Multitask clustering tries to improve the clustering performance of multiple tasks simultaneously by taking their relationship into account. Most existing multitask clustering algorithms fall into the type of generative clustering, and none are formulated as convex optimization problems. In this paper, we propose two convex Discriminative Multitask Clustering (DMTC) algorithms to address the problems. Specifically, we first propose a Bayesian DMTC framework. Then, we propose two convex DMTC objectives within the framework. The first one, which can be seen as a technical combination of the convex multitask feature learning and the convex Multiclass Maximum Margin Clustering (M3C), aims to learn a shared feature representation. The second one, which can be seen as a combination of the convex multitask relationship learning and M3C, aims to learn the task relationship. The two objectives are solved in a uniform procedure by the efficient cutting-plane algorithm. Experimental results on a toy problem and two benchmark datasets demonstrate the effectiveness of the proposed algorithms.
In @cite_12 , Argyriou proposed to minimize the empirical risk of all tasks with a Frobenius norm penalty on the differences of the task-specific models, which is a non-convex optimization problem. Then, they proved that the problem is equivalent to a convex optimization problem -- Multitask Feature Learning (MTFL). In @cite_45 Best Paper Award of , Zhang and Yeung first tried to learn the task covariance matrix of the multivariate Gaussian prior in the regularization framework. Because the concave function with respect to the covariance matrix variable makes the objective non-convex, they further replaced the concave function by two convex constraints, which results in a convex optimization problem, named MTRL.
@cite_45 @cite_12
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We present a convex framework to learn sequential decisions and apply it to the problem of learning under a budget. We consider the structure proposed in [1], where sensor measurements are acquired in a sequence. The goal after acquiring each new measurement is to make a decision whether to stop and classify or to pay the cost of using the next sensor in the sequence. We introduce a novel formulation of an empirical risk objective for the multi stage sequential decision problem. This objective naturally lends itself to a non-convex multilinear formulation. Nevertheless, we derive a novel perspective that leads to a tight convex objective. This is accomplished by expressing the empirical risk in terms of linear superposition of indicator functions. We then derive an LP formulation by utilizing hinge loss surrogates. Our LP achieves or exceeds the empirical performance of the nonconvex alternating algorithm that requires a large number of random initializations. Consequently, the LP has the advantage of guaranteed convergence, global optimality, repeatability and computation eciency. We consider the problem of multi-task reinforcement learning (MTRL) in multiple partially observable stochastic environments. We introduce the regionalized policy representation (RPR) to characterize the agent's behavior in each environment. The RPR is a parametric model of the conditional distribution over current actions given the history of past actions and observations; the agent's choice of actions is directly based on this conditional distribution, without an intervening model to characterize the environment itself. We propose off-policy batch algorithms to learn the parameters of the RPRs, using episodic data collected when following a behavior policy, and show their linkage to policy iteration. We employ the Dirichlet process as a nonparametric prior over the RPRs across multiple environments. The intrinsic clustering property of the Dirichlet process imposes sharing of episodes among similar environments, which effectively reduces the number of episodes required for learning a good policy in each environment, when data sharing is appropriate. The number of distinct RPRs and the associated clusters (the sharing patterns) are automatically discovered by exploiting the episodic data as well as the nonparametric nature of the Dirichlet process. We demonstrate the effectiveness of the proposed RPR as well as the RPR-based MTRL framework on various problems, including grid-world navigation and multi-aspect target classification. The experimental results show that the RPR is a competitive reinforcement learning algorithm in partially observable domains, and the MTRL consistently achieves better performance than single task reinforcement learning. In multi-task learning several related tasks are considered simultaneously, with the hope that by an appropriate sharing of information across tasks, each task may benefit from the others. In the context of learning linear functions for supervised classification or regression, this can be achieved by including a priori information about the weight vectors associated with the tasks, and how they are expected to be related to each other. In this paper, we assume that tasks are clustered into groups, which are unknown beforehand, and that tasks within a group have similar weight vectors. We design a new spectral norm that encodes this a priori assumption, without the prior knowledge of the partition of tasks into groups, resulting in a new convex optimization formulation for multi-task learning. We show in simulations on synthetic examples and on the IEDB MHC-I binding dataset, that our approach outperforms well-known convex methods for multi-task learning, as well as related non convex methods dedicated to the same problem. We present a framework based on convex optimization and spectral regularization to perform learning when feature observations are multidimensional arrays (tensors). We give a mathematical characterization of spectral penalties for tensors and analyze a unifying class of convex optimization problems for which we present a provably convergent and scalable template algorithm. We then specialize this class of problems to perform learning both in a transductive as well as in an inductive setting. In the transductive case one has an input data tensor with missing features and, possibly, a partially observed matrix of labels. The goal is to both infer the missing input features as well as predict the missing labels. For induction, the goal is to determine a model for each learning task to be used for out of sample prediction. Each training pair consists of a multidimensional array and a set of labels each of which corresponding to related but distinct tasks. In either case the proposed technique exploits precise low multilinear rank assumptions over unknown multidimensional arrays; regularization is based on composite spectral penalties and connects to the concept of Multilinear Singular Value Decomposition. As a by-product of using a tensor-based formalism, our approach allows one to tackle the multi-task case in a natural way. Empirical studies demonstrate the merits of the proposed methods.