# Learning Models of Dynamical Systems from Finite Observations

SYSID, July 2024

## enabled by machine learning

$$\to$$

historical interactions

probability of new interaction

## Outline

1. Modern Finite-Sample Perspectives

2. Towards SysId for Personalization

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## Outline

1. Modern Finite-Sample Perspectives

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### iii) Extensions

• Traditional perspective: asymptotic statistics & consistency
• 90s-00s: beginnings of non-asymptotic analysis
• Last 5-10 years: modern analyses with a learning theory perspective

## Finite-sample Perspectives on SysId

Anastasios Tsiamis, Ingvar Ziemann, Nikolai Matni, George J. Pappas IEEE Control Systems Magazine

Sample Complexity: How much data is necessary to learn a system?

• Given error $$\epsilon$$ and failure probability $$\delta$$, how many samples are necessary to ensure $$\mathbb P(\text{estimation err.}\geq \epsilon)\leq\delta$$?

## Sample Complexity of LQR

Work with Horia Mania, Nikolai Matni, Ben Recht, and Stephen Tu in 2017

Motivation: foundation for understanding RL & ML-enabled control

Classic RL setting: discrete problems and inspired by games

RL techniques applied to continuous systems interacting with the physical world

Simplest problem: linear dynamics, quadratic cost, Gaussian process noise

minimize $$\mathbb{E}\left[ \displaystyle\lim_{T\to\infty}\frac{1}{T}\sum_{t=0}^T x_t^\top Q x_t + u_t^\top R u_t\right]$$

s.t.  $$x_{t+1} = Ax_t+Bu_t+w_t$$

$$u_t = \underbrace{-(R+B^\top P B)^{-1} B^\top P A}_{K_\star}x_t$$

where $$P=\text{DARE}(A,B,Q,R)$$ also defines the value function $$V(x) = x^\top P x$$

Static feedback controller is optimal and can be computed in closed-form:

## Sample Complexity Problem

How many observations are necessary to control unknown system?

### Main Result (Informal):

As long as $$N$$ is large enough, then with probability at least $$1-\delta$$,
$$\mathrm{rel.~error~of}~\widehat{\mathbf K}\lesssim \frac{\mathrm{size~of~noise}}{\mathrm{size~of~excitation}} \sqrt{\frac{\mathrm{dimension}}{N} \log(1/\delta)} \cdot\mathrm{robustness~of~}K_\star$$

excitation

$$(A_\star, B_\star)$$

$$N$$ observations

## Approach: Coarse-ID Control

1. Collect $$N$$ observations and estimate $$\widehat A,\widehat B$$ and confidence intervals

2. Use estimates to synthesize robust controller $$\widehat{\mathbf{K}}$$

$$(A_\star, B_\star)$$

$$\widehat{\mathbf{K}}$$

$$(A_\star, B_\star)$$

$$\|\hat A-A_\star\|\leq \epsilon_A$$

$$\|\hat B-B_\star\|\leq \epsilon_B$$

Main Control Result (Informal):

rel. error of $$\widehat{\mathbf K}\lesssim (\epsilon_A+\epsilon_B\|K_\star\|) \|\mathscr{R}_{A_\star+B_\star K_\star}\|_{\mathcal H_\infty}$$

robustness of $$K_\star$$

## System Identification

Least squares estimate: $$(\widehat A, \widehat B) \in \arg\min \sum_{\ell=1}^N \|Ax_{T}^{(\ell)} +B u_{T}^{(\ell)} - x_{T+1}^{(\ell)}\|^2$$

