Stephen Wright

Jeongyeol Kwon, Dohyun Kwon, Stephen Wright et al.

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an $ε$-stationary solution of the bilevel problem after $ε^{-7/2}, ε^{-5/2}$, and $ε^{-3/2}$ iterations (each iteration using $O(1)$ samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to $ε^{-5/2}, ε^{-4/2}$, and $ε^{-3/2}$, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

Rudy Geelen, Laura Balzano, Stephen Wright et al.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Mao Ye, Bo Liu, Stephen Wright et al.

Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta-learning, continual learning, and reinforcement learning. Conventional BO methods need to differentiate through the low-level optimization process with implicit differentiation, which requires expensive calculations related to the Hessian matrix. There has been a recent quest for first-order methods for BO, but the methods proposed to date tend to be complicated and impractical for large-scale deep learning applications. In this work, we propose a simple first-order BO algorithm that depends only on first-order gradient information, requires no implicit differentiation, and is practical and efficient for large-scale non-convex functions in deep learning. We provide non-asymptotic convergence analysis of the proposed method to stationary points for non-convex objectives and present empirical results that show its superior practical performance.

Charles Dickens, Connor Pryor, Changyu Gao et al.

The field of Neural-Symbolic (NeSy) systems is growing rapidly. Proposed approaches show great promise in achieving symbiotic unions of neural and symbolic methods. However, a unifying framework is needed to organize common NeSy modeling patterns and develop general learning approaches. In this paper, we introduce Neural-Symbolic Energy-Based Models (NeSy-EBMs), a unifying mathematical framework for discriminative and generative NeSy modeling. Importantly, NeSy-EBMs allow the derivation of general expressions for gradients of prominent learning losses, and we introduce a suite of four learning approaches that leverage methods from multiple domains, including bilevel and stochastic policy optimization. Finally, we ground the NeSy-EBM framework with Neural Probabilistic Soft Logic (NeuPSL), an open-source NeSy-EBM library designed for scalability and expressivity, facilitating the real-world application of NeSy systems. Through extensive empirical analysis across multiple datasets, we demonstrate the practical advantages of NeSy-EBMs in various tasks, including image classification, graph node labeling, autonomous vehicle situation awareness, and question answering.

Like Hui, Mikhail Belkin, Stephen Wright

Nearly all practical neural models for classification are trained using cross-entropy loss. Yet this ubiquitous choice is supported by little theoretical or empirical evidence. Recent work (Hui & Belkin, 2020) suggests that training using the (rescaled) square loss is often superior in terms of the classification accuracy. In this paper we propose the "squentropy" loss, which is the sum of two terms: the cross-entropy loss and the average square loss over the incorrect classes. We provide an extensive set of experiments on multi-class classification problems showing that the squentropy loss outperforms both the pure cross entropy and rescaled square losses in terms of the classification accuracy. We also demonstrate that it provides significantly better model calibration than either of these alternative losses and, furthermore, has less variance with respect to the random initialization. Additionally, in contrast to the square loss, squentropy loss can typically be trained using exactly the same optimization parameters, including the learning rate, as the standard cross-entropy loss, making it a true "plug-and-play" replacement. Finally, unlike the rescaled square loss, multiclass squentropy contains no parameters that need to be adjusted.

Jeongyeol Kwon, Dohyun Kwon, Stephen Wright et al.

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter $σ> 0$. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be $O(σ)$-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as $O(σ)$-approximation of the original BO, we propose first-order algorithms that find an $ε$-stationary solution by optimizing the penalty formulation with $σ= O(ε)$. When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an $ε$-stationary point of the penalty function, using in total $O(ε^{-3})$ and $O(ε^{-7})$ accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(ε^{-3})$ and $O(ε^{-5})$, respectively.

Shubham Kumar Bharti, Stephen Wright, Adish Singla et al.

