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+# Constrained Efficient Global Optimization of Expensive Black-box Functions
+
+Wenjie Xu  $^{1,2}$  Yuning Jiang  $^{1}$  Bratislav Svetozarevic  $^{2}$  Colin N. Jones
+
+# Abstract
+
+We study the problem of constrained efficient global optimization, where both the objective and constraints are expensive black-box functions that can be learned with Gaussian processes. We propose CONFIG (CONstrained efFicient Global Optimization), a simple and effective algorithm to solve it. Under certain regularity assumptions, we show that our algorithm enjoys the same cumulative regret bound as that in the unconstrained case and similar cumulative constraint violation upper bounds. For commonly used Mátern and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem. In addition, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible. Numerical experiments on sampled instances from the Gaussian process, artificial numerical problems, and a black-box building controller tuning problem all demonstrate the competitive performance of our algorithm. Compared to the other state-of-the-art methods, our algorithm significantly improves the theoretical guarantees while achieving competitive empirical performance.
+
+# 1. Introduction
+
+Global optimization of expensive black-box functions is a pervasive problem in science and engineering. In black-box optimization problems, we typically need to evaluate different candidate solutions sequentially to find the optimal one without explicit gradient information. For example, hyperparameter tuning for machine learning models (Bergstra & Bengio, 2012; Snoek et al., 2015), control system performance optimization (Bansal et al., 2017; Xu et al., 2022b) and drug discovery or materials design (Negoescu et al.,
+
+$^{1}$ Automatic Control Laboratory, EPFL, Lausanne, Switzerland  $^{2}$ Urban Energy Systems Laboratory, Empa, Zurich, Switzerland. Correspondence to: Yuning Jiang <yuning.jiang@ieee.org>.
+
+Proceedings of the  $40^{th}$  International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
+
+2011; Frazier & Wang, 2016) can all be formulated as black-box optimization problems.
+
+In these applications, we typically know little about the shape of the black-box functions, which can be non-convex and multi-modal. As a result, a global optimization method is needed. In many applications, e.g., tuning the hyperparameters of a deep neural network, each function evaluation can take a large number of resources such as energy, time, or GPU computation, so we want to use as few function evaluations as possible. Standard global optimization methods, such as genetic algorithms or simulated annealing (Pardalos & Romeijn, 2013), typically require a large number of samples and thus can not be applied to the optimization of expensive black-box functions. Therefore, Gaussian process based efficient global optimization $^{1}$  (Jones et al., 1998; Shahrriari et al., 2015), as a sample-efficient method to solve black-box optimization problems (Xu et al., 2022a), has recently been becoming more and more popular. The general idea of efficient global optimization is to build a surrogate model, typically by Gaussian process regression, of the black box function from samples and then to choose a search direction using this model. This problem, when formulated as a (kernelized) multi-armed bandit problem, is widely studied in the literatures (Audibert et al., 2010; Srinivas et al., 2012; Soare et al., 2014; Chowdhury & Gopalan, 2017a; Amani et al., 2019; Zhou & Ji, 2022).
+
+A key challenge in many applications is black-box constraints. For example, when tuning the temperature controller parameters of a building, we need to minimize the energy consumption while keeping the occupants' comfort at user-defined levels, where both the energy and the comfort are black-box functions of the controller parameters. To handle the unknown black-box constraints, a variety of efficient global optimization methods with constraints have been developed. Intuitively, there is a non-trivial tradeoff over optimization, learning and violations. We can roughly classify the existing methods into different groups based on whether constraint violations are allowed during the optimization process. For the setting where no constraint violations are tolerated, a group of safe optimization meth
+
+ods has been developed (Sui et al., 2015; Wu et al., 2016; Sui et al., 2018; Turchetta et al., 2020; Moradipari et al., 2020; Amani et al., 2020). However, these safe algorithms require a set of feasible solutions known a priori, while in practice, we may not get access to such feasible solutions or even do not know the feasibility of the problem. Furthermore, due to these hard safety-critical constraints, these algorithms may become stuck at a local minimum. The other extreme is where constraint violations are allowed to occur during the optimization process without penalty, for which there exists generic constrained Bayesian (efficient global) optimization methods (Schonlau, 1997; Sasena et al., 2002; Basudhar et al., 2012; Bagheri et al., 2017; Zhang et al., 2021). Along this line of research, a group of popular methods (Gardner et al., 2014; Gelbart et al., 2014) encodes the constraint information into the acquisition function (e.g., Constrained Expected Improvement). However, there are no theoretical guarantees on optimality, constraint violation incurred during optimization, or convergence rates for this class of methods. A recent work of (Inatsu et al., 2022) considers Bayesian optimization for distributionally robust chance-constrained problems but can not provide cumulative regret/violation bounds.
+
+More recently, a third group of approaches has been studied (e.g., (Xu et al., 2022b; Zhou & Ji, 2022)), where constraint violations are allowed, but limits are placed on the total violation incurred during the optimization process. In this setting, two more recent works adopt a penalty-function approach (Lu & Paulson, 2022) and a primal-dual approach (Zhou & Ji, 2022) to solve constrained efficient global optimization problems. The common ideas behind these two approaches are the addition of a penalizing term of the constraint violation in the objective and the transformation of the constrained problem to an unconstrained one. However, both of these methods require choosing values for some critical parameters (e.g., penalty coefficient (Lu & Paulson, 2022) and dual update step size (Zhou & Ji, 2022)). The performance of these methods relies on the chosen parameters heavily, and hence these methods can take significant parameter tuning effort for implementation. Additionally, the cumulative violations considered in (Zhou & Ji, 2022) are the violations of the cumulative constraint values, not the real total violations. Such a form of cumulative violation bounds is weak and can not rule out the case where points with severe violations and small constraint values are alternatingly sampled while keeping a low cumulative constraint value. The work of (Lu & Paulson, 2022) only provides a penalty-based regret bound, which is defined as the regret plus a penalty weight times violations, which does not directly lead to separate bounds on regret and violation since the penalty weight is unknown beforehand. We also notice that a concurrent work (Guo et al., 2022) extends the fixed penalty to an increasing penalty sequence. However, it
+
+requires a delicate design of the penalty sequence and may result in numerical issues when the penalty is large. Last but not least, all the existing methods (Sui et al., 2015; Gelbart et al., 2014; Zhou & Ji, 2022; Lu & Paulson, 2022), either require an initial feasible solution to be given or lack the scheme to detect and report infeasibility when the original problem is infeasible. This is key, as, in practice, black-box optimization problems may be infeasible, and a scheme to detect and report infeasibility is desired.
+
+Our contributions. We propose CONFIG, a simple and effective algorithm for the constrained efficient global optimization of expensive black-box functions by exploiting the principle of optimism in the face of uncertainty. Specifically,
+
+1. CONFIG dynamically maintains an optimistic estimate of the feasible set as demonstrated in Fig. 1 and solves an auxiliary problem with lower confidence bound (LCB) surrogates as both objective and constraints to get the next sample in each step. Intuitively, CONFIG trades potential violations for faster learning of both the objective function and the constraints.  
+2. Under certain regularity assumptions, we show that the algorithm enjoys the same cumulative regret bound as that in the unconstrained case. In addition, we show that the cumulative constraint violation has an upper bound in terms of maximum information gain, which is similar to the cumulative regret bounds. For the commonly used Matern and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem.  
+3. Furthermore, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible.  
+4. Numerical results show that our algorithm achieves performance competitive with the state-of-the-art algorithms in terms of both optimality and constraint violations.
+
+In sharp contrast, existing algorithms for constrained Bayesian optimization, including the popular CEI method (Gardner et al., 2014) and the recently introduced primal-dual method (Zhou & Ji, 2022; Xu et al., 2023), can only provide no or weak guarantees on the cumulative regret/violation bounds or on the convergence to the optimal solution (See Tab. 1 for details). Additionally, none of the existing methods can detect infeasibility if the original problem is infeasible. As such, our algorithm significantly improves the theoretical guarantees while achieving competitive empirical performance compared to the existing state-of-the-art constrained global/Bayesian optimization methods.
+
+![](images/5b7438597248e50b07fb9e3cef7a4cdcae49b8f31793dcf9391e55ab24f69fa1.jpg)  
+Figure 1. A simple demonstration of different ways to construct the estimate of the feasible set, where we minimize an unknown function subject to that it is non-positive. The pessimistic estimate can guarantee feasibility with high probability. However, it is tiny and can easily be empty if the initial points are infeasible. With the certainty-equivalent estimate, the algorithm may easily get stuck due to a lack of exploration. In contrast, our optimistic estimate of the feasible set contains the ground-truth feasible set with high probability.
+
+Table 1. The comparison of our method to existing state-of-the-art constrained efficient global optimization (or kernelized multi-armed bandits with constraints) methods.  
+
+<table><tr><td>Works</td><td>Cumulative Regret Bound</td><td>Cumulative Violation Bound</td><td>Optimality Guarantee</td><td>Infeasibility Declaration</td></tr><tr><td>SafeOPT (Sui et al., 2015), etc.</td><td>X</td><td>No violation</td><td>Local convergence</td><td>X</td></tr><tr><td>Constrained EI (Gardner et al., 2014), etc.</td><td>X</td><td>X</td><td>X</td><td>X</td></tr><tr><td>Primal-Dual (Zhou &amp; Ji, 2022), etc.</td><td>✓</td><td>weak-form</td><td>X</td><td>X</td></tr><tr><td>Penalty function method (Lu &amp; Paulson, 2022), etc.</td><td colspan="2">Penalty-based Regret Bound</td><td>Global convergence with penalty-dependent rate</td><td>X</td></tr><tr><td>CONFIG (This paper)</td><td>✓</td><td>✓</td><td>Global convergence</td><td>✓</td></tr></table>
+
+# 2. Notation and Preliminaries
+
+This section introduces the class of constrained global optimization problems studied in this paper and recaps existing results from the field of efficient global optimization.
+
+# 2.1. Problem Statement
+
+We consider the following black-box optimization problem,
+
+$$
+\begin{array}{c c c} \min  _ {x \in \mathcal {X}} & f (x) , \\ \text {s u b j e c t t o} & g _ {i} (x) \leq 0 , \quad \forall i \in [ N ], \end{array} \tag {1a}
+$$
+
+where  $\mathcal{X} \subset \mathbb{R}^d$  is a known set of candidate solutions (e.g., a hyperbox) with dimension  $d$  and  $[N] := \{1, 2, \dots, N\} \subset \mathbb{Z}$  denotes the set of integers from 1 to  $N$ . Both  $f$  and  $g_i, i \in [N]$  are unknown black-box functions.
+
+Remark 1. In Prob. 1, we only consider inequality constraints because an equality constraint  $g(x) = 0$  can be equivalently transformed into two inequality constraints  $g(x) \leq 0, -g(x) \leq 0$ .
+
+To proceed, we make the following commonly used assumptions in the field of efficient global optimization.
+
+Assumption 2.1. Set  $\mathcal{X}$  is compact.
+
+Assumption 2.1 is naturally satisfied in a large set of applications. For example, when tuning the hyperparameters of a machine learning model, we can restrict the parameters to lie inside a hyperbox based on prior knowledge.
+
+Assumption 2.2.  $f \in \mathcal{H}_{k_0}$ ,  $g_i \in \mathcal{H}_{k_i}$ ,  $i \in [N]$ , where  $k_i: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}, i \in \{0\} \cup [N]$  are symmetric, positive-semidefinite kernel functions and  $\mathcal{H}_{k_i}$  are their corresponding reproducing kernel Hilbert spaces (RKHSs, see (Scholkopf et al., 2001)). Furthermore, we assume  $\| f \|_{k_0} \leq B_0$  and  $\| g_i \|_{k_i} \leq B_i$ ,  $i \in [N]$ , where  $\| \cdot \|_{k_i}$  is the norm induced by the inner product in the corresponding reproducing kernel Hilbert space.
+
+Assumption 2.2 requires that the underlying functions are regular in the sense of having a bounded norm in an RKHS. In existing literature of efficient global optimization (also known as Bayesian optimization or Gaussian process optimization), having a bounded norm in an RKHS is a common assumption (e.g., (Srinivas et al., 2012; Zhou & Ji, 2022)).
+
+Assumption 2.3. Prob. (1) is feasible and has an optimal solution  $x^{*}$  with the optimal value  $f^{*} = f(x^{*})$ .
+
+With Assumption 2.3, we focus on the case that the original black-box optimization problem is feasible. We will separately discuss how our algorithm can handle infeasible
+
+problem instances later in Sec. 5.
+
+We consider a zero-order feedback setting, where the algorithm can sequentially query noisy evaluations of different candidate solutions to learn the optimal feasible solution. At step  $t$ , with query point  $x_{t}$ , we get noisy evaluations,
+
+$$
+y _ {0, t} = f \left(x _ {t}\right) + \xi_ {0, t}, \tag {2a}
+$$
+
+$$
+y _ {i, t} = g _ {i} \left(x _ {t}\right) + \xi_ {i, t}, \quad i \in [ N ], \tag {2b}
+$$
+
+where  $\xi_{i,t}, i \in \{0\} \cup [N], t \geq 1$  are i.i.d  $\sigma$ -sub-Gaussian noise with fixed  $\sigma > 0$ . We also use  $X_{t} \coloneqq (x_{1}, x_{2}, \dots, x_{t})$  to denote the sequence of sampled points up to step  $t$ .
+
+# 2.2. Gaussian Process Surrogates
+
+To design our algorithm, we use a Gaussian process surrogate model to learn the unknown functions. As in (Chowdhury & Gopalan, 2017a), we artificially introduce a Gaussian process  $\mathcal{GP}(0,k(\cdot ,\cdot))$  for the surrogate modeling of the unknown black-box function  $f$ . We also adopt an i.i.d Gaussian zero-mean noise model with noise variance  $\lambda >0$ .
+
+Remark 2. Note that this Gaussian process model is only used to derive posterior mean functions, covariance functions, and maximum information gain for the purpose of algorithm design. We introduce the artificial Gaussian process only to make the algorithm more interpretable. It does not change our set-up that  $f$  is a deterministic function and that the noise is only assumed to be sub-Gaussian. Indeed, we are considering the agnostic setting introduced in (Srinivas et al., 2012).
+
+We introduce the following functions for  $x, x'$ ,
+
+$$
+\begin{array}{l} \mu_ {0, t} (x) = k _ {0} \left(x _ {1: t}, x\right) ^ {\top} \left(K _ {0, t} + \lambda I\right) ^ {- 1} y _ {0, 1: t}, (3a) \\ k _ {0, t} \left(x, x ^ {\prime}\right) = k _ {0} \left(x, x ^ {\prime}\right) (3b) \\ - k _ {0} \left(x _ {1: t}, x\right) ^ {\top} \left(K _ {0, t} + \lambda I\right) ^ {- 1} k _ {0} \left(x _ {1: t}, x ^ {\prime}\right), \\ \sigma_ {0, t} ^ {2} (x) = k _ {0, t} (x, x), (3c) \\ \end{array}
+$$
+
+where  $k_{0}(x_{1:t},x) = [k_{0}(x_{1},x),k_{0}(x_{2},x),\dots ,k_{0}(x_{t},x)]^{\top}$ $K_{0,t} = (k_0(x,x'))_{x,x'\in X_t}$  and  $y_{0,1:t} =$ $[y_{0,1},y_{0,2},\dots ,y_{0,t}]^{\top}$  Similarly, we can derive  $\mu_{i,t}(\cdot),k_{i,t}(\cdot ,\cdot),\sigma_{i,t}(\cdot),\forall i\in [N]$  for the constraints. We also introduce the maximum information gain for the objective  $f$  as in (Srinivas et al., 2012),
+
+$$
+\gamma_ {0, t} := \max  _ {X \subset \mathcal {X}; | X | = t} \frac {1}{2} \log \left| I + \lambda^ {- 1} K _ {0, X} \right|, \tag {4}
+$$
+
+where  $K_{0,X} = (k_0(x,x'))_{x,x'\in X}$ . Similarly, we can introduce the maximum information gain  $\gamma_{i,t},\forall i\in [N]$  for the constraints.
+
+Lemma 2.4 (Theorem 2, (Chowdhury & Gopalan, 2017a)). Let Assumptions 2.1 and 2.2 hold. For any  $\delta \in (0,1)$ , with
+
+probability at least  $1 - \delta /(N + 1)$  , the following holds for all  $x\in \mathcal{X}$  and  $1\leq t\leq T$ $T\in \mathbb{N}$
+
+$$
+\begin{array}{l} \left| \mu_ {0, t - 1} (x) - f (x) \right| \tag {5} \\ \leq \left(B _ {0} + \sigma \sqrt {2 \left(\gamma_ {0 , t - 1} + 1 + \ln ((N + 1) / \delta)\right)}\right) \sigma_ {0, t - 1} (x), \\ \end{array}
+$$
+
+where  $\mu_{0,t - 1}(x),\sigma_{0,t - 1}^2 (x)$  and  $\gamma_{0,t - 1}$  are as given in Eq. (3) and Eq. (4), and  $\lambda$  set to be  $1 + \frac{2}{T}$ . ( $\mu_{0,0}$  and  $\sigma_{0,0}$  are the prior mean and standard deviation, and  $\gamma_{0,0} = 0$ .)
+
+We remark that we replace the  $\delta$  in Theorem 2 of (Chowdhury & Gopalan, 2017a) by  $\frac{\delta}{N + 1}$ , which will be useful to derive the confidence interval with  $1 - \delta$  probability in Corollary 2.6. We can derive similar confidence intervals for the constraint functions. To facilitate the following algorithm design and discussion, we introduce the lower confidence and upper confidence bound functions,
+
+Definition 2.5. For  $i\in \{0\} \cup [N]$ $x\in \mathcal{X}$  and  $t\in [T]$
+
+$$
+l _ {i, t} (x) := \mu_ {i, t - 1} (x) - \beta_ {i, t} \sigma_ {i, t - 1} (x), \tag {6a}
+$$
+
+$$
+u _ {i, t} (x) := \mu_ {i, t - 1} (x) + \beta_ {i, t} \sigma_ {i, t - 1} (x), \tag {6b}
+$$
+
+where  $\beta_{i,t} = B_i + \sigma \sqrt{2(\gamma_{i,t - 1} + 1 + \ln((N + 1) / \delta))}$ .
+
+We then have the following corollary,
+
+Corollary 2.6. Let Assumptions 2.1 and 2.2 hold. With probability at least  $1 - \delta$ , the following holds for all  $x \in \mathcal{X}$  and  $1 \leq t \leq T$ ,
+
+$$
+g _ {i} (x) \in \left[ l _ {i, t} (x), u _ {i, t} (x) \right], \quad \forall i \in [ N ] \tag {7}
+$$
+
+$$
+a n d \quad f (x) \in [ l _ {0, t} (x), u _ {0, t} (x) ]. \tag {8}
+$$
+
+# 2.3. Performance Metric
+
+Before we design our algorithm, we introduce the performance metric. We are interested in minimizing the gap of  $f(x_{t})$  to the optimal value  $f^{*} = \min_{x\in \mathcal{X},g_{i}(x)\leq 0,\forall i\in [N]}f(x)$ , i.e., the instantaneous regret. Instantaneous regret at step  $t$  is defined as,
+
+$$
+r _ {t} = f \left(x _ {t}\right) - f ^ {*}, \tag {9}
+$$
+
+where  $x_{t}$  is the point queried by our algorithm in step  $t$ . Since  $f^{*}$  is the constrained optimal value, we may sample some infeasible points with even smaller objective values than  $f^{*}$ , we further define the positive regret as,
+
+$$
+r _ {t} ^ {+} = \left[ f \left(x _ {t}\right) - f ^ {*} \right] ^ {+}, \tag {10}
+$$
+
+where  $[\cdot ]^{+}\coloneqq \max \{0,\cdot \}$ . We are also interested in the constraint violation, defined as,
+
+$$
+v _ {i, t} = \left[ g _ {i} \left(x _ {t}\right) \right] ^ {+}. \tag {11}
+$$
+
+We then introduce cumulative regret that measures the extra cost incurred during the running of the algorithm, as in existing kernelized multi-armed bandit literature (e.g., (Srinivas et al., 2012; Chowdhury & Gopalan, 2017a)).
+
+Definition 2.7 (Cumulative-Regret). We define the cumulative-regret as
+
+$$
+R _ {T} = \sum_ {t = 1} ^ {T} \left(f \left(x _ {t}\right) - f ^ {*}\right). \tag {12}
+$$
+
+To facilitate the following discussion, we further define the cumulative positive regret,
+
+$$
+R _ {T} ^ {+} = \sum_ {t = 1} ^ {T} [ f (x _ {t}) - f ^ {*} ] ^ {+}. \tag {13}
+$$
+
+We further define the cumulative violation.
+
+Definition 2.8 (Cumulative-Violation). We define the cumulative-violation as
+
+$$
+\mathcal {V} _ {i, T} = \sum_ {t = 1} ^ {T} \left[ g _ {i} \left(x _ {t}\right) \right] ^ {+}, \forall i \in [ N ]. \tag {14}
+$$
+
+Cumulative regret/violations measure the performance during the running of the algorithm. Besides them, we are also interested in the convergence speed to the constrained global optimum. Intuitively, small average regret/violations indicate the existence of a sample with small regret/violation.
+
+# 3. Algorithm
+
+We propose the lower confidence bound based CONstrained efFicient Global (CONFIG) optimization algorithm to solve our problem as shown in Alg. 1<sup>2</sup>. Our algorithm
+
+Algorithm 1 Lower confidence bounds based CONstrained efFcient Global (CONFIG) optimization algorithm.
+
+1: for  $t\in [T]$  do  
+2: if  $\exists i\in [N]$ , such that  $\min_{x\in \mathcal{X}}l_{i,t}(x) > 0$  then  
+3: Declare infeasibility.  
+4: end if  
+5: Find  $x_{t} \in \arg \min_{x \in \mathcal{X}} l_{0,t}(x)$
+
+subject to  $l_{i,t}(x)\leq 0,\forall i\in [N]$
+
+6: Make noisy evaluations of  $f$  and  $g_{i}$ ,  $i \in [N]$  at  $x_{t}$ .  
+7: Update Gaussian process posterior mean and covariance with the new evaluations added.
+
+8: end for
+
+is conceptually simple based on the optimism in the face of uncertainty and only requires one to solve an auxiliary problem with the black-box objective and constraints replaced by their lower confidence bound surrogates as shown in line 1 of Alg. 1. Since we consider problems with expensive-to-evaluate functions, the cost of solving the auxiliary problem,
+
+which may itself be non-convex, is much smaller compared to solving the original black-box optimization problem when the dimension of the input space is small to medium size. To solve the auxiliary problem, we can use a pure grid search method when the dimension is small (e.g.,  $\leq 5$ ). For larger dimensions, one can, for example, start from multiple different points and apply gradient-based methods.
+
+Note that in line 1 of Alg. 1, we may declare infeasibility if the auxiliary problem in line 1 is infeasible. When the original problem (1) is feasible, in some very rare cases, we may find the auxiliary problem infeasible due to the potentially unbounded noise. Fortunately, we will show that with guaranteed high probability, line 1 of Alg. 1 is never declared when the original problem is feasible and Assumptions 2.1 and 2.2 hold.
+
+# 4. Analysis
+
+We now analyze the performance of Alg. 1. All the proofs to the results are put in the appendix. We first introduce a lemma to bound the instantaneous regret and violation,
+
+Lemma 4.1. Let Assumptions 2.1, 2.2, and 2.3 hold. With probability at least  $1 - \delta$ , we have first that infeasibility is never declared in line 1 of the Alg. 1, and for all  $1 \leq t \leq T$
+
+$$
+r _ {t} \leq r _ {t} ^ {+} \leq 2 \beta_ {0, t} \sigma_ {0, t - 1} \left(x _ {t}\right), \tag {15}
+$$
+
+$$
+v _ {i, t} \leq 2 \beta_ {i, t} \sigma_ {i, t - 1} \left(x _ {t}\right), \quad i \in [ N ]. \tag {16}
+$$
+
+We can also bound the cumulative sum of the posterior standard deviation,
+
+Lemma 4.2 (Lemma 4, (Chowdhury & Gopalan, 2017b)). If  $x_{1}, x_{2}, \dots, x_{T}$  are the points selected by Alg. 1, then,
+
+$$
+\sum_ {t = 1} ^ {T} \sigma_ {i, t - 1} \left(x _ {t}\right) \leq \sqrt {4 (T + 2) \gamma_ {i , T}}. \tag {17}
+$$
+
+The bound in Eq. (17) is in terms of maximum information gain. To derive a kernel-specific bound, we need to estimate the maximum information gain. A set of upper bounds for the maximum information gains were given by (Srinivas et al., 2012; Vakili et al., 2021), which we restate in the maximum information gain column of Tab. 2. We can now state our main theorem,
+
+Theorem 4.3. Under Assumptions 2.1, 2.2 and 2.3, we have, with probability at least  $1 - \delta$ , the sample points of Algorithm 1 satisfy,
+
+$$
+R _ {T} \leq R _ {T} ^ {+} \leq 4 \beta_ {0, T} \sqrt {(T + 2) \gamma_ {0 , T}} = \mathcal {O} (\gamma_ {0, T} \sqrt {T}), \tag {18a}
+$$
+
+$$
+\mathcal {V} _ {i, T} \leq 4 \beta_ {i, T} \sqrt {(T + 2) \gamma_ {i , T}} = \mathcal {O} \left(\gamma_ {i, T} \sqrt {T}\right), \quad \forall i \in [ N ]. \tag {18b}
+$$
+
+Table 2. Kernel-specific cumulative regret/violation bounds and the convergence rate to the optimal solution. See Appendix A for the specific kernel functions. The column of maximum information gain shows the results of (Srinivas et al., 2012; Vakili et al., 2021). In the table,  $d$  represents the dimension of the input space, and  $\nu$  represents the smoothness parameter of the Matern kernel.  
+
+<table><tr><td>Kernel</td><td>Maximum Information Gain</td><td>Cumulative Regret/Violation</td><td>Convergence Rate</td></tr><tr><td>Linear</td><td>O(d log T)</td><td>O(d log T√T)</td><td>O((N+1)d log T/√T)</td></tr><tr><td>Squared Exponential</td><td>O((log T)d+1)</td><td>O((log T)d+1√T)</td><td>O((N+1)(log T)d+1/√T)</td></tr><tr><td>Mátern (ν&gt; d/2)</td><td>O(Td/2ν+d log 2ν/2ν+d(T))</td><td>O(T2ν+3d/4ν+2d log 2ν/2ν+d(T))</td><td>O((N+1)T-2ν-d/4ν+2d log 2ν/2ν+d(T))</td></tr></table>
+
+Interestingly, the regret bound in Thm. 4.3 is exactly the same as the regret bound in the unconstrained case shown in (Chowdhury & Gopalan, 2017b). Furthermore, the constraint violation bound is similar to the regret bound. Based on Thm. 4.3, we can derive a convergence rate to the optimal feasible solution.
+
+Theorem 4.4. Under Assumptions 2.1, 2.2 and 2.3, we have, with probability at least  $1 - \delta$ , that there exists  $\tilde{x}_T \in \{x_1, x_2, \dots, x_T\}$ , such that,  $\forall j \in [N]$
+
+$$
+\begin{array}{l} f \left(\tilde {x} _ {T}\right) - f ^ {*} \leq \frac {4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i , T} \sqrt {\gamma_ {i , T}}}{T} = \mathcal {O} \left(\frac {\sum_ {i = 0} ^ {N} \gamma_ {i , T}}{\sqrt {T}}\right), (19a) \\ \left[ g _ {j} \left(\tilde {x} _ {T}\right) \right] ^ {+} \leq \frac {4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i , T} \sqrt {\gamma_ {i , T}}}{T} = \mathcal {O} \left(\frac {\sum_ {i = 0} ^ {N} \gamma_ {i , T}}{\sqrt {T}}\right). (19b) \\ \end{array}
+$$
+
+Thm. 4.4 gives the convergence rate to the constrained optimal value in terms of maximum information gain for general kernels. We can now apply Thm. 4.3 and Thm. 4.4 to derive kernel-specific results. Specifically, if  $k_{i}, i \in \{0\} \cup [N]$  share the same kernel types, we can list the kernel-specific bounds and the convergence rates in Tab. 2.
+
+# 5. Infeasibility Declaration
+
+Thm. 4.3 and Thm. 4.4 are derived under the assumption that the problem is feasible. However, the underlying problem may indeed sometimes be infeasible and therefore, we need an infeasibility detection scheme. In the following, we show that if the original problem is infeasible, we are able to declare infeasibility in line 1 of Alg. 1 and detect the original problem's infeasibility with high probability.
+
+For infeasibility declaration, we try to avoid 'false positives', where a feasible problem is declared as infeasible, and 'false negatives', where an infeasible problem is not declared to be infeasible. Lem. 4.1 already shows that if the original black-box optimization problem is feasible, then with high probability infeasibility is not declared. The following theorem complements Lem. 4.1 by showing that if the original problem is infeasible, our algorithm is guaranteed to declare infeasibility within finite steps with high probability.
+
+Theorem 5.1. Let Assumptions 2.1 and 2.2 hold. Assume that Prob. (1) is infeasible, that is, there exists  $i \in [N]$ , such that
+
+$$
+\min  _ {x \in \mathcal {X}} g _ {i} (x) = \epsilon > 0,
+$$
+
+and  $\lim_{T\to \infty}\frac{\gamma_{i,T}}{\sqrt{T}} = 0$ . Given a desired confidence level  $\delta \in (0,1)$ , Alg. 1 will declare infeasibility on line 1 within a number of steps equal to
+
+$$
+\bar {T} = \min  _ {T \in \mathbb {N} _ {+}} \left\{T \left| \frac {\gamma_ {i , T}}{\sqrt {T}} \leq C \epsilon \right. \right\}, \tag {20}
+$$
+
+with probability at least  $1 - \delta$ , where  $C$  is a constant independent of  $T$ .
+
+Remark 3. The proposed algorithm design and analysis in this paper focuses on the case where  $T$  is given. In cases where  $T$  is not given or not large enough to declare infeasibility, we can apply the 'doubling trick' (Besson & Kaufmann, 2018). The basic idea is that we can start running a round of steps with  $T = 1$  and double  $T$  after each round. As  $T$  grows larger and larger, it will finally be larger than  $\overline{T}$  and enough for infeasibility declaration.
+
+Remark 4. Note that in Thm. 5.1, we assume that  $\lim_{T\to \infty}\frac{\gamma_i,T}{\sqrt{T}} = 0$ . From Tab. 2, we can see that for linear and squared exponential kernels, it is satisfied and for Mátern kernel, we require that  $\nu >\frac{d}{2}$ .
+
+# 6. Experiments
+
+To demonstrate the effectiveness of CONFIG, we run experiments over sampled instances from the Gaussian process, artificial numerical instances, and a room temperature controller tuning problem. We compare our method to the state-of-the-art Gaussian process optimization baseline methods with constraints consideration. The baselines include SafeOPT (Sui et al., 2015), CEI (Constrained Expected Improvement) based method (Gelbart et al., 2014; Gardner et al., 2014), primal-dual algorithm (Zhou & Ji, 2022) and EPBO (exact penalty Bayesian optimization) (Lu & Paulson, 2022). The experiments are implemented in python, based on the package GPy (GPy, since 2012).
+
+Choice of Parameters. Definition 2.5 gives a rigorous method for selecting the coefficient  $\beta_{i,t}$ . Since  $B_{i}$  and  $\sigma$
+
+may be hard to obtain in practice,  $\beta_{i,t}$  can usually be set as a constant. The choice of  $\beta_{i,t}$  is a key factor that impacts the performance of different algorithms. Empirically, if  $\beta_{i,t}$  is set too small, exploration may be insufficient and we may miss the global optimal solution. If  $\beta_{i,t}$  is set too large, the algorithm overly explores, converges more slowly, and may suffer from high cumulative regret/violation before converging to the optimal solution. In our experiments shown in this section, manually setting  $\beta_{i,t} = 3$  works well. We also set  $\lambda = 0.05^2$  as the noise variance for the Gaussian process modelling. We use the common squared exponential kernel functions. For Sec. 6.2 and Sec. 6.3, we randomly sample a few points and maximize the likelihood function to get the hyperparameters of the kernel.
+
+Computational Time. In our experiments, all the problems have low-dimensional input  $(\leq 3)$ . So we use pure grid search to solve the auxiliary problem for different algorithms and therefore, the computational time is almost the same for different algorithms. With the explicit expression of the lower confidence bounds, the computation cost of solving the auxiliary problems is much cheaper than evaluating the expensive functions, e.g., energy consumption and discomfort of a simulated building in Sec. 6.3.
+
+# 6.1. Numerical Results over Sampled Instances from Gaussian Process
+
+In this section, we consider the constrained problem with both the objective and the constraint sampled from a Gaussian process. We use the squared exponential kernel as shown in (21).
+
+$$
+k (x, y) = \sigma_ {\mathrm {S E}} ^ {2} \exp \left\{- \frac {\| x - y \| ^ {2}}{l ^ {2}} \right\}, \tag {21}
+$$
+
+where  $\sigma_{\mathrm{SE}}^2 = 2.0$  and  $l = 1.0$ . We remove those instances with empty feasible sets. We sampled 48 instances in total.
+
+As shown in Fig. 2, we observe that SafeOPT has almost linear cumulative regret, although achieving close-to-zero violation. This is due to the possibility of getting stuck in a local minimum for SafeOPT. In contrast, the primal-dual method can achieve competitive performance with the state-of-the-art constrained expected improvement (CEI) method in terms of cumulative regret but suffers from almost linear cumulative violation. The EPBO method may also suffer from almost linear cumulative violation if the penalty term is not set large enough. The proposed CONFIG algorithm can achieve performance competitive with the state-of-the-art constrained expected improvement method in terms of both cumulative regret and cumulative violation.
+
+We further show the evolution of the best-observed suboptimality plus violation up to the current step  $t$  in Fig. 3, which measures the empirical convergence speed to the op
+
+![](images/3c43ceed514f767eb59c4929ef5031a17a803835f78b1f2358abbbd8031a3d17.jpg)
+
+![](images/ce08a70344498ffb48453e6dfe98caa5390a217707f963c5da08f7662ffa3022.jpg)
+
+![](images/982ec5c9fa7ca42c3f0df3d9b4d94b6a0add8b4e40cbdfac1aeb98b19934e950.jpg)  
+Figure 2. Cumulative regret and violation of different algorithms. The shaded area represents  $\pm 0.1$  standard deviation and EPBO- $\rho$  represents EPBO with penalty  $\rho$ .
+
+![](images/3a9694ecb2e50cb91ead4e5292ccc7defc80d2fb3619578a72b2b8ca47fdb88a.jpg)  
+Figure 3. The best suboptimality plus violation up to step  $t$ , that is  $\min_{\tau \in [t]}[f(x_{\tau}) - f^{*}]^{+} + [g(x_{\tau})]^{+}$ , of different algorithms. The shaded area represents  $\pm 0.1$  standard deviation. Here, EPBO-  $\rho$  represents the EPBO algorithm with penalty term  $\rho$ . The SafeOPT has a higher initial value due to the restriction of starting from a feasible solution.
+
+![](images/7a3e92fcb8aa85ff354996109060a740dd5a9f97df5653edee0dce1b0f64a5e7.jpg)
+
+![](images/3b57105d65cdcfb2629a7bef7490c90bd9551f66cdab8c2a5fdffc55b7a69e7e.jpg)
+
+![](images/86b735c0116882c0346c6d36c6e76eb4e1e906c7efd8a9b3b036c70458d9c2e8.jpg)
+
+![](images/0dac94d686cc520f05139a20abfe146d293f9bca73c66ffa18f0f7c3da829a2a.jpg)
+
+Figure 4. Convergence of constrained regret for the collections of problems in Tab. 3.  
+![](images/0a69c5a4f4c78ed4f7bcc51d15f3f05010f7444ee2ff61eb8a4f8ef903043ba0.jpg)  
+CEI SafeOPT Primal-Dual EPBO 0.2 EPBO 3.0 CONFIG
+
+![](images/f6ddc48ef67613302d801c9d4107cce8935b0248307d637e8f9faf8942abbdc3.jpg)
+
+![](images/ace36c2b459b5d86d18d96ac349ba126d2fb1fef3899dfccf055cbfbb55e821b.jpg)
+
+timal feasible solution. We notice that our algorithm again achieves empirical convergence speed competitive with the state-of-the-art CEI method.