### Learning Result:

As long as $$N\gtrsim n+p+\log(1/\delta)$$, then with probability at least $$1-\delta$$,

$$\|\widehat A - A_\star\| \lesssim \frac{\sigma_w}{\sqrt{\lambda_{\min}(\sigma_u^2 G_T G_T^\top + \sigma_w^2 F_T F_T^\top )}} \sqrt{\frac{n+p }{N} \log(1/\delta)}$$,        $$\|\widehat B - B_\star\| \lesssim \frac{\sigma_w}{\sigma_u} \sqrt{\frac{n+p }{N} \log(1/\delta)}$$

with controllability Grammians defined as

$$G_T = \begin{bmatrix}A_\star^{T-1}B_\star&A_\star^{T-2}B_\star&\dots&B_\star\end{bmatrix} \qquad F_T = \begin{bmatrix}A_\star^{T-1}&A_\star^{T-2}&\dots&I\end{bmatrix}$$

$$(A_\star, B_\star)$$

Excitation

$$u_t^{(\ell)} \sim \mathcal{N}(0, \sigma_u^2)$$

Observe states $$\{x_t^{(\ell)}\}$$

## I.i.d. Random Matrix Analysis

have independent Gaussian entries

$$\begin{bmatrix} x_{T}^{(\ell)} \\u_{T}^{(\ell)} \end{bmatrix} \sim \mathcal{N}\left(0, \begin{bmatrix}\sigma_u^2 G_TG_T^\top + \sigma_w^2 F_TF_T^\top &\\ & \sigma_u^2 I\end{bmatrix}\right)$$

$$w_t^{(\ell)} \sim \mathcal{N}\left(0, \sigma_w^2\right)$$

The least-squares estimate is

$$\arg \min \|Z_N \begin{bmatrix} A & B\end{bmatrix} ^\top - X_N\|^2_F = (Z_N^\top Z_N)^\dagger Z_N^\top X_N$$

$$= \begin{bmatrix} A_\star & B_\star \end{bmatrix} ^\top + (Z_N^\top Z_N)^\dagger Z_N^\top W_N$$

Data and noise matrices

$$X_N = \begin{bmatrix} x_{T+1}^{(1)} & \dots & x_{T+1}^{(N)} \end{bmatrix}^\top$$

$$Z_N = \begin{bmatrix} x_{T}^{(1)} & \dots & x_{T}^{(N)} \\u_{T}^{(1)} & \dots & u_{T}^{(N)} \end{bmatrix}^\top$$

$$W_N = \begin{bmatrix} w_{T}^{(1)} & \dots & w_{T}^{(N)} \end{bmatrix}^\top$$

lower bound minimum singular value,

or compute data-dependent bound

upper bound inner products

## Single Trajectory Identification

• Multi-trajectory setting (previous slide): excitable & less stable systems are easier to learn
• Classic mixing time analysis: more stable systems are easier to learn because data is closer to being independent
• Followup work by Simchowitz et al. (2018) on Learning without Mixing show that strict stability is not required!$$\Big\|\begin{bmatrix} \widehat A - A_\star \\ \widehat B - B_\star\end{bmatrix}\Big\| \lesssim \frac{\sigma_w}{C_u \sigma_u}\sqrt{\frac{n+p }{T} \log(d/\delta)}$$

$$(A_\star, B_\star)$$

## Extension: Online Control

1. Excite system for $$N$$ steps and estimate $$\widehat A,\widehat B$$

2. Run controller $$\widehat{\mathbf{K}}$$ for remaining time  $$T-N$$

$$(A_\star, B_\star)$$

$$\widehat{\mathbf{K}}$$

$$(A_\star, B_\star)$$

• Exploration vs. Excitation
• Average sub-optimality of Robust Adaptive Control is $$\mathcal O(T^{-1/3})$$ [DMMRT18]
• For Certainty Equivalent Adaptive Control, $$\mathcal O(T^{-1/2})$$ (Mania et al., 2019)

## Further Extensions

1. Safety constraints [DTMR19]
• Robust controller compensates for bounded exploration noise to keep system safe during exploration
2. Perception-based control [DMRY20,DR21]
• Nonlinear observation model with $$y_t = g(x_t)$$ for invertible $$g$$
3. Any many others! Non-stochastic control, partial observation & LQG, set membership identification

