Near-optimal Offline and Streaming Algorithms for Learning Non-Linear Dynamical Systems

We consider the setting of vector valued non-linear dynamical systems $X_{t+1} = \phi(A^* X_t) + \eta_t$, where $\eta_t$ is unbiased noise and $\phi : \mathbb{R} \to \mathbb{R}$ is a known link function that satisfies certain {\em expansivity property}. The goal is to learn $A^*$ from a single trajectory $X_1,\cdots,X_T$ of {\em dependent or correlated} samples. While the problem is well-studied in the linear case, where $\phi$ is identity, with optimal error rates even for non-mixing systems, existing results in the non-linear case hold only for mixing systems. In this work, we improve existing results for learning nonlinear systems in a number of ways: a) we provide the first offline algorithm that can learn non-linear dynamical systems without the mixing assumption, b) we significantly improve upon the sample complexity of existing results for mixing systems, c) in the much harder one-pass, streaming setting we study a SGD with Reverse Experience Replay ($\mathsf{SGD-RER}$) method, and demonstrate that for mixing systems, it achieves the same sample complexity as our offline algorithm, d) we justify the expansivity assumption by showing that for the popular ReLU link function -- a non-expansive but easy to learn link function with i.i.d. samples -- any method would require exponentially many samples (with respect to dimension of $X_t$) from the dynamical system. We validate our results via. simulations and demonstrate that a naive application of SGD can be highly sub-optimal. Indeed, our work demonstrates that for correlated data, specialized methods designed for the dependency structure in data can significantly outperform standard SGD based methods.

Sample Efficient Linear Meta-Learning by Alternating Minimization

Meta-learning synthesizes and leverages the knowledge from a given set of tasks to rapidly learn new tasks using very little data. Meta-learning of linear regression tasks, where the regressors lie in a low-dimensional subspace, is an extensively-studied fundamental problem in this domain. However, existing results either guarantee highly suboptimal estimation errors, or require $\Omega(d)$ samples per task (where $d$ is the data dimensionality) thus providing little gain over separately learning each task. In this work, we study a simple alternating minimization method (MLLAM), which alternately learns the low-dimensional subspace and the regressors. We show that, for a constant subspace dimension MLLAM obtains nearly-optimal estimation error, despite requiring only $\Omega(\log d)$ samples per task. However, the number of samples required per task grows logarithmically with the number of tasks. To remedy this in the low-noise regime, we propose a novel task subset selection scheme that ensures the same strong statistical guarantee as MLLAM, even with bounded number of samples per task for arbitrarily large number of tasks.

Streaming Linear System Identification with Reverse Experience Replay

We consider the problem of estimating a stochastic linear time-invariant (LTI) dynamical system from a single trajectory via streaming algorithms. The problem is equivalent to estimating the parameters of vector auto-regressive (VAR) models encountered in time series analysis (Hamilton (2020)). A recent sequence of papers (Faradonbeh et al., 2018; Simchowitz et al., 2018; Sarkar and Rakhlin, 2019) show that ordinary least squares (OLS) regression can be used to provide optimal finite time estimator for the problem. However, such techniques apply for offline setting where the optimal solution of OLS is available apriori. But, in many problems of interest as encountered in reinforcement learning (RL), it is important to estimate the parameters on the go using gradient oracle. This task is challenging since standard methods like SGD might not perform well when using stochastic gradients from correlated data points (Gy\"orfi and Walk, 1996; Nagaraj et al., 2020). In this work, we propose a novel algorithm, SGD with Reverse Experience Replay (SGD-RER), that is inspired by the experience replay (ER) technique popular in the RL literature (Lin, 1992). SGD-RER divides data into small buffers and runs SGD backwards on the data stored in the individual buffers. We show that this algorithm exactly deconstructs the dependency structure and obtains information theoretically optimal guarantees for both parameter error and prediction error for standard problem settings. Thus, we provide the first - to the best of our knowledge - optimal SGD-style algorithm for the classical problem of linear system identification aka VAR model estimation. Our work demonstrates that knowledge of dependency structure can aid us in designing algorithms which can deconstruct the dependencies between samples optimally in an online fashion.

