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Question: An important problem that arises in reinforcement learning and Monte Carlo methods is estimating quantities defined by the stationary distribution of a Markov chain. In many real-world applications, access to the underlying transition operator is limited to a fixed set of data that has already been collected, without additional interaction with the environment being available. We show that consistent estimation remains possible in this scenario, and that effective estimation can still be achieved in important applications. Our approach is based on estimating a ratio that corrects for the discrepancy between the stationary and empirical distributions, derived from fundamental properties of the stationary distribution, and exploiting constraint reformulations based on variational divergence minimization. The ing algorithm, GenDICE, is straightforward and effective. We prove the consistency of the method under general conditions, provide a detailed error analysis, and demonstrate strong empirical performance on benchmark tasks, including off-line PageRank and off-policy policy evaluation. Estimation of quantities defined by the stationary distribution of a Markov chain lies at the heart of many scientific and engineering problems. Famously, the steady-state distribution of a random walk on the World Wide Web provides the foundation of the PageRank algorithm . In many areas of machine learning, Markov chain Monte Carlo (MCMC) methods are used to conduct approximate Bayesian inference by considering Markov chains whose equilibrium distribution is a desired posterior . An example from engineering is queueing theory, where the queue lengths and waiting time under the limiting distribution have been extensively studied . As we will also see below, stationary distribution quantities are of fundamental importance in reinforcement learning (RL) (e.g.,). Classical algorithms for estimating stationary distribution quantities rely on the ability to sample next states from the current state by directly interacting with the environment (as in on-line RL or MCMC), or even require the transition probability distribution to be given explicitly (as in PageRank). Unfortunately, these classical approaches are inapplicable when direct access to the environment is not available, which is often the case in practice. There are many practical scenarios where a collection of sampled trajectories is available, having been collected off-line by an external mechanism that chose states and recorded the subsequent next states. Given such data, we still wish to estimate a stationary quantity. One important example is off-policy policy evaluation in RL, where we wish to estimate the value of a policy different from that used to collect experience. Another example is off-line PageRank (OPR), where we seek to estimate the relative importance of webpages given a sample of the web graph. Motivated by the importance of these off-line scenarios, and by the inapplicability of classical methods, we study the problem of off-line estimation of stationary values via a stationary distribution corrector. Instead of having access to the transition probabilities or a next-state sampler, we assume only access to a fixed sample of state transitions, where states have been sampled from an unknown distribution and next-states are sampled according to the Markov chain's transition operator. This off-line setting is distinct from that considered by most MCMC or on-line RL methods, where it is assumed that new observations can be continually sampled by demand from the environment. The off-line setting is indeed more challenging than its more traditional on-line counterpart, given that one must infer an asymptotic quantity from finite data. Nevertheless, we develop techniques that still allow consistent estimation under general conditions, and provide effective estimates in practice. The main contributions of this work are: • We formalize the problem of off-line estimation of stationary quantities, which captures a wide range of practical applications. • We propose a novel stationary distribution estimator, GenDICE, for this task. The ing algorithm is based on a new dual embedding formulation for divergence minimization, with a carefully designed mechanism that explicitly eliminates degenerate solutions. • We theoretically establish consistency and other statistical properties of GenDICE, and empirically demonstrate that it achieves significant improvements on several behavior-agnostic offpolicy evaluation benchmarks and an off-line version of PageRank. The methods we develop in this paper fundamentally extend recent work in off-policy policy evaluation by introducing a new formulation that leads to a more general, and as we will show, more effective estimation method. We first introduce off-line PageRank (OPR) and off-policy policy evaluation (OPE) as two motivating domains, where the goal is to estimate stationary quantities given only off-line access to a set of sampled transitions from an environment. The celebrated PageRank algorithm defines the ranking of a web page in terms of its asymptotic visitation probability under a random walk on the (augmented) directed graph specified by the hyperlinks. If we denote the World Wide Web by a directed graph G = (V, E) with vertices (web pages) v ∈ V and edges (hyperlinks) (v, u) ∈ E, PageRank considers the random walk defined by the Markov transition operator v → u: P (u|v) = (1−η) where |v| denotes the out-degree of vertex v and η ∈ is a probability of "teleporting" to any page uniformly. Define d t (v):= P (s t = v|s 0 ∼ µ 0, ∀i < t, s i+1 ∼ P(·|s i)), where µ 0 is the initial distribution over vertices, then the original PageRank algorithm explicitly iterates for the limit The classical version of this problem is solved by tabular methods that simulate Equation 1. However, we are interested in a more scalable off-line version of the problem where the transition model is not explicitly given. Instead, consider estimating the rank of a particular web page v from a large web graph, given only a sample D = {(v, u) from a random walk on G as specified above. We would still like to estimate d(v) based on this data. First, note that if one knew the distribution p by which any vertex v appeared in D, the target quantity could be re-expressed by a simple importance ratio d Policy Evaluation (Preliminaries) An important generalization of this stationary value estimation problem arises in RL in the form of policy evaluation. Consider a Markov Decision Process (MDP) M = S, A, P, R, γ, µ 0 , where S is a state space, A is an action space, P (s |s, a) denotes the transition dynamics, R is a reward function, γ ∈ is a discounted factor, and µ 0 is the initial state distribution. Given a policy, which chooses actions in any state s according to the probability distribution π(·|s), a trajectory β = (s 0, a 0, r 0, s 1, a 1, r 1, . . .) is generated by first sampling the initial state s 0 ∼ µ 0, and then for t ≥ 0, a t ∼ π(·|s t), r t ∼ R(s t, a t), and s t+1 ∼ P(·|s t, a t). The value of a policy π is the expected per-step reward defined as: Here the expectation is taken with respect to the randomness in the state-action pair P (s |s, a) π (a |s) and the reward R (s t, a t). Without loss of generality, we assume the limit exists for the average case, and hence R(π) is finite. Behavior-agnostic Off-Policy Evaluation (OPE) A natural version of policy evaluation that often arises in practice is to estimate, where p (s, a) is an unknown distribution induced by multiple unknown behavior policies. This problem is different from the classical form of OPE, where it is assumed that a known behavior policy π b is used to collect transitions; in the behavior-agnostic scenario we are considering here, standard importance sampling (IS) estimators cannot be applied. Let d π t (s, a) = P (s t = s, a t = a|s 0 ∼ µ 0, ∀i < t, a i ∼ π (·|s i), s i+1 ∼ P(·|s i, a i)). The stationary distribution can then be defined as From this definition note that R(π) and R γ (π) can be equivalently re-expressed as Here we see once again that if we had the correction ratio function τ (s, a) = is an empirical estimate of p (s, a). In this way, the behavior-agnostic OPE problem can be reduced to estimating the correction ratio function τ, as above. We note that and also exploit Equation 5 to reduce OPE to stationary distribution correction, but these prior works are distinct from the current proposal in different ways. First, the inverse propensity score (IPS) method of assumes the transitions are sampled from a single behavior policy, which must be known beforehand; hence that approach is not applicable in behavior-agnostic OPE setting. Second, the recent DualDICE algorithm is also a behavior-agnostic OPE estimator, but its derivation relies on a change-of-variable trick that is only valid for γ < 1. This previous formulation becomes unstable when γ → 1, as shown in Section 6 and Appendix E. The behavior-agnostic OPE estimator we derive below in Section 3 is applicable both when γ = 