Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of work has begun to analyze this paradigm, existing bounds leave open the question of how sharp current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage $M$-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the \emph{intrinsic} parameters of the pre-training representation, which we link with the downstream predictor through a notion of \emph{orbit-invariance}, precisely characterizing the limiting distribution of the downstream test risk. We apply our results to spectral pre-training, factor models, and Gaussian mixture models, obtaining substantial improvements in problem-specific factors over prior art when applicable.

Diffusion-based generative models provide a powerful framework for learning to sample from a complex target distribution. The remarkable empirical success of these models applied to high-dimensional signals, including images and video, stands in stark contrast to classical results highlighting the curse of dimensionality for distribution recovery. In this work, we take a step towards understanding this gap through a careful analysis of learning diffusion models over the Barron space of single hidden layer neural networks. In particular, we show that these shallow models provably adapt to simple forms of low-dimensional structure, such as an unknown linear subspace or hidden independence, thereby avoiding the curse of dimensionality. We combine our results with recent analyses of sampling with diffusions to provide an end-to-end sample complexity bound for learning to sample from structured distributions. Importantly, our results do not require specialized architectures tailored to particular latent structures, and instead rely on the low-index structure of the Barron space to adapt to the underlying distribution.

Backward stochastic differential equation (BSDE)-based deep learning methods provide an alternative to Physics-Informed Neural Networks (PINNs) for solving high-dimensional partial differential equations (PDEs), offering potential algorithmic advantages in settings such as stochastic optimal control, where the PDEs of interest are tied to an underlying dynamical system. However, standard BSDE-based solvers have empirically been shown to underperform relative to PINNs in the literature. In this paper, we identify the root cause of this performance gap as a discretization bias introduced by the standard Euler-Maruyama (EM) integration scheme applied to one-step self-consistency BSDE losses, which shifts the optimization landscape off target. We find that this bias cannot be satisfactorily addressed through finer step-sizes or multi-step self-consistency losses. To properly handle this issue, we propose a Stratonovich-based BSDE formulation, which we implement with stochastic Heun integration. We show that our proposed approach completely eliminates the bias issues faced by EM integration. Furthermore, our empirical results show that our Heun-based BSDE method consistently outperforms EM-based variants and achieves competitive results with PINNs across multiple high-dimensional benchmarks. Our findings highlight the critical role of integration schemes in BSDE-based PDE solvers, an algorithmic detail that has received little attention thus far in the literature.

In this work, we study statistical learning with dependent data and square loss in a hypothesis class with tail decay in Orlicz space: $\mathscr{F}\subset L_{\Psi_p}$. Our inquiry is motivated by the search for a sharp noise interaction term, or variance proxy, in learning with dependent (e.g. $\beta$-mixing) data. Typical non-asymptotic results exhibit variance proxies that are deflated multiplicatively in the mixing time of the underlying covariates process. We show that whenever the topologies of $L^2$ and $\Psi_p$ are comparable on our hypothesis class $\mathscr{F}$, the empirical risk minimizer achieves a rate that only depends on the complexity of the class and second order statistics in its leading term. We refer to this as a near mixing-free rate, since direct dependence on mixing is relegated to an additive higher order term. Our approach, reliant on mixed tail generic chaining, allows us to obtain sharp, instance-optimal rates. Examples that satisfy our framework include for instance sub-Gaussian linear regression and bounded smoothness classes.

