arXiv subjects

Haihao Lu

Publications and source records attributed to Haihao Lu.

At least 55 records · Page 3Linked to original sources

arXiv2021

Dual Mirror Descent for Online Allocation Problems

We consider online allocation problems with concave revenue functions and resource constraints, which are central problems in revenue management and online advertising. In these settings, requests arrive sequentially during a finite horizon and, for each request, a decision maker needs to choose an action that consumes a certain amount of resources and generates revenue. The revenue function and resource consumption of each request are drawn independently and at random from a probability distribution that is unknown to the decision maker. The objective is to maximize cumulative revenues subject to a constraint on the total consumption of resources. We design a general class of algorithms that achieve sub-linear expected regret compared to the hindsight optimal allocation. Our algorithms operate in the Lagrangian dual space: they maintain a dual multiplier for each resource that is updated using online mirror descent. By choosing the reference function accordingly, we recover dual sub-gradient descent and dual exponential weights algorithm. The resulting algorithms are simple, efficient, and shown to attain the optimal order of regret when the length of the horizon and the initial number of resources are scaled proportionally. We discuss applications to online bidding in repeated auctions with budget constraints and online proportional matching with high entropy.

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arXiv2021

Regularized Online Allocation Problems: Fairness and Beyond

Online allocation problems with resource constraints have a rich history in operations research. In this paper, we introduce the \emph{regularized online allocation problem}, a variant that includes a non-linear regularizer acting on the total resource consumption. In this problem, requests repeatedly arrive over time and, for each request, a decision maker needs to take an action that generates a reward and consumes resources. The objective is to simultaneously maximize additively separable rewards and the value of a non-separable regularizer subject to the resource constraints. Our primary motivation is allowing decision makers to trade off separable objectives such as the economic efficiency of an allocation with ancillary, non-separable objectives such as the fairness or equity of an allocation. We design an algorithm that is simple, fast, and attains good performance with both stochastic i.i.d.~and adversarial inputs. In particular, our algorithm is asymptotically optimal under stochastic i.i.d. input models and attains a fixed competitive ratio that depends on the regularizer when the input is adversarial. Furthermore, the algorithm and analysis do not require convexity or concavity of the reward function and the consumption function, which allows more model flexibility. Numerical experiments confirm the effectiveness of the proposed algorithm and of regularization in an internet advertising application.

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arXiv2021

The Best of Many Worlds: Dual Mirror Descent for Online Allocation Problems

Online allocation problems with resource constraints are central problems in revenue management and online advertising. In these problems, requests arrive sequentially during a finite horizon and, for each request, a decision maker needs to choose an action that consumes a certain amount of resources and generates reward. The objective is to maximize cumulative rewards subject to a constraint on the total consumption of resources. In this paper, we consider a data-driven setting in which the reward and resource consumption of each request are generated using an input model that is unknown to the decision maker. We design a general class of algorithms that attain good performance in various input models without knowing which type of input they are facing. In particular, our algorithms are asymptotically optimal under independent and identically distributed inputs as well as various non-stationary stochastic input models, and they attain an asymptotically optimal fixed competitive ratio when the input is adversarial. Our algorithms operate in the Lagrangian dual space: they maintain a dual multiplier for each resource that is updated using online mirror descent. By choosing the reference function accordingly, we recover the dual sub-gradient descent and dual multiplicative weights update algorithm. The resulting algorithms are simple, fast, and do not require convexity in the revenue function, consumption function and action space, in contrast to existing methods for online allocation problems. We discuss applications to network revenue management, online bidding in repeated auctions with budget constraints, online proportional matching with high entropy, and personalized assortment optimization with limited inventory.

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arXiv2021

Frank-Wolfe Methods with an Unbounded Feasible Region and Applications to Structured Learning

The Frank-Wolfe (FW) method is a popular algorithm for solving large-scale convex optimization problems appearing in structured statistical learning. However, the traditional Frank-Wolfe method can only be applied when the feasible region is bounded, which limits its applicability in practice. Motivated by two applications in statistical learning, the $\ell_1$ trend filtering problem and matrix optimization problems with generalized nuclear norm constraints, we study a family of convex optimization problems where the unbounded feasible region is the direct sum of an unbounded linear subspace and a bounded constraint set. We propose two new Frank-Wolfe methods: unbounded Frank-Wolfe method (uFW) and unbounded Away-Step Frank-Wolfe method (uAFW), for solving a family of convex optimization problems with this class of unbounded feasible regions. We show that under proper regularity conditions, the unbounded Frank-Wolfe method has a $O(1/k)$ sublinear convergence rate, and unbounded Away-Step Frank-Wolfe method has a linear convergence rate, matching the best-known results for the Frank-Wolfe method when the feasible region is bounded. Furthermore, computational experiments indicate that our proposed methods appear to outperform alternative solvers.

