We prove that, under low noise assumptions, the support vector machine with $N\ll m$ random features (RFSVM) can achieve the learning rate faster than $O(1/\sqrt{m})$ on a training set with $m$ samples when an optimized feature map is used. Our work extends the previous fast rate analysis of random features method from least square loss to 0-1 loss. We also show that the reweighted feature selection method, which approximates the optimized feature map, helps improve the performance of RFSVM in experiments on a synthetic data set.

Inverse reinforcement learning (IRL) is the problem of finding a reward function that generates a given optimal policy for a given Markov Decision Process. This paper looks at an algorithmic-independent geometric analysis of the IRL problem with finite states and actions. A L1-regularized Support Vector Machine formulation of the IRL problem motivated by the geometric analysis is then proposed with the basic objective of the inverse reinforcement problem in mind: to find a reward function that generates a specified optimal policy. The paper further analyzes the proposed formulation of inverse reinforcement learning with $n$ states and $k$ actions, and shows a sample complexity of $O(d^2 \log (nk))$ for transition probability matrices with at most $d$ non-zeros per row, for recovering a reward function that generates a policy that satisfies Bellman's optimality condition with respect to the true transition probabilities.

Distribution shift (DS) may have two levels: the distribution itself changes, and the support (i.e., the set where the probability density is non-zero) also changes. When considering the support change between the training and test distributions, there can be four cases: (i) they exactly match; (ii) the training support is wider (and thus covers the test support); (iii) the test support is wider; (iv) they partially overlap. Existing methods are good at cases (i) and (ii), while cases (iii) and (iv) are more common nowadays but still under-explored. In this paper, we generalize importance weighting (IW), a golden solver for cases (i) and (ii), to a universal solver for all cases. Specifically, we first investigate why IW might fail in cases (iii) and (iv); based on the findings, we propose generalized IW (GIW) that could handle cases (iii) and (iv) and would reduce to IW in cases (i) and (ii). In GIW, the test support is split into an in-training (IT) part and an out-of-training (OOT) part, and the expected risk is decomposed into a weighted classification term over the IT part and a standard classification term over the OOT part, which guarantees the risk consistency of GIW. Then, the implementation of GIW consists of three components: (a) the split of validation data is carried out by the one-class support vector machine, (b) the first term of the empirical risk can be handled by any IW algorithm given training data and IT validation data, and (c) the second term just involves OOT validation data. Experiments demonstrate that GIW is a universal solver for DS problems, outperforming IW methods in cases (iii) and (iv).

The growing literature on benign overfitting in overparameterized models has been mostly restricted to regression or binary classification settings; however, most success stories of modern machine learning have been recorded in multiclass settings. Motivated by this discrepancy, we study benign overfitting in multiclass linear classification. Specifically, we consider the following popular training algorithms on separable data: (i) empirical risk minimization (ERM) with cross-entropy loss, which converges to the multiclass support vector machine (SVM) solution; (ii) ERM with least-squares loss, which converges to the min-norm interpolating (MNI) solution; and, (iii) the one-vs-all SVM classifier. Our first key finding is that under a simple sufficient condition, all three algorithms lead to classifiers that interpolate the training data and have equal accuracy. When the data is generated from Gaussian mixtures or a multinomial logistic model, this condition holds under high enough effective overparameterization. Second, we derive novel error bounds on the accuracy of the MNI classifier, thereby showing that all three training algorithms lead to benign overfitting under sufficient overparameterization. Ultimately, our analysis shows that good generalization is possible for SVM solutions beyond the realm in which typical margin-based bounds apply.

In many applications of machine learning, a large number of variables are considered. Motivated by machine learning of interacting particle systems, we consider the situation when the number of input variables goes to infinity. First, we continue the recent investigation of the mean field limit of kernels and their reproducing kernel Hilbert spaces, completing the existing theory. Next, we provide results relevant for approximation with such kernels in the mean field limit, including a representer theorem. Finally, we use these kernels in the context of statistical learning in the mean field limit, focusing on Support Vector Machines. In particular, we show mean field convergence of empirical and infinite-sample solutions as well as the convergence of the corresponding risks. On the one hand, our results establish rigorous mean field limits in the context of kernel methods, providing new theoretical tools and insights for large-scale problems. On the other hand, our setting corresponds to a new form of limit of learning problems, which seems to have not been investigated yet in the statistical learning theory literature.

