Given a set of observations generated by an optimization process, the goal of inverse optimization is to determine likely parameters of that process. We cast inverse optimization as a form of deep learning. Our method, called deep inverse optimization, is to unroll an iterative optimization process and then use backpropagation to learn parameters that generate the observations. We demonstrate that by backpropagating through the interior point algorithm we can learn the coefficients determining the cost vector and the constraints, independently or jointly, for both non-parametric and parametric linear programs, starting from one or multiple observations. With this approach, inverse optimization can leverage concepts and algorithms from deep learning.
Recent work in machine learning @cite_27 @cite_23 @cite_33 views inverse optimization through the lens of online learning, where new observations appear over time rather than as one batch. Our approach may be applicable in online settings, but we focus on generality in the batch setting and do not investigate real-time cases.
@cite_27 @cite_33 @cite_23
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Inverse optimization is a powerful paradigm for learning preferences and restrictions that explain the behavior of a decision maker, based on a set of external signal and the corresponding decision pairs. However, most inverse optimization algorithms are designed specifically in batch setting, where all the data is available in advance. As a consequence, there has been rare use of these methods in an online setting suitable for real-time applications. In this paper, we propose a general framework for inverse optimization through online learning. Specifically, we develop an online learning algorithm that uses an implicit update rule which can handle noisy data. Moreover, under additional regularity assumptions in terms of the data and the model, we prove that our algorithm converges at a rate of @math and is statistically consistent. In our experiments, we show the online learning approach can learn the parameters with great accuracy and is very robust to noises, and achieves a dramatic improvement in computational efficacy over the batch learning approach. While deep learning methods have achieved state-of-theart performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach, each inverse problem requires its own dedicated network. In scenarios where we need to solve a wide variety of problems, e.g., on a mobile camera, it is inefficient and expensive to use these problem-specific networks. On the other hand, traditional methods using analytic signal priors can be used to solve any linear inverse problem; this often comes with a performance that is worse than learning-based methods. In this work, we provide a middle ground between the two kinds of methods — we propose a general framework to train a single deep neural network that solves arbitrary linear inverse problems. We achieve this by training a network that acts as a quasi-projection operator for the set of natural images and show that any linear inverse problem involving natural images can be solved using iterative methods. We empirically show that the proposed framework demonstrates superior performance over traditional methods using wavelet sparsity prior while achieving performance comparable to specially-trained networks on tasks including compressive sensing and pixel-wise inpainting. While deep learning methods have achieved state-of-the-art performance in many challenging inverse problems like image inpainting and super-resolution, they invariably involve problem-specific training of the networks. Under this approach, different problems require different networks. In scenarios where we need to solve a wide variety of problems, e.g., on a mobile camera, it is inefficient and costly to use these specially-trained networks. On the other hand, traditional methods using signal priors can be used in all linear inverse problems but often have worse performance on challenging tasks. In this work, we provide a middle ground between the two kinds of methods --- we propose a general framework to train a single deep neural network that solves arbitrary linear inverse problems. The proposed network acts as a proximal operator for an optimization algorithm and projects non-image signals onto the set of natural images defined by the decision boundary of a classifier. In our experiments, the proposed framework demonstrates superior performance over traditional methods using a wavelet sparsity prior and achieves comparable performance of specially-trained networks on tasks including compressive sensing and pixel-wise inpainting. While classic machine learning paradigms assume training and test data are generated from the same process, domain adaptation addresses the more realistic setting in which the learner has large quantities of labeled data from some source task but limited or no labeled data from the target task it is attempting to learn. In this work, we give the first formal analysis showing that using active learning for domain adaptation yields a way to address the statistical challenges inherent in this setting. We propose a novel nonparametric algorithm, ANDA, that combines an active nearest neighbor querying strategy with nearest neighbor prediction. We provide analyses of its querying behavior and of finite sample convergence rates of the resulting classifier under covariate shift. Our experiments show that ANDA successfully corrects for dataset bias in multiclass image categorization. Recent work has shown that machine learning can automate and in some cases outperform hand crafted compiler optimizations. Central to such an approach is that machine