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arXiv2021

An $O(s^r)$-Resolution ODE Framework for Understanding Discrete-Time Algorithms and Applications to the Linear Convergence of Minimax Problems

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a \emph{suitable} ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a DTA and its corresponding ODEs. In this paper, we propose a new machinery -- an $O(s^r)$-resolution ODE framework -- for analyzing the behavior of a generic DTA, which (partially) answers the above two questions. The framework contains three steps: 1. To obtain a suitable ODE from a given DTA, we define a hierarchy of $O(s^r)$-resolution ODEs of a DTA parameterized by the degree $r$, where $s$ is the step-size of the DTA. We present a principal approach to construct the unique $O(s^r)$-resolution ODEs from a DTA; 2. To analyze the resulting ODE, we propose the $O(s^r)$-linear-convergence condition of a DTA with respect to an energy function, under which the $O(s^r)$-resolution ODE converges linearly to an optimal solution; 3. To bridge the convergence properties of a DTA and its corresponding ODEs, we define the properness of an energy function and show that the linear convergence of the $O(s^r)$-resolution ODE with respect to a proper energy function can automatically guarantee the linear convergence of the DTA. To better illustrate this machinery, we utilize it to study three classic algorithms -- gradient descent ascent (GDA), proximal point method (PPM) and extra-gradient method (EGM) -- for solving the unconstrained minimax problem $\min_{x\in\RR^n} \max_{y\in \RR^m} L(x,y)$.

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arXiv2021

The Landscape of the Proximal Point Method for Nonconvex-Nonconcave Minimax Optimization

Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties this poses. In this paper, we study the classic proximal point method (PPM) applied to nonconvex-nonconcave minimax problems. We find that a classic generalization of the Moreau envelope by Attouch and Wets provides key insights. Critically, we show this envelope not only smooths the objective but can convexify and concavify it based on the level of interaction present between the minimizing and maximizing variables. From this, we identify three distinct regions of nonconvex-nonconcave problems. When interaction is sufficiently strong, we derive global linear convergence guarantees. Conversely when the interaction is fairly weak, we derive local linear convergence guarantees with a proper initialization. Between these two settings, we show that PPM may diverge or converge to a limit cycle.

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arXiv2021

Limiting Behaviors of Nonconvex-Nonconcave Minimax Optimization via Continuous-Time Systems

Unlike nonconvex optimization, where gradient descent is guaranteed to converge to a local optimizer, algorithms for nonconvex-nonconcave minimax optimization can have topologically different solution paths: sometimes converging to a solution, sometimes never converging and instead following a limit cycle, and sometimes diverging. In this paper, we study the limiting behaviors of three classic minimax algorithms: gradient descent ascent (GDA), alternating gradient descent ascent (AGDA), and the extragradient method (EGM). Numerically, we observe that all of these limiting behaviors can arise in Generative Adversarial Networks (GAN) training and are easily demonstrated for a range of GAN problems. To explain these different behaviors, we study the high-order resolution continuous-time dynamics that correspond to each algorithm, which results in the sufficient (and almost necessary) conditions for the local convergence by each method. Moreover, this ODE perspective allows us to characterize the phase transition between these different limiting behaviors caused by introducing regularization as Hopf Bifurcations.

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arXiv2021

Infeasibility detection with primal-dual hybrid gradient for large-scale linear programming

We study the problem of detecting infeasibility of large-scale linear programming problems using the primal-dual hybrid gradient method (PDHG) of Chambolle and Pock (2011). The literature on PDHG has mostly focused on settings where the problem at hand is assumed to be feasible. When the problem is not feasible, the iterates of the algorithm do not converge. In this scenario, we show that the iterates diverge at a controlled rate towards a well-defined ray. The direction of this ray is known as the infimal displacement vector $v$. The first contribution of our work is to prove that this vector recovers certificates of primal and dual infeasibility whenever they exist. Based on this fact, we propose a simple way to extract approximate infeasibility certificates from the iterates of PDHG. We study three different sequences that converge to the infimal displacement vector: the difference of iterates, the normalized iterates, and the normalized average. All of them are easy to compute, and thus the approach is suitable for large-scale problems. Our second contribution is to establish tight convergence rates for these sequences. We demonstrate that the normalized iterates and the normalized average achieve a convergence rate of $O(1/k)$, improving over the known rate of $O(1/\sqrt{k})$. This rate is general and applies to any fixed-point iteration of a nonexpansive operator. Thus, it is a result of independent interest since it covers a broad family of algorithms, including, for example, ADMM, and can be applied settings beyond linear programming, such as quadratic and semidefinite programming. Further, in the case of linear programming we show that, under nondegeneracy assumptions, the iterates of PDHG identify the active set of an auxiliary feasible problem in finite time, which ensures that the difference of iterates exhibits eventual linear convergence to the infimal displacement vector.