+
+We then test our algorithms on a set of infeasible instances, where we shift the sampled constraint function  $g(x)$  by  $\epsilon - \min_{x \in \mathcal{X}} g(x)$  with  $\epsilon = 0.1$ . Over 50 such infeasible instances, our algorithm consistently declares infeasibility, within 16.3 steps on average.
+
+# 6.2. Artificial Numerical Instances
+
+In this section, we present the results of a set of global optimization test problems. We consider the global optimization problem in two-dimensional space with  $\mathcal{X} = [-10, 10]^2$ . We then introduce a set of potentially non-convex and multimodal functions as in Tab. 4 in the Appendix G. Using these functions as the objective or constraint, we construct a set of global optimization benchmark problems as shown in Tab. 3 (Similar problems in (Vinod et al., 2022)). We use the common squared exponential kernel for all the experiments.
+
+We use constrained regret defined as
+
+$$
+\min  _ {\tau \in [ t ]} [ f (x _ {\tau}) - f ^ {*} ] ^ {+} + [ g (x _ {\tau}) ] ^ {+}, \tag {22}
+$$
+
+to measure the quality of the solution found by different algorithms up to step  $t$ . It can be seen that the smaller the constrained regret is, the closer to the constrained optimum our solution is in terms of optimality and violations.
+
+Fig. 4 demonstrates the convergence of constrained regret
+
+Table 3. Artificial numerical problems constructed using the functions from Tab. 4, where  $\mathbf{Qr}_{\mathcal{X}}(h) := \frac{3}{4}\min_{x\in \mathcal{X}}h(x) + \frac{1}{4}\max_{x\in \mathcal{X}}h(x)$ .  
+
+<table><tr><td>Problem</td><td>Objective f</td><td>Constraint g</td></tr><tr><td>P1</td><td>Br</td><td>SinQ - QrX (SinQ)</td></tr><tr><td>P2</td><td>MBr</td><td>SinQ - QrX (SinQ)</td></tr><tr><td>P3</td><td>Br</td><td>InvBowl - QrX (InvBowl)</td></tr><tr><td>P4</td><td>MBr</td><td>InvBowl - QrX (InvBowl)</td></tr><tr><td>P5</td><td>Br</td><td>Bowl - QrX (Bowl)</td></tr><tr><td>P6</td><td>MBr</td><td>Bowl - QrX (Bowl)</td></tr></table>
+
+for different algorithms. The results show that the CONFIG algorithm consistently achieves a superior or comparable speed of finding the constrained optimal solution, as compared to the other state-of-the-art baselines. We can also notice that in Problem 3 and Problem 4, SafeOPT outperforms both CEI and our method. Intuitively, it happens when the constrained global optimum lies in the same local feasible set as the initial points. In such a case, our method may spend lots of samples exploring globally, while SafeOPT identifies the local (and global) optimal solution quickly by restricting to the local feasible set.
+
+# 6.3. Tuning the P Controller of a Building
+
+In this section, we consider a building temperature controller tuning problem. We use Energym (Scharnhorst et al., 2021) with a lower level Modelica (Fritzson & Engelson, 1998)
+
+model with a single thermal zone as our simulator. We consider a P controller controlling the temperature. The mathematical expression of the P controller is given as
+
+$$
+u = K _ {p} (\text {s e t - p o i n t - t e m p e r a t u r e}),
+$$
+
+where  $K_{p}$  is the proportional gain and  $u$  is the heating control. We use  $\theta$  to denote the tuning parameters, which include the proportional gain, daytime set-point for temperature, and the switching time from nighttime set-point to daytime set-point. Here, the nighttime set-point is fixed. Our objective is to minimize energy consumption while keeping the average temperature deviation below a given threshold. Our problem is formulated as in Eq. (23).
+
+$$
+\min  _ {\theta \in \Theta} J (\theta) \quad \text {s u b j e c t} \quad g (\theta) - g _ {\mathrm {t h r}} \leq 0, \tag {23}
+$$
+
+where  $J(\theta)$  represents energy consumption in this example,  $g(\theta)$  represents average temperature deviation and  $g_{\mathrm{thr}}$  is a given average temperature deviation threshold. In each step, our algorithm selects a set of promising controller parameters. We use this set of parameters to run a closed-loop building simulation over a period of 1 day.
+
+Based on the closed-loop trajectory, we calculate the key performance indicators, that is, energy consumption and average temperature deviation. We then update the Gaussian process regression with the new data added. Since energy consumption and temperature deviations are in two different scales, we use normalized energy consumption  $\frac{J}{\sigma_J}$  plus normalized average temperature deviation  $\frac{[g - g_{\mathrm{thr}}]^+}{\sigma_g}$  to measure the quality of the solution, where  $\sigma_J, \sigma_g$  are standard deviations in a set of randomly sampled data.
+
+Fig. 5 gives the best-normalized energy consumption plus average temperature deviation up to the current step. We observe that within 15 steps, our algorithm identifies the solution with minimized normalized energy consumption plus normalized average temperature deviation, verifying the competitiveness of our algorithm with the currently popular constrained expected improvement method again. In contrast, the SafeOPT method can be overly cautious and fails to find a solution with performance comparable to ours.
+
+# 7. Conclusion
+
+In this paper, we have presented CONFIG, a simple and effective algorithm, for constrained efficient global optimization of expensive black-box functions by exploiting the principle of optimism in the face of uncertainty. Specifically, our algorithm solves an auxiliary constrained optimization problem with the lower confidence bound (LCB) surrogates as objective and constraints to get the next sample. Theoretically, we show that our algorithm enjoys the same cumulative regret bound as that in the unconstrained case. In
+
+![](images/c92295be8bb1993d24c0c91b3dc27c8bd165267035adfd847ef978db2e60896e.jpg)  
+Figure 5. Best normalized energy consumption plus normalized temperature deviation up to the current step. For SafeOPT, the initial required feasible solution is obtained based on domain knowledge.
+
+![](images/f323a4b582ce737a03f76a68c8f834f2ef0492d715d10fae28601485b097936f.jpg)
+
+addition, we show that the cumulative constraint violations have upper bounds in terms of maximum information gain, which are similar to the cumulative regret bounds of the objective function. For commonly used Mátern  $(\nu > \frac{d}{2})$  and Squared Exponential kernels, our bounds are sublinear and allow us to derive a convergence rate to the optimal solution of the original constrained problem. Furthermore, our method naturally provides a scheme to declare infeasibility when the original black-box optimization problem is infeasible. Numerical experiments over sampled instances, artificial problems, and a building controller tuning problem, all corroborate the effectiveness of our algorithm, which is competitive with the popular CEI method. As compared to existing state-of-the-art constrained efficient global/Bayesian optimization methods, CONFIG significantly improves the theoretical guarantees, while achieving competitive empirical performance.
+
+One limitation of our work is that the auxiliary problem may be infeasible with misspecified hyperparameters, which needs to be addressed in future work.
+
+# Acknowledgements
+
+This research was supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF40_180545, and in part by the Swiss Data Science Center, grant agreement C20-13.
+
+# References
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+
+# A. Examples of kernel functions
+
+- Linear:
+
+$$
+k (x, y) = x ^ {T} y.
+$$
+
+- Squared Exponential:
+
+$$
+k (x, y) = \exp \left\{- \frac {\| x - y \| ^ {2}}{\ell^ {2}} \right\}.
+$$
+
+- Matern:
+
+$$
+\begin{array}{l} k (x, y) = \\ \frac {2 ^ {1 - \nu}}{\Gamma (\nu)} \left(\sqrt {2 \nu} \frac {\| x - y \|}{\rho}\right) ^ {\nu} K _ {\nu} \left(\sqrt {2 \nu} \frac {\| x - y \|}{\rho}\right), \\ \end{array}
+$$
+
+where  $\rho$  and  $\nu$  are positive parameters of the kernel function,  $\Gamma$  is the gamma function, and  $K_{\nu}$  is the modified Bessel function of the second kind.
+
+# B. Proof of Corollary 2.6
+
+Proof. Note that  $l_{i,t}(x)$  and  $u_{i,t}(x)$  are random variables since they depend on the observations  $y_{i,t}$ 's, which are corrupted by random noise. The following events definitions are with respect to the randomness from noise. For simplicity of notation, we use  $\{l_{i,t}(x) \leq y \leq u_{i,t}(x)\}$  to denote the intersection event that the random variable  $l_{i,t}(x) \leq y$  and  $u_{i,t}(x) \geq y$ , where  $y \in \mathbb{R}$ ,  $i \in \{0\} \cup [N]$  and  $t \geq 1$ . We define the event  $\mathcal{E}_i \triangleq \cap_{x \in \mathcal{X}} \cap_{T \geq t \geq 1} \{l_{i,t}(x) \leq g_i(x) \leq u_{i,t}(x)\}, \forall i \in [N]$  and the event  $\mathcal{E}_0 \triangleq \cap_{x \in \mathcal{X}} \cap_{T \geq t \geq 1} \{l_{0,t}(x) \leq f(x) \leq u_{0,t}(x)\}$ . We have,
+
+$$
+\begin{array}{l} \mathbb {P} \left(\cap_ {i = 0} ^ {N} \mathcal {E} _ {i}\right) = 1 - \mathbb {P} \left(\overline {{\cap_ {i = 0} ^ {N} \mathcal {E} _ {i}}}\right) (24a) \\ = 1 - \mathbb {P} \left(\cup_ {i = 0} ^ {N} \overline {{\mathcal {E} _ {i}}}\right) (24b) \\ \geq 1 - \sum_ {i = 0} ^ {N} \mathbb {P} \left(\overline {{\mathcal {E} _ {i}}}\right) (24c) \\ \geq 1 - \sum_ {i = 0} ^ {N} \frac {\delta}{N + 1} (24d) \\ = 1 - \delta , (24e) \\ \end{array}
+$$
+
+where the inequality (24c) follows by union bound of probability and the inequality (24d) follows by Lemma 2.4. The conclusion then follows.
+
+![](images/8c4d5b25bec8b6280484380851e4cc1cc5edd55bf156615480c94fb5ac3b578c.jpg)
+
+# C. Proof of Lemma 4.1
+
+Proof. By Corollary 2.6, with probability at least  $1 - \delta$ , we have, for all  $x \in \mathcal{X}$  and  $1 \leq t \leq T$ ,
+
+$$
+\begin{array}{l} f (x) \quad \in [ l _ {0, t} (x), u _ {0, t} (x) ], (25a) \\ g _ {i} (x) \quad \in [ l _ {i, t} (x), u _ {i, t} (x) ], \quad \forall i \in [ N ]. (25b) \\ \end{array}
+$$
+
+All the following statements are conditioned on the above joint event happening, which has a probability of at least  $1 - \delta$ .
+
+By setting  $x = x^{*}$  in (25b) and the feasibility of the optimal solution  $x^{*}$ , we have
+
+$$
+l _ {i, t} \left(x ^ {*}\right) \leq g _ {i} \left(x ^ {*}\right) \leq 0, \quad \forall i \in [ N ]. \tag {26}
+$$
+
+Therefore, infeasibility is not declared in line 1 of the Alg. 1, and  $x^{*}$  is a feasible solution for the auxiliary problem in line 1 of the Alg. 1. Therefore, we have
+
+$$
+l _ {0, t} \left(x _ {t}\right) \leq l _ {0, t} \left(x ^ {*}\right). \tag {27}
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} r _ {t} = f \left(x _ {t}\right) - f ^ {*} (28a) \\ \leq r _ {t} ^ {+} (28b) \\ = \left[ f \left(x _ {t}\right) - l _ {0, t} \left(x _ {t}\right) + l _ {0, t} \left(x _ {t}\right) - f ^ {*} \right] ^ {+} (28c) \\ \leq \left[ f \left(x _ {t}\right) - l _ {0, t} \left(x _ {t}\right) \right] ^ {+} + \left[ l _ {0, t} \left(x _ {t}\right) - f ^ {*} \right] ^ {+} (28d) \\ \leq \left[ u _ {0, t} \left(x _ {t}\right) - l _ {0, t} \left(x _ {t}\right) \right] ^ {+} + \left[ l _ {0, t} \left(x _ {t}\right) - l _ {0, t} \left(x ^ {*}\right) \right] ^ {+} (28e) \\ = u _ {0, t} \left(x _ {t}\right) - l _ {0, t} \left(x _ {t}\right) (28f) \\ = 2 \beta_ {0, t} \sigma_ {0, t - 1} (x _ {t}), (28g) \\ \end{array}
+$$
+
+where the inequality (28e) follows by the inequalities (25a), the inequality (28d) follows by the fact that  $[a + b]^+ \leq [a]^+ + [b]^+, \forall a, b \in \mathbb{R}$ , and the equality (28f) follows by the inequality (27). We now consider the constraint violation,
+
+$$
+\begin{array}{l} v _ {i, t} = \left[ g _ {i} \left(x _ {t}\right) \right] ^ {+} (29a) \\ = \left[ g _ {i} \left(x _ {t}\right) - l _ {i, t} \left(x _ {t}\right) + l _ {i, t} \left(x _ {t}\right) \right] ^ {+} (29b) \\ \leq \left[ g _ {i} \left(x _ {t}\right) - l _ {i, t} \left(x _ {t}\right) \right] ^ {+} + \left[ l _ {i, t} \left(x _ {t}\right) \right] ^ {+} (29c) \\ = \left[ g _ {i} \left(x _ {t}\right) - l _ {i, t} \left(x _ {t}\right) \right] ^ {+} (29d) \\ \leq \left[ u _ {i, t} \left(x _ {t}\right) - l _ {i, t} \left(x _ {t}\right) \right] ^ {+} (29e) \\ = 2 \beta_ {i, t} \sigma_ {i, t - 1} (x _ {t}), (29f) \\ \end{array}
+$$
+
+where the equality (29d) follows by the feasibility of  $x_{t}$  for the auxiliary problem in line 1 of Alg. 1.
+
+# D. Proof of Theorem 4.3
+
+Proof. By the monotonicity of  $\beta_{i,t}$  and Lem. 17, we have,
+
+$$
+\begin{array}{l} R _ {T} = \sum_ {t = 1} ^ {T} r _ {t} (30a) \\ \leq \sum_ {t = 1} ^ {T} r _ {t} ^ {+} = R _ {T} ^ {+} (30b) \\ \leq \sum_ {t = 1} ^ {T} 2 \beta_ {0, t} \sigma_ {0, t} \left(x _ {t}\right) (30c) \\ \leq 2 \beta_ {0, T} \sum_ {t = 1} ^ {T} \sigma_ {0, t} \left(x _ {t}\right) (30d) \\ \leq 4 \beta_ {0, T} \sqrt {(T + 2) \gamma_ {0 , T}}, (30e) \\ \end{array}
+$$
+
+and  $\forall i\in [N]$
+
+$$
+\begin{array}{l} \mathcal {V} _ {i, T} = \sum_ {t = 1} ^ {T} v _ {i, t} (31a) \\ \leq \sum_ {t = 1} ^ {T} 2 \beta_ {i, t} \sigma_ {i, t} \left(x _ {t}\right) (31b) \\ \leq 2 \beta_ {i, T} \sum_ {t = 1} ^ {T} \sigma_ {i, t} \left(x _ {t}\right) (31c) \\ \leq 4 \beta_ {i, T} \sqrt {(T + 2) \gamma_ {i , T}}, (31d) \\ \end{array}
+$$
+
+Combining with the expression of  $\beta_{i,t}$  in Definition 2.5, we can derive
+
+$$
+R _ {T} \leq R _ {T} ^ {+} \leq 4 \beta_ {0, T} \sqrt {(T + 2) \gamma_ {0 , T}} = \mathcal {O} (\gamma_ {0, T} \sqrt {T}),
+$$
+
+$$
+\mathcal {V} _ {i, T} \leq 4 \beta_ {i, T} \sqrt {(T + 2) \gamma_ {i , T}} = \mathcal {O} (\gamma_ {i, T} \sqrt {T}), \quad \forall i \in [ N ],
+$$
+
+which completes the proof.
+
+![](images/cfc5dad11dc51e1820f5940230577a283f71cba97e27fa69a23459150e4c5f5a.jpg)
+
+# E. Proof of Theorem 4.4
+
+Proof. Consider the sum,
+
+$$
+\begin{array}{l} \sum_ {t = 1} ^ {T} \left(\left[ f \left(x _ {t}\right) - f ^ {*} \right] ^ {+} + \sum_ {i = 1} ^ {N} \left[ g _ {i} \left(x _ {t}\right) \right] ^ {+}\right) \\ = R _ {T} ^ {+} + \sum_ {i = 1} ^ {N} \mathcal {V} _ {i, T} \\ \leq 4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i, T} \sqrt {\gamma_ {i , T}}, \\ \end{array}
+$$
+
+where the last inequality follows by the inequalities (30e) and (31d). Therefore,
+
+$$
+\begin{array}{l} \min  _ {t \in [ T ]} \left(\left[ f \left(x _ {t}\right) - f ^ {*} \right] ^ {+} + \sum_ {i = 1} ^ {N} \left[ g _ {i} \left(x _ {t}\right) \right] ^ {+}\right) \\ \leq \frac {1}{T} \sum_ {t = 1} ^ {T} \left(\left[ f \left(x _ {t}\right) - f ^ {*} \right] ^ {+} + \sum_ {i = 1} ^ {N} \left[ g _ {i} \left(x _ {t}\right) \right] ^ {+}\right) \\ \leq \frac {4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i , T} \sqrt {\gamma_ {i , T}}}{T}. \\ \end{array}
+$$
+
+So there exists  $\tilde{x}_T\in \{x_1,x_2,\dots ,x_T\}$  , such that,
+
+$$
+[ f (\tilde {x} _ {T}) - f ^ {*} ] ^ {+} + \sum_ {i = 1} ^ {N} [ g _ {i} (\tilde {x} _ {T}) ] ^ {+} \leq \frac {4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i , T} \sqrt {\gamma_ {i , T}}}{T}.
+$$
+
+Since  $[f(\tilde{x}_T) - f^*]^{+}$  and  $[g_i(\tilde{x}_T)]^{+}$  are non-negative, we have
+
+$$
+f (\tilde {x} _ {T}) - f ^ {*} \leq [ f (\tilde {x} _ {T}) - f ^ {*} ] ^ {+} \leq \frac {4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i , T} \sqrt {\gamma_ {i , T}}}{T}
+$$
+
+$$
+[ g _ {j} (\tilde {x} _ {T}) ] ^ {+} \leq \frac {4 \sqrt {T + 2} \sum_ {i = 0} ^ {N} \beta_ {i , T} \sqrt {\gamma_ {i , T}}}{T}, \quad \forall j \in [ N ].
+$$
+
+The desired conclusion then follows.
+
+![](images/4f2b2cd4098c66d9eff85cf4a99bb75f0e802bc36ec3896cdd2e5cf3ac98a098.jpg)
+
+# F. Proof of Theorem 5.1
+
+Proof. Suppose up to step  $T$ , infeasibility has not been declared. We have,
+
+$$
+l _ {i, t} (x _ {t}) \leq 0
+$$
+
+and
+
+$$
+u _ {i, t} \left(x _ {t}\right) \geq g _ {i} \left(x _ {t}\right) \geq \epsilon
+$$
+
+with probability at least  $1 - \delta$
+
+Accordingly, we have
+
+$$
+2 \sum_ {t = 1} ^ {T} \beta_ {i, t} \sigma_ {i, t - 1} (x _ {t}) = \sum_ {t = 1} ^ {T} \left(u _ {i, t} (x _ {t}) - l _ {i, t} (x _ {t})\right) \geq \sum_ {t = 1} ^ {T} \epsilon = T \epsilon .
+$$
+
+Meanwhile,
+
+$$
+\begin{array}{l} 2 \sum_ {t = 1} ^ {T} \beta_ {i, t} \sigma_ {i, t - 1} (x _ {t}) \leq 2 \beta_ {i, T} \sum_ {t = 1} ^ {T} \sigma_ {i, t - 1} (x _ {t}) \\ \leq 2 \beta_ {i, T} \sqrt {4 (T + 2) \gamma_ {i , T}}, \\ \end{array}
+$$
+
+where the last inequality follows by Lem. 17. Therefore,
+
+$$
+2 \beta_ {i, T} \sqrt {4 (T + 2) \gamma_ {i , T}} \geq T \epsilon ,
+$$
+
+which implies  $\epsilon \leq \frac{2\beta_{i,T}\sqrt{4(T + 2)\gamma_{i,T}}}{T}$ . By Def. 2.5, we have
+
+$$
+\frac {2 \beta_ {i , T} \sqrt {4 (T + 2) \gamma_ {i , T}}}{T} = \mathcal {O} \left(\frac {\gamma_ {i , T}}{\sqrt {T}}\right).
+$$
+
+Hence,  $\exists \tilde{C} > 0$ , such that,
+
+$$
+\epsilon \leq \tilde {C} \frac {\gamma_ {i , T}}{\sqrt {T}}. \tag {32}
+$$
+
+That is
+
+$$
+C \epsilon \leq \frac {\gamma_ {i , T}}{\sqrt {T}}, \tag {33}
+$$
+
+where  $C = \frac{1}{\tilde{C}}$ . However, since  $\lim_{T \to \infty} \frac{\gamma_i T}{\sqrt{T}} = 0$ , the inequality (33) will be violated when  $T$  is large enough. So infeasibility will be declared on or before the first time the above inequality is violated, which is  $\overline{T}$ .
+
+# G. Explicit functions for the artificial numerical instances
+
+<table><tr><td>Name</td><td>Function</td></tr><tr><td>Branin (Br)</td><td>Br(x1,x2)=(x2-5.1/4π2x1+5/πx1-6)2+10(1-1/8π)cos(x1)+10.</td></tr><tr><td>Modified Branin (MBr)</td><td>MBr(x1,x2)=Br(x1,x2)+20x1-30x2.</td></tr><tr><td>Bowl (Bowl)</td><td>Bowl(x1,x2)=1/2(||x-cbowl||2-R2bowl) where Rbowl=10 and cbowl=[-3,-3].</td></tr><tr><td>Inverted bowl (InvBowl)</td><td>InvBowl(x1,x2)=-Bowl(x1,x2).</td></tr><tr><td>Sine-quadratic (SinQ)</td><td>SinQ(x1,x2)=sin(x12/x2/10).</td></tr></table>
+
+Table 4. Explicit functions used to construct the artificial numerical instances.

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+# Davor Runje<sup>12</sup> Sharath M Shankaranarayana<sup>1</sup>
+
+# Abstract
+
+Wider adoption of neural networks in many critical domains such as finance and healthcare is being hindered by the need to explain their predictions and to impose additional constraints on them. Monotonicity constraint is one of the most requested properties in real-world scenarios and is the focus of this paper. One of the oldest ways to construct a monotonic fully connected neural network is to constrain signs on its weights. Unfortunately, this construction does not work with popular non-saturated activation functions as it can only approximate convex functions. We show this shortcoming can be fixed by constructing two additional activation functions from a typical unsaturated monotonic activation function and employing each of them on the part of neurons. Our experiments show this approach of building monotonic neural networks has better accuracy when compared to other state-of-the-art methods, while being the simplest one in the sense of having the least number of parameters, and not requiring any modifications to the learning procedure or post-learning steps. Finally, we prove it can approximate any continuous monotone function on a compact subset of  $\mathbb{R}^n$ .
+
+# 1. Introduction
+
+Deep Learning has witnessed widespread adoption in many critical real-world domains such as finance, healthcare, etc (LeCun et al., 2015). Incorporating prior knowledge such as monotonicity in trained models helps in improving the performance and generalization ability of the trained models (Mitchell, 1980; Dugas et al., 2000). The introduction of structural biases such as monotonicity makes models also more data-efficient, enabling a leap in predictive power on
+
+$^{1}$ Airt Research, Zagreb, Croatia  $^{2}$ Algebra University College, Zagreb, Croatia. Correspondence to: Davor Runje <davor@airt.ai>, Sharath M Shankaranarayana <sharithms@alumni.iitm.ac.in>.
+
+Proceedings of the  $40^{th}$  International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
+
+smaller datasets (Veličković, 2019). Apart from the requirements of having models with high accuracy, there is also a need for transparency and interpretability, and monotonicity helps in partially achieving the above requirements (Gupta et al., 2016). Due to legal, ethical and/or safety concerns, monotonicity of predictive models with respect to some input or all the inputs is required in numerous domains such as financial (house pricing, credit scoring, insurance risk), healthcare (medical diagnosis, patient medication) and legal (criminal sentencing) to list just a few. For example, when using machine learning to predict admission decisions, it may seem unfair to select student X over student Y, if Y has a higher score than X, while all other aspects of the two are identical (Liu et al., 2020). In another example, one would expect an individual with a higher salary to have a higher loan amount approved, all else being equal (Sivaraman et al., 2020). A model without such a monotonic property would not, and certainly should not, be trusted by society to provide a basis for such important decisions.
+
+Monotonicity has been an active area of research and the existing methods on the subject can be broadly categorized into two types:
+
+1. Monotonic architectures by construction: neural architectures guaranteeing monotonicity by construction (Archer & Wang, 1993; Sill, 1997; Daniels & Velikova, 2010; Milani Fard et al., 2016; You et al., 2017).
+
+2. Monotonicity by regularization: enforcing monotonicity in neural networks during training by employing a modified loss function or a heuristic regularization term (Sill & Abu-Mostafa, 1996; Gupta et al., 2019).
+
+The simplest method to achieve monotonicity by construction is to constrain the weights of the fully connected neural network to have only non-negative (for non-decreasing variables) or only non-positive values (for non-ascending) variables when used in conjunction with a monotonic activation function, a technique known for 30 years (Archer & Wang, 1993). When used in conjunction with saturated (bounded) activation functions such as the sigmoid and hyperbolic tangent, these models are difficult to train, i.e. they do not converge to a good solution. On the other hand, when used with non-saturated (unbounded) convex activation functions such as ReLU (Nair & Hinton, 2010), the resulting models
+
+are always convex (Liu et al., 2020), severely limiting the applicability of the method in practice.
+
+Our main contribution is a modification of the method above which, in conjunction with non-saturated activation functions, is capable of approximating non-convex functions as well: when the original activation function is used with additional two monotonic activation functions constructed from it in a neural network with constrained weights, it can approximate any monotone continuous functions.
+
+The resulting model is guaranteed to be monotonic, can be used in conjunction with popular convex monotonic non-saturated activation function, doesn't have any additional parameters compared to a non-monotonic fully-connected network for the same task, and can be trained without any additional requirements on the learning procedure. Experimental results show it is exceeding the performance of all other state-of-the-art methods, all while being both simpler (in the number of parameters) and easier to train.
+
+Our contributions can be summarized as follows:
+
+1. A modification to an existing constrained neural network layer enabling it to model arbitrary monotonic function when used with non-saturated monotone convex activation functions such as ReLU, ELU, SELU, and alike.  
+2. Experimental comparisons with other recent works showing that the proposed architecture can yield equal or better results than the previous state-of-the-art and with significantly fewer parameters.  
+3. A proof showing that the proposed architecture can approximate any monotone continuous function on a compact subset of  $\mathbb{R}^n$  for a large class of non-saturated activation functions.
+
+# 2. Related work
+
+# 2.1. Activation functions
+
+Right from its inception in perceptron (Rosenblatt, 1958), non-linear activation functions have historically been one of the most important components of neural networks. Previously, the saturated functions such as the sigmoid (Rumelhart et al., 1986), the hyperbolic tangent (Neal, 1992), and its variants were the most common choice of activation functions. Currently, one of the most important factors for state-of-the-art results accomplished by modern neural networks is the use of non-saturated activation functions. The use of Rectified Linear Unit (ReLU) (Nair & Hinton, 2010; Glorot et al., 2011) as activation function was instrumental in achieving good performance in newer architectures. The ReLU has since become a de facto choice of activation in most practical implementations and continues to be
+
+widely used because of its advantages such as simple computation, representational sparsity, and linearity. Later, a number of activation functions were proposed to deal with solving problems of dead neurons and aid in faster convergence (Maas et al., 2013), (Clevert et al., 2016) (He et al., 2015), (Zheng et al., 2015), (Hendrycks & Gimpel, 2016), (Ramachandran et al., 2017), (Klambauer et al., 2017).
+
+The idea of using both the original activation function and its point reflection in the same layer has been proposed in (Shang et al., 2016) where both outputs of ReLU and the negative value of its point reflection were used in the construction of concatenated ReLU (CReLU) activation function. The proposed modification outputs two values instead of one and therefore increases the number of parameters. In (Zagoruyko & Komodakis, 2017), the authors propose negative concatenated ReLU (NCReLU) flip the sign and use the point reflection directly. In (Eidnes & Nøkland, 2018), the authors propose bipolar ReLU which consists of using ReLU on the half of the neurons in the layer and the point reflection of ReLU on the other half.
+
+# 2.2. Monotonicity by construction
+
+Apart from the approaches mentioned in the introduction, another approach to building monotonic neural architecture is Min-Max networks where monotonic linear embedding and max-min-pooling are used (Sill, 1997). In (Daniels & Velikova, 2010), authors generalized this approach to handle functions that are partially monotonic and proved that the resulting networks have the universal approximation property. However, such networks are very difficult to train and not used in practice.
+
+Deep lattice networks (DLN) (You et al., 2017) use a combination of linear calibrators and lattices (Milani Fard et al., 2016) for learning monotonic functions. This is the most widely used method in practice today, but not without its limits. Lattices are structurally rigid thereby restricting the hypothesis space significantly. They also require a very large number of parameters to obtain good performance.
+
+Given a model with a convex output function, it is possible to use backpropagation (Rumelhart et al., 1986) to make a monotonic model by computing the derivation of the output function. One simple way to construct a convex function is to use an unsaturated monotonic activation function in a fully connected layer as mentioned above or a more elaborate architecture such as the input convex neural networks (Amos et al., 2017). These constructions are computationally more complex than the simple solution proposed here.
+
+# 2.3. Monotonicity by regularization
+
+Monotonicity can be enforced during the training by modifying the loss function or adding a regularization term.
+
+![](images/9d91082eb032ad604709eefcd5b02ef44c566d3a5f70c4f58cefcfa31514a0f7.jpg)  
+(a) Unconstrained ReLU
+
+![](images/72e770c5f99e46ec764fd8325ea8a649fd044fa4d750594e914c04f397c1f1d6.jpg)  
+(b) Constrained ReLU
+
+![](images/e47b7d2a3624a1458b9f515729d3a86460661cead5e7fb7e79851ed009539198.jpg)  
+(c) Constrained ReLU based activations  
+Figure 1: Approximations of the cubic function  $f(x) = x^{3}$
+
+In (Sill & Abu-Mostafa, 1996), the authors propose a modified loss function that penalizes the non-monotonicity of the model. The algorithm models the input distribution as a joint Gaussian estimated from the training data and samples random pairs of monotonic points that are added to the training data. In (Gupta et al., 2019), the authors propose a point-wise loss function that acts as a soft monotonicity constraint. These methods are straightforward to implement and can be used with any neural network architecture, but they do not guarantee the monotonicity of the trained model.
+
+Recently, there is an increasing number of proposed methods to certify or verify monotonicity obtained by regularization methods. In (Liu et al., 2020), the authors propose an optimization-based technique for mathematically verifying, or rejecting, the monotonicity of an arbitrary piece-wise linear (e.g., ReLU) neural network. The method consists of transforming the monotonicity verification problem into a mixed integer linear programming (MILP) problem that can be solved using an off-the-shelf MILP solver.
+
+In (Sivaraman et al., 2020), the authors propose an approach that finds counterexamples (defined as the pair of points where the monotonicity constraint is violated) by employing satisfiability modulo theories (SMT) solver (Barrett & Tinelli, 2018). To satisfy the monotonicity constraints, these counterexamples are included in the training data with adjustments to their target values to enforce the next iterations of the model to be monotonic.
+
+Both methods (Liu et al., 2020; Sivaraman et al., 2020) have been shown to support ReLU as the activation function only and there is no obvious way how to extend them to other activation functions. More precisely, they rely on piecewise linearity of ReLU to work, the property not satisfied by other variants such as ELU, SELU, GELU, etc. Last but not least, the procedure for certifying/verifying using MILP or SMT solvers is computationally very costly. These approaches also require multiple reruns or iterations to arrive at certified/verified monotonic networks.
+
+# 3. Constrained neural networks
+
+Most of the commonly used activation functions such as ReLU, ELU, SELU, etc. are monotonically increasing zero-centred, convex, lower-bounded non-polynomial functions. When used in a fully-connected, feed-forward neural network with at least one hidden layer and with unconstrained weights, they can approximate any continuous function on a compact subset. The simplest way to construct a monotonic neural network is to constrain its weights when used in conjunction with a monotone activation function. However, when the activation function is convex as well, the constrained neural network is not able to approximate nonconvex functions.
+
+To better illustrate this, and to propose a simple solution in this particular example, we refer the readers to Figure 1 where the goal is to approximate a simple cubic function  $x^3$  using a neural network with a single hidden layer with either 2 or 32 neurons and with ReLU activation. A cubic function is apt for our illustration since it is concave in the considered interval  $[-1,0]$  and convex in the interval  $[0,1]$ :
+
+1a. An unconstrained ReLU network with  $n$  neurons can approximate both concave and convex segments of the cubic function using at most  $n + 1$  piecewise linear segments. Increasing the number of neurons will provide a better fit with the function being approximated. Notice that even though the cubic function is monotone, there is no guarantee that the trained model will be monotone as well.  
+1b. If we constrain the weights of the network to be non-negative while still employing ReLU activation, the resulting model is monotone and convex. We can no longer approximate non-convex segments such as the cubic function on  $[-1,0]$  in the figure, and increasing the number of neurons from 2 to 32 does not yield any significant improvement in the approximation.  
+1c. Our proposed solution uses a combination of three ac
+
+![](images/b30aeed45803b85110b4a116473035ad913435c11400b65a6a8c758f6b69cc30.jpg)  
+(a) ReLU based activations
+
+![](images/1f8136b11f4daab61d8b0e667774b93d037c42383d406a5f8db2b2c3c1a5247e.jpg)  
+(b) ELU based activations
+
+![](images/1561e838253b7f1e1ffcdae2e4feab999897cd6301635b2a1f8bfbba56196eba.jpg)  
+(c) SELU based activations  
+Figure 2: Activation functions construction
+
+tivation functions in the hidden layer in order to gain the ability to model non-convex, monotone continuous functions. Notice that increasing the number of neurons increases the number of piecewise linear segments to approximate the cubic function. The resulting network is monotone by construction even when trained on noisy data.
+
+The schematic block diagram of our proposed solution (which we refer to as Constrained Monotone Fully Connected Layer or Monotonic Dense Unit interchangeably) is shown in the figure Fig. 3. The individual components of the proposed solution are defined and described in the subsequent subsection.
+
+# 3.1. Constrained monotone fully connected layer
+
+We say that a multivariate function  $f: \mathbb{R}^n \to \mathbb{R}$  is partially monotonically increasing with respect to  $x_i$  if
+
+$$
+x _ {i} ^ {0} > x _ {i} ^ {1} \Rightarrow f \left(x _ {1}, \dots , x _ {i} ^ {0}, \dots x _ {n}\right) \geq f \left(x _ {1}, \dots , x _ {i} ^ {1}, \dots x _ {n}\right).
+$$
+
+Similarly,  $f$  is partially monotonically decreasing with respect to  $x_{i}$  if
+
+$$
+x _ {i} ^ {0} > x _ {i} ^ {1} \Rightarrow f \left(x _ {1}, \dots , x _ {i} ^ {0}, \dots x _ {n}\right) \leq f \left(x _ {1}, \dots , x _ {i} ^ {1}, \dots x _ {n}\right).
+$$
+
+A set  $S \subseteq R$  is compact if every sequence in  $S$  has a subsequence that converges to a point in  $S$ . One can easily show that closed intervals  $[a, b]$  are compact, and compact sets can be thought of as generalizations of such closed bounded intervals.