1 and γ ∈. This connection is why we name the new estimator GenDICE, for GENeralized stationary DIstribution Correction Estimation. As noted, there are important estimation problems in the Markov chain and MDP settings that can be recast as estimating a stationary distribution correction ratio. We first outline the conditions that characterize the correction ratio function τ, upon which we construct the objective for the GenDICE estimator, and design efficient algorithm for optimization. We will develop our approach for the more general MDP setting, with the understanding that all the methods and can be easily specialized to the Markov chain setting. The stationary distribution µ π γ defined in Equation 4 can also be characterized via At first glance, this equation shares a superficial similarity to the Bellman equation, but there is a fundamental difference. The Bellman operator recursively integrates out future (s, a) pairs to characterize a current pair (s, a) value, whereas the distribution operator T defined in Equation 6 operates in the reverse temporal direction. When γ ≤ 1, Equation 6 always has a fixed-point solution. For γ = 1, in the discrete case, the fixedpoint exists as long as T is ergodic; in the continuous case, the conditions for fixed-point existence become more complicated and beyond the scope of this paper. The development below is based on a divergence D and the following default assumption. Assumption 1 (Markov chain regularity) For the given target policy π, the ing state-action transition operator T has a unique stationary distribution µ that satisfies D(T • µ µ) = 0. In the behavior-agnostic setting we consider, one does not have direct access to P for element-wise evaluation or sampling, but instead is given a fixed set of samples from P (s |s, a) p (s, a) with respect to some distribution p (s, a) over S × A. Define T p γ,µ0 to be a mixture of µ 0 π and T p; i.e., let Obviously, conditioning on (s, a, s) one could easily sample a ∼ π (a |s) to form (s, a, s, a) ∼ T p ((s, a), (s, a)); similarly, a sample (s, a) ∼ µ 0 (s) π (a |s) could be formed from s. Mixing such samples with probability γ and 1 − γ respectively yields a sample (s, a, s, a) ∼ T p γ,µ0 ((s, a), (s, a)). Based on these observations, the stationary condition for the ratio from Equation 6 can be re-expressed in terms of T p γ,µ0 as where is the correction ratio function we seek to estimate. One natural approach to estimating τ * is to match the LHS and RHS of Equation 8 with respect to some divergence D (· ·) over the empirical samples. That is, we consider estimating τ * by solving the optimization problem min Although this forms the basis of our approach, there are two severe issues with this naive formulation that first need to be rectified: • τ, which in general involves an intractable integral. Thus, evaluation of the exact objective is intractable, and neglects the assumption that we only have access to samples from T p γ,µ0 and are not able to evaluate it at arbitrary points. We address each of these two issues in a principled manner. To avoid degenerate solutions when γ = 1, we ensure that the solution is a proper density ratio; that is, the property τ ∈ Ξ:= {τ (·) ≥ 0, E p [τ] = 1} must be true of any τ that is a ratio of some density to p. This provides an additional constraint that we add to the optimization formulation min With this additional constraint, it is obvious that the trivial solution τ (s, a) = 0 is eliminated as an infeasible point of Eqn, along with other degenerate solutions τ (s, a) = cτ * (s, a) with c = 1. Unfortunately, exactly solving an optimization with expectation constraints is very complicated in general , particularly given a nonlinear parameterization for τ. The penalty method provides a much simpler alternative, where a sequence of regularized problems are solved min with λ increasing. The drawback of the penalty method is that it generally requires λ → ∞ to ensure the strict feasibility, which is still impractical, especially in stochastic gradient descent. The infinite λ may induce unbounded variance in the gradient estimator, and thus, divergence in optimization. However, by exploiting the special structure of the solution sets to Equation 11, we can show that, remarkably, it is unnecessary to increase λ. Theorem 1 For γ ∈ and any λ > 0, the solution to Equation 11 is given by τ * (s, a) = u(s,a) p(s,a). The detailed proof for Theorem 1 is given in Appendix A.1. By Theorem 1, we can estimate the desired correction ratio function τ * by solving only one optimization with an arbitrary λ > 0. The optimization in Equation 11 involves the integrals T p γ,µ0 • τ and E p [τ] inside nonlinear loss functions, hence appears difficult to solve. Moreover, obtaining unbiased gradients with a naive approach requires double sampling . Instead, we bypass both difficulties by applying a dual embedding technique (; . In particular, we assume the divergence D is in the form of an f -divergence ds da where φ (·): R + → R is a convex, lower-semicontinuous function with φ = 0. Plugging this into J (τ) in Equation 11 we can easily check the convexity of the objective Theorem 2 For an f -divergence with valid φ defining D φ, the objective J (τ) is convex w.r.t. τ. The detailed proof is provided in Appendix A.2. Recall that a suitable convex function can be represented as φ (x) = max f x·f −φ * (f), where φ * is the Fenchel conjugate of φ (·). In particular, we have the representation 1 2 x 2 = max u ux − 1 2 u 2, which allows us to re-express the objective as Applying the interchangeability principle , one can replace the inner max in the first term over scalar f to maximize over a function f (·, ·): This yields the main optimization formulation, which avoids the aforementioned difficulties and is well-suited for practical optimization as discussed in Section 3.4. In addition to f -divergence, the proposed estimator Equation 11 is compatible with other divergences, such as the integral probability metrics (IPM) (Müller, 1997;), while retaining consistency. Based on the definition of the IPM, these divergences directly lead to min-max optimizations similar to Equation 13 with the identity function as φ * (·) and different feasible sets for the dual functions. Specifically, maximum mean discrepancy (MMD) requires f H k ≤ 1 where H k denotes the RKHS with kernel k; the Dudley metric requires f BL ≤ 1 where f BL:= f ∞ + ∇f 2; and Wasserstein distance requires ∇f 2 ≤ 1. These additional requirements on the dual function might incur some extra difficulty in practice. For example, with Wasserstein distance and the Dudley metric, we might need to include an extra gradient penalty , which requires additional computation to take the gradient through a gradient. Meanwhile, the consistency of the surrogate loss under regularization is not clear. For MMD, we can obtain a closed-form solution for the dual function, which saves the cost of the inner optimization , but with the tradeoff of requiring two independent samples in each outer optimization update. Moreover, MMD relies on the condition that the dual function lies in some RKHS, which introduces additional kernel parameters to be tuned and in practice may not be sufficiently flexible compared to neural networks. We have derived a consistent stationary distribution correction estimator in the form of a min-max saddle point optimization Equation 13. Here, we present a practical instantiation of GenDICE with a concrete objective and parametrization. We choose the χ 2 -divergence, which is an f -divergence with φ (x) = (x − 1) 2 and φ There two major reasons for adopting χ 2 -divergence: i) In the behavior-agnostic OPE problem, we mainly use the ratio correction function for estimating, which is an expectation. Recall that the error between the estimate and ground-truth can then be bounded by total variation, which is a lower bound of χ 2 -divergence. ii) For the alternative divergences, the conjugate of the KL-divergence involves exp (·), which may lead to instability in optimization; while the IPM variants introduce extra constraints on dual function, which may be difficult to be optimized. The conjugate function of χ 2 -divergence en-joys suitable numerical properties and provides squared regularization. We have provided an additional empirical ablation study that investigates the alternative divergences in Appendix E.3. To parameterize the correction ratio τ and dual function f we use neural networks, τ (s, a) = nn wτ (s, a) and f (s, a) = nn w f (s, a), where w τ and w f denotes the parameters of τ and f respectively. Since the optimization requires τ to be non-negative, we add an extra positive neuron, such as exp (·), log (1 + exp (·)) or (·) 2 at the final layer of nn wτ (s, a). We empirically compare the different positive neurons in Section 6.3. For these representations, and unbiased gradient estimator ∇ (wτ,u,w f) J (τ, u, f) can be obtained straightforwardly, as shown in Appendix B. This allows us to apply stochastic gradient descent to solve the saddle-point problem Equation 14 in a scalable manner, as illustrated in Algorithm 