{"item_id": "v10-duchi09a::theorem6::A-stmt", "paper_id": "v10-duchi09a", "target_name": "theorem6", "source": "Research-level", "nl_statement": "Theorem 6 Suppose that $\\| \\pmb{w}_{t}- \\pmb{w}^{\\star}\\| \\leq D$ holds at every iteration, and that G is an upper bound on the norms of the subgradient sets $\\partial f_{t}$ and $\\partial r$. Fix an arbitrary scalar $c > 0$. Then FoBoS run with step sizes $\\eta_{t}= c / \\sqrt{t}$ has a regret bound of the form\n\n$$\nR_{f+r}(T)\\leq 2GD+\\left({\\frac{D^{2}}{2c}}+7G^{2}c\\right)\\sqrt{T}.\n$$", "context": [{"name": "Setup and notation: objective f + r, convex and nonnegative (Section 1)", "statement": "We first fix the notation used throughout the paper. Scalars are written as lower case letters and vectors as bold lower case letters, e.g. w. For two vectors u and v, $\\langle{\\pmb u},{\\pmb v}\\rangle$ is their inner product. The p-norm of a vector x is written $\\| \\pmb{x}\\|_{p}$, and ||x|| abbreviates $\\| \\pmb{x}\\|_{2}$.\n\nThe paper studies an algorithmic framework for regularized convex programming whose aim is to minimize a sum of two functions,\n\n$$\nf({\\pmb w})+r({\\pmb w}),\\tag{1}\n$$\n\nin which f and r are both convex and bounded below (so that, with no loss of generality, they are taken to map into $\\mathbb{R}_{+}$). Typically f is an empirical loss of the form $\\textstyle\\sum_{i\\in S}\\ell_{i}(w)$ built from loss functions $\\ell_{i}: \\mathbb{R}^{n}\\to \\mathbb{R}_{+}$, and $r(w)$ is a regularizer that penalizes overly complex vectors, e.g. $r(\\mathbf{w})=\\lambda\\|\\mathbf{w}\\|_{p}$. Such tasks are common in machine learning, where learning problems for decision and prediction are formulated as convex optimization problems. With this in mind, we study a general and intuitive algorithm called forward-backward splitting for minimizing Eq. (1), with particular emphasis on derivations for, and the use of, regularization functions that are not differentiable."}, {"name": "Definition: subgradient set", "statement": "For a function f and a point w, write $\\partial f ({\\boldsymbol{w}})$ for the subgradient set of f at w, that is,\n\n$$\n\\partial f(w)=\\{g\\mid\\forall v:f(v)\\geq f(w)+\\langle g,v-w\\rangle\\}.\n$$"}, {"name": "Algorithm: FoBoS update and forward-looking subgradient, Eqs. (2)-(5) (Section 2)", "statement": "Our version of Forward-Backward Splitting is driven by the wish that the iterates ${\\pmb w}_{t}$ be able to reach points where r is not differentiable. For problems such as $\\ell_{1}$-regularization, the method sidesteps the difficulties caused by non-differentiability by alternating subgradient steps with analytical minimization steps. Informally, FoBoS resembles the projected subgradient method, except that the projection step is replaced or augmented by an instantaneous minimization problem whose solution can be derived in closed form. The method is compact: every iteration is made of the two steps below,\n\n$$\n\\begin{array}{r l r}{{\\bf w}_{t + \\frac{1}{2}}}&{{}=}&{{\\bf w}_{t}- \\boldsymbol \\eta_{t}{\\bf g}_{t}^{f},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{r c l}{{\\pmb w}_{t+1}}&{=}&{\\displaystyle\\operatorname*{argmin}_{{\\pmb w}}\\left\\{\\frac{1}{2}\\left\\|{\\pmb w}-{\\pmb w}_{t+\\frac{1}{2}}\\right\\|^{2}+\\eta_{t+\\frac{1}{2}}r({\\pmb w})\\right\\}\\ .}\\end{array}\\tag{3}\n$$\n\nHere $g_{t}^{f}$ is an element of $\\partial f ({\\pmb w}_{t})$, and $\\eta_{t}$ denotes the step size used by the algorithm at time step t. What value $\\eta_{t}$ takes depends on the particular setting and analysis. The first step is thus nothing more than an unconstrained subgradient step for the function $f$. The second step produces a new vector that balances two aims: (i) remaining near the interim vector $w_{t + \\frac{1}{2}}$, and (ii) having small complexity as measured by r. Observe that r is multiplied by an interim step size, written $\\eta_{t + \\frac{1}{2}}$. The analyses later in the paper fix the particular value of $\\eta_{t + \\frac{1}{2}}$, which is either $\\eta_{t}$ or $\\eta_{t + 1}$.\n\nAn important property of the solution of Eq. (3), which also explains the name FoBoS, is the necessary optimality condition: at the optimum ${\\pmb w}_{t + 1}$, the subgradient set of the objective must contain the zero vector, i.e.\n\n$$\n\\mathbf{0}\\in\\partial\\left\\{\\frac{1}{2}\\left\\|w-w_{t+\\frac{1}{2}}\\right\\|^{2}+\\eta_{t+\\frac{1}{2}}r(w)\\right\\}\\bigg|_{w=w_{t+1}}.\n$$\n\nAs $\\pmb{w}_{t + \\frac{1}{2}}= \\pmb{w}_{t}- \\eta_{t}\\pmb{g}_{t}^{f}$, this condition reads\n\n$$\n\\mathbf{0}\\in\\pmb{w}_{t+1}-\\pmb{w}_{t}+\\eta_{t}\\pmb{g}_{t}^{f}+\\eta_{t+\\frac{1}{2}}\\partial r(\\pmb{w}_{t+1}).\\tag{4}\n$$\n\nFrom $\\mathbf{0}\\in\\pmb{w}_{t+1}-\\pmb{w}_{t}+\\eta_{t}\\pmb{g}_{t}^{f}+\\eta_{t+\\frac{1}{2}}\\partial r(\\pmb{w}_{t+1})$ it follows that, as long as ${\\pmb w}_{t + 1}$ is taken to be the minimizer of Eq. (3), there is guaranteed to be a vector $\\pmb{g}_{t+1}^{r}\\in\\partial r(\\pmb{w}_{t+1})$ with\n\n$$\n{\\bf 0}={\\pmb w}_{t + 1}-{\\pmb w}_{t}+ \\eta_{t}{\\pmb g}_{t}^{f}+ \\eta_{t + \\frac{1}{2}}{\\pmb g}_{t + 1}^{r}\\mathrm{~ . ~}\n$$\n\nThis identity may be read as an update rule in which the new weight vector ${\\pmb w}_{t + 1}$ is a linear combination of three vectors: the previous weight vector ${\\pmb w}_{t}$, an element of the subgradient set of $f$ at ${\\pmb w}_{t}$, and an element of the subgradient of r at the not-yet-known ${\\pmb w}_{t + 1}$; this is the origin of the name Forward-Looking Subgradient. In summary, ${\\pmb w}_{t + 1}$ can be written as\n\n$$\n\\pmb{w}_{t + 1}= \\pmb{w}_{t}- \\eta_{t}\\pmb{g}_{t}^{f}- \\eta_{t + \\frac{1}{2}}\\pmb{g}_{t + 1}^{r},\\tag{5}\n$$\n\nwith ${\\pmb g}_{t}^{f}\\in \\partial f ({\\pmb w}_{t})$ and $\\pmb{g}_{t+1}^{r}\\in\\partial r(\\pmb{w}_{t+1})$. Solving Eq. (3) with r as above brings two main advantages. First, algorithmically, it produces sparse solutions at almost no extra computational cost. Second, thanks to the forward-looking gradient we can reuse existing analyses and prove that the resulting framework inherits the formal convergence guarantees of many existing gradient-based and online convex programming algorithms."}, {"name": "Setup: choice of the interim step size eta_{t+1/2} (Section 3)", "statement": "At first sight FoBoS appears quite different from subgradient and online convex programming methods. Yet because FoBoS in fact uses a forward-looking subgradient of the regularizer, existing analyses carry over well. In this section we adapt known results, exploiting the forward-looking property of FoBoS, to obtain convergence rate analyses for FoBoS. This requires choosing $\\eta_{t + \\frac{1}{2}}$ appropriately. As shown below, to get convergence and low regret bounds it suffices to take $\\eta_{t + \\frac{1}{2}}$ equal to $\\eta_{t}$ or to $\\eta_{t + 1}$, according to whether the optimization is online or batch. We first analyze FoBoS in the batch setting. There we use the subgradient of $f$, put $\\eta_{t + \\frac{1}{2}}= \\eta_{t + 1}$, and move from ${\\pmb w}_{t}$ to ${\\pmb w}_{t + 1}$ as specified by Eq. (2) and Eq. (3)."}, {"name": "Assumption: bounded subgradients, Eq. (6)", "statement": "In this section $w^{\\star}$ always denotes the minimizer of $f ({\\pmb w}) + r ({\\pmb w})$. Below we set $\\begin{array}{r}{\\|\\partial f(\\pmb{w})\\|\\triangleq\\operatorname*{sup}_{\\pmb{g}\\in\\partial f(\\pmb{w})}\\|\\pmb{g}\\|}\\end{array}$. Our first convergence results are derived under the rather general assumption (see, for instance, Langford et al. 2008 or Shalev-Shwartz and Tewari 2009) that the subgradients obey the bounds\n\n$$\n\\begin{array}{r}{\\|\\partial f(\\pmb{w})\\|^{2}\\leq Af(\\pmb{w})+G^{2},\\quad\\|\\partial r(\\pmb{w})\\|^{2}\\leq Ar(\\pmb{w})+G^{2}.}\\end{array}\\tag{6}\n$$\n\nAs examples, every Lipschitz loss (for instance the logistic loss, or the hinge loss of support vector machines) meets this condition with $A = 0$ and G the Lipschitz constant, while least squares estimation meets Eq. (6) with $G = 0$ and $A = 4$. The following lemma is technical, but it is the central tool behind every convergence result in this paper."}, {"name": "Lemma 1 (Bounding Step Differences)", "statement": "Lemma 1 (Bounding Step Differences) Suppose the subgradient norms of f and r obey the bounds of Eq. (6):\n\n$$\n\\|\\partial f(\\pmb{w})\\|^{2}\\leq Af(\\pmb{w})+G^{2},\\quad\\|\\partial r(\\pmb{w})\\|^{2}\\leq Ar(\\pmb{w})+G^{2}.\n$$\n\nLet $\\eta_{t + 1}\\leq \\eta_{t + \\frac{1}{2}}\\leq \\eta_{t}$, and assume $\\eta_{t}\\le 2 \\eta_{t + \\frac{1}{2}}$. Then, when the FoBoS update of Eqs. (2) and (3) is used, for a constant $c \\leq 4$ and every vector $\\pmb{w}^{\\star}$,\n\n$$\n\\begin{array}{r l r}{{2\\boldsymbol\\eta_{t}\\big(1-cA\\boldsymbol\\eta_{t}\\big)f\\big(\\boldsymbol{w}_{t}\\big)+2\\boldsymbol\\eta_{t+\\frac{1}{2}}\\big(1-cA\\boldsymbol\\eta_{t+\\frac{1}{2}}\\big)\\boldsymbol{r}\\big(\\boldsymbol{w}_{t+1}\\big)}}\\\\ &{\\leq}&{2\\boldsymbol\\eta_{t}f\\big(\\boldsymbol{w}^{\\star}\\big)+2\\boldsymbol\\eta_{t+\\frac{1}{2}}\\boldsymbol{r}\\big(\\boldsymbol{w}^{\\star}\\big)+\\|\\boldsymbol{w}_{t}-\\boldsymbol{w}^{\\star}\\|^{2}-\\|\\boldsymbol{w}_{t+1}-\\boldsymbol{w}^{\\star}\\|^{2}+7\\boldsymbol\\eta_{t}\\boldsymbol\\eta_{t+\\frac{1}{2}}G^{2}\\quad .}\\end{array}\n$$"}, {"name": "Setup: online learning and regret R_{f+r}(T) (Section 4)", "statement": "This section gives a regret analysis of FoBoS in online settings. An online learning problem supplies a sequence of functions $f_{t}: \\mathbb{R}^{n}\\longrightarrow \\mathbb{R}$. The aim is that the predictions ${\\pmb w}_{t}$ incur low regret relative to one fixed optimal predictor $w^{\\star}$. Precisely, $f_{t}({\\boldsymbol{w}})$ is the loss incurred on the $t^{t h}$ input loss function by the predictor w. For an online algorithm whose predictors are $\\boldsymbol{w}_{1},\\ldots,\\boldsymbol{w}_{t},\\ldots$, the regret with respect to a fixed predictor $\\pmb{w}^{\\star}$ under a regularization function r is\n\n$$\nR_{f+r}(T)=\\sum_{t=1}^{T}\\left[f_{t}(\\pmb{w}_{t})+r(\\pmb{w}_{t})-\\left(f_{t}(\\pmb{w}^{\\star})+r(\\pmb{w}^{\\star})\\right)\\right].\n$$\n\nThe ideal would be zero regret against a stationary $\\pmb{w}^{\\star}$ for sequences of any length.\n\nTo get an online bound for a sequence of convex functions $f_{t}$, we adapt the arguments of Zinkevich (2003). With the bound of Lemma 1 at hand, a theorem on the online regret of FoBoS can be stated and proved directly. The boundedness assumptions in the proof could be dropped (yielding a regret analogue of the bound in Theorem 2), but we do not consider this notably more interesting. Apart from its use of Lemma 1, the proof closely follows Zinkevich's, and so it is postponed to Appendix A."}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\ndef HasSubgradientAt (f : E → ℝ) (g x : E) : Prop :=\n  ∀ y, f y ≥ f x + @inner ℝ _ _ g (y - x)\n\ndef IsFoBoSRun (f r : E → ℝ) (w gf gr : ℕ → E) (η ηhalf : ℕ → ℝ) : Prop :=\n  (∀ t, HasSubgradientAt f (gf t) (w t)) ∧\n  (∀ t, HasSubgradientAt r (gr t) (w t)) ∧\n  (∀ t, w (t + 1) = w t - η t • gf t - ηhalf t • gr (t + 1))\n\ndef IsFoBoSOnlineRun (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) :\n    Prop :=\n  (∀ t, HasSubgradientAt (f t) (gf t) (w t)) ∧\n  (∀ t, HasSubgradientAt r (gr t) (w t)) ∧\n  (∀ t, w (t + 1) = w t - η t • gf t - η t • gr (t + 1))\n\ndef Regret (f : ℕ → E → ℝ) (r : E → ℝ) (w : ℕ → E) (wstar : E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 1 T, (f t (w t) + r (w t) - (f t wstar + r wstar))\n\ndef sqL2 {n : ℕ} (x : Fin n → ℝ) : ℝ := ∑ i, (x i) ^ 2\n\nnoncomputable def linftyNorm {n : ℕ} (x : Fin n → ℝ) : ℝ := ⨆ i, |x i|\n\ndef l1Norm {n : ℕ} (x : Fin n → ℝ) : ℝ := ∑ i, |x i|\n\nnoncomputable def l2Norm {n : ℕ} (x : Fin n → ℝ) : ℝ := Real.sqrt (sqL2 x)\n\ndef IsProxNorm {n : ℕ} (N : (Fin n → ℝ) → ℝ) (lam : ℝ) (x xm : Fin n → ℝ) : Prop :=\n  IsMinOn (fun u => sqL2 (u - x) / 2 + lam * N u) Set.univ xm\n\ntheorem theorem6\n    (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G c : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hc : 0 < c)\n    (hη : ∀ t, 1 ≤ t → η t = c / Real.sqrt t)\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T ≤ 2 * G * D + (D ^ 2 / (2 * c) + 7 * G ^ 2 * c) * Real.sqrt T := by\n  sorry\n\ntheorem theorem8\n    (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G H : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hH : 0 < H)\n    (hsc : ∀ t, StrongConvexOn (Metric.closedBall wstar D) H (fun x => f t x + r x))\n    (hη : ∀ t, 1 ≤ t → η t = 1 / (H * t))\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T\n      ≤ 2 * G * D + H * D ^ 2 + (7 * G ^ 2 / (2 * H)) * (1 + Real.log T) := by\n  sorry\n\ntheorem proposition11\n    {n : ℕ} (T : ℕ) (hT : 0 < T) (w0 wstar : Fin n → ℝ) (w : ℕ → Fin n → ℝ)\n    (lam : ℕ → ℝ) (hlam : ∀ t ∈ Finset.Icc 1 T, 0 ≤ lam t)\n    (hw0 : w 0 = w0)\n    (N : (Fin n → ℝ) → ℝ)\n    (hN : N = l1Norm ∨ N = l2Norm ∨ N = linftyNorm)\n    (hseq : ∀ t ∈ Finset.Icc 1 T, IsProxNorm N (lam t) (w (t - 1)) (w t))\n    (hbatch : IsProxNorm N (∑ t ∈ Finset.Icc 1 T, lam t) w0 wstar) :\n    w T = wstar := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem6", "theorem8", "proposition11"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 1269, "end": 1888, "gold": "(f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G c : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hc : 0 < c)\n    (hη : ∀ t, 1 ≤ t → η t = c / Real.sqrt t)\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T ≤ 2 * G * D + (D ^ 2 / (2 * c) + 7 * G ^ 2 * c) * Real.sqrt T"}]}
{"item_id": "v10-duchi09a::theorem8::A-stmt", "paper_id": "v10-duchi09a", "target_name": "theorem8", "source": "Research-level", "nl_statement": "Theorem 8 As in Theorem 6, suppose that $\\| \\pmb{w}_{t}- \\pmb{w}^{\\star}\\| \\leq D$ and that G bounds $\\partial f_{t}$ and $\\partial r$ from above. In addition, suppose that for every t the function $f_{t}+ r$ is H-strongly convex on the region $\\| \\pmb{w}- \\pmb{w}^{\\star}\\| \\leq D$. Then, with the step sizes $\\eta_{t}=1/(Ht)$, FoBoS has regret\n\n$$\nR_{f+r}(T)\\leq 2GD+HD^{2}+\\frac{7G^{2}}{2H}(1+\\log T).\n$$", "context": [{"name": "Setup and notation: objective f + r, convex and nonnegative (Section 1)", "statement": "We first fix the notation used throughout the paper. Scalars are written as lower case letters and vectors as bold lower case letters, e.g. w. For two vectors u and v, $\\langle{\\pmb u},{\\pmb v}\\rangle$ is their inner product. The p-norm of a vector x is written $\\| \\pmb{x}\\|_{p}$, and ||x|| abbreviates $\\| \\pmb{x}\\|_{2}$.\n\nThe paper studies an algorithmic framework for regularized convex programming whose aim is to minimize a sum of two functions,\n\n$$\nf({\\pmb w})+r({\\pmb w}),\\tag{1}\n$$\n\nin which f and r are both convex and bounded below (so that, with no loss of generality, they are taken to map into $\\mathbb{R}_{+}$). Typically f is an empirical loss of the form $\\textstyle\\sum_{i\\in S}\\ell_{i}(w)$ built from loss functions $\\ell_{i}: \\mathbb{R}^{n}\\to \\mathbb{R}_{+}$, and $r(w)$ is a regularizer that penalizes overly complex vectors, e.g. $r(\\mathbf{w})=\\lambda\\|\\mathbf{w}\\|_{p}$. Such tasks are common in machine learning, where learning problems for decision and prediction are formulated as convex optimization problems. With this in mind, we study a general and intuitive algorithm called forward-backward splitting for minimizing Eq. (1), with particular emphasis on derivations for, and the use of, regularization functions that are not differentiable."}, {"name": "Definition: subgradient set", "statement": "For a function f and a point w, write $\\partial f ({\\boldsymbol{w}})$ for the subgradient set of f at w, that is,\n\n$$\n\\partial f(w)=\\{g\\mid\\forall v:f(v)\\geq f(w)+\\langle g,v-w\\rangle\\}.\n$$"}, {"name": "Algorithm: FoBoS update and forward-looking subgradient, Eqs. (2)-(5) (Section 2)", "statement": "Our version of Forward-Backward Splitting is driven by the wish that the iterates ${\\pmb w}_{t}$ be able to reach points where r is not differentiable. For problems such as $\\ell_{1}$-regularization, the method sidesteps the difficulties caused by non-differentiability by alternating subgradient steps with analytical minimization steps. Informally, FoBoS resembles the projected subgradient method, except that the projection step is replaced or augmented by an instantaneous minimization problem whose solution can be derived in closed form. The method is compact: every iteration is made of the two steps below,\n\n$$\n\\begin{array}{r l r}{{\\bf w}_{t + \\frac{1}{2}}}&{{}=}&{{\\bf w}_{t}- \\boldsymbol \\eta_{t}{\\bf g}_{t}^{f},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{r c l}{{\\pmb w}_{t+1}}&{=}&{\\displaystyle\\operatorname*{argmin}_{{\\pmb w}}\\left\\{\\frac{1}{2}\\left\\|{\\pmb w}-{\\pmb w}_{t+\\frac{1}{2}}\\right\\|^{2}+\\eta_{t+\\frac{1}{2}}r({\\pmb w})\\right\\}\\ .}\\end{array}\\tag{3}\n$$\n\nHere $g_{t}^{f}$ is an element of $\\partial f ({\\pmb w}_{t})$, and $\\eta_{t}$ denotes the step size used by the algorithm at time step t. What value $\\eta_{t}$ takes depends on the particular setting and analysis. The first step is thus nothing more than an unconstrained subgradient step for the function $f$. The second step produces a new vector that balances two aims: (i) remaining near the interim vector $w_{t + \\frac{1}{2}}$, and (ii) having small complexity as measured by r. Observe that r is multiplied by an interim step size, written $\\eta_{t + \\frac{1}{2}}$. The analyses later in the paper fix the particular value of $\\eta_{t + \\frac{1}{2}}$, which is either $\\eta_{t}$ or $\\eta_{t + 1}$.\n\nAn important property of the solution of Eq. (3), which also explains the name FoBoS, is the necessary optimality condition: at the optimum ${\\pmb w}_{t + 1}$, the subgradient set of the objective must contain the zero vector, i.e.\n\n$$\n\\mathbf{0}\\in\\partial\\left\\{\\frac{1}{2}\\left\\|w-w_{t+\\frac{1}{2}}\\right\\|^{2}+\\eta_{t+\\frac{1}{2}}r(w)\\right\\}\\bigg|_{w=w_{t+1}}.\n$$\n\nAs $\\pmb{w}_{t + \\frac{1}{2}}= \\pmb{w}_{t}- \\eta_{t}\\pmb{g}_{t}^{f}$, this condition reads\n\n$$\n\\mathbf{0}\\in\\pmb{w}_{t+1}-\\pmb{w}_{t}+\\eta_{t}\\pmb{g}_{t}^{f}+\\eta_{t+\\frac{1}{2}}\\partial r(\\pmb{w}_{t+1}).\\tag{4}\n$$\n\nFrom $\\mathbf{0}\\in\\pmb{w}_{t+1}-\\pmb{w}_{t}+\\eta_{t}\\pmb{g}_{t}^{f}+\\eta_{t+\\frac{1}{2}}\\partial r(\\pmb{w}_{t+1})$ it follows that, as long as ${\\pmb w}_{t + 1}$ is taken to be the minimizer of Eq. (3), there is guaranteed to be a vector $\\pmb{g}_{t+1}^{r}\\in\\partial r(\\pmb{w}_{t+1})$ with\n\n$$\n{\\bf 0}={\\pmb w}_{t + 1}-{\\pmb w}_{t}+ \\eta_{t}{\\pmb g}_{t}^{f}+ \\eta_{t + \\frac{1}{2}}{\\pmb g}_{t + 1}^{r}\\mathrm{~ . ~}\n$$\n\nThis identity may be read as an update rule in which the new weight vector ${\\pmb w}_{t + 1}$ is a linear combination of three vectors: the previous weight vector ${\\pmb w}_{t}$, an element of the subgradient set of $f$ at ${\\pmb w}_{t}$, and an element of the subgradient of r at the not-yet-known ${\\pmb w}_{t + 1}$; this is the origin of the name Forward-Looking Subgradient. In summary, ${\\pmb w}_{t + 1}$ can be written as\n\n$$\n\\pmb{w}_{t + 1}= \\pmb{w}_{t}- \\eta_{t}\\pmb{g}_{t}^{f}- \\eta_{t + \\frac{1}{2}}\\pmb{g}_{t + 1}^{r},\\tag{5}\n$$\n\nwith ${\\pmb g}_{t}^{f}\\in \\partial f ({\\pmb w}_{t})$ and $\\pmb{g}_{t+1}^{r}\\in\\partial r(\\pmb{w}_{t+1})$. Solving Eq. (3) with r as above brings two main advantages. First, algorithmically, it produces sparse solutions at almost no extra computational cost. Second, thanks to the forward-looking gradient we can reuse existing analyses and prove that the resulting framework inherits the formal convergence guarantees of many existing gradient-based and online convex programming algorithms."}, {"name": "Setup: choice of the interim step size eta_{t+1/2} (Section 3)", "statement": "At first sight FoBoS appears quite different from subgradient and online convex programming methods. Yet because FoBoS in fact uses a forward-looking subgradient of the regularizer, existing analyses carry over well. In this section we adapt known results, exploiting the forward-looking property of FoBoS, to obtain convergence rate analyses for FoBoS. This requires choosing $\\eta_{t + \\frac{1}{2}}$ appropriately. As shown below, to get convergence and low regret bounds it suffices to take $\\eta_{t + \\frac{1}{2}}$ equal to $\\eta_{t}$ or to $\\eta_{t + 1}$, according to whether the optimization is online or batch. We first analyze FoBoS in the batch setting. There we use the subgradient of $f$, put $\\eta_{t + \\frac{1}{2}}= \\eta_{t + 1}$, and move from ${\\pmb w}_{t}$ to ${\\pmb w}_{t + 1}$ as specified by Eq. (2) and Eq. (3)."}, {"name": "Assumption: bounded subgradients, Eq. (6)", "statement": "In this section $w^{\\star}$ always denotes the minimizer of $f ({\\pmb w}) + r ({\\pmb w})$. Below we set $\\begin{array}{r}{\\|\\partial f(\\pmb{w})\\|\\triangleq\\operatorname*{sup}_{\\pmb{g}\\in\\partial f(\\pmb{w})}\\|\\pmb{g}\\|}\\end{array}$. Our first convergence results are derived under the rather general assumption (see, for instance, Langford et al. 2008 or Shalev-Shwartz and Tewari 2009) that the subgradients obey the bounds\n\n$$\n\\begin{array}{r}{\\|\\partial f(\\pmb{w})\\|^{2}\\leq Af(\\pmb{w})+G^{2},\\quad\\|\\partial r(\\pmb{w})\\|^{2}\\leq Ar(\\pmb{w})+G^{2}.}\\end{array}\\tag{6}\n$$\n\nAs examples, every Lipschitz loss (for instance the logistic loss, or the hinge loss of support vector machines) meets this condition with $A = 0$ and G the Lipschitz constant, while least squares estimation meets Eq. (6) with $G = 0$ and $A = 4$. The following lemma is technical, but it is the central tool behind every convergence result in this paper."}, {"name": "Lemma 1 (Bounding Step Differences)", "statement": "Lemma 1 (Bounding Step Differences) Suppose the subgradient norms of f and r obey the bounds of Eq. (6):\n\n$$\n\\|\\partial f(\\pmb{w})\\|^{2}\\leq Af(\\pmb{w})+G^{2},\\quad\\|\\partial r(\\pmb{w})\\|^{2}\\leq Ar(\\pmb{w})+G^{2}.\n$$\n\nLet $\\eta_{t + 1}\\leq \\eta_{t + \\frac{1}{2}}\\leq \\eta_{t}$, and assume $\\eta_{t}\\le 2 \\eta_{t + \\frac{1}{2}}$. Then, when the FoBoS update of Eqs. (2) and (3) is used, for a constant $c \\leq 4$ and every vector $\\pmb{w}^{\\star}$,\n\n$$\n\\begin{array}{r l r}{{2\\boldsymbol\\eta_{t}\\big(1-cA\\boldsymbol\\eta_{t}\\big)f\\big(\\boldsymbol{w}_{t}\\big)+2\\boldsymbol\\eta_{t+\\frac{1}{2}}\\big(1-cA\\boldsymbol\\eta_{t+\\frac{1}{2}}\\big)\\boldsymbol{r}\\big(\\boldsymbol{w}_{t+1}\\big)}}\\\\ &{\\leq}&{2\\boldsymbol\\eta_{t}f\\big(\\boldsymbol{w}^{\\star}\\big)+2\\boldsymbol\\eta_{t+\\frac{1}{2}}\\boldsymbol{r}\\big(\\boldsymbol{w}^{\\star}\\big)+\\|\\boldsymbol{w}_{t}-\\boldsymbol{w}^{\\star}\\|^{2}-\\|\\boldsymbol{w}_{t+1}-\\boldsymbol{w}^{\\star}\\|^{2}+7\\boldsymbol\\eta_{t}\\boldsymbol\\eta_{t+\\frac{1}{2}}G^{2}\\quad .