We study optimal teaching of Linear Behavior Cloning (LBC) learners. In this setup, the teacher can select which states to demonstrate to an LBC learner. The learner maintains a version space of infinite linear hypotheses consistent with the demonstration. The goal of the teacher is to teach a realizable target policy to the learner using minimum number of state demonstrations. This number is known as the Teaching Dimension(TD). We present a teaching algorithm called ``Teach using Iterative Elimination(TIE)" that achieves instance optimal TD. However, we also show that finding optimal teaching set computationally is NP-hard. We further provide an approximation algorithm that guarantees an approximation ratio of $\log(|A|-1)$ on the teaching dimension. Finally, we provide experimental results to validate the efficiency and effectiveness of our algorithm.

Charles Dickens, Changyu Gao, Connor Pryor et al.

We leverage convex and bilevel optimization techniques to develop a general gradient-based parameter learning framework for neural-symbolic (NeSy) systems. We demonstrate our framework with NeuPSL, a state-of-the-art NeSy architecture. To achieve this, we propose a smooth primal and dual formulation of NeuPSL inference and show learning gradients are functions of the optimal dual variables. Additionally, we develop a dual block coordinate descent algorithm for the new formulation that naturally exploits warm-starts. This leads to over 100x learning runtime improvements over the current best NeuPSL inference method. Finally, we provide extensive empirical evaluations across 8 datasets covering a range of tasks and demonstrate our learning framework achieves up to a 16% point prediction performance improvement over alternative learning methods.

Zhiyan Ding, Shi Chen, Qin Li et al.

Finding the optimal configuration of parameters in ResNet is a nonconvex minimization problem, but first-order methods nevertheless find the global optimum in the overparameterized regime. We study this phenomenon with mean-field analysis, by translating the training process of ResNet to a gradient-flow partial differential equation (PDE) and examining the convergence properties of this limiting process. The activation function is assumed to be $2$-homogeneous or partially $1$-homogeneous; the regularized ReLU satisfies the latter condition. We show that if the ResNet is sufficiently large, with depth and width depending algebraically on the accuracy and confidence levels, first-order optimization methods can find global minimizers that fit the training data.

Zhiyan Ding, Shi Chen, Qin Li et al.

Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global optimizer with perfect fit (zero-loss) in many practical situations. We examine this phenomenon for the case of Residual Neural Networks (ResNet) with smooth activation functions in a limiting regime in which both the number of layers (depth) and the number of weights in each layer (width) go to infinity. First, we use a mean-field-limit argument to prove that the gradient descent for parameter training becomes a gradient flow for a probability distribution that is characterized by a partial differential equation (PDE) in the large-NN limit. Next, we show that under certain assumptions, the solution to the PDE converges in the training time to a zero-loss solution. Together, these results suggest that the training of the ResNet gives a near-zero loss if the ResNet is large enough. We give estimates of the depth and width needed to reduce the loss below a given threshold, with high probability.

Sinong Geng, Zhaobin Kuang, Jie Liu et al.

We study the $L_1$-regularized maximum likelihood estimator/estimation (MLE) problem for discrete Markov random fields (MRFs), where efficient and scalable learning requires both sparse regularization and approximate inference. To address these challenges, we consider a stochastic learning framework called stochastic proximal gradient (SPG; Honorio 2012a, Atchade et al. 2014,Miasojedow and Rejchel 2016). SPG is an inexact proximal gradient algorithm [Schmidtet al., 2011], whose inexactness stems from the stochastic oracle (Gibbs sampling) for gradient approximation - exact gradient evaluation is infeasible in general due to the NP-hard inference problem for discrete MRFs [Koller and Friedman, 2009]. Theoretically, we provide novel verifiable bounds to inspect and control the quality of gradient approximation. Empirically, we propose the tighten asymptotically (TAY) learning strategy based on the verifiable bounds to boost the performance of SPG.

