Are Convex Optimization Curves Convex?

In this paper, we study when we might expect the optimization curve induced by gradient descent to be \emph{convex} -- precluding, for example, an initial plateau followed by a sharp decrease, making it difficult to decide when optimization should stop. Although such undesirable behavior can certainly occur when optimizing general functions, might it also occur in the benign and well-studied case of smooth convex functions? As far as we know, this question has not been tackled in previous work. We show, perhaps surprisingly, that the answer crucially depends on the choice of the step size. In particular, for the range of step sizes which are known to result in monotonic convergence to an optimal value, there is a regime where the optimization curve will be provably convex, and there is a regime where the curve can be non-convex. We also extend our results to gradient flow, and to the closely-related but different question of whether the gradient norm decreases monotonically.

Overfitting Regimes of Nadaraya-Watson Interpolators

In recent years, there has been much interest in understanding the generalization behavior of interpolating predictors, which overfit on noisy training data. Whereas standard analyses are concerned with whether a method is consistent or not, recent observations have shown that even inconsistent predictors can generalize well. In this work, we revisit the classic interpolating Nadaraya-Watson (NW) estimator (also known as Shepard's method), and study its generalization capabilities through this modern viewpoint. In particular, by varying a single bandwidth-like hyperparameter, we prove the existence of multiple overfitting behaviors, ranging non-monotonically from catastrophic, through benign, to tempered. Our results highlight how even classical interpolating methods can exhibit intricate generalization behaviors. Numerical experiments complement our theory, demonstrating the same phenomena.

Hardness of Learning Fixed Parities with Neural Networks

Learning parity functions is a canonical problem in learning theory, which although computationally tractable, is not amenable to standard learning algorithms such as gradient-based methods. This hardness is usually explained via statistical query lower bounds [Kearns, 1998]. However, these bounds only imply that for any given algorithm, there is some worst-case parity function that will be hard to learn. Thus, they do not explain why fixed parities - say, the full parity function over all coordinates - are difficult to learn in practice, at least with standard predictors and gradient-based methods [Abbe and Boix-Adsera, 2022]. In this paper, we address this open problem, by showing that for any fixed parity of some minimal size, using it as a target function to train one-hidden-layer ReLU networks with perturbed gradient descent will fail to produce anything meaningful. To establish this, we prove a new result about the decay of the Fourier coefficients of linear threshold (or weighted majority) functions, which may be of independent interest.

The Oracle Complexity of Simplex-based Matrix Games: Linear Separability and Nash Equilibria

We study the problem of solving matrix games of the form $\max_{\mathbf{w}\in\mathcal{W}}\min_{\mathbf{p}\in\Delta}\mathbf{p}^{\top}A\mathbf{w}$, where $A$ is some matrix and $\Delta$ is the probability simplex. This problem encapsulates canonical tasks such as finding a linear separator and computing Nash equilibria in zero-sum games. However, perhaps surprisingly, its inherent complexity (as formalized in the standard framework of oracle complexity [Nemirovski and Yudin, 1983]) is not well-understood. In this work, we first identify different oracle models which are implicitly used by prior algorithms, amounting to multiplying the matrix $A$ by a vector from either one or both sides. We then prove complexity lower bounds for algorithms under both access models, which in particular imply a separation between them. Specifically, we start by proving that algorithms for linear separability based on one-sided multiplications must require $\Omega(\gamma_A^{-2})$ iterations, where $\gamma_A$ is the margin, as matched by the Perceptron algorithm. We then prove that accelerated algorithms for this task, which utilize multiplications from both sides, must require $\tilde{\Omega}(\gamma_{A}^{-2/3})$ iterations, establishing the first oracle complexity barrier for such algorithms. Finally, by adapting our lower bound to $\ell_1$ geometry, we prove that computing an $\epsilon$-approximate Nash equilibrium requires $\tilde{\Omega}(\epsilon^{-2/5})$ iterations, which is an exponential improvement over the previously best-known lower bound due to Hadiji et al. [2024].

Simple Relative Deviation Bounds for Covariance and Gram Matrices

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our results provide sharper control across the spectrum. Our analysis is based on a general-purpose theorem that allows one to convert existing uniform bounds into relative ones. The theorems and techniques emphasize simplicity and should be applicable across various settings.