+
+Our construction is preconditioned on a priori knowledge of (partial) monotonicity of a multivariate, multidimensional function  $f$ . Let  $f: K \mapsto \mathbb{R}^m$  be defined on a compact segment  $K \subseteq \mathbb{R}^n$ . Then we define its  $n$ -dimensional monotonicity indicator vector  $\mathbf{t} = [t_1, \dots, t_n]$  element-wise as
+
+follows:
+
+$$
+t _ {j} = \left\{ \begin{array}{l l} 1 & \text {i f} \frac {\partial f (\mathbf {x}) _ {i}}{\partial x _ {j}} \geq 0 \text {f o r e a c h} i \in \{1, \dots , m \} \\ - 1 & \text {i f} \frac {\partial f (\mathbf {x}) _ {i}}{\partial x _ {j}} \leq 0 \text {f o r e a c h} i \in \{1, \dots , m \} \\ 0 & \text {o t h e r w i s e} \end{array} \right. \tag {1}
+$$
+
+Given an  $(m\times n)$ -dimensional matrix  $\mathbf{M}$  and  $n$ -dimensional monotonicity indicator vector  $\mathbf{t}$ , we define the operation  $|\cdot|_t$  assigning an  $(m\times n)$ -dimensional matrix  $\mathbf{M}' = |\mathbf{M}|_t$  to  $\mathbf{M}$  element-wise as follows:
+
+$$
+m _ {j, i} ^ {\prime} = \left\{ \begin{array}{l l} \left| m _ {j, i} \right| & \text {i f} t _ {i} = 1 \\ - \left| m _ {j, i} \right| & \text {i f} t _ {i} = - 1 \\ m _ {j, i} & \text {o t h e r w i s e} \end{array} \right. \tag {2}
+$$
+
+Definition 1 (Constrained linear layer). Let  $\mathbf{W} \in \mathbb{R}^{n \times m}$ ,  $\mathbf{t} \in \{-1,0,1\}^n$ ,  $\mathbf{x} \in \mathbb{R}^n$  and  $\mathbf{b} \in \mathbb{R}^m$ . The output  $\mathbf{h} \in \mathbb{R}^m$  of the constrained linear layer with monotonicity indicator vector  $\mathbf{t}$ , weights  $\mathbf{W}$ , biases  $\mathbf{b}$  and input  $\mathbf{x}$  is:
+
+$$
+\mathbf {h} = \left| \mathbf {W} ^ {\mathbf {T}} \right| _ {\mathbf {t}} \cdot \mathbf {x} + \mathbf {b} \tag {3}
+$$
+
+Lemma 1. For each  $i\in \{1,\ldots ,n\}$  and  $j\in \{1,\dots ,m\}$  we have:
+
+- if  $t_i = 1$ , then  $\frac{\partial h_j}{\partial x_i} \geq 0$ , and  
+- if  $t_i = -1$ , then  $\frac{\partial h_j}{\partial x_i} \leq 0$ .
+
+We use  $\check{\mathcal{A}}$  to denote the set of all zero-centred, monotonically increasing, convex, lower-bounded functions.
+
+Definition 2. Let  $\breve{\rho} \in \check{\mathcal{A}}$ . Then
+
+$$
+\hat {\rho} (x) = - \check {\rho} (- x) \tag {4}
+$$
+
+$$
+\tilde {\rho} (x) = \left\{ \begin{array}{l l} \check {\rho} (x + 1) - \check {\rho} (1) & i f x <   0 \\ \hat {\rho} (x - 1) + \check {\rho} (1) & o t h e r w i s e \end{array} \right. \tag {5}
+$$
+
+![](images/86d08aeeeeb8c9fed7c24c5b95cb4ff5376ae40f083acb704cfea9efc51984c3.jpg)  
+Figure 3: Proposed Monotonic Dense Unit or Constrained Monotone Fully Connected Layer
+
+The main idea of our construction is use of three zero-centred, monotonically increasing activation functions, each applied to a part of neurons in a layer:
+
+- the original activation function  $\breve{\rho} \in \mathcal{A}$ ,  
+- concave upper-bounded function  $\check{\rho}$ , and  
+- bounded function  $\tilde{\rho}$ .
+
+Plots of such constructed activation functions for popular activation functions are given in Figure 2. Using only convex and concave activation functions is sufficient for approximating many functions such as the cubic function in Figure 1, but not for e.g. the sigmoid function. As we will show below, the saturated activation function is crucial for the universal property of our construction.
+
+Definition 3 (Combined activation function). Let  $\breve{\rho} \in \check{\mathcal{A}}$ ,  $\mathbf{h} \in \mathbb{R}^m$  and  $\mathbf{s} = (\check{s}, \hat{s}, \tilde{s}) \in \mathbb{N}^3$  such that  $\check{s} + \hat{s} + \tilde{s} = m$ . Then the output of the combined activation function  $\rho^s: \mathbb{R}^m \to \mathbb{R}^m$  is defined element-wise as follows:
+
+$$
+\rho^ {\mathbf {s}} (\mathbf {h}) _ {j} = \left\{ \begin{array}{l l} \check {\rho} (h _ {j}) & i f j \leq \check {s} \\ \hat {\rho} (h _ {j}) & i f \check {s} <   j \leq \check {s} + \hat {s} \\ \tilde {\rho} (h _ {j}) & o t h e r w i s e \end{array} \right. \tag {6}
+$$
+
+Lemma 2. Let  $\mathbf{y} = \rho^{\mathbf{s}}(\mathbf{h})$ . Then for each  $j \in \{1, \dots, m\}$  we have  $\frac{\partial y_j}{\partial h_j} \geq 0$ . Moreover
+
+- if  $\mathbf{s} = (m,0,0)$ , then  $\rho_{j}^{\mathbf{s}}$  is convex; and  
+- if  $s = (0, m, 0)$ , then  $\rho_j^s$  is concave.
+
+Definition 4 (Monotone constrained fully connected layer). Let  $n, m \in \mathbb{N}$ ,  $\check{\rho} \in \check{\mathcal{A}}$ ,  $\mathbf{t} \in \{-1,0,1\}^n$ ,  $\mathbf{s} = (\check{s},\hat{s},\tilde{s}) \in \mathbb{N}^3$  such that  $\check{s} + \hat{s} + \tilde{s} = m$ ,  $\mathbf{W} \in \mathbb{R}^{n \times m}$ ,  $\mathbf{x} \in \mathbb{R}^n$  and  $\mathbf{b} \in \mathbb{R}^m$ .
+
+Then the output  $\mathbf{y}$  of the monotone constrained fully connected layer with monotonicity indicator vector  $\mathbf{t}$ , weights  $\mathbf{W}$ , biases  $\mathbf{b}$  and input  $\mathbf{x}$  is
+
+$$
+\mathbf {y} = \rho^ {\mathbf {s}} \left(\left| \mathbf {W} ^ {\mathbf {T}} \right| _ {\mathbf {t}} \cdot \mathbf {x} + \mathbf {b}\right) \tag {7}
+$$
+
+From Lemma 1 and 2 directly follows:
+
+Corollary 3. For each  $i \in \{1, \dots, n\}$ ,  $j \in \{1, \dots, m\}$  we have:
+
+- if  $t_i = 1$ , then  $\frac{\partial y_j}{\partial x_i} \geq 0$  
+- if  $t_i = -1$ , then  $\frac{\partial y_j}{\partial x_i} \leq 0$  
+- if  $\mathbf{s} = (m,0,0)$ , then  $\mathbf{y}_j$  is convex; and  
+- if  $\mathbf{s} = (0, m, 0)$ , then  $\mathbf{y}_j$  is concave.
+
+On the layer level, we can control both monotonicity, convexity and concavity of the output with respect to chosen input variables. The following section discusses how we can use such layers to build practical neural networks with the same properties.
+
+# 3.2. Composing monotonic constrained dense layers
+
+As mentioned before, the main advantage of our proposed monotonic dense unit is its simplicity. We can build deep neural nets with different architectures by plugging in our monotonic dense blocks. Figures 4 and 5 show two examples of neural architectures that can be built using the proposed monotonic dense block.
+
+The first example shown in the Figure 4, corresponds to the standard MLP type of neural network architecture used in
+
+![](images/f6d6dffcfb8df04c4c54b7d35c6ac913787680d2aad91ec28717cfac7cc65ff5.jpg)  
+Figure 4: Neural architecture type 1
+
+general, where each of the input features is concatenated to form one single input feature vector  $\mathbf{x}$  and fed into the network, with the only difference being that instead of standard fully connected or dense layers, we employ monotonic dense units throughout. For the first (or input layer) layer, the indicator vector  $\mathbf{t}$ , is used to identify the monotonicity property of the input feature with respect to the output. Specifically,  $\mathbf{t}$  is set to 1 for those components in the input feature vector that are monotonically increasing and is set to  $-1$  for those components that are monotonically decreasing and set to 0 if the feature is non-monotonic. For the subsequent hidden layers, monotonic dense units with the indicator vector  $\mathbf{t}$  always being set to 1 are used in order to preserve monotonicity. Finally, depending on whether the problem at hand is a regression problem or a classification problem (or even a multi-task problem), an appropriate activation function (such as linear activation or sigmoid or softmax) to obtain the final output.
+
+Figure 5 shows another example of a neural network architecture that can be built employing proposed monotonic dense blocks. The difference when compared to the architecture described above lies in the way input features are fed into the hidden layers of neural network architecture. Instead of concatenating the features directly, this architecture provides flexibility to employ any form of complex feature extractors for the non-monotonic features and use the extracted feature vectors as inputs. Another difference is that each monotonic input is passed through separate monotonic dense units. This provides an advantage since depending on whether the input is completely concave or convex or both, we can adjust the activation selection vector s appropriately along with an appropriate value for the indicator vector t. Thus, each of the monotonic input features has a separate monotonic dense layer associated with it. Thus as the major difference to the above-mentioned architecture, we concatenate the feature vectors instead of concatenating the inputs directly. The subsequent parts of the network are similar to the architecture described above wherein for the rest of the hidden monotonic dense units, the indicator vector t is
+
+always set to 1 to preserve monotonicity.
+
+# 3.3. Universal approximation
+
+The classical Universal Approximation Theorem (Cybenko, 1989; Hornik, 1991; Pinkus, 1999) states that any continuous function on a closed interval can be approximated with a feed-forward neural network with one hidden layer if and only if its activation function is nonpolynomial. In (Kidger & Lyons, 2020), authors prove the approximation property holds for arbitrary deep neural networks with bounded number of neurons in each layer holds if the activation function is nonaffine and differential at at least one point.
+
+In (Daniels & Velikova, 2010), authors show the universal approximation property for constrained multivariate neural networks using sigmoid as the activation functions: any multivariate continuous monotone function on a compact subset of  $\mathbb{R}^k$  can be approximated with a constrained neural network with the sigmoid activation function of at most  $k$  layers (Theorem 3.1), reproduced here for completeness:
+
+Theorem 4. For any continuous monotone nondecreasing function  $f: K \to \mathbb{R}$ , where  $K$  is a compact subset of  $\mathbb{R}^k$ , there exists a feedforward neural network using the sigmoid as the activation function with at most  $k$  hidden layers, positive weights, and output  $O$  such that  $|O_x - f(x)| < \epsilon$  for any  $x \in K$  and  $\epsilon > 0$ .
+
+However, the proof of the theorem uses only the fact that the Heaviside function  $\mathbf{H}$  defined as
+
+$$
+H (x) = \left\{ \begin{array}{l l} 1 & \text {i f} x \geq 0 \\ 0 & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+can be approximated with the sigmoid function on a closed interval (since  $\lim_{a\to \infty}\sigma (ax) = H(x)$ ).
+
+By construction, we have:
+
+Lemma 5. Let  $\breve{\rho} \in \check{\mathcal{A}}$ . Then the Heavyside function can be approximated with  $\tilde{\rho}_H$  on  $\mathbb{R}$ , where
+
+$$
+\tilde {\rho} _ {H} (x) = \alpha \tilde {\rho} (x) + \beta
+$$
+
+![](images/740e12159802be730488d2acdc8ab29402a90c906806e89f9e7a7c477f93112c.jpg)  
+Figure 5: Neural architecture type 2
+
+for some  $\alpha, \beta \in \mathbb{R}$  and  $\alpha > 0$ .
+
+Lemma 6. Let  $\tilde{\rho}_{\alpha,\beta}$  be an activation function for some  $\alpha, \beta \in \mathbb{R}, \alpha > 0$  such that for every  $x \in \mathbb{R}$
+
+$$
+\tilde {\rho} _ {\alpha , \beta} (x) = \alpha \tilde {\rho} (x) + \beta .
+$$
+
+Then for every constrained monotone neural network  $\mathcal{N}_{\alpha,\beta}$  using  $\tilde{\rho}_{\alpha,\beta}$  as an activation function ( $s = (0,0,\tilde{s})$ ), there is a constrained monotone neural network  $\mathcal{N}$  using  $\tilde{\rho}$  as an activation function such that for every  $\mathbf{x} \in \mathbb{R}^n$ :
+
+$$
+\mathcal {N} (\mathbf {x}) = \mathcal {N} _ {\alpha , \beta} (\mathbf {x}).
+$$
+
+Finally, from Theorem 4, Lemma 5 and Lemma 6, we have the universal approximation property:
+
+Theorem 7. Let  $\breve{\rho} \in \check{\mathcal{A}}$ . Then any multivariate continuous monotone function on a compact subset of  $\mathbb{R}^k$  can be approximated with a monotone constrained neural network of at most  $k$  layers using  $\rho$  as the activation function.
+
+The Theorem 7 gives us the upper bound on the number of layers needed for approximating an arbitrary function. In practice, the number and width of layers for a given function and given dataset are found by hyperparameter search. The best results in our experiments were achieved by neural networks with a significantly smaller number of layers. The proof shows that the saturated activation functions are sufficient for the universal approximation property. However, experimental results show that the networks with predominately unsaturated activation functions are easier to train and achieve better results.
+
+# 4. Experiments
+
+In order to analyze the practical utility of the proposed method, we experiment with various datasets and compare them with the recent state-of-the-art. For the first set of
+
+experiments, we use the datasets employed by authors in (Liu et al., 2020) and use the exact train and test split for proper comparison. We perform experiments on 3 datasets: COMPAS (J. Angwin & Kirchner, 2016), which is a classification dataset with 13 features of which 4 are monotonic; Blog Feedback Regression (Buza, 2014), which is a regression dataset with 276 features of which 8 are monotonic; Loan Defaulter<sup>1</sup>, which is a classification dataset with 28 features of which 5 are monotonic. The dataset contains half a million data points. For comparison with other methods, we compare with Certified monotonic networks (Certified) (Liu et al., 2020) and other methods described in it.
+
+For the second set of experiments, we use 2 datasets: Auto MPG (which is a regression dataset with 3 monotonic features) and Heart Disease (which is a classification dataset with 2 monotonic features) as employed in the work (Sivaraman et al., 2020) and once again use the exact train and test split for proper comparison. We compare with the method COMET described in (Sivaraman et al., 2020) along with Min-Max Net (Daniels & Velikova, 2010) and Deep Lattice Network (DLN) (You et al., 2017) as described in (Sivaraman et al., 2020), and also more details regarding the datasets have been provided in the supplementary.
+
+We use cross-entropy for the classification tasks and we use mean-squared-error for the regression tasks as loss functions. We employ Bayesian optimization tuning with Gaussian process (Snoek et al., 2012) to find the optimal hyperparameters such as the number of neurons, network depth or layers, initial learning rate etc.
+
+The code for experiments was written in the Keras framework (Chollet et al., 2015) and KerasTuner (O'Malley et al., 2019) via integration from the Tensorflow framework, version 2.11 (Abadi et al., 2015). All experiments were performed using a Google Colaboratory instance with NVidia
+
+<table><tr><td rowspan="2">Method</td><td>COMPAS</td><td>Blog Feedback</td><td>Loan Defaulter</td></tr><tr><td>Parameters Test Acc ↑</td><td>Parameters RMSE ↓</td><td>Parameters Test Acc ↑</td></tr><tr><td>Isotonic</td><td>N.A. 67.6%</td><td>N.A. 0.203</td><td>N.A. 62.1%</td></tr><tr><td>XGBoost (Chen &amp; Guestrin, 2016)</td><td>N.A. 68.5% ± 0.1%</td><td>N.A. 0.176 ± 0.005</td><td>N.A. 63.7% ± 0.1%</td></tr><tr><td>Crystal (Milani Fard et al., 2016)</td><td>25840 66.3% ± 0.1%</td><td>15840 0.164 ± 0.002</td><td>16940 65.0% ± 0.1%</td></tr><tr><td>DLN (You et al., 2017)</td><td>31403 67.9% ± 0.3%</td><td>27903 0.161 ± 0.001</td><td>29949 65.1% ± 0.2%</td></tr><tr><td>Min-Max Net (Daniels &amp; Velikova, 2010)</td><td>42000 67.8% ± 0.1%</td><td>27700 0.163 ± 0.001</td><td>29000 64.9% ± 0.1%</td></tr><tr><td>Non-Neg-DNN</td><td>23112 67.3% ± 0.9%</td><td>8492 0.168 ± 0.001</td><td>8502 65.1% ± 0.1%</td></tr><tr><td>Certified (Liu et al., 2020)</td><td>23112 68.8% ± 0.2%</td><td>8492 0.158 ± 0.001</td><td>8502 65.2% ± 0.1%</td></tr><tr><td>Ours</td><td>2317 69.2% ± 0.2%</td><td>1101 0.156 ± 0.001</td><td>177 65.3% ± 0.01%</td></tr></table>
+
+Table 1: Comparison of our method with other methods described in (Liu et al., 2020)  
+
+<table><tr><td rowspan="2">Method</td><td>Auto MPG</td><td>Heart Disease</td></tr><tr><td>MSE ↓</td><td>Test Acc ↑</td></tr><tr><td>Min-Max Net (Daniels &amp; Velikova, 2010)</td><td>10.14 ± 1.54</td><td>0.75 ± 0.04</td></tr><tr><td>DLN (You et al., 2017)</td><td>13.34 ± 2.42</td><td>0.86 ± 0.02</td></tr><tr><td>COMET (Sivaraman et al., 2020)</td><td>8.81 ± 1.81</td><td>0.86 ± 0.03</td></tr><tr><td>Ours</td><td>8.37 ± 0.08</td><td>0.89 ± 0.00</td></tr></table>
+
+Table 2: Comparison of our method with other methods described in (Sivaraman et al., 2020)
+
+Tesla T4 GPU (Bisong, 2019). The code is publicly available at (Runje & Shankaranarayana, 2023a), while the preprocessed datasets for experiments are available at (Runje & Shankaranarayana, 2023b).
+
+# 4.1. Results
+
+The results for the datasets above are summarized in Tables 1 and 2. It shows that our method outperforms other methods in terms of test accuracy for classification tasks and mean squared error and root mean squared error (MSE and RMSE) for regression tasks. For each of the datasets, we run the experiments ten times after finding the optimal hyperparameters and report the mean and standard deviation of the best five results. Experiment results show that networks learned by our method can achieve better results with fewer parameters, than the best-known algorithms for monotonic neural networks, such as Min-Max Network (Daniels & Velikova, 2010) and Deep Lattice Network (You et al., 2017). It should be noted that the recent state-of-the-art works- Certified (Liu et al., 2020) and COMET (Sivaraman et al., 2020) require multiple runs in order to even satisfy monotonic constraints whereas monotonicity is guaranteed by simply employing the proposed monotonic dense units.
+
+The most important advantage of our solution is simplicity and computational complexity. Our models have slightly better performance on all datasets we tested them on, but it is important to note they have significantly fewer parameters and the simplest training procedure. As such, they have the
+
+potential to significantly reduce the carbon footprint when used at scale:
+
+- The number of parameters of our model is an order or even two orders of magnitude smaller than alternatives. At prediction time, this translates roughly into 1-2 orders of magnitude less computation (most parameters are used in multiplications)  
+- At training time, we do not use any additional computationally expensive procedures apart from gradient descent, unlike alternative approaches such as COMET (Sivaraman et al., 2020) and Certified (Liu et al., 2020).
+
+# 5. Conclusion
+
+In this paper, we proposed a simple and elegant solution to build constrained monotonic networks which can approximate any continuous partially monotonic function. Specifically, we introduced a constrained monotone fully connected layer which can be used as a drop-in replacement for a fully connected layer to enforce monotonicity. We then employed our constrained monotone fully connected layer to build neural network models and showed that we can achieve better results to the recent state-of-the-art (Sivaraman et al., 2020; Liu et al., 2020) in addition to the well-known works such as Min-Max networks (Daniels & Velikova, 2010) and DLNs (You et al., 2017). However, the main advantage of the proposed solution is not higher accuracy but its computational and memory complexity: we use orders of magnitude
+
+fewer parameters and computation which makes the resulting neural networks more energy efficient. Last but not least, we proved such networks can approximate any multivariate monotonic function.
+
+Extending these results to other types of neural network layers is straightforward, but we leave it for future work. E.g. convolutional layer can be made monotonic with respect to a subset of features by replacing its fully-connected filters with monotone constrained fully connected layers. As long as such layers are combined with other monotonic convolutional and fully connected layers and monotone activation functions, the resulting neural network would be monotone.
+
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+
+K. (eds.), Advances in Neural Information Processing Systems, volume 25. Curran Associates, Inc., 2012. URL https://proceedings.neurips.cc/paper_files/paper/2012/file/05311655a15b75fab86956663e1819cd-Paper.pdf.  
+Veličković, P. The resurgence of structure in deep neural networks. PhD thesis, University of Cambridge, 2019.  
+You, S., Ding, D., Canini, K., Pfeifer, J., and Gupta, M. Deep lattice networks and partial monotonic functions. Advances in neural information processing systems, 30, 2017.  
+Zagoruyko, S. and Komodakis, N. Diracnets: Training very deep neural networks without skip-connections. CoRR, abs/1706.00388, 2017. URL https://arxiv.org/abs/1706.00388.  
+Zheng, H., Yang, Z., Liu, W., Liang, J., and Li, Y. Improving deep neural networks using softplus units. In 2015 International Joint Conference on Neural Networks (IJCNN), pp. 1-4. IEEE, 2015.
+
+# A. Detailed proofs
+
+We restate all lemmas from the main text here are give detailed proofs of them.
+
+The following is well known result, proved here for completeness only:
+
+Lemma 1. For each  $i\in \{1,\dots ,n\}$  and  $j\in \{1,\ldots ,m\}$  we have:
+
+- if  $t_i = 1$ , then  $\frac{\partial h_j}{\partial x_i} \geq 0$ , and  
+- if  $t_i = -1$ , then  $\frac{\partial h_j}{\partial x_i} \leq 0$ .
+
+Proof. From equation 3, we have
+
+$$
+\mathbf {h} = \left| \mathbf {W} ^ {T} \right| _ {\mathbf {t}} \cdot \mathbf {x} + \mathbf {b} \tag {8}
+$$
+
+$$
+h _ {j} = \sum_ {i} w _ {i, j} ^ {\prime} x _ {i} + b _ {j} \tag {9}
+$$
+
+$$
+\frac {\partial h _ {j}}{\partial x _ {i}} = w _ {i, j} ^ {\prime} \tag {10}
+$$
+
+Finally, from equation 2 we have
+
+$$
+\frac {\partial h _ {j}}{\partial x _ {i}} = \left\{ \begin{array}{l l} | w _ {i, j} | \geq 0 & \text {i f} t _ {i} = 1 \\ - | w _ {i, j} | \leq 0 & \text {i f} t _ {i} = - 1 \end{array} \right.
+$$
+
+Lemma 2. Let  $\mathbf{y} = \rho^{\mathbf{s}}(\mathbf{h})$ . Then for each  $j \in \{1, \ldots, m\}$  we have  $\frac{\partial y_j}{\partial h_j} \geq 0$ . Moreover
+
+- if  $\mathbf{s} = (m,0,0)$ , then  $\rho_{j}^{\mathbf{s}}$  is convex; and  
+- if  $\mathbf{s} = (0, m, 0)$ , then  $\rho_j^{\mathbf{s}}$  is concave.
+
+Proof. From equation 4,5 and 6:
+
+$$
+\hat {\rho} (x) = - \check {\rho} (- x)
+$$
+
+$$
+\tilde {\rho} (x) = \left\{ \begin{array}{l l} \breve {\rho} (x + 1) - \breve {\rho} (1) & \text {i f} x <   0 \\ \hat {\rho} (x - 1) - \hat {\rho} (1) & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+$$
+\rho^ {\mathbf {s}} (h _ {j}) = \left\{ \begin{array}{l l} \check {\rho} (h _ {j}) & \text {i f} j \leq \check {s} \\ \hat {\rho} (h _ {j}) & \text {i f} \check {s} <   j \leq \hat {s} \\ \tilde {\rho} (h _ {j}) & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+we have:
+
+$$
+\frac {\partial y _ {j}}{\partial h _ {j}} = \left\{ \begin{array}{l l} \check {\rho} ^ {\prime} (h _ {j}) \geq 0 & \text {i f} j \leq \check {s} \\ \check {\rho} ^ {\prime} (- h _ {j}) \geq 0 & \text {i f} \check {s} <   j \leq \hat {s} \\ \check {\rho} ^ {\prime} (h _ {j} + 1) \geq 0 & \text {i f} \check {s} + \hat {s} <   j \text {a n d} h _ {j} <   0 \\ \check {\rho} ^ {\prime} (1 - h _ {j}) \geq 0 & \text {i f} \check {s} + \hat {s} <   j \text {a n d} h _ {j} \geq 0 \end{array} \right.
+$$
+
+if  $\mathbf{s} = (m,0,0)$ , we have:
+
+$$
+\rho^ {\mathbf {s}} \left(h _ {j}\right) = \check {\rho} \left(h _ {j}\right)
+$$
+
+which is a convex function.
+
+Similarly, if  $\mathbf{s} = (0,m,0)$ , we have:
+
+$$
+\rho^ {\mathbf {s}} (h _ {j}) = \hat {\rho} (h _ {j})
+$$
+
+which is a concave function.
+
+Corollary 3. For each  $i\in \{1,\ldots ,n\}$ $j\in \{1,\dots ,m\}$  we have:
+
+- if  $t_i = 1$ , then  $\frac{\partial y_j}{\partial x_i} \geq 0$  
+if  $t_i = -1$ , then  $\frac{\partial y_j}{\partial x_i} \leq 0$  
+- if  $\mathbf{s} = (m,0,0)$ , then  $\mathbf{y}_j$  is convex; and  
+- if  $\mathbf{s} = (0, m, 0)$ , then  $\mathbf{y}_j$  is concave.
+
+For completeness, we repeat the Theorem 3.1 from (Daniels & Velikova, 2010) and its proof here:
+
+Theorem 4. For any continuous monotone nondecreasing function  $f: K \to \mathbb{R}$ , where  $K$  is a compact subset of  $\mathbb{R}^k$ , there exists a feedforward neural network using the sigmoid as the activation function with at most  $k$  hidden layers, positive weights, and output  $O$  such that  $|O_x - f(x)| < \epsilon$ , for any  $x \in K$  and  $\epsilon > 0$ .
+
+Proof. The proof is derived by induction on the number of input variables  $k$ . Without loss of generality, we may assume that  $f > 0$  (otherwise, we add a constant  $C$  and approximate  $f + C$  with the network output  $O$ , then modify  $O$  with a negative bias at the output node). First, we assume that  $f$  is strictly increasing and  $C^\infty$ . In case of  $k = 1$ , we write
+
+$$
+f (x) = \int_ {0} ^ {\infty} \mathbf {H} (f (x) - u) d u \tag {11}
+$$
+
+where  $\mathbf{H}$  is the Heaviside function:
+
+$$
+H (x) = \left\{ \begin{array}{l l} 1 & \text {i f} x \geq 0 \\ 0 & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+Since  $f$  is continuous and increasing, it is invertible and therefore the right-hand side of 11 can be written as
+
+$$
+f (x) = \int_ {0} ^ {\infty} \mathbf {H} \left(x - f ^ {- 1} (v)\right) d v \tag {12}
+$$
+
+The integral can be approximated arbitrarily well by a Riemann sum
+
+$$
+\sum_ {i = 1} ^ {N} \left(v _ {i + 1} - v i\right) \mathbf {H} \left(x - f ^ {- 1} \left(v _ {i}\right)\right) \tag {13}
+$$
+
+where  $[v_i]_{i=1}^N$  is a partition of the interval  $[f(a), f(b)]$ . This expression corresponds to a neural network with input  $x$ , one hidden layer with  $N$  neurons all connected to the input with weight of 1, bias term in the hidden neurons  $f^{-1}(v_i)$ , and the weights connecting the hidden layer with the output  $v_{i+1} - v_i > 0$ . Note that the Heaviside function  $\mathbf{H}$  can be replaced by a sigmoid activation function using a standard approximation argument.
+
+Assume that Theorem 3.1 holds for  $k - 1$  input variables. We now combine the integral representation in 11 with the induction assumption. For a given  $v$ , we may solve the equation of the level set corresponding to  $v$  for  $x_{k}$
+
+$$
+f (x _ {1}, \dots , x _ {k}) = v.
+$$
+
+By the implicit function theorem, there exists a function  $g_v$  such that
+
+$$
+f \left(x _ {1}, \dots , g _ {v} \left(x _ {1}, \dots , x _ {k - 1}\right)\right) = v. \tag {14}
+$$
+
+Note that  $g_{v}$  is decreasing in all arguments  $x_{i}$ . This can be seen by taking the partial derivative of 14 with respect to  $x_{i}$ . We will now show that
+
+$$
+\mathbf {H} \left(f (x) - v\right) = \mathbf {H} \left(x _ {k} - g _ {v} \left(x _ {1}, \dots , x _ {k - 1}\right)\right)
+$$
+
+analogously to 12 for the 1-D case. Note that it is sufficient to show that
+
+$$
+f (x) <   v \text {i f a n d o n l y i f} x _ {k} <   g _ {v} \left(x _ {1}, \dots , x _ {k - 1}\right)
+$$
+
+and
+
+$$
+f (x) > v \text {i f a n d o n l y i f} x _ {k} > g _ {v} \left(x _ {1}, \dots , x _ {k - 1}\right).
+$$
+
+But this follows easily from 14 and the fact that  $f$  is increasing in all its arguments. We now approximate the integral in 11 with a Riemann sum, leading to the following equation analogously to 13:
+
+$$
+R = \sum_ {i = 1} ^ {N} \left(v _ {i - 1} - v _ {i}\right) \mathbf {H} \left(x _ {k} - g _ {v _ {i}} \left(x _ {1}, \dots , x _ {k - 1}\right)\right) \tag {15}
+$$
+
+Since  $g_{v_i}$  is decreasing in all its arguments  $-g_{v_i}$  is increasing. By the induction assumption, we can approximate  $-g_{v_i}$  with a feedforward neural network  $O_i$  with  $x_1, \ldots, x_{k-1}$  as inputs,  $k-1$  hidden layers, and nonnegative weights, such that
+
+$$
+\left| \sum_ {i = 1} ^ {N} \left(v _ {i - 1} - v _ {i}\right) \mathbf {H} \left(x _ {k} - O _ {i} \left(x _ {1}, \dots , x _ {k - 1}\right)\right) - R \right| <   \epsilon
+$$
+
+because the sum is finite. Expression 15 corresponds to a feedforward neural network with  $k$  inputs and  $k$  hidden layers. Here  $k - 1$  hidden layers are needed to represent  $-g_{v_i}$  and the  $k$ -th hidden layer is needed to combine  $N$  neural networks with outputs  $O_i$  and the input  $x_k$ . The weights on the connections between the last hidden layer and the final output are  $(v_{i + 1} - v_i) > 0$ . The input  $x_k$  is directly (skip-layer) connected to the  $k$ -th hidden layer.
+
+We can now easily generalize the proof to continuous nondecreasing functions. For continuous functions, we define the convolution of  $f$  with a mollifier  $K_{\delta}$  by
+
+$$
+f _ {\delta} = f \otimes K _ {\delta}
+$$
+
+Then,  $f_{\delta}$  is  $C^\infty$  and  $f_{\delta} \longrightarrow f$  as  $\delta \downarrow 0$  uniform on compact subsets. Furthermore,  $f_{\delta}$  is also increasing since  $K_{\delta} > 0$ . Now choose  $\delta$  such that  $|f - f_{\delta}| < \frac{\epsilon}{2}$  and approximate  $f_{\delta}$  with a feedforward neural network  $O$  such that  $|f_{\delta} - O| < \frac{\epsilon}{2}$ . Then,  $|f - O| < \epsilon$ .
+
+If  $f$  is nondecreasing, then approximate  $f$  by  $f_{\delta}$
+
+$$
+f _ {\delta} = f + \delta \left(x _ {1} + \dots + x _ {k}\right)
+$$
+
+which is strictly increasing and let  $\delta \downarrow 0$
+
+Lemma 5. Let  $\breve{\rho} \in \check{\mathcal{A}}$ . Then the Heavyside function can be approximated with  $\tilde{\rho}_H$  on  $\mathbb{R}$ , where
+
+$$
+\tilde {\rho} _ {H} (x) = \alpha \tilde {\rho} (x) + \beta
+$$
+
+for some  $\alpha, \beta \in \mathbb{R}$  and  $\alpha > 0$ .
+
+Proof. Since  $\check{\rho}$  is bounded from below, then it has a limit
+
+$$
+c = \lim  _ {x \rightarrow - \infty} \check {\rho} (x) = - \lim  _ {x \rightarrow \infty} \hat {\rho} (x) <   0
+$$
+
+From equation 5, we have
+
+$$
+\lim  _ {x \rightarrow - \infty} \tilde {\rho} (x) = \lim  _ {x \rightarrow - \infty} \breve {\rho} (x) - \breve {\rho} (1) = c - \breve {\rho} (1)
+$$
+
+$$
+\lim  _ {x \rightarrow + \infty} \tilde {\rho} (x) = \lim  _ {x \rightarrow \infty} \hat {\rho} (x) + \breve {\rho} (1) = - (c - \breve {\rho} (1))
+$$
+
+Let  $\tilde{\rho}_H$  be defined as follows:
+
+$$
+\tilde {\rho} _ {H} (x) = \frac {\tilde {\rho} (x) - c + \breve {\rho} (1)}{2 (- c + \breve {\rho} (1))}
+$$
+
+Then
+
+$$
+\lim  _ {a \to \infty} \tilde {\rho} _ {H} (a \cdot x) = H (x)
+$$
+
+![](images/dd6a8ff7408d132809c671b23c138c8562fd88d74c6e6573c54f47147ec6eedc.jpg)
+
+Lemma 6. Let  $\tilde{\rho}_{\alpha,\beta}$  be an activation function for some  $\alpha, \beta \in \mathbb{R}, \alpha > 0$  such that for every  $x \in \mathbb{R}$
+
+$$
+\tilde {\rho} _ {\alpha , \beta} (x) = \alpha \tilde {\rho} (x) + \beta .
+$$
+
+Then for every constrained monotone neural network  $\mathcal{N}_{\alpha,\beta}$  using  $\tilde{\rho}_{\alpha,\beta}$  as an activation function ( $s = (0,0,\tilde{s})$ ), there is a constrained monotone neural network  $\mathcal{N}$  using  $\tilde{\rho}$  as an activation function such that for every  $\mathbf{x} \in \mathbb{R}^n$ :
+
+$$
+\mathcal {N} (\mathbf {x}) = \mathcal {N} _ {\alpha , \beta} (\mathbf {x}).