We initiate a study of supervised learning from many independent sequences ("trajectories") of non-independent covariates, reflecting tasks in sequence modeling, control, and reinforcement learning. Conceptually, our multi-trajectory setup sits between two traditional settings in statistical learning theory: learning from independent examples and learning from a single auto-correlated sequence. Our conditions for efficient learning generalize the former setting---trajectories must be non-degenerate in ways that extend standard requirements for independent examples. Notably, we do not require that trajectories be ergodic, long, nor strictly stable. For linear least-squares regression, given $n$-dimensional examples produced by $m$ trajectories, each of length $T$, we observe a notable change in statistical efficiency as the number of trajectories increases from a few (namely $m \lesssim n$) to many (namely $m \gtrsim n$). Specifically, we establish that the worst-case error rate of this problem is $\Theta(n / m T)$ whenever $m \gtrsim n$. Meanwhile, when $m \lesssim n$, we establish a (sharp) lower bound of $\Omega(n^2 / m^2 T)$ on the worst-case error rate, realized by a simple, marginally unstable linear dynamical system. A key upshot is that, in domains where trajectories regularly reset, the error rate eventually behaves as if all of the examples were independent, drawn from their marginals. As a corollary of our analysis, we also improve guarantees for the linear system identification problem.

In reinforcement learning (RL), state representations are key to dealing with large or continuous state spaces. While one of the promises of deep learning algorithms is to automatically construct features well-tuned for the task they try to solve, such a representation might not emerge from end-to-end training of deep RL agents. To mitigate this issue, auxiliary objectives are often incorporated into the learning process and help shape the learnt state representation. Bootstrapping methods are today's method of choice to make these additional predictions. Yet, it is unclear which features these algorithms capture and how they relate to those from other auxiliary-task-based approaches. In this paper, we address this gap and provide a theoretical characterization of the state representation learnt by temporal difference learning (Sutton, 1988). Surprisingly, we find that this representation differs from the features learned by Monte Carlo and residual gradient algorithms for most transition structures of the environment in the policy evaluation setting. We describe the efficacy of these representations for policy evaluation, and use our theoretical analysis to design new auxiliary learning rules. We complement our theoretical results with an empirical comparison of these learning rules for different cumulant functions on classic domains such as the four-room domain (Sutton et al, 1999) and Mountain Car (Moore, 1990).

A common pipeline in learning-based control is to iteratively estimate a model of system dynamics, and apply a trajectory optimization algorithm - e.g. $\mathtt{iLQR}$ - on the learned model to minimize a target cost. This paper conducts a rigorous analysis of a simplified variant of this strategy for general nonlinear systems. We analyze an algorithm which iterates between estimating local linear models of nonlinear system dynamics and performing $\mathtt{iLQR}$-like policy updates. We demonstrate that this algorithm attains sample complexity polynomial in relevant problem parameters, and, by synthesizing locally stabilizing gains, overcomes exponential dependence in problem horizon. Experimental results validate the performance of our algorithm, and compare to natural deep-learning baselines.

We consider how to most efficiently leverage teleoperator time to collect data for learning robust image-based value functions and policies for sparse reward robotic tasks. To accomplish this goal, we modify the process of data collection to include more than just successful demonstrations of the desired task. Instead we develop a novel protocol that we call Visual Backtracking Teleoperation (VBT), which deliberately collects a dataset of visually similar failures, recoveries, and successes. VBT data collection is particularly useful for efficiently learning accurate value functions from small datasets of image-based observations. We demonstrate VBT on a real robot to perform continuous control from image observations for the deformable manipulation task of T-shirt grasping. We find that by adjusting the data collection process we improve the quality of both the learned value functions and policies over a variety of baseline methods for data collection. Specifically, we find that offline reinforcement learning on VBT data outperforms standard behavior cloning on successful demonstration data by 13 % when both methods are given equal-sized datasets of 60 minutes of data from the real robot.

We derive upper bounds for random design linear regression with dependent ($\beta$-mixing) data absent any realizability assumptions. In contrast to the strictly realizable martingale noise regime, no sharp \emph{instance-optimal} non-asymptotics are available in the literature. Up to constant factors, our analysis correctly recovers the variance term predicted by the Central Limit Theorem---the noise level of the problem---and thus exhibits graceful degradation as we introduce misspecification. Past a burn-in, our result is sharp in the moderate deviations regime, and in particular does not inflate the leading order term by mixing time factors.

Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These observations have important implications for modern deep learning, but remain poorly understood due to the difficulty of the associated nonconvex optimization. Towards an understanding of this phenomenon, we analyze a family of pseudogradient methods for learning generalized linear models under the square loss – a simplified problem containing both nonlinearity in the model parameters and nonconvexity of the optimization which admits a single neuron as a special case. We prove non-asymptotic bounds on the generalization error that sharply characterize how the interplay between the optimization geometry and the feature space geometry sets the out-of-sample performance of the learned model. Experimentally, selecting the optimization geometry as suggested by our theory leads to improved performance in generalized linear model estimation problems such as nonlinear and nonconvex variants of sparse vector recovery and low-rank matrix sensing.

In reinforcement learning, state representations are used to tractably deal with large problem spaces. State representations serve both to approximate the value function with few parameters, but also to generalize to newly encountered states. Their features may be learned implicitly (as part of a neural network) or explicitly (for example, the successor representation of Dayan(1993). While the approximation properties of representations are reasonably well-understood, a precise characterization of how and when these representations generalize is lacking. In this work, we address this gap and provide an informative bound on the generalization error arising from a specific state representation. This bound is based on the notion of effective dimension which measures the degree to which knowing the value at one state informs the value at other states. Our bound applies to any state representation and quantifies the natural tension between representations that generalize well and those that approximate well. We complement our theoretical results with an empirical survey of classic representation learning methods from the literature and results on the Arcade Learning Environment, and find that the generalization behaviour of learned representations is well-explained by their effective dimension.

We propose Taylor Series Imitation Learning (TaSIL), a simple augmentation to standard behavior cloning losses in the context of continuous control. TaSIL penalizes deviations in the higher-order Tayler series terms between the learned and expert policies. We show that experts satisfying a notion of incremental input-to-state stability are easy to learn, in the sense that that a small TaSIL-augmented imitation loss over expert trajectories guarantees a small imitation loss over trajectories generated by the learned policy. We provide sample-complexity bounds for TaSIL that scale as $\tilde{\mathcal{O}}(1/n)$ in the realizable setting, for $n$ the number of expert demonstrations. Finally, we demonstrate experimentally the relationship between the robustness of the expert policy and the order of Taylor expansion required in TaSIL, and compare standard Behavior Cloning, DART, and DAgger with TaSIL-loss-augmented variants. In all cases, we show significant improvement over baselines across a variety of MuJoCo tasks.

We study square loss in a realizable time-series framework with martingale difference noise. Our main result is a fast rate excess risk bound which shows that whenever a trajectory hypercontractivity condition holds, the risk of the least-squares estimator on dependent data matches the iid rate order-wise after a burn-in time. In comparison, many existing results in learning from dependent data have rates where the effective sample size is deflated by a factor of the mixing-time of the underlying process, even after the burn-in time. Furthermore, our results allow the covariate process to exhibit long range correlations which are substantially weaker than geometric ergodicity. We call this phenomenon learning with little mixing, and present several examples for when it occurs: bounded function classes for which the $L^2$ and $L^{2+\epsilon}$ norms are equivalent, finite state irreducible and aperiodic Markov chains, various parametric models, and a broad family of infinite dimensional $\ell^2(\mathbb{N})$ ellipsoids. By instantiating our main result to system identification of nonlinear dynamics with generalized linear model transitions, we obtain a nearly minimax optimal excess risk bound after only a polynomial burn-in time.