}\\end{array}\n$$"}, {"name": "Setup: online learning and regret R_{f+r}(T) (Section 4)", "statement": "This section gives a regret analysis of FoBoS in online settings. An online learning problem supplies a sequence of functions $f_{t}: \\mathbb{R}^{n}\\longrightarrow \\mathbb{R}$. The aim is that the predictions ${\\pmb w}_{t}$ incur low regret relative to one fixed optimal predictor $w^{\\star}$. Precisely, $f_{t}({\\boldsymbol{w}})$ is the loss incurred on the $t^{t h}$ input loss function by the predictor w. For an online algorithm whose predictors are $\\boldsymbol{w}_{1},\\ldots,\\boldsymbol{w}_{t},\\ldots$, the regret with respect to a fixed predictor $\\pmb{w}^{\\star}$ under a regularization function r is\n\n$$\nR_{f+r}(T)=\\sum_{t=1}^{T}\\left[f_{t}(\\pmb{w}_{t})+r(\\pmb{w}_{t})-\\left(f_{t}(\\pmb{w}^{\\star})+r(\\pmb{w}^{\\star})\\right)\\right].\n$$\n\nThe ideal would be zero regret against a stationary $\\pmb{w}^{\\star}$ for sequences of any length.\n\nTo get an online bound for a sequence of convex functions $f_{t}$, we adapt the arguments of Zinkevich (2003). With the bound of Lemma 1 at hand, a theorem on the online regret of FoBoS can be stated and proved directly. The boundedness assumptions in the proof could be dropped (yielding a regret analogue of the bound in Theorem 2), but we do not consider this notably more interesting. Apart from its use of Lemma 1, the proof closely follows Zinkevich's, and so it is postponed to Appendix A."}, {"name": "Definition: H-strong convexity and Eq. (17)", "statement": "FoBoS also admits an improved regret bound when the loss functions $f_{t}(\\cdot)$ or the function $r(\\cdot)$ are strongly convex. Hazan et al. (2006) showed that, for the projected gradient method with strongly convex functions, regret of order ${\\cal O}(\\log T)$ is attainable by exploiting the curvature of the functions $f_{t}$ instead of relying only on convexity and linearity as in Theorems 2 and 6. These results extend to FoBoS whenever $f_{t}({\\pmb w}) + r ({\\pmb w})$ is strongly convex, at least on the region $\\| \\pmb{w}- \\pmb{w}^{\\star}\\| \\leq D$. For the sake of completeness we restate some definitions and give the logarithmic regret bound for FoBoS. A function f is said to be H-strongly convex when\n\n$$\nf(\\pmb{w})\\geq f(\\pmb{w}_{t})+\\langle\\nabla f(\\pmb{w}_{t}),\\pmb{w}-\\pmb{w}_{t}\\rangle+\\frac{H}{2}\\|\\pmb{w}-\\pmb{w}_{t}\\|^{2}.\n$$\n\nHence, when r and the functions $f_{t}$ are strongly convex with constants $H_{r}\\geq 0$ and $H_{f}\\geq 0$ respectively, we get $H = H_{f}+ H_{r}$, and H-strong convexity yields\n\n$$\nf_{t}({\\boldsymbol{w}}_{t}) - f ({\\boldsymbol{w}}^{\\star}) + r ({\\boldsymbol{w}}_{t}) - r ({\\boldsymbol{w}}^{\\star}) \\leq \\left\\langle{\\boldsymbol{g}}_{t}^{f}+{\\boldsymbol{g}}_{t}^{r},{\\boldsymbol{w}}_{t}-{\\boldsymbol{w}}^{\\star}\\right\\rangle - \\frac{H}{2}\\|{\\boldsymbol{w}}_{t}-{\\boldsymbol{w}}^{\\star}\\|^{2}~ .\\tag{17}\n$$\n\nIt is not necessary for both $f_{t}$ and r to be strongly convex; it is enough that at least one of the two has a positive strong convexity constant. For instance, with $\\begin{array}{r}{r ({\\bf w}) = \\frac{\\lambda}{2}\\|{\\bf w}\\|^{2}}\\end{array}$ we have $H \\geq \\lambda$ provided the functions $f_{t}$ are convex. Using Eq. (17), we can build directly on Hazan et al. (2006) to prove a stronger regret bound for the online learning case. Because the proof resembles that of Hazan et al., it too is deferred to Appendix A."}, {"name": "Eq. (36), from the proof of Theorem 6", "statement": "Proof of Theorem 6 When $\\| \\partial f \\|$ and $\\| \\partial r \\|$ are bounded by G, Lemma 1 gives at once\n\n$$\nf_{t}(\\boldsymbol{w}_{t})-f_{t}(\\boldsymbol{w}^{\\star})+r(\\boldsymbol{w}_{t+1})-r(\\boldsymbol{w}^{\\star})\\leq\\frac{1}{2\\eta_{t}}\\left(\\|\\boldsymbol{w}_{t}-\\boldsymbol{w}^{\\star}\\|^{2}-\\|\\boldsymbol{w}_{t+1}-\\boldsymbol{w}^{\\star}\\|^{2}\\right)+\\frac{7}{2}G^{2}\\eta_{t}.\\tag{36}\n$$"}, {"name": "Theorem 6 (another result of the same paper)", "statement": "Theorem 6 Suppose that $\\| \\pmb{w}_{t}- \\pmb{w}^{\\star}\\| \\leq D$ holds at every iteration, and that G is an upper bound on the norms of the subgradient sets $\\partial f_{t}$ and $\\partial r$. Fix an arbitrary scalar $c > 0$. Then FoBoS run with step sizes $\\eta_{t}= c / \\sqrt{t}$ has a regret bound of the form\n\n$$\nR_{f+r}(T)\\leq 2GD+\\left({\\frac{D^{2}}{2c}}+7G^{2}c\\right)\\sqrt{T}.\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\ndef HasSubgradientAt (f : E → ℝ) (g x : E) : Prop :=\n  ∀ y, f y ≥ f x + @inner ℝ _ _ g (y - x)\n\ndef IsFoBoSRun (f r : E → ℝ) (w gf gr : ℕ → E) (η ηhalf : ℕ → ℝ) : Prop :=\n  (∀ t, HasSubgradientAt f (gf t) (w t)) ∧\n  (∀ t, HasSubgradientAt r (gr t) (w t)) ∧\n  (∀ t, w (t + 1) = w t - η t • gf t - ηhalf t • gr (t + 1))\n\ndef IsFoBoSOnlineRun (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) :\n    Prop :=\n  (∀ t, HasSubgradientAt (f t) (gf t) (w t)) ∧\n  (∀ t, HasSubgradientAt r (gr t) (w t)) ∧\n  (∀ t, w (t + 1) = w t - η t • gf t - η t • gr (t + 1))\n\ndef Regret (f : ℕ → E → ℝ) (r : E → ℝ) (w : ℕ → E) (wstar : E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 1 T, (f t (w t) + r (w t) - (f t wstar + r wstar))\n\ndef sqL2 {n : ℕ} (x : Fin n → ℝ) : ℝ := ∑ i, (x i) ^ 2\n\nnoncomputable def linftyNorm {n : ℕ} (x : Fin n → ℝ) : ℝ := ⨆ i, |x i|\n\ndef l1Norm {n : ℕ} (x : Fin n → ℝ) : ℝ := ∑ i, |x i|\n\nnoncomputable def l2Norm {n : ℕ} (x : Fin n → ℝ) : ℝ := Real.sqrt (sqL2 x)\n\ndef IsProxNorm {n : ℕ} (N : (Fin n → ℝ) → ℝ) (lam : ℝ) (x xm : Fin n → ℝ) : Prop :=\n  IsMinOn (fun u => sqL2 (u - x) / 2 + lam * N u) Set.univ xm\n\ntheorem theorem6\n    (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G c : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hc : 0 < c)\n    (hη : ∀ t, 1 ≤ t → η t = c / Real.sqrt t)\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T ≤ 2 * G * D + (D ^ 2 / (2 * c) + 7 * G ^ 2 * c) * Real.sqrt T := by\n  sorry\n\ntheorem theorem8\n    (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G H : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hH : 0 < H)\n    (hsc : ∀ t, StrongConvexOn (Metric.closedBall wstar D) H (fun x => f t x + r x))\n    (hη : ∀ t, 1 ≤ t → η t = 1 / (H * t))\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T\n      ≤ 2 * G * D + H * D ^ 2 + (7 * G ^ 2 / (2 * H)) * (1 + Real.log T) := by\n  sorry\n\ntheorem proposition11\n    {n : ℕ} (T : ℕ) (hT : 0 < T) (w0 wstar : Fin n → ℝ) (w : ℕ → Fin n → ℝ)\n    (lam : ℕ → ℝ) (hlam : ∀ t ∈ Finset.Icc 1 T, 0 ≤ lam t)\n    (hw0 : w 0 = w0)\n    (N : (Fin n → ℝ) → ℝ)\n    (hN : N = l1Norm ∨ N = l2Norm ∨ N = linftyNorm)\n    (hseq : ∀ t ∈ Finset.Icc 1 T, IsProxNorm N (lam t) (w (t - 1)) (w t))\n    (hbatch : IsProxNorm N (∑ t ∈ Finset.Icc 1 T, lam t) w0 wstar) :\n    w T = wstar := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem6", "theorem8", "proposition11"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 1925, "end": 2636, "gold": "(f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G H : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hH : 0 < H)\n    (hsc : ∀ t, StrongConvexOn (Metric.closedBall wstar D) H (fun x => f t x + r x))\n    (hη : ∀ t, 1 ≤ t → η t = 1 / (H * t))\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T\n      ≤ 2 * G * D + H * D ^ 2 + (7 * G ^ 2 / (2 * H)) * (1 + Real.log T)"}]}
{"item_id": "v10-duchi09a::proposition11::A-stmt", "paper_id": "v10-duchi09a", "target_name": "proposition11", "source": "Research-level", "nl_statement": "Proposition 11 For $t=1,\\ldots,T$, solve one after another the T self-similar optimization problems\n\n$$\n\\mathcal{P}.1:~{\\pmb w}_{t}=\\operatorname*{argmin}_{{\\pmb w}}\\frac{1}{2}\\|{\\pmb w}-{\\pmb w}_{t-1}\\|^{2}+\\lambda_{t}\\|{\\pmb w}\\|_{q}\\enspace .\\tag{28}\n$$\n\nand let $\\pmb{w}_{T}$ be the vector obtained at the end. Denote by $\\pmb{w}^{\\star}$ the optimal solution of the problem\n\n$$\n\\mathcal{P}. 2:\\pmb{w}^{\\star}=\\underset{\\pmb{w}}{\\mathrm{argmin}}\\frac{1}{2}\\|\\pmb{w}-\\pmb{w}_{0}\\|^{2}+\\left(\\sum_{t=1}^{T}\\lambda_{t}\\right)\\|\\pmb{w}\\|_{q}.\n$$\n\nThen, for $q\\in\\{1,2,\\infty\\}$, the two vectors $\\pmb{w}_{T}$ and $\\pmb{w}^{\\star}$ coincide.", "context": [{"name": "Setup and notation: objective f + r, convex and nonnegative (Section 1)", "statement": "We first fix the notation used throughout the paper. Scalars are written as lower case letters and vectors as bold lower case letters, e.g. w. For two vectors u and v, $\\langle{\\pmb u},{\\pmb v}\\rangle$ is their inner product. The p-norm of a vector x is written $\\| \\pmb{x}\\|_{p}$, and ||x|| abbreviates $\\| \\pmb{x}\\|_{2}$.\n\nThe paper studies an algorithmic framework for regularized convex programming whose aim is to minimize a sum of two functions,\n\n$$\nf({\\pmb w})+r({\\pmb w}),\\tag{1}\n$$\n\nin which f and r are both convex and bounded below (so that, with no loss of generality, they are taken to map into $\\mathbb{R}_{+}$). Typically f is an empirical loss of the form $\\textstyle\\sum_{i\\in S}\\ell_{i}(w)$ built from loss functions $\\ell_{i}: \\mathbb{R}^{n}\\to \\mathbb{R}_{+}$, and $r(w)$ is a regularizer that penalizes overly complex vectors, e.g. $r(\\mathbf{w})=\\lambda\\|\\mathbf{w}\\|_{p}$. Such tasks are common in machine learning, where learning problems for decision and prediction are formulated as convex optimization problems. With this in mind, we study a general and intuitive algorithm called forward-backward splitting for minimizing Eq. (1), with particular emphasis on derivations for, and the use of, regularization functions that are not differentiable."}, {"name": "Definition: subgradient set", "statement": "For a function f and a point w, write $\\partial f ({\\boldsymbol{w}})$ for the subgradient set of f at w, that is,\n\n$$\n\\partial f(w)=\\{g\\mid\\forall v:f(v)\\geq f(w)+\\langle g,v-w\\rangle\\}.\n$$"}, {"name": "Algorithm: FoBoS update and forward-looking subgradient, Eqs. (2)-(5) (Section 2)", "statement": "Our version of Forward-Backward Splitting is driven by the wish that the iterates ${\\pmb w}_{t}$ be able to reach points where r is not differentiable. For problems such as $\\ell_{1}$-regularization, the method sidesteps the difficulties caused by non-differentiability by alternating subgradient steps with analytical minimization steps. Informally, FoBoS resembles the projected subgradient method, except that the projection step is replaced or augmented by an instantaneous minimization problem whose solution can be derived in closed form. The method is compact: every iteration is made of the two steps below,\n\n$$\n\\begin{array}{r l r}{{\\bf w}_{t + \\frac{1}{2}}}&{{}=}&{{\\bf w}_{t}- \\boldsymbol \\eta_{t}{\\bf g}_{t}^{f},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{r c l}{{\\pmb w}_{t+1}}&{=}&{\\displaystyle\\operatorname*{argmin}_{{\\pmb w}}\\left\\{\\frac{1}{2}\\left\\|{\\pmb w}-{\\pmb w}_{t+\\frac{1}{2}}\\right\\|^{2}+\\eta_{t+\\frac{1}{2}}r({\\pmb w})\\right\\}\\ .}\\end{array}\\tag{3}\n$$\n\nHere $g_{t}^{f}$ is an element of $\\partial f ({\\pmb w}_{t})$, and $\\eta_{t}$ denotes the step size used by the algorithm at time step t. What value $\\eta_{t}$ takes depends on the particular setting and analysis. The first step is thus nothing more than an unconstrained subgradient step for the function $f$. The second step produces a new vector that balances two aims: (i) remaining near the interim vector $w_{t + \\frac{1}{2}}$, and (ii) having small complexity as measured by r. Observe that r is multiplied by an interim step size, written $\\eta_{t + \\frac{1}{2}}$. The analyses later in the paper fix the particular value of $\\eta_{t + \\frac{1}{2}}$, which is either $\\eta_{t}$ or $\\eta_{t + 1}$.\n\nAn important property of the solution of Eq. (3), which also explains the name FoBoS, is the necessary optimality condition: at the optimum ${\\pmb w}_{t + 1}$, the subgradient set of the objective must contain the zero vector, i.e.\n\n$$\n\\mathbf{0}\\in\\partial\\left\\{\\frac{1}{2}\\left\\|w-w_{t+\\frac{1}{2}}\\right\\|^{2}+\\eta_{t+\\frac{1}{2}}r(w)\\right\\}\\bigg|_{w=w_{t+1}}.\n$$\n\nAs $\\pmb{w}_{t + \\frac{1}{2}}= \\pmb{w}_{t}- \\eta_{t}\\pmb{g}_{t}^{f}$, this condition reads\n\n$$\n\\mathbf{0}\\in\\pmb{w}_{t+1}-\\pmb{w}_{t}+\\eta_{t}\\pmb{g}_{t}^{f}+\\eta_{t+\\frac{1}{2}}\\partial r(\\pmb{w}_{t+1}).\\tag{4}\n$$\n\nFrom $\\mathbf{0}\\in\\pmb{w}_{t+1}-\\pmb{w}_{t}+\\eta_{t}\\pmb{g}_{t}^{f}+\\eta_{t+\\frac{1}{2}}\\partial r(\\pmb{w}_{t+1})$ it follows that, as long as ${\\pmb w}_{t + 1}$ is taken to be the minimizer of Eq. (3), there is guaranteed to be a vector $\\pmb{g}_{t+1}^{r}\\in\\partial r(\\pmb{w}_{t+1})$ with\n\n$$\n{\\bf 0}={\\pmb w}_{t + 1}-{\\pmb w}_{t}+ \\eta_{t}{\\pmb g}_{t}^{f}+ \\eta_{t + \\frac{1}{2}}{\\pmb g}_{t + 1}^{r}\\mathrm{~ . ~}\n$$\n\nThis identity may be read as an update rule in which the new weight vector ${\\pmb w}_{t + 1}$ is a linear combination of three vectors: the previous weight vector ${\\pmb w}_{t}$, an element of the subgradient set of $f$ at ${\\pmb w}_{t}$, and an element of the subgradient of r at the not-yet-known ${\\pmb w}_{t + 1}$; this is the origin of the name Forward-Looking Subgradient. In summary, ${\\pmb w}_{t + 1}$ can be written as\n\n$$\n\\pmb{w}_{t + 1}= \\pmb{w}_{t}- \\eta_{t}\\pmb{g}_{t}^{f}- \\eta_{t + \\frac{1}{2}}\\pmb{g}_{t + 1}^{r},\\tag{5}\n$$\n\nwith ${\\pmb g}_{t}^{f}\\in \\partial f ({\\pmb w}_{t})$ and $\\pmb{g}_{t+1}^{r}\\in\\partial r(\\pmb{w}_{t+1})$. Solving Eq. (3) with r as above brings two main advantages. First, algorithmically, it produces sparse solutions at almost no extra computational cost. Second, thanks to the forward-looking gradient we can reuse existing analyses and prove that the resulting framework inherits the formal convergence guarantees of many existing gradient-based and online convex programming algorithms."}, {"name": "Setup: proximal step notation, Eq. (18) and [z]_+ (Section 5)", "statement": "We begin by adjusting the notation. To make the derivations simpler, we write v for the vector $w_{t + \\frac{1}{2}}= w_{t}- \\eta_{t}g_{t}^{f}$ and write $\\tilde{\\lambda}$ for $\\eta_{t + \\frac{1}{2}}\\cdot \\lambda$. In this notation the problem of Eq. (3) becomes\n\n$$\n\\operatorname*{minimize}_{\\pmb{w}}\\frac{1}{2}\\|\\pmb{w}-\\pmb{v}\\|^{2}+\\widetilde{\\lambda}r(\\pmb{w}).\\tag{18}\n$$\n\nWe use $[ z ]_{+}$ for $\\max\\{0,z\\}$. Although full derivations are given for every regularization function we consider, to save space we do not formally state well-known technical lemmas. Many of the results below were stated implicitly by Wright et al. (2009)."}, {"name": "Update: l1 regularization (soft thresholding), Eq. (19) (Section 5.1)", "statement": "Taking $r(\\pmb{w})=\\lambda\\|\\pmb{w}\\|_{1}$ leads to a simple and intuitive update. To begin with, the objective decomposes, since Eq. (18) can be rewritten as\n\n$$\n\\operatorname*{minimize}_{\\boldsymbol{w}}\\ \\sum_{j=1}^{n}\\left(\\frac{1}{2}(w_{j}-v_{j})^{2}+\\tilde{\\lambda}|w_{j}|\\right).\n$$\n\nFix one coordinate of w and, to shorten notation, drop the index $j$. Let $w^{\\star}$ be the minimizer of $\\frac{1}{2}(w-v)^{2}+\\tilde{\\lambda}|w|$. Plainly $w^{\\star}\\cdot v \\geq 0$. Otherwise $w^{\\star}\\cdot v < 0$, but then ${\\scriptstyle{\\frac{1}{2}}}v^{2}<$ $\\begin{array}{r}{\\frac{1}{2}v^{2}-w^{\\star}\\cdot v+\\frac{1}{2}(w^{\\star})^{2}<\\frac{1}{2}(v-w^{\\star})^{2}+\\tilde{\\lambda}|w^{\\star}|}\\end{array}$, which contradicts the assumed optimality of $w^{\\star}$. Since the objective is symmetric in v, we may also take $v \\geq 0$; it then remains to minimize $\\frac{1}{2}\\left(w-v\\right)^{2}+\\tilde{\\lambda}w$ subject to $w \\ge 0$. Attaching a Lagrange multiplier $\\beta \\geq 0$ to this constraint gives the Lagrangian $\\frac{1}{2}(w-v)^{2}+\\tilde{\\lambda}w-\\beta w$. Differentiating the Lagrangian in w and equating to zero shows that the optimum is $w^{\\star}= v - \\tilde{\\lambda}+ \\beta$. When $w^{\\star}> 0$, complementary slackness, which requires the optimal pair $w^{\\star}$ and β to satisfy $w^{\\star}\\beta = 0$ (Boyd and Vandenberghe, 2004), forces $\\beta = 0$, hence $w^{\\star}= v - \\tilde{\\lambda}$. When $v < \\tilde{\\lambda}$, we have $v - \\tilde{\\lambda}< 0$, so β must be positive and, again by complementary slackness, $w^{\\star}= 0$. The case $v \\leq 0$ is the same up to a change of signs. Putting this together and returning to the full notation, the coordinates of the optimal solution $\\pmb{w}^{\\star}= \\pmb{w}_{t + 1}$ are obtained from $w_{t + \\frac{1}{2}}$ by\n\n$$\nw_{t+1,j}=\\mathrm{sign}\\Big(w_{t+\\frac{1}{2},j}\\Big)\\left[|w_{t+\\frac{1}{2},j}| -\\widetilde{\\lambda}\\right]_{+}=\\mathrm{sign}\\Big(w_{t,j}-\\eta_{t}g_{t,j}^{f}\\Big)\\left[\\Big|w_{t,j}-\\eta_{t}g_{t,j}^{f}\\Big| -\\eta_{t+\\frac{1}{2}}\\cdot\\lambda\\right]_{+}.\\tag{19}\n$$\n\nThis update can produce sparse solutions: any component of ${\\pmb w}_{t +{\\frac{1}{2}}}$ whose absolute value is below $\\tilde{\\lambda}$ yields a zero in the corresponding component of ${\\pmb w}_{t + 1}$. Eq. (19) therefore provides a simple online and offline method for minimizing a convex $f$ under $\\ell_{1}$ regularization."}, {"name": "Update: l2^2 regularization, Eq. (20) (Section 5.2)", "statement": "With $\\begin{array}{r}{r(\\pmb{w})=\\frac{\\lambda}{2}\\|\\pmb{w}\\|_{2}^{2}}\\end{array}$ the optimization problem is very simple:\n\n$$\n\\operatorname*{minimize}_{\\boldsymbol{w}}\\frac{1}{2}\\|\\boldsymbol{w}-\\boldsymbol{v}\\|^{2}+\\frac{1}{2}\\widetilde{\\lambda}\\|\\boldsymbol{w}\\|^{2},\n$$\n\nwhere, to make the solution more concise, $\\tilde{\\lambda}$ has been replaced by $\\frac{1}{2}\\tilde{\\lambda}$. Setting the derivative of this objective to zero gives $\\pmb{w}^{\\star}- v + \\tilde{\\lambda}\\pmb{w}^{\\star}= 0$, and in the original notation this is the update\n\n$$\n\\pmb{w}_{t + 1}= \\frac{\\pmb{w}_{t}- \\pmb{\\eta}_{t}\\pmb{g}_{t}^{f}}{1 + \\tilde{\\lambda}}.\\tag{20}\n$$"}, {"name": "Update: l2 regularization (Section 5.3)", "statement": "The $\\ell_{2}$ norm of the weight vector is a less commonly used choice of regularizer. Taking $r(\\boldsymbol{w})=\\tilde{\\lambda}\\|\\boldsymbol{w}\\|$, we arrive at the problem\n\n$$\n\\operatorname*{minimize}_{\\boldsymbol{w}}\\frac{1}{2}\\|\\boldsymbol{w}-\\boldsymbol{v}\\|^{2}+\\widetilde{\\lambda}\\|\\boldsymbol{w}\\|.\\tag{21}\n$$\n\nAny solution of Eq. (21) points in the direction of v, i.e. it has the form $\\pmb{w}^{\\star}= s \\pmb{v}$ with $s \\geq 0$. To verify this form, suppose for contradiction that $\\pmb{w}^{\\star}= s \\pmb{v}+ \\pmb{u}$, where u lies in the null space of v (should u have components parallel to v, these can be added to v, leaving an orthogonal $\\pmb{u}^{\\prime}$) and s is allowed to be negative. As u is orthogonal to $v$, the objective can be written in terms of $s$, $v$ and u as\n\n$$\n\\frac 12(1-s)^{2}\\|{\\pmb v}\\|^{2}+\\frac 12\\|{\\pmb u}\\|^{2}+\\widetilde{\\lambda}(\\|{\\pmb v}\\|+\\|{\\pmb u}\\|)\\ge\\frac 12(1-s)^{2}\\|{\\pmb v}\\|^{2}+\\widetilde{\\lambda}\\|{\\pmb v}\\|.\n$$\n\nSo u has to be the zero vector, ${\\pmb u}={\\bf 0}$, and the optimization problem can be written as\n\n$$\n\\operatorname*{minimize}_{s}\\frac{1}{2}(1-s)^{2}\\|\\pmb{v}\\|^{2}+\\widetilde{\\lambda}s\\|\\pmb{v}\\|\\enspace .\n$$\n\nMoreover, no negative s can be optimal. Indeed, when $s < 0$,\n\n$$\n\\frac{1}{2}(1-s)^{2}\\|\\pmb{v}\\|^{2}+\\tilde{\\lambda}s\\|\\pmb{v}\\|<\\frac{1}{2}\\|\\pmb{v}\\|^{2}.\n$$\n\nHence the choice $s = 0$ attains a smaller objective value, which rules out negative s as an optimal solution. We are thus led once more to a constrained scalar problem, $\\operatorname*{minimize}_{s\\geq 0}\\frac{1}{2}(1-s)^{2}\\|\\pmb{v}\\|^{2}+\\tilde{\\lambda}s\\|\\pmb{v}\\|$, whose Lagrangian is\n\n$$\n\\frac{1}{2}(1-s)^{2}\\|\\pmb{v}\\|^{2}+\\tilde{\\lambda}s\\|\\pmb{v}\\|-\\beta s\\ ,\n$$\n\nwith $\\beta \\geq 0$. Differentiating the Lagrangian in s and setting the derivative to zero gives $(s-1)\\|{\\pmb v}\\|^{2}+\\tilde{\\lambda}\\|{\\pmb v}\\|-{\\beta}=0$, and hence the closed form $s = 1 - \\tilde{\\lambda}/ \\| v \\| + \\beta / \\| v \\|^{2}$. If $s > 0$, complementary slackness gives $\\beta = 0$, and s simplifies to $s = 1 - \\tilde{\\lambda}/ \\| v \\|$. This quantity is positive if and only if $\\| v \\| > \\tilde{\\lambda}$. If $\\| v \\| < \\tilde{\\lambda}$, then β has to be positive and complementary slackness yields $s = 0$.\n\nIn summary, with $\\ell_{2}$ regularization the second step of the FoBoS update is\n\n$$\nw_{t+1}=\\left[1-\\frac{\\tilde{\\lambda}}{\\lVert w_{t+\\frac{1}{2}}\\rVert}\\right]_{+}w_{t+\\frac{1}{2}}=\\left[1-\\frac{\\tilde{\\lambda}}{\\lVert w_{t}-\\eta_{t}g_{t}^{f}\\rVert}\\right]_{+}(w_{t}-\\eta_{t}g_{t}^{f})\\ .\n$$\n\nConsequently, $\\ell_{2}$ regularization gives the zero weight vector exactly under the condition $\\| \\pmb{w}_{t}- \\pmb{\\eta}_{t}\\pmb{g}_{t}^{f}\\| \\le \\tilde{\\lambda}$. As a requirement for sparsity this is considerably stronger than the one for $\\ell_{1}$ (there a single weight becomes sparse depending only on its own value, whereas here sparsity occurs only when the whole weight vector has $\\ell_{2}$-norm below $\\tilde{\\lambda}$), so in high dimensions it is unlikely to be met. It is nonetheless a key building block when a mixed $\\ell_{1}/ \\ell_{2}$-norm serves as the regularizer, as shown later."}, {"name": "Update: l-infinity regularization via its dual (l1-ball projection), Eqs. (22)-(25) (Section 5.4)", "statement": "Next we consider a regularizer that has received less attention, the $\\ell_{\\infty}$ norm of w. On its own this regularizer cannot offer a strong guarantee against overfitting, because many of the weights of w may go unpenalized. There are, however, settings in which one wants to treat blocks of variables as a group, as in $\\ell_{1}/ \\ell_{\\infty}$ regularization. We postpone the discussion of mixed norms once more and concentrate on the $\\ell_{\\infty}$ norm alone, since it is a building block. Concretely, we want an efficient solution of the problem\n\n$$\n\\operatorname*{minimize}_{\\boldsymbol{w}}\\frac{1}{2}\\|\\boldsymbol{w}-\\boldsymbol{v}\\|^{2}+\\tilde{\\lambda}\\|\\boldsymbol{w}\\|_{\\infty}\\textrm{.}\\tag{22}\n$$\n\nAn efficient algorithm for the minimizer of Eq. (22) could be derived from properties of the subgradient set of $\\| \\boldsymbol{w}\\|_{\\infty}$. The dual form of Eq. (22), however, has a well-known solution. Since the conjugate of a quadratic function is quadratic and the conjugate of the $\\ell_{\\infty}$ norm is the $\\ell_{1}$ barrier function, the dual of the problem in Eq. (22) is immediately seen to be\n\n$$\n\\operatorname*{maximize}_{\\alpha}-\\frac{1}{2}\\left\\|\\alpha-\\pmb{v}\\right\\|_{2}^{2}\\mathrm{s . t .}\\left\\|\\pmb{\\alpha}\\right\\|_{1}\\leq\\tilde{\\lambda}\\mathrm{~.