arXiv subjects

Yossi Arjevani

Publications and source records attributed to Yossi Arjevani.

At least 19 recordsLinked to original sources

arXiv2026

Hidden Minima in Two-Layer ReLU Networks

We consider the optimization problem associated with training two-layer ReLU networks with $d$ inputs under the squared loss, where the labels are generated by a target network. Recent work has identified two distinct classes of infinite families of minima: one whose training loss vanishes in the high-dimensional limit, and another whose loss remains bounded away from zero. The latter family is empirically avoided by stochastic gradient descent, hence \emph{hidden}, motivating the search for analytic criteria that distinguish hidden from non-hidden minima. A key challenge is that prior analyses have shown the Hessian spectra at hidden and non-hidden minima to coincide up to terms of order $O(d^{-1/2})$, seemingly limiting the discriminative power of spectral methods. We therefore take a different route, studying instead certain curves along which the loss is locally minimized. Our main result shows that arcs emanating from hidden minima exhibit distinctive structural and symmetry properties, arising precisely from $Ω(d^{-1/2})$ eigenvalue contributions that are absent from earlier analyses.

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arXiv2025

Geometry and Optimization of Shallow Polynomial Networks

We study shallow neural networks with monomial activations and output dimension one. The function space for these models can be identified with a set of symmetric tensors with bounded rank. We describe general features of these networks, focusing on the relationship between width and optimization. We then consider teacher-student problems, which can be viewed as problems of low-rank tensor approximation with respect to non-standard inner products that are induced by the data distribution. In this setting, we introduce a teacher-metric data discriminant which encodes the qualitative behavior of the optimization as a function of the training data distribution. Finally, we focus on networks with quadratic activations, presenting an in-depth analysis of the optimization landscape. In particular, we present a variation of the Eckart-Young Theorem characterizing all critical points and their Hessian signatures for teacher-student problems with quadratic networks and Gaussian training data.

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arXiv2025

Symmetry & Critical Points for Symmetric Tensor Decomposition Problems

We consider the nonconvex optimization problem associated with the decomposition of a real symmetric tensor into a sum of rank-one terms. Use is made of the rich symmetry structure to construct infinite families of critical points represented by Puiseux series in the problem dimension, and so obtain precise analytic estimates on the objective function value and the Hessian spectrum. The results enable an analytic characterization of various obstructions to local optimization methods, revealing, in particular, a complex array of saddles and minima that differ in their symmetry, structure, and analytic properties. A notable phenomenon, observed for all critical points considered, concerns the index of the Hessian increasing with the objective function value.

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arXiv2024

Symmetry & Critical Points

Critical points of an invariant function may or may not be symmetric. We prove, however, that if a symmetric critical point exists, those adjacent to it are generically symmetry breaking. This mathematical mechanism is shown to carry important implications for our ability to efficiently minimize invariant nonconvex functions, in particular those associated with neural networks.

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arXiv2023

On the Principle of Least Symmetry Breaking in Shallow ReLU Models

We consider the optimization problem associated with fitting two-layer ReLU networks with respect to the squared loss, where labels are assumed to be generated by a target network. Focusing first on standard Gaussian inputs, we show that the structure of spurious local minima detected by stochastic gradient descent (SGD) is, in a well-defined sense, the \emph{least loss of symmetry} with respect to the target weights. A closer look at the analysis indicates that this principle of least symmetry breaking may apply to a broader range of settings. Motivated by this, we conduct a series of experiments which corroborate this hypothesis for different classes of non-isotropic non-product distributions, smooth activation functions and networks with a few layers.

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arXiv2023

Symmetry Breaking in Symmetric Tensor Decomposition

In this note, we consider the highly nonconvex optimization problem associated with computing the rank decomposition of symmetric tensors. We formulate the invariance properties of the loss function and show that critical points detected by standard gradient based methods are \emph{symmetry breaking} with respect to the target tensor. The phenomena, seen for different choices of target tensors and norms, make possible the use of recently developed analytic and algebraic tools for studying nonconvex optimization landscapes exhibiting symmetry breaking phenomena of similar nature.

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arXiv2022

Annihilation of Spurious Minima in Two-Layer ReLU Networks

We study the optimization problem associated with fitting two-layer ReLU neural networks with respect to the squared loss, where labels are generated by a target network. Use is made of the rich symmetry structure to develop a novel set of tools for studying the mechanism by which over-parameterization annihilates spurious minima. Sharp analytic estimates are obtained for the loss and the Hessian spectrum at different minima, and it is proved that adding neurons can turn symmetric spurious minima into saddles; minima of lesser symmetry require more neurons. Using Cauchy's interlacing theorem, we prove the existence of descent directions in certain subspaces arising from the symmetry structure of the loss function. This analytic approach uses techniques, new to the field, from algebraic geometry, representation theory and symmetry breaking, and confirms rigorously the effectiveness of over-parameterization in making the associated loss landscape accessible to gradient-based methods. For a fixed number of neurons and inputs, the spectral results remain true under symmetry breaking perturbation of the target.

