No Arabic abstract
In recent years, federated learning (FL) has been widely applied for supporting decentralized collaborative learning scenarios. Among existing FL models, federated logistic regression (FLR) is a widely used statistic model and has been used in various industries. To ensure data security and user privacy, FLR leverages homomorphic encryption (HE) to protect the exchanged data among different collaborative parties. However, HE introduces significant computational overhead (i.e., the cost of data encryption/decryption and calculation over encrypted data), which eventually becomes the performance bottleneck of the whole system. In this paper, we propose HAFLO, a GPU-based solution to improve the performance of FLR. The core idea of HAFLO is to summarize a set of performance-critical homomorphic operators (HO) used by FLR and accelerate the execution of these operators through a joint optimization of storage, IO, and computation. The preliminary results show that our acceleration on FATE, a popular FL framework, achieves a 49.9$times$ speedup for heterogeneous LR and 88.4$times$ for homogeneous LR.
Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logisitc bandits that are used in problems with binary rewards: for instance, when the learner/agent tries to maximize the profit over a user that can select one of two possible outcomes (e.g., `click vs `no-click). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., `click, `show me later, `never show again, `no click). In this paper, we study such an extension. We use multinomial logit (MNL) to model the probability of each one of $K+1geq 2$ possible outcomes (+1 stands for the `not click outcome): we assume that for a learners action $mathbf{x}_t$, the user selects one of $K+1geq 2$ outcomes, say outcome $i$, with a multinomial logit (MNL) probabilistic model with corresponding unknown parameter $bar{boldsymboltheta}_{ast i}$. Each outcome $i$ is also associated with a revenue parameter $rho_i$ and the goal is to maximize the expected revenue. For this problem, we present MNL-UCB, an upper confidence bound (UCB)-based algorithm, that achieves regret $tilde{mathcal{O}}(dKsqrt{T})$ with small dependency on problem-dependent constants that can otherwise be arbitrarily large and lead to loose regret bounds. We present numerical simulations that corroborate our theoretical results.
We consider the setting of online logistic regression and consider the regret with respect to the 2-ball of radius B. It is known (see [Hazan et al., 2014]) that any proper algorithm which has logarithmic regret in the number of samples (denoted n) necessarily suffers an exponential multiplicative constant in B. In this work, we design an efficient improper algorithm that avoids this exponential constant while preserving a logarithmic regret. Indeed, [Foster et al., 2018] showed that the lower bound does not apply to improper algorithms and proposed a strategy based on exponential weights with prohibitive computational complexity. Our new algorithm based on regularized empirical risk minimization with surrogate losses satisfies a regret scaling as O(B log(Bn)) with a per-round time-complexity of order O(d^2).
We present ADMM-Softmax, an alternating direction method of multipliers (ADMM) for solving multinomial logistic regression (MLR) problems. Our method is geared toward supervised classification tasks with many examples and features. It decouples the nonlinear optimization problem in MLR into three steps that can be solved efficiently. In particular, each iteration of ADMM-Softmax consists of a linear least-squares problem, a set of independent small-scale smooth, convex problems, and a trivial dual variable update. Solution of the least-squares problem can be be accelerated by pre-computing a factorization or preconditioner, and the separability in the smooth, convex problem can be easily parallelized across examples. For two image classification problems, we demonstrate that ADMM-Softmax leads to improved generalization compared to a Newton-Krylov, a quasi Newton, and a stochastic gradient descent method.
Coresets are one of the central methods to facilitate the analysis of large data sets. We continue a recent line of research applying the theory of coresets to logistic regression. First, we show a negative result, namely, that no strongly sublinear sized coresets exist for logistic regression. To deal with intractable worst-case instances we introduce a complexity measure $mu(X)$, which quantifies the hardness of compressing a data set for logistic regression. $mu(X)$ has an intuitive statistical interpretation that may be of independent interest. For data sets with bounded $mu(X)$-complexity, we show that a novel sensitivity sampling scheme produces the first provably sublinear $(1pmvarepsilon)$-coreset. We illustrate the performance of our method by comparing to uniform sampling as well as to state of the art methods in the area. The experiments are conducted on real world benchmark data for logistic regression.
In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $ell_0$ pseudo norm is able to better induce sparsity than the commonly used $ell_1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding sparse logistic regression problem, and study its local optimality conditions and the choice of the regularization parameter to exclude trivial solutions. Despite the nonconvexity, a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression, and its convergence behavior is studied theoretically. Then the general framework is applied to a specific weakly convex function, and a necessary and sufficient local optimality condition is provided. The solution method is instantiated in this case as an iterative firm-shrinkage algorithm, and its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets.