Prateek Jain

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.

Prateek Jain, Soumyabrata Pal

We study the problem of {\em online} low-rank matrix completion with $\mathsf{M}$ users, $\mathsf{N}$ items and $\mathsf{T}$ rounds. In each round, the algorithm recommends one item per user, for which it gets a (noisy) reward sampled from a low-rank user-item preference matrix. The goal is to design a method with sub-linear regret (in $\mathsf{T}$) and nearly optimal dependence on $\mathsf{M}$ and $\mathsf{N}$. The problem can be easily mapped to the standard multi-armed bandit problem where each item is an {\em independent} arm, but that leads to poor regret as the correlation between arms and users is not exploited. On the other hand, exploiting the low-rank structure of reward matrix is challenging due to non-convexity of the low-rank manifold. We first demonstrate that the low-rank structure can be exploited using a simple explore-then-commit (ETC) approach that ensures a regret of $O(\mathsf{polylog} (\mathsf{M}+\mathsf{N}) \mathsf{T}^{2/3})$. That is, roughly only $\mathsf{polylog} (\mathsf{M}+\mathsf{N})$ item recommendations are required per user to get a non-trivial solution. We then improve our result for the rank-$1$ setting which in itself is quite challenging and encapsulates some of the key issues. Here, we propose \textsc{OCTAL} (Online Collaborative filTering using iterAtive user cLustering) that guarantees nearly optimal regret of $O(\mathsf{polylog} (\mathsf{M}+\mathsf{N}) \mathsf{T}^{1/2})$. OCTAL is based on a novel technique of clustering users that allows iterative elimination of items and leads to a nearly optimal minimax rate.

Devvrit Khatri, Pranamya Kulkarni, Nilesh Gupta et al.

Large Language Models (LLMs) have been shown to be able to learn different tasks without explicit finetuning when given many input-output examples / demonstrations through In-Context Learning (ICL). Increasing the number of examples, called ``shots'', improves downstream task performance but incurs higher memory and computational costs. In this work, we study an approach to improve the memory and computational efficiency of ICL inference by compressing the many-shot prompts. Given many shots comprising t tokens, our goal is to generate a m soft-token summary, where m < t. We first show that existing prompt compression methods are ineffective for many-shot compression, and simply using fewer shots as a baseline is surprisingly strong. To achieve effective compression, we find that: (a) a stronger compressor model with more trainable parameters is necessary, and (b) compressing many-shot representations at each transformer layer enables more fine-grained compression by providing each layer with its own compressed representation. Based on these insights, we propose MemCom, a layer-wise compression method. We systematically evaluate various compressor models and training approaches across different model sizes (2B and 7B), architectures (Gemma and Mistral), many-shot sequence lengths (3k-6k tokens), and compression ratios (3x to 8x). MemCom outperforms strong baselines across all compression ratios on multiple classification tasks with large label sets. Notably, while baseline performance degrades sharply at higher compression ratios, often by over 20-30%, MemCom maintains high accuracy with minimal degradation, typically dropping by less than 10%.

Yashas Samaga, Varun Yerram, Spandana Raj Babbula et al.

We consider the Top-$K$ selection problem, which aims to identify the largest-$K$ elements from an array. Top-$K$ selection arises in many machine learning algorithms and often becomes a bottleneck on accelerators, which are optimized for dense matrix multiplications. To address this problem, \citet{chern2022tpuknnknearestneighbor} proposed a fast two-stage \textit{approximate} Top-$K$ algorithm: (i) partition the input array and select the top-$1$ element from each partition, (ii) sort this \textit{smaller subset} and return the top $K$ elements. In this paper, we consider a generalized version of this algorithm, where the first stage selects top-$K'$ elements, for some $1 \leq K' \leq K$, from each partition. Our contributions are as follows: (i) we derive an expression for the expected recall of this generalized algorithm and show that choosing $K' > 1$ with fewer partitions in the first stage reduces the input size to the second stage more effectively while maintaining the same expected recall as the original algorithm, (ii) we derive a bound on the expected recall for the original algorithm in \citet{chern2022tpuknnknearestneighbor} that is provably tighter by a factor of $2$ than the one in that paper, and (iii) we implement our algorithm on Cloud TPUv5e and achieve around an order of magnitude speedups over the original algorithm without sacrificing recall on real-world tasks.

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.

Aadirupa Saha, Nagarajan Natarajan, Praneeth Netrapalli et al.

