Learning Linear Dynamical Systems with Semi-Parametric Least Squares

We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, semi-parametric noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle inequality which demonstrates that this procedure provably mitigates the variance introduced by long-term dependencies. We then demonstrate that prefiltered least squares yields, to our knowledge, the first algorithm that provably estimates the parameters of partially-observed linear systems that attains rates which do not not incur a worst-case dependence on the rate at which these dependencies decay. The algorithm is provably consistent even for systems which satisfy the weaker marginal stability condition obeyed by many classical models based on Newtonian mechanics. In this context, our semi-parametric framework yields guarantees for both stochastic and worst-case noise

Best-of-K Bandits

This paper studies the Best-of-K Bandit game: At each time the player chooses a subset S among all N-choose-K possible options and observes reward max(X(i) : i in S) where X is a random vector drawn from a joint distribution. The objective is to identify the subset that achieves the highest expected reward with high probability using as few queries as possible. We present distribution-dependent lower bounds based on a particular construction which force a learner to consider all N-choose-K subsets, and match naive extensions of known upper bounds in the bandit setting obtained by treating each subset as a separate arm. Nevertheless, we present evidence that exhaustive search may be avoided for certain, favorable distributions because the influence of high-order order correlations may be dominated by lower order statistics. Finally, we present an algorithm and analysis for independent arms, which mitigates the surprising non-trivial information occlusion that occurs due to only observing the max in the subset. This may inform strategies for more general dependent measures, and we complement these result with independent-arm lower bounds

The Simulator: Understanding Adaptive Sampling in the Moderate-Confidence Regime

We propose a novel technique for analyzing adaptive sampling called the {\em Simulator}. Our approach differs from the existing methods by considering not how much information could be gathered by any fixed sampling strategy, but how difficult it is to distinguish a good sampling strategy from a bad one given the limited amount of data collected up to any given time. This change of perspective allows us to match the strength of both Fano and change-of-measure techniques, without succumbing to the limitations of either method. For concreteness, we apply our techniques to a structured multi-arm bandit problem in the fixed-confidence pure exploration setting, where we show that the constraints on the means imply a substantial gap between the moderate-confidence sample complexity, and the asymptotic sample complexity as $\delta \to 0$ found in the literature. We also prove the first instance-based lower bounds for the top-k problem which incorporate the appropriate log-factors. Moreover, our lower bounds zero-in on the number of times each \emph{individual} arm needs to be pulled, uncovering new phenomena which are drowned out in the aggregate sample complexity. Our new analysis inspires a simple and near-optimal algorithm for the best-arm and top-k identification, the first {\em practical} algorithm of its kind for the latter problem which removes extraneous log factors, and outperforms the state-of-the-art in experiments

On the Gap Between Strict-Saddles and True Convexity: An Omega(log d) Lower Bound for Eigenvector Approximation

We prove a \emph{query complexity} lower bound on rank-one principal component analysis (PCA). We consider an oracle model where, given a symmetric matrix $M \in \mathbb{R}^{d \times d}$, an algorithm is allowed to make $T$ \emph{exact} queries of the form $w^{(i)} = Mv^{(i)}$ for $i \in \{1,\dots,T\}$, where $v^{(i)}$ is drawn from a distribution which depends arbitrarily on the past queries and measurements $\{v^{(j)},w^{(j)}\}_{1 \le j \le i-1}$. We show that for a small constant $\epsilon$, any adaptive, randomized algorithm which can find a unit vector $\widehat{v}$ for which $\widehat{v}^{\top}M\widehat{v} \ge (1-\epsilon)\|M\|$, with even small probability, must make $T = \Omega(\log d)$ queries. In addition to settling a widely-held folk conjecture, this bound demonstrates a fundamental gap between convex optimization and "strict-saddle" non-convex optimization of which PCA is a canonical example: in the former, first-order methods can have dimension-free iteration complexity, whereas in PCA, the iteration complexity of gradient-based methods must necessarily grow with the dimension. Our argument proceeds via a reduction to estimating the rank-one spike in a deformed Wigner model. We establish lower bounds for this model by developing a "truncated" analogue of the $\chi^2$ Bayes-risk lower bound of Chen et al

Social security and HIV/AIDS: assessing “disability” in the context of ARV treatment

Despite its less-than-stellar implementation, the South African government’s 2003 commitment to providing free antiretroviral therapy to those with AIDS has both provided hope to the many infected while at the same time highlighting the gross inadequacies of the current welfare system’s design. Examining circumstances in the Western Cape is a useful way of exploring the relationship between poverty and HIV/AIDS, as well as the role of government welfare programmes in influencing the success or failure of prevention and treatment interventions. This paper attempts to outline the shortfalls of the current social safety net in South Africa and the particular effects of those inadequacies on people suffering with HIV and AIDS. It focuses specifically on the disability grant in the Western Cape province, arguing that, in the absence of comprehensive unemployment benefits or a universal basic income grant, a broader redefinition of disability is needed that takes into account social factors in addition to a medical diagnosis. Finally, future legislation is evaluated, and potential solutions are suggested and critiqued