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arXiv2022

Lower Bounds for Non-Convex Stochastic Optimization

We lower bound the complexity of finding $ε$-stationary points (with gradient norm at most $ε$) using stochastic first-order methods. In a well-studied model where algorithms access smooth, potentially non-convex functions through queries to an unbiased stochastic gradient oracle with bounded variance, we prove that (in the worst case) any algorithm requires at least $ε^{-4}$ queries to find an $ε$ stationary point. The lower bound is tight, and establishes that stochastic gradient descent is minimax optimal in this model. In a more restrictive model where the noisy gradient estimates satisfy a mean-squared smoothness property, we prove a lower bound of $ε^{-3}$ queries, establishing the optimality of recently proposed variance reduction techniques.

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arXiv2021

Analytic Study of Families of Spurious Minima in Two-Layer ReLU Neural Networks: A Tale of Symmetry II

We study the optimization problem associated with fitting two-layer ReLU neural networks with respect to the squared loss, where labels are generated by a target network. We make use of the rich symmetry structure to develop a novel set of tools for studying families of spurious minima. In contrast to existing approaches which operate in limiting regimes, our technique directly addresses the nonconvex loss landscape for a finite number of inputs $d$ and neurons $k$, and provides analytic, rather than heuristic, information. In particular, we derive analytic estimates for the loss at different minima, and prove that modulo $O(d^{-1/2})$-terms the Hessian spectrum concentrates near small positive constants, with the exception of $Θ(d)$ eigenvalues which grow linearly with~$d$. We further show that the Hessian spectrum at global and spurious minima coincide to $O(d^{-1/2})$-order, thus challenging our ability to argue about statistical generalization through local curvature. Lastly, our technique provides the exact \emph{fractional} dimensionality at which families of critical points turn from saddles into spurious minima. This makes possible the study of the creation and the annihilation of spurious minima using powerful tools from equivariant bifurcation theory.

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arXiv2021

Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of $S_n$

Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient equivariant dynamics related to neural nets. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on $n$ symbols, $S_n$; in particular, the standard representation of $S_n$. It is shown that spurious minima do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that can be encoded by a generic $S_n$-equivariant bifurcation. We describe minimal models for forced symmetry breaking that give a lower bound on the dynamic complexity involved in the creation of spurious minima when there is no symmetry. Results on generic bifurcation when there are quadratic equivariants are also proved; this work extends and clarifies results of Ihrig & Golubitsky and Chossat, Lauterback & Melbourne on the instability of solutions when there are quadratic equivariants.

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arXiv2021

Symmetry & critical points for a model shallow neural network

We consider the optimization problem associated with fitting two-layer ReLU networks with $k$ hidden neurons, where labels are assumed to be generated by a (teacher) neural network. We leverage the rich symmetry exhibited by such models to identify various families of critical points and express them as power series in $k^{-\frac{1}{2}}$. These expressions are then used to derive estimates for several related quantities which imply that not all spurious minima are alike. In particular, we show that while the loss function at certain types of spurious minima decays to zero like $k^{-1}$, in other cases the loss converges to a strictly positive constant. The methods used depend on symmetry, the geometry of group actions, bifurcation, and Artin's implicit function theorem.

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arXiv2020

Analytic Characterization of the Hessian in Shallow ReLU Models: A Tale of Symmetry

We consider the optimization problem associated with fitting two-layers ReLU networks with respect to the squared loss, where labels are generated by a target network. We leverage the rich symmetry structure to analytically characterize the Hessian at various families of spurious minima in the natural regime where the number of inputs $d$ and the number of hidden neurons $k$ is finite. In particular, we prove that for $d\ge k$ standard Gaussian inputs: (a) of the $dk$ eigenvalues of the Hessian, $dk - O(d)$ concentrate near zero, (b) $Ω(d)$ of the eigenvalues grow linearly with $k$. Although this phenomenon of extremely skewed spectrum has been observed many times before, to our knowledge, this is the first time it has been established {rigorously}. Our analytic approach uses techniques, new to the field, from symmetry breaking and representation theory, and carries important implications for our ability to argue about statistical generalization through local curvature.

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arXiv2020

Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations

We design an algorithm which finds an $ε$-approximate stationary point (with $\|\nabla F(x)\|\le ε$) using $O(ε^{-3})$ stochastic gradient and Hessian-vector products, matching guarantees that were previously available only under a stronger assumption of access to multiple queries with the same random seed. We prove a lower bound which establishes that this rate is optimal and---surprisingly---that it cannot be improved using stochastic $p$th order methods for any $p\ge 2$, even when the first $p$ derivatives of the objective are Lipschitz. Together, these results characterize the complexity of non-convex stochastic optimization with second-order methods and beyond. Expanding our scope to the oracle complexity of finding $(ε,γ)$-approximate second-order stationary points, we establish nearly matching upper and lower bounds for stochastic second-order methods. Our lower bounds here are novel even in the noiseless case.

