AdrienTaylor / Potential-functions-for-first-order-methods

Codes associated to the work "Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions"

Potential functions for first-order methods

This code comes jointly with the following reference (alternatively, it is on arXiv)

[1] Adrien Taylor, and Francis Bach. "Stochastic first-order methods: non-asymptotic and computer-aided analyses via potential functions," Proceedings of the International Conference on Learning Theory (COLT), 2019.

Date: February 4, 2019

Authors

• Adrien Taylor
• Francis Bach

Note: This code requires YALMIP along with a suitable SDP solver (e.g., Sedumi, SDPT3, Mosek).

Organization of the code

Deterministic first-order methods

• A_GradientDescent Code for reproducing the result of Section 3.2.1, Theorem 3 and Figure 1 (gradient method).
• B_ProximalGradientDescent Code for reproducing the result presented in Appendix C.2, Theorem 9 and Figure 1-like results (proximal gradient method).
• C_StepsizeSelection_FirstAcceleratedMethod Code for reproducing the first result of Appendix C.3, Theorem 10 and Figure 3 (first accelerated gradient method).
• D_StepsizeSelection_SecondAcceleratedMethod Code for reproducing the second result of Appendix C.3, Theorem 11 and Figure 4 (second accelerated gradient method).

Stochastic first-order methods: unbiased oracles with bounded variance

• A_StochasticGradientDescent Code for reproducing the first result of Section 3.2.2, Theorem 5 and Figure 2 (stochastic gradient descent).
• B_StochasticGradientDescentWithAveraging Code for reproducing the second result of Section 3.2.2, Theorem 6 (stochastic gradient descent with averaging).
• C_StochasticGradientDescentWithPrimalAveraging Code for reproducing the third result of Section 3.2.2, Theorem 7 (stochastic gradient descent with primal averaging).
• D_StochasticGradientDescent_EvaluationAtAveragedPoint Code for reproducing the result of Appendix D.4, Theorem 12 (stochastic gradient descent with evaluation at the averaged iterate).

Stochastic first-order methods: unbiased oracles arising from sampling in expectations of smooth convex functions

Over-parametrized models

• A_ParameterSelection Code for reproducing the result of Section 4 and Appendix E, Theorem 8 and Figure 5 (obtaining stochastic gradient descent with primal averaging, using the parameter selection technique).

Bounded variance at optimum

• A_ParameterSelection Code for recovering the result of Appendix G, Theorem 15 (stochastic gradient descent with primal averaging from the parameter selection technique).

Stochastic first-order methods: unbiased oracles under weak growth conditions

• A_StochasticGradientDescentWithPrimalAveraging Code for verifying the result of Appendix F, Theorem 15 (stochastic gradient descent with primal averaging, parameter selection technique).