Motivation: Online Optimal Control

Offline (hindsight) control problem

$$\min_{\pi} \sum_{t=1}^T c_t(s_t,a_t) \quad\text{s.t.}\quad s_{t+1}=f(s_t,a_t,w_t),~~a_t=\pi(s_t)$$

Outline

Online Convex Optimization (OCO) with Unbounded Memory is a framework which directly addresses these challenges

1. Problem Setting & Examples
2. Main Results: Regret Bounds
3. Applications & Conclusion

Problem Setting

Problem Setting

Online interaction protocol

• In round $t=1,2,...,T$:
  • Learner chooses decision $x_t\in\mathcal X$
  • History updates as $h_t = Ah_{t-1}+Bx_t$
  • Learner suffers loss $f_t(h_t)$ and observes $f_t$

$f_t$

$x_t$

Example: loss depends arbitrarily on $m$ past decisions (Anava et al., 2015)

• History space $\mathcal H = \mathcal X\times\dots\times\mathcal X = \mathcal X^m$
• Linear operators $A=\begin{bmatrix} & I \\ && \ddots \\ &&& I \\  &\end{bmatrix},\quad B = \begin{bmatrix} I \\ 0 \\ \vdots \end{bmatrix}$

$h_t$

Problem Setting

Online interaction protocol

• In round $t=1,2,...,T$:
  • Learner chooses decision $x_t\in\mathcal X$
  • History updates as $h_t = Ah_{t-1}+Bx_t$
  • Learner suffers loss $f_t(h_t)$ and observes $f_t$

$f_t$

$x_t$

Example: loss depends on all past decisions with $\rho$-discount factor

• History space $\mathcal H$ contains $T$ length sequences over $\mathcal X$
• Linear operators $$A(x_0, x_1, \dots)=(0, \rho x_0, \rho x_1, \dots ),\quad B x = (x,0,\dots )$$

$h_t$

Regret Minimization

$f_t$

$x_t$

Goal: perform well compared to the best fixed decision in hindsight

$h_t$

Assumptions & Definitions

$$\min_{a} \sum_{t=1}^T c_t(s_t,a) \quad\text{s.t.}\quad s_{t+1} = Fs_t+Ga + w_t$$

• History combines "noiseless" state with action $$\bar s_{t+1} = F\bar s_t + G a_t,\quad h_t = \begin{bmatrix} \bar s_t \\a_t \end{bmatrix}$$
• Linear operators defined by dynamics $$A=\begin{bmatrix} F\\ & \end{bmatrix},\quad B=\begin{bmatrix}G\\ I\end{bmatrix}$$
• Loss functions defined by cost & disturbance $$f_t(h_t) = c_t\left (\bar s_t + \sum_{k=1}^t F^k G w_{t-k}, a_t\right )=c_t(s_t, a_t)$$

Example: constant input linear control

• Theorem: there are algorithms such that the regret of an OCO with unbounded memory problem is at most $$O\left(\sqrt{T}\sqrt{H_p}\sqrt{L\tilde L}\right)$$
  • effective memory capacity & Lipschitz constants
• Theorem: there exists an OCO with unbounded memory problem with regret at least $$\Omega \left(\sqrt{T}\sqrt{H_p}\sqrt{L\tilde L}\right)$$

Main Results

• total loss from playing $x$ every round
• strongly convex regularizer
• step size $\eta = (T\tilde L(LH_p+\tilde L))^{-1/2}$

Upper Bound

Upper Bound

Algorithm:   Follow-the-Regularized-Leader on $\tilde f_t$

• For $t=1,\dots,T$ $$x_{t+1} = \min_{x\in\mathcal X} \sum_{k=1}^t \tilde f_k(x) + \frac{R(x)}{\eta}$$

Lower Bound

Application: Online Linear Control

$$\min_{K} \sum_{t=1}^T c_t(s_t,a_t) \quad\text{s.t.}\quad s_{t+1} = Fs_t+Ga + w_t ,~~a_t=Ks_t$$

• Convex lifting to disturbance-action controllers (Youla et al., 1976; Anderson et al., 2019; Agarwal et al., 2019) $$\textstyle a_t = \sum_{k=1}^{t+1}M^{[k]} w_{t-k},\quad X_t = (M^{[k]})_{k\in[t]}$$
• History contains (weighted) sequences of controllers $$H_t = (X_t, GX_{t-1}, FGX_{t-2},F^2 GX_{t-3},\dots )$$

Application: Online Linear Control

$$\min_{K} \sum_{t=1}^T c_t(s_t,a_t) \quad\text{s.t.}\quad s_{t+1} = Fs_t+Ga + w_t ,~~a_t=Ks_t$$

• Convex lifting to disturbance-action controllers (Youla et al., 1976; Anderson et al., 2019; Agarwal et al., 2019) $$\textstyle a_t = \sum_{k=1}^{t+1}M^{[k]} w_{t-k},\quad X_t = (M^{[k]})_{k\in[t]}$$
• History contains (weighted) sequences of controllers $$H_t = (X_t, GX_{t-1}, FGX_{t-2},F^2 GX_{t-3},\dots )$$
• Linear operators defined by linear dynamics $$A\left((Y_0,Y_1,\dots)\right) = (0, GY_0, F Y_1,\dots),\quad BX=(X,0,.\dots)$$
• States & actions are linear in history & decisions, so loss functions are defined by cost & disturbance $$f_t(H_t) = c_t\big ( \underbrace{\langle  H_t, w_{1:t} \rangle}_{(s_t,a_t)}\big)=c_t(s_t, a_t)$$

Application: Online Linear Control

$$\min_{K} \sum_{t=1}^T c_t(s_t,a_t) \quad\text{s.t.}\quad s_{t+1} = Fs_t+Ga + w_t ,~~a_t=Ks_t$$

• Approach: translate usual control assumptions on dynamics, controllers, and costs into the quantities
  $H_2$, $L$, and $\tilde L$
• No truncation analysis is necessary
• Improve existing upper bounds (Agarwal et al., 2019) by a factor of dimension and stability radius

Conclusion & Discussion

• Key takeaways for OCO with Unbounded Memory
  1. General framework which captures online control
  2. Matching upper & (worst case) lower regret bounds highlight fundamental quantity: effective memory capacity
• Open directions for future work
  1. Unknown dynamics $A$ and $B$
  2. "Bandit" feedback of $f_t(h_t)$
  3. Nonlinearly evolving history
  4. Nonconvex optimization

Thank you!

Online Convex Optimization with Unbounded Memory

https://arxiv.org/abs/2210.09903

Raunak Kumar    Sarah Dean    Robert Kleinberg

References:

• Agarwal, Bullins, Hazan, Kakade, Singh. "Online control with adversarial disturbances." ICML, 2019.​
• Anava, Hazan, Mannor. "Online learning for adversaries with memory: Price of past mistakes." NeurIPS, 2015.​
• Anderson, Doyle, Low, Matni. "System level synthesis." Annual Reviews in Control, 2019.
• Youla, Jabr, Bongiorno. "Modern Wiener-Hopf design of optimal controllers--Part II: The multivariable case." IEEE Transactions on Automatic Control, 1976.