+$$
+
+Proof. Let  $\mathbf{h}_k$  be the output of the  $k$ -th constrained linear layer of  $\mathcal{N}_{\alpha,\beta}$  evaluated on input  $\mathbf{x}$ . For every  $k > 1$ , we have
+
+$$
+\mathbf {h} _ {1} = | \mathbf {W} _ {1} | _ {\mathbf {t} _ {1}} \cdot \mathbf {x} + \mathbf {b} _ {1}
+$$
+
+$$
+\mathbf {h} _ {k} = \left| \mathbf {W} _ {k} \right| _ {\mathbf {t} _ {k}} \cdot \mathbf {y} _ {k - 1} + \mathbf {b} _ {k}
+$$
+
+$$
+\mathbf {y} _ {k} = \tilde {\rho} _ {\alpha , \beta} (\mathbf {h} _ {k})
+$$
+
+$$
+\mathbf {y} = \mathbf {h} _ {l}
+$$
+
+where  $l$  is the total number of layers of  $\mathcal{N}_{\alpha, \beta}$ . We can rewrite  $\mathbf{h}_k$  as follows:
+
+$$
+\begin{array}{l} \mathbf {h} _ {k} = \left| \mathbf {W} _ {k} \right| _ {\mathbf {t} _ {k}} \cdot \mathbf {y} _ {k - 1} + \mathbf {b} _ {k} \\ = \left| \mathbf {W} _ {k} \right| _ {\mathbf {t} _ {k}} \cdot \tilde {\rho} _ {\alpha , \beta} \left(\mathbf {h} _ {k - 1}\right) + \mathbf {b} _ {k} \\ = \left| \mathbf {W} _ {k} \right| _ {\mathbf {t} _ {k}} \cdot \left(\alpha \tilde {\rho} \left(\mathbf {h} _ {k - 1}\right) + \beta \mathbf {1}\right) + \mathbf {b} _ {k} \quad (\text {w i t h} | \mathbf {1} | = | \mathbf {h} _ {k - 1} |) \\ = \alpha | \mathbf {W} _ {k} | _ {\mathbf {t} _ {k}} \cdot \tilde {\rho} (\mathbf {h} _ {k - 1}) + \beta | \mathbf {W} _ {k} | _ {\mathbf {t} _ {k}} \mathbf {1} + \mathbf {b} _ {k} \\ = | \alpha \mathbf {W} _ {k} | _ {\mathbf {t} _ {k}} \cdot \tilde {\rho} (\mathbf {h} _ {k - 1}) + \beta | \mathbf {W} _ {k} | _ {t _ {k}} \mathbf {1} + \mathbf {b} _ {k} \quad \left(\text {f r o m} \alpha > 0\right) \\ = \left| \mathbf {W} _ {k} ^ {\prime} \right| _ {t _ {k}} \cdot \tilde {\rho} \left(\mathbf {h} _ {k - 1}\right) + \mathbf {b} _ {k} ^ {\prime} \\ \end{array}
+$$
+
+for  $\mathbf{W}_k^{\prime} = \alpha \mathbf{W}_k$  and  $\mathbf{b}_k^\prime = \beta |\mathbf{W}_k|_{t_k}\mathbf{1} + \mathbf{b}_k$
+
+Hence, for every  $\mathbf{x} \in \mathbb{R}^n$ , the output of the neural network  $\mathcal{N}$  with weights  $\mathbf{W}_1, \mathbf{W}_2', \ldots, \mathbf{W}_l'$  and biases  $\mathbf{b}_1, \mathbf{b}_2', \ldots, \mathbf{b}_l'$  is:
+
+$$
+\mathcal {N} (\mathbf {x}) = \mathcal {N} _ {\alpha , \beta} (\mathbf {x})
+$$
+
+Theorem 7. Let  $\breve{\rho} \in \check{\mathcal{A}}$ . Then any multivariate continuous monotone function  $f$  on a compact subset of  $\mathbb{R}^k$  can be approximated with a monotone constrained neural network of at most  $k$  layers using  $\rho$  as the activation function.
+
+Proof. The proof of the Theorem 4 uses only the fact that the Heaviside function  $\mathbf{H}$  defined as
+
+$$
+H (x) = \left\{ \begin{array}{l l} 1 & \text {i f} x \geq 0 \\ 0 & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+can be approximated with the sigmoid function on a closed interval (since  $\lim_{a\to \infty}\sigma (ax) = H(x)$ ).
+
+From the Theorem 4 and the Lemma 5, any continuous monotone function  $f$  on a compact subset of  $\mathbb{R}^k$  can be approximated with a monotone constrained neural network with at most  $k$  layers using  $\tilde{\rho}_H$  as the activation function.
+
+From Lemma 6, we can replace  $\tilde{\rho}_H$  with  $\tilde{\rho}$ . Hence, any continuous monotone function  $f$  on a compact subset of  $\mathbb{R}^k$  can be approximated with a monotone constrained neural network with at most  $k$  layers using  $\rho$  as the activation function and  $\mathbf{s}_i = (0,0,\tilde{s}_i)$ .
+
+![](images/9e3be79e1a3222622a5c6adfe5e5f6623ec263f33c89725ce9aa5531573e68c2.jpg)
+
+# B. Datasets Description
+
+The descriptions of datasets used for comparison are detailed below. As mentioned in the section 4, the datasets are chosen from (Liu et al., 2020) and (Sivaraman et al., 2020) for proper evaluation. The train-test splits of  $80\% - 20\%$  are used for all comparison experiments.
+
+1. COMPAS (J. Angwin & Kirchner, 2016) is a binary classification dataset, where the task is to predict risk score of an individual committing crime again two years, based on the criminal records of individuals arrested in Florida. The risk score needs to be monotonically increasing with respect to the following attributes: number of prior adult convictions, number of juvenile felony, number of juvenile misdemeanor, and number of other convictions. It should be noted that there have been ethical concerns with the dataset (J. Angwin & Kirchner, 2016; Rudin et al., 2020)  
+2. Blog Feedback (Buza, 2014) is a regression dataset where the task is to predict the number of comments in the upcoming 24 hours from a feature set containing 276 features of which 8 (A51, A52, A53, A54, A56, A57, A58, A59) are monotonic features. The readers are suggested to refer to link  ${}^{2}$  for more details. As mentioned by the authors of (Liu et al., 2020), only the data points with targets smaller than the 90th percentile are used since the outliers could dominate the mean-squared-error metric.  
+3. Lending club loan data<sup>3</sup> is a classification dataset, where the task is to predict whether the individual would default on loan, from a feature set having 28 features containing data such as the current loan status, latest payment information etc.. The probability of default should be non-decreasing with respect to number of public record bankruptcies, Debt-to-Income ratio, and non-increasing with respect to credit score, length of employment, annual income.  
+4. Auto  $\mathbf{MPG}^4$  (Quinlan, 1993) is a regression dataset where the task is to predict city-cycle fuel consumption in miles per gallon (MPG) from a feature set containing 7 features of which the monotonic features are weight (W), displacement (D), and horse-power (HP)  
+5. Heart Disease $^{5}$  (Gennari et al., 1989) is a classification dataset, where the task is to predict the presence of heart disease from a feature set containing 13 features of which the risk associated with heart disease needs to be monotonically increasing with respect to the features trestbps (T), cholestrol (C))

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+# Context Consistency Regularization for Label Sparsity in Time Series
+
+Yooju Shin $^{1}$  Susik Yoon $^{2}$  Hwanjun Song $^{3}$  Dongmin Park $^{1}$  Byunghyun Kim $^{4}$  Jae-Gil Lee $^{4}$  Byung Suk Lee $^{5}$
+
+# Abstract
+
+Labels are typically sparse in real-world time series due to the high annotation cost. Recently, consistency regularization techniques have been used to generate artificial labels from unlabeled augmented instances. To fully exploit the sequential characteristic of time series in consistency regularization, we propose a novel method of data augmentation called context-attached augmentation, which adds preceding and succeeding instances to a target instance to form its augmented instance. Unlike the existing augmentation techniques that modify a target instance by directly perturbing its attributes, the context-attached augmentation generates instances augmented with varying contexts while maintaining the target instance. Based on our augmentation method, we propose a context consistency regularization framework, which first adds different contexts to a target instance sampled from a given time series and then shares unitary reliability-based cross-window labels across the augmented instances to maintain consistency. We demonstrate that the proposed framework outperforms the existing state-of-the-art consistency regularization frameworks through comprehensive experiments on real-world time-series datasets.
+
+# 1 Introduction
+
+A time series is a collection of consecutive data points, often annotated with temporally coherent timestamp labels. As the boundaries of coherent labels are unknown in practice, a classifier is trained to predict the label of each timestamp (Farha & Gall, 2019). However, due to the length of and complexity in a time series, labeling every times-
+
+<sup>1</sup>Graduate School of Data Science, KAIST, Korea <sup>2</sup>Department of Computer Science, University of Illinois at Urbana-Champaign, USA <sup>3</sup>AWS AI Labs, USA <sup>4</sup>School of Computing, KAIST, Korea <sup>5</sup>Department of Computer Science, University of Vermont, USA. Correspondence to: Jae-Gil Lee <jaegil@kaist.ac.kr>.
+
+Proceedings of the  $40^{th}$  International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
+
+tamp in the time series requires prohibitively high cost, and therefore, in reality, a lot of time series are only sparsely labeled (Moltisanti et al., 2019; Ma et al., 2020; Yoon et al., 2021; Shin et al., 2022). In this regard, consistency regularization is used as a promising way to train a model from sparse labels, by leveraging the model's output to infer new labels for unlabeled data points (Laine & Aila, 2017; Rizve et al., 2021). Consistency regularization enforces the consistency among the outputs of a model for a pair of augmented data instances, and thus data augmentation plays a key role in consistency regularization. Recent state-of-the-art consistency regularization methods, mostly developed for image data, necessitate domain-specific data augmentation such as color transformation and image rotation (Cubuk et al., 2020; Sohn et al., 2020; Zhang et al., 2021; Kim & Lee, 2022).
+
+Such conventional data augmentation generates multiple different instances from a target instance (i.e., an instance for pseudo-labeling) by way of data perturbation. If data instances are independent of one another as in image data, there is no other way than to perturb the target instance itself. In contrast, using the sequential nature of time series, where data instances (segments or data points) are temporally correlated, it is feasible to generate multiple different instances from a target instance without perturbing it but by attaching its surrounding sequence (i.e., context). As shown in Figure 1(a), given a target instance in a time series, contexts of varying lengths are attached to the target instance to generate different pairs of "augmented" instances. We call this type of data augmentation the context-attached augmentation.
+
+The key property of context-attached augmentation is to achieve the effect of data augmentation without perturbing a target instance. Being free of data perturbation brings several benefits. First, consistency between augmented instances can be enforced more reliably because a target instance itself is exactly the same among its augmented instances. Second, a sufficient number of augmented instances can be easily obtained by varying contexts. Third, it is computationally inexpensive, only requiring the retrieval of a sub-sequence from a time series. Moreover, context-attached augmentation can be used together with conventional data augmentation such as jittering and scaling. Thus, the novel concept of context-attached augmentation opens a new direction of context consistency regularization.
+
+![](images/87383fbfb6689cf547247a6ae45c1019d02d4fd19659cf85989bbe305d4ef1b7.jpg)  
+(a) Context-attached augmentation.  
+Figure 1. Key technique and overall procedure of CrossMatch.
+
+![](images/4088dd30b0e52ba060522ace794b79249c8639b03dbdcbd2bd0df59369f87b71.jpg)  
+(b) Model update with cross-window soft labels.
+
+Despite its time-series-savvy concept and big benefits, applying context-attached augmentation for consistency regularization is challenging. First, it requires determining the proper number and length of contexts based on the trade-off between the expected performance improvement and training cost. Note that varying the context length in augmented instances incurs different complexity for a downstream task; intuitively, compared with the conventional data augmentation, a short context gives weak augmentation and a long context gives strong augmentation considering the intensity of perturbation. Second, a new consistency regularization method is needed to fully benefit from context-attached augmentation which does not perturb a target instance unlike the conventional data augmentation.
+
+In this work, we propose a novel context consistency regularization framework with context-attached augmentation for time series, called CrossMatch. Given a target instance, CrossMatch generates a pair of augmented instances by attaching contexts to the preceding and succeeding positions of the target instance, respectively (Figure 1(a)). We theoretically and empirically show that such a minimal context-attached augmentation is sufficient to take a full advantage of the context consistency regularization. Then, CrossMatch conducts reliability-weighted mixing to generate cross-window soft labels from the augmented instances for regularizing the inference of the target instance (Figure 1(b)). In existing consistency regularization methods such as MixMatch (Berthelot et al., 2019) and FixMatch (Sohn et al., 2020), an artificial label is created as a form of a hard label; because the model's outputs for augmented instances could be biased by the perturbation of the target instance, the most confident label is chosen by averaging and sharpening in MixMatch and weak augmentation and thresholding in FixMatch. In CrossMatch, on the other hand, the model's
+
+outputs from the two augmented instances are crossed in the form of a soft label; due to its target-preserving property, the outputs with different contexts can be considered equally meaningful. As shown in Figure 1(b), a pair of augmented instances generated from a target instance is fed to a model to get the two softmax outputs of the target instance. Then, a single set of cross-window soft labels is derived and enforced to each output for consistency regularization. These augmentation and regularization procedures are repeated in each training iteration.
+
+Overall, the main contributions are summarized as follows:
+
+- Proposing a novel context consistency regularization framework equipped with context-attached augmentation;  
+- Analyzing the impact of varying contexts theoretically and empirically to explain why CrossMatch works;  
+- Achieving higher classification accuracy than existing state-of-the-art methods (the FixMatch-style methods with jittering and scaling) by up to 23 percentage points.
+
+# 2 Related Work
+
+# 2.1 Data Augmentation
+
+Data augmentation perturbs given data instances to generate diverse and sufficient data instances to prevent overfitting (Shorten & Khoshgoftaar, 2019). The techniques used in data augmentation usually manipulate features and can be classified into two categories: (1) inner-augmentation that changes the features within a data instance and (2) inter-augmentation that exploits features across multiple data instances. Rotation, flipping, scaling, cutout, and random erasing are the examples of inner-augmentation (DeVries & Taylor, 2017; Cubuk et al., 2019). Mix-up, cut-mix, and copy-paste are the representatives of inter-augmentation that mixes two full or partial images (Zhang et al., 2018; Yun
+
+et al., 2019; Ghiasi et al., 2021). However, these studies have not considered context-additive augmentation because they deal with independent data instances such as images.
+
+Time-series augmentation techniques have been devised in recent literature (Wen et al., 2021). They include jittering, scaling, window warping, window cropping, and the Fourier transform specialized in the time and frequency domains, and belong to the inner-augmentation category (Um et al., 2017; Eldele et al., 2021; Yue et al., 2022; Chen et al., 2022). The inter-augmentation category is considered to be ineffective because temporal patterns could be lost after mixing two time-series segments (Iwana & Uchida, 2021). These studies also consider a set of already segmented time series as candidate augmentation targets and assume that the instances in each segment have the same labels. Thus, they are not directly applicable to a single continuous time series with sparse labels, which is a more practical and challenging setting. Moreover, all of them perturb the target instances, following the common trend of existing data augmentation, whereas our method preserves the target instances.
+
+# 2.2 Semi-Supervised Learning Based on Augmentations
+
+Semi-supervised learning (SSL) trains a model with both labeled and unlabeled data instances (Van Engelen & Hoos, 2020). Unlabeled data instances are harnessed by two popular approaches: (1) self-supervised learning which is mostly based on contrastive learning or pretext tasks (Chen et al., 2020; Grill et al., 2020; Singh et al., 2021; Yoon et al., 2023) and (2) self-training which produces artificial labels for unlabeled data instances from model predictions (Lee, 2013; Chen et al., 2018; Xie et al., 2020; Pham et al., 2021). We focus on self-training owing to its simplicity and effectiveness demonstrated in recent studies (Yang et al., 2022).
+
+State-of-the-art self-training methods, usually based on consistency regularization, force consistency in model predictions from multiple augmentations of a data instance. MixMatch (Berthelot et al., 2019) averages out the predictions from multiple augmentations and sharpens the averaged prediction to reduce the entropy in the pseudo-label. ReMixMatch (Berthelot et al., 2020) generates a sharpened pseudolabel from a weak augmentation and matches it against the predictions from multiple strong augmentations. FixMatch (Sohn et al., 2020) generates a one-hot pseudo-label by choosing a single class above a fixed confidence threshold. FlexMatch (Zhang et al., 2021) is a variation of FixMatch, which uses a dynamic confidence threshold to adapt to different learning speeds among different classes. Propagation regularizer (Kim & Lee, 2022) also reduces confidence in incorrect predictions to make FixMatch robust to a more sparse label setting. However, all these methods heavily rely on domain-specific augmentation and lack for consideration of time series.
+
+There are several time-series semi-supervised learning methods in the literature, but most of them are based on self-supervised learning. In their settings, a model is first pre-trained with time-series self-supervision and then finetuned with initial labels. The examples of time-series self-supervisions are (1) pretext tasks such as forecasting and temporal relation prediction (Jawed et al., 2020; Fan et al., 2021), (2) contrastive learning with the aforementioned time-series augmentation techniques (Singh et al., 2021; Xiao et al., 2022), and (3) clustering results (Singhania et al., 2022). These methods target an already-segmented time series, which cannot deal with continuous time series with sparse labels (Ma et al., 2021; Xu et al., 2022).
+
+# 3 CrossMatch: Cross-Window Consistency Regularization
+
+# 3.1 Preliminaries and Problem Setting
+
+Dataset and Model: Let  $\mathcal{D} = \mathcal{X}\times \mathcal{Y} = \left\{\left(\mathbf{x}_t,y_t\right)\mid t\in \mathcal{T}\right\}$  be a time series, where  $\mathcal{T}$  is an index set of timestamps,  $\mathbf{x}_t\in \mathbb{R}^d$  is a  $d$  -dimensional data point at timestamp  $t$  ,and  $y_{t}$  is a corresponding class label if  $\mathbf{x}_t$  is labeled or null otherwise. Let  $\mathcal{T}_L$  be the index set of labeled timestamps and  $\mathcal{T}_U$  be the index set of unlabeled timestamps, where  $\mathcal{T}_L\cup$ $\mathcal{T}_U = \mathcal{T}$  and  $|\mathcal{T}_L|\ll |\mathcal{T}_U|$  . Here,  $\mathcal{T}_L$  is sparse, meaning that its members are few and scattered. In this work, multiple consecutive timestamps, referred to as a segment instance (simply, an instance), are processed in a batch. An instance  $X = \{\mathbf{x}_t\mid t\in [m - w:m + w)\}$  is a list of consecutive  $2w$  data points (timelines) centered at timestamp  $m$  ,where  $[m - w:m + w)$  represents an integer interval from  $m - w$  through  $m + w - 1$  . Likewise,  $Y = \{y_{t}\mid t\in [m - w:$ $m + w)\}$  is a set of the corresponding class labels, where  $y_{t}\in \{1,\dots ,K\}$  and  $K$  is the number of classes. A model  $f_{\theta}$  predicts the sequential softmax probabilities of  $X$  ,i.e.,  $f_{\theta}(X)\in [0,1]^{2w\times K}$  . The classification loss for training the model given  $X$  and  $Y$  is formulated as
+
+$$
+\ell (X, Y) = \frac {1}{2 w} \sum_ {t} \mathbf {1} _ {y _ {t} \neq \mathrm {n u l l}} H \left(f _ {\theta} (X) _ {t,:}, y _ {t}\right), \tag {1}
+$$
+
+where  $H(\cdot, \cdot)$  is sparse categorical cross-entropy and  $\cdot_{t,:}$  means indexing at timestamp  $t$ .
+
+Pseudo-Labeling: For each instance  $X$ , using the maximum softmax probabilities conditioned on a confidence threshold  $\tau$ , a pseudo-label  $\hat{y}_t$  at each timestamp  $t \in [m - w : m + w)$  is derived by
+
+$$
+\hat {y} _ {t} = \left\{ \begin{array}{l l} \arg \max  _ {k \in \{1, \dots , K \}} f _ {\theta} (X) _ {t, k} & \text {i f} f _ {\theta} (X) _ {t, k} > \tau \\ \text {n u l l} & \text {o t h e r w i s e .} \end{array} \right. \tag {2}
+$$
+
+A set  $\hat{Y} = \{\hat{y}_t\mid t\in [m - w:m + w)\}$  of the pseudo-labels for  $X$  is constructed by Equation (2). Then, the classification
+
+![](images/02a1554cad16d7d5632ad1f756b04eb97d1f9ba67b653bda5440c4ea87ef9c72.jpg)  
+Figure 2. Intuition of the context-attached augmentation.
+
+loss for an instance  $X$  and its set  $\hat{Y}$  of the pseudo-labels obtained is formulated as  $\ell(X, \hat{Y})$  with Equation (1). Note that the pseudo-labels are discarded once the model is updated with the losses calculated from the sampled instances.
+
+Consistency Regularization: Recent consistency regularization methods force consistency between the model outputs of augmentations assuming that the class information is preserved across augmentations. For example, FixMatch(Sohn et al., 2020) matches the pseudolabel from a weak augmentation against the prediction from a strong augmentation. That is,  $\ell(X, \hat{Y}) = \frac{1}{2w} \sum_{t} \mathbf{1}_{\max_k f_\theta(\alpha(X))_{t,k} > \tau} H(f_\theta(\mathcal{A}(X))_t, \hat{y}_t)$ , where  $\hat{y}_t = \arg \max_k f_\theta(\alpha(X))_{t,k}$ , and  $\alpha$  and  $\mathcal{A}$  represent weak and strong augmentations, respectively. The goal of consistency regularization is to offer informative supervision to update the model by diverse augmentations and reliable pseudo-labels. In FixMatch, pseudo-labels from weak augmentations are reliable due to weak perturbation, and strong augmentations become diverse due to strong perturbation.
+
+# 3.2 Context-Attached Augmentation
+
+We first define the context-attached augmentation in Definition 3.1 and then discuss its design principle.
+
+Definition 3.1. (CONTEXT-ATTACHED AUGMENTATION) A  $c$ -length context-attached augmentation  $\mathcal{A}$  of a target instance  $\dot{X} = \{\mathbf{x}_t|t\in [m - w:m + w)\}$  is attaching  $c$  surrounding data points of  $\dot{X}$  such that  $\mathcal{A}^i (\dot{X}) = \{\mathbf{x}_t|t\in [m - w - c + i:m + w + i)\}$ , where  $i\in [0:c]$ .
+
+There are  $c + 1$  possible choices of context-attached augmentation according to where we put the context (see Figure 2). Choosing augmented instances with a larger difference in the embedding space induces stronger supervision by consistency regularization (Wang et al., 2022; Jiang et al., 2022). Thus, it is preferable to have two augmented instances whose pairwise distance is the farthest in the embedding space. In Theorem 3.2, we show that a pair of augmented instances is more likely to have a larger distance in the embedding space if they have a less number of overlapping data points under the martingale assumption.
+
+Theorem 3.2. (EMBEDDING DISTANCE) Let  $\mathbf{z}^i$  denote an embedding of  $\mathcal{A}^i(\dot{X})$  at a certain timestamp  $t \in [m - w : m + w)$ . Under the martingale embedding with a bounded difference assumption, as the overlap between  $\mathcal{A}^i(\dot{X})$  and  $\mathcal{A}^j(\dot{X})$  ( $i > j$ ) decreases, i.e.,  $(i - j)$  increases, the upper-bound of the probability of the Euclidean distance between  $\mathbf{z}^i$  and  $\mathbf{z}^j$  being farther than  $\epsilon$  increases. Formally,
+
+$$
+\Pr \left[ \| \mathbf {z} ^ {i} - \mathbf {z} ^ {j} \| _ {2} \geq \epsilon \right] \leq 2 e ^ {2} e ^ {- \epsilon^ {2} / 2 s ^ {2} (i - j)}, \tag {3}
+$$
+
+where  $0 \leq j < i \leq c$ , and  $\epsilon$  and  $s$  are positive real constants.
+
+Proof. Under the martingale embedding with a bounded difference assumption, Equation (3) is simply derived by Azuma's inequality (Hayes, 2005). Refer to Appendix A for the complete proof.  $\square$
+
+Corollary 3.3. (MAXIMUM DIFFERENCE) Given a context length  $c$ , picking the pair of  $\mathcal{A}^0(\dot{X})$  and  $\mathcal{A}^c(\dot{X})$  maximizes the upperbound of the probability of the Euclidean distance between their embeddings being farther than  $\epsilon$ , among all possible pairs of  $\mathcal{A}^j(\dot{X})$  and  $\mathcal{A}^i(\dot{X})$  ( $0 \leq j < i \leq c$ ).
+
+Proof. When  $j = 0$  and  $i = c$ ,  $(i - j)$  becomes the largest. Then, Equation (3) in Theorem 3.2 concludes the proof.
+
+By Corollary 3.3, the consistency regularization between the furthest embeddings  $\mathbf{z}^0$  and  $\mathbf{z}^c$  would make the whole embedding space shrink toward its center of a class maximally, compared to other pairs (Van Engelen & Hoos, 2020). Therefore, the effect of consistency regularization can be maximized by picking two augmented instances  $\mathcal{A}^0 (\dot{X})$  and  $\mathcal{A}^c (\dot{X})$ . Accordingly, a context of length  $c$  is attached on the left and right sides of a target instance, respectively, and thus the two augmented instances,  $\mathcal{A}^0 (\dot{X}) = X^{\mathrm{left}} = \{\mathbf{x}_t|t\in [m - w - c:m + w)\}$  and  $\mathcal{A}^c (\dot{X}) = X^{\mathrm{right}} = \{\mathbf{x}_t|t\in [m - w:m + w + c)\}$ , are generated as shown in Figure 3. Each augmented instance is fed to a model, and pseudo-labels are generated from every timestamp of only the target instance (i.e., the overlap between the two augmented instances) by Equation (2). As a result, two pseudo-label sets  $\hat{Y}^{\mathrm{left}}$  and  $\hat{Y}^{\mathrm{right}}$  for the left and right augmented instances are obtained. Reliability-weighted mixing will be introduced in Section 3.3 to merge  $\hat{Y}^{\mathrm{left}}$  and  $\hat{Y}^{\mathrm{right}}$  while considering the reliability of each pseudo-label based on the left and right side lengths.
+
+Another important factor is the context length  $c$ , which directly affects the supervision quality of consistency regularization. If  $c$  is too short, then two augmented instances become too similar so that the model barely changes after matching the embeddings of augmented instances (Wang et al., 2022). On the other hand, if  $c$  is too long, the embeddings diverge so that the matching would weld the representation of instances from different classes. We empirically study this trade-off and suggest a heuristic in Section 4.4.
+
+![](images/0baae43dee27ffc87223a41487544208e8d2f3dffe41de7587bb359ec1e1928a.jpg)  
+Figure 3. A target instance (yellow) and its context (blue).
+
+# 3.3 Reliability-Weighted Mixing
+
+Though all pseudo-labels generated for a target instance satisfy the confidence threshold in Equation (2), we treat them differently based on the reliability of a pseudo-label. Our rationale is that the pseudo-label becomes more reliable if (1) the model  $f_{\theta}$  receives more data points on the left and right sides of the pseudo-label and (2) the number of data points is balanced between the two sides so that the prediction is not biased to the preceding or succeeding interval. We design a reliability function that follows our rationale and compute reliability of each pseudo-label. Using the reliability as a weight, we mix the pseudo-labels into a single cross-window label, which will be matched against two softmax probabilities from both of augmentations.
+
+Let's consider two pseudo-labels  $\hat{y}_t^{\mathrm{left}}$  and  $\hat{y}_t^{\mathrm{right}}$  generated from  $f_{\theta}(X^{\mathrm{left}})_t$  and  $f_{\theta}(X^{\mathrm{right}})_t$  shown in Figure 3. For each timestamp  $t \in [m - w : m + w)$  of a target instance, the length  $n_t^{loc, + }$  of the left side and the length  $n_t^{loc, - }$  of the right side along each augmented instance are easily calculated by
+
+$$
+n _ {t} ^ {\text {l o c , +}} = \left\{ \begin{array}{l l} t - m + w + c & \text {i f l o c = l e f t} \\ t - m + w & \text {i f l o c = r i g h t} \end{array} \right. \tag {4}
+$$
+
+$$
+n _ {t} ^ {\text {l o c , -}} = \left\{ \begin{array}{l l} m + w - t & \text {i f l o c = l e f t} \\ m + w + c - t & \text {i f l o c = r i g h t}, \end{array} \right. \tag {5}
+$$
+
+where loc indicates the location of context attachment—either left or right.
+
+![](images/865efc4684835cf4f2916283bb61630a9a3a8be5916119e7562da866ba75ea5b.jpg)  
+Figure 4. Visualization of Equation (6).
+
+Per our design rationale, the reliability score becomes higher as  $n_t^{loc, + }$  and  $n_t^{loc, - }$  are larger and  $n_t^{loc, + }$  and  $n_t^{loc, - }$  are more similar with each other. This requirement can be achieved by exploiting a bell-shaped function,
+
+$$
+r (p) = \sqrt {2 p - p ^ {2}} + \sqrt {1 - p ^ {2}}, \tag {6}
+$$
+
+where  $0 \leq p \leq 1$ . Here,  $p$  is a normalization of  $n_t^{loc, + }$  with re
+
+spect to the full length of an augmented instance, i.e.,  $p^{loc}(t) = n^{loc, + } / (2w + c)$ . That is, using Equation (6),
+
+we obtain two reliability scores  $r(p^{\mathrm{left}}(t))$  and  $r(p^{\mathrm{right}}(t))$  for each pseudo-label from the two augmented instances  $X_{left}$  and  $X_{right}$ . Due to the bell shape in Figure 4, a timestamp with a sufficiently long side length on both sides has an adequately high reliability in an augmented instance. If a side length on either side is too long (i.e., biased to one side), the side length on the other side is too short, and the reliability score is too low. Overall, the reliability score is maximized at the center of a given augmented instance.
+
+Last, in order to treat each pseudo-label differently based on the reliability scores on the two sides,  $r(p^{\mathrm{left}}(t))$  and  $r(p^{\mathrm{right}}(t))$ , the weight of each pseudo-label is assigned as  $\beta_t^{loc} = r(p^{loc}(t)) / (r(p^{\mathrm{left}}(t)) + r(p^{\mathrm{right}}(t)))$  by normalizing the two reliability scores. The final cross-window soft label for the target instances in the left and right instances is
+
+$$
+\bar {y} _ {t} = \beta_ {t} ^ {\text {l e f t}} \cdot \text {o n e h o t} (\hat {y} _ {t} ^ {\text {l e f t}}) + \beta_ {t} ^ {\text {r i g h t}} \cdot \text {o n e h o t} (\hat {y} _ {t} ^ {\text {r i g h t}}), \tag {7}
+$$
+
+where the onehot function converts a scalar to one-hot encoded vector. In other words, reliability-weighted mixing generates a soft label by adding the two weighted pseudolabels. Using the cross-window label set and two target instance outputs, the classification loss  $\ell_{u}(X^{\mathrm{left}}, X^{\mathrm{right}}, \bar{Y})$  is computed as
+
+$$
+\frac {1}{4 w} \sum_ {t} \mathbf {1} _ {\bar {y} _ {t} \neq \text {n u l l}} \left(h \left(\mathbf {z} _ {t} ^ {\text {l e f t}}, \bar {y} _ {t}\right) + h \left(\mathbf {z} _ {t} ^ {\text {r i g h t}}, \bar {y} _ {t}\right)\right), \tag {8}
+$$
+
+where  $\mathbf{z} = f_{\theta}(X)_{t,:}$ ,  $h$  is soft cross-entropy, and  $\bar{Y} = \{\bar{y}_t|t\in [m - w:m + w)\}$ . By matching a single cross-window soft label to both augmented instances, we can reduce the variance of the label to be predicted, leading to better consistency regularization (Wang et al., 2022). In addition, consistency regularization with soft labeling is more effective than with hard labeling since soft labeling can be robust to erroneous pseudo-labels when at least one pseudo-label is correct (Müller et al., 2019).
+
+# 3.4 Overall Procedure of CrossMatch
+
+Figure 1 summarizes the overall procedure of CrossMatch using context-attached augmentation and reliability-weighted mixing. Given a time series, CrossMatch first takes a target instance sampled from the time series as an input and generates a pair of augmented instances with context-attached augmentation (see Figure 1(a)). Then, a model  $f_{\theta}$  infers the pair of augmented instances  $X^{\mathrm{left}}$  and  $X^{\mathrm{right}}$  to produce two independent softmax probabilities. The pseudo-labels respectively generated from the two softmax probabilities are merged  $(\bigoplus)$  to give cross-window soft labels  $\bar{Y}$  with reliability-weighted mixing. Finally, the cross-entropy losses are computed from the cross-window soft labels  $\bar{Y}$  as well as the true labels  $Y$  and used to update the model  $f_{\theta}$  (see Figure 1(b)). These steps are repeated for randomly sampled target instances with varying  $c$  until  $I$  iterations. CrossMatch in Appendix B details each step.
+
+Table 1. Datasets and configurations.  
+
+<table><tr><td></td><td>|T|</td><td>Length</td><td>#class</td><td>d</td><td>w</td><td>cmax</td><td>v</td><td>I</td></tr><tr><td>HAPT</td><td>408K</td><td>967</td><td>6</td><td>6</td><td>512</td><td>256</td><td>0.1%</td><td>25K</td></tr><tr><td>mHealth</td><td>343K</td><td>2933</td><td>12</td><td>23</td><td>384</td><td>256</td><td>1.0%</td><td>50K</td></tr><tr><td>Opportunity</td><td>190K</td><td>109</td><td>17</td><td>113</td><td>512</td><td>64</td><td>1.0%</td><td>30K</td></tr></table>
+
+# 4 Evaluation
+
+# 4.1 Experiment Setting
+
+Datasets: We use three widely-used benchmark datasets in Table 1. HAPT is a sensor time-series dataset tracking human movements in a laboratory sampled with the frequency of  $50\mathrm{Hz}$  (Anguita et al., 2013). mHealth is a similar action recognition dataset recorded with more wearable sensors, such as 3D accelerometers, 3D gyroscopes, 3D magnetometers, and electrodes, whose sampling frequency is  $50\mathrm{Hz}$  (Banos et al., 2014). Opportunity is a collection of sensor recordings at  $100\mathrm{Hz}$  capturing daily natural human activities with wearable, object, and ambient sensors (Roggen et al., 2010). For each originally fully-labeled dataset, we randomly sample the same number of timestamps for each class and drop the labels from the rest of the timestamps to generate a sparsely-labeled time series, leaving multiple labeled time spans located randomly. The sampled timestamps become a labeled timestamp set  $\mathcal{T}_L$  for the given ratio of labeled timestamps  $v = |\mathcal{T}_L| / |\mathcal{T}|$ . Table 1 summarizes the statistics of datasets and the data-specific parameters, specifying the number of timestamps, the average length of a segment with a single class, the number of classes, the dimensionality of a data point, the half length of a target instance  $w$ , the maximum context length  $c_{\mathrm{max}}$ , the default ratio of labeled timestamps  $v$ , and the maximum number of iterations  $I$ . In particular,  $c_{\mathrm{max}}$  is set to be sufficiently smaller than the average length of a segment.
+
+The value of  $w$ , half of the length of a target instance, needs to be carefully chosen for each data set. Too large  $w$  may include semantically irrelevant data points, while too small  $w$  may not give enough temporal information for prediction. Either way leads to incorrect pseudo-labels and ultimately degrades the performance. Thus, we set  $2w$ , the length of a target instance, as a value lower than the mean length of a label-coherent segment, which could be known in advance or estimated by a given set  $T_L$  of labeled timestamps.
+
+Implementation Details: We use a widely-used multi-stage temporal convolutional network named MS-TCN (Farha & Gall, 2019), which is applicable to our problem setting because it gives softmax probabilities for each timestamp in an instance  $X$ . We follow the same hyperparameter and configuration in the original MS-TCN, except the learning rate and an optimizer adjusted for consistency regularization with sparse labels. See Appendix C for more details.
+
+For CrossMatch, we set the confidence threshold  $\tau$  to 0.95 and the weight of the unlabeled loss  $\lambda$  to 1. The model is first trained without any pseudo-labels, i.e., only using the labeled batches. We start to update a model with pseudolabels after the number of pseudo-labels in each class for a batch is balanced. Formally, this condition is satisfied when the entropy of the numbers of pseudo-labels per class is above 0.99 for the last 100 iterations; it is enforced to prevent early confirmation bias in consistency regularization (Kim & Lee, 2022). If a data point  $x_{t}$  in an instance  $X$  has a true label (i.e.,  $t\in \mathcal{T}_L$ ), CrossMatch uses the true label instead of the generated pseudo-label.
+
+For every experiment, we use Intel(R) Xeon(R) Gold 6226R CPU @ 2.90GHz and NVIDIA RTX 3090. The source code is provided at https://github.com/kaist-dmlab/CrossMatch.
+
+Evaluation Metrics: We measure timestamp accuracy and segmental F1 score with five-fold cross validation and report the average value with standard deviation of five runs. For sequential classifiers such as MS-TCN, timestamp accuracy (denoted as TS accuracy) and segmental F1 score (denoted as F1@25) measure the performance of classification at each timestamp and segment respectively (Li et al., 2021; Kumar et al., 2022). To evaluate pseudo-labeling performance, we report pseudo-label F1 score (denoted as PLF) as well. We define each metric in detail in Appendix D.
+
+Baselines: We compare CrossMatch with three state-of-the-art consistency regularization methods: FixMatch (Sohn et al., 2020), FlexMatch (Zhang et al., 2021), and PropReg (Kim & Lee, 2022). As discussed in Section 2, these methods require inner-instance augmentation for consistency regularization. We use two popular time-series augmentations: jittering and scaling, where weak augmentation  $\alpha(X) = \text{jittering}(X)$  and strong augmentation  $\mathcal{A}(X) = \text{jittering}(\text{scaling}(X))$  (Um et al., 2017). Throughout the experiments, we use the same hyperparameters for all methods except the instance length due to the context-attached augmentation. We also compare CrossMatch with the variants of the compared methods supported by our context-attached augmentation.