1. We provide a theoretical analysis for the proposed GenDICE algorithm, following a similar learning setting and assumptions to . The main is summarized in the following theorem. A formal statement, together with the proof, is given in Appendix C. Theorem 3 (Informal) Under mild conditions, with learnable F and H, the error in the objective between the GenDICE estimate,τ, to the solution τ where E [·] is w.r.t. the randomness in D and in the optimization algorithms, opt is the optimization error, and approx (F, H) is the approximation induced by (F, H) for parametrization of (τ, f). The theorem shows that the suboptimality of GenDICE's solution, measured in terms of the objective function value, can be decomposed into three terms: the approximation error approx, which is controlled by the representation flexibility of function classes; the estimation error due to sample randomness, which decays at the order of 1/ √ N; and the optimization error, which arises from the suboptimality of the solution found by the optimization algorithm. As discussed in Appendix C, in special cases, this suboptimality can be bounded below by a divergence betweenτ and τ *, and therefore directly bounds the error in the estimated policy value. There is also a tradeoff between these three error terms. With more flexible function classes (e.g., neural networks) for F and H, the approximation error approx becomes smaller. However, it may increase the estimation error (through the constant in front of 1/ √ N) and the optimization error (by solving a harder optimization problem). On the other hand, if F and H are linearly parameterized, estimation and optimization errors tend to be smaller and can often be upper-bounded explicitly in Appendix C.3. However, the corresponding approximation error will be larger. Off-policy evaluation with importance sampling (IS) has has been explored in the contextual bandits (; Dudík et al., 2011;), and episodic RL settings , achieving many empirical successes (e.g., ; Dudík et al., 2011;). Unfortunately, IS-based methods suffer from exponential variance in long-horizon problems, known as the "curse of horizon" . A few variancereduction techniques have been introduced, but still cannot eliminate this fundamental issue (; ;). By rewriting the accumulated reward as an expectation w.r.t. a stationary distribution,; recast OPE as estimating a correction ratio function, which significantly alleviates variance. However, these methods still require the off-policy data to be collected by a single and known behavior policy, which restricts their practical applicability. The only published algorithm in the literature, to the best of our knowledge, that solves agnostic-behavior off-policy evaluation is DualDICE . However, DualDICE was developed for discounted problems and its become unstable when the discount factor approaches 1 (see below). By contrast, GenDICE can cope with the more challenging problem of undiscounted reward estimation in the general behavior-agnostic setting. Note that standard model-based methods , which estimate the transition and reward models directly then calculate the expected reward based on the learned model, are also applicable to the behavior-agnostic setting considered here. Unfortunately, model-based methods typically rely heavily on modeling assumptions about rewards and transition dynamics. In practice, these assumptions do not always hold, and the evaluation can become unreliable. For more related work on MCMC, density ratio estimation and PageRank, please refer to Appendix F. In this section, we evaluate GenDICE on OPE and OPR problems. For OPE, we use one or multiple behavior policies to collect a fixed number of trajectories at some fixed trajectory length. This data is used to recover a correction ratio function for a target policy π that is then used to estimate the average reward in two different settings: i) average reward; and ii) discounted reward. In both settings, we compare with a model-based approach and step-wise weighted IS . We also compare to (referred to as "IPS" here) in the Taxi domain with a learned behavior policy 1. We specifically compare to DualDICE in the discounted reward setting, which is a direct and current state-of-the-art baseline. For OPR, the main comparison is with the model-based method, where the transition operator is empirically estimated and stationary distribution recovered via an exact solver. We validate GenDICE in both tabular and continuous cases, and perform an ablation study to further demonstrate its effectiveness. All are based on 20 random seeds, with mean and standard deviation plotted. Our code will be publicly available for reproduction. Offline PageRank on Graphs One direct application of GenDICE is off-line PageRank (OPR). We test GenDICE on a Barabasi-Albert (BA) graph (synthetic), and two realworld graphs, Cora and Citeseer. Details of the graphs are given in Appendix D. We use the log KL-divergence between estimated stationary distribution and the ground truth as the evaluation metric, with the ground truth computed by an exact solver based on the exact transition operator of the graphs. We compared GenDICE with modelbased methods in terms of the sample efficiency. From the in Figure 1, GenDICE outperforms the modelbased method when limited data is given. Even with 20k samples for a BA graph with 100 nodes, where a transition matrix has 10k entries, GenDICE still shows better performance in the offline setting. This is reasonable since Gen-DICE directly estimates the stationary distribution vector or ratio, while the model-based method needs to learn an entire transition matrix that has many more parameters. We use a similar taxi domain as in , where a grid size of 5 × 5 yields 2000 states in total (25 × 16 × 5, corresponding to 25 taxi locations, 16 passenger appearance status and 5 taxi status). We set the target policy to a final policy π after running tabular Q-learning for 1000 iterations, and set another policy π + after 950 iterations as the base policy. The behavior policy is a mixture controlled by α as π b = (1 − α)π + απ +. For the model-based method, we use a tabular representation for the reward and transition functions, whose entries are estimated from behavior data. For IS and IPS, we fit a policy via behavior cloning to estimate the policy ratio. In this specific setting, our methods achieve better compared to IS, IPS and the model-based method. Interestingly, with longer horizons, IS cannot improve as much as other methods even with more data, while GenDICE consistently improve and achieves much better than the baselines. DualDICE only works with γ < 1. GenDICE is more stable than DualDICE when γ becomes larger (close to 1), while still showing competitive performance for smaller discount factors γ. We further test our method for OPE on three control tasks: a discrete-control task Cartpole and two continuous-control tasks Reacher and HalfCheetah. In these tasks, observations (or states) are continuous, thus we use neural network function approximators and stochastic optimization. Since DualDICE has shown the state-of-the-art performance on discounted OPE, we mainly compare with it in the discounted reward case. We also compare to IS with a learned policy via behavior cloning and a neural model-based method, similar to the tabular case, but with neural network as the function approximator. All neural networks are feed-forward with two hidden layers of dimension 64 and tanh activations. More details can be found in Appendix D. Due to limited space, we put the discrete control in Appendix E and focus on the more challenging continuous control tasks. Here, the good performance of IS and model-based methods in Section 6.1 quickly deteriorates as the environment becomes complex, i.e., with a continuous action space. Note that GenDICE is able to maintain good performance in this scenario, even when using function approximation and stochastic optimization. This is reasonable because of the difficulty of fitting to the coupled policy-environment dynamics with a continuous action space. Here we also empirically validate GenDICE with off-policy data collected by multiple policies. As illustrated in Figure 3, all methods perform better with longer trajectory length or more trajectories. When α becomes larger, i.e., the behavior policies are closer to the target policy, all methods performs better, as expected. Here, GenDICE demonstrates good performance both on averagereward and discounted reward cases in different settings. The right two figures in each row show the log MSE curve versus optimization steps, where GenDICE achieves the smallest loss. In the discounted reward case, GenDICE shows significantly better and more stable performance than the strong baseline, DualDICE. Figure 4 also shows better performance of GenDICE than all baselines in the more challenging HalfCheetah domain. Each plot in the second row shows the average reward case. Finally, we conduct an ablation study on GenDICE to study its robustness and implementation sensitivities. We investigate the effects of learning