A key assumption in the theory of nonlinear adaptive control is that the uncertainty of the system can be expressed in the linear span of a set of known basis functions. While this assumption leads to efficient algorithms, it limits applications to very specific classes of systems. We introduce a novel nonparametric adaptive algorithm that estimates an infinite-dimensional density over parameters online to learn an unknown dynamics in a reproducing kernel Hilbert space. Surprisingly, the resulting control input admits an analytical expression that enables its implementation despite its underlying infinite-dimensional structure. While this adaptive input is rich and expressive -- subsuming, for example, traditional linear parameterizations -- its computational complexity grows linearly with time, making it comparatively more expensive than its parametric counterparts. Leveraging the theory of random Fourier features, we provide an efficient randomized implementation that recovers the complexity of classical parametric methods while provably retaining the expressivity of the nonparametric input. In particular, our explicit bounds only depend polynomially on the underlying parameters of the system, allowing our proposed algorithms to efficiently scale to high-dimensional systems. As an illustration of the method, we demonstrate the ability of the randomized approximation algorithm to learn a predictive model of a 60-dimensional system consisting of ten point masses interacting through Newtonian gravitation. By reinterpretation as a gradient flow on a specific loss, we conclude with a natural extension of our kernel-based adaptive algorithms to deep neural networks. We show empirically that the extra expressivity afforded by deep representations can lead to improved performance at the expense of the closed-loop stability that is rigorously guaranteed and consistently observed for kernel machines.

The effectiveness of model-based versus model-free methods is a long-standing question in reinforcement learning (RL). Motivated by recent empirical success of RL on continuous control tasks, we study the sample complexity of popular model-based and model-free algorithms on the Linear Quadratic Regulator (LQR). We show that for policy evaluation, a simple model-based plugin method requires asymptotically less samples than the classical least-squares temporal difference (LSTD) estimator to reach the same quality of solution; the sample complexity gap between the two methods can be at least a factor of state dimension. For policy optimization, we study a simple family of problem instances and show that nominal (certainty equivalence principle) control also requires several factors of state and input dimension fewer samples than the policy gradient method to reach the same level of control performance on these instances. Furthermore, the gap persists even when employing baselines commonly used in practice. To the best of our knowledge, this is the first theoretical result which demonstrates a separation in the sample complexity between model-based and model-free methods on a continuous control task.

We study the sample complexity of approximate policy iteration (PI) for the Linear Quadratic Regulator (LQR), building on a recent line of work using LQR as a testbed to understand the limits of reinforcement learning (RL) algorithms on continuous control tasks. Our analysis quantifies the tension between policy improvement and policy evaluation, and suggests that policy evaluation is the dominant factor in terms of sample complexity. Specifically, we show that to obtain a controller that is within $\varepsilon$ of the optimal LQR controller, each step of policy evaluation requires at most $(n+d)^3/\varepsilon^2$ samples, where $n$ is the dimension of the state vector and $d$ is the dimension of the input vector. On the other hand, only $\log(1/\varepsilon)$ policy improvement steps suffice, resulting in an overall sample complexity of $(n+d)^3 \varepsilon^{-2} \log(1/\varepsilon)$. We furthermore build on our analysis and construct a simple adaptive procedure based on $\varepsilon$-greedy exploration which relies on approximate PI as a sub-routine and obtains $T^{2/3}$ regret, improving upon a recent result of Abbasi-Yadkori et al. 2019.

We study the performance of the certainty equivalent controller on Linear Quadratic (LQ) control problems with unknown transition dynamics. We show that for both the fully and partially observed settings, the sub-optimality gap between the cost incurred by playing the certainty equivalent controller on the true system and the cost incurred by using the optimal LQ controller enjoys a fast statistical rate, scaling as the square of the parameter error. To the best of our knowledge, our result is the first sub-optimality guarantee in the partially observed Linear Quadratic Gaussian (LQG) setting. Furthermore, in the fully observed Linear Quadratic Regulator (LQR), our result improves upon recent work by Dean et al., who present an algorithm achieving a sub-optimality gap linear in the parameter error. A key part of our analysis relies on perturbation bounds for discrete Riccati equations. We provide two new perturbation bounds, one that expands on an existing result from Konstantinov, and another based on a new elementary proof strategy.

