Variance-based regularization with convex objectives (1610.02581v3)

Abstract: We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.

• The paper introduces a novel convex surrogate for variance that enables robust risk minimization beyond traditional empirical risk minimization.
• It employs φ-divergences, notably the χ²-divergence, to construct a local neighborhood around the empirical distribution, ensuring the optimization remains convex.
• Empirical results show faster convergence and improved classification accuracy, particularly in imbalanced settings, compared to standard ERM techniques.

An Expert Overview of "Variance-based Regularization with Convex Objectives"

The paper by John Duchi and Hongseok Namkoong introduces a novel approach to risk minimization and stochastic optimization, focusing on variance-based regularization with convex objectives. At its core, the research formulates a convex surrogate for variance, allowing for effective trade-offs between approximation and estimation errors. This method leverages techniques from distributionally robust optimization and Owen's empirical likelihood, providing theoretical performance guarantees and empirical evidence of improved out-of-sample performance over standard empirical risk minimization (ERM) on classification problems.

Key Contributions and Methodology

The authors propose a robust approach that automatically balances bias and variance, challenging the traditional ERM paradigm, which primarily focuses on minimizing empirical risk without explicitly addressing variance. The paper leverages $\phi$-divergences, specifically the $\chi^2$-divergence, to construct a local neighborhood of distributions around the empirical distribution, enabling the formulation of the robustly regularized risk. This risk is shown to be convex whenever the loss function is convex, making the ensuing optimization problem tractable.

The methodology hinges on the expansion of the robust risk into its principal components, systematically analyzing the variance penalty term: $\risk_n(\theta, \mc{P}_n) = \E_\emp[\loss(\theta, \statrv)] + \sqrt{\frac{2 \tol \var_\emp(\loss(\theta, \statrv))}{n} + \varepsilon_n(\theta)},$ where $\varepsilon_n(\theta) \le 0$ represents a minor error term. This expansion facilitates the strategic handling of bias and variance, optimizing for classification problems where variance traditionally hampers out-of-sample performance.

Theoretical and Empirical Results

The paper provides comprehensive theoretical guarantees, including finite-sample convergence rates and asymptotic behavior. A notable conclusion is that the robust solution can attain faster convergence rates under specific growth conditions in the risk landscape. This theoretical robustness makes the approach particularly appealing in scenarios where variance is not uniformly bounded by the risk, offering advantages over classical ERM techniques.

Empirically, the paper evidences the method's superiority across several benchmark tasks. Experiments indicate that robustly regularized models yield better classification accuracy, especially in cases of class imbalance, where traditional methods often underperform on rare classes. This result is attributed to the robust solution's ability to adjust for variance induced by harder-to-classify instances, thereby improving model reliability and performance.

Practical Implications and Future Directions

Practically, this research could reshape how regularization is approached in various fields, such as machine learning, where dealing with high variance and ensuring model generalization are key concerns. It provides a compelling alternative to regularizers like $\ell_1$ or elastic net, primarily focusing on variance penalization.

Speculatively, this approach might spark further investigations into variance-sensitive learning, particularly in developing models robust to distributional shifts or outliers. Future work could explore more efficient computational solutions to the convex optimization problems posed, potentially addressing scalability and application to large datasets or high-dimensional spaces.

In conclusion, Duchi and Namkoong's contribution to variance-aware risk minimization allows for a nuanced understanding of the complexities in balancing bias and variance. It stands as a meaningful advancement in the stochastic optimization landscape, proposing a shift in how regularization is conceptualized and applied in practice.