arXiv subjects

Yair Carmon

Publications and source records attributed to Yair Carmon.

At least 37 records · Page 2Linked to original sources

arXiv2022

Optimal and Adaptive Monteiro-Svaiter Acceleration

We develop a variant of the Monteiro-Svaiter (MS) acceleration framework that removes the need to solve an expensive implicit equation at every iteration. Consequently, for any $p\ge 2$ we improve the complexity of convex optimization with Lipschitz $p$th derivative by a logarithmic factor, matching a lower bound. We also introduce an MS subproblem solver that requires no knowledge of problem parameters, and implement it as either a second- or first-order method by solving linear systems or applying MinRes, respectively. On logistic regression our method outperforms previous second-order momentum methods, but under-performs Newton's method; simply iterating our first-order adaptive subproblem solver performs comparably to L-BFGS.

math.OC↗

arXiv2022

Scaling Laws Under the Microscope: Predicting Transformer Performance from Small Scale Experiments

Neural scaling laws define a predictable relationship between a model's parameter count and its performance after training in the form of a power law. However, most research to date has not explicitly investigated whether scaling laws can be used to accelerate model development. In this work, we perform such an empirical investigation across a wide range of language understanding tasks, starting from models with as few as 10K parameters, and evaluate downstream performance across 9 language understanding tasks. We find that scaling laws emerge at finetuning time in some NLP tasks, and that they can also be exploited for debugging convergence when training large models. Moreover, for tasks where scaling laws exist, they can be used to predict the performance of larger models, which enables effective model selection. However, revealing scaling laws requires careful hyperparameter tuning and multiple runs for the purpose of uncertainty estimation, which incurs additional overhead, partially offsetting the computational benefits.

cs.CL↗

arXiv2022

Gradient Descent Finds the Cubic-Regularized Non-Convex Newton Step

We consider the minimization of non-convex quadratic forms regularized by a cubic term, which exhibit multiple saddle points and poor local minima. Nonetheless, we prove that, under mild assumptions, gradient descent approximates the $\textit{global minimum}$ to within $\varepsilon$ accuracy in $O(\varepsilon^{-1}\log(1/\varepsilon))$ steps for large $\varepsilon$ and $O(\log(1/\varepsilon))$ steps for small $\varepsilon$ (compared to a condition number we define), with at most logarithmic dependence on the problem dimension. When we use gradient descent to approximate the cubic-regularized Newton step, our result implies a rate of convergence to second-order stationary points of general smooth non-convex functions.

math.OC↗

arXiv2022

Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time

The conventional recipe for maximizing model accuracy is to (1) train multiple models with various hyperparameters and (2) pick the individual model which performs best on a held-out validation set, discarding the remainder. In this paper, we revisit the second step of this procedure in the context of fine-tuning large pre-trained models, where fine-tuned models often appear to lie in a single low error basin. We show that averaging the weights of multiple models fine-tuned with different hyperparameter configurations often improves accuracy and robustness. Unlike a conventional ensemble, we may average many models without incurring any additional inference or memory costs -- we call the results "model soups." When fine-tuning large pre-trained models such as CLIP, ALIGN, and a ViT-G pre-trained on JFT, our soup recipe provides significant improvements over the best model in a hyperparameter sweep on ImageNet. The resulting ViT-G model, which attains 90.94% top-1 accuracy on ImageNet, achieved a new state of the art. Furthermore, we show that the model soup approach extends to multiple image classification and natural language processing tasks, improves out-of-distribution performance, and improves zero-shot performance on new downstream tasks. Finally, we analytically relate the performance similarity of weight-averaging and logit-ensembling to flatness of the loss and confidence of the predictions, and validate this relation empirically. Code is available at https://github.com/mlfoundations/model-soups.

cs.LG↗

arXiv2022

RECAPP: Crafting a More Efficient Catalyst for Convex Optimization

The accelerated proximal point algorithm (APPA), also known as "Catalyst", is a well-established reduction from convex optimization to approximate proximal point computation (i.e., regularized minimization). This reduction is conceptually elegant and yields strong convergence rate guarantees. However, these rates feature an extraneous logarithmic term arising from the need to compute each proximal point to high accuracy. In this work, we propose a novel Relaxed Error Criterion for Accelerated Proximal Point (RECAPP) that eliminates the need for high accuracy subproblem solutions. We apply RECAPP to two canonical problems: finite-sum and max-structured minimization. For finite-sum problems, we match the best known complexity, previously obtained by carefully-designed problem-specific algorithms. For minimizing $\max_y f(x,y)$ where $f$ is convex in $x$ and strongly-concave in $y$, we improve on the best known (Catalyst-based) bound by a logarithmic factor.