+
+# 4.2 Comparison with State-of-the-Art Methods
+
+Overall Comparison: Table 2 shows the timestamp accuracy and F1@25 score on three datasets with varying label ratios; each value is obtained by averaging over the last 20 iterations for reliable results. Compared with other methods, CrossMatch achieves the best classification performance with a statistical significance of 0.05 using independent (unpaired) t-test for all datasets except HAPT  $10 \times$ . This is mainly because consistency regularization using context-attached augmentation is more informative than inner-instance augmentation used in other consistency regu
+
+Table 2. Timestamp accuracy and F1@25 score averaged over the last 20 iterations with adjusting the label ratio  $v$  (  $1 \times$  ,  $5 \times$  ,and  ${10} \times$  of a default value). The best values are marked in bold.  
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Ratio</td><td colspan="2">FixMatch</td><td colspan="2">FlexMatch</td><td colspan="2">PropReg</td><td colspan="2">CrossMatch</td></tr><tr><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td></tr><tr><td rowspan="3">HAPT</td><td>1×</td><td>0.78 ± 0.01</td><td>0.55 ± 0.04</td><td>0.78 ± 0.01</td><td>0.54 ± 0.04</td><td>0.79 ± 0.01</td><td>0.52 ± 0.04</td><td>0.84 ± 0.01</td><td>0.75 ± 0.02</td></tr><tr><td>5×</td><td>0.85 ± 0.01</td><td>0.51 ± 0.04</td><td>0.84 ± 0.01</td><td>0.50 ± 0.04</td><td>0.82 ± 0.01</td><td>0.49 ± 0.04</td><td>0.88 ± 0.01</td><td>0.70 ± 0.02</td></tr><tr><td>10×</td><td>0.96 ± 0.00</td><td>0.87 ± 0.01</td><td>0.96 ± 0.00</td><td>0.90 ± 0.01</td><td>0.96 ± 0.00</td><td>0.89 ± 0.01</td><td>0.95 ± 0.01</td><td>0.84 ± 0.02</td></tr><tr><td rowspan="3">mHealth</td><td>1×</td><td>0.77 ± 0.01</td><td>0.23 ± 0.01</td><td>0.77 ± 0.01</td><td>0.13 ± 0.01</td><td>0.81 ± 0.01</td><td>0.43 ± 0.03</td><td>0.85 ± 0.01</td><td>0.45 ± 0.02</td></tr><tr><td>5×</td><td>0.91 ± 0.01</td><td>0.65 ± 0.02</td><td>0.91 ± 0.01</td><td>0.62 ± 0.03</td><td>0.91 ± 0.01</td><td>0.67 ± 0.03</td><td>0.94 ± 0.01</td><td>0.74 ± 0.03</td></tr><tr><td>10×</td><td>0.91 ± 0.00</td><td>0.70 ± 0.02</td><td>0.90 ± 0.00</td><td>0.66 ± 0.02</td><td>0.90 ± 0.01</td><td>0.70 ± 0.02</td><td>0.95 ± 0.01</td><td>0.84 ± 0.03</td></tr><tr><td rowspan="3">Opportunity</td><td>1×</td><td>0.61 ± 0.04</td><td>0.65 ± 0.05</td><td>0.59 ± 0.03</td><td>0.63 ± 0.05</td><td>0.63 ± 0.03</td><td>0.65 ± 0.05</td><td>0.67 ± 0.02</td><td>0.73 ± 0.04</td></tr><tr><td>5×</td><td>0.73 ± 0.03</td><td>0.73 ± 0.04</td><td>0.72 ± 0.03</td><td>0.75 ± 0.05</td><td>0.73 ± 0.03</td><td>0.74 ± 0.05</td><td>0.78 ± 0.02</td><td>0.82 ± 0.03</td></tr><tr><td>10×</td><td>0.75 ± 0.03</td><td>0.75 ± 0.04</td><td>0.73 ± 0.03</td><td>0.77 ± 0.04</td><td>0.75 ± 0.03</td><td>0.76 ± 0.04</td><td>0.83 ± 0.02</td><td>0.86 ± 0.02</td></tr></table>
+
+![](images/939a1495b89dc29db51a2224c72af903e19de1b4636914b80b2e5f74bde67e26.jpg)  
+Figure 5. Training curve of the classification performance (the first two rows) and pseudo-labeling performance (the last row) over training iterations in Algorithm 1. The full  $y$ -axis is shown in Figure 7 of Appendix E.
+
+larization methods. In particular, CrossMatch exhibits much better performance than others especially when the initial label ratio is low. For instance, with only  $0.1\%$  of the labeled timestamps in HAPT, CrossMatch outperforms FixMatch by 20 percentage points  $(\% p)$ , FlexMatch by  $21\% p$ , and PropReg by  $23\% p$  in F1@25 (see the first row in HAPT of Table 2). Therefore, the performance dominance of CrossMatch indicates the context-attached augmentation with reliability-weighted mixing indeed helps the model select more reliable pseudo-labels even when softmax probabilities fluctuate due to label scarcity and insufficient training.
+
+Training Curve Analysis: Figure 5 shows the training curves of classification and PLF over the entire training iteration. Please refer to Appendix E for the same results
+
+with standard deviation and other metrics related to pseudolabeling. CrossMatch shows much higher performance than the other methods with respect to timestamp accuracy and F1@25 even in the early stage of training, reaching the highest performance in most cases (see the first two rows in Figure 5). This is attributed to its robustness in reliability-weighted mixing in the early stage of consistency regularization and its credibility in enforcing consistency between two augmented instances from context-attached augmentation. For example, as shown in mHealth of Figure 5(b), other methods suffer from the low-quality pseudo-labels at the iteration of around 17,000 when the warm-up period ends, thereby showing a temporary drop in timestamp accuracy. Although the accuracy recovers gradually, the final accuracy is far behind the accuracy CrossMatch achieves.
+
+Table 3. Timestamp accuracy and F1@25 score averaged over the last 20 iterations with varying context attachment for context-attached augmentation (CAA). The best values are marked in bold.  
+
+<table><tr><td>Variations</td><td>Metrics</td><td>One-sided CAA</td><td>Random-pair CAA</td><td>Multiple-2 CAA</td><td>Multiple-3 CAA</td><td>Multiple-5 CAA</td><td>Multiple-7 CAA</td></tr><tr><td rowspan="2">HAPT</td><td>TS Accuracy</td><td>0.72 ± 0.01</td><td>0.67 ± 0.02</td><td>0.84 ± 0.01</td><td>0.85 ± 0.01</td><td>0.85 ± 0.01</td><td>0.86 ± 0.01</td></tr><tr><td>F1@25</td><td>0.65 ± 0.03</td><td>0.44 ± 0.03</td><td>0.75 ± 0.02</td><td>0.70 ± 0.02</td><td>0.74 ± 0.02</td><td>0.75 ± 0.04</td></tr><tr><td rowspan="2">mHealth</td><td>TS Accuracy</td><td>0.77 ± 0.02</td><td>0.75 ± 0.01</td><td>0.85 ± 0.01</td><td>0.86 ± 0.02</td><td>0.86 ± 0.02</td><td>0.87 ± 0.01</td></tr><tr><td>F1@25</td><td>0.41 ± 0.04</td><td>0.21 ± 0.01</td><td>0.45 ± 0.02</td><td>0.40 ± 0.04</td><td>0.42 ± 0.04</td><td>0.44 ± 0.02</td></tr><tr><td rowspan="2">Opportunity</td><td>TS Accuracy</td><td>0.61 ± 0.07</td><td>0.61 ± 0.05</td><td>0.67 ± 0.02</td><td>0.67 ± 0.05</td><td>0.70 ± 0.04</td><td>0.69 ± 0.02</td></tr><tr><td>F1@25</td><td>0.64 ± 0.07</td><td>0.68 ± 0.03</td><td>0.73 ± 0.04</td><td>0.78 ± 0.08</td><td>0.77 ± 0.07</td><td>0.78 ± 0.03</td></tr></table>
+
+Table 4. Timestamp accuracy and F1@25 score averaged over the last 20 iterations with varying context lengths using CrossMatch. The best values are marked in bold.  
+
+<table><tr><td rowspan="2">Cmax</td><td colspan="2">0.25×</td><td colspan="2">0.5×</td><td colspan="2">1.00×</td><td colspan="2">2.00×</td><td colspan="2">4.00×</td></tr><tr><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td><td>TS Accuracy</td><td>F1@25</td></tr><tr><td>HAPT</td><td>0.65±0.01</td><td>0.46±0.02</td><td>0.74±0.01</td><td>0.63±0.02</td><td>0.84±0.01</td><td>0.75±0.02</td><td>0.79±0.02</td><td>0.65±0.03</td><td>0.81±0.01</td><td>0.70±0.02</td></tr><tr><td>mHealth</td><td>0.69±0.01</td><td>0.29±0.02</td><td>0.79±0.02</td><td>0.37±0.02</td><td>0.85±0.01</td><td>0.45±0.02</td><td>0.82±0.01</td><td>0.38±0.04</td><td>0.77±0.02</td><td>0.28±0.03</td></tr><tr><td>Opportunity</td><td>0.65±0.03</td><td>0.72±0.04</td><td>0.69±0.03</td><td>0.73±0.04</td><td>0.67±0.02</td><td>0.73±0.04</td><td>0.66±0.02</td><td>0.71±0.03</td><td>0.55±0.02</td><td>0.56±0.04</td></tr></table>
+
+The effectiveness of CrossMatch is also supported by its PLF computed from precision and recall (see the last row in Figure 5 and also see Figure 8 in Appendix E). Precision is the ratio of the number of correctly predicted timestamps to the number of pseudo-labeled timestamps, while recall is the ratio of the number of correctly predicted timestamps to the length of a target instance (refer to Appendix D for more details). We averaged PLF over the target instances in a batch. CrossMatch reaches the highest PLF continuously in most cases due to the reliability of our cross-window labels.
+
+# 4.3 Analysis of Varying Context Attachment
+
+We study the effect of varying the ways of context attachment in the context-attached augmentation. Given all augmented instance candidates,  $\mathcal{A}^0 (\dot{X}),\mathcal{A}^1 (\dot{X}),\ldots ,\mathcal{A}^c (\dot{X})$  for consistency regularization, one-sided CAA picks either  $\mathcal{A}^0 (\dot{X})$  (i.e.,  $X^{\mathrm{left}}$ ) or  $\mathcal{A}^c (\dot{X})$  (i.e.,  $X^{\mathrm{right}}$ ), Random-pair CAA randomly picks two augmented instances, and Multiple-N CAA includes both  $\mathcal{A}^0 (\dot{X})$  and  $\mathcal{A}^c (\dot{X})$  and randomly picks  $N - 2$  augmented instances; that is, Multiple-2 CAA is the default of CrossMatch. As shown in Table 3, one-sided CAA and Random-pair CAA perform worse than Multiple-N CAA, because of ineffective consistency regularization caused by the lack of variations in their augmented instances. While some cases Multiple-  $N > 2$  CAA variations achieve higher performances than Multiple-2 CAA, the performance gain is marginal, i.e., less than  $5\% p$ . This result indeed demonstrates that our choice of context attachment in Section 3.2 is practical, considering the computational cost for preparing augmented instances.
+
+# 4.4 Analysis of Varying Context Lengths
+
+Table 4 summarizes the timestamp accuracy and F1@25 score of CrossMatch by adjusting the default context length
+
+$c_{\mathrm{max}}$  (in Table 1) from  $0.25 \times$  to  $4.0 \times$ . If the context length is too large, augmented instances bear too much perturbation to generate high-quality cross-window labels. On the other hand, if the context length is too small, informative consistency regularization between two augmented instances becomes trivial since they show high similarity. We found out that the optimal context length is highly correlated with the average segment length of each dataset. For instance, the best timestamp accuracy of the Opportunity dataset is achieved with a relatively smaller context length (i.e., 32) than that of mHealth data (i.e., 256) because Opportunity has a much shorter mean segment length; as can be seen in Table 1, Opportunity and mHealth exhibit the shortest and the longest mean segment length among the three datasets. Therefore, the best timestamp accuracy and F1@25 score of each dataset are achieved with different context lengths. Our heuristic to find the best  $c_{\mathrm{max}}$  is selecting  $c_{\mathrm{max}}$  within  $1/5$  and  $1/2$  of the average length of segments. The average length of segments in a specific dataset is often available.
+
+# 4.5 Extensions with Additional Augmentations
+
+We further investigate three possible extensions: (1) Cross-Match to combine the two inner-instance augmentations (+IA) of jittering and scaling, (2) the variant of existing consistency regularization methods to combine our proposed context-attached augmentation (+CAA), and (3) the variant of existing consistency regularization methods to use the cut-based augmentation (+CUT). For the first variant, we perform jittering and scaling before applying the context-attached augmentation. For the second variant, we modify the existing methods in support of the context-attached augmentation. The two augmented instances (the left and right instances in Figure 3) are respectively treated as weakly and strongly augmented instances; the instance with a higher
+
+Table 5. Timestamp accuracy and F1@25 score of CrossMatch and the compared methods with various augmentation methods. The best values are marked in bold.  
+
+<table><tr><td rowspan="2">Variations</td><td rowspan="2">Metrics</td><td>CrossMatch</td><td colspan="2">FixMatch</td><td colspan="2">FlexMatch</td><td colspan="2">PropReg</td></tr><tr><td>+IA</td><td>+CAA</td><td>+CUT</td><td>+CAA</td><td>+CUT</td><td>+CAA</td><td>+CUT</td></tr><tr><td rowspan="2">HAPT</td><td>TS Accuracy</td><td>0.88 ± 0.01</td><td>0.62 ± 0.02</td><td>0.78 ± 0.01</td><td>0.64 ± 0.02</td><td>0.81 ± 0.01</td><td>0.68 ± 0.01</td><td>0.81 ± 0.01</td></tr><tr><td>F1@25</td><td>0.77 ± 0.02</td><td>0.39 ± 0.03</td><td>0.66 ± 0.02</td><td>0.49 ± 0.03</td><td>0.67 ± 0.02</td><td>0.45 ± 0.02</td><td>0.69 ± 0.02</td></tr><tr><td rowspan="2">mHealth</td><td>TS Accuracy</td><td>0.89 ± 0.01</td><td>0.76 ± 0.04</td><td>0.74 ± 0.02</td><td>0.82 ± 0.01</td><td>0.76 ± 0.02</td><td>0.77 ± 0.03</td><td>0.75 ± 0.02</td></tr><tr><td>F1@25</td><td>0.67 ± 0.03</td><td>0.35 ± 0.04</td><td>0.42 ± 0.04</td><td>0.31 ± 0.03</td><td>0.38 ± 0.03</td><td>0.30 ± 0.03</td><td>0.47 ± 0.04</td></tr><tr><td rowspan="2">Opportunity</td><td>TS Accuracy</td><td>0.75 ± 0.02</td><td>0.49 ± 0.02</td><td>0.35 ± 0.01</td><td>0.36 ± 0.01</td><td>0.34 ± 0.02</td><td>0.45 ± 0.01</td><td>0.39 ± 0.02</td></tr><tr><td>F1@25</td><td>0.85 ± 0.03</td><td>0.53 ± 0.03</td><td>0.43 ± 0.01</td><td>0.37 ± 0.02</td><td>0.40 ± 0.02</td><td>0.45 ± 0.02</td><td>0.42 ± 0.01</td></tr></table>
+
+reliability score becomes the weakly augmented instance, and that with a lower reliability score becomes the strongly augmented instance. For the third variant, weak augmentation eliminates  $1\%$  of data points in an instance, while strong augmentation eliminates  $10\%$ .
+
+Table 5 summarizes the performances of the extensions of compared methods for all datasets. Note that we use the default label ratios of  $0.1\%$ ,  $1.0\%$ , and  $1.0\%$  for the datasets, respectively. The extension of CrossMatch (+IA) shows additional performance gains of  $4 - 8\% p$  in times-amp accuracy and  $2 - 22\% p$  in F1@25 compared with the original CrossMatch. However, in general, the extensions of existing methods (+CAA) rather suffer from a significant performance drop because they failed to reliably deal with diverse augmented instances. The cut-based augmentation (+CUT) also does not improve existing consistency regularization methods in most cases, which reaffirms that inner-instance augmentation alone is not suitable for time-series. This result demonstrates that CrossMatch can be further enhanced with additional inner-instance augmentation techniques (e.g., jittering and scaling) under the proposed consistency regularization framework.
+
+# 4.6 Discussion on Versatility and Validity
+
+Context-attached augmentation in CrossMatch can be applied to any recent, popular architectures such as temporal convolutional networks and Transformers, where information is conveyed bidirectionally and output sequential softmax probabilities (Farha & Gall, 2019; Chen et al., 2022). If a model is unidirectional (e.g., from left to right as in the vanilla RNN and WaveNet), the context attached to the left side of a target instance would be only meaningful (Oord et al., 2016); thus, CrossMatch needs to consider a target instance and its augmented instance with only the left context in order to accommodate a unidirectional model.
+
+One may argue that our augmentation will not work when a context has different labels than the target instance. However, such cases may occur very infrequently (i.e., only  $6.4\%$  of the target instances in the mHealth dataset) with a proper setting of the hyper-parameter  $c_{\mathrm{max}}$ , the maximum length
+
+of a context, because of temporal coherence inherent in time series. Even though a context contains different labels, our augmentation will work well in general. It is known that a set of transition patterns tend to appear repeatedly in many time-series datasets (Roggen et al., 2010; Stein & McKenna, 2013; Banos et al., 2014); for example, suppose that a transition pattern, standing up  $\rightarrow$  walking, repeats in a human activity recognition dataset. Then, a different label (e.g., standing up) in the context is definitely helpful to predict the label (e.g., walking) in the target instance. Although the transition patterns do not repeat, there is a safety net, where pseudo-labels are not generated for uncertain target instances by confidence thresholding in CrossMatch. Therefore, CrossMatch hardly suffers from such transition between a target instance and its context.
+
+# 5 Conclusion
+
+In this paper, we propose CrossMatch, a novel consistency regularization framework with context-attached augmentation, for classifying time series with sparse labels. CrossMatch adds contexts to a target instance on either its left or right sides to generate two augmented instances with a maximal difference. Then, the inference of the target instance is regularized by reliability-weighted mixing of the two pseudo-labels from the two augmented instances. Our extensive experiments demonstrate that CrossMatch achieves considerably higher accuracy than state-of-the-art consistency regularization methods—by up to  $23\mathrm{p}\%$  even with only  $0.1\%$  of labels. CrossMatch introduces a new direction of data augmentation for sequential data and has the potential to be applied to various time-series applications.
+
+# Acknowledgements
+
+This work was partly supported by Samsung Electronics Co., Ltd. (IO201211-08051-01) and Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No. 2020-0-00862, DB4DL: High-Usability and Performance In-Memory Distributed DBMS for Deep Learning).
+
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+
+Singhania, D., Rahaman, R., and Yao, A. Iterative contrast-classify for semi-supervised temporal action segmentation. In AAAI, pp. 2262-2270, 2022.  
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+
+# A Proof of Theorem 3.2
+
+We prove Theorem 3.2 in Section 3.2. Let  $\mathcal{A}^i(\dot{X})$  be the context-attached augmented instances from Definition 3.1. A feature extractor  $f_{\theta}(\cdot)_t$  generates an embedding for a timestamp  $t$  within the target instance, i.e.,  $\mathbf{z}^i = f_{\theta}(\mathcal{A}^i(\dot{X}))_t$ , where  $\mathbf{z}^i \in \mathbb{R}^k$  and  $k$  is the dimensionality of the embedding.
+
+Comparing  $\mathbf{z}^i$  and  $\mathbf{z}^{i + 1}$ , their inputs share  $2w + i - 1$  timestamps and differ only at the timestamp  $t = m + w + i - 1$ . The data point at that timestamp would not deviate from the previous data points used to infer  $\mathbf{z}^i$  due to temporal coherence. Thus, the next embedding would change in the vicinity centered at the current embedding (if we know it in advance), like a random walk, although the global distribution of embeddings is not known. This behavior leads to the martingale assumption in the embedding sequence, which is frequently used in time-series analysis (Ho, 2005; Hansen et al., 2019; Neufeld & Sester, 2022). In addition, the range of vicinity would also be limited because the change comes from only a single input timestamp.
+
+To justify the martingale embedding assumption, let us conduct a martingale hypothesis test (Park & Whang, 2005) on the embedding sequence generated by a trained model of CrossMatch. The null hypothesis of the martingale hypothesis test for a specific dimension of  $\mathbf{z}^i$  is
+
+$$
+\mathrm {H} _ {0}: \Pr \left(\mathbb {E} \left[ z ^ {i + 1} \mid z ^ {i} \right] = z ^ {i}\right) = 1 \text {w h e r e} z ^ {i} \in \mathbb {R}. \tag {9}
+$$
+
+A Kolmogorov-Smirnov type statistic is given by
+
+$$
+K S (z) = \sup  _ {\delta \in \mathbb {R}} \left| \frac {1}{\sqrt {c + 1}} \sum_ {i = 0} ^ {c} \left(z ^ {i + 1} - z ^ {i}\right) \mathbf {1} _ {z ^ {i} \leq \delta} \right|. \tag {10}
+$$
+
+The maximum value of  $KS(z)$  from every dimension of the softmax probabilities or penultimate layer output is 1.36, 1.45, and 1.43 for the HAPT, mHealth, and Opportunity datasets, respectively, when  $\delta$  is searched from  $-10$  to 10 with a step size of 0.001. With the significance level set to be 0.05, the critical value of  $KS(z)$  is 2.338 (Park & Whang, 2005), which is much higher than these values that we computed. Therefore, we do not reject the null hypothesis for every dimension of the embedding sequence, meaning that the embedding vector sequence is martingale.
+
+When the embedding (vector) sequence is martingale and its consecutive differences are bounded, for a pair of consecutive embeddings  $\mathbf{z}^i$  and  $\mathbf{z}^{i + 1}$ , the following properties hold by definition: (1)  $\mathbb{E}[\mathbf{z}^{i + 1}|\mathbf{z}^i] = \mathbf{z}^i$ , and (2)  $\| \mathbf{z}^{i + 1} - \mathbf{z}^i \|_2 \leq s$ , where  $s$  is a sufficiently small positive real constant. Then, the following variant of Azuma's inequality holds for  $0 \leq j < i \leq c$ , and see (Hayes, 2005) for the detailed derivation.
+
+$$
+\Pr \left[ \| \mathbf {z} ^ {i} - \mathbf {z} ^ {j} \| _ {2} \geq \epsilon \right] \leq 2 e ^ {2} e ^ {- \epsilon^ {2} / 2 s ^ {2} (i - j)}. \tag {11}
+$$
+
+Equation (11) says that the upperbound for the probability that the Euclidean distance between two embeddings  $\mathbf{z}^i$  and  $\mathbf{z}^j$  is larger than a positive real constant  $\epsilon$  increases as the corresponding augmented instances  $\mathcal{A}^i(\dot{X})$  and  $\mathcal{A}^j(\dot{X})$  are more distant in terms of  $(i - j)$ .
+
+# B Overall Training Algorithm
+
+Algorithm 1 describes how CrossMatch works in time-series consistency regularization. For each iteration, there are two steps for batch initialization: center timestamp sampling from  $\mathcal{T}_L$  and  $\mathcal{T}_U$  (Line 2) and context length sampling from a uniform distribution (Line 3). Note that we assign  $q$  as  $w + c$  after the sampling (Line 4). From labeled center timestamps, instances are sliced from  $\mathcal{X}$  with length  $2q$  to construct a labeled batch,  $\mathcal{X}_l$  and  $\mathcal{Y}_l$ , where  $\mathcal{X}_{[m - q:m + q)} = \{\mathbf{x}_t \mid t \in [m - q:m + q)\}$  and  $\mathcal{Y}_{[m - q:m + q)} = \{y_t \mid t \in [m - q:m + q)\}$  (Line 6). The classification loss for a labeled batch  $\mathcal{L}_l$  is computed and averaged over the batch (Line 7). From unlabeled center timestamps, instances are sliced with context-attached augmentation that generates a pair of instances whose length is  $w + q$  (Line 9). The target instance of each instance in an augmented unlabeled batch  $\mathcal{X}_u$  is then pseudo-labeled using a confidence threshold  $\tau$  (Line 10). CrossMatch softens the pseudo-labels with reliability weighting across two augmented instances with different contexts, transforming two pseudo-labels into a single cross-window label shared across the instances (Line 11). The classification loss with cross-window labels is computed for each pair of augmented instances and averaged over the batch (Line 12). Finally, the losses for labeled and unlabeled batches are then integrated into a single loss  $\mathcal{L}$  with a hyperparameter  $\lambda$ , and the model  $f_{\theta}$  is updated using its gradient (Line 13).
+
+Algorithm 1 Time-series consistency regularization with CrossMatch   
+Input: A time series with initial labels  $\mathcal{D}$  labeled timestamp set  $\mathcal{T}_L$  unlabeled timestamp set  $\mathcal{T}_U$  labeled batch size  $B_{l}$  unlabeled batch size  $B_{u}$  maximum number of iterations I, a model  $f_{\theta}$  confidence threshold  $\tau$  loss weight  $\lambda$  learning rate  $\eta$  half length of a target instance w, maximum context length  $c_{\mathrm{max}}$    
+Output: Final model  $f_{\theta}$    
+1: for each iteration up to I do   
+2:  $\mathcal{T}_l\gets$  Sample  $B_{l}$  timestamps from  $\mathcal{T}_L$  .  $\mathcal{T}_u\gets$  Sample  $B_{u}$  timestamps from  $\mathcal{T}_U$  .   
+3:  $c\gets$  Sample a context length from Uniform(2,  $c_{\mathrm{max}}$  .   
+4:  $q\gets w + c$    
+5: /\*\* Loss computation for labeled batch \*\*/   
+6:  $\mathcal{X}_l\gets \{\mathcal{X}_{[m - q:m + q)},m\in \mathcal{T}_l\} ;\mathcal{Y}_l\gets \{\mathcal{Y}_{[m - q:m + q)},t\in \mathcal{T}_l\} ;$    
+7:  $\mathcal{L}_l = \frac{1}{B_l}\sum_{X\in \mathcal{X}_l,Y\in \mathcal{Y}_l}\ell (X,Y);$  // See Equation (1)   
+8: /\*\* Loss computation for unlabeled batch \*\*/   
+9:  $\mathcal{X}_u\gets \{(\mathcal{X}_{[m - q:m + w]},\mathcal{X}_{[m - w:m + q)}),m\in \mathcal{T}_u\} ;$  // See Section 3.2   
+10:  $\hat{\mathcal{V}}_u\gets$  PSEUDOLABELING(xu,fθ,τ); // See Equation (2)   
+11:  $\bar{\mathcal{V}}\gets$  RELIABILITYWEIGHTING(yu); // See Section 3.3   
+12:  $\mathcal{L}_u = \frac{1}{B_u}\sum_{(X^{\mathrm{left}},X^{\mathrm{right}})\in \mathcal{X}_u,\bar{Y}\in \bar{Y}}\ell_u(X^{\mathrm{left}},X^{\mathrm{right}},\bar{Y})$    
+13:  $\mathcal{L}\gets \mathcal{L}_l + \lambda \mathcal{L}_u;\theta \gets \theta -\eta \nabla_\theta \mathcal{L};$    
+14: end for   
+15: return  $f_{\theta}$
+
+# C Detailed Training Setup
+
+As described in implementation details of Section 4.1, we use MS-TCN as the backbone sequential classifier (Farha & Gall, 2019). It can classify each data point in an instance  $X$ , generating sequential softmax probabilities at each timestamp. MS-TCN has four stages, and each stage is composed of eleven dilated convolution layers and a softmax output layer. The first stage takes a subsequence of the whole time series and outputs a softmax probability distribution at each timestamp. After the first stage, every stage is fed with softmax probabilities and then outputs another softmax probabilities. For all datasets, we use the same training hyperparameters and classifier as listed in Table 6. We set the labeled batch size as 4 and the unlabeled batch size as 8, use an SGD optimizer with the momentum and Nesterov method. The initial learning rate is 0.005 and is scheduled with a cosine decay function. In "Scheduling," the symbol  $i$  denotes a current iteration during training, and the symbol  $I$  denotes the total number of iterations.
+
+Table 6. Training hyperparameters.  
+
+<table><tr><td>Stage</td><td>Layer</td><td>BL</td><td>BU</td><td>Optimizer</td><td>Momentum</td><td>Nesterov</td><td>η</td><td>Scheduling</td></tr><tr><td>4</td><td>11</td><td>4</td><td>8</td><td>SGD</td><td>0.9</td><td>True</td><td>0.005</td><td>cos(7πi/ I)</td></tr></table>
+
+# D Detailed Evaluation Metrics
+
+Timestamp accuracy is the ratio of the number of timestamps with correctly predicted labels to the total number of timestamps in a times series, which is computed as
+
+$$
+\text {T S} = \frac {1}{\left| \mathcal {T} _ {\text {t e s t}} \right|} \sum_ {t \in \mathcal {T} _ {\text {t e s t}}} \mathbf {1} _ {y _ {t} = \hat {y} _ {t}}.
+$$
+
+Segmental F1 score is a performance measure for judging whether a classifier outputs correct and coherent labels for consecutive timestamps. First, segmental precision and segmental recall are computed by counting the number of correct matches between a predicted segment set  $\hat{\mathbb{Y}}$  and the true segment set  $\mathbb{Y}$  with a Jaccard similarity threshold (e.g., 0.25),
+
+$$
+\text {S e g m a n t a l P r e c i s i o n} = \frac {1}{| \hat {\mathbb {Y}} |} \sum_ {\hat {Y} \in \hat {\mathbb {Y}}} \sum_ {Y \in \mathbb {Y}} \mathbf {1} _ {\text {J a c c a r d} (\hat {Y}, Y) > 0. 2 5} \quad \text {a n d} \quad \text {S e g m a n t a l R e c a l l} = \frac {1}{| \mathbb {Y} |} \sum_ {\hat {Y} \in \hat {\mathbb {Y}}} \sum_ {Y \in \mathbb {Y}} \mathbf {1} _ {\text {J a c c a r d} (\hat {Y}, Y) > 0. 2 5}.
+$$
+
+Then, segmental F1 score is computed as  $F1@25 = \frac{2*precision*recall}{precision+recall}$ .
+
+At each iteration, we measure pseudo-labeling precision and pseudo-labeling recall from each timestamp of the target instances in a batch, which are computed as
+
+$$
+\mathrm {P L} \quad \text {P r e c i s i o n} = \frac {\text {t h e n u m b e r o f c o r r e c t P L s}}{\text {t h e n u m b e r o f P L s}} \quad \text {a n d} \quad \mathrm {P L} \quad \text {R e c a l l} = \frac {\text {t h e n u m b e r o f c o r r e c t P L s}}{\text {t h e n u m b e r o f t i m e s t a m p s}}.
+$$
+
+To count the number of correct pseudo-labels, we examine whether a pseudo-label exists for each timestamp and, if so, compare it with the true label. When there is no pseudo-label at any timestamp, we simply define the pseudolabeling precision as 0, because it is undefined otherwise due to division by zero. Likewise, pseudo-labeling F1 score is  $\mathrm{PLF} = \frac{2*\mathrm{PL\ precision*PL\ recall}}{\mathrm{PL\ precision} + \mathrm{PL\ recall}}$ . After a few iterations, the entire time series is expected to be used by pseudo-labeling the sampled target instances.
+
+# E Detailed Experiment Results
+
+Figure 6 is a detailed version of Figure 5, with standard deviation error bars added.
+
+Figure 7 is an expanded version of Figure 5, containing the entire  $y$ -axis for TS accuracy and F1@25.
+
+Figure 8 shows the pseudo-labeling metrics such as pseudo-labeling precision, pseudo-labeling recall, and the total number of pseudo-labels.
+
+Figure 9 shows the accuracy convergence trend when the context length is varied as a factor or multiple of a given length  $c$ , illustrating the accuracy curves during the training iterations.
+
+![](images/881e4167b77839de239080a0437119078bc7fae8710432549ac6537192b2e66e.jpg)  
+Figure 6. Training curve of the classification performance (the first two rows) and pseudo-labeling performance (the last row) over training iterations in Algorithm 1. Error bars represent the standard deviation of five runs.
+
+![](images/3befe405149fb4bcedf8778b04ee95442dc2bad1307dd17c3e8c22146a8156a7.jpg)  
+Figure 7. Training curve of the classification performance over training iterations in Algorithm 1, showing the entire range of each metric.
+
+![](images/556462b90c9527bccd88e82d96def603029192c6266ca29055e89f7e75288189.jpg)  
+Figure 8. Pseudo-labeling precision, pseudo-labeling recall, and the number of pseudo-labels of CrossMatch compared with those of FixMatch, FlexMatch, and PropReg.
+
+![](images/118159e71f7878b37c811e3c3f2f3cc82f5cccb9aa410731534aa804e254df83.jpg)  
+Figure 9. Classification accuracy of CrossMatch for varying the context length  $x \times c$ .

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+# Contextual Conservative Interleaving Bandits
+
+Kei Takemura
+
+# Abstract
+
+The performance of a bandit algorithm is usually measured by the cumulative rewards of the actions chosen by the algorithm. However, in many real-world applications, the rewards in each round should be good enough for reasons such as safety and fairness. In this paper, we investigate the contextual conservative interleaving bandit problem, which has a performance constraint that requires the chosen actions to be not much worse than given baseline actions in each round. This work is the first to simultaneously consider the following practical situations: (1) multiple actions are chosen in a round, (2) the feature vectors associated with given actions depend on the round, and (3) the performance constraints in each round that depend only on the actions chosen in that round. We propose a meta-algorithm, Greedy on Confidence Widths (GCW), that satisfies the performance constraints with high probability. GCW uses a standard bandit algorithm and achieves minimax optimal regret up to logarithmic factors if the algorithm used is also minimax optimal. We improve the existing analyses for the  $\mathrm{C}^2\mathrm{UCB}$  algorithm and the Thompson sampling to combine with GCW. We show that these algorithms achieve near-optimal regret when the feasible sets of given actions are the bases of a matroid. Our numerical experiments on a real-world dataset demonstrate that GCW with the standard bandit algorithms efficiently improves performance while satisfying the performance constraints.
+
+# 1. Introduction
+
+The stochastic multi-armed bandit (MAB) problem and its extensions have been studied as sequential decision-making problems under uncertainty. In the MAB problem, a learner iterates the following process  $T$  times: The learner chooses
+
+an action from given actions and then obtains a reward for the chosen action. The learner aims to maximize the sum of rewards. The central challenge of this problem is the exploration-exploitation trade-off due to the lack of prior knowledge about the rewards. The MAB problem has been generalized to many directions to model more realistic situations of real-world applications. For example, the combinatorial semi-bandit (CS) problem (Gai et al., 2012; Kveton et al., 2014; 2015; Wang & Chen, 2018) enables the learner to choose multiple actions at once and observe corresponding rewards, the contextual linear bandit (CLB) problem (Abe & Long, 1999; Auer, 2002; Abbasi-Yadkori et al., 2011; Chu et al., 2011) introduces time-dependent actions associated with feature vectors, and the contextual combinatorial semi-bandit (CCS) problem (Qin et al., 2014; Takemura & Ito, 2019; Takemura et al., 2021) generalizes both the CS and the CLB problems.
+
+In recent years, the conservative bandit problems (Wu et al., 2016; Kazerouni et al., 2017; Garcelon et al., 2020; Khezeli & Bitar, 2020; Moradipari et al., 2020; Katariya et al., 2019) have been studied, where constraints on the learner's performance are introduced. Specifically, the learner additionally observes baseline actions for given actions and has to satisfy performance constraints that require that the learner's performance is not much worse than the performance by the baseline actions. The performance constraints mitigate a drawback that the standard bandit algorithms perform poorly in early rounds.
+
+Unfortunately, these studies are not yet suitable for many real-world applications. In real-world applications such as recommendations, the following properties summarized in Table 1 should often be satisfied simultaneously: (1) the learner chooses multiple actions at once, (2) the feasible and the baseline actions depend on the round, and (3) the performance constraints guarantee the performance of each round. However, none of the existing studies addresses all of these properties. In Section 2, we will compare this work with the existing studies on the conservative bandit problems in detail.
