Qian Yu

Lin Chen, Hossein Esfandiari, Gang Fu et al.

In this paper, we study feature cross search as a fundamental primitive in feature engineering. The importance of feature cross search especially for the linear model has been known for a while, with well-known textbook examples. In this problem, the goal is to select a small subset of features, combine them to form a new feature (called the crossed feature) by considering their Cartesian product, and find feature crosses to learn an \emph{accurate} model. In particular, we study the problem of maximizing a normalized Area Under the Curve (AUC) of the linear model trained on the crossed feature column. First, we show that it is not possible to provide an $n^{1/\log\log n}$-approximation algorithm for this problem unless the exponential time hypothesis fails. This result also rules out the possibility of solving this problem in polynomial time unless $\mathsf{P}=\mathsf{NP}$. On the positive side, by assuming the \naive\ assumption, we show that there exists a simple greedy $(1-1/e)$-approximation algorithm for this problem. This result is established by relating the AUC to the total variation of the commutator of two probability measures and showing that the total variation of the commutator is monotone and submodular. To show this, we relate the submodularity of this function to the positive semi-definiteness of a corresponding kernel matrix. Then, we use Bochner's theorem to prove the positive semi-definiteness by showing that its inverse Fourier transform is non-negative everywhere. Our techniques and structural results might be of independent interest.

Lin Chen, Qian Yu, Hannah Lawrence et al.

We study the problem of switching-constrained online convex optimization (OCO), where the player has a limited number of opportunities to change her action. While the discrete analog of this online learning task has been studied extensively, previous work in the continuous setting has neither established the minimax rate nor algorithmically achieved it. In this paper, we show that $ T $-round switching-constrained OCO with fewer than $ K $ switches has a minimax regret of $ Θ(\frac{T}{\sqrt{K}}) $. In particular, it is at least $ \frac{T}{\sqrt{2K}} $ for one dimension and at least $ \frac{T}{\sqrt{K}} $ for higher dimensions. The lower bound in higher dimensions is attained by an orthogonal subspace argument. In one dimension, a novel adversarial strategy yields the lower bound of $O(\frac{T}{\sqrt{K}})$, but a precise minimax analysis including constants is more involved. To establish the tighter one-dimensional result, we introduce the \emph{fugal game} relaxation, whose minimax regret lower bounds that of switching-constrained OCO. We show that the minimax regret of the fugal game is at least $ \frac{T}{\sqrt{2K}} $ and thereby establish the optimal minimax lower bound in one dimension. To establish the dimension-independent upper bound, we next show that a mini-batching algorithm provides an $ O(\frac{T}{\sqrt{K}}) $ upper bound, and therefore conclude that the minimax regret of switching-constrained OCO is $ Θ(\frac{T}{\sqrt{K}}) $ for any $K$. This is in sharp contrast to its discrete counterpart, the switching-constrained prediction-from-experts problem, which exhibits a phase transition in minimax regret between the low-switching and high-switching regimes.

Qian Yu, Songze Li, Netanel Raviv et al.

We consider a scenario involving computations over a massive dataset stored distributedly across multiple workers, which is at the core of distributed learning algorithms. We propose Lagrange Coded Computing (LCC), a new framework to simultaneously provide (1) resiliency against stragglers that may prolong computations; (2) security against Byzantine (or malicious) workers that deliberately modify the computation for their benefit; and (3) (information-theoretic) privacy of the dataset amidst possible collusion of workers. LCC, which leverages the well-known Lagrange polynomial to create computation redundancy in a novel coded form across workers, can be applied to any computation scenario in which the function of interest is an arbitrary multivariate polynomial of the input dataset, hence covering many computations of interest in machine learning. LCC significantly generalizes prior works to go beyond linear computations. It also enables secure and private computing in distributed settings, improving the computation and communication efficiency of the state-of-the-art. Furthermore, we prove the optimality of LCC by showing that it achieves the optimal tradeoff between resiliency, security, and privacy, i.e., in terms of tolerating the maximum number of stragglers and adversaries, and providing data privacy against the maximum number of colluding workers. Finally, we show via experiments on Amazon EC2 that LCC speeds up the conventional uncoded implementation of distributed least-squares linear regression by up to $13.43\times$, and also achieves a $2.36\times$-$12.65\times$ speedup over the state-of-the-art straggler mitigation strategies.

