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The problem of saddle-point avoidance for non-convex optimization is quite challenging in large scale distributed learning frameworks, such as Federated Learning, especially in the presence of Byzantine workers. The celebrated cubic-regularized Newton method of \cite{nest} is one of the most elegant ways to avoid saddle-points in the standard centralized (non-distributed) setup. In this paper, we extend the cubic-regularized Newton method to a distributed framework and simultaneously address several practical challenges like communication bottleneck and Byzantine attacks. Note that the issue of saddle-point avoidance becomes more crucial in the presence of Byzantine machines since rogue machines may create \emph{fake local minima} near the saddle-points of the loss function, also known as the saddle-point attack. Being a second order algorithm, our iteration complexity is much lower than the first order counterparts. Furthermore we use compression (or sparsification) techniques like $\delta$-approximate compression for communication efficiency. We obtain theoretical guarantees for our proposed scheme under several settings including approximate (sub-sampled) gradients and Hessians. Moreover, we validate our theoretical findings with experiments using standard datasets and several types of Byzantine attacks, and obtain an improvement of $25\%$ with respect to first order methods in iteration complexity.

We study the sample complexity of estimating the covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{d\times d}$ of a distribution $\mathcal D$ over $\mathbb{R}^d$ given independent samples, under the assumption that $\mathbf{\Sigma}$ is graph-structured. In particular, we focus on shortest path covariance matrices, where the covariance between any two measurements is determined by the shortest path distance in an underlying graph with $d$ nodes. Such matrices generalize Toeplitz and circulant covariance matrices and are widely applied in signal processing applications, where the covariance between two measurements depends on the (shortest path) distance between them in time or space. We focus on minimizing both the vector sample complexity: the number of samples drawn from $\mathcal{D}$ and the entry sample complexity: the number of entries read in each sample. The entry sample complexity corresponds to measurement equipment costs in signal processing applications. We give a very simple algorithm for estimating $\mathbf{\Sigma}$ up to spectral norm error $\epsilon \left\|\mathbf{\Sigma}\right\|_2$ using just $O(\sqrt{D})$ entry sample complexity and $\tilde O(r^2/\epsilon^2)$ vector sample complexity, where $D$ is the diameter of the underlying graph and $r \le d$ is the rank of $\mathbf{\Sigma}$. Our method is based on extending the widely applied idea of sparse rulers for Toeplitz covariance estimation to the graph setting. In the special case when $\mathbf{\Sigma}$ is a low-rank Toeplitz matrix, our result matches the state-of-the-art, with a far simpler proof. We also give an information theoretic lower bound matching our upper bound up to a factor $D$ and discuss some directions towards closing this gap.

We develop a distributed second order optimization algorithm that is communication-efficient as well as robust against Byzantine failures of the worker machines. We propose COMRADE (COMunication-efficient and Robust Approximate Distributed nEwton), an iterative second order algorithm, where the worker machines communicate only once per iteration with the center machine. This is in sharp contrast with the state-of-the-art distributed second order algorithms like GIANT [34] and DINGO[7], where the worker machines send (functions of) local gradient and Hessian sequentially; thus ending up communicating twice with the center machine per iteration. Moreover, we show that the worker machines can further compress the local information before sending it to the center. In addition, we employ a simple norm based thresholding rule to filter-out the Byzantine worker machines. We establish the linear-quadratic rate of convergence of COMRADE and establish that the communication savings and Byzantine resilience result in only a small statistical error rate for arbitrary convex loss functions. To the best of our knowledge, this is the first work that addresses the issue of Byzantine resilience in second order distributed optimization. Furthermore, we validate our theoretical results with extensive experiments on synthetic and benchmark LIBSVM [5] data-sets and demonstrate convergence guarantees.

Recently Ermon et al. (2013) pioneered a way to practically compute approximations to large scale counting or discrete integration problems by using random hashes. The hashes are used to reduce the counting problem into many separate discrete optimization problems. The optimization problems then can be solved by an NP-oracle such as commercial SAT solvers or integer linear programming (ILP) solvers. In particular, Ermon et al. showed that if the domain of integration is $\{0,1\}^n$ then it is possible to obtain a solution within a factor of $16$ of the optimal (a 16-approximation) by this technique. In many crucial counting tasks, such as computation of partition function of ferromagnetic Potts model, the domain of integration is naturally $\{0,1,\dots, q-1\}^n, q>2$, the hypergrid. The straightforward extension of Ermon et al.'s method allows a $q^2$-approximation for this problem. For large values of $q$, this is undesirable. In this paper, we show an improved technique to obtain an approximation factor of $4+O(1/q^2)$ to this problem. We are able to achieve this by using an idea of optimization over multiple bins of the hash functions, that can be easily implemented by inequality constraints, or even in unconstrained way. Also the burden on the NP-oracle is not increased by our method (an ILP solver can still be used). We provide experimental simulation results to support the theoretical guarantees of our algorithms.

This paper considers the problem of implementing large-scale gradient descent algorithms in a distributed computing setting in the presence of {\em straggling} processors. To mitigate the effect of the stragglers, it has been previously proposed to encode the data with an erasure-correcting code and decode at the master server at the end of the computation. We, instead, propose to encode the second-moment of the data with a low density parity-check (LDPC) code. The iterative decoding algorithms for LDPC codes have very low computational overhead and the number of decoding iterations can be made to automatically adjust with the number of stragglers in the system. We show that for a random model for stragglers, the proposed moment encoding based gradient descent method can be viewed as the stochastic gradient descent method. This allows us to obtain convergence guarantees for the proposed solution. Furthermore, the proposed moment encoding based method is shown to outperform the existing schemes in a real distributed computing setup.

In this paper, we combine task-dependent reward shaping and task-independent proto-value functions to obtain reward dependent proto-value functions (RPVFs). In constructing the RPVFs we are making use of the immediate rewards which are available during the sampling phase but are not used in the PVF construction. We show via experiments that learning with an RPVF based representation is better than learning with just reward shaping or PVFs. In particular, when the state space is symmetrical and the rewards are asymmetrical, the RPVF capture the asymmetry better than the PVFs.