rate, activation function, discount factor, and the specifically designed ratio constraint. We further demonstrate the effect of the choice of divergences and the penalty weight in Appendix E.3. Model-Based Importance Sampling DualDICE GenDICE (ours) Figure 4: Results on HalfCheetah. Plots from left to the right show the log MSE of estimated average per-step reward over different truncated lengths, numbers of trajectories, and behavior policies in discounted and average reward cases. Effects of the Learning Rate Since we are using neural network as the function approximator, and stochastic optimization, it is necessary to show sensitivity to the learning rate with {0.0001, 0.0003, 0.001, 0.003}, with in Figure 5. When α = 0.33, i.e., the OPE tasks are relatively easier and GenDICE obtains better at all learning rate settings. However, when α = 0.0, i.e., the estimation becomes more difficult and only GenDICE only obtains reasonable with the larger learning rate. Generally, this ablation study shows that the proposed method is not sensitive to the learning rate, and is easy to train. We further investigate the effects of the activation function on the last layer, which ensure the non-negative outputs required for the ratio. To better understand which activation function will lead to stable trainig for the neural correction estimator, we empirically compare using i) (·) 2; ii) log(1 + exp(·)); and iii) exp(·). In practice, we use the (·) 2 since it achieves low variance and better performance in most cases, as shown in Figure 5. We vary γ ∈ {0.95, 0.99, 0.995, 0.999, 1.0} to probe the sensitivity of GenDICE. Specifically, we compare to DualDICE, and find that GenDICE is stable, while DualDICE becomes unstable when the γ becomes large, as shown in Figure 6. GenDICE is also more general than DualDICE, as it can be applied to both the average and discounted reward cases. In Section 3, we highlighted the importance of the ratio constraint. Here we investigate the trivial solution issue without the constraint. The in Figure 6 demonstrate the necessity of adding the constraint penalty, since a trivial solution prevents an accurate corrector from being recovered (green line in left two figures). In this paper, we proposed a novel algorithm GenDICE for general stationary distribution correction estimation, which can handle both the discounted and average stationary distribution given multiple behavior-agnostic samples. Empirical on off-policy evaluation and offline PageRank show the superiority of proposed method over the existing state-of-the-art methods. the existence of the stationary distribution. Our discussion is all based on this assumption. Assumption 1 Under the target policy, the ed state-action transition operator T has a unique stationary distribution in terms of the divergence D (·||·). If the total variation divergence is selected, the Assumption 1 requires the transition operator should be ergodic, as discussed in. Theorem 1 For arbitrary λ > 0, the solution to the optimization Eqn is Proof For γ ∈, there is not degenerate solutions to D T p γ,µ0 • τ ||p · τ. The optimal solution is a density ratio. Therefore, the extra penalty E p(x) [τ (x)] − 1 2 does not affect the optimality for ∀λ > 0. negative, and the the density ratio p(x) leads to zero for both terms. Then, the density ratio is a solution to J (τ). For any other non-negative function τ (x) ≥ 0, if it is the optimal solution to J (τ), then, we have We denote µ (x) = p (x) τ (x), which is clearly a density function. Then, the optimal conditions in Equation 15 imply or equivalently, µ is the stationary distribution of T. We have thus shown the optimal τ (x) = is the target density ratio. Proof Since the φ is convex, we consider the Fenchel dual representation of the f -divergence 2 is also convex, which concludes the proof. Inputs: Convex function φ and its Fenchel conjugate φ, initial state s 0 ∼ µ 0, target policy π, distribution corrector nn wτ (·, ·), nn w f (·, ·), constraint scalar u, learning rates η τ, η f, η u, number of iterations K, batch size B. Sample actions a end for Return nn wτ. We provide the unbiased gradient estimator for ∇ wτ,u,w f J (τ, u, f) in Eqn below: Then, we have the psuedo code which applies SGD for solving Eqn. For convenience, we repeat here the notation defined in the main text. The saddle-point reformulation of the objective function of GenDICE is: To avoid the numerical infinity in D φ (·||·), we induced the bounded version as is still a valid divergence, and therefore the optimal solution τ * is still the stationary density ratio p(x). We denote the J (τ, µ, f) as the empirical surrogate of with optimal (f *, u *), and We apply some optimization algorithm for J (τ, u, f) over space (H, F, R), leading to the output τ, u, f. Under Assumption 2, we need only consider τ ∞ ≤ C, then, the corresponding dual u = E p (τ) − 1 ⇒ u ∈ U:= {|u| ≤ (C + 1)}. We choose the φ * (·) is a κ-Lipschitz continuous, then, the We consider the error betweenτ and τ * using standard arguments Remark: In some special cases, the suboptimality also implies the distance betweenτ and τ *. Specifically,for γ = 1, if the transition operator P π can be represented as P π = QΛQ −1 where Q denotes the (countable) eigenfunctions and Λ denotes the diagonal matrix with eigenvalues, the largest of which is 1. We consider φ (·) as identity and f ∈ F:= span (Q), f p,2 ≤ 1, then the d (τ, τ *) will bounded from below by a metric between τ and τ *. Particularly, we have Rewrite τ = ατ * + ζ, where ζ ∈ span Q \τ *, then Recall the optimality of τ We start with the following error decomposition: • For 1, we have. We consider the terms one-by-one. By definition, we have which is induced by introducing F for dual approximation. For the third term where we define est: Therefore, we can now bound 1 as We consider the terms from right to left. For the term J (τ * H) − J (τ *), we have, which is induced by restricting the function space to H. The second term is nonpositive, due to the optimality of (u *, f *). The final inequality comes from the fact that where the second term is nonpositive, thanks to the optimality ofτ * H. Finally, for the term J (τ * H) − J (τ * H), using the same argument in Equation 21, we have Therefore, we can bound 2 by 2 ≤ C φ,C,λ approx (H) + C P π,κ,λ (F) + 2 est. In sum, we have d (τ, τ *) ≤ 4 est +ˆ opt + 2C P π,κ,λ approx (F) + C φ,C,λ approx (H). In the following sections, we will bound the est andˆ opt. In this section, we analyze the statistical error We mainly focus on the batch RL setting with, which has been studied by previous authors (e.g., ;). However, as discussed in the literature (; ; ;), using the blocking technique of , the statistical error provided here can be generalized to β-mixing samples in a single sample path. We omit this generalization for the sake of expositional simplicity. To bound the est, we follow similar arguments by; via the covering number. For completeness, the definition is given below. The Pollard's tail inequality bounds the maximum deviation via the covering number of a function class: are i.i.d. samples from some distribution. Then, for any given > 0, The covering number can then be bounded in terms of the function class's pseudo-dimension: Lemma 5 (, Corollary 3) For any set X, any points x 1:N ∈ X N, any class F of functions on X taking values in [0, M] with pseudo-dimension D F < ∞, and any > 0, The statistical error est can be bounded using these lemmas. Lemma 6 (Stochastic error) Under the Assumption 2, if φ * is κ-Lipschitz continuous and the psuedo-dimension of H and F are finite, with probability at least 1 − δ, we have Proof The proof works by verifying the conditions in Lemma 4 and computing the covering number. 2 u 2, we will apply Lemma 4 with Z = Ω × Ω, Z i = (x i, x i), and G = h H×F ×U. We check the boundedness of h ζ,u,f (x, x). Based on Assumption 2, we only consider the τ ∈ H and u ∈ U bounded by C and C + 1. We also rectify the f ∞ ≤ C. Then, we can bound the h ∞: where C φ = max t∈[−C,C] −φ * (t). Thus, by Lemma 4, we have Next, we check the covering number of G. Firstly, we bound the distance in G, Denote the pseudo-dimension of H and F as D H and D F, respectively, we have, we obtain the bound for the statistical error: In this section, we investigate the optimization error. Notice our estimator min τ ∈H max f ∈F,u∈U J (τ, u, f) is compatible with different parametrizations for (H, F) and different optimization algorithms, the optimization error will be different. For the general neural network for (τ, f), although there are several progress recently (; ;) about the convergence to a stationary point or local minimum, it remains a largely open problem to quantify the optimization error, which is out of the scope of this paper. Here, we mainly discuss the convergence rate with tabular, linear and kernel parametrization for (τ, f). Particularly, we consider the linear parametrization particularly, i.e., τ (x) = σ w τ ψ (x), f (x) = w f ψ (x), and σ (·): R → R + is convex. There are many choices of the σ (·), e.g., exp (·), log (1 + exp (·)) and (·) 