Online Convex Optimization With Unbounded Memory

Sarah Dean

ACC Workshop, May 2023

joint work with Raunak Kumar and Bobby Kleinberg

Online interaction

Choose an action $a_t$ according to policy $\pi_t$
State updates according to $f$ and $w_t$
Pay a cost $c_t(s_t,a_t)$

Motivation: Online Optimal Control

Offline (hindsight) control problem

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

loss depends on all past actions

decision variable is a function

Outline

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

Problem Setting & Examples
Main Results: Regret Bounds
Applications & Conclusion

Components of OCO with memory problem

Decision space $\mathcal X$ is closed, convex subset of Hilbert space
History space $\mathcal H$ is Banach space
Linear operators $A:\mathcal H\to\mathcal H$ and $B:\mathcal X\to \mathcal H$
Convex loss functions $f_t:\mathcal H\to\mathbb R$

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 $f_t$

$f_t$

$x_t$

$h_t$

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 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 $$A(x_0, x_1, \dots)=(0, \rho x_0, \rho x_1, \dots ),\quad B x = (x,0,\dots )$$

$h_t$

The regret of an algorithm whose decisions result in $h_1,\dots,h_T$ is $$ R_T(\mathcal A) = \sum_{t=1}^T f_t(h_t) - \min_{x\in\mathcal X} \sum_{t=1}^T \underbrace{ f_t\left(\sum_{k=1}^t A^k B x\right)}_{\tilde f_t(x)}$$

Regret Minimization

$f_t$

$x_t$

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

$h_t$

Assumptions

Learner observes the function $f_t$ after each round, knows $A$ and $B$, and $\|B\|=1$
Functions $f_t$ are differentiable, $L$-Lipschitz continuous, and convex
- implies that $\tilde f_t = f_t\circ \sum_{k=1}^t A^k B $ are diff'ble, convex, Lipschitz with $\tilde L \leq L\sum_{k=0}^\infty \|A^k \|$

Assumptions & Definitions

Definition ($p$-effective memory capacity): $\displaystyle H_p = \left( \sum_{k=0}^\infty k^p \|A^k\|^p \right)^{1/p}$

Bounds distance in history resulting from decisions whose distance grows at most linearly with time

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

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

$f_t$

$h_t$

$x_t$

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

Upper Bound

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

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

lazy

if $t \mod \frac{LH_p}{\tilde L}=0$, otherwise $x_{t+1}=x_t$

Upper Bound

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

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

Proof Sketch. Decompose regret into two terms $$R_T(\mathcal A) = \textstyle \sum_{t=1}^T f_t(h_t) - \sum_{t=1}^T \tilde f_t(x_t) + \sum_{t=1}^T \tilde f_t(x_t) - \min_{x\in\mathcal X} \sum_{t=1}^T \tilde f_t(x)$$

Standard OCO with FTRL: $\eta^{-1} + \eta T\tilde L^2$
Actual vs. idealized history: $\eta T L \tilde L H_p$

The following instance of OCO with finite memory has $$R_T(\mathcal A) \geq \Omega \left(\sqrt{T}\sqrt{H_p}\sqrt{L\tilde L}\right) = \Omega \left(\sqrt{T} m\right)\quad \forall~~\mathcal A$$

Lower Bound

Let $\mathcal X = [-1,1]$, finite memory $\mathcal H = \mathcal X^m$, Rademacher samples $w_1,\dots w_{\frac{T}{m}}$, and $$ f_t(h_t) = w_{\lceil\frac{t}{m}\rceil} m^{-1/2} (x_{t-m+1} + \dots + x_{m\lfloor\frac{t}{m}\rfloor + 1})$$

$m$ steps in the past

time of sample

$h_t$ and $f_t$

$w_i$

$w_{i-1}$

$w_{i+1}$

$\underbrace{\qquad\qquad}$

$m$

$t$

Application: Online Linear Control

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

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 $$H_t = (X_t, GX_{t-1}, FGX_{t-2},F^2 GX_{t-3},\dots )$$

instead of a loop,

system looks like a line

$(F,G)$

$K$

$\bf s$

$\bf a$

$\bf w$

$\bf s$

$\bf a$

$\bf w$

$X$

$H$

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 $$f_t(H_t) = c_t\big ( \underbrace{\langle H_t, w_{1:t} \rangle}_{(s_t,a_t)}\big)=c_t(s_t, a_t)$$

$\bf s$

$\bf a$

$\bf w$

$X_t$

$H$

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

Questions?

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.