Marco Mondelli

Diyuan Wu, Lehan Chen, Theodor Misiakiewicz et al.

It is increasingly common in machine learning to use learned models to label data and then employ such data to train more capable models. The phenomenon of weak-to-strong generalization exemplifies the advantage of this two-stage procedure: a strong student is trained on imperfect labels obtained from a weak teacher, and yet the strong student outperforms the weak teacher. In this paper, we show that the potential improvement is substantial, in the sense that it affects the scaling law followed by the test error. Specifically, we consider students and teachers trained via random feature ridge regression (RFRR). Our main technical contribution is to derive a deterministic equivalent for the excess test error of the student trained on labels obtained via the teacher. Via this deterministic equivalent, we then identify regimes in which the scaling law of the student improves upon that of the teacher, unveiling that the improvement can be achieved both in bias-dominated and variance-dominated settings. Strikingly, the student may attain the minimax optimal rate regardless of the scaling law of the teacher -- in fact, when the test error of the teacher does not even decay with the sample size.

Simone Bombari, Mohammad Hossein Amani, Marco Mondelli

The Neural Tangent Kernel (NTK) has emerged as a powerful tool to provide memorization, optimization and generalization guarantees in deep neural networks. A line of work has studied the NTK spectrum for two-layer and deep networks with at least a layer with $Ω(N)$ neurons, $N$ being the number of training samples. Furthermore, there is increasing evidence suggesting that deep networks with sub-linear layer widths are powerful memorizers and optimizers, as long as the number of parameters exceeds the number of samples. Thus, a natural open question is whether the NTK is well conditioned in such a challenging sub-linear setup. In this paper, we answer this question in the affirmative. Our key technical contribution is a lower bound on the smallest NTK eigenvalue for deep networks with the minimum possible over-parameterization: the number of parameters is roughly $Ω(N)$ and, hence, the number of neurons is as little as $Ω(\sqrt{N})$. To showcase the applicability of our NTK bounds, we provide two results concerning memorization capacity and optimization guarantees for gradient descent training.

Massimo Fornasier, Timo Klock, Marco Mondelli et al.

Artificial neural networks are functions depending on a finite number of parameters typically encoded as weights and biases. The identification of the parameters of the network from finite samples of input-output pairs is often referred to as the \emph{teacher-student model}, and this model has represented a popular framework for understanding training and generalization. Even if the problem is NP-complete in the worst case, a rapidly growing literature -- after adding suitable distributional assumptions -- has established finite sample identification of two-layer networks with a number of neurons $m=\mathcal O(D)$, $D$ being the input dimension. For the range $D<m<D^2$ the problem becomes harder, and truly little is known for networks parametrized by biases as well. This paper fills the gap by providing constructive methods and theoretical guarantees of finite sample identification for such wider shallow networks with biases. Our approach is based on a two-step pipeline: first, we recover the direction of the weights, by exploiting second order information; next, we identify the signs by suitable algebraic evaluations, and we recover the biases by empirical risk minimization via gradient descent. Numerical results demonstrate the effectiveness of our approach.

Jean Barbier, TianQi Hou, Marco Mondelli et al.

We consider the problem of estimating a rank-1 signal corrupted by structured rotationally invariant noise, and address the following question: how well do inference algorithms perform when the noise statistics is unknown and hence Gaussian noise is assumed? While the matched Bayes-optimal setting with unstructured noise is well understood, the analysis of this mismatched problem is only at its premises. In this paper, we make a step towards understanding the effect of the strong source of mismatch which is the noise statistics. Our main technical contribution is the rigorous analysis of a Bayes estimator and of an approximate message passing (AMP) algorithm, both of which incorrectly assume a Gaussian setup. The first result exploits the theory of spherical integrals and of low-rank matrix perturbations; the idea behind the second one is to design and analyze an artificial AMP which, by taking advantage of the flexibility in the denoisers, is able to "correct" the mismatch. Armed with these sharp asymptotic characterizations, we unveil a rich and often unexpected phenomenology. For example, despite AMP is in principle designed to efficiently compute the Bayes estimator, the former is outperformed by the latter in terms of mean-square error. We show that this performance gap is due to an incorrect estimation of the signal norm. In fact, when the SNR is large enough, the overlaps of the AMP and the Bayes estimator coincide, and they even match those of optimal estimators taking into account the structure of the noise.

Simone Bombari, Shayan Kiyani, Marco Mondelli

Machine learning models are vulnerable to adversarial perturbations, and a thought-provoking paper by Bubeck and Sellke has analyzed this phenomenon through the lens of over-parameterization: interpolating smoothly the data requires significantly more parameters than simply memorizing it. However, this "universal" law provides only a necessary condition for robustness, and it is unable to discriminate between models. In this paper, we address these gaps by focusing on empirical risk minimization in two prototypical settings, namely, random features and the neural tangent kernel (NTK). We prove that, for random features, the model is not robust for any degree of over-parameterization, even when the necessary condition coming from the universal law of robustness is satisfied. In contrast, for even activations, the NTK model meets the universal lower bound, and it is robust as soon as the necessary condition on over-parameterization is fulfilled. This also addresses a conjecture in prior work by Bubeck, Li and Nagaraj. Our analysis decouples the effect of the kernel of the model from an "interaction matrix", which describes the interaction with the test data and captures the effect of the activation. Our theoretical results are corroborated by numerical evidence on both synthetic and standard datasets (MNIST, CIFAR-10).

Yihan Zhang, Marco Mondelli, Ramji Venkataramanan

In a mixed generalized linear model, the goal is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two statistically independent signals in a mixed generalized linear model with Gaussian covariates. Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples $n$ needed to guarantee recovery is super-linear in the signal dimension $d$. In this paper, we develop exact asymptotics on spectral methods in the challenging proportional regime in which $n, d$ grow large and their ratio converges to a finite constant. This allows us optimize the design of the spectral method, and combine it with a simple linear estimator, to minimize the estimation error. Our characterization exploits a mix of tools from random matrices, free probability and the theory of approximate message passing algorithms. Numerical simulations for mixed linear regression and phase retrieval demonstrate the advantage enabled by our analysis over existing designs of spectral methods.

