arXiv subjects

Dheeraj Nagaraj

Publications and source records attributed to Dheeraj Nagaraj.

32 records · Page 2Linked to original sources

arXiv2023

Introspective Experience Replay: Look Back When Surprised

In reinforcement learning (RL), experience replay-based sampling techniques play a crucial role in promoting convergence by eliminating spurious correlations. However, widely used methods such as uniform experience replay (UER) and prioritized experience replay (PER) have been shown to have sub-optimal convergence and high seed sensitivity respectively. To address these issues, we propose a novel approach called IntrospectiveExperience Replay (IER) that selectively samples batches of data points prior to surprising events. Our method builds upon the theoretically sound reverse experience replay (RER) technique, which has been shown to reduce bias in the output of Q-learning-type algorithms with linear function approximation. However, this approach is not always practical or reliable when using neural function approximation. Through empirical evaluations, we demonstrate that IER with neural function approximation yields reliable and superior performance compared toUER, PER, and hindsight experience replay (HER) across most tasks.

cs.LG↗

arXiv2022

Metastable Mixing of Markov Chains: Efficiently Sampling Low Temperature Exponential Random Graphs

In this paper we consider the problem of sampling from the low-temperature exponential random graph model (ERGM). The usual approach is via Markov chain Monte Carlo, but Bhamidi et al. showed that any local Markov chain suffers from an exponentially large mixing time due to metastable states. We instead consider metastable mixing, a notion of approximate mixing relative to the stationary distribution, for which it turns out to suffice to mix only within a collection of metastable states. We show that the Glauber dynamics for the ERGM at any temperature -- except at a lower-dimensional critical set of parameters -- when initialized at $G(n,p)$ for the right choice of $p$ has a metastable mixing time of $O(n^2\log n)$ to within total variation distance $\exp(-Ω(n))$.

math.PR↗

arXiv2021

Streaming Linear System Identification with Reverse Experience Replay

We consider the problem of estimating a linear time-invariant (LTI) dynamical system from a single trajectory via streaming algorithms, which is encountered in several applications including reinforcement learning (RL) and time-series analysis. While the LTI system estimation problem is well-studied in the {\em offline} setting, the practically important streaming/online setting has received little attention. Standard streaming methods like stochastic gradient descent (SGD) are unlikely to work since streaming points can be highly correlated. In this work, we propose a novel streaming algorithm, SGD with Reverse Experience Replay ($\mathsf{SGD}-\mathsf{RER}$), that is inspired by the experience replay (ER) technique popular in the RL literature. $\mathsf{SGD}-\mathsf{RER}$ 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. Thus, we provide the first -- to the best of our knowledge -- optimal SGD-style algorithm for the classical problem of linear system identification with a first order oracle. Furthermore, $\mathsf{SGD}-\mathsf{RER}$ can be applied to more general settings like sparse LTI identification with known sparsity pattern, and non-linear dynamical systems. Our work demonstrates that the knowledge of data dependency structure can aid us in designing statistically and computationally efficient algorithms which can "decorrelate" streaming samples.

cs.LG↗

arXiv2021

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

We consider the setting of vector valued non-linear dynamical systems $X_{t+1} = ϕ(A^* X_t) + η_t$, where $η_t$ is unbiased noise and $ϕ: \mathbb{R} \to \mathbb{R}$ 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 $ϕ$ 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.

cs.LG↗

arXiv2021

The staircase property: How hierarchical structure can guide deep learning

This paper identifies a structural property of data distributions that enables deep neural networks to learn hierarchically. We define the "staircase" property for functions over the Boolean hypercube, which posits that high-order Fourier coefficients are reachable from lower-order Fourier coefficients along increasing chains. We prove that functions satisfying this property can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks -- a class of network architectures and initializations that have homogeneity properties. Our analysis shows that for such staircase functions and neural networks, the gradient-based algorithm learns high-level features by greedily combining lower-level features along the depth of the network. We further back our theoretical results with experiments showing that staircase functions are also learnable by more standard ResNet architectures with stochastic gradient descent. Both the theoretical and experimental results support the fact that staircase properties have a role to play in understanding the capabilities of gradient-based learning on regular networks, in contrast to general polynomial-size networks that can emulate any SQ or PAC algorithms as recently shown.

cs.LG↗

arXiv2021

Online Target Q-learning with Reverse Experience Replay: Efficiently finding the Optimal Policy for Linear MDPs