Multiclass extensions of the support vector machine (SVM) have been formulated in a variety of ways. A recent empirical comparison of nine such formulations [Doǧan et al. 2016] recommends the variant proposed by Weston and Watkins (WW), despite the fact that the WW-hinge loss is not calibrated with respect to the 0-1 loss. In this work we introduce a novel discrete loss function for multiclass classification, the ordered partition loss, and prove that the WW-hinge loss is calibrated with respect to this loss. We also argue that the ordered partition loss is minimally emblematic among discrete losses satisfying this property. Finally, we apply our theory to justify the empirical observation made by Doǧan et al that the WW-SVM can work well even under massive label noise, a challenging setting for multiclass SVMs.

In this paper, we provide theoretical results of estimation bounds and excess risk upper bounds for support vector machine (SVM) with sparse multi-kernel representation. These convergence rates for multi-kernel SVM are established by analyzing a Lasso-type regularized learning scheme within composite multi-kernel spaces. It is shown that the oracle rates of convergence of classifiers depend on the complexity of multi-kernels, the sparsity, a Bernstein condition and the sample size, which significantly improves on previous results even for the additive or linear cases. In summary, this paper not only provides unified theoretical results for multi-kernel SVMs, but also enriches the literature on high-dimensional nonparametric classification.

The goal in label-imbalanced and group-sensitive classification is to optimize relevant metrics such as balanced error and equal opportunity. Classical methods, such as weighted cross-entropy, fail when training deep nets to the terminal phase of training (TPT), that is training beyond zero training error. This observation has motivated recent flurry of activity in developing heuristic alternatives following the intuitive mechanism of promoting larger margin for minorities. In contrast to previous heuristics, we follow a principled analysis explaining how different loss adjustments affect margins. First, we prove that for all linear classifiers trained in TPT, it is necessary to introduce multiplicative, rather than additive, logit adjustments so that the interclass margins change appropriately. To show this, we discover a connection of the multiplicative CE modification to the cost-sensitive support-vector machines.

In this paper, we use tools from rate-distortion theory to establish new upper bounds on the generalization error of statistical distributed learning algorithms. Specifically, there are $K$ clients whose individually chosen models are aggregated by a central server. The bounds depend on the compressibility of each client's algorithm while keeping other clients' algorithms un-compressed, and leveraging the fact that small changes in each local model change the aggregated model by a factor of only $1/K$. Adopting a recently proposed approach by Sefidgaran et al., and extending it suitably to the distributed setting, enables smaller rate-distortion terms which are shown to translate into tighter generalization bounds. The bounds are then applied to the distributed support vector machines (SVM), suggesting that the generalization error of the distributed setting decays faster than that of the centralized one with a factor of $\mathcal{O}(\sqrt{\log(K)/K})$. This finding is validated also experimentally. A similar conclusion is obtained for a multiple-round federated learning setup where each client uses stochastic gradient Langevin dynamics (SGLD).

Motivated by the problem of learning with small sample sizes, this paper shows how to incorporate into support-vector machines (SVMs) those properties that have made convolutional neural networks (CNNs) successful. Particularly important is the ability to incorporate domain knowledge of invariances, e.g., translational invariance of images. Kernels based on the \textit{maximum} similarity over a group of transformations are not generally positive definite. Perhaps it is for this reason that they have not been studied theoretically. We address this lacuna and show that positive definiteness indeed holds \textit{with high probability} for kernels based on the maximum similarity in the small training sample set regime of interest, and that they do yield the best results in that regime. We also show how additional properties such as their ability to incorporate local features at multiple spatial scales, e.g., as done in CNNs through max pooling, and to provide the benefits of composition through the architecture of multiple layers, can also be embedded into SVMs. We verify through experiments on widely available image sets that the resulting SVMs do provide superior accuracy in comparison to well-established deep neural network benchmarks for small sample sizes.