learning techniques typically rely upon summaries or features of the program. The quality of these features is critical to the accuracy of the resulting machine learned algorithm; no machine learning method will work well with poorly chosen features. However, due to the size and complexity of programs, theoretically there are an infinite number of potential features to choose from. The compiler writer now has to expend effort in choosing the best features from this space. This paper develops a novel mechanism to automatically find those features which most improve the quality of the machine learned heuristic. The feature space is described by a grammar and is then searched with genetic programming and predictive modeling. We apply this technique to loop unrolling in GCC 4.3.1 and evaluate our approach on a Pentium 6. On a benchmark suite of 57 programs, GCC's hard-coded heuristic achieves only 3 of the maximum performance available, while a state of the art machine learning approach with hand-coded features obtains 59 . Our feature generation technique is able to achieve 76 of the maximum available speedup, outperforming existing approaches. Common assumption in most machine learning algorithms is that, labeled (source) data and unlabeled (target) data are sampled from the same distribution. However, many real world tasks violate this assumption: in temporal domains, feature distributions may vary over time, clinical studies may have sampling bias, or sometimes sufficient labeled data for the domain of interest does not exist, and labeled data from a related domain must be utilized. In such settings, knowing in which dimensions source and target data vary is extremely important to reduce the distance between domains and accurately transfer knowledge. In this paper, we present a novel method to identify variant and invariant features between two datasets. Our contribution is two fold: First, we present a novel transfer learning approach for domain adaptation, and second, we formalize the problem of finding differently distributed features as a convex optimization problem. Experimental studies on synthetic and benchmark real world datasets show that our approach outperform other transfer learning approaches, and it aids the prediction accuracy significantly. Training deep neural networks with Stochastic Gradient Descent, or its variants, requires careful choice of both learning rate and batch size. While smaller batch sizes generally converge in fewer training epochs, larger batch sizes offer more parallelism and hence better computational efficiency. We have developed a new training approach that, rather than statically choosing a single batch size for all epochs, adaptively increases the batch size during the training process. Our method delivers the convergence rate of small batch sizes while achieving performance similar to large batch sizes. We analyse our approach using the standard AlexNet, ResNet, and VGG networks operating on the popular CIFAR-10, CIFAR-100, and ImageNet datasets. Our results demonstrate that learning with adaptive batch sizes can improve performance by factors of up to 6.25 on 4 NVIDIA Tesla P100 GPUs while changing accuracy by less than 1 relative to training with fixed batch sizes. This paper tackles two related questions at the heart of machine learning; how can we predict if a minimum will generalize to the test set, and why does stochastic gradient descent find minima that generalize well? Our work is inspired by (2017), who showed deep networks can easily memorize randomly labeled training data, despite generalizing well when shown real labels of the same inputs. We show here that the same phenomenon occurs in small linear models. These observations are explained by evaluating the Bayesian evidence, which penalizes sharp minima but is invariant to model parameterization. We also explore the "generalization gap" between small and large batch training, identifying an optimum batch size which maximizes the test set accuracy. Interpreting stochastic gradient descent as a stochastic differential equation, we identify a "noise scale" @math , where @math is the learning rate, @math training set size and @math batch size. Consequently the optimum batch size is proportional to the learning rate and the training set size, @math . We verify these predictions empirically. Active learning can lead to a dramatic reduction in labeling effort. However, in many practical implementations (such as crowdsourcing, surveys, high-throughput experimental design), it is preferable to query labels for batches of examples to be labelled in parallel. While several heuristics have been proposed for batch-mode active learning, little is known about their theoretical performance. We consider batch mode active learning and more general information-parallel stochastic optimization problems that exhibit adaptive submodularity, a natural diminishing returns condition. We prove that for such problems, a simple greedy strategy is competitive with the optimal batch-mode policy. In some cases, surprisingly, the use of batches incurs competitively low cost, even when compared to a fully sequential strategy. We demonstrate the effectiveness of our approach on batch-mode active learning tasks, where it outperforms the state of the art, as well as the novel problem of multi-stage influence maximization in social networks. In this paper, we address the problem of incrementally optimizing constraint networks for maximum likelihood map learning. Our approach allows a robot to efficiently compute configurations of the network with small errors while the robot moves through the environment. We apply a variant of stochastic gradient descent and use a tree-based parameterization