Q-learning is a popular Reinforcement Learning (RL) algorithm which is widely used in practice with function approximation (Mnih et al., 2015). In contrast, existing theoretical results are pessimistic about Q-learning. For example, (Baird, 1995) shows that Q-learning does not converge even with linear function approximation for linear MDPs. Furthermore, even for tabular MDPs with synchronous updates, Q-learning was shown to have sub-optimal sample complexity (Li et al., 2021;Azar et al., 2013). The goal of this work is to bridge the gap between practical success of Q-learning and the relatively pessimistic theoretical results. The starting point of our work is the observation that in practice, Q-learning is used with two important modifications: (i) training with two networks, called online network and target network simultaneously (online target learning, or OTL) , and (ii) experience replay (ER) (Mnih et al., 2015). While they have been observed to play a significant role in the practical success of Q-learning, a thorough theoretical understanding of how these two modifications improve the convergence behavior of Q-learning has been missing in literature. By carefully combining Q-learning with OTL and reverse experience replay (RER) (a form of experience replay), we present novel methods Q-Rex and Q-RexDaRe (Q-Rex + data reuse). We show that Q-Rex efficiently finds the optimal policy for linear MDPs (or more generally for MDPs with zero inherent Bellman error with linear approximation (ZIBEL)) and provide non-asymptotic bounds on sample complexity -- the first such result for a Q-learning method for this class of MDPs under standard assumptions. Furthermore, we demonstrate that Q-RexDaRe in fact achieves near optimal sample complexity in the tabular setting, improving upon the existing results for vanilla Q-learning.

cs.LG↗

arXiv2021

Sharp Representation Theorems for ReLU Networks with Precise Dependence on Depth

We prove sharp dimension-free representation results for neural networks with $D$ ReLU layers under square loss for a class of functions $\mathcal{G}_D$ defined in the paper. These results capture the precise benefits of depth in the following sense: 1. The rates for representing the class of functions $\mathcal{G}_D$ via $D$ ReLU layers is sharp up to constants, as shown by matching lower bounds. 2. For each $D$, $\mathcal{G}_{D} \subseteq \mathcal{G}_{D+1}$ and as $D$ grows the class of functions $\mathcal{G}_{D}$ contains progressively less smooth functions. 3. If $D^{\prime} < D$, then the approximation rate for the class $\mathcal{G}_D$ achieved by depth $D^{\prime}$ networks is strictly worse than that achieved by depth $D$ networks. This constitutes a fine-grained characterization of the representation power of feedforward networks of arbitrary depth $D$ and number of neurons $N$, in contrast to existing representation results which either require $D$ growing quickly with $N$ or assume that the function being represented is highly smooth. In the latter case similar rates can be obtained with a single nonlinear layer. Our results confirm the prevailing hypothesis that deeper networks are better at representing less smooth functions, and indeed, the main technical novelty is to fully exploit the fact that deep networks can produce highly oscillatory functions with few activation functions.

stat.ML↗

arXiv2020

A law of robustness for two-layers neural networks

We initiate the study of the inherent tradeoffs between the size of a neural network and its robustness, as measured by its Lipschitz constant. We make a precise conjecture that, for any Lipschitz activation function and for most datasets, any two-layers neural network with $k$ neurons that perfectly fit the data must have its Lipschitz constant larger (up to a constant) than $\sqrt{n/k}$ where $n$ is the number of datapoints. In particular, this conjecture implies that overparametrization is necessary for robustness, since it means that one needs roughly one neuron per datapoint to ensure a $O(1)$-Lipschitz network, while mere data fitting of $d$-dimensional data requires only one neuron per $d$ datapoints. We prove a weaker version of this conjecture when the Lipschitz constant is replaced by an upper bound on it based on the spectral norm of the weight matrix. We also prove the conjecture in the high-dimensional regime $n \approx d$ (which we also refer to as the undercomplete case, since only $k \leq d$ is relevant here). Finally we prove the conjecture for polynomial activation functions of degree $p$ when $n \approx d^p$. We complement these findings with experimental evidence supporting the conjecture.

cs.LG↗

arXiv2020

Phase Transitions for Detecting Latent Geometry in Random Graphs

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within computer science, probability and statistics. A fundamental initial question regarding these models is: when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure, such as the Erdős-Rényi graph $\mathcal{G}(n, p)$? We address this question for two of the most well-studied models of random graphs with latent geometry -- the random intersection and random geometric graph. Our results are as follows: (1) we prove that the random intersection graph converges in total variation to $\mathcal{G}(n, p)$ when $d = \tildeω(n^3)$, and does not if $d = o(n^3)$, resolving an open problem in Fill et al. (2000), Rybarczyk (2011) and Kim et al. (2018); (2) we provide conditions under which the matrix of intersection sizes of random family of sets converges in total variation to a symmetric matrix with independent Poisson entries, yielding the first total variation convergence result for $τ$-random intersection graphs to $\mathcal{G}(n, p)$; and (3) we show that the random geometric graph on $\mathbb{S}^{d - 1}$ with edge density $p$ converges in total variation to $\mathcal{G}(n, p)$ when $d = \tildeω\left(\min\{ pn^3, p^2 n^{7/2} \} \right)$, yielding the first progress towards a conjecture of Bubeck et al. (2016). The first of these three results was obtained simultaneously and independently by Bubeck, Racz and Richey.

