Dheeraj Nagaraj

Naman Agarwal, Prateek Jain, Suhas Kowshik et al. · deepmind

In this work, we consider the problem of collaborative multi-user reinforcement learning. In this setting there are multiple users with the same state-action space and transition probabilities but with different rewards. Under the assumption that the reward matrix of the $N$ users has a low-rank structure -- a standard and practically successful assumption in the offline collaborative filtering setting -- the question is can we design algorithms with significantly lower sample complexity compared to the ones that learn the MDP individually for each user. Our main contribution is an algorithm which explores rewards collaboratively with $N$ user-specific MDPs and can learn rewards efficiently in two key settings: tabular MDPs and linear MDPs. When $N$ is large and the rank is constant, the sample complexity per MDP depends logarithmically over the size of the state-space, which represents an exponential reduction (in the state-space size) when compared to the standard ``non-collaborative'' algorithms.

Gandharv Patil, Prashanth L. A., Dheeraj Nagaraj et al.

We study the finite-time behaviour of the popular temporal difference (TD) learning algorithm when combined with tail-averaging. We derive finite time bounds on the parameter error of the tail-averaged TD iterate under a step-size choice that does not require information about the eigenvalues of the matrix underlying the projected TD fixed point. Our analysis shows that tail-averaged TD converges at the optimal $O\left(1/t\right)$ rate, both in expectation and with high probability. In addition, our bounds exhibit a sharper rate of decay for the initial error (bias), which is an improvement over averaging all iterates. We also propose and analyse a variant of TD that incorporates regularisation. From analysis, we conclude that the regularised version of TD is useful for problems with ill-conditioned features.

Abheek Ghosh, Dheeraj Nagaraj, Manish Jain et al.

We study the problem of planning restless multi-armed bandits (RMABs) with multiple actions. This is a popular model for multi-agent systems with applications like multi-channel communication, monitoring and machine maintenance tasks, and healthcare. Whittle index policies, which are based on Lagrangian relaxations, are widely used in these settings due to their simplicity and near-optimality under certain conditions. In this work, we first show that Whittle index policies can fail in simple and practically relevant RMAB settings, even when the RMABs are indexable. We discuss why the optimality guarantees fail and why asymptotic optimality may not translate well to practically relevant planning horizons. We then propose an alternate planning algorithm based on the mean-field method, which can provably and efficiently obtain near-optimal policies with a large number of arms, without the stringent structural assumptions required by the Whittle index policies. This borrows ideas from existing research with some improvements: our approach is hyper-parameter free, and we provide an improved non-asymptotic analysis which has: (a) no requirement for exogenous hyper-parameters and tighter polynomial dependence on known problem parameters; (b) high probability bounds which show that the reward of the policy is reliable; and (c) matching sub-optimality lower bounds for this algorithm with respect to the number of arms, thus demonstrating the tightness of our bounds. Our extensive experimental analysis shows that the mean-field approach matches or outperforms other baselines.

Yunfan Zhao, Nikhil Behari, Edward Hughes et al.

Restless multi-arm bandits (RMABs), a class of resource allocation problems with broad application in areas such as healthcare, online advertising, and anti-poaching, have recently been studied from a multi-agent reinforcement learning perspective. Prior RMAB research suffers from several limitations, e.g., it fails to adequately address continuous states, and requires retraining from scratch when arms opt-in and opt-out over time, a common challenge in many real world applications. We address these limitations by developing a neural network-based pre-trained model (PreFeRMAB) that has general zero-shot ability on a wide range of previously unseen RMABs, and which can be fine-tuned on specific instances in a more sample-efficient way than retraining from scratch. Our model also accommodates general multi-action settings and discrete or continuous state spaces. To enable fast generalization, we learn a novel single policy network model that utilizes feature information and employs a training procedure in which arms opt-in and out over time. We derive a new update rule for a crucial $λ$-network with theoretical convergence guarantees and empirically demonstrate the advantages of our approach on several challenging, real-world inspired problems.

Yunfan Zhao, Tonghan Wang, Dheeraj Nagaraj et al. · harvard, tsinghua

Applying Reinforcement Learning (RL) to Restless Multi-Arm Bandits (RMABs) offers a promising avenue for addressing allocation problems with resource constraints and temporal dynamics. However, classic RMAB models largely overlook the challenges of (systematic) data errors - a common occurrence in real-world scenarios due to factors like varying data collection protocols and intentional noise for differential privacy. We demonstrate that conventional RL algorithms used to train RMABs can struggle to perform well in such settings. To solve this problem, we propose the first communication learning approach in RMABs, where we study which arms, when involved in communication, are most effective in mitigating the influence of such systematic data errors. In our setup, the arms receive Q-function parameters from similar arms as messages to guide behavioral policies, steering Q-function updates. We learn communication strategies by considering the joint utility of messages across all pairs of arms and using a Q-network architecture that decomposes the joint utility. Both theoretical and empirical evidence validate the effectiveness of our method in significantly improving RMAB performance across diverse problems.