Jean Barbier, Francesco Camilli, Marco Mondelli et al.

How do statistical dependencies in measurement noise influence high-dimensional inference? To answer this, we study the paradigmatic spiked matrix model of principal components analysis (PCA), where a rank-one matrix is corrupted by additive noise. We go beyond the usual independence assumption on the noise entries, by drawing the noise from a low-order polynomial orthogonal matrix ensemble. The resulting noise correlations make the setting relevant for applications but analytically challenging. We provide the first characterization of the Bayes-optimal limits of inference in this model. If the spike is rotation-invariant, we show that standard spectral PCA is optimal. However, for more general priors, both PCA and the existing approximate message passing algorithm (AMP) fall short of achieving the information-theoretic limits, which we compute using the replica method from statistical mechanics. We thus propose a novel AMP, inspired by the theory of Adaptive Thouless-Anderson-Palmer equations, which saturates the theoretical limit. This AMP comes with a rigorous state evolution analysis tracking its performance. Although we focus on specific noise distributions, our methodology can be generalized to a wide class of trace matrix ensembles at the cost of more involved expressions. Finally, despite the seemingly strong assumption of rotation-invariant noise, our theory empirically predicts algorithmic performance on real data, pointing at remarkable universality properties.

Alexander Shevchenko, Kevin Kögler, Hamed Hassani et al.

Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.

Yihan Zhang, Hong Chang Ji, Ramji Venkataramanan et al.

We consider the problem of parameter estimation in a high-dimensional generalized linear model. Spectral methods obtained via the principal eigenvector of a suitable data-dependent matrix provide a simple yet surprisingly effective solution. However, despite their wide use, a rigorous performance characterization, as well as a principled way to preprocess the data, are available only for unstructured (i.i.d.\ Gaussian and Haar orthogonal) designs. In contrast, real-world data matrices are highly structured and exhibit non-trivial correlations. To address the problem, we consider correlated Gaussian designs capturing the anisotropic nature of the features via a covariance matrix $Σ$. Our main result is a precise asymptotic characterization of the performance of spectral estimators. This allows us to identify the optimal preprocessing that minimizes the number of samples needed for parameter estimation. Surprisingly, such preprocessing is universal across a broad set of designs, which partly addresses a conjecture on optimal spectral estimators for rotationally invariant models. Our principled approach vastly improves upon previous heuristic methods, including for designs common in computational imaging and genetics. The proposed methodology, based on approximate message passing, is broadly applicable and opens the way to the precise characterization of spiked matrices and of the corresponding spectral methods in a variety of settings.

Yizhou Xu, TianQi Hou, ShanSuo Liang et al.

We consider the problem of reconstructing the signal and the hidden variables from observations coming from a multi-layer network with rotationally invariant weight matrices. The multi-layer structure models inference from deep generative priors, and the rotational invariance imposed on the weights generalizes the i.i.d.\ Gaussian assumption by allowing for a complex correlation structure, which is typical in applications. In this work, we present a new class of approximate message passing (AMP) algorithms and give a state evolution recursion which precisely characterizes their performance in the large system limit. In contrast with the existing multi-layer VAMP (ML-VAMP) approach, our proposed AMP -- dubbed multi-layer rotationally invariant generalized AMP (ML-RI-GAMP) -- provides a natural generalization beyond Gaussian designs, in the sense that it recovers the existing Gaussian AMP as a special case. Furthermore, ML-RI-GAMP exhibits a significantly lower complexity than ML-VAMP, as the computationally intensive singular value decomposition is replaced by an estimation of the moments of the design matrices. Finally, our numerical results show that this complexity gain comes at little to no cost in the performance of the algorithm.

Diyuan Wu, Vyacheslav Kungurtsev, Marco Mondelli

The stochastic heavy ball method (SHB), also known as stochastic gradient descent (SGD) with Polyak's momentum, is widely used in training neural networks. However, despite the remarkable success of such algorithm in practice, its theoretical characterization remains limited. In this paper, we focus on neural networks with two and three layers and provide a rigorous understanding of the properties of the solutions found by SHB: \emph{(i)} stability after dropping out part of the neurons, \emph{(ii)} connectivity along a low-loss path, and \emph{(iii)} convergence to the global optimum. To achieve this goal, we take a mean-field view and relate the SHB dynamics to a certain partial differential equation in the limit of large network widths. This mean-field perspective has inspired a recent line of work focusing on SGD while, in contrast, our paper considers an algorithm with momentum. More specifically, after proving existence and uniqueness of the limit differential equations, we show convergence to the global optimum and give a quantitative bound between the mean-field limit and the SHB dynamics of a finite-width network. Armed with this last bound, we are able to establish the dropout-stability and connectivity of SHB solutions.

Teng Fu, YuHao Liu, Jean Barbier et al.

We study the performance of a Bayesian statistician who estimates a rank-one signal corrupted by non-symmetric rotationally invariant noise with a generic distribution of singular values. As the signal-to-noise ratio and the noise structure are unknown, a Gaussian setup is incorrectly assumed. We derive the exact analytic expression for the error of the mismatched Bayes estimator and also provide the analysis of an approximate message passing (AMP) algorithm. The first result exploits the asymptotic behavior of spherical integrals for rectangular matrices and of low-rank matrix perturbations; the second one relies on the design and analysis of an auxiliary AMP. The numerical experiments show that there is a performance gap between the AMP and Bayes estimators, which is due to the incorrect estimation of the signal norm.

Mohammad Hossein Amani, Simone Bombari, Marco Mondelli et al.