## Outline

1. Modern Finite-Sample Perspectives

2. Towards SysId for Personalization

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outputs

time

## Outline

2. Towards SysId for Personalization

## Setting: Personalization

$$u_t$$

$$y_t$$

Classically studied as an online decision problem (e.g. multi-armed bandits)

unknown preference

expressed preferences

recommended content

recommender policy

## Setting: Personalization

$$u_t$$

unknown preference $$x$$

expressed preferences

recommended content

recommender policy

$$\mathbb E[y_t] = x^\top C u_t$$

goal: identify $$C^\top x$$ sufficiently well to make good recommendations

Classically studied as an online decision problem (e.g. multi-armed bandits)

Algorithms: Expore-then-Commit, $$\varepsilon$$-Greedy, Upper Confidence Bound

## Setting: Preference Dynamics

$$u_t$$

However, interests may be impacted by recommended content

preference state $$x_t$$

expressed preferences

recommended content

recommender policy

$$\mathbb E[y_t] = x_t^\top C u_t$$

updates to $$x_{t+1}$$

• Simple dynamics that capture assimilation (adapted from opinion dynamics) $$x_{t+1} \propto x_t + \eta_t u_t,\qquad y_t = x_t^\top u_t + v_t$$
• If $$\eta_t$$ constant, tends to homogenization globally
• If $$\eta_t \propto x_t^\top u_t$$ (i.e. biased assimilation), tends to polarization (Hązła et al., 2019)

## Example: Preference Dynamics

Implications for personalization [DM22]

1. It is not necessary to estimate preferences to make "good" recommendations

2. Preferences "collapse" towards whatever users are often recommended

3. Non-manipulation (and other goals) can be achieved through randomization

4. Even if harmful content is never recommended, can cause harm through preference shifts [CDEIKW24]

initial preference
resulting preference
recommendation

## Problem Setting: Identification

• Unknown dynamics and measurement functions
• Observed trajectory of inputs $$u\in\mathbb R^p$$ and outputs $$y\in\mathbb R$$ $$u_0,y_0,u_1,y_1,...,u_T,y_T$$
• Goal: identify dynamics and measurement models from data
• Setting: linear/bilinear with $$A\in\mathbb R^{n\times n}$$, $$B\in\mathbb R^{n\times p}$$, $$C\in\mathbb R^{p\times n}$$ $$x_{t+1} = Ax_t + Bu_t + w_t\\ y_t = u_t^\top Cx_t + v_t$$

e.g. playlist attributes

e.g. listen time

inputs $$u_t$$



outputs $$y_t$$

## Identification Algorithm

Input: data $$(u_0,y_0,...,u_T,y_T)$$, history length $$L$$, state dim $$n$$

Step 1: Regression

$$\hat G = \arg\min_{G\in\mathbb R^{p\times pL}} \sum_{t=L}^T \big( y_t - u_t^\top \textstyle \sum_{k=1}^L G[k] u_{t-k} \big)^2$$

Step 2: Decomposition $$\hat A,\hat B,\hat C = \mathrm{HoKalman}(\hat G, n)$$

$$t$$

$$L$$

$$\underbrace{\qquad\qquad}$$

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Yahya Sattar

$$~$$

## Estimation Errors

$$\hat G = \arg\min_{G\in\mathbb R^{p\times pL}} \sum_{t=L}^T \big( y_t - u_t^\top \textstyle \sum_{k=1}^L G[k] u_{t-k} \big)^2$$

• (Biased) estimate of Markov parameters $$G =\begin{bmatrix} C B & CA B & \dots & CA^{L-1} B \end{bmatrix}$$
• Regress $$y_t$$ against $$\underbrace{ \begin{bmatrix} u_{t-1}^\top & ... & u_{t-L}^\top \end{bmatrix}}_{\bar u_{t-1}^\top } \otimes u_t^\top$$
• Data matrix: circulant-like structure $$Z = \begin{bmatrix}\bar u_{L-1}^\top \otimes u_L^\top \\ \vdots \\ \bar u_{T-1}^\top \otimes u_T^\top\end{bmatrix}$$

$$t$$

$$L$$

$$\underbrace{\qquad\qquad}$$

$$\bar u_{t-1}^\top \otimes u_t^\top \mathrm{vec}(G)$$

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$$*$$

$$=$$

$$\underbrace{\qquad\qquad}$$

$$\underbrace{\qquad\qquad}$$

$$\underbrace{\qquad\qquad}$$

$$\underbrace{\qquad\qquad\qquad}$$

$$Z = \begin{bmatrix}\bar u_{L-1}^\top \otimes u_L^\top \\ \vdots \\ \bar u_{T-1}^\top \otimes u_T^\top\end{bmatrix}$$

## Main Results

Assumptions:

1. Process and measurement noise $$w_t,v_t$$ are i.i.d., zero mean, and have bounded second moments
2. Inputs $$u_t$$ are bounded
3. The dynamics are strictly stable, i.e. $$\rho(A)<1$$

### Informal Theorem (Markov parameter estimation)

With probability at least $$1-\delta$$, $$\|G-\hat G\|_{Z^\top Z} \lesssim \sqrt{ \frac{p^2 L}{\delta} \cdot c_{\mathrm{stability,noise}} }+ \rho(A)^L\sqrt{T} c_{\mathrm{stability}}$$

## Main Results

Assumptions:

1. Process and measurement noise $$w_t,v_t$$ are i.i.d., zero mean, and have bounded second moments
2. Inputs $$u_t$$ are bounded
3. The dynamics are strictly stable, i.e. $$\rho(A)<1$$
4. (For state space recovery: $$(A,B,C)$$ are observable, controllable)

### Informal Summary Theorem

With high probabilty, $$\mathrm{est.~errors} \lesssim \sqrt{ \frac{\mathsf{poly}(\mathrm{dimension})}{\sigma_{\min}(Z^\top Z)}}$$

## Open Q: Sample Complexity

### Informal Conjecture

When $$u_t$$ are chosen i.i.d. and sub-Gaussian and $$T$$ is large enough, whp $$\sigma_{\min}({Z^\top Z} )\gtrsim T$$

### Informal Corollary

For i.i.d. and sub-Gaussian inputs, whp $$\mathrm{est.~errors} \lesssim \sqrt{ \frac{\mathsf{poly}(\mathrm{dim.})}{T}}$$

How large does $$T$$ need to be to guarantee bounded estimation error?

$$*$$

$$=$$

formal analysis involves the structured random matrix $$Z$$

$$= \begin{bmatrix}\bar u_{L-1}^\top \otimes u_L^\top \\ \vdots \\ \bar u_{T-1}^\top \otimes u_T^\top\end{bmatrix}$$

## Open Qs:

• Sample complexity (data independent)
• Idea: random design
• Challenge: dependent matrix entries
• Relax assumption on stability of $$A$$
• Idea: regress $$y_t$$ against $$u_{0:t},y_{0:t-1}$$
• Challenge: sample complexity of partially observed bilinear sysid
• Effect on preference prediction?
• Idea: Kalman filter (certainty equivalent vs. robust)
• Effect on optimal recommendation?
• Challenge: difficult optimal control problem

## Conclusion

1. Modern Finite-Sample Perspectives

2. Towards SysId for Personalization

Questions about decision-making from the learning theory perspective motivate new system identification results

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1. On the sample complexity of the linear quadratic regulator in Foundations of Computational Mathematics (2019) with Horia Mania, Nikolai Matni, Ben Recht, Stephen Tu
2. Learning Linear Dynamics from Bilinear Observations (in preparation) with Yahya Sattar

## Thanks! Questions?

• Regret bounds for robust adaptive control of the linear quadratic regulator at NeuRIPS18 () with Mania, Matni, Recht, Tu
• Preference Dynamics Under Personalized Recommendations at EC22 (arxiv:2205.13026) with Jamie Morgenstern
• Harm Mitigation in Recommender Systems under User Preference Dynamics at KDD24 (arxiv:2406.09882) with Chee, Ernala, Ioannidis, Kalyanaraman, Weinsberg
2. Anderson, Doyle, Low, Matni. "System level synthesis." Annual Reviews in Control (2019).
3. Dean, Matni, Recht, Ya. "Robust Guarantees for Perception-Based Control." L4DC, 2020.