of the nodes in the network. By integrating adaptive learning rates in the parameterization of the network, our algorithm can use previously computed solutions to determine the result of the next optimization run. Additionally, our approach updates only the parts of the network which are affected by the newly incorporated measurements and starts the optimization approach only if the new data reveals inconsistencies with the network constructed so far. These improvements yield an efficient solution for this class of online optimization problems. Our approach has been implemented and tested on simulated and on real data. We present comparisons to recently proposed online and offline methods that address the problem of optimizing constraint network. Experiments illustrate that our approach converges faster to a network configuration with small errors than the previous approaches. We introduce a method to infer a variational approximation to the posterior distribution of solutions in computational imaging inverse problems. Machine learning methods applied to computational imaging have proven very successful, but have so far largely focused on retrieving a single optimal solution for a given task. Such retrieval is arguably an incomplete description of the solution space, as in ill-posed inverse problems there may be many similarly likely reconstructions. We minimise an upper bound on the divergence between our approximate distribution and the true intractable posterior, thereby obtaining a probabilistic description of the solution space in imaging inverse problems with empirical prior. We demonstrate the advantage of our technique in quantitative simulations with the CelebA dataset and common image reconstruction tasks. We then apply our method to two of the currently most challenging problems in experimental optics: imaging through highly scattering media and imaging through multi-modal optical fibres. In both settings we report state of the art reconstructions, while providing new capabilities, such as estimation of error-bars and visualisation of multiple likely reconstructions. A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) in which the constraint matrix is revealed column by column along with the corresponding objective coefficient. In such a model, a decision variable has to be set each time a column is revealed without observing the future inputs and the goal is to maximize the overall objective function. In this paper, we provide a near-optimal algorithm for this general class of online problems under the assumption of random order of arrival and some mild conditions on the size of the LP right-hand-side input. Specifically, our learning-based algorithm works by dynamically updating a threshold price vector at geometric time intervals, where the dual prices learned from the revealed columns in the previous period are used to determine the sequential decisions in the current period. Due to the feature of dynamic learning, the competitiveness of our algorithm improves over the past study of the same problem. We also present a worst-case example showing that the performance of our algorithm is near-optimal. Learning in real-time applications, e.g., online approximation of the inverse dynamics model for model-based robot control, requires fast online regression techniques. Inspired by local learning, we propose a method to speed up standard Gaussian Process regression (GPR) with local GP models (LGP). The training data is partitioned in local regions, for each an individual GP model is trained. The prediction for a query point is performed by weighted estimation using nearby local models. Unlike other GP approximations, such as mixtures of experts, we use a distance based measure for partitioning of the data and weighted prediction. The proposed method achieves online learning and prediction in real-time. Comparisons with other nonparametric regression methods show that LGP has higher accuracy than LWPR and close to the performance of standard GPR and nu-SVR. We examine the interplay of optimization and machine learning. Great progress has been made in machine learning by cleverly reducing machine learning problems to convex optimization problems with one or more hyper-parameters. The availability of powerful convex-programming theory and algorithms has enabled a flood of new research in machine learning models and methods. But many of the steps necessary for successful machine learning models fall outside of the convex machine learning paradigm. Thus we now propose framing machine learning problems as Stackelberg games. The resulting bilevel optimization problem allows for efficient systematic search of large numbers of hyper-parameters. We discuss recent progress in solving these bilevel problems and the many interesting optimization challenges that remain. Finally, we investigate the intriguing possibility of novel machine learning models enabled by bilevel programming. Learning of inverse dynamics modeling errors is key for compliant or force control when analytical models are only rough approximations. Thus, designing real time capable function approximation algorithms has been a necessary focus towards the goal of online model learning. However, because these approaches learn a mapping from actual state and acceleration to torque, good tracking is required to observe data points on the desired path. Recently it has been shown how online gradient descent on a simple modeling error offset term to minimize tracking at acceleration level can address this issue. However, to adapt to larger errors a high learning rate of the online learner is required, resulting in reduced compliancy. Thus, here we propose to combine both approaches: The