Recently, a parametrized class of loss functions called @math -loss, @math , has been introduced for classification. This family, which includes the log-loss and the 0-1 loss as special cases, comes with compelling properties including an equivalent margin-based form which is classification-calibrated for all @math . We introduce a generalization of this family to the entire range of @math and establish how the parameter @math enables the practitioner to choose among a host of operating conditions that are important in modern machine learning tasks. We prove that smaller @math values are more conducive to faster optimization; in fact, @math -loss is convex for @math and quasi-convex for @math . Moreover, we establish bounds to quantify the degradation of the local-quasi-convexity of the optimization landscape as @math increases; we show that this directly translates to a computational slow down. On the other hand, our theoretical results also suggest that larger @math values lead to better generalization performance. This is a consequence of the ability of the @math -loss to limit the effect of less likely data as @math increases from 1, thereby facilitating robustness to outliers and noise in the training data. We provide strong evidence supporting this assertion with several experiments on benchmark datasets that establish the efficacy of @math -loss for @math in robustness to errors in the training data. Of equal interest is the fact that, for @math , our experiments show that the decreased robustness seems to counteract class imbalances in training data.
Our work is similar to @cite_3 , wherein Nguyen present a tunable sigmoid loss which can be made arbitrarily close to the @math - @math loss. In essence, their loss moves from a smooth to non-smooth loss. Our loss is always smooth and moves from convex to quasi-convex. We find that in the setting of deep neural networks, some quasi-convexity of the @math -loss smooths the empirical landscape. In particular, we provide strong experimental evidence for a narrow range of @math to be used in practice (a limit on the amount of convexity and quasi-convexity of the loss), which significantly reduces the range of hyperparameter tuning induced by @math -loss. Increasing the degree of convexity of the optimization landscape is conducive to faster optimization. Hence, our approach could serve as an alternative to other approaches whose objective is to accelerate the optimization process, e.g., the activation function tuning in @cite_21 @cite_4 @cite_16 and references therein.
@cite_21 @cite_4 @cite_3 @cite_16
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While convex losses for binary classification are attractive due to the existence of numerous (provably) efficient methods for finding their global optima, they are sensitive to outliers. On the other hand, while the nonconvex 0-1 loss is robust to outliers, it is NP-hard to optimize and thus rarely directly optimized in practice. In this paper, however, we do just that: we explore a variety of practical methods for direct (approximate) optimization of the 0-1 loss based on branch and bound search, combinatorial search, and coordinate descent on smooth, differentiable relaxations of 0-1 loss. Empirically, we compare our proposed algorithms to logistic regression, SVM, and the Bayes point machine showing that the proposed 0-1 loss optimization algorithms perform at least as well and offer a clear advantage in the presence of outliers. To this end, we believe this work reiterates the importance of 0-1 loss and its robustness properties while challenging the notion that it is difficult to directly optimize. We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. We consider neural networks with a single hidden layer and non-decreasing positively homogeneous activation functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, they lead to a convex optimization problem and we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace. Moreover, when using sparsity-inducing norms on the input weights, we show that high-dimensional non-linear variable selection may be achieved, without any strong assumption regarding the data and with a total number of variables potentially exponential in the number of observations. However, solving this convex optimization problem in infinite dimensions is only possible if the nonconvex subproblem of addition of a new unit can be solved efficiently. We provide a simple geometric interpretation for our choice of activation functions and describe simple conditions for convex relaxations of the finite-dimensional non-convex subproblem to achieve the same generalization error bounds, even when constant-factor approximations cannot be found. We were not able to find strong enough convex relaxations to obtain provably polynomialtime algorithms and leave open the existence or non-existence of such tractable algorithms with non-exponential sample complexities. In this paper we study differentially private Empirical Risk Minimization(ERM) in different settings. For smooth (strongly) convex loss function with or without (non)-smooth regularization, we give algorithms which achieve either optimal or near optimal utility bound with less gradient complexity compared with previous work. For ERM with smooth convex loss function in high-dimension( @math ) setting, we give an algorithm which achieves the upper bound with less gradient complexity than previous ones. At last, we generalize the expected excess empirical risk from convex to Polyak-Lojasiewicz condition and give a tighter upper bound of the utility comparing with the result in DBLP:journals corr ZhangZMW17 .