We prove that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory. Our upper bound relies on a generalization of Mendelson’s small-ball method to dependent data, eschewing the use of standard mixing-time arguments. Our lower bounds reveal that these upper bounds match up to logarithmic factors. In particular, we capture the correct signal-to-noise behavior of the problem, showing that \emph{more unstable} linear systems are \emph{easier} to estimate. This behavior is qualitatively different from arguments which rely on mixing-time calculations that suggest that unstable systems are more difficult to estimate. We generalize our technique to provide bounds for a more general class of linear response time-series.

Reinforcement learning (RL) has been successfully used to solve many continuous control tasks. Despite its impressive results however, fundamental questions regarding the sample complexity of RL on continuous problems remain open. We study the performance of RL in this setting by considering the behavior of the Least-Squares Temporal Difference (LSTD) estimator on the classic Linear Quadratic Regulator (LQR) problem from optimal control. We give the first finite-time analysis of the number of samples needed to estimate the value function for a fixed static state-feedback policy to within epsilon-relative error. In the process of deriving our result, we give a general characterization for when the minimum eigenvalue of the empirical covariance matrix formed along the sample path of a fast-mixing stochastic process concentrates above zero, extending a result by Koltchinskii and Mendelson in the independent covariates setting. Finally, we provide experimental evidence indicating that our analysis correctly captures the qualitative behavior of LSTD on several LQR instances.

We consider adaptive control of the Linear Quadratic Regulator (LQR), where an unknown linear system is controlled subject to quadratic costs. Leveraging recent developments in the estimation of linear systems and in robust controller synthesis, we present the first provably polynomial time algorithm that achieves sub-linear regret on this problem. We further study the interplay between regret minimization and parameter estimation by proving a lower bound on the expected regret in terms of the exploration schedule used by any algorithm. Finally, we conduct a numerical study comparing our robust adaptive algorithm to other methods from the adaptive LQR literature, and demonstrate the flexibility of our proposed method by extending it to a demand forecasting problem subject to state constraints.

Recent work by Nesterov and Stich (2016) showed that momentum can be used to accelerate the rate of convergence for block Gauss-Seidel in the setting where a fixed partitioning of the coordinates is chosen ahead of time. We show that this setting is too restrictive, constructing instances where breaking locality by running non-accelerated Gauss-Seidel with randomly sampled coordinates substantially outperforms accelerated Gauss-Seidel with any fixed partitioning. Motivated by this finding, we analyze the accelerated block Gauss-Seidel algorithm in the random coordinate sampling setting. Our analysis captures the benefit of acceleration with a new data-dependent parameter which is well behaved when the matrix sub-blocks are well-conditioned. Empirically, we show that accelerated Gauss-Seidel with random coordinate sampling provides speedups for large scale machine learning tasks when compared to non-accelerated Gauss-Seidel and the classical conjugate-gradient algorithm.

In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n1 \times n2 matrix of rank r when the number of measurements exceeds a constant times (n1 + n2)r.

Silo is a new in-memory database that achieves excellent performance and scalability on modern multicore machines. Silo was designed from the ground up to use system memory and caches efficiently. For instance, it avoids all centralized contention points, including that of centralized transaction ID assignment. Silo's key contribution is a commit protocol based on optimistic concurrency control that provides serializability while avoiding all shared-memory writes for records that were only read. Though this might seem to complicate the enforcement of a serial order, correct logging and recovery is provided by linking periodically-updated epochs with the commit protocol. Silo provides the same guarantees as any serializable database without unnecessary scalability bottlenecks or much additional latency. Silo achieves almost 700,000 transactions per second on a standard TPC-C workload mix on a 32-core machine, as well as near-linear scalability. Considered per core, this is several times higher than previously reported results.