2. Obviously, even with such nonlinear mapping, the J (τ, u, f) is still convex-concave w.r.t (w τ, w f, u) by the convex composition rule. We can bound theˆ opt by the primal-dual gap gap: With vanilla SGD, we have, where T is the optimization steps ., where the E [·] is taken w.r.t. randomness in SGD. We are now ready to state the main theorm in a precise way: Under Assumptions 2 and 1, the stationary distribution µ exists, i.e., *, and the psuedo-dimension of H and F are finite, the error between the GenDICE estimate to τ where E [·] is w.r.t. the randomness in sample D and in the optimization algorithms. opt is the optimization error, and approx (F, H) is the approximation induced by (F, H) for parametrization of (τ, f). Proof We have the total error as where approx:= 2C T,κ,λ approx (F) + C φ,C,λ approx (H). For opt, we can apply the for SGD in Appendix C.3. We can bound the E [est] by Lemma 6. Specifically, we have Plug all these bounds into Equation 25, we achieve the . For the Taxi domain, we follow the same protocol as used in. The behavior and target policies are also taken from (referred in their work as the behavior policy for α = 0). We use a similar taxi domain, where a grid size of 5×5 yields 2000 states in total (25×16×5, cor-responding to 25 taxi locations, 16 passenger appearance status and 5 taxi status). We set our target policy as the final policy π * after running Q-learning for 1000 iterations, and set another policy π + after 950 iterations as our base policy. The behavior policy is a mixture policy controlled by α as π = (1 − α)π * + απ +, i.e., the larger α is, the behavior policy is more close to the target policy. In this setting, we solve for the optimal stationary ratio τ exactly using matrix operations. perform a similar exact solve for |S| variables µ(s), for better comparison we also perform our exact solve with respect to |S| variables τ (s). Specifically, the final objective of importance sampling will require knowledge of the importance weights µ(a|s)/p(a|s). For offline PageRank, the graph statistics are illustrated in Table 1, and the degree statistics and graph visualization are shown in Figure 7. For the BarabasiAlbert (BA) Graph, it begins with an initial connected network of m 0 nodes in the network. Each new node is connected to m ≤ m 0 existing nodes with a probability that is proportional to the number of links that the existing nodes already have. Intuitively, heavily linked nodes ('hubs') tend to quickly accumulate even more links, while nodes with only a few links are unlikely to be chosen as the destination for a new link. The new nodes have a'preference' to attach themselves to the already heavily linked nodes. For two real-world graphs, it is built upon the real-world citation networks. In our experiments, the weights of the BA graph is randomly drawn from a standard Gaussian distribution with normalization to ensure the property of the transition matrix. The offline data is collected by a random walker on the graph, which consists the initial state and next state in a single trajectory. In experiments, we vary the number of off-policy samples to validate the effectiveness of GenDICE with limited offline samples provided. We use the Cartpole, Reacher and HalfCheetah tasks as given by OpenAI Gym. In importance sampling, we learn a neural network policy via behavior cloning, and use its probabilities for computing importance weights π * (a|s)/π(a|s). All neural networks are feed-forward with two hidden layers of dimension 64 and tanh activations. Discrete Control Tasks We modify the Cartpole task to be infinite horizon: We use the same dynamics as in the original task but change the reward to be −1 if the original task returns a termination (when the pole falls below some threshold) and 1 otherwise. We train a policy on this task with standard Deep Q-Learning until convergence. We then define the target policy π * as a weighted combination of this pre-trained policy (weight 0.7) and a uniformly random policy (weight 0.3). The behavior policy π for a specific 0 ≤ α ≤ 1 is taken to be a weighted combination of the pre-trained policy (weight 0.55 + 0.15α) and a uniformly random policy (weight 0.45 − 0.15α). We train each stationary distribution correction estimation method using the Adam optimizer with batches of size 2048 and learning rates chosen using a hyperparameter search from {0.0001, 0.0003, 0.001, 0.003} and choose the best one as 0.0003. For the Reacher task, we train a deterministic policy until convergence via DDPG . We define the target policy π as a Gaussian with mean given by the pre-trained policy and standard deviation given by 0.1. The behavior policy π b for a specific 0 ≤ α ≤ 1 is taken to be a Gaussian with mean given by the pre-trained policy and standard deviation given by 0.4 − 0.3α. We train each stationary distribution correction estimation method using the Adam optimizer with batches of size 2048 and learning rates chosen using a hyperparameter search from {0.0001, 0.0003, 0.001, 0.003} and the optimal learning rate found was 0.003). For the HalfCheetah task, we also train a deterministic policy until convergence via DDPG . We define the target policy π as a Gaussian with mean given by the pre-trained policy and standard deviation given by 0.1. The behavior policy π b for a specific 0 ≤ α ≤ 1 is taken to be a Gaussian with mean given by the pre-trained policy and standard deviation given by 0.2 − 0.1α. We train each stationary distribution correction estimation method using the Adam optimizer with batches of size 2048 and learning rates chosen using a hyperparameter search from {0.0001, 0.0003, 0.001, 0.003} and the optimal learning rate found was 0.003. E.1 OPE FOR DISCRETE CONTROL On the discrete control task, we modify the Cartpole task to be infinite horizon: the original dynamics is used but with a modified reward function: the agent will receive −1 if the environment returns a termination (i.e., the pole falls below some threshold) and 1 otherwise. As shown in Figure 3, our method shows competitive with IS and Model-Based in average reward case, but our proposed method finally outperforms these two methods in terms of log MSE loss. Specifically, it is relatively difficult to fit a policy with data collected by multiple policies, which renders the poor performance of IS. In this section, we show more on the continuous control tasks, i.e., HalfCheetah and Reacher. Figure 9 shows the log MSE towards training steps, and GenDICE outperforms other baselines with different behavior policies. Figure 10 better illustrates how our method beat other baselines, and can accurately estimate the reward of the target policy. Besides, Figure 11 shows GenDICE gives better reward estimation of the target policy. In these figures, the left three figures show the performance with off-policy dataset collected by single behavior policy from more difficult to easier tasks. The right two figures show the , where off-policy dataset collected by multiple behavior policies. Figure 12 shows the ablation study in terms of estimated rewards. The left two figures shows the effects of different learning rate. When α = 0.33, i.e., the OPE tasks are relatively easier, GenDICE gets relatively good in all learning rate settings. However, when α = 0.0, i.e., the estimation becomes more difficult, only GenDICE in larger learning rate gets reasonable estimation. Interestingly, we can see with larger learning rates, the performance becomes better, and when learning rate is 0.001 with α = 0.0, the variance is very high, showing some cases the estimation becomes more accurate. The right three figures show different activation functions with different behavior policy. The square and softplus function works well; while the exponential function shows poor performance under some settings. In practice, we use the square function since its low variance and better performance in most cases. Model-Based Importance Sampling DualDICE GenDICE (ours) Oracle Figure 11: Results on HalfCheetah. Each plot in the first row shows the estimated average step reward over training and different behavior policies (higher α corresponds to a behavior policy closer to the target policy. Although any valid divergence between p · τ and T p γ,µ0 • τ in our estimator is consistent, which will always lead to the stationary distribution correction ratio asymptotically, and any λ > 0 will guarantee the normalization constraint, i.e., E p [τ] = 1, as we discussed in main text, different Figure 12: Results of ablation study with different learning rates and activation functions. The plots show the estimated average step reward over training and different behavior policies. choices of the divergences and λ may incur difficulty in the numerical optimization procedure. In this section, we investigate the empirical effects of the choice of f -divergence and IPM, and the weight of constrant regularization λ. To avoid the effects of other factors in the estimator, e.g., function parametrization, we focus on the offline PageRank task on BA graph with 100 nodes and 10k offline samples. All the