~}\\tag{23}\n$$\n\nIn addition, the dual variables $\\pmb{\\alpha}$ are related to the primal by ${\\pmb{\\alpha}}={\\pmb{v}}-{\\pmb{w}}$. Solving the dual form in Eq. (23) therefore readily gives a solution of Eq. (22). Problem Eq. (23) is the same as Euclidean projection onto the $\\ell_{1}$ ball, which many authors have studied. The solution summarized here follows recent work of Duchi et al. (2008). The maximizer of Eq. (23), written $\\alpha^{\\star}$, has the form\n\n$$\n\\alpha_{j}^{\\star}=\\mathrm{sign}(v_{j})\\left[|v_{j}| -\\theta\\right]_{+}\\ ,\\tag{24}\n$$\n\nwith θ a non-negative scalar. A linear-time algorithm for computing θ is given by Duchi et al. (2008). We therefore do not analyze that algorithm and instead concentrate on the core properties that affect the solution of the problem in Eq. (22). Computing θ requires finding a pivot element of v, namely the $\\rho^{t h}$ order statistic $v_{(\\rho)}$ (with $v_{(1)}$ the entry of v of largest magnitude), characterized as follows: $v_{(\\rho)}$ is the element of v of smallest magnitude for which\n\n$$\n\\sum_{j: |v_{j}| > |v_{(\\rho)}|}\\bigl(|v_{j}| - |v_{(\\rho)}|\\bigr)<\\tilde{\\lambda}\\ .\n$$\n\nWhen every element of v (with an extra element equal to 0 appended to take care of the smallest entry) satisfies this requirement, θ = 0 is the optimal choice. In the remaining case,\n\n$$\n\\theta=\\frac{1}{\\rho}\\left(\\sum_{j: |v_{j}| > |v_{(\\rho)}|}|v_{j}| -\\tilde{\\lambda}\\right).\n$$\n\nSo θ = 0 is optimal exactly when $\\sum_{j = 1}^{n}| v_{j}| \\leq \\tilde{\\lambda}$ (not by coincidence, this is just the subgradient optimality condition for the zero vector in Eq. (22)).\n\nCombining the linear relation $\\pmb{\\alpha}= \\pmb{v}- \\pmb{w}\\Rightarrow \\pmb{w}= \\pmb{v}- \\pmb{\\alpha}$ with the dual solution provided by Eq. (24), the solution of Eq. (22) is\n\n$$\nw_{t+1,j}=\\mathrm{sign}\\Big(w_{t+\\frac{1}{2},j}\\Big)\\operatorname*{min}\\left\\{\\Big|w_{t+\\frac{1}{2},j}\\Big|\\ ,\\ \\theta\\right\\}\\ .\\tag{25}\n$$\n\nAs noted above, $\\theta = 0$ if and only if $\\| \\pmb{w}_{t + \\frac{1}{2}}\\|_{1}\\leq \\tilde{\\lambda}$, and otherwise $\\theta > 0$ and θ can be computed in $O(n)$ steps. Put in words, the components of ${\\pmb w}_{t + 1}$ are obtained by capping every component of ${\\pmb w}_{t}$ at θ, with θ equal to zero when the 1-norm of $w_{t + \\frac{1}{2}}$ is below $\\tilde{\\lambda}$. This property mirrors, through duality, the $\\ell_{2}$-regularized update, which gives a zero weight vector when the 2-norm (which is self-dual) of v is below $\\tilde{\\lambda}$. These properties can be used with mixed-norm regularization to obtain sparse solutions for complex prediction problems; this is described later, and preliminary results are given in Sec. 7. Before turning to it, we present one further norm-related regularizer."}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\ndef HasSubgradientAt (f : E → ℝ) (g x : E) : Prop :=\n  ∀ y, f y ≥ f x + @inner ℝ _ _ g (y - x)\n\ndef IsFoBoSRun (f r : E → ℝ) (w gf gr : ℕ → E) (η ηhalf : ℕ → ℝ) : Prop :=\n  (∀ t, HasSubgradientAt f (gf t) (w t)) ∧\n  (∀ t, HasSubgradientAt r (gr t) (w t)) ∧\n  (∀ t, w (t + 1) = w t - η t • gf t - ηhalf t • gr (t + 1))\n\ndef IsFoBoSOnlineRun (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) :\n    Prop :=\n  (∀ t, HasSubgradientAt (f t) (gf t) (w t)) ∧\n  (∀ t, HasSubgradientAt r (gr t) (w t)) ∧\n  (∀ t, w (t + 1) = w t - η t • gf t - η t • gr (t + 1))\n\ndef Regret (f : ℕ → E → ℝ) (r : E → ℝ) (w : ℕ → E) (wstar : E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 1 T, (f t (w t) + r (w t) - (f t wstar + r wstar))\n\ndef sqL2 {n : ℕ} (x : Fin n → ℝ) : ℝ := ∑ i, (x i) ^ 2\n\nnoncomputable def linftyNorm {n : ℕ} (x : Fin n → ℝ) : ℝ := ⨆ i, |x i|\n\ndef l1Norm {n : ℕ} (x : Fin n → ℝ) : ℝ := ∑ i, |x i|\n\nnoncomputable def l2Norm {n : ℕ} (x : Fin n → ℝ) : ℝ := Real.sqrt (sqL2 x)\n\ndef IsProxNorm {n : ℕ} (N : (Fin n → ℝ) → ℝ) (lam : ℝ) (x xm : Fin n → ℝ) : Prop :=\n  IsMinOn (fun u => sqL2 (u - x) / 2 + lam * N u) Set.univ xm\n\ntheorem theorem6\n    (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G c : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hc : 0 < c)\n    (hη : ∀ t, 1 ≤ t → η t = c / Real.sqrt t)\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T ≤ 2 * G * D + (D ^ 2 / (2 * c) + 7 * G ^ 2 * c) * Real.sqrt T := by\n  sorry\n\ntheorem theorem8\n    (f : ℕ → E → ℝ) (r : E → ℝ) (w gf gr : ℕ → E) (η : ℕ → ℝ) (D G H : ℝ)\n    (wstar : E)\n    (hconvf : ∀ t, ConvexOn ℝ Set.univ (f t)) (hconvr : ConvexOn ℝ Set.univ r)\n    (hfnn : ∀ t x, 0 ≤ f t x) (hrnn : ∀ x, 0 ≤ r x)\n    (hrun : IsFoBoSOnlineRun f r w gf gr η)\n    (hD : ∀ t, 1 ≤ t → ‖w t - wstar‖ ≤ D) (hDnn : 0 ≤ D)\n    (hGf : ∀ t x g, HasSubgradientAt (f t) g x → ‖g‖ ≤ G)\n    (hGr : ∀ x g, HasSubgradientAt r g x → ‖g‖ ≤ G) (hGnn : 0 ≤ G)\n    (hH : 0 < H)\n    (hsc : ∀ t, StrongConvexOn (Metric.closedBall wstar D) H (fun x => f t x + r x))\n    (hη : ∀ t, 1 ≤ t → η t = 1 / (H * t))\n    (T : ℕ) (hT : 0 < T) :\n    Regret f r w wstar T\n      ≤ 2 * G * D + H * D ^ 2 + (7 * G ^ 2 / (2 * H)) * (1 + Real.log T) := by\n  sorry\n\ntheorem proposition11\n    {n : ℕ} (T : ℕ) (hT : 0 < T) (w0 wstar : Fin n → ℝ) (w : ℕ → Fin n → ℝ)\n    (lam : ℕ → ℝ) (hlam : ∀ t ∈ Finset.Icc 1 T, 0 ≤ lam t)\n    (hw0 : w 0 = w0)\n    (N : (Fin n → ℝ) → ℝ)\n    (hN : N = l1Norm ∨ N = l2Norm ∨ N = linftyNorm)\n    (hseq : ∀ t ∈ Finset.Icc 1 T, IsProxNorm N (lam t) (w (t - 1)) (w t))\n    (hbatch : IsProxNorm N (∑ t ∈ Finset.Icc 1 T, lam t) w0 wstar) :\n    w T = wstar := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem6", "theorem8", "proposition11"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 2678, "end": 3066, "gold": "{n : ℕ} (T : ℕ) (hT : 0 < T) (w0 wstar : Fin n → ℝ) (w : ℕ → Fin n → ℝ)\n    (lam : ℕ → ℝ) (hlam : ∀ t ∈ Finset.Icc 1 T, 0 ≤ lam t)\n    (hw0 : w 0 = w0)\n    (N : (Fin n → ℝ) → ℝ)\n    (hN : N = l1Norm ∨ N = l2Norm ∨ N = linftyNorm)\n    (hseq : ∀ t ∈ Finset.Icc 1 T, IsProxNorm N (lam t) (w (t - 1)) (w t))\n    (hbatch : IsProxNorm N (∑ t ∈ Finset.Icc 1 T, lam t) w0 wstar) :\n    w T = wstar"}]}
{"item_id": "v23-20-411::lemma5_oblivious::A-stmt", "paper_id": "v23-20-411", "target_name": "lemma5_oblivious", "source": "Research-level", "nl_statement": "Lemma 5 (Weighted-Regret Bound for EXP3). Let $\\{\\eta_{t}\\}$ be a non-increasing step-size sequence such that $\\eta_{t}\\leq\\frac 12e^{-2}$ for all t. Let $\\left\\{l_{t}^{\\left(i\\right)}\\right\\}$ be a cost sequence such that $l_{t}^{(i)}\\in [0,1]$ for every $t,i.$ Let $\\{d_{t}\\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . Define the set $\\mathcal{M}^{\\ast}$ of all samples that are not received before round T or that were delayed by $\\begin{array}{r}{d_{t}\\geq \\frac{1}{e^{2}\\eta_{t}}- 1}\\end{array}$ . Then using EXP3 (Algorithm 3) guarantees:\n\n1. With an oblivious adversary and $\\gamma_{t}= 0$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq\\log K+\\frac{e^{2}}{2}K\\sum_{t=1}^{T}\\eta_{t}^{2}+4\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{42}\n$$\n\n2. With an adaptive adversary and $\\gamma_{t}= \\eta_{t}$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq 2+2\\log K+(1+\\frac{e^{2}}{2})K\\sum_{t=1}^{T}\\eta_{t}^{2}+4 e^{2}K\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{43}\n$$", "context": [{"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)", "statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbol{a}_{t}\\right)\\in\\left[0,1\\right]$ . We consider two types of adversaries:\n\n1. Oblivious Adversary: chooses the cost functions $l_{1},...,l_{T}$ before the game starts.\n\n2. Adaptive Adversary: chooses the cost function $l_{t}$ after observing $\\{a_{1}, \\ldots a_{t - 1}\\}$ , for each t.\n\nWith full information and no delays, the player gets to know the function $l_{t}$ immediately after playing $\\pmb{a}_{t}$ . In the bandit delayed feedback scenario, the player only gets to know the value of ${l}_{t}\\left(\\pmb{a}_{t}\\right)$ at the beginning of round $t + d_{t}$ . (i.e., after a delay of $d_{t}\\geq 1$ rounds). The adversary (oblivious or adaptive) chooses the delay sequence $\\{d_{t}\\}$ before the game starts.\n\nWe assume that the cost feedback includes the timestamp of the action that incurred this cost. This is indeed the case in many applications, such as when robots take physical actions, a recommendation is made to a customer or a treatment is given to a patient. If the delays are bounded by d, Cesa-Bianchi et al. (2018) have shown that EXP3 can still be implemented even with no timestamps, with expected regret $O(\\sqrt{dTK\\log K})$ instead of $O\\left({\\sqrt{\\left(d+K\\right)T\\log K}}\\right)$ . For our unbounded delays case, it is not clear if FKM or EXP3 can be implemented without timestamps.\n\nThe set of costs (feedback samples) received and used at round t is denoted $\\mathcal{S}_{t}$ , so $s \\in \\mathcal{S}_{t}$ means that the cost of $\\pmb{a}_{s}$ from round s is received and used at round t. Since the game lasts for $T$ rounds, all costs for which $t + d_{t}> T$ are never received. Of course, the value of $d_{t}$ does not matter as long as $t + d_{t}> T$ , and these are just samples that the adversary chose to prevent the player from receiving. We name these costs the missing samples and denote their set by $\\mathcal{M}$ .\n\nWhile the rounds of the game are indexed by t, it will be useful to our analysis to index a finer time scale that counts the steps of the algorithm for every such t. We define $s_{-}, s_{+}$ as the steps a moment before and after the algorithm uses the feedback from round s, respectively. These steps are taking place in round t if $s \\in S_{t}$"}, {"name": "Setup: multi-armed bandit special case (Section 2)", "statement": "We analyze two widely applied algorithms for the two central special cases of the scenario above:\n\n2. Multi-Armed Bandit - $\\mathcal{K}=\\{1,...,K\\},\\ l_{t}:\\{1,...,K\\}\\to [0,1].$ With no delays, the EXP3 algorithm (Auer et al., 2002) achieves an expected regret of $O \\left({\\sqrt{T K \\log K}}\\right)$ for this problem."}, {"name": "Setup: EXP3 with delays, mixed actions and the filtration F_t (eq. 41, Section 6)", "statement": "Consider a player that at each round t picks one out of K arms. Let $a_{t}$ be the arm the player chooses at round t. The cost at round t of playing arm i is $l_{t}^{(i)}\\in [0,1]$ , and let $l_{t}=\\left(l_{t}^{(1)},...,l_{t}^{(K)}\\right)$ be the cost vector. At round t, the EXP3 algorithm, detailed in Algorithm 3, chooses an arm at random using a distribution that depends on the history of the game. The variant when $\\gamma_{t}\\neq 0$ , as we use against an adaptive adversary, is known as EXP3-IX (see Neu (2015)). We denote the vector of probabilities of the player for choosing arms at round t by $\\pmb{p}_{t}\\in \\Delta^{K}$ , where $\\Delta^{K}$ denotes the K-simplex. This is also known as the mixed action of the player. We also define the following filtration\n\n$$\n\\mathcal{F}_{t}=\\sigma\\left(\\left\\{a_{s}\\vert s+d_{s}\\le t\\right\\}\\cup\\left\\{l_{s}\\vert s\\le t\\right\\}\\right)\\tag{41}\n$$\n\nwhich is generated from all the actions for which the feedback was received up to round t and all cost functions up to round t. Note that the mixed action $\\pmb{p}_{t}$ is a $\\mathcal{F}_{t}$ -measurable random variable since $\\mathcal{F}_{t}$ includes everything that could affect the algorithm up to round t. With a slight abuse of notation, we use $\\mathcal{F}_{s_{-}}$ to denote the filtration induced from all actions for which the feedback has been received and used up to step $s_{-}$ and all the cost functions up to round t."}, {"name": "Algorithm 3 (EXP3 with delays, eqs. 44-45)", "statement": "Algorithm 3 EXP3 with delays   \nInitialization: Let $\\{\\eta_{t}\\}$ and $\\{\\gamma_{t}\\}$ be non-negative non-increasing sequences such that $\\eta_{1}\\leq \\frac{e^{- 2}}{2}$ and   \nset $\\tilde{L}_{1}^{(i)}=0$ and $\\begin{array}{r}{p_{1}^{(i)}=\\frac{1}{K}}\\end{array}$ for $i=1,...,K.$   \nFor $t=1,...,T$ do   \n1. Choose an arm $a_{t}$ at random according to the distribution $\\pmb{p}_{t}$ .   \n2. Collect in $S_{t}$ all the rounds s for which $l_{s}^{(a_{s})}$ arrived at round t after a delay of $\\begin{array}{r}{d_{s}\\le \\frac{1}{e^{2}\\eta_{s}}- 1}\\end{array}$   \n3. Set $\\tilde{L}_{t}^{(i)}=\\tilde{L}_{t-1}^{(i)}$ for all i. Update the weights of arm $a_{s}$ for all $s \\in S_{t}$ using   \n$\\tilde{L}_{t}^{(a_{s})}=\\tilde{L}_{t}^{(a_{s})}+\\eta_{s}\\frac{l_{s}^{(a_{s})}}{p_{s}^{(a_{s})}+\\gamma_{s}}.$ (44)   \n4. Update the mixed action for $i=1,\\ldots,K$ using   \n$p_{t+1}^{(i)}=\\frac{e^{-\\tilde{L}_{t}^{(i)}}}{\\sum_{j=1}^{K}e^{-\\tilde{L}_{t}^{(j)}}}.$ (45)   \nEnd"}, {"name": "Lemma 8 (effect of delays, eq. 66)", "statement": "Lemma 8. Let $\\{\\eta_{t}\\}$ be a non-increasing step-size sequence. Let $\\{d_{t}\\}$ be a delay such that the cost from round t is received at round $t + d_{t}$ . Let $S_{t}$ be the set of feedback samples received and used at round t. Define the set $\\mathcal{M}^{\\ast}$ of all samples that are not received and used before round T. Then\n\n$$\n\\sum_{t=1}^{T}\\sum_{s\\in\\mathcal{S}_{t}}\\eta_{s}\\left(\\sum_{r=s}^{t-1}\\sum_{q\\in\\mathcal{S}_{r}}\\eta_{q}+\\sum_{q\\in\\mathcal{S}_{t},q<s}\\eta_{q}\\right)\\le 2\\sum_{t\\not\\in\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}.\\tag{66}\n$$"}, {"name": "Lemma 9 (ratio bounds on the mixed action)", "statement": "The following lemma generalizes Lemma 2 from Cesa-Bianchi et al. (2019) to a sequence of delays $\\{d_{t}\\}$ and a sequence of step-sizes $\\{\\eta_{t}\\}$\n\nLemma 9. Let $\\{\\eta_{t}\\}$ be a positive non-increasing step-size sequence such that $\\eta_{t}\\leq \\frac{1}{2}e^{- 2}$ for all t. Let $\\begin{array}{r}{\\mathcal{D}= \\Big \\{t | d_{t}\\geq \\frac{1}{e^{2}\\eta_{t}}- 1 \\Big \\}}\\end{array}$ . Then for every s,t such that $s \\in \\mathcal{S}_{t}$ (so $s \\notin \\mathcal{D}$ ) Algorithm 3 maintains for all $i=1,...,K$ both $\\begin{array}{r}{\\frac{p_{s_{+}}^{(i)}}{p_{s_{-}}^{(i)}}\\leq\\frac{1}{1-e^{2}\\eta_{s}}}\\end{array}$ and $\\frac{p_{s_{-}}^{(i)}}{p_{s}^{(i)}}\\leq e^{2}$"}, {"name": "Lemma 10 (softmax smoothness, eq. 107)", "statement": "The next lemma shows standard smoothness properties of the softmax function, and we provide it here for completeness.\n\nLemma 10. Let $\\begin{array}{r}{h_{i}\\left(\\pmb{x}\\right)=\\frac{e^{-x_{i}}}{\\sum_{j=1}^{K}e^{-x_{j}}}}\\end{array}$ and $h\\left(\\pmb{x}\\right)=(h_{1}\\left(\\pmb{x}\\right),...,h_{K}\\left(\\pmb{x}\\right))$ . Then $\\forall \\mathbf{x}\\in \\mathbb{R}^{K}$ and $\\forall \\Delta \\in \\mathbb{R}_{+}^{K}$\n\n$$\n\\begin{array}{r}{\\|h\\left(\\pmb{x}\\right)-h\\left(\\pmb{x}+\\Delta\\right)\\|_{1}\\leq 2\\left\\langle h\\left(\\pmb{x}\\right),\\Delta\\right\\rangle .}\\end{array}\\tag{107}\n$$"}, {"name": "Lemma 11 (Neu 2015 concentration, eq. 111)", "statement": "The next Lemma is taken from Neu (2015), and we provide a (very) slightly modified proof to verify that the same result holds even when the order of arrivals changes as a result of the delayed feedback.\n\nLemma 11. Let $\\tilde{l}_{t}^{(i)}=\\frac{l_{t}^{(i)}1_{\\{a_{t}=i\\}}}{p_{t}^{(i)}+\\gamma_{t}}$ . If $\\left\\{\\alpha_{t}^{(i)}\\right\\}$ is a non-negative sequence such that $\\alpha_{t}^{(i)}\\leq 2\\gamma_{t}$ for all t and all i, then\n\n$$\n\\mathbb{P}\\left(\\sum_{t\\notin{\\mathcal{M}}^{*}}\\sum_{i=1}^{K}\\alpha_{t}^{(i)}\\left(\\tilde{l}_{t}^{(i)}-l_{t}^{(i)}\\right)>\\log\\frac{1}{\\delta}\\right)\\leq\\delta .\\tag{111}\n$$"}], "nl_notes": "### Lemma 5\nNOTE (Lemma 5, oblivious case) — The Lean adds a hypothesis (`a_fresh`) that the paper never states as an assumption of Lemma 5 or Algorithm 3, but uses inside the proof of the oblivious case of Lemma 5 (appendix Section 13.2, headed \"Proof of Lemma 6\" in the paper): each arm a_t is drawn with fresh randomness. Precisely, for t ≤ t' < t + d_t, the conditional law of a_t given 𝓕_{t'}, together with the arms of the rounds whose feedback arrives before that of round t, is still p_t.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped Classical\nopen MeasureTheory Filter Topology\n\ndef missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t)\n\nnoncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t ∨ (d t : ℝ) ≥ 1 / (Real.exp 2 * η t) - 1)\n\nnoncomputable def weightedExpReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  haveI : Nonempty (Fin K) := ⟨⟨0, hK⟩⟩\n  ∫ ω, (∑ t ∈ Finset.Icc 1 T, η t * l t ω (a t ω) -\n    (Finset.univ : Finset (Fin K)).inf' Finset.univ_nonempty\n      (fun i => ∑ t ∈ Finset.Icc 1 T, η t * l t ω i)) ∂μ\n\nnoncomputable def expReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (l0 : ℕ → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  weightedExpReg μ T K hK (fun _ => 1) (fun t _ i => l0 t i) a\n\nnoncomputable def NoWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : Prop :=\n  limsup (fun T : ℕ => weightedExpReg μ T K hK η l a / ∑ t ∈ Finset.Icc 1 T, η t)\n    atTop ≤ 0\n\nstructure Exp3Delayed {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (K : ℕ) (hK : 0 < K)\n    (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (η γ : ℕ → ℝ) where\n  𝓕 : ℕ → MeasurableSpace Ω\n  𝓕_le : ∀ t, 𝓕 t ≤ ‹MeasurableSpace Ω›\n  𝓕_mono : Monotone 𝓕\n  a : ℕ → Ω → Fin K\n  p : ℕ → Ω → Fin K → ℝ\n  Lhat : ℕ → Ω → Fin K → ℝ\n  measurable_a : ∀ t, Measurable (a t)\n  𝓕_eq : ∀ t, 𝓕 t =\n    (⨆ s, ⨆ (_ : 1 ≤ s ∧ s + d s ≤ t), MeasurableSpace.comap (a s) inferInstance)\n      ⊔ (⨆ s, ⨆ (_ : 1 ≤ s ∧ s ≤ t), MeasurableSpace.comap (l s) inferInstance)\n  measurable_p : ∀ t i, @Measurable Ω ℝ (𝓕 t) _ (fun ω => p t ω i)\n  p_isProb : ∀ t ω, (∀ i, 0 ≤ p t ω i) ∧ ∑ i, p t ω i = 1\n  p_init : ∀ ω i, p 1 ω i = 1 / (K : ℝ)\n  a_law : ∀ t i,\n      (μ[(fun ω => if a t ω = i then (1 : ℝ) else 0) | 𝓕 t]) =ᵐ[μ] (fun ω => p t ω i)\n  Lhat_init : ∀ ω i, Lhat 0 ω i = 0\n  Lhat_step : ∀ t ω i, 1 ≤ t →\n      Lhat t ω i = Lhat (t - 1) ω i +\n        ∑ s ∈ (Finset.Icc 1 t).filter\n          (fun s => s + d s = t ∧ (d s : ℝ) ≤ 1 / (Real.exp 2 * η s) - 1 ∧ a s ω = i),\n          η s * l s ω i / (p s ω i + γ s)\n  p_update : ∀ t ω i,\n      p (t + 1) ω i = Real.exp (- Lhat t ω i) / ∑ j, Real.exp (- Lhat t ω j)\n\ntheorem lemma5_oblivious {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n\n    \n    \n    \n    \n    \n    \n    (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n            MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η (fun t _ i => l0 t i) E.a ≤\n      Real.log K + (Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem lemma5_adaptive {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η l E.a ≤\n      2 + 2 * Real.log K\n        + (1 + Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * Real.exp 2 * K * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem theorem6_regretBound :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        (K T : ℕ) (hK : 0 < K) (hT : 0 < T) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ)\n        (hd : ∀ t ≥ 1, 1 ≤ d t)\n        (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1) (η : ℕ → ℝ)\n        (hη_pos : ∀ t ≥ 1, 0 < η t)\n        (hη : η = fun _ =>\n          (Real.exp (-2) / 2) *\n            Real.sqrt (Real.log K /\n              ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ))))\n        (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n        (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n          (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n              E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n                MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)),\n        expReg μ T K hK l0 E.a ≤\n          C * (Real.sqrt (Real.log K *\n                ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ)))\n              + (missedRounds T d).card) := by\n  sorry\n\ntheorem lemma6_noWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K : ℕ) (hK : 0 < K)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t ≥ 1, 0 < η t) (hη_dec : ∀ t ≥ 1, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t ≥ 1, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ≥ 1, ∀ ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i))\n    (h1 : Tendsto (fun T : ℕ => ∑ t ∈ Finset.Icc 1 T, η t) atTop atTop)\n    (h2 : ∃ L : ℝ, Tendsto (fun t : ℕ => η t * (d t : ℝ)) atTop (nhds L))\n    (h3 : Summable (fun t : ℕ => (η t) ^ 2 * (d t : ℝ))) :\n    NoWeightedRegret μ K hK η l E.a := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma5_oblivious", "lemma5_adaptive", "theorem6_regretBound", "lemma6_noWeightedRegret"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 2424, "end": 3388, "gold": "{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n\n    \n    \n    \n    \n    \n    \n    (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n            MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η (fun t _ i => l0 t i) E.a ≤\n      Real.log K + (Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t)"}]}
