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Self-Paced Learning: an Implicit Regularization Perspective
Yanbo Fan, Ran He, Jian Liang, Bao-Gang Hu
TL;DR
Existing SPL approaches generally hand-design explicit regularizers to determine sample weights, while its underlying mechanism remains unclear. The paper derives self-paced implicit regularizers from latent robust losses and develops SPL-IR, relating its learning procedure to robust losses and half-quadratic optimization. Experiments on supervised and unsupervised tasks are reported to corroborate the framework’s correctness and effectiveness.
Problem
Existing SPL methods usually design explicit regularizers to determine sample-weighting strategies, and the underlying working mechanism remains unclear.
Source: §1 · p. 1, §6.3 · p. 11Method
SPL-IR derives implicit regularizers and their minimizer functions from latent robust loss functions using convex conjugacy, without requiring the regularizer’s analytic form.
Source: §1 · p. 2, §3.1 · p. 4Results
Experiments on supervised and unsupervised tasks corroborate SPL-IR’s correctness and effectiveness.
Source: §1 · p. 2Takeaways & Limitations
The learning procedure’s association with latent robust losses may provide theoretical insight into SPL-IR’s working mechanism, including robustness to outliers and heavy noise.
Source: §1 · p. 2, §5 · p. 9The stated conditions include a latent loss φ that is increasing, C1, concave, and sublinear at infinity.
Source: §6.1 · p. 9
Abstract
from arXiv · showhide
Self-paced learning (SPL) mimics the cognitive mechanism of humans and animals that gradually learns from easy to hard samples. One key issue in SPL is to obtain better weighting strategy that is determined by minimizer function. Existing methods usually pursue this by artificially designing the explicit form of SPL regularizer. In this paper, we focus on the minimizer function, and study a group of new regularizer, named self-paced implicit regularizer that is deduced from robust loss function. Based on the convex conjugacy theory, the minimizer function for self-paced implicit regularizer can be directly learned from the latent loss function, while the analytic form of the regularizer can be even known. A general framework (named SPL-IR) for SPL is developed accordingly. We demonstrate that the learning procedure of SPL-IR is associated with latent robust loss functions, thus can provide some theoretical inspirations for its working mechanism. We further analyze the relation between SPL-IR and half-quadratic optimization. Finally, we implement SPL-IR to both supervised and unsupervised tasks, and experimental results corroborate our ideas and demonstrate the correctness and effectiveness of implicit regularizers.
1 Introduction
Self-paced learning (SPL) updates its curriculum during optimization, but existing approaches typically hand-design explicit regularizers and its working mechanism remains unclear. This paper introduces SPL-IR, deriving implicit regularizers from robust loss functions and connecting the framework to half-quadratic optimization.
1 Introduction
- Existing SPL methods usually obtain weighting strategies by artificially designing explicit regularizers, while SPL’s underlying mechanism remains unclear.Earlier work connected alternative search to majorization minimization on a latent SPL objective for hard, linear, and mixture regularizers [19].Source: §1 · pp. 1–2
- SPL-IR derives sample-weight minimizers from latent robust loss functions, even when the implicit regularizer’s analytic form is unknown.The framework is independent of the learning objective and can be optimized with alternative search.Source: §1 · p. 2
- The paper relates SPL-IR’s learning procedure to latent robust losses and analyzes its connection to half-quadratic optimization.The authors suggest these connections may offer theoretical insight into SPL-IR’s robustness to outliers and heavy noise.Source: §1 · p. 2
- Experiments on supervised and unsupervised tasks are reported to corroborate SPL-IR’s correctness and effectiveness.The paper describes applications to matrix factorization, clustering, and classification.Source: §1 · p. 2
2 Preliminaries
The preliminaries describe SPL as joint optimization of model parameters and sample weights, with a regularizer controlling the learning pace. They also introduce half-quadratic optimization, where a nonlinear objective is iteratively handled through reformulations and a minimizer function derived from the loss.
2.1 Self-Paced Learning via Explicit Regularizers
- SPL jointly optimizes model parameters and latent sample weights by minimizing a sum of weighted sample losses and a self-paced regularizer.The regularizer penalty λ controls the learning pace, and alternative search alternates between model and weight updates.Source: §2.1 · p. 2
- Given model parameters, each sample’s optimal weight is determined by a minimizer function; increasing λ gradually includes harder samples.Explicit-regularizer methods derive that minimizer function from a specified regularizer form.Source: §2.1 · p. 2
- Table 1 presents loss functions φ(λ, t) alongside their corresponding minimizer functions σ(λ, t).λ is the hyper-parameter.Source: Table 1 · p. 3
2.2 Half-Quadratic Optimization
- Half-quadratic optimization handles nonlinear objectives through iterative half-quadratic reformulations based on convex conjugacy.Its minimizer function δ(t) determines the optimal p, and the dual potential function’s analytic form can be unknown.Source: §2.2 · p. 3
3 The Proposed Method
The method defines self-paced implicit regularizers from robust loss functions, learning their minimizer functions from latent losses without requiring an explicit regularizer form. It develops SPL-IR and relates it to half-quadratic optimization.