online adapted offset term ensures good tracking such that a nonlinear function approximator is able to learn an error model on the desired trajectory. This, in turn, reduces the load on the adaptive feedback, enabling it to use a lower learning rate. Combined this creates a controller with variable feedback and low gains, and a feedforward model that can account for larger modeling errors. We demonstrate the effectiveness of this framework, in simulation and on a real system. Since performance is not portable between platforms, engineers must fine-tune heuristics for each processor in turn. This is such a laborious task that high-profile compilers, supporting many architectures, cannot keep up with hardware innovation and are actually out-of-date. Iterative compilation driven by machine learning has been shown to be efficient at generating portable optimization models automatically. However, good quality models require costly, repetitive, and extensive training which greatly hinders the wide adoption of this powerful technique. In this work, we show that much of this cost is spent collecting training data, runtime measurements for different optimization decisions, which contribute little to the final heuristic. Current implementations evaluate randomly chosen, often redundant, training examples a pre-configured, almost always excessive, number of times – a large source of wasted effort. Our approach optimizes not only the selection of training examples but also the number of samples per example, independently. To evaluate, we construct 11 high-quality models which use a combination of optimization settings to predict the runtime of benchmarks from the SPAPT suite. Our novel, broadly applicable, methodology is able to reduce the training overhead by up to 26x compared to an approach with a fixed number of sample runs, transforming what is potentially months of work into days. Online optimization has emerged as powerful tool in large scale optimization. In this paper, we introduce efficient online algorithms based on the alternating directions method (ADM). We introduce a new proof technique for ADM in the batch setting, which yields the O(1 T) convergence rate of ADM and forms the basis of regret analysis in the online setting. We consider two scenarios in the online setting, based on whether the solution needs to lie in the feasible set or not. In both settings, we establish regret bounds for both the objective function as well as constraint violation for general and strongly convex functions. Preliminary results are presented to illustrate the performance of the proposed algorithms. We explore the problem of assigning heterogeneous tasks to workers with different, unknown skill sets in crowdsourcing markets such as Amazon Mechanical Turk. We first formalize the online task assignment problem, in which a requester has a fixed set of tasks and a budget that specifies how many times he would like each task completed. Workers arrive one at a time (with the same worker potentially arriving multiple times), and must be assigned to a task upon arrival. The goal is to allocate workers to tasks in a way that maximizes the total benefit that the requester obtains from the completed work. Inspired by recent research on the online adwords problem, we present a two-phase exploration-exploitation assignment algorithm and prove that it is competitive with respect to the optimal offline algorithm which has access to the unknown skill levels of each worker. We empirically evaluate this algorithm using data collected on Mechanical Turk and show that it performs better than random assignment or greedy algorithms. To our knowledge, this is the first work to extend the online primal-dual technique used in the online adwords problem to a scenario with unknown parameters, and the first to offer an empirical validation of an online primal-dual algorithm. Learning in real-time applications, e.g., online approximation of the inverse dynamics model for model-based robot control, requires fast online regression techniques. Inspired by local learning, we propose a method to speed up standard Gaussian process regression (GPR) with local GP models (LGP). The training data is partitioned in local regions, for each an individual GP model is trained. The prediction for a query point is performed by weighted estimation using nearby local models. Unlike other GP approximations, such as mixtures of experts, we use a distance based measure for partitioning of the data and weighted prediction. The proposed method achieves online learning and prediction in real-time. Comparisons with other non-parametric regression methods show that LGP has higher accuracy than LWPR and close to the performance of standard GPR and ν-SVR. We study few-shot learning in natural language domains. Compared to many existing works that apply either metric-based or optimization-based meta-learning to image domain with low inter-task variance, we consider a more realistic setting, where tasks are diverse. However, it imposes tremendous difficulties to existing state-of-the-art metric-based algorithms since a single metric is insufficient to capture complex task variations in natural language domain. To alleviate the problem, we propose an adaptive metric learning approach that automatically determines the best weighted combination from a set of metrics obtained from meta-training tasks for a newly seen few-shot task. Extensive quantitative evaluations on real-world sentiment analysis and dialog intent classification datasets demonstrate that the proposed method performs favorably against state-of-the-art few shot learning algorithms in terms of predictive accuracy. We make our code and data available for further study.