performances are evaluated with 20 random trials. We test the GenDICE with several other alternative divergences, e.g., Wasserstein-1 distance, Jensen-Shannon divergence, KL-divergence, Hellinger divergence, and MMD. To ensure the dual function to be 1-Lipchitz, we add the gradient penalty. We use a learned Gaussian kernel in MMD, similar to. As we can see in Figure 13 (a), with these different divergences, the proposed GenDICE estimator can always achieve significantly better performance compared with the model-based estimator, showing that the GenDICE estimator is compatible with many different divergences. Most of the divergences, with appropriate extra techniques to handle the difficulties in optimization and carefully tuning for extra parameters, can achieve similar performances, consistent with phenomena in the variants of GANs . However, KL-divergence is an outlier, performing noticeably worse, which might be caused by the ill-behaved exp (·) in its conjugate function. The χ 2 -divergence and JS-divergence are better, which achieve good performances with fewer parameters to be tuned. The effect of the penalty weight λ is illustrated the in Figure 13(b). We vary the λ ∈ [0.1, 5] with χ 2 -divergence in the GenDICE estimator. Within a large range of λ, the performances of the proposed GenDICE are quite consistent, which justifies Theorem 1. The penalty multiplies with λ. Therefore, with λ increases, the variance of the stochastic gradient estimator also increases, which explains the variance increasing in large λ in Figure 13(b). In practice, λ = 1 is a reasonable choice for general cases. Markov Chain Monte Carlo Classical MCMC aims at sampling from µ π by iteratively simulting from the transition operator. It requires continuous interaction with the transition operator and heavy computational cost to update many particles. Amor-tized SVGD and Adversarial MCMC alleviate this issue via combining with neural network, but they still interact with the transition operator directly, i.e., in an on-policy setting. The major difference of our GenDICE is the learning setting: we only access the off-policy dataset, and cannot sample from the transition operator. The proposed GenDICE leverages stationary density ratio estimation for approximating the stationary quantities, which distinct it from classical methods. Density ratio estimation is a fundamental tool in machine learning and much related work exists. Classical density ratio estimation includes moment matching (Gretton et al.), probabilistic classification , and ratio matching (; ;). These classical methods focus on estimating the ratio between two distributions with samples from both of them, while GenDICE estimates the density ratio to a stationary distribution of a transition operator, from which even one sample is difficult to obtain. developed a reverse-time RL framework for PageRank via solving a reverse Bellman equation, which is less sensitive to graph topology and shows faster adaptation with graph change. still considers the online manner, which is different with our OPR setting.

Represent this text to find a much much longer article about it with say hundreds-thousands of words

In this paper, we proposed a novel algorithm, GenDICE, for general stationary distribution correction estimation, which can handle both discounted and average off-policy evaluation on multiple behavior-agnostic samples.

It's important to consider optimization in function space, not just parameter space. We introduce a learning rule that reduces distance traveled in function space, just like SGD limits distance traveled in parameter space.


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Question: The power of neural networks lies in their ability to generalize to unseen data, yet the underlying reasons for this phenomenon remain elusive. Numerous rigorous attempts have been made to explain generalization, but available bounds are still quite loose, and analysis does not always lead to true understanding. The goal of this work is to make generalization more intuitive. Using visualization methods, we discuss the mystery of generalization, the geometry of loss landscapes, and how the curse (or, rather, the blessing) of dimensionality causes optimizers to settle into minima that generalize well. Neural networks are a powerful tool for solving classification problems. The power of these models is due in part to their expressiveness; they have many parameters that can be efficiently optimized to fit nearly any finite training set. However, the real power of neural network models comes from their ability to generalize; they often make accurate predictions on test data that were not seen during training, provided the test data is sampled from the same distribution as the training data. The generalization ability of neural networks is seemingly at odds with their expressiveness. Neural network training algorithms work by minimizing a loss function that measures model performance using only training data. Because of their flexibility, it is possible to find parameter configurations Figure 1: A minefield of bad minima: we train a neural net classifier and plot the iterates of SGD after each tenth epoch (red dots). We also plot locations of nearby "bad" minima with poor generalization (blue dots). We visualize these using t-SNE embedding. All blue dots achieve near perfect train accuracy, but with test accuracy below 53% (random chance is 50%). The final iterate of SGD (yellow star) also achieves perfect train accuracy, but with 98.5% test accuracy. Miraculously, SGD always finds its way through a landscape full of bad minima, and lands at a minimizer with excellent generalization. for neural networks that perfectly fit the training data and minimize the loss function while making mostly incorrect predictions on test data. Miraculously, commonly used optimizers reliably avoid such "bad" minima of the loss function, and succeed at finding "good" minima that generalize well. Our goal here is to develop an intuitive understanding of neural network generalization using visualizations and experiments rather than analysis. We begin with some experiments to understand why generalization is puzzling, and how over-parameterization impacts model behavior. Then, we explore how the "flatness" of minima correlates with generalization, and in particular try to understand why this correlation exists. We explore how the high dimensionality of parameter spaces biases optimizers towards landing in flat minima that generalize well. Finally, we present some counterfactual experiments to validate the intuition we develop. Code to reproduce experiments is available at https://github.com/genviz2019/genviz. Neural networks define a highly expressive model class. In fact, given enough parameters, a neural network can approximate virtually any function . But just because neural nets have the power to represent any function does not mean they have the power to learn any function from a finite amount of training data. Neural network classifiers are trained by minimizing a loss function that measures model performance using only training data. A standard classification loss has the form where p θ (x, y) is the probability that data sample x lies in class y according to a neural net with parameters θ, and D t is the training dataset of size |D t |. This loss is near zero when a model with parameters θ accurately classifies the training data. Over-parameterized neural networks (i.e., those with more parameters than training data) can represent arbitrary, even random, labeling functions on large datasets . As a , an optimizer can reliably fit an over-parameterized network to training data and achieve near zero loss (; We illustrate the difference between model fitting and generalization with an experiment. The CIFAR-10 training dataset contains 50,000 small images. We train two over-parameterized models on this dataset. The first is a neural network (ResNet-18) with 269,722 parameters (nearly 6× the number of training images). The second is a linear model with a feature set that includes pixel intensities as well as pair-wise products of pixels intensities. 