In this paper, we study the compression of a target two-layer neural network with N nodes into a compressed network with M<N nodes. More precisely, we consider the setting in which the weights of the target network are i.i.d. sub-Gaussian, and we minimize the population L_2 loss between the outputs of the target and of the compressed network, under the assumption of Gaussian inputs. By using tools from high-dimensional probability, we show that this non-convex problem can be simplified when the target network is sufficiently over-parameterized, and provide the error rate of this approximation as a function of the input dimension and N. In this mean-field limit, the simplified objective, as well as the optimal weights of the compressed network, does not depend on the realization of the target network, but only on expected scaling factors. Furthermore, for networks with ReLU activation, we conjecture that the optimum of the simplified optimization problem is achieved by taking weights on the Equiangular Tight Frame (ETF), while the scaling of the weights and the orientation of the ETF depend on the parameters of the target network. Numerical evidence is provided to support this conjecture.

Valentina Njaradi, Clémentine Dominé, Rachel Swanson et al.

Learning to generalise from limited data is a fundamental challenge for both artificial and biological systems. A common strategy is to extract reusable structure from abundant unlabelled data, enabling efficient adaptation to new tasks from limited labelled data. This two-stage paradigm is now standard in modern training pipelines, where pretraining is followed by fine-tuning or linear probing. We provide an analytical model of this process: structure extraction is formalized as principal component analysis on unlabelled data, and downstream learning as linear regression on a separate labelled dataset. In the high-dimensional regime, we derive exact expressions for training and generalisation error showcasing their dependence on representation dimensionality, unlabelled and labelled sample sizes, and task alignment. Our results show that pretrained representations strongly influence downstream generalisation, and we characterize the optimal representation size as a function of task parameters: with abundant pretraining data but scarce downstream data, maximally compressed representations are optimal, whereas with limited pretraining data, higher-dimensional representations generalise better. Furthermore, we establish an exact trade-off between pretraining and supervision, quantifying how much unlabelled data is required to replace a single labelled sample. Beyond our idealised model, we observe similar phenomenology in autoencoders and pretrained LLMs. Altogether, we highlight that optimising representation size is critical, giving conditions for when compression during pretraining improves generalisation.

Nicolas Anguita, Francesco Locatello, Andrew M. Saxe et al.

Pretraining and fine-tuning are central stages in modern machine learning systems. In practice, feature learning plays an important role across both stages: deep neural networks learn a broad range of useful features during pretraining and further refine those features during fine-tuning. However, an end-to-end theoretical understanding of how choices of initialization impact the ability to reuse and refine features during fine-tuning has remained elusive. Here we develop an analytical theory of the pretraining-fine-tuning pipeline in diagonal linear networks, deriving exact expressions for the generalization error as a function of initialization parameters and task statistics. We find that different initialization choices place the network into four distinct fine-tuning regimes that are distinguished by their ability to support feature learning and reuse, and therefore by the task statistics for which they are beneficial. In particular, a smaller initialization scale in earlier layers enables the network to both reuse and refine its features, leading to superior generalization on fine-tuning tasks that rely on a subset of pretraining features. We demonstrate empirically that the same initialization parameters impact generalization in nonlinear networks trained on CIFAR-100. Overall, our results demonstrate analytically how data and network initialization interact to shape fine-tuning generalization, highlighting an important role for the relative scale of initialization across different layers in enabling continued feature learning during fine-tuning.

Filip Kovačević, Hong Chang Ji, Denny Wu et al.

It is folklore that reusing training data more than once can improve the statistical efficiency of gradient-based learning. However, beyond linear regression, the theoretical advantage of full-batch gradient descent (GD, which always reuses all the data) over one-pass stochastic gradient descent (online SGD, which uses each data point only once) remains unclear. In this work, we consider learning a $d$-dimensional single-index model with a quadratic activation, for which it is known that one-pass SGD requires $n\gtrsim d\log d$ samples to achieve weak recovery. We first show that this $\log d$ factor in the sample complexity persists for full-batch spherical GD on the correlation loss; however, by simply truncating the activation, full-batch GD exhibits a favorable optimization landscape at $n \simeq d$ samples, thereby outperforming one-pass SGD (with the same activation) in statistical efficiency. We complement this result with a trajectory analysis of full-batch GD on the squared loss from small initialization, showing that $n \gtrsim d$ samples and $T \gtrsim\log d$ gradient steps suffice to achieve strong (exact) recovery.

Peter Súkeník, Cristina López Amado, Christoph H. Lampert et al.

This paper studies the role of sinks and diagonal patterns as attention switch and anti-oversmoothing mechanisms. We analyze geometric conditions under which sinks can be represented, showing a necessary alignment between the embedding of the sink and all other embeddings. Next, we refine the current understanding of the role of sinks in oversmoothing prevention: we specify the conditions under which dense attention provably smooths more than sparse attention, and empirically verify that such conditions are often satisfied in practice. We further prove an equivalence between sinks and hard attention switch, in which the output of the attention is identically 0. Finally, we relax the hard attention switch by allowing token self-communication: we provide a quantitative comparison of the costs of representing sinks vs.\ diagonal patterns, showing why sinks are favored in pretrained transformers. The introduction and analysis of diagonal patterns and the generalization of the attention switch close the gap between what oversmoothing prevention requires and what sinks provide, while also establishing when and why attention layers act like MLPs if token communication is not necessary.

Daniel Beaglehole, Peter Súkeník, Marco Mondelli et al.

Deep Neural Collapse (DNC) refers to the surprisingly rigid structure of the data representations in the final layers of Deep Neural Networks (DNNs). Though the phenomenon has been measured in a variety of settings, its emergence is typically explained via data-agnostic approaches, such as the unconstrained features model. In this work, we introduce a data-dependent setting where DNC forms due to feature learning through the average gradient outer product (AGOP). The AGOP is defined with respect to a learned predictor and is equal to the uncentered covariance matrix of its input-output gradients averaged over the training dataset. The Deep Recursive Feature Machine (Deep RFM) is a method that constructs a neural network by iteratively mapping the data with the AGOP and applying an untrained random feature map. We demonstrate empirically that DNC occurs in Deep RFM across standard settings as a consequence of the projection with the AGOP matrix computed at each layer. Further, we theoretically explain DNC in Deep RFM in an asymptotic setting and as a result of kernel learning. We then provide evidence that this mechanism holds for neural networks more generally. In particular, we show that the right singular vectors and values of the weights can be responsible for the majority of within-class variability collapse for DNNs trained in the feature learning regime. As observed in recent work, this singular structure is highly correlated with that of the AGOP.