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arXiv2020

IDEAL: Inexact DEcentralized Accelerated Augmented Lagrangian Method

We introduce a framework for designing primal methods under the decentralized optimization setting where local functions are smooth and strongly convex. Our approach consists of approximately solving a sequence of sub-problems induced by the accelerated augmented Lagrangian method, thereby providing a systematic way for deriving several well-known decentralized algorithms including EXTRA arXiv:1404.6264 and SSDA arXiv:1702.08704. When coupled with accelerated gradient descent, our framework yields a novel primal algorithm whose convergence rate is optimal and matched by recently derived lower bounds. We provide experimental results that demonstrate the effectiveness of the proposed algorithm on highly ill-conditioned problems.

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arXiv2020

On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions

Recent advances in randomized incremental methods for minimizing $L$-smooth $μ$-strongly convex finite sums have culminated in tight complexity of $\tilde{O}((n+\sqrt{n L/μ})\log(1/ε))$ and $O(n+\sqrt{nL/ε})$, where $μ>0$ and $μ=0$, respectively, and $n$ denotes the number of individual functions. Unlike incremental methods, stochastic methods for finite sums do not rely on an explicit knowledge of which individual function is being addressed at each iteration, and as such, must perform at least $Ω(n^2)$ iterations to obtain $O(1/n^2)$-optimal solutions. In this work, we exploit the finite noise structure of finite sums to derive a matching $O(n^2)$-upper bound under the global oracle model, showing that this lower bound is indeed tight. Following a similar approach, we propose a novel adaptation of SVRG which is both \emph{compatible with stochastic oracles}, and achieves complexity bounds of $\tilde{O}((n^2+n\sqrt{L/μ})\log(1/ε))$ and $O(n\sqrt{L/ε})$, for $μ>0$ and $μ=0$, respectively. Our bounds hold w.h.p. and match in part existing lower bounds of $\tildeΩ(n^2+\sqrt{nL/μ}\log(1/ε))$ and $\tildeΩ(n^2+\sqrt{nL/ε})$, for $μ>0$ and $μ=0$, respectively.

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arXiv2018

A Tight Convergence Analysis for Stochastic Gradient Descent with Delayed Updates

We provide tight finite-time convergence bounds for gradient descent and stochastic gradient descent on quadratic functions, when the gradients are delayed and reflect iterates from $τ$ rounds ago. First, we show that without stochastic noise, delays strongly affect the attainable optimization error: In fact, the error can be as bad as non-delayed gradient descent ran on only $1/τ$ of the gradients. In sharp contrast, we quantify how stochastic noise makes the effect of delays negligible, improving on previous work which only showed this phenomenon asymptotically or for much smaller delays. Also, in the context of distributed optimization, the results indicate that the performance of gradient descent with delays is competitive with synchronous approaches such as mini-batching. Our results are based on a novel technique for analyzing convergence of optimization algorithms using generating functions.

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arXiv2017

Limitations on Variance-Reduction and Acceleration Schemes for Finite Sum Optimization

We study the conditions under which one is able to efficiently apply variance-reduction and acceleration schemes on finite sum optimization problems. First, we show that, perhaps surprisingly, the finite sum structure by itself, is not sufficient for obtaining a complexity bound of $\tilde{\cO}((n+L/μ)\ln(1/ε))$ for $L$-smooth and $μ$-strongly convex individual functions - one must also know which individual function is being referred to by the oracle at each iteration. Next, we show that for a broad class of first-order and coordinate-descent finite sum algorithms (including, e.g., SDCA, SVRG, SAG), it is not possible to get an `accelerated' complexity bound of $\tilde{\cO}((n+\sqrt{n L/μ})\ln(1/ε))$, unless the strong convexity parameter is given explicitly. Lastly, we show that when this class of algorithms is used for minimizing $L$-smooth and convex finite sums, the optimal complexity bound is $\tilde{\cO}(n+L/ε)$, assuming that (on average) the same update rule is used in every iteration, and $\tilde{\cO}(n+\sqrt{nL/ε})$, otherwise.

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arXiv2017

Oracle Complexity of Second-Order Methods for Smooth Convex Optimization

Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth convex functions, or equivalently, the worst-case number of iterations required to optimize such functions to a given accuracy. In particular, these bounds indicate when such methods can or cannot improve on gradient-based methods, whose oracle complexity is much better understood. We also provide generalizations of our results to higher-order methods.

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