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

Prateek Jain, Praneeth Netrapalli

Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this problem. However, the sample complexity in all these results is sub-optimal in its dependence on the rank, condition number and the desired accuracy. In this paper, we present a fast iterative algorithm that solves the matrix completion problem by observing $O(nr^5 \log^3 n)$ entries, which is independent of the condition number and the desired accuracy. The run time of our algorithm is $O(nr^7\log^3 n\log 1/ε)$ which is near linear in the dimension of the matrix. To the best of our knowledge, this is the first near linear time algorithm for exact matrix completion with finite sample complexity (i.e. independent of $ε$). Our algorithm is based on a well known projected gradient descent method, where the projection is onto the (non-convex) set of low rank matrices. There are two key ideas in our result: 1) our argument is based on a $\ell_{\infty}$ norm potential function (as opposed to the spectral norm) and provides a novel way to obtain perturbation bounds for it. 2) we prove and use a natural extension of the Davis-Kahan theorem to obtain perturbation bounds on the best low rank approximation of matrices with good eigen-gap. Both of these ideas may be of independent interest.

Praneeth Netrapalli, U N Niranjan, Sujay Sanghavi et al.

We propose a new method for robust PCA -- the task of recovering a low-rank matrix from sparse corruptions that are of unknown value and support. Our method involves alternating between projecting appropriate residuals onto the set of low-rank matrices, and the set of sparse matrices; each projection is {\em non-convex} but easy to compute. In spite of this non-convexity, we establish exact recovery of the low-rank matrix, under the same conditions that are required by existing methods (which are based on convex optimization). For an $m \times n$ input matrix ($m \leq n)$, our method has a running time of $O(r^2mn)$ per iteration, and needs $O(\log(1/ε))$ iterations to reach an accuracy of $ε$. This is close to the running time of simple PCA via the power method, which requires $O(rmn)$ per iteration, and $O(\log(1/ε))$ iterations. In contrast, existing methods for robust PCA, which are based on convex optimization, have $O(m^2n)$ complexity per iteration, and take $O(1/ε)$ iterations, i.e., exponentially more iterations for the same accuracy. Experiments on both synthetic and real data establishes the improved speed and accuracy of our method over existing convex implementations.

Srinadh Bhojanapalli, Prateek Jain, Sujay Sanghavi

In this work, we propose a new randomized algorithm for computing a low-rank approximation to a given matrix. Taking an approach different from existing literature, our method first involves a specific biased sampling, with an element being chosen based on the leverage scores of its row and column, and then involves weighted alternating minimization over the factored form of the intended low-rank matrix, to minimize error only on these samples. Our method can leverage input sparsity, yet produce approximations in {\em spectral} (as opposed to the weaker Frobenius) norm; this combines the best aspects of otherwise disparate current results, but with a dependence on the condition number $κ= σ_1/σ_r$. In particular we require $O(nnz(M) + \frac{nκ^2 r^5}{ε^2})$ computations to generate a rank-$r$ approximation to $M$ in spectral norm. In contrast, the best existing method requires $O(nnz(M)+ \frac{nr^2}{ε^4})$ time to compute an approximation in Frobenius norm. Besides the tightness in spectral norm, we have a better dependence on the error $ε$. Our method is naturally and highly parallelizable. Our new approach enables two extensions that are interesting on their own. The first is a new method to directly compute a low-rank approximation (in efficient factored form) to the product of two given matrices; it computes a small random set of entries of the product, and then executes weighted alternating minimization (as before) on these. The sampling strategy is different because now we cannot access leverage scores of the product matrix (but instead have to work with input matrices). The second extension is an improved algorithm with smaller communication complexity for the distributed PCA setting (where each server has small set of rows of the matrix, and want to compute low rank approximation with small amount of communication with other servers).

Srinadh Bhojanapalli, Prateek Jain

The problem of low-rank matrix completion has recently generated a lot of interest leading to several results that offer exact solutions to the problem. However, in order to do so, these methods make assumptions that can be quite restrictive in practice. More specifically, the methods assume that: a) the observed indices are sampled uniformly at random, and b) for every new matrix, the observed indices are sampled afresh. In this work, we address these issues by providing a universal recovery guarantee for matrix completion that works for a variety of sampling schemes. In particular, we show that if the set of sampled indices come from the edges of a bipartite graph with large spectral gap (i.e. gap between the first and the second singular value), then the nuclear norm minimization based method exactly recovers all low-rank matrices that satisfy certain incoherence properties. Moreover, we also show that under certain stricter incoherence conditions, $O(nr^2)$ uniformly sampled entries are enough to recover any rank-$r$ $n\times n$ matrix, in contrast to the $O(nr\log n)$ sample complexity required by other matrix completion algorithms as well as existing analyses of the nuclear norm method.