Do Input Gradients Highlight Discriminative Features?

Interpretability methods that seek to explain instance-specific model predictions [Simonyan et al. 2014, Smilkov et al. 2017] are often based on the premise that the magnitude of input-gradient -- gradient of the loss with respect to input -- highlights discriminative features that are relevant for prediction over non-discriminative features that are irrelevant for prediction. In this work, we introduce an evaluation framework to study this hypothesis for benchmark image classification tasks, and make two surprising observations on CIFAR-10 and Imagenet-10 datasets: (a) contrary to conventional wisdom, input gradients of standard models (i.e., trained on the original data) actually highlight irrelevant features over relevant features; (b) however, input gradients of adversarially robust models (i.e., trained on adversarially perturbed data) starkly highlight relevant features over irrelevant features. To better understand input gradients, we introduce a synthetic testbed and theoretically justify our counter-intuitive empirical findings. Our observations motivate the need to formalize and verify common assumptions in interpretability, while our evaluation framework and synthetic dataset serve as a testbed to rigorously analyze instance-specific interpretability methods.

Optimal Regret Algorithm for Pseudo-1d Bandit Convex Optimization

We study online learning with bandit feedback (i.e. learner has access to only zeroth-order oracle) where cost/reward functions $\f_t$ admit a "pseudo-1d" structure, i.e. $\f_t(\w) = \loss_t(\pred_t(\w))$ where the output of $\pred_t$ is one-dimensional. At each round, the learner observes context $\x_t$, plays prediction $\pred_t(\w_t; \x_t)$ (e.g. $\pred_t(\cdot)=\langle \x_t, \cdot\rangle$) for some $\w_t \in \mathbb{R}^d$ and observes loss $\loss_t(\pred_t(\w_t))$ where $\loss_t$ is a convex Lipschitz-continuous function. The goal is to minimize the standard regret metric. This pseudo-1d bandit convex optimization problem (\SBCO) arises frequently in domains such as online decision-making or parameter-tuning in large systems. For this problem, we first show a lower bound of $\min(\sqrt{dT}, T^{3/4})$ for the regret of any algorithm, where $T$ is the number of rounds. We propose a new algorithm \sbcalg that combines randomized online gradient descent with a kernelized exponential weights method to exploit the pseudo-1d structure effectively, guaranteeing the {\em optimal} regret bound mentioned above, up to additional logarithmic factors. In contrast, applying state-of-the-art online convex optimization methods leads to $\tilde{O}\left(\min\left(d^{9.5}\sqrt{T},\sqrt{d}T^{3/4}\right)\right)$ regret, that is significantly suboptimal in $d$.

Projection Efficient Subgradient Method and Optimal Nonsmooth Frank-Wolfe Method

We consider the classical setting of optimizing a nonsmooth Lipschitz continuous convex function over a convex constraint set, when having access to a (stochastic) first-order oracle (FO) for the function and a projection oracle (PO) for the constraint set. It is well known that to achieve $\epsilon$-suboptimality in high-dimensions, $\Theta(\epsilon^{-2})$ FO calls are necessary. This is achieved by the projected subgradient method (PGD). However, PGD also entails $O(\epsilon^{-2})$ PO calls, which may be computationally costlier than FO calls (e.g. nuclear norm constraints). Improving this PO calls complexity of PGD is largely unexplored, despite the fundamental nature of this problem and extensive literature. We present first such improvement. This only requires a mild assumption that the objective function, when extended to a slightly larger neighborhood of the constraint set, still remains Lipschitz and accessible via FO. In particular, we introduce MOPES method, which carefully combines Moreau-Yosida smoothing and accelerated first-order schemes. This is guaranteed to find a feasible $\epsilon$-suboptimal solution using only $O(\epsilon^{-1})$ PO calls and optimal $O(\epsilon^{-2})$ FO calls. Further, instead of a PO if we only have a linear minimization oracle (LMO, a la Frank-Wolfe) to access the constraint set, an extension of our method, MOLES, finds a feasible $\epsilon$-suboptimal solution using $O(\epsilon^{-2})$ LMO calls and FO calls---both match known lower bounds, resolving a question left open since White (1993). Our experiments confirm that these methods achieve significant speedups over the state-of-the-art, for a problem with costly PO and LMO calls.