+
+This paper investigates the contextual conservative interleaving bandit (CCIB) problem, which satisfies the three properties above. Specifically, the CCIB problem is the CCS problem with stage-wise performance constraints. As
+
+Table 1. Conservative bandit problems.  
+
+<table><tr><td>Algorithm</td><td>Combinatorial action set</td><td>Time-dependent feasible and baseline actions</td><td>Stage-wise constraints</td></tr><tr><td>Conservative UCB (Wu et al., 2016)</td><td></td><td></td><td></td></tr><tr><td>CLUCB (Kazerouni et al., 2017)</td><td></td><td>✓</td><td></td></tr><tr><td>CLUCB2 (Garcelon et al., 2020)</td><td></td><td>✓</td><td></td></tr><tr><td>SEGE (Khezeli &amp; Bitar, 2020)</td><td></td><td></td><td>✓</td></tr><tr><td>SCLUCB &amp; SCLTS (Moradipari et al., 2020)</td><td></td><td></td><td>✓</td></tr><tr><td>iUCB (Katariya et al., 2019)</td><td>✓</td><td></td><td>✓</td></tr><tr><td>Algorithm 1 (This work)</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+in previous studies on bandit algorithms, we measure the performance of an algorithm by its regret, which is the difference between the sum of the rewards of the optimal actions and that of the algorithm's actions.
+
+Our main contribution is to propose an algorithm that is minimax optimal up to logarithmic factors. Specifically, the proposed algorithm achieves, with probability at least  $1 - \delta$ ,  $\widetilde{O}(\min(\sqrt{dkT}, d / \Delta) + dk + \min(k\sqrt{dT/n}, dk/(n\Delta)))$  regret while satisfying the constraints, where  $d$  is the dimension of the feature vectors,  $k$  is the number of actions to be chosen in a round,  $T$  is the number of rounds,  $n$  is a parameter that controls how conservative the algorithm is,  $\Delta$  is a sub-optimality gap, and  $\widetilde{O}(\cdot)$  ignores the logarithmic factors in  $d$ ,  $k$ ,  $T$ , and  $1/\delta$ . To show the proposed algorithm is almost optimal, we show matching lower bounds by reducing the CCIB problem to the conservative MAB problem.
+
+The proposed algorithm is a meta-algorithm using an algorithm for the CCS problem. Our algorithm interleaves the baseline actions with the actions that the given algorithm recommends to satisfy the performance constraints. The regret by the proposed algorithm can be represented as the sum of the regret due to choosing the baseline actions and that due to choosing the given algorithm's actions. In terms of these high-level ideas, our algorithm and analysis are similar to the iUCB algorithm (Katariya et al., 2019) and its regret analysis for the conservative interleaving bandit (CIB) problem, respectively.<sup>1</sup> However, we cannot use the analysis for the iUCB algorithm to bound the regret of choosing the baseline actions due to the time-dependent actions.
+
+The time-dependent actions make it difficult to bound the regret suffered by the baseline actions. This regret can be bounded by the widths of the confidence intervals of the reward estimates for the baseline and the given algorithm's actions. Since the feasible sets of actions are fixed in the
+
+CIB problem, the iUCB algorithm can choose all of the baseline and the given algorithm's actions over multiple rounds. Thus, we can use a standard analysis for UCB-type algorithms to bound the regret. However, we cannot take this approach for the CCIB problem since a feature vector in a round may never appear in future rounds. Therefore, it is necessary to bound the confidence widths for the actions that are not chosen by the proposed algorithm since the proposed algorithm cannot choose all the actions of the baseline and the given algorithm's actions.
+
+To overcome this difficulty, the proposed algorithm preferentially chooses actions with large confidence widths from the actions of the baseline and the given algorithm. This technique allows us to bound the confidence widths for the actions that are not chosen by the proposed algorithm using those for the the proposed algorithm's actions. Consequently, we can use existing analyses to bound the regret due to satisfying the performance constraints. Since the proposed algorithm chooses the actions with large confidence widths, a sophisticated analysis is needed to bound the sum of these confidence widths. To the best of our knowledge, our analysis is the first to bound the regret by the confidence widths for the actions not chosen by the algorithm. We believe that our technique is useful for sibling problems.
+
+We consider the  $\mathrm{C^2UCB}$  algorithm (Qin et al., 2014) and the (round-wise) Thompson sampling (Takemura & Ito, 2019) as the algorithm that the proposed algorithm uses. We show that these algorithms achieve near-optimal regret bounds for the CCS problem with the feasible actions characterized by any matroid. Moreover, we show the first gap-dependent regret bound for the  $\mathrm{C^2UCB}$  algorithm. Specifically, the  $\mathrm{C^2UCB}$  algorithm and the Thompson sampling achieve  $\widetilde{O} (\min (\sqrt{dkT},d / \Delta) + dk)$  and  $\widetilde{O} (d^{3 / 2}\sqrt{kT} +dk)$  regrets, respectively. The first term of the regret bound of the Thompson sampling has a gap from the optimal bound. However, we can also see this gap in the regret bound of Thompson sampling for the CLB problem (Agrawal & Goyal, 2013; Abeille & Lazaric, 2017). Note that the proposed algorithm is minimax optimal if it uses the  $\mathrm{C^2UCB}$  algorithm.
+
+We evaluate the proposed algorithm through numerical experiments on a real-world dataset. We conduct our experiments in two cases. One is the case where the feasible and the baseline actions are fixed across the rounds. The other is the case where the given actions depend on the round. In the first case, we compare the proposed algorithm with the iUCB algorithm and its variant that uses the feature vectors to estimate the rewards and the confidence intervals. We also compare our algorithm with the baseline actions of the proposed algorithm and some standard bandit algorithms in both cases. In the first case, the proposed algorithm outperforms other algorithms except for the variant of the iUCB algorithm. The variant of the iUCB algorithm is compatible with the proposed algorithm in terms of regret. However, the variant of the iUCB algorithm is unstable in terms of performance constraints. In the second case, the proposed algorithm behaves similarly to the first. In other words, we observe that our algorithm is robust to the changes in the given actions.
+
+# 2. Related Work
+
+# 2.1. Conservative Bandits
+
+The concept of the conservative bandit problems was proposed by Wu et al. (2016). They studied the MAB problem with constraints that in each round, the learner has to satisfy that the cumulative reward until this round is not much worse than the cumulative reward by the baseline. This work was generalized to the CLB problem (Kazerouni et al., 2017; Garcelon et al., 2020). While these studies consider the constraints on cumulative rewards, the performance constraints should guarantee the performance of each round in real-world applications. For instance, let us consider a recommender system. If the system recommends items that do not match the target user's preferences at all due to exploration by the bandit algorithms, this user may never use the system again. Furthermore, it is unfair to recommend unfavorable items to one user due to the exploration but favorable items to another.
+
+Khezeli & Bitar (2020) and Moradipari et al. (2020) introduced stage-wise constraints that consider the rewards of each round to the linear bandit problem with a time-independent convex action set. While the stage-wise constraints solve the performance issue in early rounds, these studies strongly rely on the assumption that the action set is time-independent and convex. In other words, we cannot apply these studies to many real-world applications (e.g., the recommender system considered above).
+
+Katariya et al. (2019) studied the CIB problem, which is the CS problem with baseline actions and stage-wise performance constraints. They simultaneously handled the non-convex action set and the stage-wise constraints by in
+
+troducing the problem in which the learner chooses multiple actions in a round. Moreover, the learner must often choose multiple actions simultaneously in real-world applications. Note that the CIB and CCIB problems denote the problems of maximizing the sum of rewards, while the CS and CCS problems cover some classes of non-linear functions of the rewards.
+
+While the CIB problem assumes that the feasible and the baseline actions do not depend on the round, the feasible and the baseline actions often depend on the round in real-world applications. In a news article recommendation, for example, news articles would change over the rounds as old news articles are deleted or the latest news articles are added (Li et al., 2010; Wang et al., 2017). The given actions correspond to the news articles in this example. Note that the feasible and the baseline actions could depend on the round, even if the given actions are fixed over the rounds.
+
+# 2.2. Contextual Combinatorial Semi-Bandits
+
+Qin et al. (2014) proposed the CCS problem and the  $\mathrm{C}^2\mathrm{UCB}$  algorithm. Takemura & Ito (2019) and Takemura et al. (2021) improved the regret bound by the  $\mathrm{C}^2\mathrm{UCB}$  algorithm. In particular, Takemura et al. (2021) showed that the  $\mathrm{C}^2\mathrm{UCB}$  algorithm is minimax optimal up to logarithmic factors when the partition matroid characterizes the feasible actions. Our analysis of the  $\mathrm{C}^2\mathrm{UCB}$  algorithm is an extension of the analysis by Takemura et al. (2021).
+
+Takemura & Ito (2019) proposed Thompson sampling algorithms for the CCS problem was proposed and showed that this algorithm achieves  $\widetilde{O} (\max (\sqrt{d},\sqrt{k})d\sqrt{kT})$  regret. Unlike the  $\mathrm{C^2UCB}$  algorithm, the Thompson sampling algorithms in the CCS problem with the feasible actions characterized by matroids have not been studied. In addition, a gap-dependent bound of the Thompson sampling algorithms for the CCS problem is not known.
+
+# 3. Problem Setting
+
+# 3.1. Contextual Conservative Interleaving Bandits
+
+We formally define the CCIB problem. This problem consists of  $T$  rounds, and a learner chooses a set of  $k$  actions from a given family of action sets in each round. Each action, called arm, is associated with a  $d$ -dimensional feature vector. Let  $N$  denote the number of given arms, and each arm is indexed by an integer in  $[N] := \{1, 2, \dots, N\}$ . Note that the learner knows the above parameters in advance.
+
+The learner progresses through each round as follows. At the beginning of the  $t$ -th round, a learner observes the feature vectors  $\{x_{t}(i)\}_{i\in [N]}\subseteq \mathbb{R}^{d}$  of the arms. In addition, the learner observes a family  $\mathcal{S}_t\subseteq 2^{[N]}$  of feasible sets of arms and a feasible set  $B_{t}\in \mathcal{S}_{t}$  called baseline arms. Then, the
+
+learner chooses a feasible set  $I_{t} \in S_{t}$ . We assume that each feasible set of arms is of size  $k$ , i.e.,  $\forall I \in S_{t}, |I| = k$ . At the end of the round, the learner obtains the rewards  $\{r_{t}(i)\}_{i \in I_{t}}$ , where for all  $i \in I_{t}$ ,  $r_{t}(i) = \theta^{\top} x_{t}(i) + \eta_{t}(i)$  for some unknown parameter  $\theta \in \mathbb{R}^{d}$  and a random noise  $\eta_{t}(i) \in \mathbb{R}$ . We will introduce the assumption on  $\eta_{t}(i)$  in Section 3.2.
+
+We evaluate the performance of the learner by the regret  $R(T)$ , which is defined as
+
+$$
+R (T) = \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in I _ {t}} \mu_ {t} (i)\right),
+$$
+
+where  $\mu_t(i) = \theta^\top x_t(i)$  for all  $i \in [N]$  and  $t \in [T]$ , and  $I_t^* \in \operatorname{argmax}_{I \in \mathcal{S}_t} \sum_{i \in I} \mu_t(i)$  for all  $t \in [T]$ . The learner aims to minimize the regret, which is equivalent to maximizing  $\sum_{t \in [T]} \sum_{i \in I_t} \mu_t(i)$ .
+
+Compared to the standard bandit problems, the CCIB problem additionally introduces stage-wise performance constraints as in the CIB problem (Katariya et al., 2019). These constraints require that chosen arms' performance in each round should not be much worse than the baseline arms' performance of the round. More precisely, with high probability, there exists a bijection  $\rho_t: I_t \to B_t$  such that
+
+$$
+\sum_ {i \in I _ {t}} \mathbb {1} \left(\mu_ {t} (i) <   \mu_ {t} \left(\rho_ {t} (i)\right)\right) \leq m \tag {1}
+$$
+
+for all  $t \in [T]$ , where  $m$  is a non-negative parameter that represents how  $I_{t}$  should be conservative. Our constraint allows aggressive explorations when  $m$  is large. Note that choosing  $B_{t}$  satisfies this constraint.
+
+Several existing studies (Wu et al., 2016; Kazerouni et al., 2017; Garcelon et al., 2020; Moradipari et al., 2020) define a performance constraint using the sum of obtained rewards. An adaptation of this constraint to our problem is that
+
+$$
+\sum_ {i \in I _ {t}} \mu_ {t} (i) \geq (1 - \alpha) \sum_ {i \in B _ {t}} \mu_ {t} (i) \tag {2}
+$$
+
+for a fixed  $\alpha \in [0,1]$  for all  $t\in [T]$ , with high probability. In general, this constraint is not directly comparable to our constraint. We show this fact in Appendix B.1. However, we show that constraint (1) implies constraint (2) under a reasonable assumption, which will be discussed in Section 3.2.
+
+# 3.2. Assumptions
+
+We introduce several assumptions on the CCIB problem and define a few parameters of the problem.
+
+First, we discuss the assumptions of the feature vectors, the unknown parameter, the rewards, and their noises. Recall that  $\mu_t(i) = \theta^\top x_t(i)$  and  $r_t(i) = \mu_t(i) + \eta_t(i)$ . We assume the following standard assumptions in the CCS problem (Qin et al., 2014; Takemura & Ito, 2019; Takemura et al., 2021) and the conservative bandit problems (Wu et al., 2016; Kazerouni et al., 2017; Garcelon et al., 2020):
+
+Assumption 3.1. The feature vectors  $\{x_{t}(i)\}_{i\in [N]}$  are determined by oblivious adversary.
+
+Assumption 3.2. The learner knows  $\| \theta \|_2 \leq M$  and  $\| x_t(i) \|_2 \leq L$  for all  $i \in [N]$  and  $t \in [T]$ .
+
+Assumption 3.3. The mean reward  $\mu_t(i)$  satisfies  $\mu_t(i) \in [0,1]$  for all  $i \in [N]$  and  $t \in [T]$ .
+
+Assumption 3.4. The noise sequence  $\{\eta_t(i)\}_{i\in I_t,t\in [T]}$  is conditionally  $R$  -sub-Gaussian, i.e,
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \exp (\lambda \eta_ {t} (i)) \mid \{x _ {s} (j) \} _ {j \in I _ {s}, s \in [ t ]}, \{\eta_ {s} (j) \} _ {j \in I _ {s}, s \in [ t - 1 ]} \right] \\ \leq \exp (\lambda^ {2} R ^ {2} / 2) \\ \end{array}
+$$
+
+for all  $\lambda \in \mathbb{R}$ ,  $i\in I_t$  and  $t\in [T]$ .
+
+Note that Assumption 3.4 implies that the mean of  $\eta_t(i)$  is zero for all  $i\in I_t$  and  $t\in [T]$ .
+
+Next, we define the requirements for the algorithm used by the proposed meta-algorithm.
+
+Assumption 3.5. The algorithm used by the proposed meta-algorithm works for the CCS problem (i.e., the CCIB problem without the performance constraints). The algorithm has an internal model for the estimation of rewards. In a round  $t$ , the algorithm runs as follows:
+
+1. It estimates the rewards of given arms based on observed feature vectors.  
+2. It chooses arms by solving  $\operatorname{argmax}_{I\in S_t}r_t'(i)$ , where  $\{r_t'(i)\}_{i\in [N]}$  is estimation of the rewards.  
+3. It plays the chosen arms and updates the internal model using the feature vectors, the chosen arms, and the observed rewards.
+
+Furthermore, the algorithm can retry the second step with another set  $S_{t}^{\prime}$  of feasible actions as long as it is before executing the third step.
+
+Note that the standard bandit algorithms such as the UCB-type algorithms, the Thompson sampling, and the  $(\varepsilon -)$  greedy algorithm satisfy Assumption 3.5.
+
+Next, we discuss the family of feasible sets of arms, i.e.,  $S_{t}$ . As in the CIB problem (Katariya et al., 2019), we focus on exchangeable set:
+
+Definition 3.6 (Exchangeable set). Given a set  $E$ , a family  $S \subseteq 2^{E}$  is exchangeable if for any two sets  $I_{1}, I_{2} \in S$ , there exists a bijection  $\rho : I_{1} \to I_{2}$  such that
+
+$$
+\forall J \subseteq I _ {1}, (I _ {1} \backslash J) \cup \{\rho (i) \mid i \in J \} \in \mathcal {S}. \tag {3}
+$$
+
+Assumption 3.7. The family of feasible sets of arms,  $\mathcal{S}_t$ , is exchangeable for all  $t \in [T]$ .
+
+Assumption 3.8. For any  $t \in [T]$  and  $I_1, I_2 \in S_t$ , the learner knows a bijection  $\rho: I_1 \to I_2$  that satisfies (3).
+
+The set  $E$  in Definition 3.6 corresponds to  $[N]$  in our problem. Note that if  $S_{t}$  is the bases of a uniform matroid or a partition matroid, it is known how to construct the bijection that satisfies (3) (Katariya et al., 2019).
+
+We discuss how large the class of exchangeable sets is. Katariya et al. (2019) pointed out that the set of the bases of a strongly base-orderable matroid is exchangeable, but the converse remains an open question. This paper solves this question in the affirmative, i.e., we show that every exchangeable set is the set of the bases of some strongly base-orderable matroid. We defer our proof to Appendix C. It is known that the strongly base-orderable matroid includes the uniform matroid, the partition matroid, and the transversal matroid.
+
+Next, we discuss the lower bound of the parameter  $m$ . Intuitively, small  $m$  makes the CCIB problem difficult. In fact, when  $m = 0$ , no algorithm can achieve sub-linear regret while satisfying our performance constraint. We defer our proof to Appendix E.1. To obtain sub-linear regret, we assume the following:
+
+Assumption 3.9. The parameter  $m$  satisfies  $m \geq 1$ .
+
+Finally, we introduce a reasonable assumption for a reduction from the constraint (2) to constraint (1). We defer our proof to Appendix B.2.
+
+Assumption 3.10. Let  $r_{\ell} = \min_{t\in [T]}\min_{i\in B_t}\mu_t(i)$ . Then, the parameters  $k, m, \alpha$  and  $r_{\ell}$  satisfy  $\alpha r_{\ell} \geq m / (k - m)$ .
+
+Note that we do not need Assumption 3.10 to satisfy constraint (1). Since we have a reduction, we will focus on constraint (1) in the rest of this paper.
+
+# 4. Proposed Algorithm
+
+In this section, we propose an algorithm for the CCIB problem. Our algorithm, described in Algorithm 1, is the first to deal with time-dependent feature vectors and stage-wise performance constraints, as discussed in Section 1. Since our algorithm is a meta-algorithm, it requires a base algorithm  $\mathcal{A}$  for the CCS problem as input.
+
+At the beginning of the  $t$ -th round, our algorithm observes the feature vectors  $\{x_{t}(i)\}_{i\in [N]}$ , feasible sets  $\mathcal{S}_t$ , and baseline arms  $B_{t}$  (line 3). Then, our algorithm receives the arms
+
+# Algorithm 1 Greedy on confidence widths
+
+Input:  $\lambda > 0, \delta \in (0,1), n \in \mathbb{N}$ , and an algorithm  $\mathcal{A}$ .
+
+1:  $V_{0}\gets \lambda I$  and  $b_0\gets 0$  
+2: for  $t = 1,2,\ldots ,T$  do
+
+3: Observe  $\{x_{t}(i)\}_{i\in [N]},S_{t}$  , and  $B_{t}$  
+4:  $\hat{I}_t \in \operatorname{argmax}_{I \in \mathcal{S}_t} \sum_{i \in I} r_t'(i)$ , where  $\{r_t'(i)\}_{i \in [N]}$  is the estimation of rewards by  $\mathcal{A}$ .  
+5: Define  $\bar{r}_t(i)$  for all  $i\in [N]$  according to (4).  
+6:  $\bar{I}_t\in \mathrm{argmax}_{I\in S_t}\sum_{i\in I}\bar{r}_t(i)$  
+7: Let  $\bar{\rho}_t:\bar{I}_t\to \hat{I}_t$  be the bijection that satisfies (3).  
+8:  $\bar{I}_t^0\gets \bar{I}_t$  and  $J_{t}^{0}\leftarrow \emptyset$  
+9: for  $\ell = 1,2,\ldots ,n$  do  
+0:  $i_t^\ell \in \mathrm{argmax}_{i\in \bar{l}_t^{\ell -1}}\max (c_t(i),c_t(\bar{\rho}_t(i))).$  
+11:  $\bar{I}_t^\ell \gets \bar{I}_t^{\ell -1}\setminus \{i_t^\ell \} .$  
+12:  $j_{t}^{\ell}\gets \mathrm{argmax}_{j\in \{i_{t}^{\ell},\hat{\rho}_{t}(i_{t}^{\ell})\}}c_{t}(j).$  
+13:  $J_{t}^{\ell}\gets J_{t}^{\ell -1}\cup \{j_{t}^{\ell}\} .$  
+14: end for  
+15: Choose  $I_{t} \in S_{t}$  such that  $|I_{t} \setminus \bar{I}_{t}| \leq n$ ,  $J_{t}^{n} \subseteq I_{t} \subseteq \hat{I}_{t} \cup \bar{I}_{t}$ , and  $i \in I_{t}$  if and only if  $\bar{\rho}_{t}(i) \notin I_{t}$  for all  $i \in \bar{I}_{t} \setminus \hat{I}_{t}$ .  
+16: Play  $I_{t} \in S_{t}$  and observe the rewards  $\{r_{t}(i)\}_{i \in I_{t}}$ .  
+17: Feed  $\hat{I}_t \cap I_t$  and the corresponding rewards to  $\mathcal{A}$ , and update  $\mathcal{A}$  as if it has chosen  $\hat{I}_t \cap I_t$ .  
+18:  $V_{t}\gets V_{t - 1} + \sum_{i\in I_{t}}x_{t}(i)x_{t}(i)^{\top}.$  
+19:  $b_{t}\gets b_{t - 1} + \sum_{i\in I_{t}}r_{t}(i)x_{t}(i).$  
+20: end for
+
+$\hat{I}_t$  that the base algorithm  $\mathcal{A}$  recommends (line 4). In the following and our analysis, we call  $\hat{I}_t$  recommended choice.
+
+Then, the proposed algorithm starts to interleave the recommended arms with the baseline arms. At the first step of our interleaving, the proposed algorithm obtains safe arms  $\bar{I}_t$  such that there exists a bijection  $\rho_t: \bar{I}_t \to B_t$  that satisfies  $\mu_t(i) \geq \mu_t(\rho_t(i))$  for all  $i \in \bar{I}_t$  with high probability. To obtain the safe arms, we define the following rewards and solve the maximization problem (lines 5 and 6):
+
+$$
+\bar {r} _ {t} (i) = \left\{ \begin{array}{l l} \hat {r} _ {t} (i) & \text {i f} i \in B _ {t} \\ \check {r} _ {t} (i) & \text {i f} i \in \hat {I} _ {t} \backslash B _ {t} \\ - \infty & \text {o t h e r w i s e} \end{array} , \right. \tag {4}
+$$
+
+where  $\hat{r}_t(i) = \hat{\theta}_t^\top x_t(i) + c_t(i)$ ,  $\check{r}_t(i) = \hat{\theta}_t^\top x_t(i) - c_t(i)$ ,  $\hat{\theta}_t = V_{t-1}^{-1}b_{t-1}$ ,  $c_t(i) = \beta_t(\delta)\sqrt{x_t(i)^\top V_{t-1}^{-1}x_t(i)}$ , and  $\beta_t(\delta) = \widetilde{O}(\sqrt{\min(\log(N),d)} + M\sqrt{\lambda})$  for all  $i \in [N]$ . The formal definition of  $\beta_t(\delta)$  is described in Appendix D.1. Note that  $\bar{I}_t \subseteq \hat{I}_t \cup B_t$  because  $B_t$  is a feasible solution with a finite objective value. Note also that since  $S_t$  is the bases of a matroid, this optimization problem can be solved efficiently by the greedy algorithm (Korte & Vygen, 2018).
+
+Then, the proposed algorithm chooses the arms from  $\hat{I}_t\cup \bar{I}_t$  for the exploration. Our algorithm needs a positive integer
+
+$n$ , which controls the maximum number of arms for the exploration. The algorithm preferentially chooses  $n$  arms with large confidence widths as  $J_{t}^{n}$  (lines 8-14). Then, the algorithm chooses the playing arms  $\bar{I}_{t}$  that includes  $J_{t}^{n}$  (line 15). Note that one can choose any set of arms that satisfies all conditions in line 17 of Algorithm 1. For example, the set  $(\bar{I}_t\setminus \{j\in J_t^n\cap (\hat{I}_t\setminus \bar{I}_t)\mid \bar{\rho}_t^{-1}(j)\})\cup (J_t^n\cap (\hat{I}_t\setminus \bar{I}_t))$  satisfies the conditions. If  $|J_{t}^{n}\cap (\hat{I}_{t}\setminus \bar{I}_{t})| < n$  and  $\hat{I}_t\backslash (J_t^n\cup \bar{I}_t)\neq \emptyset$  we can further exchange the arms in  $\hat{I}_t\setminus (J_t^n\cup \bar{I}_t)$  for the corresponding baseline arms.
+
+At the end of the  $t$ -th round, the proposed algorithm and the base algorithm  $\mathcal{A}$  update their reward estimation by the observed rewards (lines 17-19). This update of  $\mathcal{A}$  can be seen as a retry of choosing arms by  $\mathcal{A}$ . In fact, it holds that  $\hat{I}_t \cap I_t \in \operatorname{argmax}_{I \in S_t'} \sum_{i \in [N]} r_t'(i)$  for some  $S_t'$  (see our proof of Lemma 5.2 for details). Therefore, the base algorithm  $\mathcal{A}$  runs on an instance of the CCS problem. In other words, we can use existing regret bounds of  $\mathcal{A}$  to bound the regret by  $\hat{I}_t \cap I_t$ .
+
+To bound the regret by the proposed algorithm, the central part of our analysis (and the analysis by Katariya et al. (2019)) is to bound the regret by  $\{I_t\backslash \hat{I}_t\}_{t\in [T]}$ , i.e., bounding the penalty by satisfying our performance constraint. We call the set  $I_{t}\backslash \hat{I}_{t}$  conservative choice. Note that  $I_{t}\backslash \hat{I}_{t}\subseteq B_{t}$ . The regret by the conservative choices can be bounded by the confidence widths for the conservative choices and those for the recommended choices.
+
+For the CIB problem (i.e.,  $\{x_{i}(i)\}_{i\in [N]}$ ,  $S_{t}$ , and  $B_{t}$  are fixed over the round)<sup>3</sup>, Katariya et al. (2019) showed that the confidence widths by the iUCB algorithm for the conservative choices can be bounded by those for the recommended choices. Since  $S_{t}$  fixed, the iUCB algorithm chooses the conservative and the recommended arms using multiple rounds and it is possible to use a standard analysis for UCB-type algorithms to bound the regret. However, we cannot use this approach for the CCIB problem because of the contextual setting. First, we cannot bound the confidence width for the conservative choices by that for the recommended choices since the baseline arms may change. Second, our algorithm may not be able to choose the recommended choices in other rounds because these feature vectors may not appear in other rounds.
+
+To solve these problems, our algorithm chooses  $J_{t}^{n}$ . As discussed in Section 1, choosing  $J_{t}^{n}$  enables us to bound the confidence widths for the arms that are not chosen by our algorithm using those for the playing arms  $I_{t}$ . Then, we can utilize the existing analyses for the standard bandit algorithms. We formally show this fact in the next section.
+
+# 5. Regret Analysis
+
+In this section, we introduce and prove our regret bound of the proposed algorithm. Let  $S_{t}^{*} = \operatorname{argmax}_{I\in S_{t}}\sum_{i\in I}\mu_{t}(i)$ . Throughout our analysis, we fix  $I_{t}^{*}\in S_{t}^{*}$  arbitrarily for all  $t\in [T]$  and assume that  $\forall t\in [T],S_{t}\neq S_{t}^{*}$  for ease of presentation. Note that we have no regret when  $S_{t} = S_{t}^{*}$ . Then, we define the following suboptimality gaps:  $\Delta_t(i,j) = \mu_t(j) - \mu_t(i)$  and  $\Delta = \min_{t\in [T],I\in S_t\backslash S_t^*,i\in I,i^*\in I_t^*: \Delta_t(i,i^*) > 0}\Delta_t(i,i^*)$
+
+Using our gap  $\Delta$ , we introduce our regret bound:
+
+Theorem 5.1. Let  $\kappa = \min (\log (N),d)$ $\nu = \log (kT)$  and  $\iota = \min (\log (NkT / \delta),d\log (kT / \delta))$  . If  $\lambda = (R / M)^2\kappa$  and  $n\leq m$  , with probability at least  $1 - \delta$  Algorithm 1 satisfies performance constraint (1) for all  $t\in [T]$  and achieves the following regret bound:
+
+$$
+\begin{array}{l} R (T) = R _ {\mathcal {A}} (T) + O \left(\min  \left(\sqrt {d k T \nu \iota}, d \nu \iota / \Delta\right) + d k \nu \right. \\ \left. + \min \left(k \sqrt {d T \nu \iota / n}, d k \nu \iota / (n \Delta)\right)\right), \\ \end{array}
+$$
+
+where  $R_{\mathcal{A}}(T)$  is the regret bound of the base algorithm  $\mathcal{A}$  for the CCS problem.
+
+Our regret bound consists of the regret bound of the base algorithm  $\mathcal{A}$  and the penalty for satisfying the performance constraints. We will show in Section 6 that the penalty term is minimax optimal up to logarithmic factors. Furthermore, the penalty term matches the regret bound of the iUCB algorithm by Katariya et al. (2019) when we apply the proposed algorithm to the CIB problem.
+
+In the remainder of this section, we prove Theorem 5.1 and the regret bounds for the two algorithms used as the base algorithm  $\mathcal{A}$ . We defer all missing proofs to Appendix D. Recall that  $\hat{I}_t$  is the recommended choice (line 4 of Algorithm 1), and  $I_t \setminus \hat{I}_t$  is the conservative choice. Let  $\hat{\rho}_t^* : \hat{I}_t \to I_t^*$  be the bijection satisfying (3). Let  $\rho_t^* : I_t \to I_t^*$  denote  $\rho_t^*(i) = \hat{\rho}_t^*(i)$  if  $i \in \hat{I}_t$  and  $\rho_t^*(i) = \hat{\rho}_t^*(\bar{\rho}_t(i))$  otherwise. Let  $R_t(i) = \mu_t(\rho_t^*(i)) - \mu_t(i)$  for  $i \in I_t \setminus \hat{I}_t$ .
+
+Our analysis decomposes the regret as follows and bounds them separately:
+
+$$
+R (T) = \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} \cap \hat {I} _ {t}} R _ {t} (i) + \sum_ {i \in I _ {t} \backslash \hat {I} _ {t}} R _ {t} (i)\right). \tag {5}
+$$
+
+The former is the regret of the recommended choices, and the latter is the regret of the conservative choices. We have the following lemma for the regret by the recommended choices:
+
+Lemma 5.2. We have  $\sum_{t\in [T]}\sum_{i\in I_t\cap \hat{I}_t}R_t(i)\leq R_{\mathcal{A}}(T).$
+
+As a common tool for our analysis, we bound the estimation errors of the rewards as in previous work (Qin et al., 2014; Takemura & Ito, 2019; Takemura et al., 2021). Let  $\| x \|_A$  denote  $\sqrt{x^\top A x}$  for all  $x \in \mathbb{R}^d$  and  $A \in \mathbb{R}^{d \times d}$  such that  $A$  is positive definite. Then, we have the following:
+
+Lemma 5.3. Suppose that  $\delta \in (0,1)$ . We have, with probability at least  $1 - \delta$ , for all  $t \in [T]$  and  $i \in [N]$ ,
+
+$$
+\left| \mu_ {t} (i) - \hat {\theta} _ {t} ^ {\top} x _ {t} (i) \right| \leq \beta_ {t} (\delta) \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}}. \tag {6}
+$$
+
+In our proof of Theorem 5.1, we assume that (6) holds.
+
+# 5.1. Regret by Conservative Choices
+
+Similar to the analysis by Katariya et al. (2019), we can bound the regret of an arm in the conservative choices by the sum of the confidence widths for that arm and the corresponding arm in the recommended choices:
+
+Lemma 5.4. For any  $t \in [T]$  and any  $i \in I_t \setminus \hat{I}_t$ , we have  $\mu_t(\rho_t^*(i)) - \mu_t(i) \leq 2(c_t(i) + c_t(\bar{\rho}_t(i)))$ .
+
+Since the confidence widths of  $J_{t}^{n}$  is larger than those of  $(\bar{I}_{t} \cup \hat{I}_{t}) \setminus J_{t}^{n}$ , we can bound the regret by the sum of the confidence widths of  $J_{t}^{n}$ . However, to bound such large confidence widths, we need two additional theoretical tools.
+
+First, we show that the number of times the confidence widths for  $J_{t}^{n}$  are greater than the sub-optimality gap can be bounded. Recall that  $j_{t}^{\ell}$  is defined in line 15 of Algorithm 1 for all  $\ell \in [n]$ .
+
+Lemma 5.5. Let  $\gamma_{\ell} = \sqrt{\ell}\Delta /(4\beta_{T}(\delta))$  for all  $\ell \in [n]$ . Then, for any  $\ell \in [n]$ , we have  $\sum_{t\in [T]}\mathbb{1}(c_t(j_t^\ell)\geq \Delta /4)\leq 2d\log (1 + L^2 kT / (d\lambda)) / \min (\gamma_\ell^2,1)$
+
+Second, we show that the conservative choice suffers no regret if their confidence widths are smaller than the suboptimality gap. Recall that  $\hat{\rho}_t^*:\hat{I}_t\to I_t^*$  and  $\bar{\rho}_t(i):\bar{I}_t\rightarrow \hat{I}_t$  are the bijections satisfying (3).
+
+Lemma 5.6. For any  $i\in \bar{I}_t$  , if  $c_{t}(i) + c_{t}(\bar{\rho}_{t}(i)) <   \Delta /2$  we have  $\mu_t(i) = \mu_t(\bar{\rho}_t(i)) = \mu_t(\hat{\rho}_t^* (\bar{\rho}_t(i)))$
+
+We are ready to bound the regret due to the conservative choices. The regret can be rewritten as
+
+$$
+\begin{array}{l} \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash \hat {I} _ {t}} R _ {t} (i) = \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash (\hat {I} _ {t} \cup J _ {t} ^ {n})} R _ {t} (i) (7) \\ + \sum_ {t \in [ T ]} \sum_ {i \in J _ {t} ^ {n} \backslash \hat {I} _ {t}} R _ {t} (i). (8) \\ \end{array}
+$$
+
+First, we bound the regret due to the right-hand side of (7). Let  $\bar{I}_t' = I_t \setminus (\hat{I}_t \cup J_t^n)$ . Since  $\beta_T(\delta) \geq \beta_t(\delta)$  for all  $t \in [T]$
+
+and  $c_{t}(i) + c_{t}(\bar{\rho}_{t}(i)) \leq 2c_{t}(j_{t}^{n})$  for all  $I_{t} \setminus J_{t}^{n}$  and  $t \in [T]$ , by Lemma 5.4 and Lemma 5.6, we obtain
+
+$$
+\begin{array}{l} \sum_ {t \in [ T ]} \sum_ {i \in \bar {I} _ {t} ^ {\prime}} R _ {t} (i) \leq \sum_ {t \in \Phi} \sum_ {i \in \bar {I} _ {t} ^ {\prime}} R _ {t} (i) + \sum_ {t \in \Psi} \sum_ {i \in \bar {I} _ {t} ^ {\prime}} R _ {t} (i) \\ \leq 2 k \beta_ {T} (\delta) \sum_ {t \in \Phi} \| x _ {t} (j _ {t} ^ {n}) \| _ {V _ {t} ^ {- 1}} + k | \Psi |, \\ \end{array}
+$$
+
+where  $\Phi = \{t\in [T]\mid c_t(j_t^n)\geq \frac{\Delta}{4},\| x_t(j_t^n)\|_{\widetilde{V}_{t - 1,n}^{-1}}^2\leq \frac{1}{n}\}$ $\Psi = \{t\in [T]\mid \| x_t(j_t^n)\|_{\widetilde{V}_{t - 1,n}^{-1}}^2 >\frac{1}{n}\}$  , and  $\widetilde{V}_{t,\ell} = \lambda I+$ $\sum_{s\in [t]}\sum_{j\in J_t^\ell}x_s(j)x_s(j)^\top$  for all  $\ell \in [n]$  and  $t\in [T]$  . We consider the rounds in  $\Phi$  . Let  $\xi = \log (1 + L^{2}kT / (d\lambda))$  Then, we have
+
+$$
+\begin{array}{l} \sum_ {t \in \Phi} \| x _ {t} \left(j _ {t} ^ {n}\right) \| _ {V _ {t - 1} ^ {- 1}} \leq \frac {1}{n} \sum_ {t \in \Phi} \sum_ {j \in J _ {t} ^ {n}} \min  \left(\frac {1}{\sqrt {n}}, \| x _ {t} (j) \| _ {\widetilde {V} _ {t - 1, n} ^ {- 1}}\right) \\ \leq \frac {1}{n} \sqrt {n | \Phi | \sum_ {t \in \Phi} \sum_ {j \in J _ {t} ^ {n}} \min  \left(\frac {1}{n} , \| x _ {t} (j) \| _ {\tilde {V} _ {t - 1 , n}} ^ {2}\right)} \\ \leq \min \left(\sqrt {\frac {2 d T \xi}{n}}, \frac {2 d}{\sqrt {n} \min (\gamma_ {n} , 1)} \xi\right), \\ \end{array}
+$$
+
+where the first inequality is derived from the facts that  $V_{t} \succeq \widetilde{V}_{t,n}$  and  $c_{t}(j_{t}^{n}) \leq c_{t}(j_{t}^{\ell})$  for all  $\ell \in [n]$  and  $t \in [T]$ , the second inequality holds by the Cauchy-Schwarz inequality, and the last inequality is obtained by Lemma A.3 (Lemma 5 by Takemura et al. (2021)) and Lemma 5.5.
