Kushal Chakrabarti

Kushal Chakrabarti, Nikhil Chopra

Adaptive gradient methods have become popular in optimizing deep neural networks; recent examples include AdaGrad and Adam. Although Adam usually converges faster, variations of Adam, for instance, the AdaBelief algorithm, have been proposed to enhance Adam's poor generalization ability compared to the classical stochastic gradient method. This paper develops a generic framework for adaptive gradient methods that solve non-convex optimization problems. We first model the adaptive gradient methods in a state-space framework, which allows us to present simpler convergence proofs of adaptive optimizers such as AdaGrad, Adam, and AdaBelief. We then utilize the transfer function paradigm from classical control theory to propose a new variant of Adam, coined AdamSSM. We add an appropriate pole-zero pair in the transfer function from squared gradients to the second moment estimate. We prove the convergence of the proposed AdamSSM algorithm. Applications on benchmark machine learning tasks of image classification using CNN architectures and language modeling using LSTM architecture demonstrate that the AdamSSM algorithm improves the gap between generalization accuracy and faster convergence than the recent adaptive gradient methods.

Kushal Chakrabarti, Mayank Baranwal

Adaptive gradient-descent optimizers are the standard choice for training neural network models. Despite their faster convergence than gradient-descent and remarkable performance in practice, the adaptive optimizers are not as well understood as vanilla gradient-descent. A reason is that the dynamic update of the learning rate that helps in faster convergence of these methods also makes their analysis intricate. Particularly, the simple gradient-descent method converges at a linear rate for a class of optimization problems, whereas the practically faster adaptive gradient methods lack such a theoretical guarantee. The Polyak-Łojasiewicz (PL) inequality is the weakest known class, for which linear convergence of gradient-descent and its momentum variants has been proved. Therefore, in this paper, we prove that AdaGrad and Adam, two well-known adaptive gradient methods, converge linearly when the cost function is smooth and satisfies the PL inequality. Our theoretical framework follows a simple and unified approach, applicable to both batch and stochastic gradients, which can potentially be utilized in analyzing linear convergence of other variants of Adam.

Mayank Baranwal, Kushal Chakrabarti

Optimizing problems in a distributed manner is critical for systems involving multiple agents with private data. Despite substantial interest, a unified method for analyzing the convergence rates of distributed optimization algorithms is lacking. This paper introduces an energy conservation approach for analyzing continuous-time dynamical systems in dilated coordinates. Instead of directly analyzing dynamics in the original coordinate system, we establish a conserved quantity, akin to physical energy, in the dilated coordinate system. Consequently, convergence rates can be explicitly expressed in terms of the inverse time-dilation factor. Leveraging this generalized approach, we formulate a novel second-order distributed accelerated gradient flow with a convergence rate of $O\left(1/t^{2-ε}\right)$ in time $t$ for $ε>0$. We then employ a semi second-order symplectic Euler discretization to derive a rate-matching algorithm with a convergence rate of $O\left(1/k^{2-ε}\right)$ in $k$ iterations. To the best of our knowledge, this represents the most favorable convergence rate for any distributed optimization algorithm designed for smooth convex optimization. Its accelerated convergence behavior is benchmarked against various state-of-the-art distributed optimization algorithms on practical, large-scale problems.

Kushal Chakrabarti, Mayank Baranwal

This paper considers solving distributed optimization problems in peer-to-peer multi-agent networks. The network is synchronous and connected. By using the proportional-integral (PI) control strategy, various algorithms with fixed stepsize have been developed. Two notable among them are the PI algorithm and the PI consensus algorithm. Although the PI algorithm has provable linear or exponential convergence without the standard requirement of (strong) convexity, a similar guarantee for the PI consensus algorithm is unavailable. In this paper, using Lyapunov theory, we guarantee exponential convergence of the PI consensus algorithm for global cost functions that satisfy the restricted secant inequality, with rate-matching discretization, without requiring convexity. To accelerate the PI consensus algorithm, we incorporate local pre-conditioning in the form of constant positive definite matrices and numerically validate its efficiency compared to the prominent distributed convex optimization algorithms. Unlike classical pre-conditioning, where only the gradients are multiplied by a pre-conditioner, the proposed pre-conditioning modifies both the gradients and the consensus terms, thereby controlling the effect of the communication graph on the algorithm.

Van Sy Mai, Kushal Chakrabarti, Richard J. La et al.

