Stochastic gradient descent with discontinuity across a manifold
Stochastic gradient descent for a loss function discontinuous across lower dimensional manifolds is analyzed by studying its differential equation limit.
stat.ML↗
arXiv subjects
Vivek S. Borkar
Publications and source records attributed to Vivek S. Borkar.
Adynamical systems view of training generativemodels and the memorization phenomenon
Using recent works of one of the authors (VSB) on collapse in generative models and two time scale dynamics in stochastic gradient descent in high dimensions, we give a system theoretic explanation of the memorization phenomenon in generative models. This relies purely on the dynamic aspects of the training phase. Specifically, we use a result of Austin [2016] to motivate a stylized model for the loss function for stochastic gradient descent (SGD) wherein the loss function has a strong dependence on some variables and weak dependence on the rest in a precise sense. This naturally leads to two distinct time scales in the constant step size SGD that is commonly used in machine learning. This fact has been used to explain the double descent phenomenon in SGD in Borkar [2026]. In conjunction with a mathematical model for collapse phenomenon in SGD developed in Borkar [2025a], we analyze the constant step size SGD using the recent results of Azizian et al. [2024] in order to explain the phenomenon of memorization wherein a generative model that is concurrently being tuned yields the same or similar outputs for significant stretches of time. This gives a novel perspective on the aforementioned phenomena reported in machine learning literature and their interrelationships, using a dynamical systems viewpoint.
cs.LG↗
Regret and Sample Complexity of Online Q-Learning via Concentration of Stochastic Approximation with Time-Inhomogeneous Markov Chains
We present the first regret bound for classical online Q-learning in infinite-horizon discounted Markov decision processes (MDPs), without relying on optimism or bonus terms. We first analyze Boltzmann Q-learning with decaying temperature and show that its regret depends critically on the suboptimality gap of the MDP: for sufficiently large gaps, the regret is sublinear, while for small gaps it deteriorates and can approach linear growth. To address this limitation, we study a Smoothed $ε_n$-Greedy exploration scheme that combines $ε_n$-greedy and Boltzmann exploration, for which we prove a gap-robust regret bound of near-$\tilde{O}(N^{9/10})$. We also obtain sample complexity guarantees, with both regret and sample complexity bounds holding with high probability. To analyze these algorithms, we develop a high-probability concentration bound for contractive Markovian stochastic approximation with iterate- and time-dependent transition dynamics. This bound may be of independent interest as the contraction factor in our framework is allowed to converge to one asymptotically.
cs.LG↗
Small noise asymptotics for a class of jump-diffusions with heavy tails for large times
In this work, we investigate positive recurrent Lévy diffusions driven by appropriately scaled Brownian motion and $α$-stable process (with $1<α<2$) in the small noise regime. Supposing that in the vanishing noise limit, our Lévy diffusion approaches a deterministic system with a unique asymptotically stable fixed point, we show that the limiting behavior of the one-dimensional marginal distribution at large times is dictated by the optimal value of a deterministic control problem, just as in the classical case of diffusions driven by small variance Brownian motion. In our case, the control is allowed to have two parts: continuous control and impulse control.
math.PR↗
A Concentration Bound for TD(0) with Function Approximation
We derive uniform all-time concentration bound of the type 'for all $n \geq n_0$ for some $n_0$' for TD(0) with linear function approximation. We work with online TD learning with samples from a single sample path of the underlying Markov chain. This makes our analysis significantly different from offline TD learning or TD learning with access to independent samples from the stationary distribution of the Markov chain. We treat TD(0) as a contractive stochastic approximation algorithm, with both martingale and Markov noises. Markov noise is handled using the Poisson equation and the lack of almost sure guarantees on boundedness of iterates is handled using the concept of relaxed concentration inequalities.
cs.LG↗
Lagrangian Index Policy for Restless Bandits with Average Reward
We study the Lagrangian Index Policy (LIP) for restless multi-armed bandits with long-run average reward. In particular, we compare the performance of LIP with the performance of the Whittle Index Policy (WIP), both heuristic policies known to be asymptotically optimal under certain natural conditions. Even though in most cases their performances are very similar, in the cases when WIP shows bad performance, LIP continues to perform very well. We then propose reinforcement learning algorithms, both tabular and NN-based, to obtain online learning schemes for LIP in the model-free setting. The proposed reinforcement learning schemes for LIP require significantly less memory than the analogous schemes for WIP. We calculate analytically the Lagrangian index for the restart model, which applies to the optimal web crawling and the minimization of the weighted age of information. We also give a new proof of asymptotic optimality in case of homogeneous arms as the number of arms goes to infinity, based on exchangeability and de Finetti's theorem.