Learning Minimax Estimators via Online Learning

We consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for constructing such estimators. We view the problem of designing minimax estimators as finding a mixed strategy Nash equilibrium of a zero-sum game. By leveraging recent results in online learning with non-convex losses, we provide a general algorithm for finding a mixed-strategy Nash equilibrium of general non-convex non-concave zero-sum games. Our algorithm requires access to two subroutines: (a) one which outputs a Bayes estimator corresponding to a given prior probability distribution, and (b) one which computes the worst-case risk of any given estimator. Given access to these two subroutines, we show that our algorithm outputs both a minimax estimator and a least favorable prior. To demonstrate the power of this approach, we use it to construct provably minimax estimators for classical problems such as estimation in the finite Gaussian sequence model, and linear regression.

Least Squares Regression with Markovian Data: Fundamental Limits and Algorithms

We study the problem of least squares linear regression where the data-points are dependent and are sampled from a Markov chain. We establish sharp information theoretic minimax lower bounds for this problem in terms of $\tau_{\mathsf{mix}}$, the mixing time of the underlying Markov chain, under different noise settings. Our results establish that in general, optimization with Markovian data is strictly harder than optimization with independent data and a trivial algorithm (SGD-DD) that works with only one in every $\tilde{\Theta}(\tau_{\mathsf{mix}})$ samples, which are approximately independent, is minimax optimal. In fact, it is strictly better than the popular Stochastic Gradient Descent (SGD) method with constant step-size which is otherwise minimax optimal in the regression with independent data setting. Beyond a worst case analysis, we investigate whether structured datasets seen in practice such as Gaussian auto-regressive dynamics can admit more efficient optimization schemes. Surprisingly, even in this specific and natural setting, Stochastic Gradient Descent (SGD) with constant step-size is still no better than SGD-DD. Instead, we propose an algorithm based on experience replay--a popular reinforcement learning technique--that achieves a significantly better error rate. Our improved rate serves as one of the first results where an algorithm outperforms SGD-DD on an interesting Markov chain and also provides one of the first theoretical analyses to support the use of experience replay in practice.

The Pitfalls of Simplicity Bias in Neural Networks

Several works have proposed Simplicity Bias (SB)---the tendency of standard training procedures such as Stochastic Gradient Descent (SGD) to find simple models---to justify why neural networks generalize well [Arpit et al. 2017, Nakkiran et al. 2019, Valle-Perez et al. 2019]. However, the precise notion of simplicity remains vague. Furthermore, previous settings that use SB to justify why neural networks generalize well do not simultaneously capture the brittleness of neural networks---a widely observed phenomenon in practice [Goodfellow et al. 2014, Jo and Bengio 2017]. To this end, we introduce a collection of piecewise-linear and image-based datasets that (a) naturally incorporate a precise notion of simplicity and (b) capture the subtleties of neural networks trained on real datasets. Through theory and experiments on these datasets, we show that SB of SGD and variants is extreme: neural networks rely exclusively on the simplest feature and remain invariant to all predictive complex features. Consequently, the extreme nature of SB explains why seemingly benign distribution shifts and small adversarial perturbations significantly degrade model performance. Moreover, contrary to conventional wisdom, SB can also hurt generalization on the same data distribution, as SB persists even when the simplest feature has less predictive power than the more complex features. We also demonstrate that common approaches for improving generalization and robustness---ensembles and adversarial training---do not mitigate SB and its shortcomings. Given the central role played by SB in generalization and robustness, we hope that the datasets and methods in this paper serve as an effective testbed to evaluate novel algorithmic approaches aimed at avoiding the pitfalls of extreme SB.

Follow the Perturbed Leader: Optimism and Fast Parallel Algorithms for Smooth Minimax Games

We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ worst-case regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is predictable, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of $O(T^{-1/2})$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making $O(T^{1/2})$ parallel calls to the optimization oracle.