Recent progress in model selection raises the question of the fundamental limits of these techniques. Under specific scrutiny has been model selection for general contextual bandits with nested policy classes, resulting in a COLT2020 open problem. It asks whether it is possible to obtain simultaneously the optimal single algorithm guarantees over all policies in a nested sequence of policy classes, or if otherwise this is possible for a trade-off $\alpha\in[\frac{1}{2},1)$ between complexity term and time: $\ln(|\Pi_m|)^{1-\alpha}T^\alpha$. We give a disappointing answer to this question. Even in the purely stochastic regime, the desired results are unobtainable. We present a Pareto frontier of up to logarithmic factors matching upper and lower bounds, thereby proving that an increase in the complexity term $\ln(|\Pi_m|)$ independent of $T$ is unavoidable for general policy classes.As a side result, we also resolve a COLT2016 open problem concerning second-order bounds in full-information games.

We consider the problem of online learning with dynamics, where a learner interacts with a stateful environment over multiple rounds. In each round of the interaction, the learner selects a policy to deploy and incurs a cost that depends on both the chosen policy and current state of the world. The state-evolution dynamics and the costs are allowed to be time-varying, in a possibly adversarial way. In this setting, we study the problem of minimizing policy regret and provide non-constructive upper bounds on the minimax rate for the problem. Our main results provide sufficient conditions for online learnability for this setup with corresponding rates. The rates are characterized by: 1) a complexity term capturing the expressiveness of the underlying policy class under the dynamics of state change, and 2) a dynamics stability term measuring the deviation of the instantaneous loss from a certain counterfactual loss. Further, we provide matching lower bounds which show that both the complexity terms are indeed necessary. Our approach provides a unifying analysis that recovers regret bounds for several well studied problems including online learning with memory, online control of linear quadratic regulators, online Markov decision processes, and tracking adversarial targets. In addition, we show how our tools help obtain tight regret bounds for a new problems (with non-linear dynamics and non-convex losses) for which such bounds were not known prior to our work.

Neural Information Processing Systems http://nips.cc/

We consider an adversarial formulation of the problem of predicting a time series with square loss. The aim is to predict an arbitrary sequence of vectors almost as well as the best smooth comparator sequence in retrospect. Our approach allows natural measures of smoothness such as the squared norm of increments. More generally, we consider a linear time series model and penalize the compara-tor sequence through the energy of the implied driving noise terms. We derive the minimax strategy for all problems of this type and show that it can be implemented efficiently. The optimal predictions are linear in the previous observations. We obtain an explicit expression for the regret in terms of the parameters defining the problem. For typical, simple definitions of smoothness, the computation of the optimal predictions involves only sparse matrices. In the case of norm-constrained data, where the smoothness is defined in terms of the squared norm of the com-parator's increments, we show that the regret grows as T/ λ

Given such a setup, a natural question to ask is how does one measure the performance of the learner? Classical online learning studies one such notion of performance known as regret.

Deep learning methods are known to generalize well from training to future data, even in an overparametrized regime, where they could easily overfit. One explanation for this phenomenon is that even when their *ambient dimensionality*, (i.e. the number of parameters) is large, the models' *intrinsic dimensionality* is small, i.e. their learning takes place in a small subspace of all possible weight configurations. In this work, we confirm this phenomenon in the setting of *deep multi-task learning*. We introduce a method to parametrize multi-task network directly in the low-dimensional space, facilitated by the use of *random expansions* techniques. We then show that high-accuracy multi-task solutions can be found with much smaller intrinsic dimensionality (fewer free parameters) than what single-task learning requires. Subsequently, we show that the low-dimensional representations in combination with *weight compression* and *PAC-Bayesian* reasoning lead to the first *non-vacuous generalization bounds* for deep multi-task networks.

We consider the problem of online learning with dynamics, where a learner interacts with a stateful environment over multiple rounds. In each round of the interaction, the learner selects a policy to deploy and incurs a cost that depends on both the chosen policy and current state of the world. The state-evolution dynamics and the costs are allowed to be time-varying, in a possibly adversarial way. In this setting, we study the problem of minimizing policy regret and provide non-constructive upper bounds on the minimax rate for the problem. Our main results provide sufficient conditions for online learnability for this setup with corresponding rates.

In this work we aim at extending the theoretical foundations of lifelong learning. Previous work analyzing this scenario is based on the assumption that learning tasks are sampled i.i.d.

We consider an adversarial formulation of the problem of predicting a time series with square loss. The aim is to predict an arbitrary sequence of vectors almost as well as the best smooth comparator sequence in retrospect. Our approach allows natural measures of smoothness such as the squared norm of increments. More generally, we consider a linear time series model and penalize the comparator sequence through the energy of the implied driving noise terms. We derive the minimax strategy for all problems of this type and show that it can be implemented efficiently. The optimal predictions are linear in the previous observations. We obtain an explicit expression for the regret in terms of the parameters defining the problem. For typical, simple definitions of smoothness, the computation of the optimal predictions involves only sparse matrices. In the case of norm-constrained data, where the smoothness is defined in terms of the squared norm of the comparator's increments, we show that the regret grows as T/ λ