We consider stochastic gradient methods under the interpolation regime where a perfect fit can be obtained (minimum loss at each observation). While previous work highlighted the implicit regularization of such algorithms, we consider an explicit regularization framework as a minimum Bregman divergence convex feasibility problem. Using convex duality, we propose randomized Dykstra-style algorithms based on randomized dual coordinate ascent. For non-accelerated coordinate descent, we obtain an algorithm which bears strong similarities with (non-averaged) stochastic mirror descent on specific functions, as it is is equivalent for quadratic objectives, and equivalent in the early iterations for more general objectives. It comes with the benefit of an explicit convergence theorem to a minimum norm solution. For accelerated coordinate descent, we obtain a new algorithm that has better convergence properties than existing stochastic gradient methods in the interpolating regime. This leads to accelerated versions of the perceptron for generic $\ell_p$-norm regularizers, which we illustrate in experiments.
Trust-region (TR) type method, based on a quadratic model such as the trust-region subproblem (TRS) and $ p $-regularization subproblem ($p$RS), is arguably one of the most successful methods for unconstrained minimization. In this paper, we study a general regularized subproblem (named $ ρ$RS), which covers TRS and $p$RS as special cases. We derive a strong duality theorem for $ ρ$RS, and also its necessary and sufficient optimality condition under general assumptions on the regularization term. We then define the Rendl-Wolkowicz (RW) dual problem of $ ρ$RS, which is a maximization problem wh…
It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $β$ play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?". To answer this question, first, we show that the discrete H.B momentum update (GD+M) follows a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. Then, we show that this implicit regularizer for (GD+M) is stronger than that of (GD) by factor …
University of Oxford（牛津大学）；Instadeep Ltd（Instadeep有限公司）；University of Southern California（南加州大学）
We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervis…
Given a convex optimization problem and its dual, there are many possible first-order algorithms. In this paper, we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method. This is done through convex duality, and implies notably that for certain problems, such as for supervised machine learning problems with non-smooth losses or problems regularized by non-smooth regularizers, the primal subgradient method and the dual conditional gradient method are formally equivalent. The dual interpretation leads to a form of line search for mirro…
We provide a simple proof for the Fenchel duality between strong convexity and Lipschitz continuous gradient. To this end, we first establish equivalent conditions of convexity for a general function that may not be differentiable. By utilizing these equivalent conditions, we can directly obtain equivalent conditions for strong convexity and Lipschitz continuous gradient. Based on these results, we can easily prove Fenchel duality. Beside this main result, we also identify several conditions that are implied by strong convexity or Lipschitz continuous gradient, but are not necessarily equivale…
We study a hybrid conditional gradient - smoothing algorithm (HCGS) for solving composite convex optimization problems which contain several terms over a bounded set. Examples of these include regularization problems with several norms as penalties and a norm constraint. HCGS extends conditional gradient methods to cases with multiple nonsmooth terms, in which standard conditional gradient methods may be difficult to apply. The HCGS algorithm borrows techniques from smoothing proximal methods and requires first-order computations (subgradients and proximity operations). Unlike proximal methods…
We show that the iterates generated by a generic first-order meta-algorithm satisfy a canonical perturbed Fenchel duality inequality. The latter in turn readily yields a unified derivation of the best known convergence rates for various popular first-order algorithms including the conditional gradient method as well as the main kinds of Bregman proximal methods: subgradient, gradient, fast gradient, and universal gradient methods.
We suggest simple implementable modifications of conditional gradient and gradient projection methods for smooth convex optimization problems in Hilbert spaces. Usually, the custom methods attain only weak convergence. We prove strong convergence of the new versions and establish their complexity estimates, which appear similar to the convergence rate of the weakly convergent versions.
University of Pennsylvania（宾夕法尼亚大学）；Stanford University（斯坦福大学）
Acceleration and momentum are the de facto standard in modern applications of machine learning and optimization, yet the bulk of the work on implicit regularization focuses instead on unaccelerated methods. In this paper, we study the statistical risk of the iterates generated by Nesterov's accelerated gradient method and Polyak's heavy ball method, when applied to least squares regression, drawing several connections to explicit penalization. We carry out our analyses in continuous-time, allowing us to make sharper statements than in prior work, and revealing complex interactions between earl…
Stanford University（斯坦福大学）；University of Pennsylvania（宾夕法尼亚大学）；Carnegie Mellon University（卡内基梅隆大学）
We study the implicit regularization of mini-batch stochastic gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time stochastic differential equation having the same moments as stochastic gradient descent, which we call stochastic gradient flow. We give a bound on the excess risk of stochastic gradient flow at time $t$, over ridge regression with tuning parameter $λ= 1/t$. The bound may be computed from explicit constants (e.g., the mini-batch size, step size, number of iterations), revealing precisely how these quantities drive the…
UW-Madison（威斯康星大学麦迪逊分校）；University of Chicago（芝加哥大学）
This note provides a novel, simple analysis of the method of conjugate gradients for the minimization of convex quadratic functions. In contrast with standard arguments, our proof is entirely self-contained and does not rely on the existence of Chebyshev polynomials. Another advantage of our development is that it clarifies the relation between the method of conjugate gradients and general accelerated methods for smooth minimization by unifying their analyses within the framework of the Approximate Duality Gap Technique that was introduced by the authors.