This paper explores the use of server learning for enhancing the robustness of federated learning against malicious attacks even when clients' training data are not independent and identically distributed. We propose a heuristic algorithm that uses server learning and client update filtering in combination with geometric median aggregation. We demonstrate via experiments that this approach can achieve significant improvement in model accuracy even when the fraction of malicious clients is high, even more than $50\%$ in some cases, and the dataset utilized by the server is small and could be synthetic with its distribution not necessarily close to that of the clients' aggregated data.

Dipankar Maity, Kushal Chakrabarti

In this study, we investigate the protection offered by federated learning algorithms against eavesdropping adversaries. In our model, the adversary is capable of intercepting model updates transmitted from clients to the server, enabling it to create its own estimate of the model. Unlike previous research, which predominantly focuses on safeguarding client data, our work shifts attention protecting the client model itself. Through a theoretical analysis, we examine how various factors, such as the probability of client selection, the structure of local objective functions, global aggregation at the server, and the eavesdropper's capabilities, impact the overall level of protection. We further validate our findings through numerical experiments, assessing the protection by evaluating the model accuracy achieved by the adversary. Finally, we compare our results with methods based on differential privacy, underscoring their limitations in this specific context.

Kushal Chakrabarti, Nirmal Balachundhar

Language models continue to hallucinate despite increases in parameters, compute, and data. We propose neural diversity -- decorrelated parallel representations -- as a principled mechanism that reduces hallucination rates at fixed parameter and data budgets. Inspired by portfolio theory, where uncorrelated assets reduce risk by $\sqrt{P}$, we prove hallucination probability is bounded by representational correlation: $P(H) \leq f(σ^2((1-ρ(P))/P + ρ(P)), μ^2)$, which predicts that language models need an optimal amount of neurodiversity. To validate this, we introduce ND-LoRA (Neural Diversity Low-Rank Adaptation), combining parallel LoRA adapters with Barlow Twins regularization, and demonstrate that ND-LoRA reduces hallucinations by up to 25.6% (and 14.6% on average) without degrading general accuracy. Ablations show LoRA adapters and regularization act synergistically, causal interventions prove neurodiversity as the mediating factor and correlational analyses indicate scale: a 0.1% neural correlation increase is associated with a 3.8% hallucination increase. Finally, task-dependent optimality emerges: different tasks require different amounts of optimal neurodiversity. Together, our results highlight neural diversity as a third axis of scaling -- orthogonal to parameters and data -- to improve the reliability of language models at fixed budgets.

Kushal Chakrabarti, Nirupam Gupta, Nikhil Chopra

This paper studies a distributed multi-agent convex optimization problem. The system comprises multiple agents in this problem, each with a set of local data points and an associated local cost function. The agents are connected to a server, and there is no inter-agent communication. The agents' goal is to learn a parameter vector that optimizes the aggregate of their local costs without revealing their local data points. In principle, the agents can solve this problem by collaborating with the server using the traditional distributed gradient-descent method. However, when the aggregate cost is ill-conditioned, the gradient-descent method (i) requires a large number of iterations to converge, and (ii) is highly unstable against process noise. We propose an iterative pre-conditioning technique to mitigate the deleterious effects of the cost function's conditioning on the convergence rate of distributed gradient-descent. Unlike the conventional pre-conditioning techniques, the pre-conditioner matrix in our proposed technique updates iteratively to facilitate implementation on the distributed network. In the distributed setting, we provably show that the proposed algorithm converges linearly with an improved rate of convergence than the traditional and adaptive gradient-descent methods. Additionally, for the special case when the minimizer of the aggregate cost is unique, our algorithm converges superlinearly. We demonstrate our algorithm's superior performance compared to prominent distributed algorithms for solving real logistic regression problems and emulating neural network training via a noisy quadratic model, thereby signifying the proposed algorithm's efficiency for distributively solving non-convex optimization. Moreover, we empirically show that the proposed algorithm results in faster training without compromising the generalization performance.

Kushal Chakrabarti, Nikhil Chopra

Accelerated gradient-based methods are being extensively used for solving non-convex machine learning problems, especially when the data points are abundant or the available data is distributed across several agents. Two of the prominent accelerated gradient algorithms are AdaGrad and Adam. AdaGrad is the simplest accelerated gradient method, which is particularly effective for sparse data. Adam has been shown to perform favorably in deep learning problems compared to other methods. In this paper, we propose a new fast optimizer, Generalized AdaGrad (G-AdaGrad), for accelerating the solution of potentially non-convex machine learning problems. Specifically, we adopt a state-space perspective for analyzing the convergence of gradient acceleration algorithms, namely G-AdaGrad and Adam, in machine learning. Our proposed state-space models are governed by ordinary differential equations. We present simple convergence proofs of these two algorithms in the deterministic settings with minimal assumptions. Our analysis also provides intuition behind improving upon AdaGrad's convergence rate. We provide empirical results on MNIST dataset to reinforce our claims on the convergence and performance of G-AdaGrad and Adam.