math.PR↗

arXiv2020

A Corrective View of Neural Networks: Representation, Memorization and Learning

We develop a corrective mechanism for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second group approximates the error in approximation produced by the first group and corrects it, the third group approximates the error produced by the first and second groups together and so on. This technique yields several new representation and learning results for neural networks. First, we show that two-layer neural networks in the random features regime (RF) can memorize arbitrary labels for arbitrary points under under Euclidean distance separation condition using $\tilde{O}(n)$ ReLUs which is optimal in $n$ up to logarithmic factors. Next, we give a powerful representation result for two-layer neural networks with ReLUs and smoothed ReLUs which can achieve a squared error of at most $ε$ with $O(C(a,d)ε^{-1/(a+1)})$ for $a \in \mathbb{N}\cup\{0\}$ when the function is smooth enough (roughly when it has $Θ(ad)$ bounded derivatives). In certain cases $d$ can be replaced with effective dimension $q \ll d$. Previous results of this type implement Taylor series approximation using deep architectures. We also consider three-layer neural networks and show that the corrective mechanism yields faster representation rates for smooth radial functions. Lastly, we obtain the first $O(\mathrm{subpoly}(1/ε))$ upper bound on the number of neurons required for a two layer network to learn low degree polynomials up to squared error $ε$ via gradient descent. Even though deep networks can express these polynomials with $O(\mathrm{polylog}(1/ε))$ neurons, the best learning bounds on this problem require $\mathrm{poly}(1/ε)$ neurons.

cs.LG↗

arXiv2020

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 $τ_{\mathsf{mix}}$, 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 $\tildeΘ(τ_{\mathsf{mix}})$ 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.

cs.LG↗

arXiv2020

SGD without Replacement: Sharper Rates for General Smooth Convex Functions

We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement} \cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strongly-convex} functions. In particular, we show that \sgdwor converges at a rate of $O(1/K^2)$ while \sgd is known to converge at $O(1/K)$ rate, where $K$ denotes the number of passes over data and is required to be {\em large enough}. Existing results for \sgdwor in this setting require additional {\em Hessian Lipschitz assumption} \cite{gurbuzbalaban2015random,haochen2018random}. For {\em small} $K$, we show \sgdwor can achieve same convergence rate as \sgd for {\em general smooth strongly-convex} functions. Existing results in this setting require $K=1$ and hold only for generalized linear models \cite{shamir2016without}. In addition, by careful analysis of the coupling, for both large and small $K$, we obtain better dependence on problem dependent parameters like condition number.

math.OC↗

arXiv2019

Making the Last Iterate of SGD Information Theoretically Optimal

Stochastic gradient descent (SGD) is one of the most widely used algorithms for large scale optimization problems. While classical theoretical analysis of SGD for convex problems studies (suffix) \emph{averages} of iterates and obtains information theoretically optimal bounds on suboptimality, the \emph{last point} of SGD is, by far, the most preferred choice in practice. The best known results for last point of SGD \cite{shamir2013stochastic} however, are suboptimal compared to information theoretic lower bounds by a $\log T$ factor, where $T$ is the number of iterations. \cite{harvey2018tight} shows that in fact, this additional $\log T$ factor is tight for standard step size sequences of $\OTheta{\frac{1}{\sqrt{t}}}$ and $\OTheta{\frac{1}{t}}$ for non-strongly convex and strongly convex settings, respectively. Similarly, even for subgradient descent (GD) when applied to non-smooth, convex functions, the best known step-size sequences still lead to $O(\log T)$-suboptimal convergence rates (on the final iterate). The main contribution of this work is to design new step size sequences that enjoy information theoretically optimal bounds on the suboptimality of \emph{last point} of SGD as well as GD. We achieve this by designing a modification scheme, that converts one sequence of step sizes to another so that the last point of SGD/GD with modified sequence has the same suboptimality guarantees as the average of SGD/GD with original sequence. We also show that our result holds with high-probability. We validate our results through simulations which demonstrate that the new step size sequence indeed improves the final iterate significantly compared to the standard step size sequences.

math.OC↗

arXiv2018

Optimal Single Sample Tests for Structured versus Unstructured Network Data

We study the problem of testing, using only a single sample, between mean field distributions (like Curie-Weiss, Erdős-Rényi) and structured Gibbs distributions (like Ising model on sparse graphs and Exponential Random Graphs). Our goal is to test without knowing the parameter values of the underlying models: only the \emph{structure} of dependencies is known. We develop a new approach that applies to both the Ising and Exponential Random Graph settings based on a general and natural statistical test. The test can distinguish the hypotheses with high probability above a certain threshold in the (inverse) temperature parameter, and is optimal in that below the threshold no test can distinguish the hypotheses. The thresholds do not correspond to the presence of long-range order in the models. By aggregating information at a global scale, our test works even at very high temperatures. The proofs are based on distributional approximation and sharp concentration of quadratic forms, when restricted to Hamming spheres. The restriction to Hamming spheres is necessary, since otherwise any scalar statistic is useless without explicit knowledge of the temperature parameter. At the same time, this restriction radically changes the behavior of the functions under consideration, resulting in a much smaller variance than in the independent setting; this makes it hard to directly apply standard methods (i.e., Stein's method) for concentration of weakly dependent variables. Instead, we carry out an additional tensorization argument using a Markov chain that respects the symmetry of the Hamming sphere.

math.ST↗