1 This linear model has 298, 369 parameters, which is comparable to the neural network, and both are trained using SGD. On the left of Figure 2, we see that overparameterization causes both models to achieve perfect accuracy on training data. But the linear model achieves only 49% test accuracy, while ResNet-18 achieves 92%. The excellent performance of the neural network model raises the question of whether bad minima exist at all. Maybe deep networks generalize because bad minima are rare and lie far away from the region of parameter space where initialization takes place? We can confirm the existence of bad minima by incorporating a loss term that explicitly promotes poor generalization, by discouraging performance on unseen data drawn from the same distribution. We do this by minimizing where D t is the training set, and D d is a set of unseen examples sampled from the same distribution. D d could be obtained via a GAN or additional data collection (note that it is not the test set). Here, β parametrizes the amount of "anti-generalization" we wish to achieve. The first term in is the standard cross entropy loss on the training set D t, and is minimized when the training data are classified correctly. The second term is the reverse cross entropy loss on D d, and is minimized when D d is classified incorrectly. With a sufficiently over-parameterized network, gradient descent on drives both terms to zero, and we find a parameter vector that minimizes the original training set loss while failing to generalize. In other words, the minima found by are stationary points with comparable true objective function values (Eq.), indicating that it's quite possible to land in one of these "bad" minima in a normal training routine if initialized within the loss basin. Sec. 5 will show that the likelihood of this occurring is negligible. When we use the anti-generalization loss to search for bad minima near the optimization trajectory, we see that bad minima are everywhere. We visualize the distribution of bad minima in Figure 1. We run a standard SGD optimizer on the swissroll and trace out the path it takes from a random initialization to a minimizer. We plot the iterate after every tenth epoch as a red dot with opacity proportional to its epoch number. Starting from these iterates, we run the anti-generalization optimizer to find nearby bad minima. We project the iterates and bad minima into a 2D plane for visualization using a t-SNE embedding 2. Our anti-generalization optimizer easily finds minima with poor generalization within close proximity to every SGD iterate. Yet SGD avoids these bad minima, carving out a path towards a parameter configuration that generalizes well. Figure 1 illustrates that neural network optimizers are inherently biased towards good minima, a behavior commonly known as "implicit regularization." To see how the choice of optimizer affects generalization, we trained a simple neural network (VGG13) on 11 different gradient methods and 2 non-gradient methods in Figure 2 (right). This includes LBFGS (a second-order method), and ProxProp (which chooses search directions by solving least-squares problems rather than using the gradient). Interestingly, all of these methods generalize far better than the linear model. While there are undeniably differences between the performance of different optimizers, the presence of implicit regularization for virtually any optimizer strongly indicates that implicit regularization may be caused in part by the geometry of the loss function, rather than the choice of optimizer alone. Later on, we visually explore the relationship between the loss function's geometry and generalization, and how the high dimensionality of parameter space is one source of implicit regularization for optimizers. Classical PAC learning theory balances model complexity (the expressiveness of a model class) against data volume. When a model class is too expressive relative to the volume of training data, it has the ability to ace the training data while flunking the test data, and learning fails. Classical theory fails to explain generalization in over-parameterized neural nets, as the complexity of networks is often large (exponential in depth (; ;) or linear in the number of parameters (; ;) ). Therefore classical bounds become too loose or even vacuous in the over-parameterized setting that we are interested in studying. To explain this mismatch between empirical observation and classical theory, a number of recent works propose new metrics that characterize the capacity of neural networks. Most of these appeal to the PAC framework to characterize the generalization ability of a model class Θ (e.g., neural nets of a shared architecture) through a high probability upper bound: with probability at least 1 − δ, where R(θ) is generalization risk (true error) of a net with parameters θ ∈ Θ,R S (θ) denotes empirical risk (training error) with training sample S. We explain B under different metrics below. Model space complexity. This line of work takes B to be proportional to the complexity of the model class being trained, and efforts have been put into finding tight characterizations of this complexity.; built on prior works to produce bounds where model class complexity depends on the spectral norm of the weight matrices without having an exponential dependence on the depth of the network. Such bounds can improve the model class complexity provided that weight matrices adhere to some structural constraints (e.g. sparsity or eigenvalue concentration). Stability and robustness. This line of work considers B to be proportional to the stability of the model; ), which is a measure of how much changing a data point in S changes the output of the model . However it is nontrivial to characterize the stability of a neural network. Robustness, while producing insightful and effective generalization bounds, still suffers from the curse of the dimensionality on the priori-known fixed input manifold. PAC-Bayes and margin theory. PAC-Bayes bounds (; 1999; ; ; ;), provide generalization guarantees for randomized predictors drawn from a learned distribution that depends on the training data, as opposed to a learned single predictor. These bounds often yield sample complexity bounds worse than naive parameter counting, however show that PAC-Bayes theory does provide meaningful generalization bounds for "flat" minima. Model compression. Most recent theoretical work can be understood through the lens of "model compression" . Clearly, it is impossible to generalize when the model class is too big; in this case, many different parameter choices explain the data perfectly while having wildly different predictions on test data. The idea of model compression is that neural network model classes are effectively much smaller than they seem to be because optimizers are only willing to settle into a very selective set of minima. When we restrict ourselves to only the narrow set of models that are acceptable to an optimizer, we end up with a smaller model class on which learning is possible. While our focus is on gaining insights through visualizations, the intuitive arguments below can certainly be linked back to theory. The class of models representable by a network architecture has extremely high complexity, but experiments suggest that most of these models are effectively removed from consideration by the optimizer, which has an extremely strong bias towards "flat" minima, ing in a reduced effective model complexity. Over-parameterization is not specific to neural networks. A traditional approach to coping with over-parameterization for linear models is to use regularization (aka "priors") to bias the optimizer towards good minima. For linear classification, a common regularizer is the wide margin penalty (which appears in the form of an 2 regularizer on the parameters of a support vector machine). When used with linear classifiers, wide margin priors choose the linear classifier that maximizes Euclidean distance to the class boundaries while still classifying data correctly. Neural networks replace the classical wide margin regularization with an implicit regulation that promotes the closely related notion of "flatness." In this section, we explain the relationship between flat minima and wide margin classifiers, and provide intuition for why flatness is a good prior. Many have observed links between flatness and generalization. first proposed that flat minima tend to generalize well. This idea was reinvigorated by , who showed that large batch sizes yield sharper minima, and that sharp minima generalize poorly. This correlation was subsequently observed for a range of optimizers by Izmailov et al. Flatness is a measure of how sensitive network performance is to perturbations in parameters. Consider a parameter vector that minimizes the loss (i.e., it correctly classifies most if not all training data). If small perturbations to this parameter vector cause a lot of data misclassification, the minimizer is sharp; a small movement away from the optimal parameters causes a large increase in the loss function. In contrast, flat minima have training accuracy that remains nearly constant under small parameter perturbations. The stability of flat minima to parameter perturbations can be seen as a wide margin condition. When we add random perturbations to network parameters, it causes the class boundaries to wiggle around in space. If the minimizer is flat, then training data lies a safe distance from the class boundary, and perturbing the class boundaries does not change the classification of nearby data points. In contrast, sharp minima have class boundaries that pass close to training data, putting those nearby points at risk of misclassification when the boundaries are perturbed. We visualize the impact of sharpness on neural networks in Figure 3. We train a 6-layer fully connected neural network on the swiss roll dataset using regular SGD, and also using the anti-generalization loss to find a minimizer that does not