Dheeraj Baby, Aniket Das, Dheeraj Nagaraj et al.

We consider the problem of heteroscedastic linear regression, where, given $n$ samples $(\mathbf{x}_i, y_i)$ from $y_i = \langle \mathbf{w}^{*}, \mathbf{x}_i \rangle + ε_i \cdot \langle \mathbf{f}^{*}, \mathbf{x}_i \rangle$ with $\mathbf{x}_i \sim N(0,\mathbf{I})$, $ε_i \sim N(0,1)$, we aim to estimate $\mathbf{w}^{*}$. Beyond classical applications of such models in statistics, econometrics, time series analysis etc., it is also particularly relevant in machine learning when data is collected from multiple sources of varying but apriori unknown quality. Our work shows that we can estimate $\mathbf{w}^{*}$ in squared norm up to an error of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2 \cdot \left(\frac{1}{n} + \left(\frac{d}{n}\right)^2\right)\right)$ and prove a matching lower bound (upto log factors). This represents a substantial improvement upon the previous best known upper bound of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2\cdot \frac{d}{n}\right)$. Our algorithm is an alternating minimization procedure with two key subroutines 1. An adaptation of the classical weighted least squares heuristic to estimate $\mathbf{w}^{*}$, for which we provide the first non-asymptotic guarantee. 2. A nonconvex pseudogradient descent procedure for estimating $\mathbf{f}^{*}$ inspired by phase retrieval. As corollaries, we obtain fast non-asymptotic rates for two important problems, linear regression with multiplicative noise and phase retrieval with multiplicative noise, both of which are of independent interest. Beyond this, the proof of our lower bound, which involves a novel adaptation of LeCam's method for handling infinite mutual information quantities (thereby preventing a direct application of standard techniques like Fano's method), could also be of broader interest for establishing lower bounds for other heteroscedastic or heavy-tailed statistical problems.

Aniket Das, Dheeraj Nagaraj, Anant Raj

We consider stochastic approximations of sampling algorithms, such as Stochastic Gradient Langevin Dynamics (SGLD) and the Random Batch Method (RBM) for Interacting Particle Dynamcs (IPD). We observe that the noise introduced by the stochastic approximation is nearly Gaussian due to the Central Limit Theorem (CLT) while the driving Brownian motion is exactly Gaussian. We harness this structure to absorb the stochastic approximation error inside the diffusion process, and obtain improved convergence guarantees for these algorithms. For SGLD, we prove the first stable convergence rate in KL divergence without requiring uniform warm start, assuming the target density satisfies a Log-Sobolev Inequality. Our result implies superior first-order oracle complexity compared to prior works, under significantly milder assumptions. We also prove the first guarantees for SGLD under even weaker conditions such as Hölder smoothness and Poincare Inequality, thus bridging the gap between the state-of-the-art guarantees for LMC and SGLD. Our analysis motivates a new algorithm called covariance correction, which corrects for the additional noise introduced by the stochastic approximation by rescaling the strength of the diffusion. Finally, we apply our techniques to analyze RBM, and significantly improve upon the guarantees in prior works (such as removing exponential dependence on horizon), under minimal assumptions.

Ramnath Kumar, Kushal Majmundar, Dheeraj Nagaraj et al.

We present Re-weighted Gradient Descent (RGD), a novel optimization technique that improves the performance of deep neural networks through dynamic sample re-weighting. Leveraging insights from distributionally robust optimization (DRO) with Kullback-Leibler divergence, our method dynamically assigns importance weights to training data during each optimization step. RGD is simple to implement, computationally efficient, and compatible with widely used optimizers such as SGD and Adam. We demonstrate the effectiveness of RGD on various learning tasks, including supervised learning, meta-learning, and out-of-domain generalization. Notably, RGD achieves state-of-the-art results on diverse benchmarks, with improvements of +0.7% on DomainBed, +1.44% on tabular classification, \textcolor{blue}+1.94% on GLUE with BERT, and +1.01% on ImageNet-1K with ViT.

Ramnath Kumar, Dheeraj Nagaraj

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.

Nikhil Behari, Edwin Zhang, Yunfan Zhao et al.