4. Dean, Tu, Matni, Recht. "Safely learning to control the constrained linear quadratic regulator." ACC, 2019

5. Gaitonde, Kleinberg, Tardos, 2021. Polarization in geometric opinion dynamics. EC.

6. Hązła, Jin, Mossel, Ramnarayan, 2019. A geometric model of opinion polarization. Mathematics of Operations Research.

7. Mania, Tu, Recht. "Certainty Equivalence is Efficient for Linear Quadratic Control." NeuRIPS, 2019.

8. Omyak & Ozay, 2019. Non-asymptotic Identification of LTI Systems from a Single Trajectory. ACC.

9. Simchowitz, Mania, Tu, Jordan, Recht. "Learning without mixing: Towards a sharp analysis of linear system identification." COLT, 2018.

## Main LQR Result

As long as $$N\gtrsim (n+p)\sigma_w^2\|\mathscr R_{A_\star+B_\star K_\star}\|_{\mathcal H_\infty}(1/\lambda_G + \|K_\star\|^2/\sigma_u^2)\log(1/\delta)$$, then for robustly synthesized $$\widehat{\mathbf{K}}$$, with probability at least $$1-\delta$$,

rel. error of $$\widehat{\mathbf K}$$ $$\lesssim \sigma_w (\frac{1}{\sqrt{\lambda_G}} + \frac{\|K_\star\|}{\sigma_u}) \|\mathscr{R}_{A_\star+B_\star K_\star}\|_{\mathcal H_\infty} \sqrt{\frac{n+p }{T} \log(1/\delta)}$$

robustness of optimal closed-loop

excitability of system

optimal controller gain

vs.

## Experimental Results

more details on affinity maximization, preference stationarity, and mode collapse

## Proof Sketch

$$\hat G = \arg\min_{G\in\mathbb R^{p\times pL}} \sum_{t=L}^T \big( y_t - \bar u_{t-1}^\top \otimes u_t^\top \mathrm{vec}(G) \big)^2$$

• Claim: this is a biased estimate of Markov parameters $$G_\star =\begin{bmatrix} C B & CA B & \dots & CA^{L-1} B \end{bmatrix}$$
• Observe that $$x_t = \sum_{k=1}^L CA^{k-1} B (u_{t-k} + w_{t-k}) + A^L x_{t-L}$$
• Hence, $$y_t= \bar u_{t-1}^\top \otimes u_t^\top \mathrm{vec}(G_\star) +u_t^\top \textstyle \sum_{k=1}^L CA^{k-1} w_{t-k} + u_t CA^L x_{t-L} + v_t$$
• Least squares: for $$y_t = z_t^\top \theta + n_t$$, the estimate  $$\hat\theta=\arg \min\sum_t (z_t^\top \theta - y_t)^2$$ $$= \textstyle\arg \min \|Z \theta - Y\|^2_2 = (Z^\top Z)^\dagger Z^\top Y= \theta_\star + (Z^\top Z)^\dagger Z^\top N$$
• Estimation errors are therefore $$\|G_\star -\hat G\|_{\tilde U^\top\tilde U} = \|\tilde U^\top N\|$$

## Main Results

Assumptions:

1. Process and measurement noise $$w_t,v_t$$ are i.i.d., zero mean, and have bounded second moments
2. Inputs $$u_t$$ are bounded
3. The dynamics $$(A,B,C)$$ are observable, controllable, and strictly stable, i.e. $$\rho(A)<1$$

### Informal Theorem (system identification)

Suppose $$L$$ is sufficiently large. Then, there exists a nonsingular matrix $$S$$ (i.e. a similarity transform) such that

$$\|A-S\hat AS^{-1}\|_{Z^\top Z}$$

$$\| B-S\hat B\|_{Z^\top Z}$$

$$\| C-\hat CS^{-1}\|_{Z^\top Z}$$

$$\lesssim c_{\mathrm{contr,obs,dim}} \|G-\hat G\|_{Z^\top Z}$$

$$\underbrace{\qquad\qquad}$$

By Sarah Dean

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