{"item_id": "v23-20-411::lemma5_adaptive::A-stmt", "paper_id": "v23-20-411", "target_name": "lemma5_adaptive", "source": "Research-level", "nl_statement": "Lemma 5 (Weighted-Regret Bound for EXP3). Let $\\{\\eta_{t}\\}$ be a non-increasing step-size sequence such that $\\eta_{t}\\leq\\frac 12e^{-2}$ for all t. Let $\\left\\{l_{t}^{\\left(i\\right)}\\right\\}$ be a cost sequence such that $l_{t}^{(i)}\\in [0,1]$ for every $t,i.$ Let $\\{d_{t}\\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . Define the set $\\mathcal{M}^{\\ast}$ of all samples that are not received before round T or that were delayed by $\\begin{array}{r}{d_{t}\\geq \\frac{1}{e^{2}\\eta_{t}}- 1}\\end{array}$ . Then using EXP3 (Algorithm 3) guarantees:\n\n1. With an oblivious adversary and $\\gamma_{t}= 0$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq\\log K+\\frac{e^{2}}{2}K\\sum_{t=1}^{T}\\eta_{t}^{2}+4\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{42}\n$$\n\n2. With an adaptive adversary and $\\gamma_{t}= \\eta_{t}$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq 2+2\\log K+(1+\\frac{e^{2}}{2})K\\sum_{t=1}^{T}\\eta_{t}^{2}+4 e^{2}K\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{43}\n$$", "context": [{"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)", "statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbol{a}_{t}\\right)\\in\\left[0,1\\right]$ . We consider two types of adversaries:\n\n1. Oblivious Adversary: chooses the cost functions $l_{1},...,l_{T}$ before the game starts.\n\n2. Adaptive Adversary: chooses the cost function $l_{t}$ after observing $\\{a_{1}, \\ldots a_{t - 1}\\}$ , for each t.\n\nWith full information and no delays, the player gets to know the function $l_{t}$ immediately after playing $\\pmb{a}_{t}$ . In the bandit delayed feedback scenario, the player only gets to know the value of ${l}_{t}\\left(\\pmb{a}_{t}\\right)$ at the beginning of round $t + d_{t}$ . (i.e., after a delay of $d_{t}\\geq 1$ rounds). The adversary (oblivious or adaptive) chooses the delay sequence $\\{d_{t}\\}$ before the game starts.\n\nWe assume that the cost feedback includes the timestamp of the action that incurred this cost. This is indeed the case in many applications, such as when robots take physical actions, a recommendation is made to a customer or a treatment is given to a patient. If the delays are bounded by d, Cesa-Bianchi et al. (2018) have shown that EXP3 can still be implemented even with no timestamps, with expected regret $O(\\sqrt{dTK\\log K})$ instead of $O\\left({\\sqrt{\\left(d+K\\right)T\\log K}}\\right)$ . For our unbounded delays case, it is not clear if FKM or EXP3 can be implemented without timestamps.\n\nThe set of costs (feedback samples) received and used at round t is denoted $\\mathcal{S}_{t}$ , so $s \\in \\mathcal{S}_{t}$ means that the cost of $\\pmb{a}_{s}$ from round s is received and used at round t. Since the game lasts for $T$ rounds, all costs for which $t + d_{t}> T$ are never received. Of course, the value of $d_{t}$ does not matter as long as $t + d_{t}> T$ , and these are just samples that the adversary chose to prevent the player from receiving. We name these costs the missing samples and denote their set by $\\mathcal{M}$ .\n\nWhile the rounds of the game are indexed by t, it will be useful to our analysis to index a finer time scale that counts the steps of the algorithm for every such t. We define $s_{-}, s_{+}$ as the steps a moment before and after the algorithm uses the feedback from round s, respectively. These steps are taking place in round t if $s \\in S_{t}$"}, {"name": "Setup: multi-armed bandit special case (Section 2)", "statement": "We analyze two widely applied algorithms for the two central special cases of the scenario above:\n\n2. Multi-Armed Bandit - $\\mathcal{K}=\\{1,...,K\\},\\ l_{t}:\\{1,...,K\\}\\to [0,1].$ With no delays, the EXP3 algorithm (Auer et al., 2002) achieves an expected regret of $O \\left({\\sqrt{T K \\log K}}\\right)$ for this problem."}, {"name": "Setup: EXP3 with delays, mixed actions and the filtration F_t (eq. 41, Section 6)", "statement": "Consider a player that at each round t picks one out of K arms. Let $a_{t}$ be the arm the player chooses at round t. The cost at round t of playing arm i is $l_{t}^{(i)}\\in [0,1]$ , and let $l_{t}=\\left(l_{t}^{(1)},...,l_{t}^{(K)}\\right)$ be the cost vector. At round t, the EXP3 algorithm, detailed in Algorithm 3, chooses an arm at random using a distribution that depends on the history of the game. The variant when $\\gamma_{t}\\neq 0$ , as we use against an adaptive adversary, is known as EXP3-IX (see Neu (2015)). We denote the vector of probabilities of the player for choosing arms at round t by $\\pmb{p}_{t}\\in \\Delta^{K}$ , where $\\Delta^{K}$ denotes the K-simplex. This is also known as the mixed action of the player. We also define the following filtration\n\n$$\n\\mathcal{F}_{t}=\\sigma\\left(\\left\\{a_{s}\\vert s+d_{s}\\le t\\right\\}\\cup\\left\\{l_{s}\\vert s\\le t\\right\\}\\right)\\tag{41}\n$$\n\nwhich is generated from all the actions for which the feedback was received up to round t and all cost functions up to round t. Note that the mixed action $\\pmb{p}_{t}$ is a $\\mathcal{F}_{t}$ -measurable random variable since $\\mathcal{F}_{t}$ includes everything that could affect the algorithm up to round t. With a slight abuse of notation, we use $\\mathcal{F}_{s_{-}}$ to denote the filtration induced from all actions for which the feedback has been received and used up to step $s_{-}$ and all the cost functions up to round t."}, {"name": "Algorithm 3 (EXP3 with delays, eqs. 44-45)", "statement": "Algorithm 3 EXP3 with delays   \nInitialization: Let $\\{\\eta_{t}\\}$ and $\\{\\gamma_{t}\\}$ be non-negative non-increasing sequences such that $\\eta_{1}\\leq \\frac{e^{- 2}}{2}$ and   \nset $\\tilde{L}_{1}^{(i)}=0$ and $\\begin{array}{r}{p_{1}^{(i)}=\\frac{1}{K}}\\end{array}$ for $i=1,...,K.$   \nFor $t=1,...,T$ do   \n1. Choose an arm $a_{t}$ at random according to the distribution $\\pmb{p}_{t}$ .   \n2. Collect in $S_{t}$ all the rounds s for which $l_{s}^{(a_{s})}$ arrived at round t after a delay of $\\begin{array}{r}{d_{s}\\le \\frac{1}{e^{2}\\eta_{s}}- 1}\\end{array}$   \n3. Set $\\tilde{L}_{t}^{(i)}=\\tilde{L}_{t-1}^{(i)}$ for all i. Update the weights of arm $a_{s}$ for all $s \\in S_{t}$ using   \n$\\tilde{L}_{t}^{(a_{s})}=\\tilde{L}_{t}^{(a_{s})}+\\eta_{s}\\frac{l_{s}^{(a_{s})}}{p_{s}^{(a_{s})}+\\gamma_{s}}.$ (44)   \n4. Update the mixed action for $i=1,\\ldots,K$ using   \n$p_{t+1}^{(i)}=\\frac{e^{-\\tilde{L}_{t}^{(i)}}}{\\sum_{j=1}^{K}e^{-\\tilde{L}_{t}^{(j)}}}.$ (45)   \nEnd"}, {"name": "Lemma 8 (effect of delays, eq. 66)", "statement": "Lemma 8. Let $\\{\\eta_{t}\\}$ be a non-increasing step-size sequence. Let $\\{d_{t}\\}$ be a delay such that the cost from round t is received at round $t + d_{t}$ . Let $S_{t}$ be the set of feedback samples received and used at round t. Define the set $\\mathcal{M}^{\\ast}$ of all samples that are not received and used before round T. Then\n\n$$\n\\sum_{t=1}^{T}\\sum_{s\\in\\mathcal{S}_{t}}\\eta_{s}\\left(\\sum_{r=s}^{t-1}\\sum_{q\\in\\mathcal{S}_{r}}\\eta_{q}+\\sum_{q\\in\\mathcal{S}_{t},q<s}\\eta_{q}\\right)\\le 2\\sum_{t\\not\\in\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}.\\tag{66}\n$$"}, {"name": "Lemma 9 (ratio bounds on the mixed action)", "statement": "The following lemma generalizes Lemma 2 from Cesa-Bianchi et al. (2019) to a sequence of delays $\\{d_{t}\\}$ and a sequence of step-sizes $\\{\\eta_{t}\\}$\n\nLemma 9. Let $\\{\\eta_{t}\\}$ be a positive non-increasing step-size sequence such that $\\eta_{t}\\leq \\frac{1}{2}e^{- 2}$ for all t. Let $\\begin{array}{r}{\\mathcal{D}= \\Big \\{t | d_{t}\\geq \\frac{1}{e^{2}\\eta_{t}}- 1 \\Big \\}}\\end{array}$ . Then for every s,t such that $s \\in \\mathcal{S}_{t}$ (so $s \\notin \\mathcal{D}$ ) Algorithm 3 maintains for all $i=1,...,K$ both $\\begin{array}{r}{\\frac{p_{s_{+}}^{(i)}}{p_{s_{-}}^{(i)}}\\leq\\frac{1}{1-e^{2}\\eta_{s}}}\\end{array}$ and $\\frac{p_{s_{-}}^{(i)}}{p_{s}^{(i)}}\\leq e^{2}$"}, {"name": "Lemma 10 (softmax smoothness, eq. 107)", "statement": "The next lemma shows standard smoothness properties of the softmax function, and we provide it here for completeness.\n\nLemma 10. Let $\\begin{array}{r}{h_{i}\\left(\\pmb{x}\\right)=\\frac{e^{-x_{i}}}{\\sum_{j=1}^{K}e^{-x_{j}}}}\\end{array}$ and $h\\left(\\pmb{x}\\right)=(h_{1}\\left(\\pmb{x}\\right),...,h_{K}\\left(\\pmb{x}\\right))$ . Then $\\forall \\mathbf{x}\\in \\mathbb{R}^{K}$ and $\\forall \\Delta \\in \\mathbb{R}_{+}^{K}$\n\n$$\n\\begin{array}{r}{\\|h\\left(\\pmb{x}\\right)-h\\left(\\pmb{x}+\\Delta\\right)\\|_{1}\\leq 2\\left\\langle h\\left(\\pmb{x}\\right),\\Delta\\right\\rangle .}\\end{array}\\tag{107}\n$$"}, {"name": "Lemma 11 (Neu 2015 concentration, eq. 111)", "statement": "The next Lemma is taken from Neu (2015), and we provide a (very) slightly modified proof to verify that the same result holds even when the order of arrivals changes as a result of the delayed feedback.\n\nLemma 11. Let $\\tilde{l}_{t}^{(i)}=\\frac{l_{t}^{(i)}1_{\\{a_{t}=i\\}}}{p_{t}^{(i)}+\\gamma_{t}}$ . If $\\left\\{\\alpha_{t}^{(i)}\\right\\}$ is a non-negative sequence such that $\\alpha_{t}^{(i)}\\leq 2\\gamma_{t}$ for all t and all i, then\n\n$$\n\\mathbb{P}\\left(\\sum_{t\\notin{\\mathcal{M}}^{*}}\\sum_{i=1}^{K}\\alpha_{t}^{(i)}\\left(\\tilde{l}_{t}^{(i)}-l_{t}^{(i)}\\right)>\\log\\frac{1}{\\delta}\\right)\\leq\\delta .\\tag{111}\n$$"}], "nl_notes": "### Lemma 5\nNOTE (Lemma 5, adaptive case) — The Lean adds a hypothesis (`a_law_adaptive`) that the paper never states as an assumption of Lemma 5 or Algorithm 3, but uses inside the proof of Lemma 11 (\"given 𝒢_t, a_t = i with probability p_t^(i)\"), which the proof of the adaptive case of Lemma 5 applies (appendix Section 13.1, headed \"Proof of Lemma 6\" in the paper). Precisely, the conditional law of a_t given the arms a_1, …, a_{t-1} of the earlier rounds (the σ-algebra 𝒢_t of that proof) is p_t.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped Classical\nopen MeasureTheory Filter Topology\n\ndef missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t)\n\nnoncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t ∨ (d t : ℝ) ≥ 1 / (Real.exp 2 * η t) - 1)\n\nnoncomputable def weightedExpReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  haveI : Nonempty (Fin K) := ⟨⟨0, hK⟩⟩\n  ∫ ω, (∑ t ∈ Finset.Icc 1 T, η t * l t ω (a t ω) -\n    (Finset.univ : Finset (Fin K)).inf' Finset.univ_nonempty\n      (fun i => ∑ t ∈ Finset.Icc 1 T, η t * l t ω i)) ∂μ\n\nnoncomputable def expReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (l0 : ℕ → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  weightedExpReg μ T K hK (fun _ => 1) (fun t _ i => l0 t i) a\n\nnoncomputable def NoWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : Prop :=\n  limsup (fun T : ℕ => weightedExpReg μ T K hK η l a / ∑ t ∈ Finset.Icc 1 T, η t)\n    atTop ≤ 0\n\nstructure Exp3Delayed {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (K : ℕ) (hK : 0 < K)\n    (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (η γ : ℕ → ℝ) where\n  𝓕 : ℕ → MeasurableSpace Ω\n  𝓕_le : ∀ t, 𝓕 t ≤ ‹MeasurableSpace Ω›\n  𝓕_mono : Monotone 𝓕\n  a : ℕ → Ω → Fin K\n  p : ℕ → Ω → Fin K → ℝ\n  Lhat : ℕ → Ω → Fin K → ℝ\n  measurable_a : ∀ t, Measurable (a t)\n  𝓕_eq : ∀ t, 𝓕 t =\n    (⨆ s, ⨆ (_ : 1 ≤ s ∧ s + d s ≤ t), MeasurableSpace.comap (a s) inferInstance)\n      ⊔ (⨆ s, ⨆ (_ : 1 ≤ s ∧ s ≤ t), MeasurableSpace.comap (l s) inferInstance)\n  measurable_p : ∀ t i, @Measurable Ω ℝ (𝓕 t) _ (fun ω => p t ω i)\n  p_isProb : ∀ t ω, (∀ i, 0 ≤ p t ω i) ∧ ∑ i, p t ω i = 1\n  p_init : ∀ ω i, p 1 ω i = 1 / (K : ℝ)\n  a_law : ∀ t i,\n      (μ[(fun ω => if a t ω = i then (1 : ℝ) else 0) | 𝓕 t]) =ᵐ[μ] (fun ω => p t ω i)\n  Lhat_init : ∀ ω i, Lhat 0 ω i = 0\n  Lhat_step : ∀ t ω i, 1 ≤ t →\n      Lhat t ω i = Lhat (t - 1) ω i +\n        ∑ s ∈ (Finset.Icc 1 t).filter\n          (fun s => s + d s = t ∧ (d s : ℝ) ≤ 1 / (Real.exp 2 * η s) - 1 ∧ a s ω = i),\n          η s * l s ω i / (p s ω i + γ s)\n  p_update : ∀ t ω i,\n      p (t + 1) ω i = Real.exp (- Lhat t ω i) / ∑ j, Real.exp (- Lhat t ω j)\n\ntheorem lemma5_oblivious {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n\n    \n    \n    \n    \n    \n    \n    (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n            MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η (fun t _ i => l0 t i) E.a ≤\n      Real.log K + (Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem lemma5_adaptive {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η l E.a ≤\n      2 + 2 * Real.log K\n        + (1 + Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * Real.exp 2 * K * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem theorem6_regretBound :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        (K T : ℕ) (hK : 0 < K) (hT : 0 < T) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ)\n        (hd : ∀ t ≥ 1, 1 ≤ d t)\n        (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1) (η : ℕ → ℝ)\n        (hη_pos : ∀ t ≥ 1, 0 < η t)\n        (hη : η = fun _ =>\n          (Real.exp (-2) / 2) *\n            Real.sqrt (Real.log K /\n              ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ))))\n        (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n        (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n          (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n              E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n                MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)),\n        expReg μ T K hK l0 E.a ≤\n          C * (Real.sqrt (Real.log K *\n                ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ)))\n              + (missedRounds T d).card) := by\n  sorry\n\ntheorem lemma6_noWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K : ℕ) (hK : 0 < K)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t ≥ 1, 0 < η t) (hη_dec : ∀ t ≥ 1, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t ≥ 1, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ≥ 1, ∀ ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i))\n    (h1 : Tendsto (fun T : ℕ => ∑ t ∈ Finset.Icc 1 T, η t) atTop atTop)\n    (h2 : ∃ L : ℝ, Tendsto (fun t : ℕ => η t * (d t : ℝ)) atTop (nhds L))\n    (h3 : Summable (fun t : ℕ => (η t) ^ 2 * (d t : ℝ))) :\n    NoWeightedRegret μ K hK η l E.a := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma5_oblivious", "lemma5_adaptive", "theorem6_regretBound", "lemma6_noWeightedRegret"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 3428, "end": 4421, "gold": "{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η l E.a ≤\n      2 + 2 * Real.log K\n        + (1 + Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * Real.exp 2 * K * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t)"}]}
{"item_id": "v23-20-411::theorem6_regretBound::A-stmt", "paper_id": "v23-20-411", "target_name": "theorem6_regretBound", "source": "Research-level", "nl_statement": "Theorem 6. Let $\\left\\{l_{t}^{\\left(i\\right)}\\right\\}$ be a cost sequence such that $l_{t}^{(i)}\\in [0,1]$ for every $t,i$ Let $\\{d_{t}\\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . Define the set $\\mathcal{M}=$ $\\{t|t+d_{t}>T,t\\in [1,T]\\}$ of all samples that are not received before round T. Let us choose the fixed step-size $\\begin{array}{r}{\\eta = \\frac{e^{- 2}}{2}\\sqrt{\\frac{\\log K}{K T + \\sum_{t \\notin \\mathcal{M}}d_{t}}}}\\end{array}$ . Then the expected regret of EXP3 (Algorithm 3 with $\\gamma_{t}= 0$ for all t) against an oblivious adversary satisfies\n\n$$\n\\mathbb{E}\\left\\{R\\left(T\\right)\\right\\}=\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\langle l_{t},p_{t}\\rangle-\\operatorname*{min}_{i}\\sum_{t=1}^{T}l_{t}^{\\left(i\\right)}\\right\\}=O\\left(\\sqrt{\\log K\\left(K T+\\sum_{t\\notin\\mathcal{M}}d_{t}\\right)}+\\left|\\mathcal{M}\\right|\\right).\\tag{46}\n$$", "context": [{"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)", "statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbol{a}_{t}\\right)\\in\\left[0,1\\right]$ . We consider two types of adversaries:\n\n1. Oblivious Adversary: chooses the cost functions $l_{1},...,l_{T}$ before the game starts.\n\n2. Adaptive Adversary: chooses the cost function $l_{t}$ after observing $\\{a_{1}, \\ldots a_{t - 1}\\}$ , for each t.\n\nWith full information and no delays, the player gets to know the function $l_{t}$ immediately after playing $\\pmb{a}_{t}$ . In the bandit delayed feedback scenario, the player only gets to know the value of ${l}_{t}\\left(\\pmb{a}_{t}\\right)$ at the beginning of round $t + d_{t}$ . (i.e., after a delay of $d_{t}\\geq 1$ rounds). The adversary (oblivious or adaptive) chooses the delay sequence $\\{d_{t}\\}$ before the game starts.\n\nWe assume that the cost feedback includes the timestamp of the action that incurred this cost. This is indeed the case in many applications, such as when robots take physical actions, a recommendation is made to a customer or a treatment is given to a patient. If the delays are bounded by d, Cesa-Bianchi et al. (2018) have shown that EXP3 can still be implemented even with no timestamps, with expected regret $O(\\sqrt{dTK\\log K})$ instead of $O\\left({\\sqrt{\\left(d+K\\right)T\\log K}}\\right)$ . For our unbounded delays case, it is not clear if FKM or EXP3 can be implemented without timestamps.\n\nThe set of costs (feedback samples) received and used at round t is denoted $\\mathcal{S}_{t}$ , so $s \\in \\mathcal{S}_{t}$ means that the cost of $\\pmb{a}_{s}$ from round s is received and used at round t. Since the game lasts for $T$ rounds, all costs for which $t + d_{t}> T$ are never received. Of course, the value of $d_{t}$ does not matter as long as $t + d_{t}> T$ , and these are just samples that the adversary chose to prevent the player from receiving. We name these costs the missing samples and denote their set by $\\mathcal{M}$ .\n\nWhile the rounds of the game are indexed by t, it will be useful to our analysis to index a finer time scale that counts the steps of the algorithm for every such t. We define $s_{-}, s_{+}$ as the steps a moment before and after the algorithm uses the feedback from round s, respectively. These steps are taking place in round t if $s \\in S_{t}$"}, {"name": "Setup: multi-armed bandit special case (Section 2)", "statement": "We analyze two widely applied algorithms for the two central special cases of the scenario above:\n\n2. Multi-Armed Bandit - $\\mathcal{K}=\\{1,...,K\\},\\ l_{t}:\\{1,...,K\\}\\to [0,1].$ With no delays, the EXP3 algorithm (Auer et al., 2002) achieves an expected regret of $O \\left({\\sqrt{T K \\log K}}\\right)$ for this problem."}, {"name": "Definition 1 (expected regret, eq. 1)", "statement": "The player wants to have a learning algorithm that uses past observations to make good decisions over time. The performance of the player's algorithm is measured using the regret. The expected regret is the total expected cost over a horizon of $T$ rounds, compared to the total expected cost that results from playing the best fixed action in hindsight in all rounds:\n\nDefinition 1. Let $\\begin{array}{r}{\\pmb{a}^{*}=\\underset{\\pmb{a}\\in\\mathcal{K}}{\\arg\\operatorname*{min}}\\sum_{t=1}^{T}l_{t}\\left(\\pmb{a}\\right)}\\end{array}$ . The expected regret is defined as\n\n$$\n\\mathbb{E}\\left\\{R\\left(T\\right)\\right\\}\\triangleq\\sum_{t=1}^{T}\\mathbb{E}\\left\\{l_{t}\\left(\\pmb{a}_{t}\\right)-l_{t}\\left(\\pmb{a}^{*}\\right)\\right\\}\\tag{1}\n$$\n\nwhere $\\mathbb{E}$ is the expectation over the (possibly) random actions $\\pmb{a}_{1},\\ldots,\\pmb{a}_{T}$ of the player."}, {"name": "Setup: EXP3 with delays, mixed actions and the filtration F_t (eq. 41, Section 6)", "statement": "Consider a player that at each round t picks one out of K arms. Let $a_{t}$ be the arm the player chooses at round t. The cost at round t of playing arm i is $l_{t}^{(i)}\\in [0,1]$ , and let $l_{t}=\\left(l_{t}^{(1)},...,l_{t}^{(K)}\\right)$ be the cost vector. At round t, the EXP3 algorithm, detailed in Algorithm 3, chooses an arm at random using a distribution that depends on the history of the game. The variant when $\\gamma_{t}\\neq 0$ , as we use against an adaptive adversary, is known as EXP3-IX (see Neu (2015)). We denote the vector of probabilities of the player for choosing arms at round t by $\\pmb{p}_{t}\\in \\Delta^{K}$ , where $\\Delta^{K}$ denotes the K-simplex. This is also known as the mixed action of the player. We also define the following filtration\n\n$$\n\\mathcal{F}_{t}=\\sigma\\left(\\left\\{a_{s}\\vert s+d_{s}\\le t\\right\\}\\cup\\left\\{l_{s}\\vert s\\le t\\right\\}\\right)\\tag{41}\n$$\n\nwhich is generated from all the actions for which the feedback was received up to round t and all cost functions up to round t. Note that the mixed action $\\pmb{p}_{t}$ is a $\\mathcal{F}_{t}$ -measurable random variable since $\\mathcal{F}_{t}$ includes everything that could affect the algorithm up to round t. With a slight abuse of notation, we use $\\mathcal{F}_{s_{-}}$ to denote the filtration induced from all actions for which the feedback has been received and used up to step $s_{-}$ and all the cost functions up to round t."}, {"name": "Algorithm 3 (EXP3 with delays, eqs. 44-45)", "statement": "Algorithm 3 EXP3 with delays   \nInitialization: Let $\\{\\eta_{t}\\}$ and $\\{\\gamma_{t}\\}$ be non-negative non-increasing sequences such that $\\eta_{1}\\leq \\frac{e^{- 2}}{2}$ and   \nset $\\tilde{L}_{1}^{(i)}=0$ and $\\begin{array}{r}{p_{1}^{(i)}=\\frac{1}{K}}\\end{array}$ for $i=1,...,K.$   \nFor $t=1,...,T$ do   \n1. Choose an arm $a_{t}$ at random according to the distribution $\\pmb{p}_{t}$ .   \n2. Collect in $S_{t}$ all the rounds s for which $l_{s}^{(a_{s})}$ arrived at round t after a delay of $\\begin{array}{r}{d_{s}\\le \\frac{1}{e^{2}\\eta_{s}}- 1}\\end{array}$   \n3. Set $\\tilde{L}_{t}^{(i)}=\\tilde{L}_{t-1}^{(i)}$ for all i. Update the weights of arm $a_{s}$ for all $s \\in S_{t}$ using   \n$\\tilde{L}_{t}^{(a_{s})}=\\tilde{L}_{t}^{(a_{s})}+\\eta_{s}\\frac{l_{s}^{(a_{s})}}{p_{s}^{(a_{s})}+\\gamma_{s}}.$ (44)   \n4. Update the mixed action for $i=1,\\ldots,K$ using   \n$p_{t+1}^{(i)}=\\frac{e^{-\\tilde{L}_{t}^{(i)}}}{\\sum_{j=1}^{K}e^{-\\tilde{L}_{t}^{(j)}}}.$ (45)   \nEnd"}, {"name": "Lemma 5 (another result of the same paper)", "statement": "Lemma 5 (Weighted-Regret Bound for EXP3). Let $\\{\\eta_{t}\\}$ be a non-increasing step-size sequence such that $\\eta_{t}\\leq\\frac 12e^{-2}$ for all t. Let $\\left\\{l_{t}^{\\left(i\\right)}\\right\\}$ be a cost sequence such that $l_{t}^{(i)}\\in [0,1]$ for every $t,i.$ Let $\\{d_{t}\\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . Define the set $\\mathcal{M}^{\\ast}$ of all samples that are not received before round T or that were delayed by $\\begin{array}{r}{d_{t}\\geq \\frac{1}{e^{2}\\eta_{t}}- 1}\\end{array}$ . Then using EXP3 (Algorithm 3) guarantees:\n\n1. With an oblivious adversary and $\\gamma_{t}= 0$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq\\log K+\\frac{e^{2}}{2}K\\sum_{t=1}^{T}\\eta_{t}^{2}+4\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{42}\n$$\n\n2. With an adaptive adversary and $\\gamma_{t}= \\eta_{t}$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq 2+2\\log K+(1+\\frac{e^{2}}{2})K\\sum_{t=1}^{T}\\eta_{t}^{2}+4 e^{2}K\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{43}\n$$"}], "nl_notes": "### Theorem 6\nNOTE (Theorem 6) — As in the oblivious case of Lemma 5, the Lean assumes (`a_fresh`) that each arm a_t is drawn with fresh randomness: for t ≤ t' < t + d_t, the conditional law of a_t given 𝓕_{t'}, together with the arms of the rounds whose feedback arrives before that of round t, is still p_t. The paper never states this as an assumption; it is used inside the proof of the oblivious case of Lemma 5.