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arXiv2020

Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming

We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the \emph{Exponential Time Hypothesis} fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem. Finally, we present computational results, showcasing that the MIP formulation, along with its LP relaxation, are able to achieve superior in- and out-of-sample performance, as compared to state-of-the-art algorithms on both real and synthetic datasets. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems.

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arXiv2020

Randomized Gradient Boosting Machine

Gradient Boosting Machine (GBM) introduced by Friedman is a powerful supervised learning algorithm that is very widely used in practice---it routinely features as a leading algorithm in machine learning competitions such as Kaggle and the KDDCup. In spite of the usefulness of GBM in practice, our current theoretical understanding of this method is rather limited. In this work, we propose Randomized Gradient Boosting Machine (RGBM) which leads to substantial computational gains compared to GBM, by using a randomization scheme to reduce search in the space of weak-learners. We derive novel computational guarantees for RGBM. We also provide a principled guideline towards better step-size selection in RGBM that does not require a line search. Our proposed framework is inspired by a special variant of coordinate descent that combines the benefits of randomized coordinate descent and greedy coordinate descent; and may be of independent interest as an optimization algorithm. As a special case, our results for RGBM lead to superior computational guarantees for GBM. Our computational guarantees depend upon a curious geometric quantity that we call Minimal Cosine Angle, which relates to the density of weak-learners in the prediction space. On a series of numerical experiments on real datasets, we demonstrate the effectiveness of RGBM over GBM in terms of obtaining a model with good training and/or testing data fidelity with a fraction of the computational cost.

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arXiv2020

Accelerating Gradient Boosting Machine

Gradient Boosting Machine (GBM) is an extremely powerful supervised learning algorithm that is widely used in practice. GBM routinely features as a leading algorithm in machine learning competitions such as Kaggle and the KDDCup. In this work, we propose Accelerated Gradient Boosting Machine (AGBM) by incorporating Nesterov's acceleration techniques into the design of GBM. The difficulty in accelerating GBM lies in the fact that weak (inexact) learners are commonly used, and therefore the errors can accumulate in the momentum term. To overcome it, we design a "corrected pseudo residual" and fit best weak learner to this corrected pseudo residual, in order to perform the z-update. Thus, we are able to derive novel computational guarantees for AGBM. This is the first GBM type of algorithm with theoretically-justified accelerated convergence rate. Finally we demonstrate with a number of numerical experiments the effectiveness of AGBM over conventional GBM in obtaining a model with good training and/or testing data fidelity.

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arXiv2020

Ordered SGD: A New Stochastic Optimization Framework for Empirical Risk Minimization

We propose a new stochastic optimization framework for empirical risk minimization problems such as those that arise in machine learning. The traditional approaches, such as (mini-batch) stochastic gradient descent (SGD), utilize an unbiased gradient estimator of the empirical average loss. In contrast, we develop a computationally efficient method to construct a gradient estimator that is purposely biased toward those observations with higher current losses. On the theory side, we show that the proposed method minimizes a new ordered modification of the empirical average loss, and is guaranteed to converge at a sublinear rate to a global optimum for convex loss and to a critical point for weakly convex (non-convex) loss. Furthermore, we prove a new generalization bound for the proposed algorithm. On the empirical side, the numerical experiments show that our proposed method consistently improves the test errors compared with the standard mini-batch SGD in various models including SVM, logistic regression, and deep learning problems.

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arXiv2019

Generalized Stochastic Frank-Wolfe Algorithm with Stochastic "Substitute" Gradient for Structured Convex Optimization

The stochastic Frank-Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the guaranteed convergence rate for stochastic Frank-Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank-Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with a certain type of ``linear prediction'' property (formally defined in the paper), which is typically present loss minimization problems in practice. Our method also introduces the notion of a ``substitute gradient'' that is a not-necessarily-unbiased sample of the gradient. We show that our new method is equivalent to a particular randomized coordinate mirror descent algorithm applied to the dual problem, which in turn provides a new interpretation of randomized dual coordinate descent in the primal space. Also, in the special case of a strongly convex regularizer our generalized stochastic Frank-Wolfe method (as well as the randomized dual coordinate descent method) exhibits linear convergence. Furthermore, we present computational experiments that indicate that our method outperforms other stochastic Frank-Wolfe methods consistent with the theory developed herein.