M. Emrullah Ildiz, Halil Alperen Gozeten, Ege Onur Taga et al.

A growing number of machine learning scenarios rely on knowledge distillation where one uses the output of a surrogate model as labels to supervise the training of a target model. In this work, we provide a sharp characterization of this process for ridgeless, high-dimensional regression, under two settings: (i) model shift, where the surrogate model is arbitrary, and (ii) distribution shift, where the surrogate model is the solution of empirical risk minimization with out-of-distribution data. In both cases, we characterize the precise risk of the target model through non-asymptotic bounds in terms of sample size and data distribution under mild conditions. As a consequence, we identify the form of the optimal surrogate model, which reveals the benefits and limitations of discarding weak features in a data-dependent fashion. In the context of weak-to-strong (W2S) generalization, this has the interpretation that (i) W2S training, with the surrogate as the weak model, can provably outperform training with strong labels under the same data budget, but (ii) it is unable to improve the data scaling law. We validate our results on numerical experiments both on ridgeless regression and on neural network architectures.

Peter Súkeník, Marco Mondelli, Christoph Lampert

Deep neural networks (DNNs) exhibit a surprising structure in their final layer known as neural collapse (NC), and a growing body of works has currently investigated the propagation of neural collapse to earlier layers of DNNs -- a phenomenon called deep neural collapse (DNC). However, existing theoretical results are restricted to special cases: linear models, only two layers or binary classification. In contrast, we focus on non-linear models of arbitrary depth in multi-class classification and reveal a surprising qualitative shift. As soon as we go beyond two layers or two classes, DNC stops being optimal for the deep unconstrained features model (DUFM) -- the standard theoretical framework for the analysis of collapse. The main culprit is a low-rank bias of multi-layer regularization schemes: this bias leads to optimal solutions of even lower rank than the neural collapse. We support our theoretical findings with experiments on both DUFM and real data, which show the emergence of the low-rank structure in the solution found by gradient descent.

Simone Bombari, Marco Mondelli

Understanding the reasons behind the exceptional success of transformers requires a better analysis of why attention layers are suitable for NLP tasks. In particular, such tasks require predictive models to capture contextual meaning which often depends on one or few words, even if the sentence is long. Our work studies this key property, dubbed word sensitivity (WS), in the prototypical setting of random features. We show that attention layers enjoy high WS, namely, there exists a vector in the space of embeddings that largely perturbs the random attention features map. The argument critically exploits the role of the softmax in the attention layer, highlighting its benefit compared to other activations (e.g., ReLU). In contrast, the WS of standard random features is of order $1/\sqrt{n}$, $n$ being the number of words in the textual sample, and thus it decays with the length of the context. We then translate these results on the word sensitivity into generalization bounds: due to their low WS, random features provably cannot learn to distinguish between two sentences that differ only in a single word; in contrast, due to their high WS, random attention features have higher generalization capabilities. We validate our theoretical results with experimental evidence over the BERT-Base word embeddings of the imdb review dataset.

Halil Alperen Gozeten, M. Emrullah Ildiz, Xuechen Zhang et al.

Test-time training (TTT) methods explicitly update the weights of a model to adapt to the specific test instance, and they have found success in a variety of settings, including most recently language modeling and reasoning. To demystify this success, we investigate a gradient-based TTT algorithm for in-context learning, where we train a transformer model on the in-context demonstrations provided in the test prompt. Specifically, we provide a comprehensive theoretical characterization of linear transformers when the update rule is a single gradient step. Our theory (i) delineates the role of alignment between pretraining distribution and target task, (ii) demystifies how TTT can alleviate distribution shift, and (iii) quantifies the sample complexity of TTT including how it can significantly reduce the eventual sample size required for in-context learning. As our empirical contribution, we study the benefits of TTT for TabPFN, a tabular foundation model. In line with our theory, we demonstrate that TTT significantly reduces the required sample size for tabular classification (3 to 5 times fewer) unlocking substantial inference efficiency with a negligible training cost.

Simone Bombari, Marco Mondelli

Learning models have been shown to rely on spurious correlations between non-predictive features and the associated labels in the training data, with negative implications on robustness, bias and fairness. In this work, we provide a statistical characterization of this phenomenon for high-dimensional regression, when the data contains a predictive core feature $x$ and a spurious feature $y$. Specifically, we quantify the amount of spurious correlations $C$ learned via linear regression, in terms of the data covariance and the strength $λ$ of the ridge regularization. As a consequence, we first capture the simplicity of $y$ through the spectrum of its covariance, and its correlation with $x$ through the Schur complement of the full data covariance. Next, we prove a trade-off between $C$ and the in-distribution test loss $L$, by showing that the value of $λ$ that minimizes $L$ lies in an interval where $C$ is increasing. Finally, we investigate the effects of over-parameterization via the random features model, by showing its equivalence to regularized linear regression. Our theoretical results are supported by numerical experiments on Gaussian, Color-MNIST, and CIFAR-10 datasets.

Yihan Zhang, Marco Mondelli

We study the matrix denoising problem of estimating the singular vectors of a rank-$1$ signal corrupted by noise with both column and row correlations. Existing works are either unable to pinpoint the exact asymptotic estimation error or, when they do so, the resulting approaches (e.g., based on whitening or singular value shrinkage) remain vastly suboptimal. On top of this, most of the literature has focused on the special case of estimating the left singular vector of the signal when the noise only possesses row correlation (one-sided heteroscedasticity). In contrast, our work establishes the information-theoretic and algorithmic limits of matrix denoising with doubly heteroscedastic noise. We characterize the exact asymptotic minimum mean square error, and design a novel spectral estimator with rigorous optimality guarantees: under a technical condition, it attains positive correlation with the signals whenever information-theoretically possible and, for one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical experiments demonstrate the significant advantage of our theoretically principled method with the state of the art. The proofs draw connections with statistical physics and approximate message passing, departing drastically from standard random matrix theory techniques.