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped Classical\nopen MeasureTheory Filter Topology\n\ndef missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t)\n\nnoncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t ∨ (d t : ℝ) ≥ 1 / (Real.exp 2 * η t) - 1)\n\nnoncomputable def weightedExpReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  haveI : Nonempty (Fin K) := ⟨⟨0, hK⟩⟩\n  ∫ ω, (∑ t ∈ Finset.Icc 1 T, η t * l t ω (a t ω) -\n    (Finset.univ : Finset (Fin K)).inf' Finset.univ_nonempty\n      (fun i => ∑ t ∈ Finset.Icc 1 T, η t * l t ω i)) ∂μ\n\nnoncomputable def expReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (l0 : ℕ → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  weightedExpReg μ T K hK (fun _ => 1) (fun t _ i => l0 t i) a\n\nnoncomputable def NoWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : Prop :=\n  limsup (fun T : ℕ => weightedExpReg μ T K hK η l a / ∑ t ∈ Finset.Icc 1 T, η t)\n    atTop ≤ 0\n\nstructure Exp3Delayed {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (K : ℕ) (hK : 0 < K)\n    (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (η γ : ℕ → ℝ) where\n  𝓕 : ℕ → MeasurableSpace Ω\n  𝓕_le : ∀ t, 𝓕 t ≤ ‹MeasurableSpace Ω›\n  𝓕_mono : Monotone 𝓕\n  a : ℕ → Ω → Fin K\n  p : ℕ → Ω → Fin K → ℝ\n  Lhat : ℕ → Ω → Fin K → ℝ\n  measurable_a : ∀ t, Measurable (a t)\n  𝓕_eq : ∀ t, 𝓕 t =\n    (⨆ s, ⨆ (_ : 1 ≤ s ∧ s + d s ≤ t), MeasurableSpace.comap (a s) inferInstance)\n      ⊔ (⨆ s, ⨆ (_ : 1 ≤ s ∧ s ≤ t), MeasurableSpace.comap (l s) inferInstance)\n  measurable_p : ∀ t i, @Measurable Ω ℝ (𝓕 t) _ (fun ω => p t ω i)\n  p_isProb : ∀ t ω, (∀ i, 0 ≤ p t ω i) ∧ ∑ i, p t ω i = 1\n  p_init : ∀ ω i, p 1 ω i = 1 / (K : ℝ)\n  a_law : ∀ t i,\n      (μ[(fun ω => if a t ω = i then (1 : ℝ) else 0) | 𝓕 t]) =ᵐ[μ] (fun ω => p t ω i)\n  Lhat_init : ∀ ω i, Lhat 0 ω i = 0\n  Lhat_step : ∀ t ω i, 1 ≤ t →\n      Lhat t ω i = Lhat (t - 1) ω i +\n        ∑ s ∈ (Finset.Icc 1 t).filter\n          (fun s => s + d s = t ∧ (d s : ℝ) ≤ 1 / (Real.exp 2 * η s) - 1 ∧ a s ω = i),\n          η s * l s ω i / (p s ω i + γ s)\n  p_update : ∀ t ω i,\n      p (t + 1) ω i = Real.exp (- Lhat t ω i) / ∑ j, Real.exp (- Lhat t ω j)\n\ntheorem lemma5_oblivious {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n\n    \n    \n    \n    \n    \n    \n    (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n            MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η (fun t _ i => l0 t i) E.a ≤\n      Real.log K + (Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem lemma5_adaptive {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η l E.a ≤\n      2 + 2 * Real.log K\n        + (1 + Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * Real.exp 2 * K * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem theorem6_regretBound :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        (K T : ℕ) (hK : 0 < K) (hT : 0 < T) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ)\n        (hd : ∀ t ≥ 1, 1 ≤ d t)\n        (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1) (η : ℕ → ℝ)\n        (hη_pos : ∀ t ≥ 1, 0 < η t)\n        (hη : η = fun _ =>\n          (Real.exp (-2) / 2) *\n            Real.sqrt (Real.log K /\n              ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ))))\n        (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n        (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n          (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n              E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n                MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)),\n        expReg μ T K hK l0 E.a ≤\n          C * (Real.sqrt (Real.log K *\n                ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ)))\n              + (missedRounds T d).card) := by\n  sorry\n\ntheorem lemma6_noWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K : ℕ) (hK : 0 < K)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t ≥ 1, 0 < η t) (hη_dec : ∀ t ≥ 1, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t ≥ 1, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ≥ 1, ∀ ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i))\n    (h1 : Tendsto (fun T : ℕ => ∑ t ∈ Finset.Icc 1 T, η t) atTop atTop)\n    (h2 : ∃ L : ℝ, Tendsto (fun t : ℕ => η t * (d t : ℝ)) atTop (nhds L))\n    (h3 : Summable (fun t : ℕ => (η t) ^ 2 * (d t : ℝ))) :\n    NoWeightedRegret μ K hK η l E.a := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma5_oblivious", "lemma5_adaptive", "theorem6_regretBound", "lemma6_noWeightedRegret"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 4466, "end": 5519, "gold": ":\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        (K T : ℕ) (hK : 0 < K) (hT : 0 < T) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ)\n        (hd : ∀ t ≥ 1, 1 ≤ d t)\n        (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1) (η : ℕ → ℝ)\n        (hη_pos : ∀ t ≥ 1, 0 < η t)\n        (hη : η = fun _ =>\n          (Real.exp (-2) / 2) *\n            Real.sqrt (Real.log K /\n              ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ))))\n        (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n        (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n          (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n              E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n                MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)),\n        expReg μ T K hK l0 E.a ≤\n          C * (Real.sqrt (Real.log K *\n                ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ)))\n              + (missedRounds T d).card)"}]}
{"item_id": "v23-20-411::lemma6_noWeightedRegret::A-stmt", "paper_id": "v23-20-411", "target_name": "lemma6_noWeightedRegret", "source": "Research-level", "nl_statement": "Lemma 6. EXP3 (Algorithm 3 with $\\gamma_{t}= \\eta_{t}$ for all t, against an adaptive adversary) with a non-increasing and positive step-size sequence $\\{\\eta_{t}\\}$ such that $\\eta_{t}\\leq\\frac 12e^{-2}$ for all t has no weighted-regret with respect to the sequence of delays $\\{d_{t}\\}$ and $\\{\\eta_{t}\\}$ as the weight sequence, if the following two conditions hold:\n\n1. $\\textstyle \\sum_{t = 1}^{\\infty}\\eta_{t}= \\infty .$\n\n2. $\\operatorname*{lim}_{t\\to\\infty}\\eta_{t}d_{t}<\\infty$ and $\\textstyle \\sum_{t = 1}^{\\infty}\\eta_{t}^{2}d_{t}< \\infty$", "context": [{"name": "Setup: problem formulation, adversaries, delays, missing samples (Section 2)", "statement": "Consider a player that in each round t from 1 to $T$ picks an action $\\pmb{a}_{t}\\in \\mathcal{K}$ from a set $\\mathcal{K}$ . The cost at round t from playing $\\pmb{a}_{t}$ is $l_{t}\\left(\\boldsymbol{a}_{t}\\right)\\in\\left[0,1\\right]$ . We consider two types of adversaries:\n\n1. Oblivious Adversary: chooses the cost functions $l_{1},...,l_{T}$ before the game starts.\n\n2. Adaptive Adversary: chooses the cost function $l_{t}$ after observing $\\{a_{1}, \\ldots a_{t - 1}\\}$ , for each t.\n\nWith full information and no delays, the player gets to know the function $l_{t}$ immediately after playing $\\pmb{a}_{t}$ . In the bandit delayed feedback scenario, the player only gets to know the value of ${l}_{t}\\left(\\pmb{a}_{t}\\right)$ at the beginning of round $t + d_{t}$ . (i.e., after a delay of $d_{t}\\geq 1$ rounds). The adversary (oblivious or adaptive) chooses the delay sequence $\\{d_{t}\\}$ before the game starts.\n\nWe assume that the cost feedback includes the timestamp of the action that incurred this cost. This is indeed the case in many applications, such as when robots take physical actions, a recommendation is made to a customer or a treatment is given to a patient. If the delays are bounded by d, Cesa-Bianchi et al. (2018) have shown that EXP3 can still be implemented even with no timestamps, with expected regret $O(\\sqrt{dTK\\log K})$ instead of $O\\left({\\sqrt{\\left(d+K\\right)T\\log K}}\\right)$ . For our unbounded delays case, it is not clear if FKM or EXP3 can be implemented without timestamps.\n\nThe set of costs (feedback samples) received and used at round t is denoted $\\mathcal{S}_{t}$ , so $s \\in \\mathcal{S}_{t}$ means that the cost of $\\pmb{a}_{s}$ from round s is received and used at round t. Since the game lasts for $T$ rounds, all costs for which $t + d_{t}> T$ are never received. Of course, the value of $d_{t}$ does not matter as long as $t + d_{t}> T$ , and these are just samples that the adversary chose to prevent the player from receiving. We name these costs the missing samples and denote their set by $\\mathcal{M}$ .\n\nWhile the rounds of the game are indexed by t, it will be useful to our analysis to index a finer time scale that counts the steps of the algorithm for every such t. We define $s_{-}, s_{+}$ as the steps a moment before and after the algorithm uses the feedback from round s, respectively. These steps are taking place in round t if $s \\in S_{t}$"}, {"name": "Setup: multi-armed bandit special case (Section 2)", "statement": "We analyze two widely applied algorithms for the two central special cases of the scenario above:\n\n2. Multi-Armed Bandit - $\\mathcal{K}=\\{1,...,K\\},\\ l_{t}:\\{1,...,K\\}\\to [0,1].$ With no delays, the EXP3 algorithm (Auer et al., 2002) achieves an expected regret of $O \\left({\\sqrt{T K \\log K}}\\right)$ for this problem."}, {"name": "Definition 2 (no weighted-regret, eq. 2) and remarks", "statement": "Our key observation is that with delayed feedback, it is not the regret that matters for the game dynamics but rather what we call the weighted-regret. The weighted-regret weighs the costs in different rounds according to a given non-increasing sequence $\\eta_{t}$ so it coincides with the regret when $\\eta_{t}=\\eta,\\forall t$ . We define \"no weighted-regret\" to replace the traditional no-regret property:\n\nDefinition 2. Let $\\{l_{t}:\\mathcal{K}\\to [0,1]\\}_{t}$ be a sequence of cost functions, chosen by an adaptive adversary. Let $\\pmb{a}^{*}=\\arg\\operatorname*{min}_{\\pmb{a}\\in\\mathcal{K}}\\sum_{t=1}^{T}\\eta_{t}l_{t}\\left(\\pmb{a}\\right)$ . Let $\\{d_{t}\\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . We say that an algorithm that produces the random sequence of (single-agent) actions $\\left\\{\\pmb{a}_{t}\\right\\}$ has no weighted-regret with respect to $\\{d_{t}\\}$ and the non-increasing weight\n\nsequence $\\{\\eta_{t}\\}$ if\n\n$$\n\\operatorname*{lim}_{T\\to\\infty}\\mathbb{E}\\left\\{\\frac{\\sum_{t=1}^{T}\\eta_{t}\\left(l_{t}\\left(\\pmb{a}_{t}\\right)-l_{t}\\left(\\pmb{a}^{*}\\right)\\right)}{\\sum_{t=1}^{T}\\eta_{t}}\\right\\}=0\\tag{2}\n$$\n\nwhere the expectation is with respect to the random $\\left\\{\\pmb{a}_{t}\\right\\}$ generated by the algorithm.\n\nHaving no weighted-regret is only non-trivial when $\\textstyle \\sum_{t = 1}^{\\infty}\\eta_{t}\\ = \\infty$ When $\\textstyle\\sum_{t=1}^{\\infty}\\eta_{t}<\\infty,$ the feedback from the last $\\textstyle{\\frac{T}{2}}$ rounds can be discarded without affecting (2). When $\\textstyle \\sum_{t = 1}^{\\infty}\\eta_{t}= \\infty$ ,there is no round t after which we can discard all feedback and still maintain (2).\n\nWe define no weighted-regret with respect to an adaptive adversary since, in a non-cooperative game with cost functions $\\{u_{n}\\}$ , the \"adversarial\" cost function of player n is $l_{t}({\\bf{a}}_{n})=u_{n}({\\bf{a}}_{n},{\\bf{a}}_{-n,t})$ so it is determined by the actions of the other players. In turn, the actions of the other players depend on the past actions of player n. Hence, the cost function of player n depends on her past actions, as with an adaptive adversary. In general, the equilibrium does not consist of absolute strategies such as \"min-max\" in zero-sum games. Hence, regret guarantees against an adaptive adversary allow proving convergence to the set of CCE for general non-cooperative games.\n\nWhen taking the limit $T\\to\\infty,$ it is important to emphasize that we do not change the infinite sequence of delays $\\{d_{t}\\}$ , but only reveal more elements in this sequence. In other words, we are looking at the same game but over a longer time horizon. Therefore, while $\\begin{array}{r}{d_{t}= \\frac{T}{2}}\\end{array}$ makes sense for a constant $T,$ it is misleading when taking $T \\to \\infty$ since it represents a delay that occurred at time t but changes with the limit, so the limit is no longer of the \"same game\"."}, {"name": "Setup: EXP3 with delays, mixed actions and the filtration F_t (eq. 41, Section 6)", "statement": "Consider a player that at each round t picks one out of K arms. Let $a_{t}$ be the arm the player chooses at round t. The cost at round t of playing arm i is $l_{t}^{(i)}\\in [0,1]$ , and let $l_{t}=\\left(l_{t}^{(1)},...,l_{t}^{(K)}\\right)$ be the cost vector. At round t, the EXP3 algorithm, detailed in Algorithm 3, chooses an arm at random using a distribution that depends on the history of the game. The variant when $\\gamma_{t}\\neq 0$ , as we use against an adaptive adversary, is known as EXP3-IX (see Neu (2015)). We denote the vector of probabilities of the player for choosing arms at round t by $\\pmb{p}_{t}\\in \\Delta^{K}$ , where $\\Delta^{K}$ denotes the K-simplex. This is also known as the mixed action of the player. We also define the following filtration\n\n$$\n\\mathcal{F}_{t}=\\sigma\\left(\\left\\{a_{s}\\vert s+d_{s}\\le t\\right\\}\\cup\\left\\{l_{s}\\vert s\\le t\\right\\}\\right)\\tag{41}\n$$\n\nwhich is generated from all the actions for which the feedback was received up to round t and all cost functions up to round t. Note that the mixed action $\\pmb{p}_{t}$ is a $\\mathcal{F}_{t}$ -measurable random variable since $\\mathcal{F}_{t}$ includes everything that could affect the algorithm up to round t. With a slight abuse of notation, we use $\\mathcal{F}_{s_{-}}$ to denote the filtration induced from all actions for which the feedback has been received and used up to step $s_{-}$ and all the cost functions up to round t."}, {"name": "Algorithm 3 (EXP3 with delays, eqs. 44-45)", "statement": "Algorithm 3 EXP3 with delays   \nInitialization: Let $\\{\\eta_{t}\\}$ and $\\{\\gamma_{t}\\}$ be non-negative non-increasing sequences such that $\\eta_{1}\\leq \\frac{e^{- 2}}{2}$ and   \nset $\\tilde{L}_{1}^{(i)}=0$ and $\\begin{array}{r}{p_{1}^{(i)}=\\frac{1}{K}}\\end{array}$ for $i=1,...,K.$   \nFor $t=1,...,T$ do   \n1. Choose an arm $a_{t}$ at random according to the distribution $\\pmb{p}_{t}$ .   \n2. Collect in $S_{t}$ all the rounds s for which $l_{s}^{(a_{s})}$ arrived at round t after a delay of $\\begin{array}{r}{d_{s}\\le \\frac{1}{e^{2}\\eta_{s}}- 1}\\end{array}$   \n3. Set $\\tilde{L}_{t}^{(i)}=\\tilde{L}_{t-1}^{(i)}$ for all i. Update the weights of arm $a_{s}$ for all $s \\in S_{t}$ using   \n$\\tilde{L}_{t}^{(a_{s})}=\\tilde{L}_{t}^{(a_{s})}+\\eta_{s}\\frac{l_{s}^{(a_{s})}}{p_{s}^{(a_{s})}+\\gamma_{s}}.$ (44)   \n4. Update the mixed action for $i=1,\\ldots,K$ using   \n$p_{t+1}^{(i)}=\\frac{e^{-\\tilde{L}_{t}^{(i)}}}{\\sum_{j=1}^{K}e^{-\\tilde{L}_{t}^{(j)}}}.$ (45)   \nEnd"}, {"name": "Section 6.2 introduction (conditions for no weighted-regret)", "statement": "In this subsection, we provide conditions for EXP3 to have no weighted-regret with respect to the delay sequence and its step-size sequence as the weight sequence of Section 3. As discussed in Section 3, $\\textstyle \\sum_{t = 1}^{\\infty}\\eta_{t}= \\infty$ , is necessary for the no weighted-regret property to be non-trivial. All other conditions of Lemma 6 are as tight as the bound of Lemma 5."}, {"name": "Lemma 5 (another result of the same paper)", "statement": "Lemma 5 (Weighted-Regret Bound for EXP3). Let $\\{\\eta_{t}\\}$ be a non-increasing step-size sequence such that $\\eta_{t}\\leq\\frac 12e^{-2}$ for all t. Let $\\left\\{l_{t}^{\\left(i\\right)}\\right\\}$ be a cost sequence such that $l_{t}^{(i)}\\in [0,1]$ for every $t,i.$ Let $\\{d_{t}\\}$ be a delay sequence such that the cost from round t is received at round $t + d_{t}$ . Define the set $\\mathcal{M}^{\\ast}$ of all samples that are not received before round T or that were delayed by $\\begin{array}{r}{d_{t}\\geq \\frac{1}{e^{2}\\eta_{t}}- 1}\\end{array}$ . Then using EXP3 (Algorithm 3) guarantees:\n\n1. With an oblivious adversary and $\\gamma_{t}= 0$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq\\log K+\\frac{e^{2}}{2}K\\sum_{t=1}^{T}\\eta_{t}^{2}+4\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{42}\n$$\n\n2. With an adaptive adversary and $\\gamma_{t}= \\eta_{t}$ for all t:\n\n$$\n\\mathbb{E}\\left\\{\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(a_{t})}-\\operatorname*{min}_{i}\\sum_{t=1}^{T}\\eta_{t}l_{t}^{(i)}\\right\\}\\leq 2+2\\log K+(1+\\frac{e^{2}}{2})K\\sum_{t=1}^{T}\\eta_{t}^{2}+4 e^{2}K\\sum_{t\\notin\\mathcal{M}^{*}}\\eta_{t}^{2}d_{t}+\\sum_{t\\in\\mathcal{M}^{*}}\\eta_{t}.\\tag{43}\n$$"}], "nl_notes": "### Lemma 6\nLemma 6 concludes that EXP3 \"has no weighted-regret\", which Definition 2 / eq. (2) defines as the TWO-SIDED limit `lim_{T->inf} E{ sum eta_t (l_t(a_t) - l_t(a*)) / sum eta_t } = 0`.\nThe ratio lies in [-1,1], so `lim = 0` is equivalent to `limsup <= 0` AND `0 <= liminf`. 'The limit exists' adds nothing else. The formalization keeps exactly the proved half: `NoWeightedRegret := limsup (...) atTop <= 0`, NOT `Tendsto (...) atTop (nhds 0)`.\nNOTE (Lemma 6) — As in the adaptive case of Lemma 5, which its proof applies, the Lean adds a hypothesis (`a_law_adaptive`): the conditional law of a_t given the arms a_1, …, a_{t-1} of the earlier rounds is p_t. The paper never states this as an assumption; it is used inside the proof of Lemma 11.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped Classical\nopen MeasureTheory Filter Topology\n\ndef missedRounds (T : ℕ) (d : ℕ → ℕ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t)\n\nnoncomputable def MStar (T : ℕ) (d : ℕ → ℕ) (η : ℕ → ℝ) : Finset ℕ :=\n  (Finset.Icc 1 T).filter (fun t => T < t + d t ∨ (d t : ℝ) ≥ 1 / (Real.exp 2 * η t) - 1)\n\nnoncomputable def weightedExpReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  haveI : Nonempty (Fin K) := ⟨⟨0, hK⟩⟩\n  ∫ ω, (∑ t ∈ Finset.Icc 1 T, η t * l t ω (a t ω) -\n    (Finset.univ : Finset (Fin K)).inf' Finset.univ_nonempty\n      (fun i => ∑ t ∈ Finset.Icc 1 T, η t * l t ω i)) ∂μ\n\nnoncomputable def expReg {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (T K : ℕ) (hK : 0 < K) (l0 : ℕ → Fin K → ℝ) (a : ℕ → Ω → Fin K) : ℝ :=\n  weightedExpReg μ T K hK (fun _ => 1) (fun t _ i => l0 t i) a\n\nnoncomputable def NoWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    (K : ℕ) (hK : 0 < K) (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (a : ℕ → Ω → Fin K) : Prop :=\n  limsup (fun T : ℕ => weightedExpReg μ T K hK η l a / ∑ t ∈ Finset.Icc 1 T, η t)\n    atTop ≤ 0\n\nstructure Exp3Delayed {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (K : ℕ) (hK : 0 < K)\n    (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (η γ : ℕ → ℝ) where\n  𝓕 : ℕ → MeasurableSpace Ω\n  𝓕_le : ∀ t, 𝓕 t ≤ ‹MeasurableSpace Ω›\n  𝓕_mono : Monotone 𝓕\n  a : ℕ → Ω → Fin K\n  p : ℕ → Ω → Fin K → ℝ\n  Lhat : ℕ → Ω → Fin K → ℝ\n  measurable_a : ∀ t, Measurable (a t)\n  𝓕_eq : ∀ t, 𝓕 t =\n    (⨆ s, ⨆ (_ : 1 ≤ s ∧ s + d s ≤ t), MeasurableSpace.comap (a s) inferInstance)\n      ⊔ (⨆ s, ⨆ (_ : 1 ≤ s ∧ s ≤ t), MeasurableSpace.comap (l s) inferInstance)\n  measurable_p : ∀ t i, @Measurable Ω ℝ (𝓕 t) _ (fun ω => p t ω i)\n  p_isProb : ∀ t ω, (∀ i, 0 ≤ p t ω i) ∧ ∑ i, p t ω i = 1\n  p_init : ∀ ω i, p 1 ω i = 1 / (K : ℝ)\n  a_law : ∀ t i,\n      (μ[(fun ω => if a t ω = i then (1 : ℝ) else 0) | 𝓕 t]) =ᵐ[μ] (fun ω => p t ω i)\n  Lhat_init : ∀ ω i, Lhat 0 ω i = 0\n  Lhat_step : ∀ t ω i, 1 ≤ t →\n      Lhat t ω i = Lhat (t - 1) ω i +\n        ∑ s ∈ (Finset.Icc 1 t).filter\n          (fun s => s + d s = t ∧ (d s : ℝ) ≤ 1 / (Real.exp 2 * η s) - 1 ∧ a s ω = i),\n          η s * l s ω i / (p s ω i + γ s)\n  p_update : ∀ t ω i,\n      p (t + 1) ω i = Real.exp (- Lhat t ω i) / ∑ j, Real.exp (- Lhat t ω j)\n\ntheorem lemma5_oblivious {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n\n    \n    \n    \n    \n    \n    \n    (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n            MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η (fun t _ i => l0 t i) E.a ≤\n      Real.log K + (Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem lemma5_adaptive {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K T : ℕ) (hK : 0 < K) (hT : 0 < T)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t, 0 < η t) (hη_dec : ∀ t, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i)) :\n    weightedExpReg μ T K hK η l E.a ≤\n      2 + 2 * Real.log K\n        + (1 + Real.exp 2 / 2) * K * (∑ t ∈ Finset.Icc 1 T, (η t) ^ 2)\n        + 4 * Real.exp 2 * K * (∑ t ∈ (Finset.Icc 1 T) \\ MStar T d η, (η t) ^ 2 * (d t : ℝ))\n        + (∑ t ∈ MStar T d η, η t) := by\n  sorry\n\ntheorem theorem6_regretBound :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        (K T : ℕ) (hK : 0 < K) (hT : 0 < T) (l0 : ℕ → Fin K → ℝ) (d : ℕ → ℕ)\n        (hd : ∀ t ≥ 1, 1 ≤ d t)\n        (hl : ∀ t i, l0 t i ∈ Set.Icc (0 : ℝ) 1) (η : ℕ → ℝ)\n        (hη_pos : ∀ t ≥ 1, 0 < η t)\n        (hη : η = fun _ =>\n          (Real.exp (-2) / 2) *\n            Real.sqrt (Real.log K /\n              ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ))))\n        (E : Exp3Delayed μ K hK (fun t _ i => l0 t i) d η (fun _ => 0))\n        (a_fresh : ∀ t t' i, t ≤ t' → t' < t + d t →\n          (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n              E.𝓕 t' ⊔ ⨆ s ∈ {s : ℕ | s + d s < t + d t ∨ (s + d s = t + d t ∧ s < t)},\n                MeasurableSpace.comap (E.a s) inferInstance]) =ᵐ[μ] (fun ω => E.p t ω i)),\n        expReg μ T K hK l0 E.a ≤\n          C * (Real.sqrt (Real.log K *\n                ((K : ℝ) * T + ∑ t ∈ (Finset.Icc 1 T) \\ missedRounds T d, (d t : ℝ)))\n              + (missedRounds T d).card) := by\n  sorry\n\ntheorem lemma6_noWeightedRegret {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K : ℕ) (hK : 0 < K)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t ≥ 1, 0 < η t) (hη_dec : ∀ t ≥ 1, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t ≥ 1, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ≥ 1, ∀ ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i))\n    (h1 : Tendsto (fun T : ℕ => ∑ t ∈ Finset.Icc 1 T, η t) atTop atTop)\n    (h2 : ∃ L : ℝ, Tendsto (fun t : ℕ => η t * (d t : ℝ)) atTop (nhds L))\n    (h3 : Summable (fun t : ℕ => (η t) ^ 2 * (d t : ℝ))) :\n    NoWeightedRegret μ K hK η l E.a := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma5_oblivious", "lemma5_adaptive", "theorem6_regretBound", "lemma6_noWeightedRegret"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 5567, "end": 6540, "gold": "{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)\n    [IsProbabilityMeasure μ] (K : ℕ) (hK : 0 < K)\n    (η : ℕ → ℝ) (l : ℕ → Ω → Fin K → ℝ) (d : ℕ → ℕ) (hd : ∀ t ≥ 1, 1 ≤ d t)\n    (hη_pos : ∀ t ≥ 1, 0 < η t) (hη_dec : ∀ t ≥ 1, η (t + 1) ≤ η t)\n    (hη_bound : ∀ t ≥ 1, η t ≤ (1 / 2) * Real.exp (-2))\n    (hl : ∀ t ≥ 1, ∀ ω i, l t ω i ∈ Set.Icc (0 : ℝ) 1)\n    (E : Exp3Delayed μ K hK l d η η)\n    (hl_adapted : ∀ t i, @Measurable Ω ℝ\n      (⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance) _\n      (fun ω => l t ω i))\n    (a_law_adaptive : ∀ t i,\n      (μ[(fun ω => if E.a t ω = i then (1 : ℝ) else 0) |\n          ⨆ τ ∈ Finset.Ico 1 t, MeasurableSpace.comap (E.a τ) inferInstance]) =ᵐ[μ]\n        (fun ω => E.p t ω i))\n    (h1 : Tendsto (fun T : ℕ => ∑ t ∈ Finset.Icc 1 T, η t) atTop atTop)\n    (h2 : ∃ L : ℝ, Tendsto (fun t : ℕ => η t * (d t : ℝ)) atTop (nhds L))\n    (h3 : Summable (fun t : ℕ => (η t) ^ 2 * (d t : ℝ))) :\n    NoWeightedRegret μ K hK η l E.a"}]}
