9. Adaptive methods

9.1. Polyak steps in distance to optimum 

PEPit.examples.adaptive_methods.wc_polyak_steps_in_distance_to_optimum(L, mu, gamma, wrapper='cvxpy', solver=None, verbose=1)[source]

Consider the minimization problem

\[f_\star \triangleq \min_x f(x),\]

where \(f\) is \(L\)-smooth and \(\mu\)-strongly convex, and \(x_\star=\arg\min_x f(x)\).

This code computes a worst-case guarantee for a variant of a gradient method relying on Polyak step-sizes (PS). That is, it computes the smallest possible \(\tau(L, \mu, \gamma)\) such that the guarantee

\[\|x_{t+1} - x_\star\|^2 \leqslant \tau(L, \mu, \gamma) \|x_{t} - x_\star\|^2\]

is valid, where \(x_t\) is the output of the gradient method with PS and \(\gamma\) is the effective value of the step-size of the gradient method with PS.

In short, for given values of \(L\), \(\mu\), and \(\gamma\), \(\tau(L, \mu, \gamma)\) is computed as the worst-case value of \(\|x_{t+1} - x_\star\|^2\) when \(\|x_{t} - x_\star\|^2 \leqslant 1\).

Algorithm: Gradient descent is described by

\[x_{t+1} = x_t - \gamma \nabla f(x_t),\]

where \(\gamma\) is a step-size. The Polyak step-size rule under consideration here corresponds to choosing of \(\gamma\) satisfying:

\[\gamma \|\nabla f(x_t)\|^2 = 2 (f(x_t) - f_\star).\]

Theoretical guarantee: The gradient method with the variant of Polyak step-sizes under consideration enjoys the tight theoretical guarantee [1, Proposition 1]:

\[\|x_{t+1} - x_\star\|^2 \leqslant \tau(L, \mu, \gamma) \|x_{t} - x_\star\|^2,\]

where \(\gamma\) is the effective step-size used at iteration \(t\) and

\begin{eqnarray} \tau(L, \mu, \gamma) & = & \left\{\begin{array}{ll} \frac{(\gamma L-1)(1-\gamma \mu)}{\gamma(L+\mu)-1} & \text{if } \gamma\in[\tfrac{1}{L},\tfrac{1}{\mu}],\\ 0 & \text{otherwise.} \end{array}\right. \end{eqnarray}

References:

[1] M. Barré, A. Taylor, A. d’Aspremont (2020). Complexity guarantees for Polyak steps with momentum. In Conference on Learning Theory (COLT).

Parameters:

L (float) – the smoothness parameter.
mu (float) – the strong convexity parameter.
gamma (float) – the step-size.
wrapper (str) – the name of the wrapper to be used.
solver (str) – the name of the solver the wrapper should use.
verbose (int) –
level of information details to print.
- -1: No verbose at all.
- 0: This example’s output.
- 1: This example’s output + PEPit information.
- 2: This example’s output + PEPit information + solver details.

Returns:

pepit_tau (float) – worst-case value
theoretical_tau (float) – theoretical value

Example

>>> L = 1
>>> mu = 0.1
>>> gamma = 2 / (L + mu)
>>> pepit_tau, theoretical_tau = wc_polyak_steps_in_distance_to_optimum(L=L, mu=mu, gamma=gamma, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 4x4
(PEPit) Setting up the problem: performance measure is the minimum of 1 element(s)
(PEPit) Setting up the problem: Adding initial conditions and general constraints ...
(PEPit) Setting up the problem: initial conditions and general constraints (2 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 1 function(s)
                        Function 1 : Adding 6 scalar constraint(s) ...
                        Function 1 : 6 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 0 function(s)
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.6694214876445734
(PEPit) Primal feasibility check:
                The solver found a Gram matrix that is positive semi-definite
                All the primal scalar constraints are verified up to an error of 1.765730096858764e-11
(PEPit) Dual feasibility check:
                The solver found a residual matrix that is positive semi-definite
                All the dual scalar values associated with inequality constraints are nonnegative
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 3.680247484013155e-10
(PEPit) Final upper bound (dual): 0.6694214876573649 and lower bound (primal example): 0.6694214876445734
(PEPit) Duality gap: absolute: 1.2791434578218741e-11 and relative: 1.91081923934451e-11
*** Example file: worst-case performance of Polyak steps ***
        PEPit guarantee:         ||x_1 - x_*||^2 <= 0.669421 ||x_0 - x_*||^2
        Theoretical guarantee:   ||x_1 - x_*||^2 <= 0.669421 ||x_0 - x_*||^2

9.2. Polyak steps in function value 

PEPit.examples.adaptive_methods.wc_polyak_steps_in_function_value(L, mu, gamma, wrapper='cvxpy', solver=None, verbose=1)[source]

Consider the minimization problem

\[f_\star \triangleq \min_x f(x),\]

where \(f\) is \(L\)-smooth and \(\mu\)-strongly convex, and \(x_\star=\arg\min_x f(x)\).

This code computes a worst-case guarantee for a variant of a gradient method relying on Polyak step-sizes. That is, it computes the smallest possible \(\tau(L, \mu, \gamma)\) such that the guarantee

\[f(x_{t+1}) - f_\star \leqslant \tau(L, \mu, \gamma) (f(x_t) - f_\star)\]

is valid, where \(x_t\) is the output of the gradient method with PS and \(\gamma\) is the effective value of the step-size of the gradient method.

In short, for given values of \(L\), \(\mu\), and \(\gamma\), \(\tau(L, \mu, \gamma)\) is computed as the worst-case value of \(f(x_{t+1})-f_\star\) when \(f(x_t)-f_\star \leqslant 1\).

Algorithm: Gradient descent is described by

\[x_{t+1} = x_t - \gamma \nabla f(x_t),\]

where \(\gamma\) is a step-size. The Polyak step-size rule under consideration here corresponds to choosing of \(\gamma\) satisfying:

\[\|\nabla f(x_t)\|^2 = 2 L (2 - L \gamma) (f(x_t) - f_\star).\]

Theoretical guarantee: The gradient method with the variant of Polyak step-sizes under consideration enjoys the tight theoretical guarantee [1, Proposition 2]:

\[f(x_{t+1})-f_\star \leqslant \tau(L,\mu,\gamma) (f(x_{t})-f_\star),\]

where \(\gamma\) is the effective step-size used at iteration \(t\) and

\begin{eqnarray} \tau(L,\mu,\gamma) & = & \left\{\begin{array}{ll} (\gamma L - 1) (L \gamma (3 - \gamma (L + \mu)) - 1) & \text{if } \gamma\in[\tfrac{1}{L},\tfrac{2L-\mu}{L^2}],\\ 0 & \text{otherwise.} \end{array}\right. \end{eqnarray}