Machine learning pipelines often rely on optimization procedures to make discrete decisions (e.g. sorting, picking closest neighbors, finding shortest paths or optimal matchings). Although these discrete decisions are easily computed in a forward manner, they cannot be used to modify model parameters using first-order optimization techniques because they break the back-propagation of computational graphs. In order to expand the scope of learning problems that can be solved in an end-to-end fashion, we propose a systematic method to transform a block that outputs an optimal discrete decision into a differentiable operation. Our approach relies on stochastic perturbations of these parameters, and can be used readily within existing solvers without the need for ad hoc regularization or smoothing. These perturbed optimizers yield solutions that are differentiable and never locally constant. The amount of smoothness can be tuned via the chosen noise amplitude, whose impact we analyze. The derivatives of these perturbed solvers can be evaluated efficiently. We also show how this framework can be connected to a family of losses developed in structured prediction, and describe how these can be used in unsupervised and supervised learning, with theoretical guarantees. We demonstrate the performance of our approach on several machine learning tasks in experiments on synthetic and real data.

We address the problem of training models with black-box and hard-to-optimize metrics by expressing the metric as a monotonic function of a small number of easy-to-optimize surrogates. We pose the training problem as an optimization over a relaxed surrogate space, which we solve by estimating local gradients for the metric and performing inexact convex projections. We analyze gradient estimates based on finite differences and local linear interpolations, and show convergence of our approach under smoothness assumptions with respect to the surrogates. Experimental results on classification and ranking problems verify the proposal performs on par with methods that know the mathematical formulation, and adds notable value when the form of the metric is unknown.

We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of machines typically yields poor performance. It often leads to the dependence of convergence rate on maximum Lipschitz constant of gradients across the devices. In this paper, we propose a novel \emph{adaptive} sampling of machines specially catered to these settings. Our method relies on an adaptive estimate of local Lipschitz constants base on the information of past gradients. We show that the new way improves the dependence of convergence rate from maximum Lipschitz constant to \emph{average} Lipschitz constant across machines, thereby, significantly accelerating the convergence. Our experiments demonstrate that our method indeed speeds up the convergence of the standard SVRG algorithm in heterogeneous environments.

Gaussian mixture reduction (GMR) is the problem of approximating a finite Gaussian mixture by one with fewer components. It is widely used in density estimation, nonparametric belief propagation, and Bayesian recursive filtering. Although optimization and clustering-based algorithms have been proposed for GMR, they are either computationally expensive or lacking in theoretical supports. In this work, we propose to perform GMR by minimizing the entropic regularized composite transportation distance between two mixtures. We show our approach provides a unified framework for GMR that is both interpretable and computationally efficient. Our work also bridges the gap between optimization and clustering-based approaches for GMR. A Majorization-Minimization algorithm is developed for our optimization problem and its theoretical convergence is also established in this paper. Empirical experiments are also conducted to show the effectiveness of GMR. The effect of the choice of transportation cost on the performance of GMR is also investigated.

In this paper, we provide a unified convergence analysis for a class of shuffling-type gradient methods for solving a well-known finite-sum minimization problem commonly used in machine learning. This algorithm covers various variants such as randomized reshuffling, single shuffling, and cyclic/incremental gradient schemes. We consider two different settings: strongly convex and non-convex problems. Our main contribution consists of new non-asymptotic and asymptotic convergence rates for a general class of shuffling-type gradient methods to solve both non-convex and strongly convex problems. While our rate in the non-convex problem is new (i.e. not known yet under standard assumptions), the rate on the strongly convex case matches (up to a constant) the best-known results. However, unlike existing works in this direction, we only use standard assumptions such as smoothness and strong convexity. Finally, we empirically illustrate the effect of learning rates via a non-convex logistic regression and neural network examples.

Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. Yet, so far, such methods have been mostly analyzed from an optimization perspective, without addressing the problem of exploration, or by making strong assumptions on the interaction with the environment. In this paper we consider model-based RL in the tabular finite-horizon MDP setting with unknown transitions and bandit feedback. For this setting, we propose an optimistic trust region policy optimization (TRPO) algorithm for which we establish $\tilde O(\sqrt{S^2 A H^4 K})$ regret for stochastic rewards. Furthermore, we prove $\tilde O( \sqrt{ S^2 A H^4 } K^{2/3} ) $ regret for adversarial rewards. Interestingly, this result matches previous bounds derived for the bandit feedback case, yet with known transitions. To the best of our knowledge, the two results are the first sub-linear regret bounds obtained for policy optimization algorithms with unknown transitions and bandit feedback.

Partial-label learning is one of the important weakly supervised learning problems, where each training example is equipped with a set of candidate labels that contains the true label. Most existing methods elaborately designed learning objectives as constrained optimizations that must be solved in specific manners, making their computational complexity a bottleneck for scaling up to big data. The goal of this paper is to propose a novel framework of partial-label learning without implicit assumptions on the model or optimization algorithm. More specifically, we propose a general estimator of the classification risk, theoretically analyze the classifier-consistency, and establish an estimation error bound. We then explore a progressive identification method for approximately minimizing the proposed risk estimator, where the update of the model and identification of true labels are conducted in a seamless manner. The resulting algorithm is model-independent and loss-independent, and compatible with stochastic optimization. Thorough experiments demonstrate it sets the new state of the art.

Most product-development tasks are complex optimization problems. Design teams approach them iteratively, refining an initial best guess through rounds of engineering analysis, interpretation, and refinement. But each such iteration takes time and money, and teams may achieve only a handful of iterations within the development timeline. Because teams rarely have the opportunity to explore alternative solutions that depart significantly from their base-case assumptions, too often the final design is suboptimal. Today's technology offers an alternative.

We consider an extension of the rollout algorithm that applies to constrained deterministic dynamic programming, including challenging combinatorial optimization problems. The algorithm relies on a suboptimal policy, called base heuristic. Under suitable assumptions, we show that if the base heuristic produces a feasible solution, the rollout algorithm has a cost improvement property: it produces a feasible solution, whose cost is no worse than the base heuristic's cost. We then focus on multiagent problems, where the control at each stage consists of multiple components (one per agent), which are coupled either through the cost function or the constraints or both. We show that the cost improvement property is maintained with an alternative implementation that has greatly reduced computational requirements, and makes possible the use of rollout in problems with many agents. We demonstrate this alternative algorithm by applying it to layered graph problems that involve both a spatial and a temporal structure. We consider in some detail a prominent example of such problems: multidimensional assignment, where we use the auction algorithm for 2-dimensional assignment as a base heuristic. This auction algorithm is particularly well-suited for our context, because through the use of prices, it can advantageously use the solution of an assignment problem as a starting point for solving other related assignment problems, and this can greatly speed up the execution of the rollout algorithm.

We consider practical data characteristics underlying federated learning, where unbalanced and non-i.i.d. data from clients have a block-cyclic structure: each cycle contains several blocks, and each client's training data follow block-specific and non-i.i.d. distributions. Such a data structure would introduce client and block biases during the collaborative training: the single global model would be biased towards the client or block specific data. To overcome the biases, we propose two new distributed optimization algorithms called multi-model parallel SGD (MM-PSGD) and multi-chain parallel SGD (MC-PSGD) with a convergence rate of $O(1/\sqrt{NT})$, achieving a linear speedup with respect to the total number of clients. In particular, MM-PSGD adopts the block-mixed training strategy, while MC-PSGD further adds the block-separate training strategy. Both algorithms create a specific predictor for each block by averaging and comparing the historical global models generated in this block from different cycles. We extensively evaluate our algorithms over the CIFAR-10 dataset. Evaluation results demonstrate that our algorithms significantly outperform the conventional federated averaging algorithm in terms of test accuracy, and also preserve robustness for the variance of critical parameters.