In this paper, we introduce a new projection-free algorithm for Online Convex Optimization (OCO) with a state-of-the-art regret guarantee among separation-based algorithms. Existing projection-free methods based on the classical Frank-Wolfe algorithm achieve a suboptimal regret bound of $O(T^{3/4})$, while more recent separation-based approaches guarantee a regret bound of $O(κ\sqrt{T})$, where $κ$ denotes the asphericity of the feasible set, defined as the ratio of the radii of the containing and contained balls. However, for ill-conditioned sets, $κ$ can be arbitrarily large, potentially leading to poor performance. Our algorithm achieves a regret bound of $\widetilde{O}(\sqrt{dT} + κd)$, while requiring only $\widetilde{O}(1)$ calls to a separation oracle per round. Crucially, the main term in the bound, $\widetilde{O}(\sqrt{d T})$, is independent of $κ$, addressing the limitations of previous methods. Additionally, as a by-product of our analysis, we recover the $O(κ\sqrt{T})$ regret bound of existing OCO algorithms with a more straightforward analysis and improve the regret bound for projection-free online exp-concave optimization. Finally, for constrained stochastic convex optimization, we achieve a state-of-the-art convergence rate of $\widetilde{O}(σ/\sqrt{T} + κd/T)$, where $σ$ represents the noise in the stochastic gradients, while requiring only $\widetilde{O}(1)$ calls to a separation oracle per iteration.
We note that known methods achieving the optimal oracle complexity for first order convex optimization require quadratic memory, and ask whether this is necessary, and more broadly seek to characterize the minimax number of first order queries required to optimize a convex Lipschitz function subject to a memory constraint.
In this note, we consider the complexity of optimizing a highly smooth (Lipschitz $k$-th order derivative) and strongly convex function, via calls to a $k$-th order oracle which returns the value and first $k$ derivatives of the function at a given point, and where the dimension is unrestricted. Extending the techniques introduced in Arjevani et al. [2019], we prove that the worst-case oracle complexity for any fixed $k$ to optimize the function up to accuracy $ε$ is on the order of $\left(\frac{μ_k D^{k-1}}λ\right)^{\frac{2}{3k+1}}+\log\log\left(\frac{1}ε\right)$ (in sufficiently high dimensi…
In constrained convex optimization, existing methods based on the ellipsoid or cutting plane method do not scale well with the dimension of the ambient space. Alternative approaches such as Projected Gradient Descent only provide a computational benefit for simple convex sets such as Euclidean balls, where Euclidean projections can be performed efficiently. For other sets, the cost of the projections can be too high. To circumvent these issues, alternative methods based on the famous Frank-Wolfe algorithm have been studied and used. Such methods use a Linear Optimization Oracle at each iterati…
We consider the nonsmooth convex composition optimization problem where the objective is a composition of two finite-sum functions and analyze stochastic compositional variance reduced gradient (SCVRG) methods for them. SCVRG and its variants have recently drawn much attention given their edge over stochastic compositional gradient descent (SCGD); but the theoretical analysis exclusively assumes strong convexity of the objective, which excludes several important examples such as Lasso, logistic regression, principle component analysis and deep neural nets. In contrast, we prove non-asymptotic …
University of Washington（华盛顿大学）；Massachusetts Institute of Technology（麻省理工学院）
Many fundamental problems in machine learning can be formulated by the convex program \[ \min_{θ\in R^d}\ \sum_{i=1}^{n}f_{i}(θ), \] where each $f_i$ is a convex, Lipschitz function supported on a subset of $d_i$ coordinates of $θ$. One common approach to this problem, exemplified by stochastic gradient descent, involves sampling one $f_i$ term at every iteration to make progress. This approach crucially relies on a notion of uniformity across the $f_i$'s, formally captured by their condition number. In this work, we give an algorithm that minimizes the above convex formulation to $ε$-accuracy…
Department of Computer Science, University of Rochester（计算机科学系，罗切斯特大学）；Goergen Institute for Data Science and Artificial Intelligence (GIDS-AI)（数据科学与人工智能研究所（GIDS-AI））
Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(ε^{-4/7})$ Second-Order Oracle Complexity
Tsinghua University（清华大学）；Shanghai Qizhi Institute（上海期智研究院）；The Chinese University of Hong Kong（香港中文大学）；Fudan University（复旦大学）；Shanghai AI Lab（上海人工智能实验室）
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Institute for Information Transmission Problems RAS（俄罗斯科学院信息传输问题研究所）
In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $Ω\left(ε^{-2/(3p+1)}\right)$ on the number of the $p$-th order oracle calls required by an algorithm to find an $ε$-accurate solution to the problem, where the $p$-th order oracle stands for the computation of the objective function value and the derivatives up to the order $p$. However, the existing state-of-the-art high-order methods of Gasnikov et al. (2019b); Bubeck et al. (2019); Jiang et al. …
We give nearly matching upper and lower bounds on the oracle complexity of finding $ε$-stationary points ($\| \nabla F(x) \| \leqε$) in stochastic convex optimization. We jointly analyze the oracle complexity in both the local stochastic oracle model and the global oracle (or, statistical learning) model. This allows us to decompose the complexity of finding near-stationary points into optimization complexity and sample complexity, and reveals some surprising differences between the complexity of stochastic optimization versus learning. Notably, we show that in the global oracle/statistical le…
University of Pennsylvania（宾夕法尼亚大学）；University of Minnesota（明尼苏达大学）
