Multi-Task Feature Learning Via Efficient l2,1-Norm Minimization

arXiv.org Machine Learning

The problem of joint feature selection across a group of related tasks has applications in many areas including biomedical informatics and computer vision. We consider the l2,1-norm regularized regression model for joint feature selection from multiple tasks, which can be derived in the probabilistic framework by assuming a suitable prior from the exponential family. One appealing feature of the l2,1-norm regularization is that it encourages multiple predictors to share similar sparsity patterns. However, the resulting optimization problem is challenging to solve due to the non-smoothness of the l2,1-norm regularization. In this paper, we propose to accelerate the computation by reformulating it as two equivalent smooth convex optimization problems which are then solved via the Nesterov's method-an optimal first-order black-box method for smooth convex optimization. A key building block in solving the reformulations is the Euclidean projection. We show that the Euclidean projection for the first reformulation can be analytically computed, while the Euclidean projection for the second one can be computed in linear time. Empirical evaluations on several data sets verify the efficiency of the proposed algorithms.


Differentiable Convex Optimization Layers

arXiv.org Machine Learning

Recent work has shown how to embed differentiable optimization problems (that is, problems whose solutions can be backpropagated through) as layers within deep learning architectures. This method provides a useful inductive bias for certain problems, but existing software for differentiable optimization layers is rigid and difficult to apply to new settings. In this paper, we propose an approach to differentiating through disciplined convex programs, a subclass of convex optimization problems used by domain-specific languages (DSLs) for convex optimization. We introduce disciplined parametrized programming, a subset of disciplined convex programming, and we show that every disciplined parametrized program can be represented as the composition of an affine map from parameters to problem data, a solver, and an affine map from the solver's solution to a solution of the original problem (a new form we refer to as affine-solver-affine form). We then demonstrate how to efficiently differentiate through each of these components, allowing for end-to-end analytical differentiation through the entire convex program. We implement our methodology in version 1.1 of CVXPY, a popular Python-embedded DSL for convex optimization, and additionally implement differentiable layers for disciplined convex programs in PyTorch and TensorFlow 2.0. Our implementation significantly lowers the barrier to using convex optimization problems in differentiable programs. We present applications in linear machine learning models and in stochastic control, and we show that our layer is competitive (in execution time) compared to specialized differentiable solvers from past work.


Differentiable Convex Optimization Layers

Neural Information Processing Systems

Recent work has shown how to embed differentiable optimization problems (that is, problems whose solutions can be backpropagated through) as layers within deep learning architectures. This method provides a useful inductive bias for certain problems, but existing software for differentiable optimization layers is rigid and difficult to apply to new settings. In this paper, we propose an approach to differentiating through disciplined convex programs, a subclass of convex optimization problems used by domain-specific languages (DSLs) for convex optimization. We introduce disciplined parametrized programming, a subset of disciplined convex programming, and we show that every disciplined parametrized program can be represented as the composition of an affine map from parameters to problem data, a solver, and an affine map from the solver's solution to a solution of the original problem (a new form we refer to as affine-solver-affine form). We then demonstrate how to efficiently differentiate through each of these components, allowing for end-to-end analytical differentiation through the entire convex program.


Convex Techniques for Model Selection

arXiv.org Machine Learning

We develop a robust convex algorithm to select the regularization parameter in model selection. In practice this would be automated in order to save practitioners time from having to tune it manually. In particular, we implement and test the convex method for $K$-fold cross validation on ridge regression, although the same concept extends to more complex models. We then compare its performance with standard methods.


Projection-Free Bandit Convex Optimization

arXiv.org Machine Learning

In this paper, we propose the first computationally efficient projection-free algorithm for the bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $ O(nT^{4/5}) $ (where $ T $ is the horizon and $ n $ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against prior art on both synthetic and real data sets for portfolio selection and multiclass classification problems.