Self-paced learning (SPL) is a recently raised methodology designed through simulating the learning principle of humans animals. A variety of SPL realization schemes have been designed for different computer vision and pattern recognition tasks, and empirically substantiated to be effective in these applications. However, the investigation on its theoretical insight is still a blank. To this issue, this study attempts to provide some new theoretical understanding under the SPL scheme. Specifically, we prove that the solving strategy on SPL accords with a majorization minimization algorithm implemented on a latent objective function. Furthermore, we find that the loss function contained in this latent objective has a similar configuration with non-convex regularized penalty (NSPR) known in statistics and machine learning. Such connection inspires us discovering more intrinsic relationship between SPL regimes and NSPR forms, like SCAD, LOG and EXP. The robustness insight under SPL can then be finely explained. We also analyze the capability of SPL on its easy loss prior embedding property, and provide an insightful interpretation to the effectiveness mechanism under previous SPL variations. Besides, we design a group-partial-order loss prior, which is especially useful to weakly labeled large-scale data processing tasks. Through applying SPL with this loss prior to the FCVID dataset, which is currently one of the biggest manually annotated video dataset, our method achieves state-of-the-art performance beyond previous methods, which further helps supports the proposed theoretical arguments.
NCRP has been demonstrated to have attractive properties in sparse estimation (as a penalty term) @cite_0 @cite_21 @cite_13 and robust learning (as a loss term) @cite_5 @cite_17 both theoretically and practically, and attracted much attention in machine learning and statistics in recent years. Various NCRP realizations have also been proposed. Typical ones include the capped-norm based penalty (CNP) @cite_0 @cite_2 @cite_12 , the minimax concave plus penalty (MCP) @cite_21 , the smoothly clipped absolute deviation penalty (SCAD) @cite_13 ,the logarithmic penalty (LOG) @cite_38 and the nonconvex exponential penalty (EXP) @cite_32 . The mathematical forms of these NCRP terms in one dimension cases are listed as follows @cite_27 @cite_9 : . SCAD: p_ , ^ SCAD (t)= . LOG: p_ , ^ LOG (t)= 1 (1+ |t|) EXP: p_ , ^ EXP (t)= 1 (1- (- |t|)).
@cite_38 @cite_9 @cite_21 @cite_32 @cite_0 @cite_27 @cite_2 @cite_5 @cite_13 @cite_12 @cite_17
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We propose MC+, a fast, continuous, nearly unbiased and accurate method of penalized variable selection in high-dimensional linear regression. The LASSO is fast and continuous, but biased. The bias of the LASSO may prevent consistent variable selection. Subset selection is unbiased but computationally costly. The MC+ has two elements: a minimax concave penalty (MCP) and a penalized linear unbiased selection (PLUS) algorithm. The MCP provides the convexity of the penalized loss in sparse regions to the greatest extent given certain thresholds for variable selection and unbiasedness. The PLUS computes multiple exact local minimizers of a possibly nonconvex penalized loss function in a certain main branch of the graph of critical points of the penalized loss. Its output is a continuous piecewise linear path encompassing from the origin for infinite penalty to a least squares solution for zero penalty. We prove that at a universal penalty level, the MC+ has high probability of matching the signs of the unknowns, and thus correct selection, without assuming the strong irrepresentable condition required by the LASSO. This selection consistency applies to the case of p » n, and is proved to hold for exactly the MC+ solution among possibly many local minimizers. We prove that the MC+ attains certain minimax convergence rates in probability for the estimation of regression coefficients in l r balls. We use the SURE method to derive degrees of freedom and C p -type risk estimates for general penalized LSE, including the LASSO and MC+ estimators, and prove their unbiasedness. Based on the estimated degrees of freedom, we propose an estimator of the noise level for proper choice of the penalty level. For full rank designs and general sub-quadratic penalties, we provide necessary and sufficient conditions for the continuity of the penalized LSE. Simulation results overwhelmingly support our claim of superior variable selection properties and demonstrate the computational efficiency of the proposed method. Extracting useful information from high-dimensional data is an important focus of today's statistical research and practice. Penalized loss function minimization has been shown to be effective for this task both theoretically and empirically. With the virtues of both regularization and sparsity, the L 1 -penalized squared error minimization method Lasso has been popular in regression models and beyond. In this paper, we combine different norms including L 1 to form an intelligent penalty in order to add side information to the fitting of a regression or classification model to obtain reasonable estimates. Specifically, we introduce the Composite Absolute Penalties (CAP) family, which allows given grouping and hierarchical relationships between the predictors to be expressed. CAP penalties are built by defining groups and combining the properties of norm penalties at the across-group and within-group levels. Grouped selection occurs for nonoverlapping groups. Hierarchical variable selection is reached by defining groups with particular overlapping patterns. We propose using the BLASSO and cross-validation to compute CAP estimates in general. For a subfamily of CAP estimates involving only the L 1 and L ∞ norms, we introduce the iCAP algorithm to trace the entire regularization path for the grouped selection problem. Within this subfamily, unbiased estimates of the degrees of freedom (df) are derived so that the regularization parameter is selected without cross-validation. CAP is shown to improve on the predictive performance of the LASSO in a series of simulated experiments, including cases with p » n and possibly mis-specified groupings. When the complexity of a model is properly calculated, iCAP is seen to be parsimonious in the experiments. We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then an optimal solution of the Arora-Rao-Vazirani relaxation (or of any stronger version of it) can be almost entirely covered by a small number of balls of bounded radius. Then, we show that, if k is the number of balls, a solution of this form can be rounded with an approximation factor of O(sqrt log k ) in the case of the Arora-Rao-Vazirani relaxation, and with a constant-factor approximation in the case of the k-th round of the Sherali-Adams hierarchy starting at the Arora-Rao-Vazirani relaxation. The structure theorem and the rounding scheme combine to prove the following result, where G=(V,E) is a graph of expansion (G), is the k-th smallest eigenvalue of the normalized Laplacian of G, and (G) = disjoint S_1,...,S_k 1 > log^ 2.5 k phi(G) or k (G) >> log k sqrt log n loglog n (G), then the Arora-Rao-Vazirani relaxation can be rounded in polynomial time with an approximation ratio O(sqrt log k ). Stronger approximation guarantees are achievable in time exponential in k via relaxations in the Lasserre hierarchy. Guruswami and Sinop [GS13] and Arora, Ge and Sinop [AGS13] prove that 1+eps approximation is achievable in time 2^ O(k) poly(n) if either > (G) poly(eps), or if SSE_ n k > sqrt log k log n (G) poly(eps), where SSE_s is the minimal expansion of sets of size at most s. 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.