Restless multi-armed bandits (RMAB) have demonstrated success in optimizing resource allocation for large beneficiary populations in public health settings. Unfortunately, RMAB models lack flexibility to adapt to evolving public health policy priorities. Concurrently, Large Language Models (LLMs) have emerged as adept automated planners across domains of robotic control and navigation. In this paper, we propose a Decision Language Model (DLM) for RMABs, enabling dynamic fine-tuning of RMAB policies in public health settings using human-language commands. We propose using LLMs as automated planners to (1) interpret human policy preference prompts, (2) propose reward functions as code for a multi-agent RMAB environment, and (3) iterate on the generated reward functions using feedback from grounded RMAB simulations. We illustrate the application of DLM in collaboration with ARMMAN, an India-based non-profit promoting preventative care for pregnant mothers, that currently relies on RMAB policies to optimally allocate health worker calls to low-resource populations. We conduct a technology demonstration in simulation using the Gemini Pro model, showing DLM can dynamically shape policy outcomes using only human prompts as input.

Rishikesh Srinivasan, Dheeraj Nagaraj

We study the problem of sampling from strongly log-concave distributions over $\mathbb{R}^d$ using the Poisson midpoint discretization (a variant of the randomized midpoint method) for overdamped/underdamped Langevin dynamics. We prove its convergence in the 2-Wasserstein distance ($W_2$), achieving a cubic speedup in dependence on the target accuracy ($ε$) over the Euler-Maruyama discretization, surpassing existing bounds for randomized midpoint methods. Notably, in the case of underdamped Langevin dynamics, we demonstrate the complexity of $W_2$ convergence is much smaller than the complexity lower bounds for convergence in $L^2$ strong error established in the literature.

Aniket Das, Dheeraj Nagaraj, Soumyabrata Pal et al.

We consider the problem of high-dimensional heavy-tailed statistical estimation in the streaming setting, which is much harder than the traditional batch setting due to memory constraints. We cast this problem as stochastic convex optimization with heavy tailed stochastic gradients, and prove that the widely used Clipped-SGD algorithm attains near-optimal sub-Gaussian statistical rates whenever the second moment of the stochastic gradient noise is finite. More precisely, with $T$ samples, we show that Clipped-SGD, for smooth and strongly convex objectives, achieves an error of $\sqrt{\frac{\mathsf{Tr}(Σ)+\sqrt{\mathsf{Tr}(Σ)\|Σ\|_2}\log(\frac{\log(T)}δ)}{T}}$ with probability $1-δ$, where $Σ$ is the covariance of the clipped gradient. Note that the fluctuations (depending on $\frac{1}δ$) are of lower order than the term $\mathsf{Tr}(Σ)$. This improves upon the current best rate of $\sqrt{\frac{\mathsf{Tr}(Σ)\log(\frac{1}δ)}{T}}$ for Clipped-SGD, known only for smooth and strongly convex objectives. Our results also extend to smooth convex and lipschitz convex objectives. Key to our result is a novel iterative refinement strategy for martingale concentration, improving upon the PAC-Bayes approach of Catoni and Giulini.

Syamantak Kumar, Dheeraj Nagaraj, Purnamrita Sarkar

Diffusion models generate samples by estimating the score function of the target distribution at various noise levels. The model is trained using samples drawn from the target distribution by progressively adding noise. Previous sample complexity bounds have polynomial dependence on the dimension $d$, apart from a $\log(|\mathcal{H}|)$ term, where $\mathcal{H}$ is the hypothesis class. In this work, we establish the first (nearly) dimension-free sample complexity bounds, modulo the $\log(|\mathcal{H}|)$ dependence, for learning these score functions, achieving a double exponential improvement in the dimension over prior results. A key aspect of our analysis is the use of a single function approximator to jointly estimate scores across noise levels, a practical feature that enables generalization across time steps. We introduce a martingale-based error decomposition and sharp variance bounds, enabling efficient learning from dependent data generated by Markov processes, which may be of independent interest. Building on these insights, we propose Bootstrapped Score Matching (BSM), a variance reduction technique that leverages previously learned scores to improve accuracy at higher noise levels. These results provide insights into the efficiency and effectiveness of diffusion models for generative modeling.

Gautham Govind Anil, Shaan Ul Haque, Nithish Kannen et al.