+
+For the rounds in  $\Psi$ , by a similar discussion above, we have  $k|\Psi| \leq \frac{k}{n} \sum_{t \in [T]} \sum_{j \in J_t^n} \mathbb{1}(\|x_t(j)\|_{\widetilde{V}_{t-1,n}^{-1}}^2 > \frac{1}{n}) \leq 2dk\xi$ , where the last inequality follows from Lemma A.3.
+
+Next, we bound the regret due to (8). Using Lemma 5.4, we have  $R_{t}(j)\leq 2\beta_{t}(\delta)\| x_{t}(j)\|_{V_{t - 1}^{-1}}$  for all  $j\in J_t^n$  and  $t\in [T]$  . Thus, by a similar analysis for the  $\mathbf{C}^2$  UCB algorithm, we have the following bound (see Appendix D.7 for details):  $\sum_{t\in [T]}\sum_{i\in J_t^n\setminus \hat{I}_t}R_t(i) = \tilde{O} (\min (\sqrt{dnT\nu},\frac{d\nu}{\Delta}) + dn\nu).$  Finally, combining the above discussions, we obtain the desired result.
+
+# 5.2. Performance Guarantee
+
+The rest of our proof of Theorem 5.1 is to show that the set  $\{I_t\}_{t \in [T]}$  of the playing arms satisfies our performance constraint (1) with high probability. To show this performance guarantee, we bound the number of arms to be chosen for the exploration. To prove the desired result, we use the following lemma:
+
+Lemma 5.7. Let  $E$  be the ground set of a matroid,  $\mathcal{A}$  be the set of the bases of the matroid, and  $A\in \mathcal{A}$  be a basis. Let  $r:E\to \mathbb{R}$  be a reward function and  $A^{*}$  be a basis such that  $A^{*}\in \operatorname{argmax}_{I\in \mathcal{A}}\sum_{i\in I}r(i)$ . Then, there exists
+
+a bijection  $\rho : A^{*} \to A$  such that  $r(i) \geq r(\rho(i))$  for all  $i \in A^{*}$ . In addition, we have  $\rho(i) = i$  for all  $i \in A^{*} \cap A$ .
+
+We fix  $t \in [T]$  arbitrarily. Let  $\rho_t': \bar{I}_t \to B_t$  be a bijection by Lemma 5.7 with  $\{\bar{r}_t(i)\}_{i \in [N]}$ . Then, since (6) holds, for any  $i \in \bar{I}_t$ , we have  $\mu_t(i) \geq \bar{r}_t(i) \geq \bar{r}_t(\rho_t'(i)) \geq \mu_t(\rho_t'(i))$ . Hence, it is sufficient to compare  $I_t$  with  $\bar{I}_t$ . By the construction of  $I_t$ , we always have  $|I_t \setminus \bar{I}_t| \leq n$ . This fact and  $n \leq m$  imply that  $I_t$  satisfies the constraint.
+
+# 5.3. Near-Optimal Regret Bounds for the CCS Problem
+
+In this subsection, we briefly introduce our analysis for the  $\mathrm{C^2UCB}$  algorithm and the Thompson sampling. A full proof is provided in Appendix G and Appendix H. Note that we consider the CCS problem and the definition of regret is the same as the regret of the CCIB problem.
+
+The key component of our proof is the bijection  $\rho_t^* : I_t \to I_t^*$  by Lemma 5.7 with the reward estimates. These estimates  $\{r_t'(i)\}_{i \in [N]}$  correspond to the upper confidence bounds and the rewards calculated by the sampled parameter in the C²UCB algorithm and the Thompson sampling, respectively. Recall that  $R_t(i) = \mu_t(\rho^*(i)) - \mu_t(i)$ . Then, we can decompose the regret as follows:  $R(T) = \sum_{t \in [T]} \left( \sum_{i \in I_t \setminus J_t} R_t(i) + \sum_{i \in J_t} R_t(i) \right)$ , where  $J_t = \{ i \in I_t \mid \| x_t(i) \|_{V_{t-1}^{-1}} \geq 1 / \sqrt{k} \}$ . The former term can be bounded by a variant of the existing analyses because we have  $r_t'(i) \geq r_t'(\rho_t^*(i))$  for all  $i \in I_t$ . The latter term can be bounded by the fact that  $\sum_{t \in [T]} |J_t| = \widetilde{O}(dk)$ .
+
+# 6. Lower Bounds
+
+In this section, we show the following theorem:
+
+Theorem 6.1. Suppose that there exist an algorithm and some  $m \leq k/(4e + 2)$  such that the algorithm satisfies the constraint (1) in expectation for any environment. Then, for any sub-optimality gap  $\Delta \in \left[\sqrt{\frac{d - 1}{4mT}}, \sqrt{\frac{d - 1}{4m}}\right]$ , there is some environment such that  $\mathbb{E}\left[R_T\right] > \frac{(d - 1)k}{(32e + 16)m\Delta}$ .
+
+We defer the omitted proofs in this section to Appendix E.2. Since substituting  $\Delta = \Theta (\sqrt{d / (mT)})$  leads to a gap-independent bound, we focus on the gap-dependent bound.
+
+To prove Theorem 6.1, we consider instances of the CCIB problem constructed by these of the conservative MAB (CMAB) problem. More precisely, we consider  $k$  rounds of the CMAB problem as a round of the CCIB problem. Using the standard basis  $\{e_i\}_{i\in [d]}$  as the feature vectors  $\{x_{t}(i)\}_{i\in [d]}$ , the CCIB problem reduces to the  $d$ -armed CMAB problem. To represent  $k$  rounds of the CMAB problem as a round of the CCIB problem, it is sufficient to prepare  $dk$  arms as  $x_{t}(i + dj) = e_{i}$  for  $i\in [d]$  and  $j\in [k]$ .
+
+We show that an algorithm for the CCIB problem can be applied to the CMAB problem. The algorithm knows the baseline arms of all rounds in advance because the baseline arm of the CMAB problem is fixed over the rounds. The algorithm can choose arbitrary  $k$  arms in a round. Then, one can play the chosen arms in any order in the CMAB problem. Finally, the algorithm observes  $k$  rewards and proceeds to the next round.
+
+The remainder of our proof of Theorem 6.1 is to obtain a lower bound of the CMAB problem. Using a variant of the technique by Wu et al. (2016) for the lower bound, we obtain the following bound:
+
+Theorem 6.2. Let  $R_{T'}^{\mathrm{CMAB}}$  denote the regret of the  $N$ -armed CMAB problem with  $T'$  rounds. Let  $i^{\mathrm{base}}$  be the baseline arm and  $i_t$  be the arm chosen in round  $t$  by an algorithm. Assume that some algorithm and some positive integer  $k$  satisfy that  $\mathbb{E}\left[\sum_{t' \in [kt]} \mathbb{1}\left(\bar{r}(i^{\mathrm{base}}) > \bar{r}(i_{t'})\right)\right] \leq \alpha kt$  for all positive integer  $t$  such that  $kt \leq T'$  for any environment. Then, for any sub-optimality gap  $\Delta \in \left[\sqrt{\frac{N - 1}{4\alpha T'}}\right]$ , there is some environment such that  $\mathbb{E}\left[R_{T'}^{\mathrm{CMAB}}\right] > \frac{N - 1}{(32e + 16)\alpha\Delta}$ .
+
+Substituting  $T' = kT$  and  $\alpha = m / k$  in Theorem 6.2, we finish the proof of Theorem 6.1.
+
+We discuss our lower bounds. In contrast to our high-probability upper bound, our lower bounds assume that the algorithm satisfies the performance constraints in expectation. However, these lower bounds cover the proposed algorithm. In fact, the proposed algorithm satisfies the performance constraints in expectation if  $n = m / 2$  and  $\delta = \min (n / k,1 / T)$ . Note that the expected regret, in this case, has the same regret bound in Theorem 5.1 up to logarithmic factors. Therefore, our lower bounds imply that the proposed algorithm is almost optimal.
+
+# 7. Numerical Experiments
+
+In this section, we experimentally evaluate the performance of the proposed algorithm compared to some existing algorithms using the Book-Crossing dataset (Ziegler et al., 2005). We describe the details of the experimental setup in Appendix F. We conducted our experiments in two settings. First, we considered the case where the feature vectors, the feasible sets of given arms, and the baseline arms are fixed over the rounds. Second, we considered the case where these three components change two times.
+
+Our experiments used the proposed algorithm combined with the  $C^2$ UCB algorithm (Qin et al., 2014) (GCW +  $C^2$ UCB) and that combined with the Thompson sampling (Takemura & Ito, 2019) (GCW + TS). We compared the proposed algorithm with the weighted regularized matrix factorization (Hu et al., 2008) as the baseline, the  $\varepsilon$ -greedy
+
+![](images/dd3513937b3d4c7ed2004d10e5c054a1793126e7c088e89c86be3aca8d919ae5.jpg)
+
+![](images/fb31abbd7afc254597187aecd4c783f9dd5639bb4b12e5c0f1eaa8ff07f645ee.jpg)
+
+![](images/0b3ee971348c0482d595a9c4901fa46a499ae65f1c9a60ff05dd2c87f3631d84.jpg)  
+Figure 1. Average of the per-step regret (i.e.,  $R(t) / t$ ) (top) and average of the smallest  $m$  satisfying our performance constraint (1) (bottom). The error bar represents the standard error. Left: the case where  $\{x_{t}(i)\}_{i\in [N]}$ ,  $S_{t}$ , and  $B_{t}$  are fixed. Right: the case where  $\{x_{t}(i)\}_{i\in [N]}$  and  $B_{t}$  change two times.
+
+![](images/022bb704c689c100e49e5b07d9aa1feeaefcb2b0edaaf985ff0ce6576a1f922a.jpg)
+
+algorithm, the  $\mathrm{C^2UCB}$  algorithm, and the Thompson sampling in both settings. We additionally compared the iUCB algorithm (Katariya et al., 2019) and its variant (Lin-iUCB algorithm) when they can be applied (i.e., in the first case). The Lin-iUCB used the same reward estimates and confidence widths as those by the proposed algorithm.
+
+Figure 1(left) shows the experimental results in the first case. The proposed algorithms outperformed the standard bandit algorithms and the iUCB algorithm. Specifically, the proposed algorithms suffered small regrets in early rounds and outperformed the baseline because of our interleaving technique. The standard bandit algorithms needed large  $m$  to satisfy the constraints in early rounds, while these algorithms improved faster than the conservative algorithms by the exploration without the performance constraints. The regret by the Lin-iUCB algorithm was smaller than that by  $\mathrm{GCW} + \mathrm{TS}$  and is compatible with the regret by  $\mathrm{GCW} + \mathrm{C}^2\mathrm{UCB}$ . However, the Lin-iUCB algorithm was unstable in terms of performance constraints.
+
+Figure 1 (right) shows that the proposed algorithms outperformed the existing ones and the baseline in the second case. In addition,  $\mathrm{GCW} + \mathrm{C}^2\mathrm{UCB}$  outperformed  $\mathrm{GCW} + \mathrm{TS}$  as in the first case. Compared to the results of the first case, all algorithms improved more slowly due to the changing environment. We can see from the numerical results on performance constraints that the quality of recommendations by the bandit algorithms deteriorates temporarily when the environment changes. However, the degradation of the proposed algorithms are smaller than that of other algorithms. This fact suggests that our algorithms are robust to the changes in the given actions.
+
+# 8. Conclusion
+
+We investigated the CCIB problem, which is the CCS problem with stage-wise performance constraints. We proposed the first meta-algorithm for this problem, which preferentially chooses arms with large confidence widths in the baseline arms and those chosen by a given algorithm for the CCS problem. We showed that the proposed algorithm achieves  $\widetilde{O}(\min(\sqrt{dkT}, d / \Delta) + dk + \min(k\sqrt{dT/n}, dk / (n\Delta)))$  regret while satisfying the constraints. We also showed a gap-dependent and a gap-independent lower bounds through a reduction to the conservative MAB problem. These lower bounds imply that the proposed algorithm achieves minimax optimal up to logarithmic factors. Our numerical experiments demonstrated that the proposed algorithm improves more efficiently than the iUCB algorithm and outperforms existing algorithms.
+
+There are several open questions about the CCIB problem and its extensions. First, an efficient algorithm for the learner to obtain a bijection that satisfies (3) has not been obtained. Second, it is open whether a similar regret bound can be obtained for the CCIB problem with feasible arms characterized by any matroid. Note that we cannot use the exchange property (3) in this case. Finally, this paper does not consider the problem where the reward of the learner is a non-linear function with respect to the rewards of arms chosen by the learner.
+
+# Acknowledgements
+
+Kei Takemura would like to thank Shinji Ito and Tatsuya Matsuoka for helpful discussions on matroids.
+
+# References
+
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+Agrawal, S. and Goyal, N. Thompson sampling for contextual bandits with linear payoffs. In Proceedings of the Thirtieth International Conference on International Conference on Machine Learning, pp. 127-135. PMLR, 2013.  
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+Takemura, K., Ito, S., Hatano, D., Sumita, H., Fukunaga, T., Kakimura, N., and Kawarabayashi, K.-i. Near-optimal regret bounds for contextual combinatorial semi-bandits with linear payoff functions. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 9791-9798, 2021.  
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+
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+
+# A. Known Results
+
+Our proofs use the following known results:
+
+Theorem A.1 (Theorem 13.9 by Korte & Vygen (2018)). Let  $E$  be a finite set and  $\mathcal{B} \in 2^{E}$ .  $\mathcal{B}$  is the set of bases of some matroid  $(E, \mathcal{F})$  if and only if the following holds:
+
+(B1)  $\mathcal{B} \neq \emptyset$ ;
+
+(B2) For any  $B_{1}, B_{2} \in \mathcal{B}$  and  $x \in B_{1} \setminus B_{2}$ , there exists a  $y \in B_{2} \setminus B_{1}$  with  $(B_{1} \setminus \{x\}) \cup \{y\} \in \mathcal{B}$ .
+
+Theorem A.2 (Theorem 2 by Abbasi-Yadkori et al. (2011)). Let  $\{F_t\}_{t=0}^{\infty}$  be a filtration,  $\{X_t\}_{t=1}^{\infty}$  be an  $\mathbb{R}^d$ -valued stochastic process such that  $X_t$  is  $F_{t-1}$ -measurable,  $\{\eta_t\}_{t=1}^{\infty}$  be a real-valued stochastic process such that  $\eta_t$  is  $F_t$ -measurable. Let  $V = \lambda I$  be a positive definite matrix,  $V_t = V + \sum_{s \in [t]} X_s X_s^\top$ ,  $Y_t = \sum_{s \in [t]} \theta^*^\top X_s + \eta_s$  and  $\hat{\theta}_t = V_{t-1}^{-1} Y_t$ . Assume for all  $t$  that  $\eta_t$  is conditionally  $R$ -sub-Gaussian for some  $R > 0$  and  $\| \theta^* \|_2 \leq S$ . Then, for any  $\delta > 0$ , with probability at least  $1 - \delta$ , for any  $t \geq 1$ ,
+
+$$
+\| \hat {\theta} _ {t} - \theta^ {*} \| _ {V _ {t - 1}} \leq R \sqrt {2 \log \left(\frac {\det  (V _ {t - 1}) ^ {1 / 2} \det  (\lambda I) ^ {- 1 / 2}}{\delta}\right)} + \sqrt {\lambda} S.
+$$
+
+Furthermore, if  $\| X_t\| _2\leq L$  for all  $t\geq 1$ , then with probability at least  $1 - \delta$ , for all  $t\geq 1$
+
+$$
+\| \hat {\theta} _ {t} - \theta^ {*} \| _ {V _ {t - 1}} \leq R \sqrt {d \log \left(\frac {1 + (t - 1) L ^ {2} / \lambda}{\delta}\right)} + \sqrt {\lambda} S.
+$$
+
+Lemma A.3 (Lemma 5 by Takemura et al. (2021)). Let  $\{\{x_{t}(i)\}_{i\in [k]}\}_{t\in [T]}$  be any sequence such that  $x_{t}(i)\in \mathbb{R}^{d}$  and  $\| x_{t}(i)\|_{2}\leq L$  for all  $i\in [k]$  and  $t\in [T]$ . Let  $V_{t} = \lambda I + \sum_{s\in [t]}\sum_{i\in [k]}x_{s}(i)x_{s}(i)^{\top}$  with  $\lambda >0$ . Then, we have
+
+$$
+\sum_ {t \in [ T ]} \sum_ {i \in [ k ]} \min  \left(\frac {1}{k}, \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} ^ {2}\right) \leq 2 d \log (1 + L ^ {2} k T / (d \lambda)).
+$$
+
+Lemma A.4 (Lemma 6 by Takemura et al. (2021)). Let  $\{\{x_{t}(i)\}_{i\in [k]}\}_{t\in [T]}$  be any sequence such that  $x_{t}(i)\in \mathbb{R}^{d}$  and  $\| x_{t}(i)\|_{2}\leq L$  for all  $i\in [k]$  and  $t\in [T]$ . Let  $V_{t} = \lambda I + \sum_{s\in [t]}\sum_{i\in [k]}x_{s}(i)x_{s}(i)^{\top}$  with  $\lambda >0$ . Then, we have
+
+$$
+\sum_ {t \in [ T ]} \sum_ {i \in [ k ]} \mathbb {1} \left(\| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} > 1 / \sqrt {k}\right) \leq 2 d k \log (1 + L ^ {2} k T / (d \lambda)).
+$$
+
+# B. Discussions of Performance Constraints
+
+# B.1. Comparison of Constraint (1) with Constraint (2)
+
+We discuss the fact that the two types of constraints are incomparable. More precisely, we show that each constraint does not include the other constraint.
+
+First, we show that constraint (1) does not include constraint (2).
+
+Lemma B.1. Assume that  $m \geq 1$ . Then, there exists a set of rewards such that the rewards satisfy the constraint (1) while they do not satisfy the constraint (2) for any  $\alpha \in [0,1)$ .
+
+Proof. Suppose that  $\mu_t(i) = 0$  for  $i \in I_t$ . Suppose also that  $\mu_t(i) = 1$  for some  $i \in B_t$  and  $\mu_t(i) = 0$  otherwise. Then, for any bijection  $\rho_t : I_t \to B_t$ , the constraint (1) is satisfied with  $m = 1$ . However, since  $\sum_{i \in I_t} \mu_t(i) = 0$  and  $\sum_{i \in B_t} \mu_t(i) = 1$ , the constraint (2) is not satisfied for any  $\alpha \in [0,1)$ .
+
+Next, we show the inverse direction.
+
+Lemma B.2. Assume that  $\alpha \in (0,1]$ . Then, there exists a set of rewards such that the rewards satisfy the constraint (2) while they do not satisfy the constraint (1) for any  $m < k$ .
+
+Proof. We fix  $\alpha \in (0,1]$  arbitrarily. Suppose that  $\mu_t(i) = 1 - \alpha$  for  $i \in I_t$  and  $\mu_t(b) = 1$  for  $b \in B_t$ . These rewards satisfy the constraint (2) with  $\alpha$ . Since any reward of arm  $b \in B_t$  is strictly larger than any reward of arm  $i \in I_t$ , we have  $\sum_{i \in I_t} \mathbb{1}(\mu_t(i) \geq \mu_t(\rho_t(i))) = 0$  for any bijection  $\rho_t : I_t \to B_t$ .
+
+# B.2. Reduction from Constraint (2) to Constraint (1)
+
+We introduce a reduction from constraint (2) to constraint (1). Note that while Assumption 3.10 requires  $r_{\ell} > 0$  due to Assumption 3.9, most of existing studies of conservative bandit problems also require this condition (Wu et al., 2016; Kazerouni et al., 2017; Garcelon et al., 2020; Khezeli & Bitar, 2020; Moradipari et al., 2020).
+
+Lemma B.3. Under Assumption 3.10, satisfying constraint (1) implies satisfying constraint (2).
+
+Proof. Suppose that the sequence  $\{I_t\}_{t \in [T]}$  satisfies the constraints (1). Recall that  $\rho_t : I_t \to B_t$  is a bijection. Let  $I_t' = \{i \in I_t \mid \mu_t(i) \geq \mu_t(\rho_t(i))\}$ . Note that since  $\{I_t\}_{t \in [T]}$  satisfies the constraints (1),  $|I_t'| \geq k - m$ . Rewriting the performance constraints (2), we have
+
+$$
+\sum_ {i \in I _ {t} ^ {\prime}} (\mu_ {t} (i) - (1 - \alpha) \mu_ {t} (\rho_ {t} (i))) \geq \sum_ {i \in I _ {t} \setminus I _ {t} ^ {\prime}} ((1 - \alpha) \mu_ {t} (\rho_ {t} (i)) - \mu_ {t} (i)).
+$$
+
+Recall that  $r_{\ell} = \min_{t\in [T]}\min_{i\in B_t}\mu_t(i)$ . From the definition of  $I_t^\prime$  and  $r_\ell$ , and the fact that  $\mu_t(i)\in [0,1]$  for all  $i\in [N]$ , we have
+
+$$
+\begin{array}{l} \sum_ {i \in I _ {t} ^ {\prime}} \left(\mu_ {t} (i) - (1 - \alpha) \mu_ {t} \left(\rho_ {t} (i)\right)\right) \geq \alpha \sum_ {i \in I _ {t} ^ {\prime}} \mu_ {t} (\rho (i)) \geq \alpha (k - m) r _ {\ell} \quad \text {a n d} \\ \sum_ {i \in I _ {t} \backslash I _ {t} ^ {\prime}} ((1 - \alpha) \mu_ {t} (\rho_ {t} (i)) - \mu_ {t} (i)) \leq m. \\ \end{array}
+$$
+
+Thus, a sufficient condition of the constraints (2) is  $(k - m)\alpha r_{\ell} \geq m$ , which is Assumption 3.10.
+
+![](images/77508a35530c989f73c5cdcbc228e971836ddfaa51d528fb3ff47b9d7dae6ca8.jpg)
+
+# C. Properties of Matroid
+
+# C.1. Strongly Base-Orderable Matroid
+
+First, we introduce the definition of strongly base-orderable matroid and exchangeable set.
+
+Definition C.1. A matroid is strongly base-orderable if for any two bases  $B_{1}$  and  $B_{2}$ , there is a bijection  $f: B_{1} \to B_{2}$  with the property that  $(B_{1} \setminus X) \cup f(X)$  is a base for any  $X \subseteq B_{1}$ .
+
+Then, we show an equivalence between exchangeable set and strongly base-orderable matroid.
+
+Lemma C.2. A set  $S$  is exchangeable if and only if the set  $S$  is the bases of a strongly base-orderable matroid.
+
+Proof. If the set  $S$  is the bases of a strongly base-orderable matroid, we have  $S$  is exchangeable by Definition C.1. Suppose that  $S$  is exchangeable. We fix  $B_1, B_2 \in S$  arbitrarily. Let  $\rho : B_1 \to B_2$  be the bijection defined in (3). Since  $S$  is exchangeable, we have  $(B_1 \setminus \{i\}) \cup \{\rho(i)\} \in S$  for any  $i \in B_1 \setminus B_2$ . Thus, from Theorem A.1, there exists a matroid such that  $S$  is the set of the bases. By Definition 3.6 and Definition C.1, we have the desired result.
+
+# C.2. B bijection between Two Bases
+
+Proof of Lemma 5.7. Let  $(E, \mathcal{F})$  be the matroid and  $k$  be the rank of the matroid. Let  $a_1^*, \ldots, a_k^*$  be a sequence such that  $\{a_i^*\}_{i \in [k]} = A^*$  and  $r(a_i^*) \leq r(a_j^*)$  for  $i \leq j$ .
+
+We construct the bijection by induction. We fix  $\ell \in [k]$  arbitrarily. Suppose that we have already defined  $\rho(a_i^*)$  for all  $i < \ell$ . Let  $A_\ell^* = \{a_i^*\}_{i=\ell,\dots,k}$  and  $A_\ell = A \setminus \{\rho(a_i^*)\}_{i \in [\ell-1]}$ . If we have  $a_\ell^* \in A_\ell$ , we define  $\rho(a_\ell^*) = a_\ell^*$ . Thus, we consider the other case, i.e.,  $a_\ell^* \notin A_\ell$ . From the augmentation property of matroid, there exists  $a \in A_\ell \setminus A_{\ell+1}^*$  such that  $A_{\ell+1}^* \cup \{a\} \in \mathcal{F}$ . Since  $A^* \in \operatorname{argmax}_{I \in \mathcal{A}} \sum_{i \in I} r(i)$ ,  $A_\ell^*$  can be regarded as the output until the  $k-\ell+1$ -th step of the greedy algorithm. Therefore, we have  $r(a_\ell^*) \geq r(a)$  and can define  $\rho(a_\ell^*) = a$ .
+
+Note that the construction in our proof of the following lemma is essentially the same to the proof of Lemma 1 of Kveton et al. (2014).
+
+# D. Missing Proofs in Section 5
+
+# D.1. High-Probability Event
+
+For  $\delta \in (0,1)$ , we define
+
+$$
+\beta_ {t} (\delta) = \min \left(R \sqrt {2 \log {(N (\pi k t) ^ {2} / (3 \delta))}}, R \sqrt {d \log {((1 + L ^ {2} k t / \lambda) / \delta)}}\right) + M \sqrt {\lambda}
+$$
+
+for all  $t\in [T]$
+
+Proof of Lemma 5.3. We consider two cases: when  $\log (N) < d$  and when  $\log (N)\geq d$ .
+
+First, we consider when  $\log (N) < d$ . We fix  $t \in [T]$  and  $i \in [N]$  arbitrarily. From the definition of  $\hat{\theta}_t$ , we have
+
+$$
+\begin{array}{l} \left(\hat {\theta} _ {t} - \theta\right) ^ {\top} x _ {t} (i) = \left(V _ {t - 1} ^ {- 1} \sum_ {s \in [ t - 1 ]} \sum_ {i \in I _ {s}} \left(\theta^ {\top} x _ {s} (i) + \eta_ {s} (i)\right) x _ {s} (i) - \theta\right) ^ {\top} x _ {t} (i) \\ = \sum_ {s \in [ t - 1 ]} \sum_ {i \in I _ {s}} x _ {t} (i) ^ {\top} V _ {t - 1} ^ {- 1} x _ {s} (i) \eta_ {s} (i) - \lambda x _ {t} (i) ^ {\top} V _ {t - 1} ^ {- 1} \theta . \\ \end{array}
+$$
+
+For the latter term, we have  $\lambda x_{t}(i)^{\top}V_{t - 1}^{-1}\theta \leq \lambda \| \theta \|_{V_{t - 1}^{-1}}\| x_{t}(i)\|_{V_{t - 1}^{-1}}\leq \sqrt{\lambda} M\| x_{t}(i)\|_{V_{t - 1}^{-1}}$ . We then bound the former term. Let  $\alpha = R\sqrt{2\log(2 / \delta')}$  for  $\delta^{\prime} > 0$ . Using (a variant of) Azuma's inequality, we obtain
+
+$$
+\begin{array}{l} \mathbb {P} \left(\left| \sum_ {s \in [ t - 1 ]} \sum_ {i \in I _ {s}} x _ {t} (i) ^ {\top} V _ {t - 1} ^ {- 1} x _ {s} (i) \eta_ {s} (i) \right| > \alpha \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}}\right) \\ \leq 2 \exp \left(\frac {\alpha^ {2} \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} ^ {2}}{2 R ^ {2} \sum_ {s \in [ t - 1 ]} \sum_ {i \in I _ {s}} \left(x _ {t} (i) ^ {\top} V _ {t - 1} ^ {- 1} x _ {s} (i)\right) ^ {2}}\right) \\ = 2 \exp \left(\frac {\alpha^ {2} \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} ^ {2}}{2 R ^ {2} x _ {t} (i) ^ {\top} V _ {t - 1} ^ {- 1} (\sum_ {s \in [ t - 1 ]} \sum_ {i \in I _ {s}} x _ {s} (i) x _ {s} (i) ^ {\top}) V _ {t - 1} ^ {- 1} x _ {t} (i)}\right) \\ \leq 2 \exp (- \alpha^ {2} / (2 R ^ {2})) \leq \delta^ {\prime}. \\ \end{array}
+$$
+
+Hence, we obtain
+
+$$
+\left| \left(\hat {\theta} _ {t} - \theta\right) ^ {\top} x _ {t} (i) \right| \leq \left(R \sqrt {2 \log \left(N (\pi k t) ^ {2} / (3 \delta)\right)} + M \sqrt {\lambda}\right) \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} \tag {9}
+$$
+
+with probability at least  $1 - 6\delta / (N(\pi kt)^2)$ . Taking union bound over the arms in all rounds, we have (9) with probability at least  $1 - \delta$ .
+
+Next, we consider when  $\log (N)\geq d$ . By the Cauchy-Schwarz inequality, we have
+
+$$
+\left| \mu_ {t} (i) - \hat {\theta} _ {t} ^ {\top} x _ {t} (i) \right| \leq \left\| \theta - \hat {\theta} _ {t} \right\| _ {V _ {t - 1}} \left\| x _ {t} (i) \right\| _ {V _ {t - 1} ^ {- 1}}
+$$
+
+for all  $t \in [T]$  and  $i \in [N]$ . Then, using Theorem A.2, we have  $\| \theta - \hat{\theta}_t \|_{V_{t-1}} \leq R \sqrt{d \log \left( \frac{1 + L^2 kt / \lambda}{\delta} \right)} + M \sqrt{\lambda}$  for all  $t \in [T]$  with probability at least  $1 - \delta$ , which completes the proof.
+
+# D.2. Analysis for the Recommended Choices
+
+Proof of Lemma 5.2. We fix  $t \in [T]$  arbitrarily. Let  $\tilde{I}_t = I_t \cap \hat{I}_t$  and  $\tilde{I}_t^* = \{\rho_t^*(i) \mid i \in \tilde{I}_t\}$ .
+
+Let  $([N],\mathcal{F}_t)$  be the matroid whose bases are  $\mathcal{S}_t$ . Let  $E_{t} = \tilde{I}_{t}\cup \tilde{I}_{t}^{*}$  and  $k_{t} = |\tilde{I}_{t}|$ . Then, we can construct a matroid  $(E_t,\mathcal{F}_t^{\prime})$  such that  $\mathcal{F}_t^\prime \subseteq \mathcal{F}_t$  and the size of bases of  $(E_t,\mathcal{F}_t^{\prime})$  is  $k_{t}$ . Let  $\mathcal{S}_t^\prime$  be the bases of  $(E_t,\mathcal{F}_t^\prime)$ . Then, we have
+
+$$
+\sum_ {i \in \tilde {I} _ {t}} \left(\mu_ {t} \left(\rho_ {t} ^ {*} (i)\right) - \mu_ {t} (i)\right) \leq \max  _ {I \in \mathcal {S} _ {t} ^ {\prime}} \sum_ {i \in I} \mu_ {t} (i) - \sum_ {i \in \tilde {I} _ {t}} \mu_ {t} (i). \tag {10}
+$$
+
+Recall that  $\{r_t'(i)\}_{i\in [N]}$  is the estimation of the rewards by the algorithm  $\mathcal{A}$ . Since  $\hat{I}_t\in \mathrm{argmax}_{I\in S_t}\sum_{i\in I}r_t'(i)$ , we have  $\tilde{I}_t\in \mathrm{argmax}_{I\in S_t'}\sum_{i\in I}r_t'(i)$ . Thus, (10) is the regret of the CCS problem in which the feasible set is  $S_t'$ . Since the algorithm  $\mathcal{A}$  uses only  $\{r_t(i)\}_{i\in \tilde{I}_t}$  as feedback in the proposed algorithm, we can bound (10) by the regret upper bound of the algorithm  $\mathcal{A}$ .
+
+# D.3. Property of the B bijection Satisfying (3)
+
+Lemma D.1. For any  $t \in [T]$  and  $I, I' \in S_t$  such that  $I \in \operatorname{argmax}_{J \in S_t} \sum_{i \in J} r(i)$  for some  $r: [N] \to \mathbb{R}$ , we have  $r(i) \geq r(\rho(i))$  for any  $i \in I$ , where  $\rho: I \to I'$  is the bijection satisfying (3).
+
+Proof of Lemma D.1. We fix  $t \in [T]$  and  $i \in I_t$  arbitrarily. Let  $([N], \mathcal{F})$  be the matroid that corresponds to the constraint  $\mathcal{S}_t$ . Let  $I^{(i)}$  be the set of arms just before the greedy algorithm chooses  $i$ . By the definition of  $\rho$ , we have  $(I \setminus \{i\}) \cup \{\rho(i)\} \in \mathcal{S}_t$ . From the property of matroid, we have  $I^{(i)} \cup \{i\} \in \mathcal{F}$  and  $I^{(i)} \cup \{\rho(i)\} \in \mathcal{F}$ , which finishes the proof.
+
+# D.4. Properties of Confidence Intervals
+
+To prove Lemma 5.6, we need the following lemma:
+
+Lemma D.2. For any  $i\in \hat{I}_t$  , if  $c_{t}(i) <   \Delta /2$  , we have  $\mu_t(i) = \mu_t(\hat{\rho}_t^* (i))$
+
+Proof. Recall that  $c_{t}(i) = \beta_{t}(\delta)\| x_{t}(i)\|_{V_{t - 1}^{-1}}$ . From Lemma D.1, we have  $\hat{r}_t(i)\geq \hat{r}_t(\hat{\rho}_t^* (i))$  for all  $i\in \hat{I}_t$ . Since (6) holds for all  $t\in [T]$  and  $i\in [N]$ , we have
+
+$$
+\mu_ {t} \left(\hat {\rho} _ {t} ^ {*} (i)\right) \leq \hat {r} _ {t} \left(\hat {\rho} _ {t} ^ {*} (i)\right) \leq \hat {r} _ {t} (i) \leq \mu_ {t} (i) + 2 c _ {t} (i)
+$$
+
+for all  $i \in \hat{I}_t$ . From the assumption of this lemma, we have  $\mu_t(\hat{\rho}_t^*(i)) - \mu_t(i) \leq 2c_t(i) < \Delta$ . Thus, from the definition of  $\Delta$ , we obtain  $\mu_t(i) = \mu_t(\hat{\rho}_t^*(i))$ .
+
+We are ready to prove Lemma 5.6.