3.1 Self-Paced Implicit Regularizer
- The minimizer function is uniquely determined by the robust loss, even when the dual potential's analytic form is unknown.The minimizer function and regularizer properties can be learned from the latent loss function.Source: §3.1 · p. 4
3.2 Self-Paced Learning via Implicit Regularizers
- SPL-IR uses the implicit regularizer in an alternating optimization framework, with sample weights determined by the minimizer function rather than an explicit regularizer form.Its learning procedure is associated with latent robust losses: sufficiently large losses can receive zero weight, while increasing λ raises weights and gradually incorporates harder samples.Source: §3.2 · pp. 4–5
3.3 SPL-IR and Half-Quadratic Optimization
- SPL-IR is closely related to HQ, but constrains the minimizer function for self-paced learning and updates λ stage by stage, unlike HQ's fixed λ.The paper uses this relation to adapt robust-loss-based minimizers as self-paced implicit regularizers.Source: §3.3 · pp. 5–6
- SPL-IR with Welsch weights sequentially optimizes robust losses with increasing λ and outperforms HQ for every λ in the synthetic matrix-factorization comparison.HQ fixes λ, so at λ < 1 some hard samples may be treated as outliers and discarded.Source: §3.3 · p. 6
4 Experiments
SPL-IR is evaluated on matrix factorization, multi-view clustering, and classification, with implicit regularizers generally improving on baselines and matching or surpassing designed SPL regularizers. The experiments also report behavior under noisy data and across iterations.
4 Experiments
- Across matrix factorization, clustering, and classification, SPL-IR outperforms baseline algorithms and achieves comparable or better performance than artificially designed SPL regularizers.The experiments include both supervised and unsupervised tasks.Source: §4 · p. 6
4.1 Matrix Factorization
- The matrix-factorization experiment uses a synthetic rank-4 matrix with missing entries, uniform noise, and Gaussian noise, comparing L1-norm matrix factorization with L2 regularization against PRMF.The evaluation reports means over 50 realizations.Source: §4.1 · p. 7
- In matrix factorization, all SPL-IR algorithms improve over PRMF, and the proposed regularizers are comparable to or better than mixture and hard schemes, especially Welsch.The tendency curves show rapid improvement in early iterations, followed by steadier gains as harder instances or outliers are included.Source: §4.1 · pp. 7–8
4.2 Multi-view Clustering
- Multi-view algorithms improve over single-view baselines, while SPL-IR further improves on Con-MC and its regularizers match or exceed the compared SPL regularizers.The comparison uses all six views of the Handwritten Digit dataset.Source: Table 3 · p. 7, §4.2 · p. 8
4.3 Classification
- For classification, SPL-IR improves on logistic regression in both clean and noisy settings; with 20% label noise, logistic regression degrades substantially while SPL methods retain relatively high accuracy, especially Welsch.The experiments use 10-fold cross-validation on three databases.Source: §4.3 · pp. 8–9, Table 5 · p. 8
5 Conclusions
The paper proposes self-paced implicit regularizers and develops SPL-IR, deriving minimizer functions from latent robust losses. It connects the framework to half-quadratic optimization and reports effectiveness on supervised and unsupervised tasks.
5 Conclusions
- The paper introduces self-paced implicit regularizers based on convex conjugacy and develops the general SPL-IR framework.The regularizer's analytic form may be unknown, while its properties and minimizer function can be learned from the latent loss.Source: §5 · p. 9
- The learning procedure is associated with latent robust loss functions, and the paper analyzes SPL-IR's relation to half-quadratic optimization.The authors present this association as a source of theoretical inspiration about the framework's working mechanisms.Source: §5 · p. 9
- Experiments on supervised and unsupervised tasks support the correctness and effectiveness of the proposed implicit regularizers.Source: §5 · p. 9
6 Appendix
The appendix derives SPL-IR weights from latent losses, establishes equivalence between two implicit-regularizer definitions, and discusses how robust-loss and half-quadratic connections relate to self-paced regularizers.
6.1 Proof of Proposition 1
- Under H1–H3, the convex-conjugate derivation uniquely determines the optimal weight as σ(t)=φ′(t), even when the analytic form of ψ(v) is unknown.The proof uses the unique minimum of a convex function to obtain the minimizer function from the latent loss.Source: §6.1 · p. 9
6.2 Definition 1 and Definition 2
- For fixed λ and the stated conditions, Proposition 2 says the minimizer function is uniquely determined by φ(λ, t), even if ψ(λ, v) is unknown.The proof derives σ(t)=φ′(t)/t under its stated assumptions.Source: §6.2 · pp. 9–10
- The appendix establishes that the two definitions yield the same latent loss and optimal weight, while model (13) optimizes its corresponding latent loss.For Definition 1, the stated latent loss is φ1(ℓ); the appendix concludes the definitions are equivalent.Source: §6.2 · p. 10
6.3 Self-Paced Regularizer
- Classical SPL weights decrease with loss and increase with pace, but designing regularizers and analyzing their properties remains nontrivial.The cited definition also gives limiting weight behavior as loss or pace approaches its extremes.Source: §6.3 · pp. 10–11
- SPL-IR relates robust latent losses, implicit regularizers, and minimizer functions, and connects to half-quadratic optimization to develop further regularizers.The appendix notes that loss functions and minimizer functions from half-quadratic optimization can be adjusted for self-paced implicit regularizers.Source: §6.3 · p. 11