Filip Kovačević, Yihan Zhang, Marco Mondelli

Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections with neural networks, they have been object of recent intensive study. In this paper, we focus on recovering the subspace spanned by the signals via spectral estimators -- a family of methods routinely used in practice, often as a warm-start for iterative algorithms. Our main technical contribution is a precise asymptotic characterization of the performance of spectral methods, when sample size and input dimension grow proportionally and the dimension $p$ of the space to recover is fixed. Specifically, we locate the top-$p$ eigenvalues of the spectral matrix and establish the overlaps between the corresponding eigenvectors (which give the spectral estimators) and a basis of the signal subspace. Our analysis unveils a phase transition phenomenon in which, as the sample complexity grows, eigenvalues escape from the bulk of the spectrum and, when that happens, eigenvectors recover directions of the desired subspace. The precise characterization we put forward enables the optimization of the data preprocessing, thus allowing to identify the spectral estimator that requires the minimal sample size for weak recovery.

Simone Bombari, Marco Mondelli

Differentially private gradient descent (DP-GD) is a popular algorithm to train deep learning models with provable guarantees on the privacy of the training data. In the last decade, the problem of understanding its performance cost with respect to standard GD has received remarkable attention from the research community, which formally derived upper bounds on the excess population risk $R_{P}$ in different learning settings. However, existing bounds typically degrade with over-parameterization, i.e., as the number of parameters $p$ gets larger than the number of training samples $n$ -- a regime which is ubiquitous in current deep-learning practice. As a result, the lack of theoretical insights leaves practitioners without clear guidance, leading some to reduce the effective number of trainable parameters to improve performance, while others use larger models to achieve better results through scale. In this work, we show that in the popular random features model with quadratic loss, for any sufficiently large $p$, privacy can be obtained for free, i.e., $\left|R_{P} \right| = o(1)$, not only when the privacy parameter $\varepsilon$ has constant order, but also in the strongly private setting $\varepsilon = o(1)$. This challenges the common wisdom that over-parameterization inherently hinders performance in private learning.

Peter Súkeník, Christoph H. Lampert, Marco Mondelli

The empirical emergence of neural collapse -- a surprising symmetry in the feature representations of the training data in the penultimate layer of deep neural networks -- has spurred a line of theoretical research aimed at its understanding. However, existing work focuses on data-agnostic models or, when data structure is taken into account, it remains limited to multi-layer perceptrons. Our paper fills both these gaps by analyzing modern architectures in a data-aware regime: we prove that global optima of deep regularized transformers and residual networks (ResNets) with LayerNorm trained with cross entropy or mean squared error loss are approximately collapsed, and the approximation gets tighter as the depth grows. More generally, we formally reduce any end-to-end large-depth ResNet or transformer training into an equivalent unconstrained features model, thus justifying its wide use in the literature even beyond data-agnostic settings. Our theoretical results are supported by experiments on computer vision and language datasets showing that, as the depth grows, neural collapse indeed becomes more prominent.

Simone Bombari, Inbar Seroussi, Marco Mondelli

Differentially private (DP) linear regression has received significant attention in the recent theoretical literature, with several works aimed at obtaining improved error rates. A common approach is to set the clipping constant much larger than the expected norm of the per-sample gradients. While simplifying the analysis, this is however in sharp contrast with what empirical evidence suggests to optimize performance. Our work bridges this gap between theory and practice: we provide sharper rates for DP stochastic gradient descent (DP-SGD) by crucially operating in a regime where clipping happens frequently. Specifically, we consider the setting where the data is multivariate Gaussian, the number of training samples $n$ is proportional to the input dimension $d$, and the algorithm guarantees constant-order zero concentrated DP. Our method relies on establishing a deterministic equivalent for the trajectory of DP-SGD in terms of a family of ordinary differential equations (ODEs). As a consequence, the risk of DP-SGD is bounded between two ODEs, with upper and lower bounds matching for isotropic data. By studying these ODEs when $n / d$ is large enough, we demonstrate the optimality of aggressive clipping, and we uncover the benefits of decaying learning rate and private noise scheduling.

Kevin Kögler, Alexander Shevchenko, Hamed Hassani et al.

Autoencoders are a prominent model in many empirical branches of machine learning and lossy data compression. However, basic theoretical questions remain unanswered even in a shallow two-layer setting. In particular, to what degree does a shallow autoencoder capture the structure of the underlying data distribution? For the prototypical case of the 1-bit compression of sparse Gaussian data, we prove that gradient descent converges to a solution that completely disregards the sparse structure of the input. Namely, the performance of the algorithm is the same as if it was compressing a Gaussian source - with no sparsity. For general data distributions, we give evidence of a phase transition phenomenon in the shape of the gradient descent minimizer, as a function of the data sparsity: below the critical sparsity level, the minimizer is a rotation taken uniformly at random (just like in the compression of non-sparse data); above the critical sparsity, the minimizer is the identity (up to a permutation). Finally, by exploiting a connection with approximate message passing algorithms, we show how to improve upon Gaussian performance for the compression of sparse data: adding a denoising function to a shallow architecture already reduces the loss provably, and a suitable multi-layer decoder leads to a further improvement. We validate our findings on image datasets, such as CIFAR-10 and MNIST.