One of the most effective algorithms for differentially private learning and optimization is objective perturbation. This technique augments a given optimization problem (e.g. deriving from an ERM problem) with a random linear term, and then exactly solves it. However, to date, analyses of this approach crucially rely on the convexity and smoothness of the objective function, limiting its generality. We give two algorithms that extend this approach substantially. The first algorithm requires nothing except boundedness of the loss function, and operates over a discrete domain. Its privacy and a…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error that can be controlled through a batch size parameter. Our proposed oracles are appealing in several practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed (i.i.d.) samples, or simulation optimization, where the function measurements are `biased' due to computational constraints. In either case, increasing the batch size reduces the estimation error. We highlight the applicability of our biased gradient oracles in a risk-s…
Toyota Technological Institute at Chicago（芝加哥丰田技术学院）
In this thesis, I study the minimax oracle complexity of distributed stochastic optimization. First, I present the "graph oracle model", an extension of the classic oracle complexity framework that can be applied to study distributed optimization algorithms. Next, I describe a general approach to proving optimization lower bounds for arbitrary randomized algorithms (as opposed to more restricted classes of algorithms, e.g., deterministic or "zero-respecting" algorithms), which is used extensively throughout the thesis. For the remainder of the thesis, I focus on the specific case of the "inter…
University of Michigan（密歇根大学）；Princeton University（普林斯顿大学）
This paper explores a surprising equivalence between two seemingly-distinct convex optimization methods. We show that simulated annealing, a well-studied random walk algorithms, is directly equivalent, in a certain sense, to the central path interior point algorithm for the the entropic universal barrier function. This connection exhibits several benefits. First, we are able improve the state of the art time complexity for convex optimization under the membership oracle model. We improve the analysis of the randomized algorithm of Kalai and Vempala by utilizing tools developed by Nesterov and …
University of Alberta（阿尔伯塔大学）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Imperial College London（伦敦帝国理工学院）
Algorithms for bandit convex optimization and online learning often rely on constructing noisy gradient estimates, which are then used in appropriately adjusted first-order algorithms, replacing actual gradients. Depending on the properties of the function to be optimized and the nature of ``noise'' in the bandit feedback, the bias and variance of gradient estimates exhibit various tradeoffs. In this paper we propose a novel framework that replaces the specific gradient estimation methods with an abstract oracle. With the help of the new framework we unify previous works, reproducing their res…
Google Research（谷歌研究院）；University of Washington（华盛顿大学）
We consider the problem of minimizing a non-convex objective while preserving the privacy of the examples in the training data. Building upon the previous variance-reduced algorithm SpiderBoost, we introduce a new framework that utilizes two different kinds of gradient oracles. The first kind of oracles can estimate the gradient of one point, and the second kind of oracles, less precise and more cost-effective, can estimate the gradient difference between two points. SpiderBoost uses the first kind periodically, once every few steps, while our framework proposes using the first oracle whenever…
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）；UC Berkeley（加州大学伯克利分校）
In the framework of online convex optimization, most iterative algorithms require the computation of projections onto convex sets, which can be computationally expensive. To tackle this problem HK12 proposed the study of projection-free methods that replace projections with less expensive computations. The most common approach is based on the Frank-Wolfe method, that uses linear optimization computation in lieu of projections. Recent work by GK22 gave sublinear adaptive regret guarantees with projection free algorithms based on the Frank Wolfe approach. In this work we give projection-free alg…
We consider the problem of minimizing a convex objective function $F$ when one can only evaluate its noisy approximation $\hat{F}$. Unless one assumes some structure on the noise, $\hat{F}$ may be an arbitrary nonconvex function, making the task of minimizing $F$ intractable. To overcome this, prior work has often focused on the case when $F(x)-\hat{F}(x)$ is uniformly-bounded. In this paper we study the more general case when the noise has magnitude $αF(x) + β$ for some $α, β> 0$, and present a polynomial time algorithm that finds an approximate minimizer of $F$ for this noise model. Previous…
Accelerated stochastic first-order method for convex optimization under heavy-tailed noise
Department of Mathematics, Linköping University, Sweden（林雪平大学数学系）；Department of Industrial and Systems Engineering, University of Minnesota, USA（明尼苏达大学工业与系统工程系）
Institute for Mathematical and Computational Engineering（数学与计算工程研究所）；Faculty of Mathematics（理学院）；School of Engineering（工学院）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
This paper considers decentralized stochastic optimization over a network of $n$ nodes, where each node possesses a smooth non-convex local cost function and the goal of the networked nodes is to find an $ε$-accurate first-order stationary point of the sum of the local costs. We focus on an online setting, where each node accesses its local cost only by means of a stochastic first-order oracle that returns a noisy version of the exact gradient. In this context, we propose a novel single-loop decentralized hybrid variance-reduced stochastic gradient method, called GT-HSGD, that outperforms the …