+
+Proof of Lemma 5.6. We fix  $i \in \bar{I}_t$  arbitrarily. For all  $j \in \hat{I}_t$ , using Lemma D.2, we have  $\mu_t(j) = \mu_t(\hat{\rho}_t^*(j))$ . Thus, we have  $\mu_t(i) = \mu_t(\bar{\rho}_t(i))$  or  $\mu_t(i) - \mu_t(\bar{\rho}_t(i)) \geq \Delta$ . If  $\mu_t(i) - \mu_t(\bar{\rho}_t(i)) \geq \Delta$ , we have
+
+$$
+\begin{array}{l} \bar {r} _ {t} (\bar {\rho} _ {t} (i)) - \bar {r} _ {t} (i) = \check {r} _ {t} (\bar {\rho} _ {t} (i)) - \hat {r} _ {t} (i) \\ \geq \left(\mu_ {t} \left(\bar {\rho} _ {t} (i)\right) - 2 c _ {t} \left(\bar {\rho} _ {t} (i)\right)\right) - \left(\mu_ {t} (i) + 2 c _ {t} (i)\right) \\ \geq \Delta - 2 \left(c _ {t} (i) + c _ {t} (\bar {\rho} _ {t} (i))\right) \\ > 0, \\ \end{array}
+$$
+
+which contradicts the property of  $\bar{\rho}_t$  obtained by Lemma D.1.
+
+# D.5. Number of Rounds in Which Conservative Choices Suffer the Regret
+
+Proof of Lemma 5.5. We fix  $\ell \in [n]$  arbitrarily. To bound the number of rounds such that  $c_{t}(j_{t}^{\ell})\geq \Delta /4$  , we consider a sufficient condition of rounds such that  $c_{t}(j_{t}^{\ell}) <   \Delta /4$  . Since  $\beta_T(\delta)\geq \beta_t(\delta)$  for all  $t\in [T]$  , a sufficient condition of  $c_{t}(j_{t}^{\ell}) <   \Delta /4$  is that
+
+$$
+\left\| x _ {t} \left(j _ {t} ^ {\ell}\right) \right\| _ {V _ {t - 1} ^ {- 1}} <   \Delta / \left(4 \beta_ {T} (\delta)\right) = \gamma_ {\ell} / \sqrt {\ell}. \tag {11}
+$$
+
+Recall that  $J_{t}^{\ell}$  is defined in line 16 of Algorithm 1. Let  $\widetilde{V}_{t}$  denote  $\lambda I + \sum_{s\in [t]}\sum_{j\in J_{s}^{\ell}}x_{s}(j)x_{s}(j)^{\top}$ . By the definition of  $V_{t}$  and  $\widetilde{V}_{t}$ , we have  $V_{t}\succeq \widetilde{V}_{t}$  for all  $t\in [T]\cup \{0\}$ . Thus, we obtain
+
+$$
+\left\| x _ {t} (i) \right\| _ {V _ {t - 1} ^ {- 1}} \leq \left\| x _ {t} (i) \right\| _ {\widetilde {V} _ {t - 1} ^ {- 1}} \tag {12}
+$$
+
+for all  $t \in [T]$  and  $i \in [N]$ . Combining (11) and (12), we obtain that a sufficient condition of rounds with  $c_{t}(j_{t}^{\ell}) < \Delta / 4$  is that  $\| x_{t}(j_{t}^{\ell}) \|_{\widetilde{V}_{t-1}^{-1}} < \gamma_{\ell} / \sqrt{\ell}$ . This implies that a necessary condition of rounds with  $c_{t}(j_{t}^{\ell}) \geq \Delta / 4$  is that  $\| x_{t}(j_{t}^{\ell}) \|_{\widetilde{V}_{t-1}^{-1}} \geq \widetilde{\gamma}_{\ell}$ , where  $\widetilde{\gamma}_{\ell} = \min(\gamma_{\ell}, 1) / \sqrt{\ell}$ . Let  $T'$  be the number of rounds such that  $c_{t}(j_{t}^{\ell}) \geq \Delta / 4$ . Then, we obtain
+
+$$
+\widetilde {\gamma} _ {\ell} ^ {2} \ell T ^ {\prime} \leq \widetilde {\gamma} _ {\ell} ^ {2} \sum_ {t \in [ T ]} \sum_ {j \in J _ {t} ^ {\ell}} \mathbb {1} \left(\| x _ {t} (j) \| _ {\widetilde {V} _ {t - 1} ^ {- 1}} \geq \widetilde {\gamma} _ {\ell}\right) \leq \sum_ {t \in [ T ]} \sum_ {j \in J _ {t} ^ {\ell}} \min \left(\widetilde {\gamma} _ {\ell} ^ {2}, \| x _ {t} (j) \| _ {\widetilde {V} _ {t - 1} ^ {- 1}} ^ {2}\right) \leq \sum_ {t \in [ T ]} \sum_ {j \in J _ {t} ^ {\ell}} \min \left(\frac {1}{\ell}, \| x _ {t} (j) \| _ {\widetilde {V} _ {t - 1} ^ {- 1}} ^ {2}\right).
+$$
+
+Applying Lemma A.3 to the last term, we finish the proof.
+
+![](images/302bc943a2ab652d1318121259ed4dd357423ddd3134957deaa6183747801806.jpg)
+
+# D.6. Regret of an Arm in Conservative Choices
+
+Proof of Lemma 5.4. We fix  $i \in I_t \setminus \hat{I}_t$  arbitrarily. Recall that  $\bar{\rho}_t : \bar{I}_t \to \hat{I}_t$  be the bijection defined in (3). Let  $i' = \bar{\rho}_t(i)$ . Since (6) holds, we have  $\mu_t(i) \geq \hat{r}_t(i) - 2c_t(i)$ . Moreover, since  $i \in B_t$  and  $i' \in \hat{I}_t \setminus B_t$ , we have  $\hat{r}_t(i) \geq \check{r}_t(i')$  using Lemma D.1. Thus, we obtain
+
+$$
+\mu_ {t} (i) \geq \hat {r} _ {t} (i) - 2 c _ {t} (i) \geq \check {r} _ {t} \left(i ^ {\prime}\right) - 2 c _ {t} (i). \tag {13}
+$$
+
+Recall that  $\hat{\rho}_t^*:\hat{I}_t\to I_t^*$  is the bijection defined in (3). From Lemma D.1, we have  $\hat{r}_t(i')\geq \hat{r}_t(\hat{\rho}_t^*(i'))$ . Thus, we have
+
+$$
+\mu_ {t} \left(\hat {\rho} _ {t} ^ {*} \left(i ^ {\prime}\right)\right) \leq \hat {r} _ {t} \left(\hat {\rho} _ {t} ^ {*} \left(i ^ {\prime}\right)\right) \leq \hat {r} _ {t} \left(i ^ {\prime}\right) = \check {r} _ {t} \left(i ^ {\prime}\right) + 2 c _ {t} \left(i ^ {\prime}\right), \tag {14}
+$$
+
+where the first inequality is derived from (6).
+
+Recall that  $\rho_t^* = \hat{\rho}_t^* \circ \bar{\rho}_t$  for  $i \in I_t \setminus \hat{I}_t$ . Thus, we have  $\tilde{\rho}_t^*(i') = \rho_t^*(i)$ . Combining (13) and (14), we finish the proof.
+
+# D.7. Regret Bound for (8)
+
+We can bound the regret as follows:
+
+$$
+\sum_ {t \in [ T ]} \sum_ {j \in J _ {t} ^ {n}} R _ {t} (j) = \sum_ {t \in [ T ]} \sum_ {j \in \tilde {J} _ {t} ^ {n}} R _ {t} (j) + \sum_ {t \in [ T ]} \sum_ {j \in \tilde {J} _ {t} ^ {n}} R _ {t} (j)
+$$
+
+where
+
+$$
+\tilde {J} _ {t} ^ {n} = \{j \in J _ {t} ^ {n} \mid \| x _ {t} (j) \| _ {\bar {V} _ {t - 1, n} ^ {- 1}} ^ {2} \leq 1 / n \} \quad \mathrm {a n d} \quad \bar {J} _ {t} ^ {n} = \{j \in J _ {t} ^ {n} \mid \| x _ {t} (j) \| _ {\bar {V} _ {t - 1, n} ^ {- 1}} ^ {2} > 1 / n \}
+$$
+
+for all  $t\in [T]$
+
+Recall that  $\widetilde{V}_{t,n} = \lambda I + \sum_{s\in [t]}\sum_{j\in J_t^n}x_s(j)x_s(j)^\top$ . Using Lemma 5.4 and the fact that  $V_{t}\succeq \widetilde{V}_{t,n}$  for all  $t\in [T]$ , we have
+
+$$
+\sum_ {t \in [ T ]} \sum_ {j \in \tilde {J} _ {t} ^ {n}} R _ {t} (j) \leq \sum_ {t \in [ T ]} \sum_ {j \in \tilde {J} _ {t} ^ {n}} 2 \beta_ {t} (\delta) \| x _ {t} (j) \| _ {\tilde {V} _ {t - 1, n} ^ {- 1}} \quad \text {a n d}
+$$
+
+$$
+\sum_ {t \in [ T ]} \sum_ {j \in \tilde {\mathcal {J}} _ {t} ^ {n}} R _ {t} (j) \leq \sum_ {t \in [ T ]} \sum_ {j \in \tilde {\mathcal {J}} _ {t} ^ {n}} R _ {t} (j) ^ {2} / \Delta \leq 2 \sum_ {t \in [ T ]} \sum_ {j \in \tilde {\mathcal {J}} _ {t} ^ {n}} \frac {4 \beta_ {t} (\delta) ^ {2}}{\Delta} \| x _ {t} (j) \| _ {\tilde {V} _ {t - 1, n} ^ {- 1}} ^ {2}.
+$$
+
+Thus, we finish the proof by following the same line of our proof of Theorem G.1.
+
+# E. Lower Bounds of Regret
+
+# E.1. Lower Bound when  $m = 0$
+
+Lemma E.1. If  $m = 0$ , for any algorithm, there is an instance of the CCIB problem such that the algorithm does not achieve sub-linear regret while satisfying the performance constraint (1).
+
+Proof of Lemma E.1. We consider two instances of the CCIB problem and show that no algorithm can obtain sub-linear regret while satisfying the constraint for the two instances simultaneously. Let  $N = 2$ ,  $d = 2$ ,  $k = 1$  and  $m = 0$ . Let  $x_{t}(1) = (1,0)^{\top}$  and  $x_{t}(2) = (0,1)^{\top}$  for all  $t \in [T]$ , i.e., two-armed bandit problem with the stage-wise performance constraints. Suppose that  $r_{t}(1) = 0.5$  and  $B_{t} = \{1\}$  for all  $t \in [T]$ . We consider two types of rewards for this setting: One is  $r_{t}(2) = 0$  and the other is  $r_{t}(2) = 1$ . In the former case, the baseline is optimal. Thus, to satisfy the constraints, an algorithm must choose the baseline (i.e., the first arm) in all rounds. In the latter case, however, such algorithm does not choose the second arm and then leads to a linear regret.
+
+# E.2. Lower Bound when  $m > 0$
+
+First, we show a gap-independent bound.
+
+Theorem E.2. Consider stochastic conservative MAB problem with  $N$  arms and sub-Gaussian noises. Let  $R_T^{\mathrm{CMAB}}$  denote the regret of  $T$  rounds. Let  $i^{\mathrm{base}}$  be the baseline arm and  $i_t$  be the arm that are chosen. Suppose that there exist an algorithm and some  $\alpha \in \left(0, \frac{1}{4e + 2}\right]$  such that the algorithm satisfies  $\mathbb{E}\left[\sum_{t \in [T]} \mathbb{1}\left(\bar{r}(i^{\mathrm{base}}) > \bar{r}(i_t)\right)\right] \leq \alpha T$  for any environment, where  $\bar{r}(i)$  is the mean reward of arm  $i$ . Then, there is some environment such that  $\mathbb{E}\left[R_T^{\mathrm{CMAB}}\right] > \sqrt{\frac{(N - 1)T}{(8e + 4)^2\alpha}}$ .
+
+Proof of Theorem E.2. Our proof largely follows the proof of Theorem 9 of Wu et al. (2016).
+
+We prove this theorem by contradiction. Thus, we assume that  $\mathbb{E}[R_T^{\mathrm{CMAB}}] \leq \sqrt{\frac{(N-1)T}{(8e+4)^2\alpha}}$  for any environment.
+
+We define two types of environments, i.e. the reward distributions of arms and the baseline arm. One is defined as follows:
+
+$$
+\mu_ {i} = \left\{ \begin{array}{l l} \mathcal {N} (\Delta , 1) & \mathrm {i f} i = 1 \\ \mathcal {N} (0, 1) & \mathrm {o t h e r w i s e} \end{array} \right.,
+$$
+
+where we abuse  $\Delta$  that is defined later. The other type of environments is defined as
+
+$$
+\mu_ {i} ^ {(j)} = \left\{ \begin{array}{l l} \mathcal {N} (\Delta , 1) & \text {i f} i = 1 \\ \mathcal {N} (2 \Delta , 1) & \text {i f} i = j \\ \mathcal {N} (0, 1) & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+for  $j \in [N] \setminus \{1\}$ . Note that these environments have the same sub-optimality gap. Suppose that the baseline arm is the first arm for all environments.
+
+Let  $P_{\mu}(\cdot)$  and  $\mathbb{E}_{\mu}[\cdot]$  denote the probability and the expectation under reward distributions  $\mu$ , respectively. Let  $T_{i}$  denote the number of times arm  $i$  was chosen in the problem. Let  $A_{i} = \{T_{i} \leq 2\alpha T\}$  and  $\Delta = \sqrt{\frac{N - 1}{4\alpha T}}$ . First, we show  $P_{\mu}(A_{i}) \geq \frac{1}{2}$  for all  $i \in [N] \setminus \{1\}$ :
+
+$$
+P _ {\mu} (A _ {i}) = 1 - P _ {\mu} (T _ {i} > 2 \alpha T) \geq 1 - \frac {\mathbb {E} _ {\mu} [ T _ {i} ]}{2 \alpha T} \geq \frac {1}{2},
+$$
+
+where the first inequality is derived from Markov's inequality, the second inequality follows from the assumption of the algorithm. Next, we show  $P_{\mu^{(i)}}(A_i) \leq \frac{1}{4e}$  for all  $i \in [N] \setminus \{1\}$ :
+
+$$
+\begin{array}{l} P _ {\mu^ {(i)}} (A _ {i}) = P _ {\mu^ {(i)}} (T - T _ {i} \geq T - 2 \alpha T) \\ \leq \frac {\mathbb {E} _ {\mu^ {(i)}} [ T - T _ {i} ]}{T - 2 \alpha T} \\ \leq \frac {\mathbb {E} _ {\mu^ {(i)}} \left[ R _ {T} ^ {\mathrm {C M A B}} \right] / \Delta}{T - 2 \alpha T} \\ \leq \frac {1}{(4 e + 2) - (8 e + 4) \alpha} \\ \leq \frac {1}{4 e}, \\ \end{array}
+$$
+
+where the first inequality is derived from Markov's inequality and the other inequalities hold due to the definition of the regret and the assumptions. Then, using Lemma 1 of Kaufmann et al. (2016), we have
+
+$$
+\mathbb {E} _ {\mu} [ T _ {i} ] \mathrm {K L} (\mu_ {i}, \mu_ {i} ^ {(i)}) \geq d (P _ {\mu} (A _ {i}), P _ {\mu^ {(i)}} (A _ {i})) \geq \frac {1}{2} \log \left(\frac {1}{4 (1 / (4 e))}\right) = \frac {1}{2},
+$$
+
+where  $d(x,y) = x\log \left(\frac{x}{y}\right) + (1 - x)\log \left(\frac{1 - x}{1 - y}\right)$  and  $\mathrm{KL}(P,Q)$  is the KL-divergence of  $P$  and  $Q$ . By the definition of  $\mu$  and  $\mu^{(i)}$ , we have  $\mathrm{KL}(\mu_i,\mu_i^{(i)}) = 2\Delta^2$ . Therefore, we obtain  $\mathbb{E}_{\mu}[T_i] > \frac{1}{4\Delta^2}$  for all  $i\in [N]\setminus \{1\}$ .
+
+Using this fact, we have  $\mathbb{E}_{\mu}[R_T^{\mathrm{CMAB}}] = \Delta \sum_{i\in [N]\backslash \{1\}}\mathbb{E}_{\mu}[T_i] > \frac{N - 1}{4\Delta} = \Delta \alpha T$  , which contradicts the assumption of the algorithm.
+
+Next, we show a gap-dependent bound.
+
+Proof of Theorem 6.1. We prove this theorem by the same line of our proof of Theorem E.2. Thus, we only discuss the difference from that proof.
+
+We assume  $\mathbb{E}\left[R_T^{\mathrm{CMAB}}\right] \leq \frac{N - 1}{(32e + 16)\alpha\Delta}$  for any environment. Let  $s$  be the largest number such that  $\Delta \leq \sqrt{\frac{N - 1}{4\alpha ks}}$ . Note that  $\mathbb{E}\left[R_{ks}^{\mathrm{CMAB}}\right] \leq \frac{N - 1}{(32e + 16)\alpha\Delta}$ . Let be  $A_i = \{ks_i \leq 2\alpha ks\}$ , where  $ks_i$  is the number of times arm  $i$  was chosen until  $ks$ -th round. We consider the same environments as in Theorem E.2. Then, one can obtain  $P_{\mu}(A_i) \geq 1/2$ . Furthermore, since  $\Delta \geq \sqrt{\frac{N - 1}{4\alpha k(s + 1)}}$ , we have
+
+$$
+\begin{array}{l} P _ {\mu^ {(i)}} \left(A _ {i}\right) = P _ {\mu^ {(i)}} \left(k s - k s _ {i} \geq k s - 2 \alpha k s\right) \\ \leq \frac {\mathbb {E} _ {\mu^ {(i)}} [ k s - k s _ {i} ]}{k s - 2 \alpha k s} \\ \leq \frac {\mathbb {E} _ {\mu^ {(i)}} \left[ R _ {k s} ^ {\mathrm {C M A B}} \right] / \Delta}{k s - 2 \alpha k s} \\ \leq \frac {N - 1}{(3 2 e + 8) \alpha \Delta^ {2} (k s - 2 \alpha k s)} \\ \leq \frac {s + 1}{(8 e + 4) (s - 2 \alpha s)} \\ \leq \frac {2 s}{(8 e + 4) (s - 2 \alpha s)} \\ = \frac {1}{(4 e + 2) - (8 e + 4) \alpha} \\ \leq \frac {1}{4 e}, \\ \end{array}
+$$
+
+where the first inequality is derived from Markov's inequality and the other inequalities hold due to the definition of the regret and the assumptions. Therefore, we obtain  $\mathbb{E}_{\mu}[ks_i] > 1 / (4\Delta^2)$ . Finally, we obtain
+
+$$
+\mathbb {E} _ {\mu} \left[ R _ {k s} ^ {\mathrm {C M A B}} \right] > \frac {N - 1}{4 \Delta} = \Delta \frac {N - 1}{4 \Delta^ {2}} \geq \Delta \alpha k s,
+$$
+
+which contradicts the assumption of the algorithm.
+
+# F. Experiments
+
+# F.1. Environments
+
+In each setting, we set  $d = 20$ ,  $T = 1000$ ,  $k = 30$ , and  $n = 10$ . Moreover, any set of  $k$  arms can be chosen in each round, i.e.,  $S_{t} = \{I \subseteq [N] \mid |I| = k\}$  for all  $t \in [T]$ .
+
+Table 2. Algorithms in Numerical Experiments  
+
+<table><tr><td>Algorithm</td><td>Parameters</td></tr><tr><td>ε-greedy</td><td>ε = 0.05 and λ = 1.</td></tr><tr><td>C2UCB</td><td>λ = 1 and δ = 0.05.</td></tr><tr><td>Thompson sampling</td><td>λ = 1 and δ = 0.05.</td></tr><tr><td>iUCB</td><td>α = n/k = 1/3.</td></tr><tr><td>Lin-iUCB</td><td>λ = 1, δ = 0.05, and α = n/k = 1/3.</td></tr><tr><td>Proposed</td><td>λ = 1, δ = 0.05 and n = 10.</td></tr></table>
+
+# Algorithm 2  $\mathbf{C}^2\mathbf{U}\mathbf{C}\mathbf{B}$
+
+Input:  $\lambda > 0$  and  $\{\alpha_{t}\}_{t\in [T]}$  s.t.  $\alpha_{t} > 0$  for all  $t\in [T]$ .
+
+1:  $V_{0}\gets \lambda I$  and  $b_0\gets 0$  
+2: for  $t = 1,2,\ldots ,T$  do  
+3: Observe  $\{x_{t}(i)\}_{i\in [N]}$  and  $S_{t}$ .  
+4:  $\hat{\theta}_t\gets V_{t - 1}^{-1}b_{t - 1}$  
+5: for  $i\in [N]$  do  
+6:  $\hat{r}_t(i) \gets \hat{\theta}_t^\top x_t(i) + \alpha_t \sqrt{x_t(i)^\top V_{t-1}^{-1} x_t(i)}.$  
+7: end for  
+8: Play a set of arms  $I_{t} \in \operatorname{argmax}_{I \in S_{t}} \sum_{i \in I} \hat{r}_{t}(i)$  and observe rewards  $\{r_{t}(i)\}_{i \in I_{t}}$ .  
+9:  $V_{t} \gets V_{t-1} + \sum_{i \in I_{t}} x_{t}(i) x_{t}(i)^{\top}$  and  $b_{t} \gets b_{t-1} + \sum_{i \in I_{t}} r_{t}(i) x_{t}(i)$ .  
+10: end for
+
+We use the Book-Crossing dataset (Ziegler et al., 2005), which consists of ratings expressed on a scale from 0 to 10 for books. We extracted users that rated at least 100 items and items that are rated by at least 3 users in the extracted users, which obtains 37, 157 ratings about 7, 068 items by 490 users.
+
+We consider the cold start problem as in Garcelon et al. (2020). An algorithm aims to learn the preference of a new user. We obtained the item vectors from the ratings by the weighted regularized matrix factorization (Hu et al., 2008) and used the item vectors as the feature vectors. The baseline is the recommendation by the matrix factorization. A user is randomly selected at the beginning of a simulation. The mean rewards are the selected user's ratings that are normalized in  $[0,1]$ . The noises of the rewards follow  $\mathcal{N}(0,0.1^2)$ . We ran 100 simulations.
+
+# F.2. Algorithms
+
+We compare the proposed algorithm with the algorithms whose parameters are described in Table 2 and the baseline. The  $\varepsilon$ -greedy algorithm has two ways to estimate the rewards of given arms: One is to use random values, and the other is to use the ridge regression where  $\lambda$  is the parameter of the regularization. The algorithm chooses the former way with probability  $\varepsilon$  and the latter way otherwise. Then, it plays a feasible set of arms that maximizes the sum of the estimated rewards. We set  $\alpha$  of the iUCB algorithms such that the iUCB and the proposed algorithms have the same condition for exploration, i.e.,  $\alpha = n / k$ . Note that  $\alpha$  for the iUCB algorithm and  $\alpha$  in (2) are different parameters.
+
+# G. C²UCB Algorithm for Matroid Constraint
+
+We consider the CCS problem defined in Takemura et al. (2021). Note that this problem assumes that  $|\mu_t(i)| \leq B$  for some parameter  $B$  but does not assume that  $\mu_t(i) \in [0,1]$ . The CCS problem coincides with the CCIB problem without the performance constraints except the above assumption.
+
+We show that the  $C^2$ UCB algorithm (Algorithm 2) is minimax optimal up to logarithmic factors for the CCS problem with any matroid constraint. We emphasize that our theorem is a generalization of Theorem 3 by Takemura et al. (2021) and that our proof is much simpler than that of the theorem by Takemura et al. (2021). While the CCS problem assumes that the
+
+number  $k$  of arms chosen in a round is fixed over the rounds, the  $\mathbf{C}^2\mathbf{U}\mathbf{C}\mathbf{B}$  algorithm works and achieves the same regret bound if the number of arms chosen in a round is bounded by  $k$  from above.
+
+Let  $\rho_t^* : I_t \to I_t^*$  be the bijection defined by Lemma 5.7 with  $\{\hat{r}_t(i)\}_{i \in [N]}$ . Then, we define our sub-optimality gap as follows:
+
+$$
+\Delta_ {t} (i, j) = \mu_ {t} (j) - \mu_ {t} (i) \quad \mathrm {a n d} \quad \Delta = \min _ {t \in [ T ]} \min _ {i \in I _ {t}: \Delta_ {t} (i, \rho_ {t} ^ {*} (i)) > 0} \Delta_ {t} (i, \rho_ {t} ^ {*} (i)).
+$$
+
+Using this sub-optimality gap, we have the following regret bound:
+
+Theorem G.1 (A generalization of Theorem 3 by Takemura et al. (2021)). Let  $\kappa = \min (\log (N),d)$ $\nu = \log (kT)$  and  $\iota = \min (\log (NkT / \delta),d\log (kT / \delta))$  . Assume that  $S_{t}$  is a set of bases of a matroid for all  $t\in [T]$  . Then, if  $\alpha_{t} = \beta_{t}(\delta)$  and  $\lambda = (R / M)^{2}\kappa$  , the  $C^2$  UCB algorithm has the following regret bound with probability  $1 - \delta$
+
+$$
+R (T) = O \left(\min \left(R \sqrt {d k T \nu \iota}, \frac {R ^ {2} d \nu \iota}{\Delta}\right) + B d k \nu\right).
+$$
+
+Proof. Let  $J_{t} = \{i\in I_{t}\mid \| x_{t}(i)\|_{V_{t - 1}^{-1}} > 1 / \sqrt{k}\}$  and  $J_{t}^{*} = \{\rho_{t}^{*}(i)\mid i\in J_{t}\}$ . Using these notations, the regret is rewritten as
+
+$$
+R (T) = \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} ^ {*} \backslash J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in I _ {t} \backslash J _ {t}} \mu_ {t} (i)\right) + \sum_ {t \in [ T ]} \left(\sum_ {i \in J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in J _ {t}} \mu_ {t} (i)\right).
+$$
+
+We bound these terms separately.
+
+First, we consider the former term. By the definition of the bijection  $\rho_t^*$ , we have  $\hat{r}_t(i) \geq \hat{r}_t(\rho_t^*(i))$  for all  $i \in I_t$ . Using Lemma 5.3, we have  $\hat{r}_t(i) \geq \mu_t(\rho_t^*(i))$  and  $\hat{r}_t(i) \leq \mu_t(i) + 2c_t(i)$  for all  $i \in I_t$ , where  $c_t(i) = \beta_t(\delta) \|x_t(i)\|_{V_{t-1}^{-1}}$ . Therefore, we obtain
+
+$$
+\sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} ^ {*} \backslash J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in I _ {t} \backslash J _ {t}} \mu_ {t} (i)\right) \leq \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} 2 c _ {t} (i) \quad \text {a n d}
+$$
+
+$$
+\begin{array}{l} \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} ^ {*} \backslash J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in I _ {t} \backslash J _ {t}} \mu_ {t} (i)\right) \leq \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} \backslash J _ {t}} \left(\mu_ {t} \left(\rho_ {t} ^ {*} (i)\right) - \mu_ {t} (i)\right) ^ {2}\right) / \Delta \\ \leq \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} 4 c _ {t} (i) ^ {2} / \Delta \\ \end{array}
+$$
+
+Since  $\beta_{T}(\delta) \geq \beta_{t}(\delta)$  for all  $t \in [T]$ , using Lemma A.3, we have
+
+$$
+\begin{array}{l} \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} c _ {t} (i) \leq \beta_ {T} (\delta) \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} \leq \beta_ {T} (\delta) \sqrt {k T \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} ^ {2}} \\ = O (R \sqrt {d k T \iota}) \quad \text {a n d} \\ \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} c _ {t} (i) ^ {2} / \Delta = O (R ^ {2} d \iota / \Delta). \\ \end{array}
+$$
+
+Next, we bound the latter term. From Lemma A.4, we have
+
+$$
+\sum_ {t \in [ T ]} | J _ {t} | = \widetilde {O} (d k).
+$$
+
+Since  $|\mu_t(i)| \leq B$ , we have
+
+$$
+\sum_ {t \in [ T ]} \left(\sum_ {i \in J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in J _ {t}} \mu_ {t} (i)\right) \leq 2 B \sum_ {t \in [ T ]} | J _ {t} | = O (B d k \nu).
+$$
+
+Algorithm 3 Thompson sampling  
+Input:  $\lambda >0$  and  $\{v_{t}\}_{t\in [T]}$  for all  $t\in [T]$    
+1:  $V_{0}\gets \lambda I$  and  $b_{0}\gets 0$    
+2: for  $t = 1,2,\ldots ,T$  do   
+3: Observe  $\{x_t(i)\}_{i\in [N]}$  and  $S_{t}$    
+4:  $\hat{\theta}_t\gets V_{t - 1}^{-1}b_{t - 1}$    
+5: Sample  $\tilde{\theta}_t$  from  $\mathcal{N}\left(\hat{\theta}_t,v_t^2 V_{t - 1}^{-1}\right)$    
+6: for  $i\in [N]$  do   
+7:  $\tilde{r}_t(i)\gets \tilde{\theta}_t^\top x_t(i)$    
+8: end for   
+9: Play a set of arms  $I_{t}\in \mathrm{argmax}_{I\in S_{t}}\sum_{i\in I}\tilde{r}_{t}(i)$  and observe rewards  $\{r_t(i)\}_{i\in I_t}$    
+10:  $V_{t}\gets V_{t - 1} + \sum_{i\in I_{t}}x_{t}(i)x_{t}(i)^{\top}$  and  $b_{t}\gets b_{t - 1} + \sum_{i\in I_{t}}r_{t}(i)x_{t}(i)$    
+11: end for
+
+# H. Thompson Sampling for Matroid Constraint
+
+We show that the Thompson sampling Algorithm 3 for the CCS problem achieves a near-optimal regret bound. Note that, similar to the  $C^2$  UCB algorithm, in the CCS problem where the number of arms chosen in  $t$ -th round depends on  $t$ , the Thompson sampling achieves the same regret bound if the number of arms chosen in a round is bounded by  $k$  from above.
+
+Theorem H.1. Let  $\kappa = \min (\log (N),d)$ $\nu = \log (kT)$  and  $\iota^{\prime} = \min (\log (NkT^{2} / \delta),d\log (kT^{2} / \delta))\log (dT / \delta)$ . Assume that  $S_{t}$  is a set of bases of a matroid for all  $t\in [T]$ . Then, if  $v_{t} = \beta_{t}(\delta /(4T))$  and  $\lambda = L / (dM)$ , the Thompson sampling has the following regret bound with probability  $1 - \delta$
+
+$$
+R (T) = O \left(\left(R \sqrt {d \kappa} + \sqrt {M L}\right) \sqrt {d k T \nu \iota^ {\prime}} + B d k \nu\right).
+$$
+
+Proof. Let  $\rho_t^* : I_t \to I_t^*$  be the bijection defined by Lemma 5.7 with  $\{\tilde{r}_t(i)\}_{i \in [N]}$ . Let  $J_t = \{i \in I_t \mid \| x_t(i) \|_{V_{t-1}^{-1}} > 1 / \sqrt{k}\}$  and  $J_t^* = \{\rho_t^*(i) \mid i \in J_t\}$ . Using these notations, the regret is rewritten as
+
+$$
+\begin{array}{l} R (T) = \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} ^ {*} \backslash J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in I _ {t} \backslash J _ {t}} \mu_ {t} (i)\right) \tag {15} \\ + \sum_ {t \in [ T ]} \left(\sum_ {i \in J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in J _ {t}} \mu_ {t} (i)\right). \\ \end{array}
+$$
+
+We bound these terms separately.
+
+First, we bound the latter term. By the same discussion in the proof of Theorem G.1, we have
+
+$$
+\sum_ {t \in [ T ]} \left(\sum_ {i \in J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in J _ {t}} \mu_ {t} (i)\right) \leq 2 B \sum_ {t \in [ T ]} | J _ {t} | = O (B d k \nu).
+$$
+
+Next, we consider the former term. If  $\mu_t(i) \geq \mu_t(\rho_t^*(i))$ , the term  $\mu_t(\rho_t^*(i)) - \mu_t(i)$  does not contribute to the regret. Thus, we can assume
+
+$$
+\forall i \in I _ {t} \backslash J _ {t}, \mu_ {t} (i) \leq \mu_ {t} \left(\rho_ {t} ^ {*} (i)\right) \tag {16}
+$$
+
+without loss of generality.
+
+We follow a similar line of proof for the Thompson sampling for the contextual linear bandit problem by Abeille & Lazaric (2017). As in Abeille & Lazaric (2017), we define the following events for all  $t \in [T]$ :
+
+$$
+\hat {E} _ {t} = \left\{\forall s \leq t, \| \hat {\theta} _ {s} - \theta \| _ {V _ {s - 1}} \leq \beta_ {s} (\delta^ {\prime}) \right\} \quad \text {a n d}
+$$
+
+$$
+\tilde {E} _ {t} = \{\forall s \leq t, \| \tilde {\theta} _ {s} - \hat {\theta} _ {s} \| _ {V _ {s - 1}} \leq \gamma_ {s} (\delta^ {\prime}) \},
+$$
+
+where  $\delta' = \delta/(4T)$  and  $\gamma_t(\delta) = \sqrt{2d\log(2d/\delta)}\beta_t(\delta)$ . From a similar argument of Lemma 1 by Abeille & Lazaric (2017), we have  $P(\hat{E}_T \cap \tilde{E}_T) \geq 1 - \delta/2$ . In the remaining of this proof, we consider the case under the event  $\hat{E}_T \cap \tilde{E}_T$ .
+
+Then, we can decompose the right-hand side of (15) as follows:
+
+$$
+\begin{array}{l} \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} ^ {*} \backslash J _ {t} ^ {*}} \mu_ {t} (i) - \sum_ {i \in I _ {t} \backslash J _ {t}} \mu_ {t} (i)\right) = \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} \backslash J _ {t}} \theta^ {\top} x _ {t} \left(\rho_ {t} ^ {*} (i)\right) - \tilde {\theta} _ {t} ^ {\top} x _ {t} (i)\right) (17) \\ + \sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} \backslash J _ {t}} \tilde {\theta} _ {t} ^ {\top} x _ {t} (i) - \theta^ {\top} x _ {t} (i)\right). (18) \\ \end{array}
+$$
+
+We can bound (18) as
+
+$$
+\sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} \backslash J _ {t}} \tilde {\theta} _ {t} ^ {\top} x _ {t} (i) - \theta^ {\top} x _ {t} (i)\right) \leq \sum_ {t \in [ T ]} \sum_ {i \in I _ {t} \backslash J _ {t}} \left(\gamma_ {t} \left(\delta^ {\prime}\right) + \beta_ {t} \left(\delta^ {\prime}\right)\right) \| x _ {t} (i) \| _ {V _ {t - 1} ^ {- 1}} = O \left((R \sqrt {\kappa} + M \sqrt {\lambda}) d \sqrt {k T \nu \iota^ {\prime}}\right).
+$$
+
+To bound (17), we introduce convex functions  $J_{t,i}(u) = \max_{j\in C_{t,i}}u^{\top}x_{t}(j)$  for some  $C_{t,i}\subseteq [N]$  such that  $J_{t,i}(\tilde{\theta}_t) = \tilde{\theta}_t^\top x_t(i)$  and  $J_{t,i}(\theta) = \theta^{\top}x_{t}(\rho_{t}^{*}(i))$ . If we construct such convex functions, by the same line of the proof by Abeille & Lazaric (2017), we can complete the proof. In fact, we have
+
+$$
+\sum_ {t \in [ T ]} \left(\sum_ {i \in I _ {t} \backslash J _ {t}} \theta^ {\top} x _ {t} \left(\rho_ {t} ^ {*} (i)\right) - \tilde {\theta} _ {t} ^ {\top} x _ {t} (i)\right) = O \left((R \sqrt {\kappa} + M \sqrt {\lambda}) d \sqrt {k T \nu \iota^ {\prime}} + \frac {L}{\sqrt {\lambda}} \sqrt {k T \log (1 / \delta)}\right)
+$$
+
+with probability at least  $1 - \delta /2$
+
+We now construct  $J_{t,i}(u)$ . By the construction of  $\rho_t^*$ , we have  $\tilde{\theta}_t^\top x_t(i) \geq \tilde{\theta}_t^\top x_t(\rho_t^*(i))$ . On the other hand, we have  $\theta^\top x_t(i) \leq \theta^\top x_t(\rho_t^*(i))$  by (16). Therefore, if we define  $C_{t,i} = \{i, \rho_t^*(i)\}$ , we have  $J_{t,i}(\tilde{\theta}_t) = \tilde{\theta}_t^\top x_t(i)$  and  $J_{t,i}(\theta) = \theta^\top x_t(\rho_t^*(i))$ .

2023/Contextual Conservative Interleaving Bandits/images.zip

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