math.OC↗

arXiv2022

Distributionally Robust Optimization via Ball Oracle Acceleration

We develop and analyze algorithms for distributionally robust optimization (DRO) of convex losses. In particular, we consider group-structured and bounded $f$-divergence uncertainty sets. Our approach relies on an accelerated method that queries a ball optimization oracle, i.e., a subroutine that minimizes the objective within a small ball around the query point. Our main contribution is efficient implementations of this oracle for DRO objectives. For DRO with $N$ non-smooth loss functions, the resulting algorithms find an $ε$-accurate solution with $\widetilde{O}\left(Nε^{-2/3} + ε^{-2}\right)$ first-order oracle queries to individual loss functions. Compared to existing algorithms for this problem, we improve complexity by a factor of up to $ε^{-4/3}$.

math.OC↗

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.

math.OC↗

arXiv2022

Unlabeled Data Improves Adversarial Robustness

We demonstrate, theoretically and empirically, that adversarial robustness can significantly benefit from semisupervised learning. Theoretically, we revisit the simple Gaussian model of Schmidt et al. that shows a sample complexity gap between standard and robust classification. We prove that unlabeled data bridges this gap: a simple semisupervised learning procedure (self-training) achieves high robust accuracy using the same number of labels required for achieving high standard accuracy. Empirically, we augment CIFAR-10 with 500K unlabeled images sourced from 80 Million Tiny Images and use robust self-training to outperform state-of-the-art robust accuracies by over 5 points in (i) $\ell_\infty$ robustness against several strong attacks via adversarial training and (ii) certified $\ell_2$ and $\ell_\infty$ robustness via randomized smoothing. On SVHN, adding the dataset's own extra training set with the labels removed provides gains of 4 to 10 points, within 1 point of the gain from using the extra labels.

stat.ML↗

arXiv2021

Stochastic Bias-Reduced Gradient Methods

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $δ$, variance $O(\log(1/δ))$, and an expected sampling cost of $O(\log(1/δ))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up to logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.

math.OC↗

arXiv2021

Accuracy on the Line: On the Strong Correlation Between Out-of-Distribution and In-Distribution Generalization

For machine learning systems to be reliable, we must understand their performance in unseen, out-of-distribution environments. In this paper, we empirically show that out-of-distribution performance is strongly correlated with in-distribution performance for a wide range of models and distribution shifts. Specifically, we demonstrate strong correlations between in-distribution and out-of-distribution performance on variants of CIFAR-10 & ImageNet, a synthetic pose estimation task derived from YCB objects, satellite imagery classification in FMoW-WILDS, and wildlife classification in iWildCam-WILDS. The strong correlations hold across model architectures, hyperparameters, training set size, and training duration, and are more precise than what is expected from existing domain adaptation theory. To complete the picture, we also investigate cases where the correlation is weaker, for instance some synthetic distribution shifts from CIFAR-10-C and the tissue classification dataset Camelyon17-WILDS. Finally, we provide a candidate theory based on a Gaussian data model that shows how changes in the data covariance arising from distribution shift can affect the observed correlations.

cs.LG↗

arXiv2021

Never Go Full Batch (in Stochastic Convex Optimization)

We study the generalization performance of $\text{full-batch}$ optimization algorithms for stochastic convex optimization: these are first-order methods that only access the exact gradient of the empirical risk (rather than gradients with respect to individual data points), that include a wide range of algorithms such as gradient descent, mirror descent, and their regularized and/or accelerated variants. We provide a new separation result showing that, while algorithms such as stochastic gradient descent can generalize and optimize the population risk to within $ε$ after $O(1/ε^2)$ iterations, full-batch methods either need at least $Ω(1/ε^4)$ iterations or exhibit a dimension-dependent sample complexity.

math.OC↗

arXiv2021

Thinking Inside the Ball: Near-Optimal Minimization of the Maximal Loss

We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(Nε^{-2})$ queries to a first-order oracle to compute an $ε$-suboptimal point and $\tilde{O}(Nε^{-1})$ queries if the $f_i$ are $O(1/ε)$-smooth. We develop methods with improved complexity bounds of $\tilde{O}(Nε^{-2/3} + ε^{-8/3})$ in the non-smooth case and $\tilde{O}(Nε^{-2/3} + \sqrt{N}ε^{-1})$ in the $O(1/ε)$-smooth case. Our methods consist of a recently proposed ball optimization oracle acceleration algorithm (which we refine) and a careful implementation of said oracle for the softmax function. We also prove an oracle complexity lower bound scaling as $Ω(Nε^{-2/3})$, showing that our dependence on $N$ is optimal up to polylogarithmic factors.