Diffusion models are widely used for generative tasks across domains. While pre-trained diffusion models effectively capture the training data distribution, it is often desirable to shape these distributions using reward functions to align with downstream applications. Policy gradient methods, such as Proximal Policy Optimization (PPO), are widely used in the context of autoregressive generation. However, the marginal likelihoods required for such methods are intractable for diffusion models, leading to alternative proposals and relaxations. In this context, we unify variants of Rejection sAmpling based Fine-Tuning (RAFT) as GRAFT, and show that this implicitly performs PPO with reshaped rewards. We then introduce P-GRAFT to shape distributions at intermediate noise levels and demonstrate empirically that this can lead to more effective fine-tuning. We mathematically explain this via a bias-variance tradeoff. Motivated by this, we propose inverse noise correction to improve flow models without leveraging explicit rewards. We empirically evaluate our methods on text-to-image(T2I) generation, layout generation, molecule generation and unconditional image generation. Notably, our framework, applied to Stable Diffusion 2, improves over policy gradient methods on popular T2I benchmarks in terms of VQAScore and shows an $8.81\%$ relative improvement over the base model. For unconditional image generation, inverse noise correction improves FID of generated images at lower FLOPs/image.

Gautham Govind Anil, Sachin Yadav, Dheeraj Nagaraj et al.

We introduce Interleaved Gibbs Diffusion (IGD), a novel generative modeling framework for discrete-continuous data, focusing on problems with important, implicit and unspecified constraints in the data. Most prior works on discrete and discrete-continuous diffusion assume a factorized denoising distribution, which can hinder the modeling of strong dependencies between random variables in such problems. We empirically demonstrate a significant improvement in 3-SAT performance out of the box by switching to a Gibbs-sampling style discrete diffusion model which does not assume factorizability. Motivated by this, we introduce IGD which generalizes discrete time Gibbs sampling type Markov chain for the case of discrete-continuous generation. IGD allows for seamless integration between discrete and continuous denoisers while theoretically guaranteeing exact reversal of a suitable forward process. Further, it provides flexibility in the choice of denoisers, allows conditional generation via state-space doubling and inference time refinement. Empirical evaluations on three challenging generation tasks - molecule structures, layouts and tabular data - demonstrate state-of-the-art performance. Notably, IGD achieves state-of-the-art results without relying on domain-specific inductive biases like equivariant diffusion or auxiliary losses. We explore a wide range of modeling, and interleaving strategies along with hyperparameters in each of these problems.

Aniket Das, Dheeraj Nagaraj

Stein Variational Gradient Descent (SVGD) is a popular variational inference algorithm which simulates an interacting particle system to approximately sample from a target distribution, with impressive empirical performance across various domains. Theoretically, its population (i.e, infinite-particle) limit dynamics is well studied but the behavior of SVGD in the finite-particle regime is much less understood. In this work, we design two computationally efficient variants of SVGD, namely VP-SVGD and GB-SVGD, with provably fast finite-particle convergence rates. We introduce the notion of virtual particles and develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, which are exactly implementable using a finite number of particles. Our algorithms can be viewed as specific random-batch approximations of SVGD, which are computationally more efficient than ordinary SVGD. We show that the $n$ particles output by VP-SVGD and GB-SVGD, run for $T$ steps with batch-size $K$, are at-least as good as i.i.d samples from a distribution whose Kernel Stein Discrepancy to the target is at most $O\left(\tfrac{d^{1/3}}{(KT)^{1/6}}\right)$ under standard assumptions. Our results also hold under a mild growth condition on the potential function, which is much weaker than the isoperimetric (e.g. Poincare Inequality) or information-transport conditions (e.g. Talagrand's Inequality $\mathsf{T}_1$) generally considered in prior works. As a corollary, we consider the convergence of the empirical measure (of the particles output by VP-SVGD and GB-SVGD) to the target distribution and demonstrate a double exponential improvement over the best known finite-particle analysis of SVGD. Beyond this, our results present the first known oracle complexities for this setting with polynomial dimension dependence.

Naman Agarwal, Syomantak Chaudhuri, Prateek Jain et al.

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.

Emmanuel Abbe, Enric Boix-Adsera, Matthew Brennan et al.

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.

Prateek Jain, Suhas S Kowshik, Dheeraj Nagaraj et al.

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.

Prateek Jain, Suhas S Kowshik, Dheeraj Nagaraj et al.

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.

Sébastien Bubeck, Yuanzhi Li, Dheeraj Nagaraj

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.

Guy Bresler, Prateek Jain, Dheeraj Nagaraj et al.

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.

Guy Bresler, Dheeraj Nagaraj

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.

Guy Bresler, Dheeraj Nagaraj

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.

Prateek Jain, Dheeraj Nagaraj, Praneeth Netrapalli

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.

Prateek Jain, Dheeraj Nagaraj, Praneeth Netrapalli

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.

Guy Bresler, Dheeraj Nagaraj

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.