Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach

Abstract

This paper presents a discrete-time passivity-based analysis of the gradient descent method for a class of functions with sector-bounded gradients. Using a loop transformation, it is shown that the gradient descent method can be interpreted as a passive controller in negative feedback with a very strictly passive system. The passivity theorem is then used to guarantee input-output stability, as well as the global convergence, of the gradient descent method. Furthermore, provided that the lower and upper sector bounds are not equal, the input-output stability of the gradient descent method is guaranteed using the weak passivity theorem for a larger choice of step size. Finally, to demonstrate the utility of this passivity-based analysis, a new variation of the gradient descent method with variable step size is proposed by gain-scheduling the input and output of the gradient.

Figures (7)

• Figure 2: The negative feedback interconnection of two systems $\bm{\mathcal{G}}$ and $\boldsymbol{\Delta}$.
• Figure 3: The negative feedback interconnection of the GD controller, $\bm{\mathcal{G}}_{\textrm{GD}}$, and the nonlinearity $\boldsymbol{\Delta}$, being the shifted gradient that maps zero inputs to zero outputs.
• Figure 4: Loop transformation of the negative feedback interconnection representation of the GD method in \ref{['fig:gd_neg_feedback']}. The loop transformation introduces a feedthrough term $D\mbf{1}$, resulting in the modified GD controller $\bar{\bm{\mathcal{G}}}_{\textrm{GD}}$, the nonlinearity $\bar{\boldsymbol{\Delta}}$, and $\bar{\mbf{r}}_2^k = \mbf{r}_2^k - D\mbf{r}_1^k$.
• Figure 5: A discrete-time system $\bar{\boldsymbol{\Delta}}$, composed of the positive feedback interconnection of a VSP system $\boldsymbol{\Delta}$, and the feedthrough term $D\mbf{1}$.
• Figure 6: Two gain-scheduling architectures resulting in the same gain-scheduled modified GD controller $\bar{\bm{\mathcal{G}}}_\textrm{GS}$ with minimal realization $\mleft( \bar{\mbf{A}}, \bar{\mbf{B}}, \bar{\mbf{C}}, \bar{\mbf{D}} \mright) = (\mbf{1}, s^k\mbf{1}, s^k\mbf{1}, (s^k)^2 D\mbf{1})$.

Theorems & Definitions (21)

• Definition 1: Truncation operator feedback_systems
• Definition 2: Truncated inner product feedback_systems
• Definition 3: $\ell_{2e}$ and $\ell_{2}$ function spaces feedback_systems
• Definition 4: $\ell_{2}$ stability feedback_systems
• Definition 5: Passivity feedback_systems
• Lemma 1: Positive real anderson
• Remark 1
• Theorem 1: Weak passivity theorem feedback_systems
• Theorem 2: Strong passivity theorem feedback_systems
• Remark 2