Diyuan Wu, Marco Mondelli

Neural Collapse is a phenomenon where the last-layer representations of a well-trained neural network converge to a highly structured geometry. In this paper, we focus on its first (and most basic) property, known as NC1: the within-class variability vanishes. While prior theoretical studies establish the occurrence of NC1 via the data-agnostic unconstrained features model, our work adopts a data-specific perspective, analyzing NC1 in a three-layer neural network, with the first two layers operating in the mean-field regime and followed by a linear layer. In particular, we establish a fundamental connection between NC1 and the loss landscape: we prove that points with small empirical loss and gradient norm (thus, close to being stationary) approximately satisfy NC1, and the closeness to NC1 is controlled by the residual loss and gradient norm. We then show that (i) gradient flow on the mean squared error converges to NC1 solutions with small empirical loss, and (ii) for well-separated data distributions, both NC1 and vanishing test loss are achieved simultaneously. This aligns with the empirical observation that NC1 emerges during training while models attain near-zero test error. Overall, our results demonstrate that NC1 arises from gradient training due to the properties of the loss landscape, and they show the co-occurrence of NC1 and small test error for certain data distributions.

Yihan Zhang, Hong Chang Ji, Ramji Venkataramanan et al.

We consider the problem of parameter estimation from a generalized linear model with a random design matrix that is orthogonally invariant in law. Such a model allows the design have an arbitrary distribution of singular values and only assumes that its singular vectors are generic. It is a vast generalization of the i.i.d. Gaussian design typically considered in the theoretical literature, and is motivated by the fact that real data often have a complex correlation structure so that methods relying on i.i.d. assumptions can be highly suboptimal. Building on the paradigm of spectrally-initialized iterative optimization, this paper proposes optimal spectral estimators and combines them with an approximate message passing (AMP) algorithm, establishing rigorous performance guarantees for these two algorithmic steps. Both the spectral initialization and the subsequent AMP meet existing conjectures on the fundamental limits to estimation -- the former on the optimal sample complexity for efficient weak recovery, and the latter on the optimal errors. Numerical experiments suggest the effectiveness of our methods and accuracy of our theory beyond orthogonally invariant data.

Edwige Cyffers, Alireza Mirrokni, Marco Mondelli

In performative learning, the data distribution reacts to the deployed model - for example, because strategic users adapt their features to game it - which creates a more complex dynamic than in classical supervised learning. One should thus not only optimize the model for the current data but also take into account that the model might steer the distribution in a new direction, without knowing the exact nature of the potential shift. We explore how regularization can help cope with performative effects by studying its impact in high-dimensional ridge regression. We show that, while performative effects worsen the test risk in the population setting, they can be beneficial in the over-parameterized regime where the number of features exceeds the number of samples. We show that the optimal regularization scales with the overall strength of the performative effect, making it possible to set the regularization in anticipation of this effect. We illustrate this finding through empirical evaluations of the optimal regularization parameter on both synthetic and real-world datasets.

Parham Rezaei, Filip Kovacevic, Francesco Locatello et al.

Despite the progress in the development of generative models, their usefulness in creating synthetic data that improve prediction performance of classifiers has been put into question. Besides heuristic principles such as "synthetic data should be close to the real data distribution", it is actually not clear which specific properties affect the generalization error. Our paper addresses this question through the lens of high-dimensional regression. Theoretically, we show that, for linear models, the covariance shift between the target distribution and the distribution of the synthetic data affects the generalization error but, surprisingly, the mean shift does not. Furthermore we prove that, in some settings, matching the covariance of the target distribution is optimal. Remarkably, the theoretical insights from linear models carry over to deep neural networks and generative models. We empirically demonstrate that the covariance matching procedure (matching the covariance of the synthetic data with that of the data coming from the target distribution) performs well against several recent approaches for synthetic data selection, across training paradigms, architectures, datasets and generative models used for augmentation.

Leonardo Iurada, Simone Bombari, Tatiana Tommasi et al.

Large-scale deep learning models are known to memorize parts of the training set. In machine learning theory, memorization is often framed as interpolation or label fitting, and classical results show that this can be achieved when the number of parameters $p$ in the model is larger than the number of training samples $n$. In this work, we consider memorization from the perspective of data reconstruction, demonstrating that this can be achieved when $p$ is larger than $dn$, where $d$ is the dimensionality of the data. More specifically, we show that, in the random features model, when $p \gg dn$, the subspace spanned by the training samples in feature space gives sufficient information to identify the individual samples in input space. Our analysis suggests an optimization method to reconstruct the dataset from the model parameters, and we demonstrate that this method performs well on various architectures (random features, two-layer fully-connected and deep residual networks). Our results reveal a law of data reconstruction, according to which the entire training dataset can be recovered as $p$ exceeds the threshold $dn$.

Diyuan Wu, Aleksandr Shevchenko, Samet Oymak et al.

Token embeddings play a crucial role in language modeling but, despite this practical relevance, their theoretical understanding remains limited. Our paper addresses the gap by characterizing the structure of embeddings obtained via gradient descent. Specifically, we consider a one-layer softmax attention model with a linear head for binary classification, i.e., $\texttt{Softmax}( p^\top E_X^\top ) E_X v = \frac{ \sum_{i=1}^T \exp(p^\top E_{x_i}) E_{x_i}^\top v}{\sum_{j=1}^T \exp(p^\top E_{x_{j}}) }$, where $E_X = [ E_{x_1} , \dots, E_{x_T} ]^\top$ contains the embeddings of the input sequence, $p$ is the embedding of the $\mathrm{\langle cls \rangle}$ token and $v$ the output vector. First, we show that, already after a single step of gradient training with the logistic loss, the embeddings $E_X$ capture the importance of tokens in the dataset by aligning with the output vector $v$ proportionally to the frequency with which the corresponding tokens appear in the dataset. Then, after training $p$ via gradient flow until convergence, the softmax selects the important tokens in the sentence (i.e., those that are predictive of the label), and the resulting $\mathrm{\langle cls \rangle}$ embedding maximizes the margin for such a selection. Experiments on real-world datasets (IMDB, Yelp) exhibit a phenomenology close to that unveiled by our theory.