{"item_id": "v23-21-1027::theorem3::A-stmt", "paper_id": "v23-21-1027", "target_name": "theorem3", "source": "Research-level", "nl_statement": "Theorem 3 (Regret bound for dual-stabilized OMD) Assume we have $\\eta_{t}\\geq \\eta_{t + 1}>$ 0 for each $t > 1$ . Define $\\gamma_{t}=\\eta_{t+1}/\\eta_{t}\\in(0,1]$ for all $t \\geq 1$ . Let $\\{f_{t}\\}_{t \\ge 1}$ be a sequence of convex functions with $f_{t}\\colon \\mathcal{X}\\to \\mathbb{R}$ for each $t \\geq 1$ . Let $\\{x_{t}\\}_{t \\ge 1}$ and $\\{\\hat{w}_{t}\\}_{t \\ge 2}$ be as in Algorithm 2. Then, for all $T > 0$ and $z \\in \\mathcal{X}$ ,\n\n$$\n\\mathrm{Regret}(T,z)\\ \\leq\\ \\sum_{t=1}^{T}\\frac{D_{\\Phi}\\bigl({}_{x_{t+1}}^{x_{t}};w_{t+1}\\bigr)}{\\eta_{t}}+\\frac{D_{\\Phi}\\bigl(z,x_{1}\\bigr)}{\\eta_{T+1}}.\\tag{8}\n$$", "context": [{"name": "Setup: online convex optimization and regret (Section 3)", "statement": "We consider the online convex optimization problem with unknown time horizon. For each time step $t\\in\\{1,2,\\ldots\\}$ the algorithm proposes a point $x_{t}$ from a closed convex set ${\\mathcal{X}}\\subseteq \\mathbb{R}^{n}$ and an adversary simultaneously picks a convex cost function $f_{t}$ which the algorithm has access to via a first-order oracle, that is, for any $x \\in \\mathcal{X}$ the algorithm can compute $f_{t}(x)$ and a subgradient $\\hat{g}\\in\\partial f_{t}(x): =\\{\\hat{g}\\in\\mathbb{R}^{n}\\mid f_{t}(z)\\geq f_{t}(x)+\\langle\\hat{g},z-x\\rangle\\forall z\\in\\mathcal{X}\\}$ . We will assume1 that all cost functions $f_{t}$ in this text are subdifferentiable on $\\mathcal{X}$ , that is, meaning that the subdifferential $\\partial f_{t}(x)$ is non-empty for all $x \\in \\mathcal{X}$\n\nThe cost of the iteration at time t is defined as $f_{t}(x_{t})$ . In this setting the goal is to produce a sequence of proposals $\\{x_{t}\\}_{t \\ge 1}$ that minimizes the regret against an unknown comparison point $z \\in \\mathcal{X}$ that has accrued up until time $T$:\n\n$$\n\\mathrm{Regret}(T,z): =\\ \\sum_{t=1}^{T}f_{t}(x_{t})-\\sum_{t=1}^{T}f_{t}(z).\n$$"}, {"name": "Standing assumptions on the mirror map Φ and hat notation (1) (Section 3)", "statement": "Both dual averaging and online mirror descent are parameterized by a special convex function $\\Phi,$ often referred as a regularizer or a mirror map (for D and X), which among other properties needs$^{2}$ to be of Legendre type (Rockafellar, 1970, Chapter 26). Formally, throughout the paper we assume that the function $\\Phi\\colon \\bar{\\mathcal{D}}\\to \\mathbb{R}$ is a closed convex function whose conjugate is differentiable on $\\mathbb{R}^{n}$ and such that $\\bar{\\mathcal{D}}$ satisfies $\\mathcal{X}\\subseteq \\bar{\\mathcal{D}}$ and $\\mathrm{int}\\,\\bar{\\mathcal{D}}\\cap \\mathrm{ri}\\,\\mathcal{X}\\neq \\varnothing$ (where $\\mathrm{ri}\\,\\mathcal{X}$ denotes the relative interior of $\\mathcal{X}$). Moreover, we also suppose that $\\Phi$ is of Legendre type, which means that $\\Phi$ is strictly convex on its domain$^{3}$ and essentially smooth, that is, for $\\mathcal{D}: =\\mathrm{int}\\bar{\\mathcal{D}}$ we have\n\n$\\mathcal{D}$ is nonempty,\n\n$\\Phi$ is differentiable on $\\mathcal{D}$, and\n\n$\\lim_{x\\to\\partial\\mathcal{D}}\\|\\nabla\\Phi(x)\\|= +\\infty$ , where ∂D is the boundary of $\\mathcal{D},\\mathrm{i.e.,}\\partial\\mathcal{D}: =\\mathrm{cl}\\mathcal{D}\\backslash\\mathcal{D}$\n\nThe gradient of the mirror map $\\nabla \\Phi :{\\mathcal{D}}\\to \\mathbb{R}^{n}$ and the gradient of its conjugate $\\nabla \\Phi^{*}: \\mathbb{R}^{n}\\to \\mathcal{D}$ are mutually inverse bijections between the primal space $\\mathcal{D}$ and the dual space $\\mathbb{R}^{n}$ . We will adopt the following notational convention. Any vector in the primal space will be written without a hat, such as $x \\in \\mathcal{D}$ . The same letter with a hat, namely $\\hat{x}$, will denote the corresponding dual vector:\n\n$$\n{\\hat{x}}: =\\nabla\\Phi(x)\\qquad{\\mathrm{and}}\\qquad x: =\\nabla\\Phi^{*}({\\hat{x}})\\qquad{\\mathrm{f o r~a l l~l e t t e r s~}}x .\\tag{1}\n$$"}, {"name": "Essential smoothness and well-posedness of the Bregman projection (Section 3)", "statement": "Essential smoothness ensures not only that $\\Phi$ is differentiable on the interior of its domain, but also that the slope of $\\Phi$ increases arbitrarily fast near the boundary of its domain. The latter guarantees, at least intuitively, that the function is increasing near and in the direction of the boundary of its domain. This property is fundamental for mirror descent to be well-defined (although not essential for dual averaging) since it ensures that the Bregman projection onto $\\mathcal{X}$ is attained by a point on $\\mathcal{D}$ where Φ is differentiable (we give more details on this on Section 7), and uniqueness is a consequence of the strict convexity of Φ. Some mirror maps we shall look at are classical cases of the OCO literature such as the negative entropy $x \\in \\mathbb{R}_{+}^{n}\\mapsto \\sum_{i = 1}^{n} x_{i}\\ln x_{i}$ and the squared 2-norm ${\\frac{1}{2}}\\|\\cdot\\|_{2}^{2}$ , and details on the reasons they are mirror maps can be found in the works of Shalev-Shwartz (2012); Bubeck (2011) and Bubeck (2015) (in particular, Bubeck, 2011, Section 5.2 discusses the properties of functions of Legendre type and why requiring the conjugate of the mirror map to be differentiable on the whole space is not necessary for mirror descent to be well-defined if one restricts the gradient steps in the dual space in some way)."}, {"name": "Definition: Bregman divergence (2), difference notation (3) and Bregman projection", "statement": "Given a mirror map $\\Phi,$ the Bregman divergence with respect to $\\Phi$ is defined as\n\n$$\nD_{\\Phi}(x,y)\\ : =\\ \\Phi(x)-\\Phi(y)-\\langle\\nabla\\Phi(y),x-y\\rangle,\\qquad\\forall x\\in\\bar{\\mathcal{D}},\\forall y\\in\\mathcal{D}.\\tag{2}\n$$\n\nThroughout this paper it will be convenient to use the notation\n\n$$\nD_{\\Phi}({}_{b}^{a};c): =D_{\\Phi}(a,c)-D_{\\Phi}(b,c)=\\Phi(a)-\\Phi(b)-\\langle\\nabla\\Phi(c),a-b\\rangle .\\tag{3}\n$$\n\nIn the important special case where $\\begin{array}{r}{\\Phi(x)=\\frac{1}{2}\\|x\\|_{2}^{2}}\\end{array}$ , the Bregman divergence relates to the Euclidean distance, i.e., $\\begin{array}{r}{D_{\\Phi}(x,y)=\\frac 12\\|x-y\\|_{2}^{2}}\\end{array}$ When $\\begin{array}{r}{\\Phi(x)=\\sum_{i=1}^{n}x_{i}\\log x_{i}}\\end{array}$ , the Bregman divergence becomes the generalized Kullback-Leibler (KL) divergence. The projection operator induced by the Bregman divergence is $\\Pi_{\\mathcal{X}}^{\\Phi}$ given by $\\{\\Pi_{\\mathcal{X}}^{\\Phi}\\}(y): =\\arg\\min\\{D_{\\Phi}(x,y)\\mid x\\in\\mathcal{X}\\}$ for any $y \\in \\mathcal{X}\\cap \\mathcal{D}$"}, {"name": "Algorithm 1: generic OMD / DA template", "statement": "A general template for optimization in the mirror descent framework is shown in Algorithm 1. The two classical algorithms, online mirror descent and dual averaging, are incarnations of this, differing only in how the dual variable $\\hat{y}_{t}$ is updated.\n\nIn this work, for a given initial point $x_{1}\\in{\\mathcal{X}}$ of the player, we are interested in the case when $\\begin{array}{r}{\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded, which still allows $\\sup_{z,x\\in{\\mathcal{X}}}D_{\\Phi}(z,x)$ to be unbounded. In fact, in the Euclidean setting $\\begin{array}{r}{(\\mathrm{i . e .,}\\Phi=\\frac{1}{2}\\|\\cdot\\|_{2}^{2}),\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded if and only if the diameter of X given by $\\sup_{x,y\\in\\mathcal{X}}\\frac{1}{2}\\Vert x-y\\Vert_{2}^{2}$ is also bounded. However, for a general mirror map $\\Phi$, assuming $\\begin{array}{r}{\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded is strictly weaker than assuming $\\scriptstyle\\operatorname*{sup}_{x,y\\in{\\mathcal{X}}}D_{\\Phi}(x,y)$ is bounded. This is the case for the well-known experts' problem, where Φ is the negative entropy, $D_{\\Phi}$ is the KL-divergence, X is the unit simplex $\\Delta_{n}: =\\{p\\in [0,1]^{n}:\\textstyle\\sum_{i=1}^{n}p_{i}=1\\}$ , and $\\begin{array}{r}{x_{1}: = \\frac{1}{n}\\vec{1},}\\end{array}$ where $\\vec{1}$ is the vector in $\\mathbb{R}^{n}$ with entries all set to 1. In this case, we have $\\sup_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})\\leq\\ln n$ while $\\begin{array}{r}{\\operatorname*{sup}_{z,x\\in\\mathcal{X}}D_{\\Phi}(z,x)= +\\infty}\\end{array}$ .\n\nTo conclude, throughout the paper we shall try to stick to the following naming convention: greek letters will denote scalars, lower-case roman letters will denote vectors, $\\mathrm{e.g.},x,y$ ,and capital calligraphic letters shall denote sets, $\\mathrm{e . g .}, \\mathcal{X}, \\mathcal{Y}$ . Any deviations from this convention (or the hat notation as in (1)) are intended to follow other conventions in the literature.\n\nAlgorithm 1 Pseudocode for both online mirror descent and dual averaging with adaptive   \nlearning rate given by $\\eta_{t}$ on iteration t. These methods differ only in how the iterate $\\hat{y}_{t + 1}$ is   \nupdated.   \nInput: $x_{1}\\in \\mathcal{X}\\cap \\mathcal{D}, \\eta : \\mathbb{N}\\to \\mathbb{R}_{> 0}$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$   \n[OMD update] $\\hat{y}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$   \n[DA update] $\\begin{array}{r}{\\hat{y}_{t + 1}= \\hat{x}_{1}- \\eta_{t}\\sum_{i \\leq t}\\hat{g}_{i}}\\end{array}$   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$   \nend for"}, {"name": "Algorithm 2: dual-stabilized OMD, eqs. (5)-(7)", "statement": "Algorithm 2 Dual-stabilized OMD (DS-OMD). The parameters $\\gamma_{t}$ control the amount of   \nstabilization.   \nInput: $x_{1}\\in\\mathcal{X}\\cap\\mathcal{D},\\eta:\\mathbb{N}\\to\\mathbb{R}_{+},\\ \\gamma:\\mathbb{N}\\to(0,1]$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$ map primal iterate to dual space   \n$\\hat{w}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$ gradient step in dual space (5)   \n$\\hat{y}_{t+1}=\\gamma_{t}\\hat{w}_{t+1}+(1-\\gamma_{t})\\hat{x}_{1}$ stabilization in dual space (6)   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$ map dual iterate to primal space   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$ project onto feasible region (7)   \nend for"}, {"name": "Proposition 38 (generalized triangle inequality)", "statement": "An immediate consequence is the \"generalized triangle inequality for Bregman divergence\", such as in Bubeck (2015, Eq. (4.1)) or in Beck and Teboulle (2003, Lemma 4.1).\n\nProposition 38 For all $a,b,d\\in{\\mathcal{D}}$\n\n$$\nD_{\\Phi}(a,d)-D_{\\Phi}(b,d)+D_{\\Phi}(b,a)\\ =\\ \\langle\\hat{a}-\\hat{d},a-b\\rangle\n$$"}, {"name": "Proposition 39", "statement": "Proposition 39 Let $a,b,c,u,v\\in\\mathbb{R}^{n}$ satisfy $\\gamma\\hat{a}+(1-\\gamma)\\hat{b}=\\hat{c}$ for some $\\gamma \\in \\mathbb{R}$ . Then\n\n$$\n\\gamma D_{\\Phi}({}_{v}^{u};a)\\ : +\\ :(1-\\gamma)D_{\\Phi}({}_{v}^{u};b)\\ : =\\ :D_{\\Phi}({}_{v}^{u};c).\n$$"}, {"name": "Proposition 40 (Pythagorean theorem for Bregman divergence)", "statement": "The following proposition is the \"Pythagorean theorem for Bregman divergence\". Recall that $\\Pi_{\\mathcal{X}}^{\\Phi}(y)=\\arg\\operatorname*{min}_{u\\in\\mathcal{X}}D_{\\Phi}(u,y)$ . A proof may be found in Bubeck (2015, Lemma 4.1).\n\nProposition 40 Let $\\mathcal{X}\\subset \\mathbb{R}^{n}$ be a convex set. Let $p \\in \\mathbb{R}^{n}$ and $\\pi=\\Pi_{\\mathcal{X}}^{\\Phi}(p)$ . Then\n\n$$\nD_{\\Phi}({}_{\\pi}^{z};p)\\geq D_{\\Phi}({}_{\\pi}^{z};\\pi)=D_{\\Phi}(z,\\pi)\\forall z\\in\\mathcal{X}.\n$$"}, {"name": "Lemma 34 (basic properties of the mirror map and Bregman divergence)", "statement": "The following lemma collects basic facts regarding the a mirror map Φ (or simply a function of Legendre type) and the Bregman divergence it induces. See Cesa-Bianchi and Lugosi (2006, Lemma 11.5 and Proposition 11.1).\n\nLemma 34 The mirror map $\\Phi$ and the Bregman divergence it induces satisfy the following properties:\n\n$D_{\\Phi}(x,y)$ is convex in x.\n\n$\\nabla\\Phi(\\nabla\\Phi^{*}(z))=z$ and $\\nabla^{*}\\Phi(\\nabla\\Phi(x))=x$ for all x and z.\n\n$D_{\\Phi}(x,y)=D_{\\Phi^{*}}\\bigl(\\nabla\\Phi(y),\\nabla\\Phi(x)\\bigr)$ for all x and $y$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\nnoncomputable def bregman (Φ : E → ℝ) (gΦ : E → E) (x y : E) : ℝ :=\n  Φ x - Φ y - (@inner ℝ E _ (gΦ y) (x - y))\n\nnoncomputable def regret (f : ℕ → E → ℝ) (x : ℕ → E) (z : E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 1 T, (f t (x t) - f t z)\n\nstructure PaperNorm (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] where\n  nrm : E → ℝ\n  nonneg : ∀ x, 0 ≤ nrm x\n  eq_zero_iff : ∀ x, nrm x = 0 ↔ x = 0\n  smul : ∀ (c : ℝ) (x : E), nrm (c • x) = |c| * nrm x\n  triangle : ∀ x y, nrm (x + y) ≤ nrm x + nrm y\n\nnoncomputable def dualNorm (N : PaperNorm E) (z : E) : ℝ :=\n  sSup {v : ℝ | ∃ x : E, N.nrm x ≤ 1 ∧ v = (@inner ℝ E _ z x)}\n\ntheorem theorem3\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hXclosed : IsClosed X)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hprojX : ∀ y : E, proj y ∈ X) (hprojD : ∀ y : E, proj y ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) =\n        proj (gΦs ((η (t + 1) / η t) • (gΦ (x t) - η t • g t)\n                    + (1 - η (t + 1) / η t) • gΦ (x 1))))\n\n    (hpyth : ∀ z ∈ X, ∀ y ∈ D,\n      bregman Φ gΦ z (proj y) ≤ bregman Φ gΦ z y - bregman Φ gΦ (proj y) y)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (x (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\ntheorem theorem5\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hy : ∀ t, 1 ≤ t → y (t + 1) = proj (w (t + 1)))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) = (η (t + 1) / η t) • y (t + 1) + (1 - η (t + 1) / η t) • x 1)\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n\n    (hDconvex : ∀ z ∈ X, ConvexOn ℝ (X ∩ D) (fun u => bregman Φ gΦ z u))\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (y (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\ntheorem theorem8\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hy1 : y 1 = x 1)\n\n    (hyupdate : ∀ t, 1 ≤ t →\n      y (t + 1) = gΦs (gΦ (x 1) - η (t + 1) • (∑ i ∈ Finset.Icc 1 t, g i)))\n\n    (hxupdate : ∀ t, 1 ≤ t → x (t + 1) = proj (y (t + 1)))\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (gΦs (gΦ (x t) - η t • g t))\n            - bregman Φ gΦ (x (t + 1)) (gΦs (gΦ (x t) - η t • g t))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem3", "theorem5", "theorem8"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 828, "end": 2491, "gold": "(Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hXclosed : IsClosed X)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hprojX : ∀ y : E, proj y ∈ X) (hprojD : ∀ y : E, proj y ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) =\n        proj (gΦs ((η (t + 1) / η t) • (gΦ (x t) - η t • g t)\n                    + (1 - η (t + 1) / η t) • gΦ (x 1))))\n\n    (hpyth : ∀ z ∈ X, ∀ y ∈ D,\n      bregman Φ gΦ z (proj y) ≤ bregman Φ gΦ z y - bregman Φ gΦ (proj y) y)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (x (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1)"}]}
{"item_id": "v23-21-1027::theorem5::A-stmt", "paper_id": "v23-21-1027", "target_name": "theorem5", "source": "Research-level", "nl_statement": "Theorem 5 (Regret bound for primal-stabilized OMD) Assume $\\eta_{t}\\geq \\eta_{t + 1}> 0$ for each $t > 1$ . Define $\\gamma_{t}=\\eta_{t+1}/\\eta_{t}\\in(0,1]$ for all $t \\geq 1$ . Let $\\{x_{t}\\}_{t \\ge 1}$ be the sequence of iterates generated by Algorithm 3. Furthermore, assume that\n\nfor all $z \\in \\mathcal{X}$ , the map $x\\mapsto D_{\\Phi}(z,x)$ is convex on $\\mathcal{X}.$\n\n(15)\n\nThen for any sequence of convex functions $\\{f_{t}\\}_{t \\ge 1}$ with each $f_{t}: \\mathcal{X}\\to \\mathbb{R}$ ,\n\n$$\n\\mathrm{Regret}(T,z)\\ \\leq\\ \\sum_{t=1}^{T}\\frac{D_{\\Phi}\\big({}_{y_{t+1}}^{x_{t}};w_{t+1}\\big)}{\\eta_{t}}+\\frac{D_{\\Phi}\\big(z,x_{1}\\big)}{\\eta_{T+1}}\\quad\\forall T>0 .\\tag{16}\n$$", "context": [{"name": "Setup: online convex optimization and regret (Section 3)", "statement": "We consider the online convex optimization problem with unknown time horizon. For each time step $t\\in\\{1,2,\\ldots\\}$ the algorithm proposes a point $x_{t}$ from a closed convex set ${\\mathcal{X}}\\subseteq \\mathbb{R}^{n}$ and an adversary simultaneously picks a convex cost function $f_{t}$ which the algorithm has access to via a first-order oracle, that is, for any $x \\in \\mathcal{X}$ the algorithm can compute $f_{t}(x)$ and a subgradient $\\hat{g}\\in\\partial f_{t}(x): =\\{\\hat{g}\\in\\mathbb{R}^{n}\\mid f_{t}(z)\\geq f_{t}(x)+\\langle\\hat{g},z-x\\rangle\\forall z\\in\\mathcal{X}\\}$ . We will assume1 that all cost functions $f_{t}$ in this text are subdifferentiable on $\\mathcal{X}$ , that is, meaning that the subdifferential $\\partial f_{t}(x)$ is non-empty for all $x \\in \\mathcal{X}$\n\nThe cost of the iteration at time t is defined as $f_{t}(x_{t})$ . In this setting the goal is to produce a sequence of proposals $\\{x_{t}\\}_{t \\ge 1}$ that minimizes the regret against an unknown comparison point $z \\in \\mathcal{X}$ that has accrued up until time $T$:\n\n$$\n\\mathrm{Regret}(T,z): =\\ \\sum_{t=1}^{T}f_{t}(x_{t})-\\sum_{t=1}^{T}f_{t}(z).\n$$"}, {"name": "Standing assumptions on the mirror map Φ and hat notation (1) (Section 3)", "statement": "Both dual averaging and online mirror descent are parameterized by a special convex function $\\Phi,$ often referred as a regularizer or a mirror map (for D and X), which among other properties needs$^{2}$ to be of Legendre type (Rockafellar, 1970, Chapter 26). Formally, throughout the paper we assume that the function $\\Phi\\colon \\bar{\\mathcal{D}}\\to \\mathbb{R}$ is a closed convex function whose conjugate is differentiable on $\\mathbb{R}^{n}$ and such that $\\bar{\\mathcal{D}}$ satisfies $\\mathcal{X}\\subseteq \\bar{\\mathcal{D}}$ and $\\mathrm{int}\\,\\bar{\\mathcal{D}}\\cap \\mathrm{ri}\\,\\mathcal{X}\\neq \\varnothing$ (where $\\mathrm{ri}\\,\\mathcal{X}$ denotes the relative interior of $\\mathcal{X}$). Moreover, we also suppose that $\\Phi$ is of Legendre type, which means that $\\Phi$ is strictly convex on its domain$^{3}$ and essentially smooth, that is, for $\\mathcal{D}: =\\mathrm{int}\\bar{\\mathcal{D}}$ we have\n\n$\\mathcal{D}$ is nonempty,\n\n$\\Phi$ is differentiable on $\\mathcal{D}$, and\n\n$\\lim_{x\\to\\partial\\mathcal{D}}\\|\\nabla\\Phi(x)\\|= +\\infty$ , where ∂D is the boundary of $\\mathcal{D},\\mathrm{i.e.,}\\partial\\mathcal{D}: =\\mathrm{cl}\\mathcal{D}\\backslash\\mathcal{D}$\n\nThe gradient of the mirror map $\\nabla \\Phi :{\\mathcal{D}}\\to \\mathbb{R}^{n}$ and the gradient of its conjugate $\\nabla \\Phi^{*}: \\mathbb{R}^{n}\\to \\mathcal{D}$ are mutually inverse bijections between the primal space $\\mathcal{D}$ and the dual space $\\mathbb{R}^{n}$ . We will adopt the following notational convention. Any vector in the primal space will be written without a hat, such as $x \\in \\mathcal{D}$ . The same letter with a hat, namely $\\hat{x}$, will denote the corresponding dual vector:\n\n$$\n{\\hat{x}}: =\\nabla\\Phi(x)\\qquad{\\mathrm{and}}\\qquad x: =\\nabla\\Phi^{*}({\\hat{x}})\\qquad{\\mathrm{f o r~a l l~l e t t e r s~}}x .\\tag{1}\n$$"}, {"name": "Essential smoothness and well-posedness of the Bregman projection (Section 3)", "statement": "Essential smoothness ensures not only that $\\Phi$ is differentiable on the interior of its domain, but also that the slope of $\\Phi$ increases arbitrarily fast near the boundary of its domain. The latter guarantees, at least intuitively, that the function is increasing near and in the direction of the boundary of its domain. This property is fundamental for mirror descent to be well-defined (although not essential for dual averaging) since it ensures that the Bregman projection onto $\\mathcal{X}$ is attained by a point on $\\mathcal{D}$ where Φ is differentiable (we give more details on this on Section 7), and uniqueness is a consequence of the strict convexity of Φ. Some mirror maps we shall look at are classical cases of the OCO literature such as the negative entropy $x \\in \\mathbb{R}_{+}^{n}\\mapsto \\sum_{i = 1}^{n} x_{i}\\ln x_{i}$ and the squared 2-norm ${\\frac{1}{2}}\\|\\cdot\\|_{2}^{2}$ , and details on the reasons they are mirror maps can be found in the works of Shalev-Shwartz (2012); Bubeck (2011) and Bubeck (2015) (in particular, Bubeck, 2011, Section 5.2 discusses the properties of functions of Legendre type and why requiring the conjugate of the mirror map to be differentiable on the whole space is not necessary for mirror descent to be well-defined if one restricts the gradient steps in the dual space in some way)."}, {"name": "Definition: Bregman divergence (2), difference notation (3) and Bregman projection", "statement": "Given a mirror map $\\Phi,$ the Bregman divergence with respect to $\\Phi$ is defined as\n\n$$\nD_{\\Phi}(x,y)\\ : =\\ \\Phi(x)-\\Phi(y)-\\langle\\nabla\\Phi(y),x-y\\rangle,\\qquad\\forall x\\in\\bar{\\mathcal{D}},\\forall y\\in\\mathcal{D}.\\tag{2}\n$$\n\nThroughout this paper it will be convenient to use the notation\n\n$$\nD_{\\Phi}({}_{b}^{a};c): =D_{\\Phi}(a,c)-D_{\\Phi}(b,c)=\\Phi(a)-\\Phi(b)-\\langle\\nabla\\Phi(c),a-b\\rangle .\\tag{3}\n$$\n\nIn the important special case where $\\begin{array}{r}{\\Phi(x)=\\frac{1}{2}\\|x\\|_{2}^{2}}\\end{array}$ , the Bregman divergence relates to the Euclidean distance, i.e., $\\begin{array}{r}{D_{\\Phi}(x,y)=\\frac 12\\|x-y\\|_{2}^{2}}\\end{array}$ When $\\begin{array}{r}{\\Phi(x)=\\sum_{i=1}^{n}x_{i}\\log x_{i}}\\end{array}$ , the Bregman divergence becomes the generalized Kullback-Leibler (KL) divergence. The projection operator induced by the Bregman divergence is $\\Pi_{\\mathcal{X}}^{\\Phi}$ given by $\\{\\Pi_{\\mathcal{X}}^{\\Phi}\\}(y): =\\arg\\min\\{D_{\\Phi}(x,y)\\mid x\\in\\mathcal{X}\\}$ for any $y \\in \\mathcal{X}\\cap \\mathcal{D}$"}, {"name": "Algorithm 1: generic OMD / DA template", "statement": "A general template for optimization in the mirror descent framework is shown in Algorithm 1. The two classical algorithms, online mirror descent and dual averaging, are incarnations of this, differing only in how the dual variable $\\hat{y}_{t}$ is updated.\n\nIn this work, for a given initial point $x_{1}\\in{\\mathcal{X}}$ of the player, we are interested in the case when $\\begin{array}{r}{\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded, which still allows $\\sup_{z,x\\in{\\mathcal{X}}}D_{\\Phi}(z,x)$ to be unbounded. In fact, in the Euclidean setting $\\begin{array}{r}{(\\mathrm{i . e .,}\\Phi=\\frac{1}{2}\\|\\cdot\\|_{2}^{2}),\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded if and only if the diameter of X given by $\\sup_{x,y\\in\\mathcal{X}}\\frac{1}{2}\\Vert x-y\\Vert_{2}^{2}$ is also bounded. However, for a general mirror map $\\Phi$, assuming $\\begin{array}{r}{\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded is strictly weaker than assuming $\\scriptstyle\\operatorname*{sup}_{x,y\\in{\\mathcal{X}}}D_{\\Phi}(x,y)$ is bounded. This is the case for the well-known experts' problem, where Φ is the negative entropy, $D_{\\Phi}$ is the KL-divergence, X is the unit simplex $\\Delta_{n}: =\\{p\\in [0,1]^{n}:\\textstyle\\sum_{i=1}^{n}p_{i}=1\\}$ , and $\\begin{array}{r}{x_{1}: = \\frac{1}{n}\\vec{1},}\\end{array}$ where $\\vec{1}$ is the vector in $\\mathbb{R}^{n}$ with entries all set to 1. In this case, we have $\\sup_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})\\leq\\ln n$ while $\\begin{array}{r}{\\operatorname*{sup}_{z,x\\in\\mathcal{X}}D_{\\Phi}(z,x)= +\\infty}\\end{array}$ .