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
We consider online learning in an adversarial, non-convex setting under the assumption that the learner has an access to an offline optimization oracle. In the general setting of prediction with expert advice, Hazan et al. (2016) established that in the optimization-oracle model, online learning requires exponentially more computation than statistical learning. In this paper we show that by slightly strengthening the oracle model, the online and the statistical learning models become computationally equivalent. Our result holds for any Lipschitz and bounded (but not necessarily convex) functio…
Technion - Israel Institute of Technology（以色列理工学院）
This paper considers a convex composite optimization problem with affine constraints, which includes problems that take the form of minimizing a smooth convex objective function over the intersection of (simple) convex sets, or regularized with multiple (simple) functions. Motivated by high-dimensional applications in which exact projection/proximal computations are not tractable, we propose a \textit{projection-free} augmented Lagrangian-based method, in which primal updates are carried out using a \textit{weak proximal oracle} (WPO). In an earlier work, WPO was shown to be more powerful than…
Recently, convex nested stochastic composite optimization (NSCO) has received considerable attention for its applications in reinforcement learning and risk-averse optimization. The current NSCO algorithms have worse stochastic oracle complexities, by orders of magnitude, than those for simpler stochastic composite optimization problems (e.g., sum of smooth and nonsmooth functions) without the nested structure. Moreover, they require all outer-layer functions to be smooth, which is not satisfied by some important applications. These discrepancies prompt us to ask: ``does the nested composition…
NTT Communication Science Laboratories（NTT通信科学实验室）
We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth loss function. As we show, adding a projection as output layer provably makes the loss smaller. We identify the marginal polytope, the output space's convex hull, as the best convex set on which to project. However, because the projection onto the marginal polytope can sometimes b…
We prove lower bounds for higher-order methods in smooth non-convex finite-sum optimization. Our contribution is threefold: We first show that a deterministic algorithm cannot profit from the finite-sum structure of the objective, and that simulating a pth-order regularized method on the whole function by constructing exact gradient information is optimal up to constant factors. We further show lower bounds for randomized algorithms and compare them with the best known upper bounds. To address some gaps between the bounds, we propose a new second-order smoothness assumption that can be seen as…
Technion - Israel Institute of Technology（以色列理工学院）
We consider variants of the classical Frank-Wolfe algorithm for constrained smooth convex minimization, that instead of access to the standard oracle for minimizing a linear function over the feasible set, have access to an oracle that can find an extreme point of the feasible set that is closest in Euclidean distance to a given vector. We first show that for many feasible sets of interest, such an oracle can be implemented with the same complexity as the standard linear optimization oracle. We then show that with such an oracle we can design new Frank-Wolfe variants which enjoy significantly …
Minimisation of Quasar-Convex Functions Using Random Zeroth-Order Oracles
School of Engineering（工程学院）；Australian National University（澳大利亚国立大学）；Department of Electrical and Electronic Engineering（电子与电气工程系）；University of Melbourne（墨尔本大学）
University of Milan（米兰大学）；Google（谷歌）；Italian Institute of Technology（意大利技术研究院）
We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(β,γ)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(β,γ…
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…
School of Mathematical Sciences, Peking University（北京大学数学科学学院）
This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for each individual component. For the strongly-convex case, we prove such an algorithm can not reach an $\varepsilon$-suboptimal point in fewer than $Ω((n+\sqrt{κn})\log(1/\varepsilon))$ iterations, where $κ$ is the condition number of the objective function. This lower bound is tighter than previous results and perfectly matches the upper bound of the existing p…
Imperial College London（帝国理工学院）；INRIA（法国国家信息与自动化研究所）；Ecole normale superieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Graz University of Technology（格拉茨工业大学）
We consider the problem of minimizing the sum of submodular set functions assuming minimization oracles of each summand function. Most existing approaches reformulate the problem as the convex minimization of the sum of the corresponding Lovász extensions and the squared Euclidean norm, leading to algorithms requiring total variation oracles of the summand functions; without further assumptions, these more complex oracles require many calls to the simpler minimization oracles often available in practice. In this paper, we consider a modified convex problem requiring constrained version of the …
University of Washington（华盛顿大学）；Stanford University（斯坦福大学）；Microsoft Research（微软研究院）
We consider sampling from composite densities on $\mathbb{R}^d$ of the form $dπ(x) \propto \exp(-f(x) - g(x))dx$ for well-conditioned $f$ and convex (but possibly non-smooth) $g$, a family generalizing restrictions to a convex set, through the abstraction of a restricted Gaussian oracle. For $f$ with condition number $κ$, our algorithm runs in $O \left(κ^2 d \log^2\tfrac{κd}ε\right)$ iterations, each querying a gradient of $f$ and a restricted Gaussian oracle, to achieve total variation distance $ε$. The restricted Gaussian oracle, which draws samples from a distribution whose negative log-lik…