Rahul Mazumder, Stephen Wright, Andrew Zheng

We consider a class of linear-programming based estimators in reconstructing a sparse signal from linear measurements. Specific formulations of the reconstruction problem considered here include Dantzig selector, basis pursuit (for the case in which the measurements contain no errors), and the fused Dantzig selector (for the case in which the underlying signal is piecewise constant). In spite of being estimators central to sparse signal processing and machine learning, solving these linear programming problems for large scale instances remains a challenging task, thereby limiting their usage in practice. We show that classic constraint- and column-generation techniques from large scale linear programming, when used in conjunction with a commercial implementation of the simplex method, and initialized with the solution from a closely-related Lasso formulation, yields solutions with high efficiency in many settings.

Zachary Charles, Shashank Rajput, Stephen Wright et al.

Adversarial training is a technique for training robust machine learning models. To encourage robustness, it iteratively computes adversarial examples for the model, and then re-trains on these examples via some update rule. This work analyzes the performance of adversarial training on linearly separable data, and provides bounds on the number of iterations required for large margin. We show that when the update rule is given by an arbitrary empirical risk minimizer, adversarial training may require exponentially many iterations to obtain large margin. However, if gradient or stochastic gradient update rules are used, only polynomially many iterations are required to find a large-margin separator. By contrast, without the use of adversarial examples, gradient methods may require exponentially many iterations to achieve large margin. Our results are derived by showing that adversarial training with gradient updates minimizes a robust version of the empirical risk at a $\mathcal{O}(\ln(t)^2/t)$ rate, despite non-smoothness. We corroborate our theory empirically.

Kwang-Sung Jun, Rebecca Willett, Stephen Wright et al.

We introduce the bilinear bandit problem with low-rank structure in which an action takes the form of a pair of arms from two different entity types, and the reward is a bilinear function of the known feature vectors of the arms. The unknown in the problem is a $d_1$ by $d_2$ matrix $\mathbfΘ^*$ that defines the reward, and has low rank $r \ll \min\{d_1,d_2\}$. Determination of $\mathbfΘ^*$ with this low-rank structure poses a significant challenge in finding the right exploration-exploitation tradeoff. In this work, we propose a new two-stage algorithm called "Explore-Subspace-Then-Refine" (ESTR). The first stage is an explicit subspace exploration, while the second stage is a linear bandit algorithm called "almost-low-dimensional OFUL" (LowOFUL) that exploits and further refines the estimated subspace via a regularization technique. We show that the regret of ESTR is $\widetilde{\mathcal{O}}((d_1+d_2)^{3/2} \sqrt{r T})$ where $\widetilde{\mathcal{O}}$ hides logarithmic factors and $T$ is the time horizon, which improves upon the regret of $\widetilde{\mathcal{O}}(d_1d_2\sqrt{T})$ attained for a naïve linear bandit reduction. We conjecture that the regret bound of ESTR is unimprovable up to polylogarithmic factors, and our preliminary experiment shows that ESTR outperforms a naïve linear bandit reduction.

Hongyi Wang, Scott Sievert, Zachary Charles et al.

Distributed model training suffers from communication overheads due to frequent gradient updates transmitted between compute nodes. To mitigate these overheads, several studies propose the use of sparsified stochastic gradients. We argue that these are facets of a general sparsification method that can operate on any possible atomic decomposition. Notable examples include element-wise, singular value, and Fourier decompositions. We present ATOMO, a general framework for atomic sparsification of stochastic gradients. Given a gradient, an atomic decomposition, and a sparsity budget, ATOMO gives a random unbiased sparsification of the atoms minimizing variance. We show that recent methods such as QSGD and TernGrad are special cases of ATOMO and that sparsifiying the singular value decomposition of neural networks gradients, rather than their coordinates, can lead to significantly faster distributed training.