\n\nTo conclude, throughout the paper we shall try to stick to the following naming convention: greek letters will denote scalars, lower-case roman letters will denote vectors, $\\mathrm{e.g.},x,y$ ,and capital calligraphic letters shall denote sets, $\\mathrm{e . g .}, \\mathcal{X}, \\mathcal{Y}$ . Any deviations from this convention (or the hat notation as in (1)) are intended to follow other conventions in the literature.\n\nAlgorithm 1 Pseudocode for both online mirror descent and dual averaging with adaptive   \nlearning rate given by $\\eta_{t}$ on iteration t. These methods differ only in how the iterate $\\hat{y}_{t + 1}$ is   \nupdated.   \nInput: $x_{1}\\in \\mathcal{X}\\cap \\mathcal{D}, \\eta : \\mathbb{N}\\to \\mathbb{R}_{> 0}$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$   \n[OMD update] $\\hat{y}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$   \n[DA update] $\\begin{array}{r}{\\hat{y}_{t + 1}= \\hat{x}_{1}- \\eta_{t}\\sum_{i \\leq t}\\hat{g}_{i}}\\end{array}$   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$   \nend for"}, {"name": "Algorithm 2: dual-stabilized OMD, eqs. (5)-(7)", "statement": "Algorithm 2 Dual-stabilized OMD (DS-OMD). The parameters $\\gamma_{t}$ control the amount of   \nstabilization.   \nInput: $x_{1}\\in\\mathcal{X}\\cap\\mathcal{D},\\eta:\\mathbb{N}\\to\\mathbb{R}_{+},\\ \\gamma:\\mathbb{N}\\to(0,1]$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$ map primal iterate to dual space   \n$\\hat{w}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$ gradient step in dual space (5)   \n$\\hat{y}_{t+1}=\\gamma_{t}\\hat{w}_{t+1}+(1-\\gamma_{t})\\hat{x}_{1}$ stabilization in dual space (6)   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$ map dual iterate to primal space   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$ project onto feasible region (7)   \nend for"}, {"name": "Proposition 38 (generalized triangle inequality)", "statement": "An immediate consequence is the \"generalized triangle inequality for Bregman divergence\", such as in Bubeck (2015, Eq. (4.1)) or in Beck and Teboulle (2003, Lemma 4.1).\n\nProposition 38 For all $a,b,d\\in{\\mathcal{D}}$\n\n$$\nD_{\\Phi}(a,d)-D_{\\Phi}(b,d)+D_{\\Phi}(b,a)\\ =\\ \\langle\\hat{a}-\\hat{d},a-b\\rangle\n$$"}, {"name": "Proposition 40 (Pythagorean theorem for Bregman divergence)", "statement": "The following proposition is the \"Pythagorean theorem for Bregman divergence\". Recall that $\\Pi_{\\mathcal{X}}^{\\Phi}(y)=\\arg\\operatorname*{min}_{u\\in\\mathcal{X}}D_{\\Phi}(u,y)$ . A proof may be found in Bubeck (2015, Lemma 4.1).\n\nProposition 40 Let $\\mathcal{X}\\subset \\mathbb{R}^{n}$ be a convex set. Let $p \\in \\mathbb{R}^{n}$ and $\\pi=\\Pi_{\\mathcal{X}}^{\\Phi}(p)$ . Then\n\n$$\nD_{\\Phi}({}_{\\pi}^{z};p)\\geq D_{\\Phi}({}_{\\pi}^{z};\\pi)=D_{\\Phi}(z,\\pi)\\forall z\\in\\mathcal{X}.\n$$"}, {"name": "Algorithm 3: primal-stabilized OMD, eqs. (11)-(14)", "statement": "Algorithm 3 Online mirror descent with primal stabilization.   \nInput: $x_{1}\\in \\mathcal{X}\\cap \\mathcal{D}, \\eta : \\mathbb{N}\\to \\mathbb{R}, \\gamma : \\mathbb{N}\\to \\mathbb{R}.$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{\\boldsymbol g}_{t}\\in\\partial f_{t}(\\boldsymbol x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$ map primal iterate to dual space   \n$\\hat{w}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$ gradient step in dual space (11)   \n$\\boldsymbol{w}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{w}}_{t+1})$ map dual iterate to primal space (12)   \n$y_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(w_{t+1})$ project onto feasible region (13)   \n$x_{t+1}=\\gamma_{t}y_{t+1}+(1-\\gamma_{t})x_{1}$ stabilization in primal space (14)   \nend for"}, {"name": "Lemma 34 (basic properties of the mirror map and Bregman divergence)", "statement": "The following lemma collects basic facts regarding the a mirror map Φ (or simply a function of Legendre type) and the Bregman divergence it induces. See Cesa-Bianchi and Lugosi (2006, Lemma 11.5 and Proposition 11.1).\n\nLemma 34 The mirror map $\\Phi$ and the Bregman divergence it induces satisfy the following properties:\n\n$D_{\\Phi}(x,y)$ is convex in x.\n\n$\\nabla\\Phi(\\nabla\\Phi^{*}(z))=z$ and $\\nabla^{*}\\Phi(\\nabla\\Phi(x))=x$ for all x and z.\n\n$D_{\\Phi}(x,y)=D_{\\Phi^{*}}\\bigl(\\nabla\\Phi(y),\\nabla\\Phi(x)\\bigr)$ for all x and $y$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\nnoncomputable def bregman (Φ : E → ℝ) (gΦ : E → E) (x y : E) : ℝ :=\n  Φ x - Φ y - (@inner ℝ E _ (gΦ y) (x - y))\n\nnoncomputable def regret (f : ℕ → E → ℝ) (x : ℕ → E) (z : E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 1 T, (f t (x t) - f t z)\n\nstructure PaperNorm (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] where\n  nrm : E → ℝ\n  nonneg : ∀ x, 0 ≤ nrm x\n  eq_zero_iff : ∀ x, nrm x = 0 ↔ x = 0\n  smul : ∀ (c : ℝ) (x : E), nrm (c • x) = |c| * nrm x\n  triangle : ∀ x y, nrm (x + y) ≤ nrm x + nrm y\n\nnoncomputable def dualNorm (N : PaperNorm E) (z : E) : ℝ :=\n  sSup {v : ℝ | ∃ x : E, N.nrm x ≤ 1 ∧ v = (@inner ℝ E _ z x)}\n\ntheorem theorem3\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hXclosed : IsClosed X)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hprojX : ∀ y : E, proj y ∈ X) (hprojD : ∀ y : E, proj y ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) =\n        proj (gΦs ((η (t + 1) / η t) • (gΦ (x t) - η t • g t)\n                    + (1 - η (t + 1) / η t) • gΦ (x 1))))\n\n    (hpyth : ∀ z ∈ X, ∀ y ∈ D,\n      bregman Φ gΦ z (proj y) ≤ bregman Φ gΦ z y - bregman Φ gΦ (proj y) y)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (x (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\ntheorem theorem5\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hy : ∀ t, 1 ≤ t → y (t + 1) = proj (w (t + 1)))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) = (η (t + 1) / η t) • y (t + 1) + (1 - η (t + 1) / η t) • x 1)\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n\n    (hDconvex : ∀ z ∈ X, ConvexOn ℝ (X ∩ D) (fun u => bregman Φ gΦ z u))\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (y (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\ntheorem theorem8\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hy1 : y 1 = x 1)\n\n    (hyupdate : ∀ t, 1 ≤ t →\n      y (t + 1) = gΦs (gΦ (x 1) - η (t + 1) • (∑ i ∈ Finset.Icc 1 t, g i)))\n\n    (hxupdate : ∀ t, 1 ≤ t → x (t + 1) = proj (y (t + 1)))\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (gΦs (gΦ (x t) - η t • g t))\n            - bregman Φ gΦ (x (t + 1)) (gΦs (gΦ (x t) - η t • g t))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem3", "theorem5", "theorem8"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 2528, "end": 4262, "gold": "(Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hy : ∀ t, 1 ≤ t → y (t + 1) = proj (w (t + 1)))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) = (η (t + 1) / η t) • y (t + 1) + (1 - η (t + 1) / η t) • x 1)\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n\n    (hDconvex : ∀ z ∈ X, ConvexOn ℝ (X ∩ D) (fun u => bregman Φ gΦ z u))\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (y (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1)"}]}
{"item_id": "v23-21-1027::theorem8::A-stmt", "paper_id": "v23-21-1027", "target_name": "theorem8", "source": "Research-level", "nl_statement": "Theorem 8 (Regret bound for dual averaging) Assume we have $\\eta_{t}\\geq \\eta_{t + 1}> 0$ for each $t \\geq 1$ . Let $\\{x_{t}\\}_{t \\ge 1}$ be the sequence of iterates generated by Algorithm 4. Then for any sequence of convex functions $\\{f_{t}\\}_{t \\ge 1}$ with each $f_{t}: \\mathcal{X}\\to \\mathbb{R}$ ,\n\n$$\n\\mathrm{Regret}(T,z)~\\le~\\sum_{t=1}^{T}\\frac{D_{\\Phi}\\bigl({{}_{x_{t+1}}^{x_{t}};\\nabla\\Phi^{*}(\\hat{x}_{t}-\\eta_{t}\\hat{g}_{t})\\bigr)}}{\\eta_{t}}+\\frac{D_{\\Phi}\\bigl(z,x_{1}\\bigr)}{\\eta_{T+1}}~\\forall T>0 .\\tag{20}\n$$", "context": [{"name": "Setup: online convex optimization and regret (Section 3)", "statement": "We consider the online convex optimization problem with unknown time horizon. For each time step $t\\in\\{1,2,\\ldots\\}$ the algorithm proposes a point $x_{t}$ from a closed convex set ${\\mathcal{X}}\\subseteq \\mathbb{R}^{n}$ and an adversary simultaneously picks a convex cost function $f_{t}$ which the algorithm has access to via a first-order oracle, that is, for any $x \\in \\mathcal{X}$ the algorithm can compute $f_{t}(x)$ and a subgradient $\\hat{g}\\in\\partial f_{t}(x): =\\{\\hat{g}\\in\\mathbb{R}^{n}\\mid f_{t}(z)\\geq f_{t}(x)+\\langle\\hat{g},z-x\\rangle\\forall z\\in\\mathcal{X}\\}$ . We will assume1 that all cost functions $f_{t}$ in this text are subdifferentiable on $\\mathcal{X}$ , that is, meaning that the subdifferential $\\partial f_{t}(x)$ is non-empty for all $x \\in \\mathcal{X}$\n\nThe cost of the iteration at time t is defined as $f_{t}(x_{t})$ . In this setting the goal is to produce a sequence of proposals $\\{x_{t}\\}_{t \\ge 1}$ that minimizes the regret against an unknown comparison point $z \\in \\mathcal{X}$ that has accrued up until time $T$:\n\n$$\n\\mathrm{Regret}(T,z): =\\ \\sum_{t=1}^{T}f_{t}(x_{t})-\\sum_{t=1}^{T}f_{t}(z).\n$$"}, {"name": "Standing assumptions on the mirror map Φ and hat notation (1) (Section 3)", "statement": "Both dual averaging and online mirror descent are parameterized by a special convex function $\\Phi,$ often referred as a regularizer or a mirror map (for D and X), which among other properties needs$^{2}$ to be of Legendre type (Rockafellar, 1970, Chapter 26). Formally, throughout the paper we assume that the function $\\Phi\\colon \\bar{\\mathcal{D}}\\to \\mathbb{R}$ is a closed convex function whose conjugate is differentiable on $\\mathbb{R}^{n}$ and such that $\\bar{\\mathcal{D}}$ satisfies $\\mathcal{X}\\subseteq \\bar{\\mathcal{D}}$ and $\\mathrm{int}\\,\\bar{\\mathcal{D}}\\cap \\mathrm{ri}\\,\\mathcal{X}\\neq \\varnothing$ (where $\\mathrm{ri}\\,\\mathcal{X}$ denotes the relative interior of $\\mathcal{X}$). Moreover, we also suppose that $\\Phi$ is of Legendre type, which means that $\\Phi$ is strictly convex on its domain$^{3}$ and essentially smooth, that is, for $\\mathcal{D}: =\\mathrm{int}\\bar{\\mathcal{D}}$ we have\n\n$\\mathcal{D}$ is nonempty,\n\n$\\Phi$ is differentiable on $\\mathcal{D}$, and\n\n$\\lim_{x\\to\\partial\\mathcal{D}}\\|\\nabla\\Phi(x)\\|= +\\infty$ , where ∂D is the boundary of $\\mathcal{D},\\mathrm{i.e.,}\\partial\\mathcal{D}: =\\mathrm{cl}\\mathcal{D}\\backslash\\mathcal{D}$\n\nThe gradient of the mirror map $\\nabla \\Phi :{\\mathcal{D}}\\to \\mathbb{R}^{n}$ and the gradient of its conjugate $\\nabla \\Phi^{*}: \\mathbb{R}^{n}\\to \\mathcal{D}$ are mutually inverse bijections between the primal space $\\mathcal{D}$ and the dual space $\\mathbb{R}^{n}$ . We will adopt the following notational convention. Any vector in the primal space will be written without a hat, such as $x \\in \\mathcal{D}$ . The same letter with a hat, namely $\\hat{x}$, will denote the corresponding dual vector:\n\n$$\n{\\hat{x}}: =\\nabla\\Phi(x)\\qquad{\\mathrm{and}}\\qquad x: =\\nabla\\Phi^{*}({\\hat{x}})\\qquad{\\mathrm{f o r~a l l~l e t t e r s~}}x .\\tag{1}\n$$"}, {"name": "Essential smoothness and well-posedness of the Bregman projection (Section 3)", "statement": "Essential smoothness ensures not only that $\\Phi$ is differentiable on the interior of its domain, but also that the slope of $\\Phi$ increases arbitrarily fast near the boundary of its domain. The latter guarantees, at least intuitively, that the function is increasing near and in the direction of the boundary of its domain. This property is fundamental for mirror descent to be well-defined (although not essential for dual averaging) since it ensures that the Bregman projection onto $\\mathcal{X}$ is attained by a point on $\\mathcal{D}$ where Φ is differentiable (we give more details on this on Section 7), and uniqueness is a consequence of the strict convexity of Φ. Some mirror maps we shall look at are classical cases of the OCO literature such as the negative entropy $x \\in \\mathbb{R}_{+}^{n}\\mapsto \\sum_{i = 1}^{n} x_{i}\\ln x_{i}$ and the squared 2-norm ${\\frac{1}{2}}\\|\\cdot\\|_{2}^{2}$ , and details on the reasons they are mirror maps can be found in the works of Shalev-Shwartz (2012); Bubeck (2011) and Bubeck (2015) (in particular, Bubeck, 2011, Section 5.2 discusses the properties of functions of Legendre type and why requiring the conjugate of the mirror map to be differentiable on the whole space is not necessary for mirror descent to be well-defined if one restricts the gradient steps in the dual space in some way)."}, {"name": "Definition: Bregman divergence (2), difference notation (3) and Bregman projection", "statement": "Given a mirror map $\\Phi,$ the Bregman divergence with respect to $\\Phi$ is defined as\n\n$$\nD_{\\Phi}(x,y)\\ : =\\ \\Phi(x)-\\Phi(y)-\\langle\\nabla\\Phi(y),x-y\\rangle,\\qquad\\forall x\\in\\bar{\\mathcal{D}},\\forall y\\in\\mathcal{D}.\\tag{2}\n$$\n\nThroughout this paper it will be convenient to use the notation\n\n$$\nD_{\\Phi}({}_{b}^{a};c): =D_{\\Phi}(a,c)-D_{\\Phi}(b,c)=\\Phi(a)-\\Phi(b)-\\langle\\nabla\\Phi(c),a-b\\rangle .\\tag{3}\n$$\n\nIn the important special case where $\\begin{array}{r}{\\Phi(x)=\\frac{1}{2}\\|x\\|_{2}^{2}}\\end{array}$ , the Bregman divergence relates to the Euclidean distance, i.e., $\\begin{array}{r}{D_{\\Phi}(x,y)=\\frac 12\\|x-y\\|_{2}^{2}}\\end{array}$ When $\\begin{array}{r}{\\Phi(x)=\\sum_{i=1}^{n}x_{i}\\log x_{i}}\\end{array}$ , the Bregman divergence becomes the generalized Kullback-Leibler (KL) divergence. The projection operator induced by the Bregman divergence is $\\Pi_{\\mathcal{X}}^{\\Phi}$ given by $\\{\\Pi_{\\mathcal{X}}^{\\Phi}\\}(y): =\\arg\\min\\{D_{\\Phi}(x,y)\\mid x\\in\\mathcal{X}\\}$ for any $y \\in \\mathcal{X}\\cap \\mathcal{D}$"}, {"name": "Algorithm 1: generic OMD / DA template", "statement": "A general template for optimization in the mirror descent framework is shown in Algorithm 1. The two classical algorithms, online mirror descent and dual averaging, are incarnations of this, differing only in how the dual variable $\\hat{y}_{t}$ is updated.\n\nIn this work, for a given initial point $x_{1}\\in{\\mathcal{X}}$ of the player, we are interested in the case when $\\begin{array}{r}{\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded, which still allows $\\sup_{z,x\\in{\\mathcal{X}}}D_{\\Phi}(z,x)$ to be unbounded. In fact, in the Euclidean setting $\\begin{array}{r}{(\\mathrm{i . e .,}\\Phi=\\frac{1}{2}\\|\\cdot\\|_{2}^{2}),\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded if and only if the diameter of X given by $\\sup_{x,y\\in\\mathcal{X}}\\frac{1}{2}\\Vert x-y\\Vert_{2}^{2}$ is also bounded. However, for a general mirror map $\\Phi$, assuming $\\begin{array}{r}{\\operatorname*{sup}_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})}\\end{array}$ is bounded is strictly weaker than assuming $\\scriptstyle\\operatorname*{sup}_{x,y\\in{\\mathcal{X}}}D_{\\Phi}(x,y)$ is bounded. This is the case for the well-known experts' problem, where Φ is the negative entropy, $D_{\\Phi}$ is the KL-divergence, X is the unit simplex $\\Delta_{n}: =\\{p\\in [0,1]^{n}:\\textstyle\\sum_{i=1}^{n}p_{i}=1\\}$ , and $\\begin{array}{r}{x_{1}: = \\frac{1}{n}\\vec{1},}\\end{array}$ where $\\vec{1}$ is the vector in $\\mathbb{R}^{n}$ with entries all set to 1. In this case, we have $\\sup_{z\\in\\mathcal{X}}D_{\\Phi}(z,x_{1})\\leq\\ln n$ while $\\begin{array}{r}{\\operatorname*{sup}_{z,x\\in\\mathcal{X}}D_{\\Phi}(z,x)= +\\infty}\\end{array}$ .\n\nTo conclude, throughout the paper we shall try to stick to the following naming convention: greek letters will denote scalars, lower-case roman letters will denote vectors, $\\mathrm{e.g.},x,y$ ,and capital calligraphic letters shall denote sets, $\\mathrm{e . g .}, \\mathcal{X}, \\mathcal{Y}$ . Any deviations from this convention (or the hat notation as in (1)) are intended to follow other conventions in the literature.\n\nAlgorithm 1 Pseudocode for both online mirror descent and dual averaging with adaptive   \nlearning rate given by $\\eta_{t}$ on iteration t. These methods differ only in how the iterate $\\hat{y}_{t + 1}$ is   \nupdated.   \nInput: $x_{1}\\in \\mathcal{X}\\cap \\mathcal{D}, \\eta : \\mathbb{N}\\to \\mathbb{R}_{> 0}$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$   \n[OMD update] $\\hat{y}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$   \n[DA update] $\\begin{array}{r}{\\hat{y}_{t + 1}= \\hat{x}_{1}- \\eta_{t}\\sum_{i \\leq t}\\hat{g}_{i}}\\end{array}$   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$   \nend for"}, {"name": "Algorithm 2: dual-stabilized OMD, eqs. (5)-(7)", "statement": "Algorithm 2 Dual-stabilized OMD (DS-OMD). The parameters $\\gamma_{t}$ control the amount of   \nstabilization.   \nInput: $x_{1}\\in\\mathcal{X}\\cap\\mathcal{D},\\eta:\\mathbb{N}\\to\\mathbb{R}_{+},\\ \\gamma:\\mathbb{N}\\to(0,1]$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\hat{x}_{t}=\\nabla\\Phi(x_{t})$ map primal iterate to dual space   \n$\\hat{w}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}$ gradient step in dual space (5)   \n$\\hat{y}_{t+1}=\\gamma_{t}\\hat{w}_{t+1}+(1-\\gamma_{t})\\hat{x}_{1}$ stabilization in dual space (6)   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$ map dual iterate to primal space   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$ project onto feasible region (7)   \nend for"}, {"name": "Proposition 39", "statement": "Proposition 39 Let $a,b,c,u,v\\in\\mathbb{R}^{n}$ satisfy $\\gamma\\hat{a}+(1-\\gamma)\\hat{b}=\\hat{c}$ for some $\\gamma \\in \\mathbb{R}$ . Then\n\n$$\n\\gamma D_{\\Phi}({}_{v}^{u};a)\\ : +\\ :(1-\\gamma)D_{\\Phi}({}_{v}^{u};b)\\ : =\\ :D_{\\Phi}({}_{v}^{u};c).\n$$"}, {"name": "Proposition 40 (Pythagorean theorem for Bregman divergence)", "statement": "The following proposition is the \"Pythagorean theorem for Bregman divergence\". Recall that $\\Pi_{\\mathcal{X}}^{\\Phi}(y)=\\arg\\operatorname*{min}_{u\\in\\mathcal{X}}D_{\\Phi}(u,y)$ . A proof may be found in Bubeck (2015, Lemma 4.1).\n\nProposition 40 Let $\\mathcal{X}\\subset \\mathbb{R}^{n}$ be a convex set. Let $p \\in \\mathbb{R}^{n}$ and $\\pi=\\Pi_{\\mathcal{X}}^{\\Phi}(p)$ . Then\n\n$$\nD_{\\Phi}({}_{\\pi}^{z};p)\\geq D_{\\Phi}({}_{\\pi}^{z};\\pi)=D_{\\Phi}(z,\\pi)\\forall z\\in\\mathcal{X}.\n$$"}, {"name": "Setup: lazy gradient step (18), stabilization (19) and Claim 7 (Section 5.3)", "statement": "In this section, we show that Nesterov's dual averaging algorithm can be obtained from a small modification to dual-stabilized online mirror descent. Furthermore, our proof of Theorem 3 can be adapted to analyze DA.\n\nThe main difference between DS-OMD and dual averaging is in the gradient step, as we now explain. In iteration t of DS-OMD, the gradient step in (5) is taken from $\\hat{x}_{t}$ ,the dual counterpart of the iterate $x_{t}\\colon$\n\n$$\n\\mathrm{D S - O M D \\ g r a d i e n t \\ s t e p}: \\quad \\hat{w}_{t + 1}= \\hat{x}_{t}- \\eta_{t}\\hat{g}_{t}.\n$$\n\nSuppose that the algorithm is modified so that the gradient step is taken from $\\hat{y}_{t}$ , the dual point from iteration t before projection onto the feasible region. (Here $\\hat{y}_{1}$ is defined to be $\\hat{x}_{1}$.) The resulting gradient step is:\n\n$$\n\\mathrm{L a z y ~ g r a d i e n t ~ s t e p}\\colon \\hat{w}_{t + 1}= \\hat{y}_{t}- \\eta_{t}\\hat{g}_{t}.\\tag{18}\n$$\n\nAs before, we set\n\n$$\n\\hat{y}_{t+1}~=~\\gamma_{t}\\hat{w}_{t+1}+(1-\\gamma_{t})\\hat{x}_{1}\\tag{19}\n$$\n\nwhere $\\gamma_{t}= \\eta_{t + 1}/ \\eta_{t}$ . Then a simple inductive proof yields the following claim.\n\nClaim 7 $\\hat{w}_{t}= \\hat{x}_{1}- \\eta_{t - 1}\\sum_{i < t}\\hat{g}_{i}$ and $\\hat{y}_{t}= \\hat{x}_{1}- \\eta_{t}\\sum_{i < t}\\hat{g}_{i}$ for all $t > 1$.\n\nThus, the algorithm with the lazy gradient step can be written as in Algorithm 4. This is equivalent to Algorithm 1 with the DA update, except that $\\eta_{t}$ in Algorithm 1 corresponds to $\\eta_{t + 1}$ in Algorithm 4. In the next theorem we analyze DA with similar techniques to the ones used in Theorems 3 and 5."}, {"name": "Algorithm 4: dual averaging with re-indexed learning rate", "statement": "Algorithm 4 Dual averaging with learning rate re-indexed as $\\eta_{2}, \\eta_{3}, \\ldots$   \nInput: $x_{1}\\in\\mathcal{X}\\cap\\mathcal{D},~\\eta:\\mathbb{N}\\to\\mathbb{R}_{+},~\\gamma:\\mathbb{N}\\to(0,1]$   \n$\\hat{y}_{1}=\\nabla\\Phi(x_{1})$   \nfor $t=1,2,\\dots$ do   \nIncur cost $f_{t}(x_{t})$ and receive $\\hat{g}_{t}\\in\\partial f_{t}(x_{t})$   \n$\\begin{array}{r}{\\hat{y}_{t + 1}= \\hat{x}_{1}- \\eta_{t + 1}\\sum_{i \\leq t}\\hat{g}_{i}}\\end{array}$ dual averaging update   \n$\\boldsymbol{y}_{t+1}=\\nabla\\Phi^{*}(\\hat{\\boldsymbol{y}}_{t+1})$ map dual iterate to primal space   \n$x_{t+1}=\\Pi_{\\mathcal{X}}^{\\Phi}(y_{t+1})$ project onto feasible region   \nend for"}, {"name": "Proposition 36 (linearity of D_Φ(^a_b; p) in p̂)", "statement": "Recall that in (3) we defined the notation\n\n$$\nD_{\\Phi}({}_{b}^{a};c): =D_{\\Phi}(a,c)-D_{\\Phi}(b,c)=\\Phi(a)-\\Phi(b)-\\langle\\nabla\\Phi(c),a-b\\rangle .\n$$\n\nThis has several useful properties, which we now discuss.\n\nProposition 36 $D_{\\Phi}({}_{b}^{a}; p)$ is linear in $\\hat{p}$. In particular,\n\n$$\nD_{\\Phi}\\big({}_{b}^{a};\\nabla\\Phi^{*}(\\widehat{p}-\\widehat{q})\\big)\\ =\\ D_{\\Phi}\\big({}_{b}^{a};p\\big)+\\big\\langle\\widehat{q},a-b\\big\\rangle\\qquad\\forall\\widehat{q}\\in\\mathbb{R}^{n}.\n$$"}, {"name": "Proposition 37", "statement": "Proposition 37 For all $a,b,c,d\\in\\mathcal{D}$\n\n$$\nD_{\\Phi}\\big({}_{b}^{a};d\\big)-D_{\\Phi}\\big({}_{b}^{a};c\\big)=\\langle\\hat{c}-\\hat{d},a-b\\rangle=D_{\\Phi}\\big({}_{b}^{a};d\\big)+D_{\\Phi}\\big({}_{a}^{b};c\\big).\n$$"}, {"name": "Proposition 41", "statement": "A generalization of the previous proposition can be obtained by using linearity.\n\nProposition 41 Let $\\mathcal{X}\\subset \\mathbb{R}^{n}$ be a convex set. Let $p \\in \\mathbb{R}^{n}$ and $\\pi=\\Pi_{\\mathcal{X}}^{\\Phi}(p)$ . Then\n\n$$\nD_{\\Phi}\\big({}_{\\pi}^{v};\\nabla\\Phi^{*}({\\widehat{p}}-{\\widehat{q}})\\big)\\ \\geq\\ D_{\\Phi}\\big({}_{\\pi}^{v};\\nabla\\Phi^{*}({\\widehat{\\pi}}-{\\widehat{q}})\\big)\\qquad\\forall v\\in\\mathcal{X},{\\widehat{q}}\\in\\mathbb{R}^{n}.\n$$"}, {"name": "Lemma 34 (basic properties of the mirror map and Bregman divergence)", "statement": "The following lemma collects basic facts regarding the a mirror map Φ (or simply a function of Legendre type) and the Bregman divergence it induces. See Cesa-Bianchi and Lugosi (2006, Lemma 11.5 and Proposition 11.1).