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arXiv2018

Approximate Leave-One-Out for High-Dimensional Non-Differentiable Learning Problems

Consider the following class of learning schemes: \begin{equation} \label{eq:main-problem1} \hat{\boldsymbolβ} := \underset{\boldsymbolβ \in \mathcal{C}}{\arg\min} \;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbolβ; y_j) + λR(\boldsymbolβ), \qquad \qquad \qquad (1) \end{equation} where $\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\rm th}$ feature and response variable respectively. Let $\ell$ and $R$ be the convex loss function and regularizer, $\boldsymbolβ$ denote the unknown weights, and $λ$ be a regularization parameter. $\mathcal{C} \subset \mathbb{R}^{p}$ is a closed convex set. Finding the optimal choice of $λ$ is a challenging problem in high-dimensional regimes where both $n$ and $p$ are large. We propose three frameworks to obtain a computationally efficient approximation of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our three frameworks are based on the primal, dual, and proximal formulations of (1). Each framework shows its strength in certain types of problems. We prove the equivalence of the three approaches under smoothness conditions. This equivalence enables us to justify the accuracy of the three methods under such conditions. We use our approaches to obtain a risk estimate for several standard problems, including generalized LASSO, nuclear norm regularization, and support vector machines. We empirically demonstrate the effectiveness of our results for non-differentiable cases.

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arXiv2018

"Relative-Continuity" for Non-Lipschitz Non-Smooth Convex Optimization using Stochastic (or Deterministic) Mirror Descent

The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter $M_f$. However, in many settings the non-differentiable convex function $f(\cdot)$ is not uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with $f(x) := \max\{0, x\} + x^2$. Herein we develop a notion of "relative continuity" that is determined relative to a user-specified "reference function" $h(\cdot)$ (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function $h(\cdot)$. We also similarly develop a notion of "relative stochastic continuity" for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization.

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arXiv2018

Approximate Leave-One-Out for Fast Parameter Tuning in High Dimensions

Consider the following class of learning schemes: $$\hat{\boldsymbolβ} := \arg\min_{\boldsymbolβ}\;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbolβ; y_j) + λR(\boldsymbolβ),\qquad\qquad (1) $$ where $\boldsymbol{x}_i \in \mathbb{R}^p$ and $y_i \in \mathbb{R}$ denote the $i^{\text{th}}$ feature and response variable respectively. Let $\ell$ and $R$ be the loss function and regularizer, $\boldsymbolβ$ denote the unknown weights, and $λ$ be a regularization parameter. Finding the optimal choice of $λ$ is a challenging problem in high-dimensional regimes where both $n$ and $p$ are large. We propose two frameworks to obtain a computationally efficient approximation ALO of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our two frameworks are based on the primal and dual formulations of (1). We prove the equivalence of the two approaches under smoothness conditions. This equivalence enables us to justify the accuracy of both methods under such conditions. We use our approaches to obtain a risk estimate for several standard problems, including generalized LASSO, nuclear norm regularization, and support vector machines. We empirically demonstrate the effectiveness of our results for non-differentiable cases.

stat.ML↗

arXiv2018

Accelerating Greedy Coordinate Descent Methods

We study ways to accelerate greedy coordinate descent in theory and in practice, where "accelerate" refers either to $O(1/k^2)$ convergence in theory, in practice, or both. We introduce and study two algorithms: Accelerated Semi-Greedy Coordinate Descent (ASCD) and Accelerated Greedy Coordinate Descent (AGCD). While ASCD takes greedy steps in the $x$-updates and randomized steps in the $z$-updates, AGCD is a straightforward extension of standard greedy coordinate descent that only takes greedy steps. On the theory side, our main results are for ASCD: we show that ASCD achieves $O(1/k^2)$ convergence, and it also achieves accelerated linear convergence for strongly convex functions. On the empirical side, we observe that both AGCD and ASCD outperform Accelerated Randomized Coordinate Descent on a variety of instances. In particular, we note that AGCD significantly outperforms the other accelerated coordinate descent methods in numerical tests, in spite of a lack of theoretical guarantees for this method. To complement the empirical study of AGCD, we present a Lyapunov energy function argument that points to an explanation for why a direct extension of the acceleration proof for AGCD does not work; and we also introduce a technical condition under which AGCD is guaranteed to have accelerated convergence. Last of all, we confirm that this technical condition holds in our empirical study.

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arXiv2017

Relatively-Smooth Convex Optimization by First-Order Methods, and Applications

The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the differentiable convex function $f(\cdot)$ is not uniformly smooth -- for example in $D$-optimal design where $f(x):=-\ln \det(HXH^T)$, or even the univariate setting with $f(x) := -\ln(x) + x^2$. Herein we develop a notion of "relative smoothness" and relative strong convexity that is determined relative to a user-specified "reference function" $h(\cdot)$ (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth with respect to a correspondingly fairly-simple reference function $h(\cdot)$. We extend two standard algorithms -- the primal gradient scheme and the dual averaging scheme -- to our new setting, with associated computational guarantees. We apply our new approach to develop a new first-order method for the $D$-optimal design problem, with associated computational complexity analysis. Some of our results have a certain overlap with the recent work \cite{bbt}.

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