Harder, Better, Faster, Stronger Convergence Rates for Least-Squares Regression (1602.05419v2)

Abstract: We consider the optimization of a quadratic objective function whose gradients are only accessible through a stochastic oracle that returns the gradient at any given point plus a zero-mean finite variance random error. We present the first algorithm that achieves jointly the optimal prediction error rates for least-squares regression, both in terms of forgetting of initial conditions in O(1/n 2), and in terms of dependence on the noise and dimension d of the problem, as O(d/n). Our new algorithm is based on averaged accelerated regularized gradient descent, and may also be analyzed through finer assumptions on initial conditions and the Hessian matrix, leading to dimension-free quantities that may still be small while the " optimal " terms above are large. In order to characterize the tightness of these new bounds, we consider an application to non-parametric regression and use the known lower bounds on the statistical performance (without computational limits), which happen to match our bounds obtained from a single pass on the data and thus show optimality of our algorithm in a wide variety of particular trade-offs between bias and variance.

• The paper presents an averaged accelerated regularized gradient descent algorithm that attains optimal bias (O(1/n^2)) and variance (O(d/n)) rates.
• It employs acceleration and averaging techniques to improve noise robustness and perform efficiently even in high-dimensional settings.
• The refined analysis under modified initial and Hessian conditions provides strong theoretical and practical insights for scalable stochastic optimization.

Convergence Rates for Least-Squares Regression

The paper "Harder, Better, Faster, Stronger Convergence Rates for Least-Squares Regression" presents an advanced analysis of least-squares regression within a stochastic optimization framework. The authors introduce a novel algorithm based on averaged accelerated regularized gradient descent, achieving optimal prediction error rates in terms of bias and variance.

Contributions and Methodology

The paper addresses the problem of optimizing a quadratic objective function with gradients accessible only through a stochastic oracle. The oracle returns the gradient at any point plus a zero-mean finite variance random error. This setting is common in stochastic approximation where the covariance matrix of the noise and the initial point deviation from the optimal solution significantly affect algorithm performance.

Key contributions of the paper include:

1. Joint Optimal Rates Achievement: The authors propose an algorithm that simultaneously achieves optimal bias and variance rates. The bias term, associated with the initial condition forgetting, converges at an improved rate proportional to $\frac{1}{n^2}$, while the variance term, dependent on problem dimension $d$ and noise $\sigma^2$, converges at $\frac{d}{n}$.
2. Algorithmic Framework: The algorithm is based on averaged accelerated regularized gradient descent. It leverages acceleration techniques and averaging, which is proven to be beneficial in noise-robustness. The algorithm remains efficient even when the dimension $d$ exceeds the number of iterations $n$, showcasing adaptability to problem conditions.
3. Improved Analysis: The work includes finer analysis under amended assumptions regarding the initial conditions and the Hessian matrix. This results in dimension-free predictions that are valid in scenarios where conventional bounds may be large.

Strong Numerical Evidence and Bold Claims

The paper substantiates its claims with strong theoretical derivations proving the near-optimal performance of the proposed algorithm. Furthermore, the authors venture into high-dimensional settings, where $d > n$, and show through their analysis that the algorithm remains effective—a claim that is bold yet backed by robust empirical and theoretical evidence.

Implications and Future Prospects

Practically, the application to non-parametric regression demonstrates the potential of single-pass efficient algorithms to achieve statistical performance bounds previously attainable only in computationally expensive setups. This has implications for large-scale machine learning applications where computational efficiency is paramount.

Theoretically, the paper suggests a paradigm where optimization and approximation are jointly considered, with regularization and early stopping seen as facets of a single conceptual framework. The research opens avenues for further inquiries into leveraging acceleration in noisy environments, especially within non-linear and non-convex settings, which remain ripe for exploration.

Conclusion

This paper advances the understanding of convergence rates in the stochastic optimization of least-squares regression by providing a rigorous analysis that blends acceleration with averaging to achieve optimal convergence rates. Both the theoretical underpinnings and empirical validation provide a comprehensive view of how current methodologies can be enhanced to meet the increasing demands of high-dimensional data processing frameworks. Future work could extend these findings into more complex machine learning models, possibly affecting the design of next-generation stochastic optimization algorithms.