Carnegie Mellon University（卡内基梅隆大学）；Princeton University（普林斯顿大学）；University of Southern California（南加州大学）
We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using …
It is widely accepted that the stepsize is of great significance to gradient method. Two efficient gradient methods with approximately optimal stepsizes mainly based on regularization models are proposed for unconstrained optimization. More exactly, if the objective function is not close to a quadratic function on the line segment between the current and latest iterates, regularization models are exploited carefully to generate approximately optimal stepsizes. Otherwise, quadratic approximation models are used. In particular, when the curvature is non-positive, special regularization models ar…
In this work, we generalized and unified two recent completely different works of~\cite{shi2015large} and~\cite{cartis2012adaptive} respectively into one by proposing the cyclic incremental Newton-type gradient descent with cubic regularization (SingCubic) method for optimizing non-convex functions. Through the iterations of SingCubic, a cubic regularized global quadratic approximation using Hessian information is kept and solved. Preliminary numerical experiments show the encouraging performance of the SingCubic algorithm when compared to basic incremental or stochastic Newton-type implementa…
In the development of first-order methods for smooth (resp., composite) convex optimization problems, where smooth functions with Lipschitz continuous gradients are minimized, the gradient (resp., gradient mapping) norm becomes a fundamental optimality measure. Under this measure, a fixed iteration algorithm with the optimal iteration complexity is known, while determining this number of iteration to obtain a desired accuracy requires the prior knowledge of the distance from the initial point to the optimal solution set. In this paper, we report an adaptive regularization approach, which attai…
Mila（米拉研究所）；DIRO（蒙特利尔大学计算机与运筹学系）；Université de Montréal（蒙特利尔大学）；INRIA（法国国家信息与自动化研究所）；École Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
When optimizing over-parameterized models, such as deep neural networks, a large set of parameters can achieve zero training error. In such cases, the choice of the optimization algorithm and its respective hyper-parameters introduces biases that will lead to convergence to specific minimizers of the objective. Consequently, this choice can be considered as an implicit regularization for the training of over-parametrized models. In this work, we push this idea further by studying the discrete gradient dynamics of the training of a two-layer linear network with the least-squares loss. Using a t…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Microsoft Research（微软研究院）
Recent work across many machine learning disciplines has highlighted that standard descent methods, even without explicit regularization, do not merely minimize the training error, but also exhibit an implicit bias. This bias is typically towards a certain regularized solution, and relies upon the details of the learning process, for instance the use of the cross-entropy loss. In this work, we show that for empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses, if the risk does not attain its infimum, then the gradient-descent path and the algorit…
In a Hilbert setting, for convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. The Tikhonov regularization parameter is assumed to tend to zero as time tends to infinity, which preserves equilibria. The presence of the Tikhonov regularization term induces a strong convexity property which vanishes asymptotically. To take advantage of the exponential convergence rates attached to the heavy ball method in the strongly convex case, we consider the inertial dynamic where the viscous damping coefficient is taken…
In this paper, two new subspace minimization conjugate gradient methods based on $p - $regularization models are proposed, where a special scaled norm in $p - $regularization model is analyzed. Different choices for special scaled norm lead to different solutions to the $p - $regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. $R - $linear convergence of the pr…
In a Hilbert framework, for convex differentiable optimization, we consider accelerated gradient methods obtained by combining temporal scaling and averaging techniques with Tikhonov regularization. We start from the continuous steepest descent dynamic with an additional Tikhonov regularization term whose coefficient vanishes asymptotically. We provide an extensive Lyapunov analysis of this first-order evolution equation. Then we apply to this dynamic the method of time scaling and averaging recently introduced by Attouch, Bot and Nguyen. We thus obtain an inertial dynamic which involves visco…
This paper concerns an optimization algorithm for unconstrained non-convex problems where the objective function has sparse connections between the unknowns. The algorithm is based on applying a dissipation preserving numerical integrator, the Itoh--Abe discrete gradient scheme, to the gradient flow of an objective function, guaranteeing energy decrease regardless of step size. We introduce the algorithm, prove a convergence rate estimate for non-convex problems with Lipschitz continuous gradients, and show an improved convergence rate if the objective function has sparse connections between u…
In this paper we propose a unified two-phase scheme for convex optimization to accelerate: (1) the adaptive cubic regularization methods with exact/inexact Hessian matrices, and (2) the adaptive gradient method, without any knowledge of the Lipschitz constants for the gradient or the Hessian. This is achieved by tuning the parameters used in the algorithm $\textit{adaptively}$ in its process of progression, which can be viewed as a relaxation over the existing algorithms in the literature. Under the assumption that the sub-problems can be solved approximately, we establish overall iteration co…
Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVRC algorithms can be further improved. Moreover, the semi-stochastic Hessian estimator adopted in existing SVRC algorithms prevents the use of Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC algorithms computationally intractable for high-dimensional p…