generalize. The "good" minimizer has a wide margin -the class boundary lies far away from the training data. The "bad" minimizer has almost zero margin, and each data point lies near the edge of class boundaries, on small class label "islands" surrounded by a different class label, or at the tips of "peninsulas" that reach from one class into the other. The class labels of most training points are unstable under perturbations to network parameters, and so we expect this minimizer to be sharp. An animation of the decision boundary under perturbation is provided at https://www.youtube.com/watch?v=4VUJyQknf4s&t=. We can visualize the sharpness of the minima in Figure 3, but we need to take some care with our metrics of sharpness. It is known that trivial definitions of sharpness can be manipulated simply by rescaling network parameters . When parameters are small (say, 0.1), a perturbation of size 1 might cause a major performance degradation. Conversely, when parameters are large (say, 100), a perturbation of size 1 might have little impact on performance. However, rescalings of network parameters are irrelevant; commonly used batch normalization layers remove the effect of parameter scaling. For this reason, it is important to define measures of sharpness that are invariant to trivial rescalings of network parameters. One such measure is local entropy , which is invariant to rescalings, but is difficult to compute. For our purposes, we use the filter-normalization scheme proposed in , which simply rescales network filters to have unit norm before plotting. The ing sharpness/flatness measures have been observed to correlate well with generalization. The bottom of Figure 3 visualizes loss function geometry around the two minima for the swiss roll. These surface plots show the loss evaluated on a random 2D plane 3 sliced out of parameter space using the method described in. We see that the instability of class labels under parameter perturbations does indeed lead to dramatically sharper minima for the bad minimizer, while the wide margin of the good minimizer produces a wide basin. To validate our observations on a more complex problem, we produce similar sharpness plots for the Street View House Number (SVHN) classification problem in Figure 4 using ResNet-18. The SVHN dataset is ideal for this experiment because, in addition to train and test data, the creators collected a large (531k) set of extra data from the same distribution that can be used for D d in Eq.. We minimize the SVHN loss function using standard training with and without penalizing for generalization (Eq.). The good, well-generalizing minimizer is flat and achieves 97.1% test accuracy, while the bad minimizer is much sharper and achieves 28.2% test accuracy. Both achieve 100% train accuracy and use identical hyperparameters (other than the β factor), network architecture, and weight initialization. We have seen that neural network loss functions are densely populated with both good and bad minima, and that good minima tend to have "flat" loss function geometry. But what causes optimizers to find these good/flat minima and avoid the bad ones? One possible explanation to the bias of stochastic optimizers towards good minima is the volume disparity between the basins around good and bad minima. Flat minima that generalize well lie in wide basins that occupy a large volume of parameter space, while sharp minima lie in narrow basins that occupy a comparatively small volume of parameter space. As a , an optimizer using random initialization is more likely to land in the attraction basin for a good minimizer than a bad one. The volume disparity between good and bad minima is magnified by the curse (or, rather, the blessing?) of dimensionality. The differences in "width" between good and bad basins does not appear too dramatic in the visualizations in Figures 3 and 4, or in sharpness visualizations for other datasets . However, the probability of colliding with a region during a random initialization does not scale with its width, but rather its volume. Network parameters live in very highdimensional spaces where small differences in sharpness between minima translate to exponentially large disparities in the volume of their surrounding basins. It should be noted that the vanishing probability of finding sets of small width in high dimensions is well studied by probabilists, and is formalized by a variety of escape theorems . To explore the effect of dimensionality on neural loss landscapes, we quantify the local volume within the low-lying basins surrounding different minima. The volume (or "horizon") of a basin is not well-defined, especially for SGD with discrete time-steps. For this experiment, we define the "basin" to be the set of points in a neighborhood of the minimizer that have loss value below a cutoff of 0.1 (Fig. 7). We chose this definition because the volume of this set can be efficiently computed. We calculate the volume of these basins using a Monte-Carlo integration method. Let r(φ) denote the radius of the basin (distance from minimizer to basin boundary) in the direction of the unit vector φ. Then the n-dimensional volume of the basin is Γ(1+n/2) is the volume of the unit n-ball, and Γ is Euler's gamma function. We estimate this expectation by calculating r(φ) for 3k random directions, as illustrated in Figure 7. In Figure 5, we visualize the combined relationship between generalization and volume for swissroll and SVHN. By varying β, we control the generalizability of each minimizer. As generalization accuracy decreases, we see the radii of the basins decrease as well, indicating that minima become sharper. Figure 5 also contains scatter plots showing a severe correlation between generalization and (log) volume for various choices of the basin cutoff value. For SVHN, the basins surrounding good minima have a volume at least 10,000 orders of magnitude larger than that of bad minima, rendering it nearly impossible to accidentally stumble upon bad minima. Finally, we visualize the decision boundaries for several levels of generalization in Figure 6. All networks achieve above 99.5% training accuracy. As the generalization gap increases, the area that belongs to the red class begins encroaching into the area that belongs to the blue class, and vice versa. The margin between the decision boundary and training points also decreases until the training points, though correctly classified, sit on "islands" or "peninsulas" as discussed above. Neural nets solve complex classification problems by finding "flat" minima with class boundaries that assign labels that are stable to parameter perturbations. Using this intuition, can we formulate a problem that neural nets can't solve? Consider the problem of separating the blue and red dots in Figure 8. When the distance between the inner rings is large, a neural network consistently finds a well-behaved circular boundary as in Fig. 8aa. The wide margin of this classifier makes the minimizer "flat," and the ing high volume makes it likely to be found by SGD. We can remove the well-behaved minima from this problem by pinching the margin between the inner red and blue rings. In this case, a network trained with random initialization is shown in Fig. 8b. Now, SGD finds networks that cherry-pick red points, and arc away from the more numerous blue points to maintain a large margin. In contrast, a simple circular decision boundary as in Fig. 8a would pass extremely close to all points on the inner rings, making such a small margin solution less stable under perturbations and unlikely to be found by SGD. We explored the connection between generalization and loss function geometry using visualizations and experiments on classification margin and loss basin volumes, the latter of which does not appear in the literature. While experiments can provide useful insights, they sometimes raise more questions than they answer. We explored why the "large margin" properties of flat minima promote generalization. But what is the precise metric for "margin" that neural networks respect? Experiments suggest that the small volume of bad minima prevents optimizers from landing in them. But what is a correct definition of "volume" in a space that is invariant to parameter re-scaling and other transforms, and how do we correctly identify the attraction basins for good minima? Finally and most importantly: how do we connect these observations back to a rigorous PAC learning framework? The goal of this study is to foster appreciation for the complex behaviors of neural networks, and to provide some intuitions for why neural networks generalize. We hope that the experiments contained here will provide inspiration for theoretical progress that leads us to rigorous and definitive answers to the deep questions raised by generalization.

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An intuitive empirical and visual exploration of the generalization properties of deep neural networks.

We propose a hypothesis for why gradient descent generalizes based on how per-example gradients interact with each other.