cs.LG↗
Strong and weak quantitative estimates in slow-fast diffusions using filtering techniques
The behavior of slow-fast diffusions as the separation of scale diverges is a well-studied problem in the literature. In this short paper, we revisit this problem and obtain a new proof of existing strong quantitative convergence estimates (in particular, $L^2$ estimates) and weak convergence estimates in terms of $n$ (the parameter associated with the separation of scales). In particular, we obtain the rate of $n^{-\frac{1}{2}}$ in the strong convergence estimates and the rate of $n^{-1}$ for weak convergence estimate which are already known to be optimal in the literature. We achieve this using nonlinear filtering theory where we represent the evolution of fast diffusion in terms of its conditional distribution given the slow diffusion. We then use the well-known Kushner-Stratanovich equation which gives the evolution of the conditional distribution of the fast diffusion given the slow diffusion and establish that this conditional distribution approaches the invariant measure of the ``frozen" diffusion (obtained by freezing the slow variable in the evolution equation of the fast diffusion). At the heart of the analysis lies a key estimate of a weighted Lipschitz distance like function between a generic one-parameter family of measures and the family of unique invariant measures (of the ``frozen" diffusion parametrized by a path). This estimate is in terms of the operator norm of the dual of the infinitesimal generator of the ``frozen" diffusion.
math.OC↗
An Actor-Critic Algorithm with Function Approximation for Risk Sensitive Cost Markov Decision Processes
In this paper, we consider the risk-sensitive cost criterion with exponentiated costs for Markov decision processes and develop a model-free policy gradient algorithm in this setting. Unlike additive cost criteria such as average or discounted cost, the risk-sensitive cost criterion is less studied due to the complexity resulting from the multiplicative structure of the resulting Bellman equation. We develop an actor-critic algorithm with function approximation in this setting and provide its asymptotic convergence analysis. We also show the results of numerical experiments that demonstrate the superiority in performance of our algorithm over other recent algorithms in the literature.
cs.LG↗
Actor-Critic or Critic-Actor? A Tale of Two Time Scales
We revisit the standard formulation of tabular actor-critic algorithm as a two time-scale stochastic approximation with value function computed on a faster time-scale and policy computed on a slower time-scale. This emulates policy iteration. We observe that reversal of the time scales will in fact emulate value iteration and is a legitimate algorithm. We provide a proof of convergence and compare the two empirically with and without function approximation (with both linear and nonlinear function approximators) and observe that our proposed critic-actor algorithm performs on par with actor-critic in terms of both accuracy and computational effort.
cs.LG↗
Whittle Index Based User Association in Dense Millimeter Wave Networks
We address the problem of user association in a dense millimeter wave (mmWave) network, in which each arriving user brings a file containing a random number of packets and each time slot is divided into multiple mini-slots. This problem is an instance of the restless multi-armed bandit problem, and is provably hard to solve. Using a technique introduced by Whittle, we relax the hard per-stage constraint that each arriving user must be associated with exactly one mmWave base station (mBS) to a long-term constraint and then use the Lagrangian multiplier technique to convert the problem into an unconstrained problem. This decouples the process governing the system into separate Markov Decision Processes at different mBSs. We prove that the problem is Whittle indexable, present a scheme for computing the Whittle indices of different mBSs, and propose an association scheme under which, each arriving user is associated with the mBS with the smallest value of the Whittle index. Using extensive simulations, we show that the proposed Whittle index based scheme outperforms several user association schemes proposed in prior work in terms of various performance metrics such as average cost, delay, throughput, and Jain's fairness index.
cs.NI↗
Approximation of Convex Envelope Using Reinforcement Learning
Oberman gave a stochastic control formulation of the problem of estimating the convex envelope of a non-convex function. Based on this, we develop a reinforcement learning scheme to approximate the convex envelope, using a variant of Q-learning for controlled optimal stopping. It shows very promising results on a standard library of test problems.