Songze Li, Seyed Mohammadreza Mousavi Kalan, Qian Yu et al.

We consider the problem of training a least-squares regression model on a large dataset using gradient descent. The computation is carried out on a distributed system consisting of a master node and multiple worker nodes. Such distributed systems are significantly slowed down due to the presence of slow-running machines (stragglers) as well as various communication bottlenecks. We propose "polynomially coded regression" (PCR) that substantially reduces the effect of stragglers and lessens the communication burden in such systems. The key idea of PCR is to encode the partial data stored at each worker, such that the computations at the workers can be viewed as evaluating a polynomial at distinct points. This allows the master to compute the final gradient by interpolating this polynomial. PCR significantly reduces the recovery threshold, defined as the number of workers the master has to wait for prior to computing the gradient. In particular, PCR requires a recovery threshold that scales inversely proportionally with the amount of computation/storage available at each worker. In comparison, state-of-the-art straggler-mitigation schemes require a much higher recovery threshold that only decreases linearly in the per worker computation/storage load. We prove that PCR's recovery threshold is near minimal and within a factor two of the best possible scheme. Our experiments over Amazon EC2 demonstrate that compared with state-of-the-art schemes, PCR improves the run-time by 1.50x ~ 2.36x with naturally occurring stragglers, and by as much as 2.58x ~ 4.29x with artificial stragglers.

Lu Duan, Haoyuan Hu, Yu Qian et al.

A 3D flexible bin packing problem (3D-FBPP) arises from the process of warehouse packing in e-commerce. An online customer's order usually contains several items and needs to be packed as a whole before shipping. In particular, 5% of tens of millions of packages are using plastic wrapping as outer packaging every day, which brings pressure on the plastic surface minimization to save traditional logistics costs. Because of the huge practical significance, we focus on the issue of packing cuboid-shaped items orthogonally into a least-surface-area bin. The existing heuristic methods for classic 3D bin packing don't work well for this particular NP-hard problem and designing a good problem-specific heuristic is non-trivial. In this paper, rather than designing heuristics, we propose a novel multi-task framework based on Selected Learning to learn a heuristic-like policy that generates the sequence and orientations of items to be packed simultaneously. Through comprehensive experiments on a large scale real-world transaction order dataset and online AB tests, we show: 1) our selected learning method trades off the imbalance and correlation among the tasks and significantly outperforms the single task Pointer Network and the multi-task network without selected learning; 2) our method obtains an average 5.47% cost reduction than the well-designed greedy algorithm which is previously used in our online production system.

Qian Yu, Mohammad Ali Maddah-Ali, A. Salman Avestimehr

We consider the problem of computing the Fourier transform of high-dimensional vectors, distributedly over a cluster of machines consisting of a master node and multiple worker nodes, where the worker nodes can only store and process a fraction of the inputs. We show that by exploiting the algebraic structure of the Fourier transform operation and leveraging concepts from coding theory, one can efficiently deal with the straggler effects. In particular, we propose a computation strategy, named as coded FFT, which achieves the optimal recovery threshold, defined as the minimum number of workers that the master node needs to wait for in order to compute the output. This is the first code that achieves the optimum robustness in terms of tolerating stragglers or failures for computing Fourier transforms. Furthermore, the reconstruction process for coded FFT can be mapped to MDS decoding, which can be solved efficiently. Moreover, we extend coded FFT to settings including computing general $n$-dimensional Fourier transforms, and provide the optimal computing strategy for those settings.