Bin Hu, Stephen Wright, Laurent Lessard

Techniques for reducing the variance of gradient estimates used in stochastic programming algorithms for convex finite-sum problems have received a great deal of attention in recent years. By leveraging dissipativity theory from control, we provide a new perspective on two important variance-reduction algorithms: SVRG and its direct accelerated variant Katyusha. Our perspective provides a physically intuitive understanding of the behavior of SVRG-like methods via a principle of energy conservation. The tools discussed here allow us to automate the convergence analysis of SVRG-like methods by capturing their essential properties in small semidefinite programs amenable to standard analysis and computational techniques. Our approach recovers existing convergence results for SVRG and Katyusha and generalizes the theory to alternative parameter choices. We also discuss how our approach complements the linear coupling technique. Our combination of perspectives leads to a better understanding of accelerated variance-reduced stochastic methods for finite-sum problems.

Gábor Braun, Sebastian Pokutta, Dan Tu et al.

We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope P, combining the Frank--Wolfe algorithm (also called conditional gradient) with gradient-based steps, different from away steps and pairwise steps, but still achieving linear convergence for strongly convex functions, along with good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, notably avoidance of projections onto P and maintenance of iterates as sparse convex combinations of a limited number of extreme points of P. The algorithm is lazy, making use of inexpensive inexact solutions of the linear programming subproblem that characterizes the conditional gradient approach. It decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the lazy conditional gradient algorithms of [arXiv:1410.8816]. We also present a streamlined version of the algorithm for the probability simplex.

Kwang-Sung Jun, Francesco Orabona, Stephen Wright et al.

A key challenge in online learning is that classical algorithms can be slow to adapt to changing environments. Recent studies have proposed "meta" algorithms that convert any online learning algorithm to one that is adaptive to changing environments, where the adaptivity is analyzed in a quantity called the strongly-adaptive regret. This paper describes a new meta algorithm that has a strongly-adaptive regret bound that is a factor of $\sqrt{\log(T)}$ better than other algorithms with the same time complexity, where $T$ is the time horizon. We also extend our algorithm to achieve a first-order (i.e., dependent on the observed losses) strongly-adaptive regret bound for the first time, to our knowledge. At its heart is a new parameter-free algorithm for the learning with expert advice (LEA) problem in which experts sometimes do not output advice for consecutive time steps (i.e., \emph{sleeping} experts). This algorithm is derived by a reduction from optimal algorithms for the so-called coin betting problem. Empirical results show that our algorithm outperforms state-of-the-art methods in both learning with expert advice and metric learning scenarios.

Kwang-Sung Jun, Francesco Orabona, Rebecca Willett et al.

This paper describes a new parameter-free online learning algorithm for changing environments. In comparing against algorithms with the same time complexity as ours, we obtain a strongly adaptive regret bound that is a factor of at least $\sqrt{\log(T)}$ better, where $T$ is the time horizon. Empirical results show that our algorithm outperforms state-of-the-art methods in learning with expert advice and metric learning scenarios.

Nikhil Rao, Parikshit Shah, Stephen Wright

In many signal processing applications, the aim is to reconstruct a signal that has a simple representation with respect to a certain basis or frame. Fundamental elements of the basis known as "atoms" allow us to define "atomic norms" that can be used to formulate convex regularizations for the reconstruction problem. Efficient algorithms are available to solve these formulations in certain special cases, but an approach that works well for general atomic norms, both in terms of speed and reconstruction accuracy, remains to be found. This paper describes an optimization algorithm called CoGEnT that produces solutions with succinct atomic representations for reconstruction problems, generally formulated with atomic-norm constraints. CoGEnT combines a greedy selection scheme based on the conditional gradient approach with a backward (or "truncation") step that exploits the quadratic nature of the objective to reduce the basis size. We establish convergence properties and validate the algorithm via extensive numerical experiments on a suite of signal processing applications. Our algorithm and analysis also allow for inexact forward steps and for occasional enhancements of the current representation to be performed. CoGEnT can outperform the basic conditional gradient method, and indeed many methods that are tailored to specific applications, when the enhancement and truncation steps are defined appropriately. We also introduce several novel applications that are enabled by the atomic-norm framework, including tensor completion, moment problems in signal processing, and graph deconvolution.