Ioannis Mitliagkas, Constantine Caramanis, Prateek Jain

We consider streaming, one-pass principal component analysis (PCA), in the high-dimensional regime, with limited memory. Here, $p$-dimensional samples are presented sequentially, and the goal is to produce the $k$-dimensional subspace that best approximates these points. Standard algorithms require $O(p^2)$ memory; meanwhile no algorithm can do better than $O(kp)$ memory, since this is what the output itself requires. Memory (or storage) complexity is most meaningful when understood in the context of computational and sample complexity. Sample complexity for high-dimensional PCA is typically studied in the setting of the {\em spiked covariance model}, where $p$-dimensional points are generated from a population covariance equal to the identity (white noise) plus a low-dimensional perturbation (the spike) which is the signal to be recovered. It is now well-understood that the spike can be recovered when the number of samples, $n$, scales proportionally with the dimension, $p$. Yet, all algorithms that provably achieve this, have memory complexity $O(p^2)$. Meanwhile, algorithms with memory-complexity $O(kp)$ do not have provable bounds on sample complexity comparable to $p$. We present an algorithm that achieves both: it uses $O(kp)$ memory (meaning storage of any kind) and is able to compute the $k$-dimensional spike with $O(p \log p)$ sample-complexity -- the first algorithm of its kind. While our theoretical analysis focuses on the spiked covariance model, our simulations show that our algorithm is successful on much more general models for the data.

Prateek Jain, Inderjit S. Dhillon

Consider a movie recommendation system where apart from the ratings information, side information such as user's age or movie's genre is also available. Unlike standard matrix completion, in this setting one should be able to predict inductively on new users/movies. In this paper, we study the problem of inductive matrix completion in the exact recovery setting. That is, we assume that the ratings matrix is generated by applying feature vectors to a low-rank matrix and the goal is to recover back the underlying matrix. Furthermore, we generalize the problem to that of low-rank matrix estimation using rank-1 measurements. We study this generic problem and provide conditions that the set of measurements should satisfy so that the alternating minimization method (which otherwise is a non-convex method with no convergence guarantees) is able to recover back the {\em exact} underlying low-rank matrix. In addition to inductive matrix completion, we show that two other low-rank estimation problems can be studied in our framework: a) general low-rank matrix sensing using rank-1 measurements, and b) multi-label regression with missing labels. For both the problems, we provide novel and interesting bounds on the number of measurements required by alternating minimization to provably converges to the {\em exact} low-rank matrix. In particular, our analysis for the general low rank matrix sensing problem significantly improves the required storage and computational cost than that required by the RIP-based matrix sensing methods \cite{RechtFP2007}. Finally, we provide empirical validation of our approach and demonstrate that alternating minimization is able to recover the true matrix for the above mentioned problems using a small number of measurements.

Praneeth Netrapalli, Prateek Jain, Sujay Sanghavi

Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of (Gerchberg and Saxton 1972) and (Fienup 1982) is still the popular choice for solving many variants of this problem. The algorithm is based on alternating minimization; i.e. it alternates between estimating the missing phase information, and the candidate solution. Despite its wide usage in practice, no global convergence guarantees for this algorithm are known. In this paper, we show that a (resampling) variant of this approach converges geometrically to the solution of one such problem -- finding a vector $\mathbf{x}$ from $\mathbf{y},\mathbf{A}$, where $\mathbf{y} = \left|\mathbf{A}^{\top}\mathbf{x}\right|$ and $|\mathbf{z}|$ denotes a vector of element-wise magnitudes of $\mathbf{z}$ -- under the assumption that $\mathbf{A}$ is Gaussian. Empirically, we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem (which are based on "lifting" to a convex matrix problem) in sample complexity and robustness to noise. However, it is much more efficient and can scale to large problems. Analytically, for a resampling version of alternating minimization, we show geometric convergence to the solution, and sample complexity that is off by log factors from obvious lower bounds. We also establish close to optimal scaling for the case when the unknown vector is sparse. Our work represents the first theoretical guarantee for alternating minimization (albeit with resampling) for any variant of phase retrieval problems in the non-convex setting.

Purushottam Kar, Bharath K Sriperumbudur, Prateek Jain et al.

In this paper, we study the generalization properties of online learning based stochastic methods for supervised learning problems where the loss function is dependent on more than one training sample (e.g., metric learning, ranking). We present a generic decoupling technique that enables us to provide Rademacher complexity-based generalization error bounds. Our bounds are in general tighter than those obtained by Wang et al (COLT 2012) for the same problem. Using our decoupling technique, we are further able to obtain fast convergence rates for strongly convex pairwise loss functions. We are also able to analyze a class of memory efficient online learning algorithms for pairwise learning problems that use only a bounded subset of past training samples to update the hypothesis at each step. Finally, in order to complement our generalization bounds, we propose a novel memory efficient online learning algorithm for higher order learning problems with bounded regret guarantees.