eess.SY↗
Decentralised Q-Learning for Multi-Agent Markov Decision Processes with a Satisfiability Criterion
In this paper, we propose a reinforcement learning algorithm to solve a multi-agent Markov decision process (MMDP). The goal, inspired by Blackwell's Approachability Theorem, is to lower the time average cost of each agent to below a pre-specified agent-specific bound. For the MMDP, we assume the state dynamics to be controlled by the joint actions of agents, but the per-stage costs to only depend on the individual agent's actions. We combine the Q-learning algorithm for a weighted combination of the costs of each agent, obtained by a gossip algorithm with the Metropolis-Hastings or Multiplicative Weights formalisms to modulate the averaging matrix of the gossip. We use multiple timescales in our algorithm and prove that under mild conditions, it approximately achieves the desired bounds for each of the agents. We also demonstrate the empirical performance of this algorithm in the more general setting of MMDPs having jointly controlled per-stage costs.
eess.SY↗
Node Cardinality Estimation in the Internet of Things Using Privileged Feature Distillation
The Internet of Things (IoT) is emerging as a critical technology to connect resource-constrained devices such as sensors and actuators as well as appliances to the Internet. In this paper, we propose a novel methodology for node cardinality estimation in wireless networks such as the IoT and Radio-Frequency IDentification (RFID) systems, which uses the privileged feature distillation (PFD) technique and works using a neural network with a teacher-student model. The teacher is trained using both privileged and regular features, and the student is trained with predictions from the teacher and regular features. We propose node cardinality estimation algorithms based on the PFD technique for homogeneous as well as heterogeneous wireless networks. We show via extensive simulations that the proposed PFD based algorithms for homogeneous as well as heterogeneous networks achieve much lower mean squared errors in the computed node cardinality estimates than state-of-the-art protocols proposed in prior work, while taking the same number of time slots for executing the node cardinality estimation process as the latter protocols.
cs.NI↗
Controlled Martingale Problems And Their Markov Mimics
In this article we prove under suitable assumptions that the marginals of any solution to a relaxed controlled martingale problem on a Polish space $E$ can be mimicked by a Markovian solution of a Markov-relaxed controlled martingale problem. We also show how such `Markov mimics' can be obtained by relative entropy minimisation. We provide many examples where the above results can be applied.
math.PR↗
Average cost optimal control under weak ergodicity hypotheses: Relative value iterations
We study Markov decision processes with Polish state and action spaces. The action space is state dependent and is not necessarily compact. We first establish the existence of an optimal ergodic occupation measure using only a near-monotone hypothesis on the running cost. Then we study the well-posedness of Bellman equation, or what is commonly known as the average cost optimality equation, under the additional hypothesis of the existence of a small set. We deviate from the usual approach which is based on the vanishing discount method and instead map the problem to an equivalent one for a controlled split chain. We employ a stochastic representation of the Poisson equation to derive the Bellman equation. Next, under suitable assumptions, we establish convergence results for the 'relative value iteration' algorithm which computes the solution of the Bellman equation recursively. In addition, we present some results concerning the stability and asymptotic optimality of the associated rolling horizon policies.
math.OC↗
Ergodic Risk-sensitive control -- A survey
Risk-sensitive control has received considerable interest since the seminal work of Howard and Matheson [120] because of its ability to account for fluctuations about the mean, its connection with $H_\infty$ control, and its application to financial mathematics. In this article, we attempt to put together a comprehensive survey on the research done on ergodic risk-sensitive control over the last four decades.
math.OC↗
A Concentration Bound for LSPE($λ$)
The popular LSPE($λ$) algorithm for policy evaluation is revisited to derive a concentration bound that gives high probability performance guarantees from some time on.
cs.LG↗
Concentration of Contractive Stochastic Approximation and Reinforcement Learning
Using a martingale concentration inequality, concentration bounds `from time $n_0$ on' are derived for stochastic approximation algorithms with contractive maps and both martingale difference and Markov noises. These are applied to reinforcement learning algorithms, in particular to asynchronous Q-learning and TD(0).
cs.LG↗