Francesco Pedrotti, Jan Maas, Marco Mondelli

Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time $T_1$ by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires $T_1\to\infty$. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as $T_1$ diverges; from a practical viewpoint, a large $T_1$ increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time $T_1$. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the $L^2$ loss on the score approximation, which is the quantity minimized in practice.

Peter Súkeník, Marco Mondelli, Christoph Lampert

Neural collapse (NC) refers to the surprising structure of the last layer of deep neural networks in the terminal phase of gradient descent training. Recently, an increasing amount of experimental evidence has pointed to the propagation of NC to earlier layers of neural networks. However, while the NC in the last layer is well studied theoretically, much less is known about its multi-layered counterpart - deep neural collapse (DNC). In particular, existing work focuses either on linear layers or only on the last two layers at the price of an extra assumption. Our paper fills this gap by generalizing the established analytical framework for NC - the unconstrained features model - to multiple non-linear layers. Our key technical contribution is to show that, in a deep unconstrained features model, the unique global optimum for binary classification exhibits all the properties typical of DNC. This explains the existing experimental evidence of DNC. We also empirically show that (i) by optimizing deep unconstrained features models via gradient descent, the resulting solution agrees well with our theory, and (ii) trained networks recover the unconstrained features suitable for the occurrence of DNC, thus supporting the validity of this modeling principle.

Simone Bombari, Marco Mondelli

Deep learning models are known to overfit and memorize spurious features in the training dataset. While numerous empirical studies have aimed at understanding this phenomenon, a rigorous theoretical framework to quantify it is still missing. In this paper, we consider spurious features that are uncorrelated with the learning task, and we provide a precise characterization of how they are memorized via two separate terms: (i) the stability of the model with respect to individual training samples, and (ii) the feature alignment between the spurious feature and the full sample. While the first term is well established in learning theory and it is connected to the generalization error in classical work, the second one is, to the best of our knowledge, novel. Our key technical result gives a precise characterization of the feature alignment for the two prototypical settings of random features (RF) and neural tangent kernel (NTK) regression. We prove that the memorization of spurious features weakens as the generalization capability increases and, through the analysis of the feature alignment, we unveil the role of the model and of its activation function. Numerical experiments show the predictive power of our theory on standard datasets (MNIST, CIFAR-10).

Ramji Venkataramanan, Kevin Kögler, Marco Mondelli

We consider the problem of signal estimation in generalized linear models defined via rotationally invariant design matrices. Since these matrices can have an arbitrary spectral distribution, this model is well suited for capturing complex correlation structures which often arise in applications. We propose a novel family of approximate message passing (AMP) algorithms for signal estimation, and rigorously characterize their performance in the high-dimensional limit via a state evolution recursion. Our rotationally invariant AMP has complexity of the same order as the existing AMP derived under the restrictive assumption of a Gaussian design; our algorithm also recovers this existing AMP as a special case. Numerical results showcase a performance close to Vector AMP (which is conjectured to be Bayes-optimal in some settings), but obtained with a much lower complexity, as the proposed algorithm does not require a computationally expensive singular value decomposition.

Alexander Shevchenko, Vyacheslav Kungurtsev, Marco Mondelli

Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.

Marco Mondelli, Ramji Venkataramanan

We study the problem of estimating a rank-$1$ signal in the presence of rotationally invariant noise-a class of perturbations more general than Gaussian noise. Principal Component Analysis (PCA) provides a natural estimator, and sharp results on its performance have been obtained in the high-dimensional regime. Recently, an Approximate Message Passing (AMP) algorithm has been proposed as an alternative estimator with the potential to improve the accuracy of PCA. However, the existing analysis of AMP requires an initialization that is both correlated with the signal and independent of the noise, which is often unrealistic in practice. In this work, we combine the two methods, and propose to initialize AMP with PCA. Our main result is a rigorous asymptotic characterization of the performance of this estimator. Both the AMP algorithm and its analysis differ from those previously derived in the Gaussian setting: at every iteration, our AMP algorithm requires a specific term to account for PCA initialization, while in the Gaussian case, PCA initialization affects only the first iteration of AMP. The proof is based on a two-phase artificial AMP that first approximates the PCA estimator and then mimics the true AMP. Our numerical simulations show an excellent agreement between AMP results and theoretical predictions, and suggest an interesting open direction on achieving Bayes-optimal performance.

Quynh Nguyen, Pierre Brechet, Marco Mondelli

The question of how and why the phenomenon of mode connectivity occurs in training deep neural networks has gained remarkable attention in the research community. From a theoretical perspective, two possible explanations have been proposed: (i) the loss function has connected sublevel sets, and (ii) the solutions found by stochastic gradient descent are dropout stable. While these explanations provide insights into the phenomenon, their assumptions are not always satisfied in practice. In particular, the first approach requires the network to have one layer with order of $N$ neurons ($N$ being the number of training samples), while the second one requires the loss to be almost invariant after removing half of the neurons at each layer (up to some rescaling of the remaining ones). In this work, we improve both conditions by exploiting the quality of the features at every intermediate layer together with a milder over-parameterization condition. More specifically, we show that: (i) under generic assumptions on the features of intermediate layers, it suffices that the last two hidden layers have order of $\sqrt{N}$ neurons, and (ii) if subsets of features at each layer are linearly separable, then no over-parameterization is needed to show the connectivity. Our experiments confirm that the proposed condition ensures the connectivity of solutions found by stochastic gradient descent, even in settings where the previous requirements do not hold.

Quynh Nguyen, Marco Mondelli, Guido Montufar

A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths. In the finite-width setting, the network architectures we consider are fairly general: we require the existence of a wide layer with roughly order of $N$ neurons, $N$ being the number of data samples; and the scaling of the remaining layer widths is arbitrary (up to logarithmic factors). To obtain our results, we analyze various quantities of independent interest: we give lower bounds on the smallest singular value of hidden feature matrices, and upper bounds on the Lipschitz constant of input-output feature maps.

Marco Mondelli, Ramji Venkataramanan

We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach.