Making Non-Stochastic Control (Almost) as Easy as Stochastic

Recent literature has made much progress in understanding \emph{online LQR}: a modern learning-theoretic take on the classical control problem in which a learner attempts to optimally control an unknown linear dynamical system with fully observed state, perturbed by i.i.d. Gaussian noise. It is now understood that the optimal regret on time horizon $T$ against the optimal control law scales as $\widetilde{\Theta}(\sqrt{T})$. In this paper, we show that the same regret rate (against a suitable benchmark) is attainable even in the considerably more general non-stochastic control model, where the system is driven by \emph{arbitrary adversarial} noise (Agarwal et al. 2019). In other words, \emph{stochasticity confers little benefit in online LQR}. We attain the optimal $\widetilde{\mathcal{O}}(\sqrt{T})$ regret when the dynamics are unknown to the learner, and $\mathrm{poly}(\log T)$ regret when known, provided that the cost functions are strongly convex (as in LQR). Our algorithm is based on a novel variant of online Newton step (Hazan et al. 2007), which adapts to the geometry induced by possibly adversarial disturbances, and our analysis hinges on generic "policy regret" bounds for certain structured losses in the OCO-with-memory framework (Anava et al. 2015). Moreover, our results accomodate the full generality of the non-stochastic control setting: adversarially chosen (possibly non-quadratic) costs, partial state observation, and fully adversarial process and observation noise

Low-rank Solutions of Linear Matrix Equations via Procrustes Flow

In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a $n_1 \times n_2$ matrix of rank $r$ when the number of measurements exceeds a constant times $(n_1+n_2)r$.Comment: Added new results for general rectangular matrice

Gradient Descent Converges to Minimizers

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.Comment: Submitted to COLT 201

Delayed Impact of Fair Machine Learning

Fairness in machine learning has predominantly been studied in static classification settings without concern for how decisions change the underlying population over time. Conventional wisdom suggests that fairness criteria promote the long-term well-being of those groups they aim to protect. We study how static fairness criteria interact with temporal indicators of well-being, such as long-term improvement, stagnation, and decline in a variable of interest. We demonstrate that even in a one-step feedback model, common fairness criteria in general do not promote improvement over time, and may in fact cause harm in cases where an unconstrained objective would not. We completely characterize the delayed impact of three standard criteria, contrasting the regimes in which these exhibit qualitatively different behavior. In addition, we find that a natural form of measurement error broadens the regime in which fairness criteria perform favorably. Our results highlight the importance of measurement and temporal modeling in the evaluation of fairness criteria, suggesting a range of new challenges and trade-offs.Comment: 37 pages, 6 figure

Interactions of bromide, iodide, and fluoride with the pathways of chloride transport and diffusion in human neutrophils

Isolated human neutrophils possess three distinct pathways by which Cl- crosses the plasma membrane of steady state cells: anion exchange, active transport, and electrodiffusion. The purpose of the present work was to investigate the selectivity of each of these separate processes with respect to other external halide ions. (a) The bulk of total anion movements represents transport through an electrically silent anion-exchange mechanism that is insensitive to disulfonic stilbenes, but which can be competitively inhibited by alpha-cyano-4-hydroxycinnamate (CHC; Ki approximately 0.3 mM). The affinity of the external translocation site of the carrier for each of the different anions was determined (i) from substrate competition between Cl- and either Br-, F-, or I-, (ii) from trans stimulation of 36Cl- efflux as a function of the external concentrations of these anions, (iii) from changes in the apparent Ki for CHC depending on the nature of the replacement anion in the bathing medium, and (iv) from activation of 82Br- and 125I- influxes by their respective ions. Each was bound and transported at roughly similar rates (Vmax values all 1.0-1.4 meq/liter cell water.min); the order of decreasing affinities is Cl- greater than Br- greater than F- greater than I- (true Km values of 5, 9, 23, and 44 mM, respectively). These anions undergo 1:1 countertransport for internal Cl-. (b) There is a minor component of total Cl- influx that constitutes an active inward transport system for the intracellular accumulation of Cl- [( Cl-]i approximately 80 meq/liter cell water), fourfold higher than expected for passive distribution. This uptake is sensitive to intracellular ATP depletion by 2-deoxy-D-glucose and can be inhibited by furosemide, ethacrynic acid, and CHC, which also blocks anion exchange. This active Cl- uptake process binds and transports other members of the halide series in the sequence Cl- greater than Br- greater than I- greater than F- (Km values of 5, 8, 15, and 41 mM, respectively). (c) Electrodiffusive fluxes are small. CHC-resistant 82Br- and 125I- influxes behave as passive leak fluxes through low-conductance ion channels: they are nonsaturable and strongly voltage dependent. These anions permeate the putative Cl- channel in the sequence I- greater than Br- greater than Cl- with relative permeability ratios of 2.2:1.4:1, respectively, where PCl approximately 5 X 10(-9) cm/s