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Modern machine learning models require a large amount of labeled data for training to perform well. A recently emerging paradigm for reducing the reliance of large model training on massive labeled data is to take advantage of abundantly available labeled data from a related source task to boost the performance of the model in a desired target task where there may not be a lot of data available. This approach, which is called transfer learning, has been applied successfully in many application domains. However, despite the fact that many transfer learning algorithms have been developed, the fundamental understanding of "when" and "to what extent" transfer learning can reduce sample complexity is still limited. In this work, we take a step towards foundational understanding of transfer learning by focusing on binary classification with linear models and Gaussian features and develop statistical minimax lower bounds in terms of the number of source and target samples and an appropriate notion of similarity between source and target tasks. To derive this bound, we reduce the transfer learning problem to hypothesis testing via constructing a packing set of source and target parameters by exploiting Gilbert-Varshamov bound, which in turn leads to a lower bound on sample complexity. We also evaluate our theoretical results by experiments on real data sets. 

Dealing with the shear size and complexity of today’s massive data sets requires computational platforms that can analyze data in a parallelized and distributed fashion. A major bottleneck that arises in such modern distributed computing environments is that some of the worker nodes may run slow. These nodes a.k.a. stragglers can significantly slow down computation as the slowest node may dictate the overall computational time. A recent computational framework, called encoded optimization, creates redundancy in the data to mitigate the effect of stragglers. In this paper we develop novel mathematical understanding for this framework demonstrating its effectiveness in much broader settings than was previously understood. We also analyze the convergence behavior of iterative encoded optimization algorithms, allowing us to characterize fundamental trade-offs between convergence rate, size of data set, accuracy, computational load (or data redundancy), and straggler toleration in this framework. 

In this paper we focus on the problem of finding the optimal weights of the shallowest of neural networks consisting of a single Rectified Linear Unit (ReLU). These functions are of the form x → max(0,⟨w,x⟩) with w ∈ Rd denoting the weight vector. We focus on a planted model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to a planted weight vector. We first show that mini-batch stochastic gradient descent when suitably initialized, converges at a geometric rate to the planted model with a number of samples that is optimal up to numerical constants. Next we focus on a parallel implementation where in each iteration the mini-batch gradient is calculated in a distributed manner across multiple processors and then broadcast to a master or all other processors. To reduce the communication cost in this setting we utilize a Quanitzed Stochastic Gradient Scheme (QSGD) where the partial gradients are quantized. Perhaps unexpectedly, we show that QSGD maintains the fast convergence of SGD to a globally optimal model while significantly reducing the communication cost. We further corroborate our numerical findings via various experiments including distributed implementations over Amazon EC2. 

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×, and also achieves a 2.36×-12.65× speedup over the state-of-the-art straggler mitigation strategies.