Finito: A Faster, Permutable Incremental Gradient Method for Big Data Problems (1407.2710v1)

Abstract: Recent advances in optimization theory have shown that smooth strongly convex finite sums can be minimized faster than by treating them as a black box "batch" problem. In this work we introduce a new method in this class with a theoretical convergence rate four times faster than existing methods, for sums with sufficiently many terms. This method is also amendable to a sampling without replacement scheme that in practice gives further speed-ups. We give empirical results showing state of the art performance.

• The paper introduces Finito, a new incremental gradient method specifically designed for minimizing smooth strongly convex finite sums common in big data machine learning.
• Finito is theoretically four times faster than methods like SAG for large datasets and uses fixed step lengths, unlike SGD which requires tuning.
• The algorithm offers practical advantages in big data scenarios due to its efficiency and robustness introduced by sampling and permutation techniques.

An Analysis of Finito: A Permutable Incremental Gradient Method for Big Data Optimization

In the field of numerical optimization, the paper "Finito: A Faster, Permutable Incremental Gradient Method for Big Data Problems" elucidates a novel algorithm designed to efficiently handle the minimization of smooth strongly convex finite sums typical in big data contexts. The authors introduce Finito, an incremental gradient method with a theoretical convergence rate purportedly four times faster than existing alternatives such as Stochastic Average Gradient (SAG) when provided with sufficiently large data sets.

Technical Summary

Finito exploits the inherent structure in optimization problems defined as sums over data points—a condition often faced in machine learning tasks involving empirical risk minimization. Traditional black-box optimization techniques do not leverage this structure effectively. However, algorithms like SAG have demonstrated substantial improvements in optimization efficiency by considering this structure, specifically for datasets where the number of data points $n$ is proportionate to the condition number of the problem $L/s$, where $L$ is the Lipschitz gradient constant and $s$ the strong convexity constant. Finito adheres to this "big data condition" and further enhances the optimization process by incrementally revisiting data points—offering a convergence rate expressed by $$E[f(\bar{\phi}^{(k)})] - f(w^*) \leq \frac{3}{4s}(1-\frac{1}{2n})^k \|f'(\bar{\phi}^{(0)})\|^2$$.

Algorithmic Insights

At each iteration $k$, Finito performs several critical steps: updating the average estimate of the parameter vector $w$, selecting a data index at random (or employing a sampling-without-replacement scheme), and recalculating gradients for that specific data point. Notably, Finito utilizes fixed step lengths, distinguishing it from stochastic gradient descent (SGD)—a method requiring meticulous parameter tuning for optimal performance.

Implications and Comparisons

Finito offers practical advantages over established methods like SAG and SDCA (Stochastic Dual Coordinate Ascent), especially in big data regimes where large datasets afford enhanced exploitation of the finite-sum structure. Empirically, Finito exhibits state-of-the-art performance with convergence rates markedly close to its theoretical predictions, contrasting the often superior empirical outcomes observed with methods such as LBFGS.

Additionally, the introduction of sampling strategies and permutation techniques in Finito’s operational schema underscores the vital role of randomness in optimizing finite sums. Such techniques infuse robustness into the optimization process by preventing order-induced stagnation, alluding to broader applications in stochastic settings.

Future Directions

The paper speculates on future developments that could further refine Finito's applicability in non-convex settings or enhance its integration into composite problems where proximal methods are ideal. There remains potential for theoretical exploration regarding sampling methods without replacement, offering room for advancing optimization paradigms through enriched stochastic characterizations.

In conclusion, Finito emerges as a significant contribution to incremental gradient methods, with its theoretical precision matched by promising empirical validations. The implications of its robustness and effectiveness in large-scale optimization tasks make it a compelling option for researchers and practitioners grappling with the nuances of big data-driven machine learning problems.