addition, even close to the critical threshold, we show that there exist clusters of linear diameter for the symmetric perceptron, as well as for the asymmetric perceptron under additional assumptions.

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