Sudeep Raja Putta

Sudeep Raja Putta, Shipra Agrawal

We consider the Scale-Free Adversarial Multi Armed Bandits(MAB) problem. At the beginning of the game, the player only knows the number of arms $n$. It does not know the scale and magnitude of the losses chosen by the adversary or the number of rounds $T$. In each round, it sees bandit feedback about the loss vectors $l_1,\dots, l_T \in \mathbb{R}^n$. The goal is to bound its regret as a function of $n$ and norms of $l_1,\dots, l_T$. We design a bandit Follow The Regularized Leader (FTRL) algorithm, that uses an adaptive learning rate and give two different regret bounds, based on the exploration parameter used. With non-adaptive exploration, our algorithm has a regret of $\tilde{\mathcal{O}}(\sqrt{nL_2} + L_\infty\sqrt{nT})$ and with adaptive exploration, it has a regret of $\tilde{\mathcal{O}}(\sqrt{nL_2} + L_\infty\sqrt{nL_1})$. Here $L_\infty = \sup_t \| l_t\|_\infty$, $L_2 = \sum_{t=1}^T \|l_t\|_2^2$, $L_1 = \sum_{t=1}^T \|l_t\|_1$ and the $\tilde{\mathcal{O}}$ notation suppress logarithmic factors. These are the first MAB bounds that adapt to the $\|\cdot\|_2$, $\|\cdot\|_1$ norms of the losses. The second bound is the first data-dependent scale-free MAB bound as $T$ does not directly appear in the regret. We also develop a new technique for obtaining a rich class of local-norm lower-bounds for Bregman Divergences. This technique plays a crucial role in our analysis for controlling the regret when using importance weighted estimators of unbounded losses. This technique could be of independent interest.

Sudeep Raja Putta, Abhishek Shetty

We study a general online linear optimization problem(OLO). At each round, a subset of objects from a fixed universe of $n$ objects is chosen, and a linear cost associated with the chosen subset is incurred. To measure the performance of our algorithms, we use the notion of regret which is the difference between the total cost incurred over all iterations and the cost of the best fixed subset in hindsight. We consider Full Information and Bandit feedback for this problem. This problem is equivalent to OLO on the $\{0,1\}^n$ hypercube. The Exp2 algorithm and its bandit variant are commonly used strategies for this problem. It was previously unknown if it is possible to run Exp2 on the hypercube in polynomial time. In this paper, we present a polynomial time algorithm called PolyExp for OLO on the hypercube. We show that our algorithm is equivalent Exp2 on $\{0,1\}^n$, Online Mirror Descent(OMD), Follow The Regularized Leader(FTRL) and Follow The Perturbed Leader(FTPL) algorithms. We show PolyExp achieves expected regret bound that is a factor of $\sqrt{n}$ better than Exp2 in the full information setting under $L_\infty$ adversarial losses. Because of the equivalence of these algorithms, this implies an improvement on Exp2's regret bound in full information. We also show matching regret lower bounds. Finally, we show how to use PolyExp on the $\{-1,+1\}^n$ hypercube, solving an open problem in Bubeck et al (COLT 2012).

Sudeep Raja Putta, Theja Tulabandhula

In several realistic situations, an interactive learning agent can practice and refine its strategy before going on to be evaluated. For instance, consider a student preparing for a series of tests. She would typically take a few practice tests to know which areas she needs to improve upon. Based of the scores she obtains in these practice tests, she would formulate a strategy for maximizing her scores in the actual tests. We treat this scenario in the context of an agent exploring a fixed-horizon episodic Markov Decision Process (MDP), where the agent can practice on the MDP for some number of episodes (not necessarily known in advance) before starting to incur regret for its actions. During practice, the agent's goal must be to maximize the probability of following an optimal policy. This is akin to the problem of Pure Exploration (PE). We extend the PE problem of Multi Armed Bandits (MAB) to MDPs and propose a Bayesian algorithm called Posterior Sampling for Pure Exploration (PSPE), which is similar to its bandit counterpart. We show that the Bayesian simple regret converges at an optimal exponential rate when using PSPE. When the agent starts being evaluated, its goal would be to minimize the cumulative regret incurred. This is akin to the problem of Reinforcement Learning (RL). The agent uses the Posterior Sampling for Reinforcement Learning algorithm (PSRL) initialized with the posteriors of the practice phase. We hypothesize that this PSPE + PSRL combination is an optimal strategy for minimizing regret in RL problems with an initial practice phase. We show empirical results which prove that having a lower simple regret at the end of the practice phase results in having lower cumulative regret during evaluation.