math.OC↗

arXiv2020

Large-Scale Methods for Distributionally Robust Optimization

We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $χ^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $χ^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $χ^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.

math.OC↗

arXiv2020

Coordinate Methods for Matrix Games

We develop primal-dual coordinate methods for solving bilinear saddle-point problems of the form $\min_{x \in \mathcal{X}} \max_{y\in\mathcal{Y}} y^\top A x$ which contain linear programming, classification, and regression as special cases. Our methods push existing fully stochastic sublinear methods and variance-reduced methods towards their limits in terms of per-iteration complexity and sample complexity. We obtain nearly-constant per-iteration complexity by designing efficient data structures leveraging Taylor approximations to the exponential and a binomial heap. We improve sample complexity via low-variance gradient estimators using dynamic sampling distributions that depend on both the iterates and the magnitude of the matrix entries. Our runtime bounds improve upon those of existing primal-dual methods by a factor depending on sparsity measures of the $m$ by $n$ matrix $A$. For example, when rows and columns have constant $\ell_1/\ell_2$ norm ratios, we offer improvements by a factor of $m+n$ in the fully stochastic setting and $\sqrt{m+n}$ in the variance-reduced setting. We apply our methods to computational geometry problems, i.e. minimum enclosing ball, maximum inscribed ball, and linear regression, and obtain improved complexity bounds. For linear regression with an elementwise nonnegative matrix, our guarantees improve on exact gradient methods by a factor of $\sqrt{\mathrm{nnz}(A)/(m+n)}$.

cs.DS↗

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.

cs.LG↗

arXiv2020

Acceleration with a Ball Optimization Oracle

Consider an oracle which takes a point $x$ and returns the minimizer of a convex function $f$ in an $\ell_2$ ball of radius $r$ around $x$. It is straightforward to show that roughly $r^{-1}\log\frac{1}ε$ calls to the oracle suffice to find an $ε$-approximate minimizer of $f$ in an $\ell_2$ unit ball. Perhaps surprisingly, this is not optimal: we design an accelerated algorithm which attains an $ε$-approximate minimizer with roughly $r^{-2/3} \log \frac{1}ε$ oracle queries, and give a matching lower bound. Further, we implement ball optimization oracles for functions with locally stable Hessians using a variant of Newton's method. The resulting algorithm applies to a number of problems of practical and theoretical import, improving upon previous results for logistic and $\ell_\infty$ regression and achieving guarantees comparable to the state-of-the-art for $\ell_p$ regression.

math.OC↗

arXiv2020

First-Order Methods for Nonconvex Quadratic Minimization

We consider minimization of indefinite quadratics with either trust-region (norm) constraints or cubic regularization. Despite the nonconvexity of these problems we prove that, under mild assumptions, gradient descent converges to their global solutions, and give a non-asymptotic rate of convergence for the cubic variant. We also consider Krylov subspace solutions and establish sharp convergence guarantees to the solutions of both trust-region and cubic-regularized problems. Our rates mirror the behavior of these methods on convex quadratics and eigenvector problems, highlighting their scalability. When we use Krylov subspace solutions to approximate the cubic-regularized Newton step, our results recover the strongest known convergence guarantees to approximate second-order stationary points of general smooth nonconvex functions.

math.OC↗

arXiv2019

Variance Reduction for Matrix Games

We present a randomized primal-dual algorithm that solves the problem $\min_{x} \max_{y} y^\top A x$ to additive error $ε$ in time $\mathrm{nnz}(A) + \sqrt{\mathrm{nnz}(A)n}/ε$, for matrix $A$ with larger dimension $n$ and $\mathrm{nnz}(A)$ nonzero entries. This improves the best known exact gradient methods by a factor of $\sqrt{\mathrm{nnz}(A)/n}$ and is faster than fully stochastic gradient methods in the accurate and/or sparse regime $ε\le \sqrt{n/\mathrm{nnz}(A)}$. Our results hold for $x,y$ in the simplex (matrix games, linear programming) and for $x$ in an $\ell_2$ ball and $y$ in the simplex (perceptron / SVM, minimum enclosing ball). Our algorithm combines Nemirovski's "conceptual prox-method" and a novel reduced-variance gradient estimator based on "sampling from the difference" between the current iterate and a reference point.

math.OC↗