Kushal Chakrabarti, Nirupam Gupta, Nikhil Chopra

This paper considers the problem of multi-agent distributed linear regression in the presence of system noises. In this problem, the system comprises multiple agents wherein each agent locally observes a set of data points, and the agents' goal is to compute a linear model that best fits the collective data points observed by all the agents. We consider a server-based distributed architecture where the agents interact with a common server to solve the problem; however, the server cannot access the agents' data points. We consider a practical scenario wherein the system either has observation noise, i.e., the data points observed by the agents are corrupted, or has process noise, i.e., the computations performed by the server and the agents are corrupted. In noise-free systems, the recently proposed distributed linear regression algorithm, named the Iteratively Pre-conditioned Gradient-descent (IPG) method, has been claimed to converge faster than related methods. In this paper, we study the robustness of the IPG method, against both the observation noise and the process noise. We empirically show that the robustness of the IPG method compares favorably to the state-of-the-art algorithms.

Kushal Chakrabarti, Nirupam Gupta, Nikhil Chopra

This paper considers the multi-agent distributed linear least-squares problem. The system comprises multiple agents, each agent with a locally observed set of data points, and a common server with whom the agents can interact. The agents' goal is to compute a linear model that best fits the collective data points observed by all the agents. In the server-based distributed settings, the server cannot access the data points held by the agents. The recently proposed Iteratively Pre-conditioned Gradient-descent (IPG) method has been shown to converge faster than other existing distributed algorithms that solve this problem. In the IPG algorithm, the server and the agents perform numerous iterative computations. Each of these iterations relies on the entire batch of data points observed by the agents for updating the current estimate of the solution. Here, we extend the idea of iterative pre-conditioning to the stochastic settings, where the server updates the estimate and the iterative pre-conditioning matrix based on a single randomly selected data point at every iteration. We show that our proposed Iteratively Pre-conditioned Stochastic Gradient-descent (IPSG) method converges linearly in expectation to a proximity of the solution. Importantly, we empirically show that the proposed IPSG method's convergence rate compares favorably to prominent stochastic algorithms for solving the linear least-squares problem in server-based networks.

Kushal Chakrabarti, Nirupam Gupta, Nikhil Chopra

This paper considers the multi-agent linear least-squares problem in a server-agent network. In this problem, the system comprises multiple agents, each having a set of local data points, that are connected to a server. The goal for the agents is to compute a linear mathematical model that optimally fits the collective data points held by all the agents, without sharing their individual local data points. This goal can be achieved, in principle, using the server-agent variant of the traditional iterative gradient-descent method. The gradient-descent method converges linearly to a solution, and its rate of convergence is lower bounded by the conditioning of the agents' collective data points. If the data points are ill-conditioned, the gradient-descent method may require a large number of iterations to converge. We propose an iterative pre-conditioning technique that mitigates the deleterious effect of the conditioning of data points on the rate of convergence of the gradient-descent method. We rigorously show that the resulting pre-conditioned gradient-descent method, with the proposed iterative pre-conditioning, achieves superlinear convergence when the least-squares problem has a unique solution. In general, the convergence is linear with improved rate of convergence in comparison to the traditional gradient-descent method and the state-of-the-art accelerated gradient-descent methods. We further illustrate the improved rate of convergence of our proposed algorithm through experiments on different real-world least-squares problems in both noise-free and noisy computation environment.

Kushal Chakrabarti, Nirupam Gupta, Nikhil Chopra

This paper considers the problem of multi-agent distributed optimization. In this problem, there are multiple agents in the system, and each agent only knows its local cost function. The objective for the agents is to collectively compute a common minimum of the aggregate of all their local cost functions. In principle, this problem is solvable using a distributed variant of the traditional gradient-descent method, which is an iterative method. However, the speed of convergence of the traditional gradient-descent method is highly influenced by the conditioning of the optimization problem being solved. Specifically, the method requires a large number of iterations to converge to a solution if the optimization problem is ill-conditioned. In this paper, we propose an iterative pre-conditioning approach that can significantly attenuate the influence of the problem's conditioning on the convergence-speed of the gradient-descent method. The proposed pre-conditioning approach can be easily implemented in distributed systems and has minimal computation and communication overhead. For now, we only consider a specific distributed optimization problem wherein the individual local cost functions of the agents are quadratic. Besides the theoretical guarantees, the improved convergence speed of our approach is demonstrated through experiments on a real data-set.