In this paper, we propose a first second-order scheme based on arbitrary non-Euclidean norms, incorporated by Bregman distances. They are introduced directly in the Newton iterate with regularization parameter proportional to the square root of the norm of the current gradient. For the basic scheme, as applied to the composite optimization problem, we establish the global convergence rate of the order $O(k^{-2})$ both in terms of the functional residual and in the norm of subgradients. Our main assumption on the smooth part of the objective is Lipschitz continuity of its Hessian. For uniformly…
The $\ell_p$ regularization problem with $0< p< 1$ has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. The proximal gradient algorithm is one of the most popular algorithms for solving the $\ell_p$ regularisation problem. In the present paper, we investigate the linear convergence issue of one inexact descent method and two inexact proximal gradient algorithms (PGA). For this purpose, an optimality condition theorem is explored to provide the equivalences among a local minimum, sec…
We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$ \mathbb{E} ~\left\| Ax_{k+1}-b\right\|^2_{2} \leq \left(1 + \frac{c_{A}}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} - \frac{2}{\|A\|_F^2} \left\|A^T A (x_k - x)\right\|^2_{2}.$$ This is a curious inequality: the last term has one more matrix applied to the residual $u_k - u$ than the remaining terms: if $x_k - x$ is mainly comprised of large singular vecto…
In this paper, we consider the exact continuous relaxation model of $l_0$ regularization problem which was given by Bian and Chen (SIAM J. Numer. Anal 58(1): 858-883, 2020) and propose a smoothing proximal gradient algorithm with extrapolation (SPGE) for this kind of problem. We show that any accumulation point of the sequence generated by SPGE algorithm is a lifted stationary point of the relaxation model for a fairly general choice of extrapolation parameter. Moreover, it is shown that the whole sequence generated by SPGE algorithm converges to a lifted stationary point of the relaxation mod…
This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak 2006]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(ε^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(ε^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima withou…
Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex op…
The paper is devoted to the regularization of linear Copositive Programming problems which consists of transforming a problem to an equivalent form, where the Slater condition is satisfied and the strong duality holds. We describe here two regularization algorithms based on the concept of immobile indices and an understanding of the important role these indices play in the feasible sets' characterization. These algorithms are compared to some regularization procedures developed for a more general case of convex problems and based on a facial reduction approach. We show that the immobile-index-…
We consider primal-dual pairs of semidefinite programs and assume that they are ill-posed, i.e., both primal and dual are either weakly feasible or weakly infeasible. Under such circumstances, strong duality may break down and the primal and dual might have a nonzero duality gap. Nevertheless, there are arbitrary small perturbations to the problem data which makes the perturbed primal-dual pair strongly feasible thus zeroing the duality gap. In this paper, we conduct an asymptotic analysis of the optimal value as the perturbation is driven to zero. Specifically, we fix two positive definite ma…
Space Applications Centre（空间应用中心）；Indian Space Research Organization（印度空间研究组织）
Despite numerous attempts sought to provide empirical evidence of adversarial regularization outperforming sole supervision, the theoretical understanding of such phenomena remains elusive. In this study, we aim to resolve whether adversarial regularization indeed performs better than sole supervision at a fundamental level. To bring this insight into fruition, we study vanishing gradient issue, asymptotic iteration complexity, gradient flow and provable convergence in the context of sole supervision and adversarial regularization. The key ingredient is a theoretical justification supported by…
In this paper, we investigate a group sparse optimization problem via $\ell_{p,q}$ regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish some oracle property and a global recovery bound of order $O(λ^\frac{2}{2-q})$ for any point in a level set of the $\ell_{p,q}$ regularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order $O(λ^2)$ for a path of local minima. In the algorithmic aspect, we apply t…
Università degli Studi di Genova（热那亚大学）；Machine Learning Genoa Center（热那亚机器学习中心）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）
We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster bias decay than gradient descent, but also suffers of a more unstable behavior. As a result acceleration cannot be in general expected to improve learning accuracy with respect to gradient descent, but rather to achieve the same accuracy with reduced computations. Our theoretical…
Natural policy gradient (NPG) methods are among the most widely used policy optimization algorithms in contemporary reinforcement learning. This class of methods is often applied in conjunction with entropy regularization -- an algorithmic scheme that encourages exploration -- and is closely related to soft policy iteration and trust region policy optimization. Despite the empirical success, the theoretical underpinnings for NPG methods remain limited even for the tabular setting. This paper develops $\textit{non-asymptotic}$ convergence guarantees for entropy-regularized NPG methods under sof…
How to induce regularization in linear models: A guide to reparametrizing gradient flow
School of Computation, Information and Technology, TU Munich（慕尼黑工业大学计算、信息与技术学院）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Department of Mathematics, LMU Munich（慕尼黑大学数学系）；Mathematical Institute for Machine Learning and Data Science (MIDS), KU Eichstätt-Ingolstadt（艾希施泰特-因戈尔施塔特天主教大学机器学习与数据科学数学研究所）