Marco Mondelli, Christos Thrampoulidis, Ramji Venkataramanan

We study the problem of recovering an unknown signal $\boldsymbol x$ given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator $\hat{\boldsymbol x}^{\rm L}$ and a spectral estimator $\hat{\boldsymbol x}^{\rm s}$. The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine $\hat{\boldsymbol x}^{\rm L}$ and $\hat{\boldsymbol x}^{\rm s}$. At the heart of our analysis is the exact characterization of the joint empirical distribution of $(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$ in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of $\hat{\boldsymbol x}^{\rm L}$ and $\hat{\boldsymbol x}^{\rm s}$, given the limiting distribution of the signal $\boldsymbol x$. When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form $θ\hat{\boldsymbol x}^{\rm L}+\hat{\boldsymbol x}^{\rm s}$ and we derive the optimal combination coefficient. In order to establish the limiting distribution of $(\boldsymbol x, \hat{\boldsymbol x}^{\rm L}, \hat{\boldsymbol x}^{\rm s})$, we design and analyze an Approximate Message Passing (AMP) algorithm whose iterates give $\hat{\boldsymbol x}^{\rm L}$ and approach $\hat{\boldsymbol x}^{\rm s}$. Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.

Quynh Nguyen, Marco Mondelli

Recent works have shown that gradient descent can find a global minimum for over-parameterized neural networks where the widths of all the hidden layers scale polynomially with $N$ ($N$ being the number of training samples). In this paper, we prove that, for deep networks, a single layer of width $N$ following the input layer suffices to ensure a similar guarantee. In particular, all the remaining layers are allowed to have constant widths, and form a pyramidal topology. We show an application of our result to the widely used LeCun's initialization and obtain an over-parameterization requirement for the single wide layer of order $N^2.$

Alexander Shevchenko, Marco Mondelli

The optimization of multilayer neural networks typically leads to a solution with zero training error, yet the landscape can exhibit spurious local minima and the minima can be disconnected. In this paper, we shed light on this phenomenon: we show that the combination of stochastic gradient descent (SGD) and over-parameterization makes the landscape of multilayer neural networks approximately connected and thus more favorable to optimization. More specifically, we prove that SGD solutions are connected via a piecewise linear path, and the increase in loss along this path vanishes as the number of neurons grows large. This result is a consequence of the fact that the parameters found by SGD are increasingly dropout stable as the network becomes wider. We show that, if we remove part of the neurons (and suitably rescale the remaining ones), the change in loss is independent of the total number of neurons, and it depends only on how many neurons are left. Our results exhibit a mild dependence on the input dimension: they are dimension-free for two-layer networks and depend linearly on the dimension for multilayer networks. We validate our theoretical findings with numerical experiments for different architectures and classification tasks.

Adel Javanmard, Marco Mondelli, Andrea Montanari

Fitting a function by using linear combinations of a large number $N$ of `simple' components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is non-convex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches. Here we consider the problem of learning a concave function $f$ on a compact convex domain $Ω\subseteq {\mathbb R}^d$, using linear combinations of `bump-like' components (neurons). The parameters to be fitted are the centers of $N$ bumps, and the resulting empirical risk minimization problem is highly non-convex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over $Ω$. Further, when the bump width $δ$ tends to $0$, this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for $N\to\infty$, $δ\to 0$. Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of $δ, N$. Explaining this phenomenon, and understanding the dependence on $δ,N$ in a quantitative manner remains an outstanding challenge.

Marco Mondelli, Andrea Montanari

We establish connections between the problem of learning a two-layer neural network and tensor decomposition. We consider a model with feature vectors $\boldsymbol x \in \mathbb R^d$, $r$ hidden units with weights $\{\boldsymbol w_i\}_{1\le i \le r}$ and output $y\in \mathbb R$, i.e., $y=\sum_{i=1}^r σ( \boldsymbol w_i^{\mathsf T}\boldsymbol x)$, with activation functions given by low-degree polynomials. In particular, if $σ(x) = a_0+a_1x+a_3x^3$, we prove that no polynomial-time learning algorithm can outperform the trivial predictor that assigns to each example the response variable $\mathbb E(y)$, when $d^{3/2}\ll r\ll d^2$. Our conclusion holds for a `natural data distribution', namely standard Gaussian feature vectors $\boldsymbol x$, and output distributed according to a two-layer neural network with random isotropic weights, and under a certain complexity-theoretic assumption on tensor decomposition. Roughly speaking, we assume that no polynomial-time algorithm can substantially outperform current methods for tensor decomposition based on the sum-of-squares hierarchy. We also prove generalizations of this statement for higher degree polynomial activations, and non-random weight vectors. Remarkably, several existing algorithms for learning two-layer networks with rigorous guarantees are based on tensor decomposition. Our results support the idea that this is indeed the core computational difficulty in learning such networks, under the stated generative model for the data. As a side result, we show that under this model learning the network requires accurate learning of its weights, a property that does not hold in a more general setting.

Marco Mondelli, Andrea Montanari

In phase retrieval we want to recover an unknown signal $\boldsymbol x\in\mathbb C^d$ from $n$ quadratic measurements of the form $y_i = |\langle{\boldsymbol a}_i,{\boldsymbol x}\rangle|^2+w_i$ where $\boldsymbol a_i\in \mathbb C^d$ are known sensing vectors and $w_i$ is measurement noise. We ask the following weak recovery question: what is the minimum number of measurements $n$ needed to produce an estimator $\hat{\boldsymbol x}(\boldsymbol y)$ that is positively correlated with the signal $\boldsymbol x$? We consider the case of Gaussian vectors $\boldsymbol a_i$. We prove that - in the high-dimensional limit - a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For $n\le d-o(d)$ no estimator can do significantly better than random and achieve a strictly positive correlation. For $n\ge d+o(d)$ a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theory arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper and lower bound generalize beyond phase retrieval to measurements $y_i$ produced according to a generalized linear model. As a byproduct of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.