Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions (1411.3803v1)

Abstract: Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., problems of the form $\min_x \mathbf{E}v [f_v\big(\mathbf{E}_w [g_w(x)]\big)]$. In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of $f_v,g{w}$ and use an auxiliary variable to track the unknown quantity $\mathbf{E}_w[g_w(x)]$. We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieve a convergence rate of $O(k^{-1/4})$ in the general case and $O(k^{-2/3})$ in the strongly convex case, after taking $k$ samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of $O(k^{-2/7})$ in the general case and $O(k^{-4/5})$ in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc.

• The paper introduces the SCGD framework that leverages noisy sample gradients and an auxiliary variable to optimize compositional objectives.
• The authors establish convergence guarantees with rates such as O(k^-1/4) for non-smooth convex problems and accelerated rates for strongly convex cases.
• The study demonstrates SCGD’s efficiency in handling complex stochastic compositions, highlighting its applicability in fields like dynamic programming and risk management.

Overview of "Stochastic Compositional Gradient Descent: Algorithms for Minimizing Compositions of Expected-Value Functions"

The paper authored by Mengdi Wang, Ethan X. Fang, and Han Liu explores the optimization of compositions of expected-value functions using stochastic gradient methodologies. The conventional stochastic gradient descent (SGD), a well-researched method for minimizing expected-value objectives, faces limitations when addressing more complex structures like the nonlinear compositions of expected functions.

Key Contributions and Results

The authors introduce the Stochastic Compositional Gradient Descent (SCGD) framework aimed at addressing these limitations. The proposed SCGD class can be viewed as a stochastic adaptation of quasi-gradient methods. The innovative approach taken in SCGD is distinct in that it utilizes noisy sample gradients in combination with an auxiliary variable that estimates unknown quantities effectively. This mechanism shows particular promise in optimizing objectives that involve function compositions such as the form min E[g (E[h (x)])], where g and h are expected-value functions themselves.

The paper presents several significant findings:

1. Convergence Analysis: The authors rigorously prove that SCGD converges almost surely to an optimal solution when the objective is convex. The convergence maintains efficacy across varying assumptions about the smoothness and convexity of the functions involved, providing convergence rates such as O(k^-1/4) for non-smooth convex problems and O(k^-2/3) for strongly convex problems.
2. Nonconvex Problems: SCGD is shown to produce stationary points in nonconvex settings, with convergence rates depending on the problem’s structure.
3. Acceleration for Smooth Optimization: The paper further proposes an accelerated version of SCGD, improving the convergence rate for smooth optimization problems to O(k^-2/7) for general convex problems and O(k^-4/5) for strongly convex scenarios. Acceleration is achieved via extrapolation steps which enhance the sampling efficiency.
4. Sample Complexity: Compared to classical SGD, the SCGD requires only sub-quadratic samples to achieve a comparable optimization error for the more complex composition problems addressed, suggesting efficiency given the additional layer of computational challenge.

Practical and Theoretical Implications

The structure of the SCGD algorithms facilitates their applicability in several domains like statistical learning, risk management, and dynamic programming, where such stochastic compositions are prevalent. The paper discusses potential applications such as sparse additive models, minimax problems, and dynamic programming contexts, demonstrating the versatility and impact of the proposed methodology.

From a theoretical standpoint, the work sets a precedent in the exploration of stochastic optimization where nonlinearity and stochasticity are intertwined. The convergence analysis and complexity benchmarks provided form a foundation for further examination into the adaptability and efficiency of stochastic optimization algorithms.

Future Directions

The paper opens up several avenues for future research. Primarily, it suggests investigating the lower bounds of sample complexity for such stochastic composition problems to validate or improve upon the current SCGD rates. Additionally, further exploration into extending SCGD algorithms to handle compositions involving more than two stochastic functions would be a significant progression, potentially broadening the range of applicable problems.

Overall, the paper provides a comprehensive insight into the complexities of optimizing compositions of expected-value functions, offering robust solutions and paving the way for enhancements in the field of stochastic optimization.