On the Hardness of Meaningful Local Guarantees in Nonsmooth Nonconvex Optimization

We study the oracle complexity of nonsmooth nonconvex optimization, with the algorithm assumed to have access only to local function information. It has been shown by Davis, Drusvyatskiy, and Jiang (2023) that for nonsmooth Lipschitz functions satisfying certain regularity and strictness conditions, perturbed gradient descent converges to local minimizers asymptotically. Motivated by this result and by other recent algorithmic advances in nonconvex nonsmooth optimization concerning Goldstein stationarity, we consider the question of obtaining a non-asymptotic rate of convergence to local minima for this problem class. We provide the following negative answer to this question: Local algorithms acting on regular Lipschitz functions cannot, in the worst case, provide meaningful local guarantees in terms of function value in sub-exponential time, even when all near-stationary points are global minima. This sharply contrasts with the smooth setting, for which it is well-known that standard gradient methods can do so in a dimension-independent rate. Our result complements the rich body of work in the theoretical computer science literature that provide hardness results conditional on conjectures such as $\mathsf{P}\neq\mathsf{NP}$ or cryptographic assumptions, in that ours holds unconditional of any such assumptions.

Open Problem: Anytime Convergence Rate of Gradient Descent

Recent results show that vanilla gradient descent can be accelerated for smooth convex objectives, merely by changing the stepsize sequence. We show that this can lead to surprisingly large errors indefinitely, and therefore ask: Is there any stepsize schedule for gradient descent that accelerates the classic $\mathcal{O}(1/T)$ convergence rate, at \emph{any} stopping time $T$?

Depth Separation in Norm-Bounded Infinite-Width Neural Networks

We study depth separation in infinite-width neural networks, where complexity is controlled by the overall squared $\ell_2$-norm of the weights (sum of squares of all weights in the network). Whereas previous depth separation results focused on separation in terms of width, such results do not give insight into whether depth determines if it is possible to learn a network that generalizes well even when the network width is unbounded. Here, we study separation in terms of the sample complexity required for learnability. Specifically, we show that there are functions that are learnable with sample complexity polynomial in the input dimension by norm-controlled depth-3 ReLU networks, yet are not learnable with sub-exponential sample complexity by norm-controlled depth-2 ReLU networks (with any value for the norm). We also show that a similar statement in the reverse direction is not possible: any function learnable with polynomial sample complexity by a norm-controlled depth-2 ReLU network with infinite width is also learnable with polynomial sample complexity by a norm-controlled depth-3 ReLU network.

Generalization in Kernel Regression Under Realistic Assumptions

It is by now well-established that modern over-parameterized models seem to elude the bias-variance tradeoff and generalize well despite overfitting noise. Many recent works attempt to analyze this phenomenon in the relatively tractable setting of kernel regression. However, as we argue in detail, most past works on this topic either make unrealistic assumptions, or focus on a narrow problem setup. This work aims to provide a unified theory to upper bound the excess risk of kernel regression for nearly all common and realistic settings. Specifically, we provide rigorous bounds that hold for common kernels and for any amount of regularization, noise, any input dimension, and any number of samples. Furthermore, we provide relative perturbation bounds for the eigenvalues of kernel matrices, which may be of independent interest. These reveal a self-regularization phenomenon, whereby a heavy tail in the eigendecomposition of the kernel provides it with an implicit form of regularization, enabling good generalization. When applied to common kernels, our results imply benign overfitting in high input dimensions, nearly tempered overfitting in fixed dimensions, and explicit convergence rates for regularized regression. As a by-product, we obtain time-dependent bounds for neural networks trained in the kernel regime.

An Algorithm with Optimal Dimension-Dependence for Zero-Order Nonsmooth Nonconvex Stochastic Optimization

We study the complexity of producing $(\delta,\epsilon)$-stationary points of Lipschitz objectives which are possibly neither smooth nor convex, using only noisy function evaluations. Recent works proposed several stochastic zero-order algorithms that solve this task, all of which suffer from a dimension-dependence of $\Omega(d^{3/2})$ where $d$ is the dimension of the problem, which was conjectured to be optimal. We refute this conjecture by providing a faster algorithm that has complexity $O(d\delta^{-1}\epsilon^{-3})$, which is optimal (up to numerical constants) with respect to $d$ and also optimal with respect to the accuracy parameters $\delta,\epsilon$, thus solving an open question due to Lin et al. (NeurIPS'22). Moreover, the convergence rate achieved by our algorithm is also optimal for smooth objectives, proving that in the nonconvex stochastic zero-order setting, nonsmooth optimization is as easy as smooth optimization. We provide algorithms that achieve the aforementioned convergence rate in expectation as well as with high probability. Our analysis is based on a simple yet powerful geometric lemma regarding the Goldstein-subdifferential set, which allows utilizing recent advancements in first-order nonsmooth nonconvex optimization.