The traditional wisdom for building disk-based relational database management systems (DBMS) is to organize data in heavily-encoded blocks stored on disk, with a main memory block cache. In order to improve performance given high disk latency, these systems use a multi-threaded architecture with dynamic record-level locking that allows multiple transactions to access the database at the same time. Previous research has shown that this results in substantial overhead for on-line transaction processing (OLTP) applications [15]. The next generation DBMSs seek to overcome these limitations with architecture based on main memory resident data. To overcome the restriction that all data fit in main memory, we propose a new technique, called anti-caching, where cold data is moved to disk in a transactionally-safe manner as the database grows in size. Because data initially resides in memory, an anti-caching architecture reverses the traditional storage hierarchy of disk-based systems. Main memory is now the primary storage device. We implemented a prototype of our anti-caching proposal in a high-performance, main memory OLTP DBMS and performed a series of experiments across a range of database sizes, workload skews, and read/write mixes. We compared its performance with an open-source, disk-based DBMS optionally fronted by a distributed main memory cache. Our results show that for higher skewed workloads the anti-caching architecture has a performance advantage over either of the other architectures tested of up to 9× for a data size 8× larger than memory.

MONOMI is a system for securely executing analytical workloads over sensitive data on an untrusted database server. MONOMI works by encrypting the entire database and running queries over the encrypted data. MONOMI introduces split client/server query execution, which can execute arbitrarily complex queries over encrypted data, as well as several techniques that improve performance for such workloads, including per-row precomputation, space-efficient encryption, grouped homomorphic addition, and pre-filtering. Since these optimizations are good for some queries but not others, MONOMI introduces a designer for choosing an efficient physical design at the server for a given workload, and a planner to choose an efficient execution plan for a given query at runtime. A prototype of MONOMI running on top of Postgres can execute most of the queries from the TPC-H benchmark with a median overhead of only 1.24× (ranging from 1.03×to 2.33×) compared to an un-encrypted Postgres database where a compromised server would reveal all data.

Scripting languages are widely used to quickly accomplish a variety of tasks because of the high productivity they enable. Among other reasons, this increased productivity results from a combination of extensive libraries, fast development cycle, dynamic typing, and polymorphism. The dynamic features of scripting languages are traditionally associated with interpreters, which is the approach used to implement most scripting languages. Although easy to implement, interpreters are generally slow, which makes scripting languages prohibitive for implementing large, CPU-intensive applications. This efficiency problem is particularly important for PHP given that it is the most commonly used language for server-side web development. This paper presents the design, implementation, and an evaluation of the HipHop compiler for PHP. HipHop goes against the standard practice and implements a very dynamic language through static compilation. After describing the most challenging PHP features to support through static compilation, this paper presents HipHop's design and techniques that support almost all PHP features. We then present a thorough evaluation of HipHop running both standard benchmarks and the Facebook web site. Overall, our experiments demonstrate that HipHop is about 5.5x faster than standard, interpreted PHP engines. As a result, HipHop has reduced the number of servers needed to run Facebook and other web sites by a factor between 4 and 6, thus drastically cutting operating costs.

Large-scale, user-facing applications are increasingly moving from relational databases to distributed key/value stores for high-request-rate, low-latency workloads. Often, this move is motivated not only by key/value stores' ability to scale simply by adding more hardware, but also by the easy to understand predictable performance they provide for all operations. For complex queries, this approach often requires onerous explicit index management and imperative data lookup by the developer. We propose PIQL, a Performance Insightful Query Language that allows developers to express many queries found on these websites while still providing strict bounds on the number of I/O operations that will be performed.

Large-scale websites are increasingly moving from relational databases to distributed key-value stores for high request rate, low latency workloads. Often this move is motivated not only by key-value stores' ability to scale simply by adding more hardware, but also by the easy to understand predictable performance they provide for all operations. While this data model works well, lookups are only done by primary key. More complex queries require onerous, explicit index management and imperative data lookups by the developer. We demonstrate PIQL, a Performance Insightful Query Language that allows developers to express many of the queries found on these websites, while still providing strict bounds on the number of I/O operations for any query.