Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Ordinal regression seeks class label predictions when the penalty incurred for mistakes increases according to an ordering over the labels. The absolute error is a canonical example. Many existing methods for this task reduce to binary classification problems and employ surrogate losses, such as the hinge loss. We instead derive uniquely defined surrogate ordinal regression loss functions by seeking the predictor that is robust to the worst-case approximations of training data labels, subject to matching certain provided training data statistics. We demonstrate the advantages of our approach over other surrogate losses based on hinge loss approximations using UCI ordinal prediction tasks.

In this paper we consider the problem of minimizing a submodular function from training data. Submodular functions can be efficiently minimized and are consequently heavily applied in machine learning. There are many cases, however, in which we do not know the function we aim to optimize, but rather have access to training data that is used to learn the function. In this paper we consider the question of whether submodular functions can be minimized in such cases. We show that even learnable submodular functions cannot be minimized within any non-trivial approximation when given access to polynomially-many samples. Specifically, we show that there is a class of submodular functions with range in [0, 1] such that, despite being PAC-learnable and minimizable in polynomial-time, no algorithm can obtain an approximation strictly better than 1/2 o(1) using polynomially-many samples drawn from any distribution. Furthermore, we show that this bound is tight using a trivial algorithm that obtains an approximation of 1/2.

For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an n by n positive definite matrix, and its derivatives---leading to prohibitive O(n^3) computations. We propose novel O(n) approaches to estimating these quantities from only fast matrix vector multiplications (MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and surrogate models, and converge quickly even for kernel matrices that have challenging spectra. We leverage these approximations to develop a scalable Gaussian process approach to kernel learning. We find that Lanczos is generally superior to Chebyshev for kernel learning, and that a surrogate approach can be highly efficient and accurate with popular kernels.

Low-rank matrix approximation (LRMA) methods have achieved excellent accuracy among today's collaborative filtering (CF) methods. In existing LRMA methods, the rank of user/item feature matrices is typically fixed, i.e., the same rank is adopted to describe all users/items. However, our studies show that submatrices with different ranks could coexist in the same user-item rating matrix, so that approximations with fixed ranks cannot perfectly describe the internal structures of the rating matrix, therefore leading to inferior recommendation accuracy. In this paper, a mixture-rank matrix approximation (MRMA) method is proposed, in which user-item ratings can be characterized by a mixture of LRMA models with different ranks. Meanwhile, a learning algorithm capitalizing on iterated condition modes is proposed to tackle the non-convex optimization problem pertaining to MRMA. Experimental studies on MovieLens and Netflix datasets demonstrate that MRMA can outperform six state-of-the-art LRMA-based CF methods in terms of recommendation accuracy.

Optimization with noisy gradients has become ubiquitous in statistics and machine learning. Reparameterization gradients, or gradient estimates computed via the ``reparameterization trick,'' represent a class of noisy gradients often used in Monte Carlo variational inference (MCVI). However, when these gradient estimators are too noisy, the optimization procedure can be slow or fail to converge. One way to reduce noise is to generate more samples for the gradient estimate, but this can be computationally expensive. Instead, we view the noisy gradient as a random variable, and form an inexpensive approximation of the generating procedure for the gradient sample. This approximation has high correlation with the noisy gradient by construction, making it a useful control variate for variance reduction. We demonstrate our approach on a non-conjugate hierarchical model and a Bayesian neural net where our method attained orders of magnitude (20-2{,}000$\times$) reduction in gradient variance resulting in faster and more stable optimization.

In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor $1/2$ approximation to the global maxima. We also study stochastic gradient methods and show that after $\mathcal{O}(1/\epsilon^2)$ iterations these methods reach solutions which achieve in expectation objective values exceeding $(\frac{\text{OPT}}{2}-\epsilon)$. An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor $1/2$ approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort.

Affine policies (or control) are widely used as a solution approach in dynamic optimization where computing an optimal adjustable solution is usually intractable. While the worst case performance of affine policies can be significantly bad, the empirical performance is observed to be near-optimal for a large class of problem instances. For instance, in the two-stage dynamic robust optimization problem with linear covering constraints and uncertain right hand side, the worst-case approximation bound for affine policies is $O(\sqrt m)$ that is also tight (see Bertsimas and Goyal (2012)), whereas observed empirical performance is near-optimal. In this paper, we aim to address this stark-contrast between the worst-case and the empirical performance of affine policies. In particular, we show that affine policies give a good approximation for the two-stage adjustable robust optimization problem with high probability on random instances where the constraint coefficients are generated i.i.d.

Adaptive stochastic gradient methods such as AdaGrad have gained popularity in particular for training deep neural networks. The most commonly used and studied variant maintains a diagonal matrix approximation to second order information by accumulating past gradients which are used to tune the step size adaptively. In certain situations the full-matrix variant of AdaGrad is expected to attain better performance, however in high dimensions it is computationally impractical. We present Ada-LR and RadaGrad two computationally efficient approximations to full-matrix AdaGrad based on randomized dimensionality reduction. They are able to capture dependencies between features and achieve similar performance to full-matrix AdaGrad but at a much smaller computational cost. We show that the regret of Ada-LR is close to the regret of full-matrix AdaGrad which can have an up-to exponentially smaller dependence on the dimension than the diagonal variant. Empirically, we show that Ada-LR and RadaGrad perform similarly to full-matrix AdaGrad. On the task of training convolutional neural networks as well as recurrent neural networks, RadaGrad achieves faster convergence than diagonal AdaGrad.