\n\nLemma 34 The mirror map $\\Phi$ and the Bregman divergence it induces satisfy the following properties:\n\n$D_{\\Phi}(x,y)$ is convex in x.\n\n$\\nabla\\Phi(\\nabla\\Phi^{*}(z))=z$ and $\\nabla^{*}\\Phi(\\nabla\\Phi(x))=x$ for all x and z.\n\n$D_{\\Phi}(x,y)=D_{\\Phi^{*}}\\bigl(\\nabla\\Phi(y),\\nabla\\Phi(x)\\bigr)$ for all x and $y$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\nnoncomputable def bregman (Φ : E → ℝ) (gΦ : E → E) (x y : E) : ℝ :=\n  Φ x - Φ y - (@inner ℝ E _ (gΦ y) (x - y))\n\nnoncomputable def regret (f : ℕ → E → ℝ) (x : ℕ → E) (z : E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 1 T, (f t (x t) - f t z)\n\nstructure PaperNorm (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] where\n  nrm : E → ℝ\n  nonneg : ∀ x, 0 ≤ nrm x\n  eq_zero_iff : ∀ x, nrm x = 0 ↔ x = 0\n  smul : ∀ (c : ℝ) (x : E), nrm (c • x) = |c| * nrm x\n  triangle : ∀ x y, nrm (x + y) ≤ nrm x + nrm y\n\nnoncomputable def dualNorm (N : PaperNorm E) (z : E) : ℝ :=\n  sSup {v : ℝ | ∃ x : E, N.nrm x ≤ 1 ∧ v = (@inner ℝ E _ z x)}\n\ntheorem theorem3\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hXclosed : IsClosed X)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hprojX : ∀ y : E, proj y ∈ X) (hprojD : ∀ y : E, proj y ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) =\n        proj (gΦs ((η (t + 1) / η t) • (gΦ (x t) - η t • g t)\n                    + (1 - η (t + 1) / η t) • gΦ (x 1))))\n\n    (hpyth : ∀ z ∈ X, ∀ y ∈ D,\n      bregman Φ gΦ z (proj y) ≤ bregman Φ gΦ z y - bregman Φ gΦ (proj y) y)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (x (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\ntheorem theorem5\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y w : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hw : ∀ t, 1 ≤ t → w (t + 1) = gΦs (gΦ (x t) - η t • g t))\n\n    (hy : ∀ t, 1 ≤ t → y (t + 1) = proj (w (t + 1)))\n\n    (hx : ∀ t, 1 ≤ t →\n      x (t + 1) = (η (t + 1) / η t) • y (t + 1) + (1 - η (t + 1) / η t) • x 1)\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n\n    (hDconvex : ∀ z ∈ X, ConvexOn ℝ (X ∩ D) (fun u => bregman Φ gΦ z u))\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (w (t + 1)) - bregman Φ gΦ (y (t + 1)) (w (t + 1))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\ntheorem theorem8\n    (Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hy1 : y 1 = x 1)\n\n    (hyupdate : ∀ t, 1 ≤ t →\n      y (t + 1) = gΦs (gΦ (x 1) - η (t + 1) • (∑ i ∈ Finset.Icc 1 t, g i)))\n\n    (hxupdate : ∀ t, 1 ≤ t → x (t + 1) = proj (y (t + 1)))\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (gΦs (gΦ (x t) - η t • g t))\n            - bregman Φ gΦ (x (t + 1)) (gΦs (gΦ (x t) - η t • g t))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem3", "theorem5", "theorem8"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 4299, "end": 5968, "gold": "(Φ : E → ℝ) (gΦ gΦs : E → E)\n    (X : Set E)\n    (D : Set E) (hDconv : Convex ℝ D)\n    (hXD : X ⊆ closure D)\n    (hDX : (D ∩ intrinsicInterior ℝ X).Nonempty)\n    (hΦstrict : StrictConvexOn ℝ D Φ)\n    (hΦgrad : ∀ a ∈ D, HasGradientWithinAt Φ (gΦ a) D a)\n    (η : ℕ → ℝ) (hηpos : ∀ t, 1 ≤ t → 0 < η t) (hηmono : ∀ t, 1 ≤ t → η (t + 1) ≤ η t)\n    (f : ℕ → E → ℝ)\n    (x y : ℕ → E) (g : ℕ → E)\n    (proj : E → E)\n\n    (hXconv : Convex ℝ X)\n\n    (hfsubdiff : ∀ t, 1 ≤ t → ∀ u ∈ X, ∃ gu : E, ∀ v ∈ X, f t v ≥ f t u + (@inner ℝ E _ gu (v - u)))\n\n    (hinv : ∀ v : E, gΦ (gΦs v) = v)\n    (hinvD : ∀ a ∈ D, gΦs (gΦ a) = a)\n\n    (hΦconv : ConvexOn ℝ X Φ)\n    (hΦsub : ∀ a ∈ D, ∀ b ∈ X, Φ b ≥ Φ a + (@inner ℝ E _ (gΦ a) (b - a)))\n    (hgΦsD : ∀ v : E, gΦs v ∈ D)\n\n    (hXclosed : IsClosed X)\n    (hprojX : ∀ v : E, proj v ∈ X) (hprojD : ∀ v : E, proj v ∈ D)\n\n    (hprojDef : ∀ v ∈ D, ∀ u ∈ X, bregman Φ gΦ (proj v) v ≤ bregman Φ gΦ u v)\n\n    (hfconv : ∀ t, 1 ≤ t → ConvexOn ℝ X (f t))\n    (hx1 : x 1 ∈ X) (hx1D : x 1 ∈ D)\n    (hsub : ∀ t, 1 ≤ t → ∀ u ∈ X, f t (x t) - f t u ≤ (@inner ℝ E _ (g t) (x t - u)))\n\n    (hy1 : y 1 = x 1)\n\n    (hyupdate : ∀ t, 1 ≤ t →\n      y (t + 1) = gΦs (gΦ (x 1) - η (t + 1) • (∑ i ∈ Finset.Icc 1 t, g i)))\n\n    (hxupdate : ∀ t, 1 ≤ t → x (t + 1) = proj (y (t + 1)))\n\n    (hpyth : ∀ z ∈ X, ∀ v ∈ D,\n      bregman Φ gΦ z (proj v) ≤ bregman Φ gΦ z v - bregman Φ gΦ (proj v) v)\n    (T : ℕ) (hT : 1 ≤ T) (z : E) (hz : z ∈ X) :\n    regret f x z T\n      ≤ (∑ t ∈ Finset.Icc 1 T,\n          (bregman Φ gΦ (x t) (gΦs (gΦ (x t) - η t • g t))\n            - bregman Φ gΦ (x (t + 1)) (gΦs (gΦ (x t) - η t • g t))) / η t)\n        + bregman Φ gΦ z (x 1) / η (T + 1)"}]}
{"item_id": "v24-19-980::lemma6::A-stmt", "paper_id": "v24-19-980", "target_name": "lemma6", "source": "Research-level", "nl_statement": "Lemma 6 For an MDP with rewards $r^{k}(s,a)$ and transition probability $\\tilde{P}(s^{\\prime}|s,a)$ such that $\\|\\tilde{P}(\\cdot|s,a)-P(\\cdot|s,a)\\|_{1}\\leq\\epsilon_{s,a},$ the Bellman error $B_{\\pi_{e}}^{\\tilde{P},k}(s,a)$ for state-action pair s, a is upper bounded as\n\n$$\n\\begin{array}{r}{B_{\\pi}^{\\tilde{P},k}(s,a)\\leq\\left\\|\\tilde{P}(\\cdot|s,a)-P(\\cdot|s,a)\\right\\|_{1}\\|h_{\\pi}^{\\tilde{P},k}(\\cdot)\\|_{\\infty},\\quad|B_{\\pi}^{\\tilde{P},k}(s,a)|\\leq\\tilde{D}\\operatorname*{min}\\left\\{\\epsilon_{s,a},2\\right\\},}\\end{array}\\tag{28}\n$$\n\nwhere $\\tilde{D}\\geq 0$ is an upper bound on the diameter of the MDP with transition probability $\\tilde{P}$ and on the span of the bias, $h_{\\pi}^{\\tilde{P},k}(s)-h_{\\pi}^{\\tilde{P},k}(s^{\\prime})\\leq\\tilde{D}$ for all $s,s^{\\prime}$", "context": [{"name": "Setup: MDP, rewards, policy and average reward (Section 3)", "statement": "We consider an infinite horizon discounted Markov decision process (MDP) $\\mathcal{M}$ defined by the tuple $\\left(\\mathcal{S},\\mathcal{A},P,r^{1},r^{2},\\cdot\\cdot\\cdot,r^{K},\\rho_{0}\\right)$ . $\\mathcal{S}$ denotes a finite set of state space of size S, and $\\mathcal{A}$ denotes a finite set of A actions. $P:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]^{S}$ denotes the probability transition distribution. K denotes the number of objectives and $[K]=\\{1,2,\\cdots,K\\}$ is the set of K objectives. Let $r^{k}:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ be the bounded reward function for objective $k \\in [ K ]$. Lastly, $\\rho_{0}:\\mathcal{S}\\to [0,1]$ is the distribution of initial state. We motivate our choice of bounds on rewards from the fact that many problems in practice require explicit reward shaping. Hence, the controller or the learner is aware of the bounds on the rewards. We consider the bounds to be [0, 1] for our case which is easy to satisfy by reward shaping.\n\nWe use a stochastic policy $\\pi:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ which returns the probability of selecting action $a \\in{\\mathcal{A}}$ for any given state $s \\in{\\mathcal{S}}$. Following policy π on the MDP, the agent observes a sequence of random variables $\\{S_{t}, A_{t}\\}_{t}$ where $S_{t}$ denotes the state of the agent at time t and $A_{t}$ denotes the action taken by the agent at time t. The expected discounted long term reward and expected per step reward of the objective k are given by $J_{\\pi}^{P,k}$ and $\\lambda_{\\pi}^{P,k}$ , respectively, when the joint policy π is followed. Formally, for discount factor $\\gamma\\in(0,1)$ , $J_{\\pi}^{P,k}$ and $\\lambda_{\\pi}^{P,k}$ are defined as\n\n$$\nJ_{\\pi}^{P,k}=\\mathbb{E}_{S_{0},A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\sum_{t=0}^{\\tau}\\gamma^{t}r^{k}(S_{t},A_{t})]\\tag{1}\n$$\n\n$$\nS_{0}\\sim\\rho_{0}(S_{0}),\\ A_{t}\\sim\\pi(A_{t}|S_{t}),\\ S_{t+1}\\sim P(\\cdot|S_{t},A_{t})\n$$\n\n$$\n\\lambda_{\\pi}^{P,k}=\\mathbb{E}_{S_{0},A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\frac{1}{\\tau}\\sum_{t=0}^{\\tau}r^{k}(S_{t},A_{t})]\\tag{2}\n$$\n\n$$\n=\\operatorname*{lim}_{\\gamma\\to 1}(1-\\gamma)J_{\\pi}^{P,k}\\tag{3}\n$$"}, {"name": "Bias Bellman equation, eq. (4)", "statement": "Equation (3) follows from the Laurent series expansion of $J_{\\pi}^{k}$ (Puterman, 1994). For brevity, in the rest of the paper $\\mathbb{E}_{S_{t}, A_{t}, S_{t + 1}; t \\geq 0}[ \\cdot ]$ will be denoted as $\\mathbb{E}_{\\rho,\\pi,P}[\\cdot]$ , where $S_{0}\\sim \\rho_{0}, \\ A_{t}\\sim$ $\\pi(\\cdot|S_{t}),\\ S_{t+1}\\sim P(\\cdot|S_{t},A_{t})$ . The expected per step reward satisfies the following Bellman equation\n\n$$\nh_{\\pi}^{P,k}(s)+\\lambda_{\\pi}^{P,k}=\\mathbb{E}_{a\\sim\\pi}\\left[r^{k}(s,a)\\right]+\\mathbb{E}_{a\\sim\\pi}\\left[\\sum_{s^{\\prime}}P(s^{\\prime}|s,a)h_{\\pi}^{P,k}(s^{\\prime})\\right]\\tag{4}\n$$\n\nwhere $h_{\\pi}^{k}(s)$ is the bias of policy π for state s."}, {"name": "Discounted value and Q-functions, eqs. (5)-(6)", "statement": "We also define the discounted value function $V_{\\gamma,\\pi}^{P,k}(s)$ and Q-value functions $Q_{\\gamma,\\pi}^{P,k}(s,a)$ as follows:\n\n$$\nV_{\\gamma,\\pi}^{P,k}(s)=\\mathbb{E}_{A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\sum_{t=0}^{\\tau}\\gamma^{t}r^{k}(S_{t},A_{t})|S_{0}=s]\\tag{5}\n$$\n\n$$\n\\begin{array}{l}{{Q_{\\gamma,\\pi}^{P,k}(s,a)=\\mathbb{E}_{S_{1},A_{1},\\cdots}[\\displaystyle\\operatorname*{lim}_{\\tau\\to\\infty}\\sum_{t=0}^{\\tau}\\gamma^{t}r^{k}(S_{t},A_{t})\\vert S_{0}=s,A_{0}=a]}}\\\\ {{\\mathrm{~}=r^{k}(s,a)+\\mathbb{E}_{a\\sim\\pi}[\\displaystyle\\sum_{s^{\\prime}}P(s^{\\prime}\\vert s,a)V_{\\gamma,\\pi}^{P,k}(s^{\\prime})]}}\\end{array}\\tag{6}\n$$"}, {"name": "Bias vs. discounted value, eq. (7)", "statement": "Further, the bias $h_{\\pi}^{P,k}(s)$ and the value function $V_{\\gamma,\\pi}^{P,k}$ are related as\n\n$$\nh_{\\pi}^{P,k}(s_{1})-h_{\\pi}^{P,k}(s_{2})=\\operatorname*{lim}_{\\gamma\\to 1}(V_{\\gamma,\\pi}^{P,k}(s_{1})-V_{\\gamma,\\pi}^{P,k}(s_{2})),{\\mathrm{~w h e r e~}}s_{1},s_{2}\\in{\\mathcal{S}}.\\tag{7}\n$$\n\nFor notation simplicity, we may drop the superscript P when discussing about variables for the true MDP."}, {"name": "Induced Markov chain and Definition 1 (Diameter), eq. (8)", "statement": "Note that each policy induces a Markov Chain on the states $\\mathcal{S}$ with transition probabilities $\\begin{array}{r}{P_{\\pi,s}(s^{\\prime})=\\sum_{a\\in\\mathcal{A}}\\pi(a|s)P(s^{\\prime}|s,a)}\\end{array}$ . After defining a policy, we can now define the diameter of the MDP $\\mathcal{M}$ as:\n\nDefinition 1 (Diameter) Consider the Markov Chain induced by the policy π on the MDP $\\mathcal{M}$. Let $T(s^{\\prime}|\\mathcal{M},\\pi,s)$ be a random variable that denotes the first time step when this Markov Chain enters state s' starting from state s. Then, the diameter of the MDP $\\mathcal{M}$ is defined as:\n\n$$\nD(\\mathcal{M})=\\operatorname*{max}_{\\pi}\\operatorname*{max}_{s^{\\prime}\\neq s}\\mathbb{E}\\left[T(s^{\\prime}|\\mathcal{M},\\pi,s)\\right]\\tag{8}\n$$"}, {"name": "Assumption 1 (ergodic MDP), eq. (9)", "statement": "Further, starting from an arbitrary initial state distribution, the state distribution may take a while to converge to the steady state distribution. For any policy π, let $P_{\\pi,s}^{t}=(P_{\\pi,s})^{t}$ be the t-step probability distribution of the states when policy π is applied to MDP with transition probabilities P starting from state s.\n\nWe are now ready to state our first assumption on the MDP. We assume that the Markov Decision Process is ergodic. This implies that: 1. for any policy all states, $s \\in{\\mathcal{S}}$ communicate with each other; 2. for any policy, the process converges to the steady state distribution exponentially fast. Formally, we have\n\nAssumption 1 The Markov Decision Process, $\\mathcal{M}$, is ergodic. Then, we have,\n\n1. The diameter, D, of $\\mathcal{M}$ is finite.\n\n2. For any policy π, for some $C > 0$ and $0 \\leq \\rho < 1$ , we have\n\n$$\n\\|P_{\\pi,s}^{t}-d_{\\pi}\\|_{TV}\\leq C\\rho^{t}\\tag{9}\n$$\n\nwhere $d_{\\pi}$ is the steady state distribution induced by policy π on the MDP."}, {"name": "Bellman error, eq. (26)", "statement": "To compute the regret incurred by the optimal policy $\\pi_{e}$ for the sampled MDP on the true MDP ${\\mathcal{M}},$ we use Bellman error. For some policy $\\pi,$ we define Bellman error $B_{\\pi}^{\\tilde{P},k}(s,a)$ for the infinite horizon MDPs as the difference between the cumulative expected rewards obtained for deviating from the system model with transition $\\tilde{P}$ for one step by taking action a in state s and then following policy π. We have:\n\n$$\n\\begin{array}{r l}&{B_{\\gamma,\\pi}^{\\tilde{P},k}(s,a)=\\left(Q_{\\gamma,\\pi}^{\\tilde{P},k}(s,a)-r(s,a)-\\gamma\\sum_{s^{\\prime}\\in\\mathcal{S}}P(s^{\\prime}|s,a)V_{\\gamma,\\pi}^{\\tilde{P},k}(s,a)\\right)}\\\\ &{B_{\\pi}^{\\tilde{P},k}(s,a)=\\underset{\\gamma\\to 1}{\\operatorname*{lim}}B_{\\gamma,\\pi}^{\\tilde{P},k}(s,a)}\\end{array}\\tag{26}\n$$"}, {"name": "Lemma 11 (another result of the same paper)", "statement": "Lemma 11 (Bounded Span of MDP) For an MDP $\\mathcal{M}$ with rewards $r^{k}(s,a)$ and transition probabilities P, for any stationary policy π with average reward $\\lambda_{\\pi}^{k}$ , the difference of bias of any two states $s,$ and $s^{\\prime},$ is upper bounded by the diameter of the MDP D as:\n\n$$\nh_{\\pi}^{P,k}(s)-h_{\\pi}^{P,k}(s^{\\prime})\\leq D~\\forall~s,s^{\\prime}\\in\\mathcal{S}.\\tag{102}\n$$"}], "nl_notes": "### Lemma 6\nNOTE (Lemma 6) — In the Lean, `bellmanError` is defined directly as the closed form Σ_{s'} (P̃ − P)(s'|s,a)·h(s') of the paper's eq. (134), so the γ → 1 limit of eq. (26) does not have to be derived. (The paper's eq. (6) omits the discount γ; its proof uses the γ-discounted Q_γ.) The goal is the first inequality of eq. (28), B ≤ ‖P̃ − P‖₁·‖h‖∞, together with |B| ≤ D̃·min{ε, 2}; the middle link ‖P̃ − P‖₁·‖h‖∞ ≤ D̃·min{ε, 2} is not part of it, because `h` is any solution of `hBellman` and need not be normalised (h + const is also a solution).\n\nThe statement also adds a hypothesis the paper does not state: the bias-span bound h(s) − h(s′) ≤ D̃ for all s, s′ (`hspan`), with D̃ ≥ 0 any upper bound on the diameter of P̃. The paper's proof uses this bound directly (it takes it from its Lemma 11), so the statement assumes it rather than requiring it to be derived.", "lean": "import Mathlib\n\nopen scoped BigOperators\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nvariable {S : Type*} {A : Type*}\n\nnoncomputable def bellmanError (Pt P : S → A → S → ℝ) (h : S → ℝ)\n    [Fintype S] (s : S) (a : A) : ℝ :=\n  ∑ s' : S, (Pt s a s' - P s a s') * h s'\n\nnoncomputable def transL1 (Pt P : S → A → S → ℝ)\n    [Fintype S] (s : S) (a : A) : ℝ :=\n  ∑ s' : S, |Pt s a s' - P s a s'|\n\nnoncomputable def biasSupNorm [Fintype S] [Nonempty S] (h : S → ℝ) : ℝ :=\n  Finset.univ.sup' Finset.univ_nonempty (fun s' => |h s'|)\n\nnoncomputable def hitTime {S Ω : Type*} (X : ℕ → Ω → S) (s' : S) (ω : Ω) : ℕ :=\n  sInf {t : ℕ | 1 ≤ t ∧ X t ω = s'}\n\ndef IsDiameterBoundOfChain {S Ω : Type*} [Fintype S] [MeasurableSpace Ω]\n    (μ : Measure Ω) (X : S → ℕ → Ω → S) (D : ℝ) : Prop :=\n  ∀ s s' : S, s ≠ s' →\n    (∀ᵐ ω ∂μ, ∃ t : ℕ, 1 ≤ t ∧ X s t ω = s') ∧\n    Integrable (fun ω => (hitTime (X s) s' ω : ℝ)) μ ∧\n    ∫ ω, (hitTime (X s) s' ω : ℝ) ∂μ ≤ D\n\ndef IsDiameterUpperBound (Ω : Type*) {S A : Type*} [Fintype S] [MeasurableSpace S]\n    [Fintype A] [MeasurableSpace Ω] (P : S → A → S → ℝ) (D : ℝ) : Prop :=\n  ∀ (μ₂ : Measure Ω), IsProbabilityMeasure μ₂ →\n  ∀ π₂ : S → A → ℝ, (∀ s, (∀ a, 0 ≤ π₂ s a) ∧ ∑ a, π₂ s a = 1) →\n  ∀ (X₂ : S → ℕ → Ω → S) (F₂ : ℕ → MeasurableSpace Ω),\n    (∀ s₀ t, Measurable (X₂ s₀ t)) →\n    (∀ s₀ : S, μ₂ {ω | X₂ s₀ 1 ω = s₀} = 1) →\n    Monotone F₂ → (∀ t, F₂ t ≤ (inferInstance : MeasurableSpace Ω)) →\n    (∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F₂ t'] (X₂ s₀ t)) →\n    (∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F₂ t] H →\n        ∀ x y : S, μ₂ (H ∩ {ω | X₂ s₀ t ω = x}) ≠ 0 →\n        μ₂ (H ∩ {ω | X₂ s₀ t ω = x} ∩ {ω | X₂ s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (∑ a : A, π₂ x a * P x a y)\n              * μ₂ (H ∩ {ω | X₂ s₀ t ω = x})) →\n    IsDiameterBoundOfChain μ₂ X₂ D\n\ndef IsDiameterBound (Ω : Type*) {S A : Type*} [Fintype S] [MeasurableSpace S]\n    [Fintype A] [MeasurableSpace Ω] (P : S → A → S → ℝ) (D : ℝ) : Prop :=\n  IsDiameterUpperBound Ω P D ∧ ∀ D' : ℝ, IsDiameterUpperBound Ω P D' → D ≤ D'\n\n\ntheorem lemma6 {Ω : Type*} [MeasurableSpace Ω] [Fintype S] [Nonempty S] [DecidableEq S] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    (Pt P : S → A → S → ℝ) (h : S → ℝ) (s : S) (a : A) (Dt ε : ℝ)\n    (hD : 0 ≤ Dt)\n    (hP_nonneg : ∀ s', 0 ≤ P s a s') (hP_sum : ∑ s', P s a s' = 1)\n    (hPt_nonneg : ∀ s', 0 ≤ Pt s a s') (hPt_sum : ∑ s', Pt s a s' = 1)\n    (hP_prob : ∀ x b, (∀ y, 0 ≤ P x b y) ∧ ∑ y, P x b y = 1)\n    (hPt_prob : ∀ x b, (∀ y, 0 ≤ Pt x b y) ∧ ∑ y, Pt x b y = 1)\n    (hε : transL1 Pt P s a ≤ ε)\n    (hspan : ∀ x y : S, h x - h y ≤ Dt)\n    (hDiam : IsDiameterUpperBound Ω Pt Dt)\n    (π : S → A → ℝ) (hπ_prob : ∀ x, (∀ a', 0 ≤ π x a') ∧ ∑ a', π x a' = 1)\n    (r : S → A → ℝ) (hr_bound : ∀ x b, r x b ∈ Set.Icc (0:ℝ) 1)\n    (Pπ : S → S → ℝ) (hPπ_def : ∀ x y, Pπ x y = ∑ a' : A, π x a' * Pt x a' y)\n    (hPπ_prob : ∀ x, (∀ y, 0 ≤ Pπ x y) ∧ ∑ y, Pπ x y = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ x, rπ x = ∑ a' : A, π x a' * r x a')\n    (lam : ℝ)\n    (hBellman : ∀ x, h x = rπ x - lam + ∑ y, Pπ x y * h y)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y) :\n    bellmanError Pt P h s a ≤ transL1 Pt P s a * biasSupNorm h\n      ∧ |bellmanError Pt P h s a| ≤ Dt * min ε 2 := by\n  sorry\n\ntheorem lemma11 {S A Ω : Type*} [Fintype S] [Nonempty S] [DecidableEq S]\n    [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (r : S → A → ℝ) (hr_bound : ∀ s a, r s a ∈ Set.Icc (0 : ℝ) 1)\n    (P : S → A → S → ℝ) (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (π : S → A → ℝ)\n    (hπ_prob : ∀ s, (∀ a, 0 ≤ π s a) ∧ ∑ a, π s a = 1)\n    (hPπ : S → S → ℝ) (hPπ_def : ∀ s s', hPπ s s' = ∑ a : A, π s a * P s a s')\n    (hPπ_prob : ∀ s, (∀ s', 0 ≤ hPπ s s') ∧ ∑ s', hPπ s s' = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ s, rπ s = ∑ a : A, π s a * r s a)\n    (lamPi : ℝ) (h : S → ℝ)\n    (hBellman : ∀ s, h s = rπ s - lamPi + ∑ s', hPπ s s' * h s')\n    (D : ℝ) (hD0 : 0 ≤ D)\n    (X : S → ℕ → Ω → S) (hXmeas : ∀ s₀ t, Measurable (X s₀ t))\n    (hXstart : ∀ s₀ : S, μ {ω | X s₀ 1 ω = s₀} = 1)\n    (F : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone F) (hFle : ∀ t, F t ≤ (inferInstance : MeasurableSpace Ω))\n    (hXAdapted : ∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F t'] (X s₀ t))\n    (hMarkov : ∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F t] H →\n        ∀ x y : S, μ (H ∩ {ω | X s₀ t ω = x}) ≠ 0 →\n        μ (H ∩ {ω | X s₀ t ω = x} ∩ {ω | X s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (hPπ x y) * μ (H ∩ {ω | X s₀ t ω = x}))\n    (hD : IsDiameterUpperBound Ω P D)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (s s' : S) :\n    h s - h s' ≤ D := by\n  sorry\n\nnoncomputable def visitCount {S A Ω : Type*} [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (t : ℕ) (s : S) (a : A) (ω : Ω) : ℕ :=\n  ∑ t' ∈ Finset.Icc 1 t, if Sp t' ω = s ∧ Ap t' ω = a then 1 else 0\n\nnoncomputable def empTrans {S A Ω : Type*} [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (t : ℕ) (s : S) (a : A) (s' : S) (ω : Ω) : ℝ :=\n  (∑ t' ∈ Finset.Icc 1 t, if Sp t' ω = s ∧ Ap t' ω = a ∧ Sp (t' + 1) ω = s'\n      then (1 : ℝ) else 0) / (visitCount Sp Ap t s a ω : ℝ)\n\ndef goodEvent {S A Ω : Type*} [Fintype S] [Fintype A] [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (P : S → A → S → ℝ) (t T : ℕ) (ω : Ω) : Prop :=\n  ∀ s : S, ∀ a : A,\n    transL1 (fun s a s' => empTrans Sp Ap t s a s' ω) P s a\n      ≤ Real.sqrt (14 * (Fintype.card S : ℝ) * Real.log (2 * (Fintype.card A : ℝ) * T)\n          / max 1 (visitCount Sp Ap t s a ω : ℝ))\n\ntheorem lemma7 {S A Ω : Type*} [Fintype S] [Nonempty S] [Fintype A] [Nonempty A]\n    [DecidableEq S] [DecidableEq A] [MeasurableSpace S] [MeasurableSpace A]\n    [MeasurableSingletonClass S] [MeasurableSingletonClass A]\n    [MeasurableSpace Ω]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A)\n    (hSpmeas : ∀ t', Measurable (Sp t')) (hApmeas : ∀ t', Measurable (Ap t'))\n    (ℱ : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone ℱ) (hFle : ∀ t', ℱ t' ≤ (inferInstance : MeasurableSpace Ω))\n    (hSpAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Sp t'))\n    (hApAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Ap t'))\n    (P : S → A → S → ℝ)\n    (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y)\n    (hMarkov : ∀ t' : ℕ, ∀ H : Set Ω, MeasurableSet[ℱ t'] H →\n        ∀ x : S, ∀ y1 : A, ∀ y2 : S,\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}) ≠ 0 →\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1} ∩ {ω | Sp (t' + 1) ω = y2})\n          = ENNReal.ofReal (P x y1 y2) * μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}))\n    (t T : ℕ) (ht : 1 ≤ t) (hT : t ≤ T) :\n    μ {ω | ¬ goodEvent Sp Ap P t T ω} ≤ ENNReal.ofReal (1 / (20 * (t : ℝ) ^ 6)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma6", "lemma11", "lemma7"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "holes": [{"start": 2060, "end": 3636, "gold": "{Ω : Type*} [MeasurableSpace Ω] [Fintype S] [Nonempty S] [DecidableEq S] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    (Pt P : S → A → S → ℝ) (h : S → ℝ) (s : S) (a : A) (Dt ε : ℝ)\n    (hD : 0 ≤ Dt)\n    (hP_nonneg : ∀ s', 0 ≤ P s a s') (hP_sum : ∑ s', P s a s' = 1)\n    (hPt_nonneg : ∀ s', 0 ≤ Pt s a s') (hPt_sum : ∑ s', Pt s a s' = 1)\n    (hP_prob : ∀ x b, (∀ y, 0 ≤ P x b y) ∧ ∑ y, P x b y = 1)\n    (hPt_prob : ∀ x b, (∀ y, 0 ≤ Pt x b y) ∧ ∑ y, Pt x b y = 1)\n    (hε : transL1 Pt P s a ≤ ε)\n    (hspan : ∀ x y : S, h x - h y ≤ Dt)\n    (hDiam : IsDiameterUpperBound Ω Pt Dt)\n    (π : S → A → ℝ) (hπ_prob : ∀ x, (∀ a', 0 ≤ π x a') ∧ ∑ a', π x a' = 1)\n    (r : S → A → ℝ) (hr_bound : ∀ x b, r x b ∈ Set.Icc (0:ℝ) 1)\n    (Pπ : S → S → ℝ) (hPπ_def : ∀ x y, Pπ x y = ∑ a' : A, π x a' * Pt x a' y)\n    (hPπ_prob : ∀ x, (∀ y, 0 ≤ Pπ x y) ∧ ∑ y, Pπ x y = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ x, rπ x = ∑ a' : A, π x a' * r x a')\n    (lam : ℝ)\n    (hBellman : ∀ x, h x = rπ x - lam + ∑ y, Pπ x y * h y)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y) :\n    bellmanError Pt P h s a ≤ transL1 Pt P s a * biasSupNorm h\n      ∧ |bellmanError Pt P h s a| ≤ Dt * min ε 2"}]}
{"item_id": "v24-19-980::lemma11::A-stmt", "paper_id": "v24-19-980", "target_name": "lemma11", "source": "Research-level", "nl_statement": "Lemma 11 (Bounded Span of MDP) For an MDP $\\mathcal{M}$ with rewards $r^{k}(s,a)$ and transition probabilities P, for any stationary policy π with average reward $\\lambda_{\\pi}^{k}$ , the difference of bias of any two states $s,$ and $s^{\\prime},$ is upper bounded by the diameter of the MDP D as:\n\n$$\nh_{\\pi}^{P,k}(s)-h_{\\pi}^{P,k}(s^{\\prime})\\leq D~\\forall~s,s^{\\prime}\\in\\mathcal{S}.\\tag{102}\n$$", "context": [{"name": "Setup: MDP, rewards, policy and average reward (Section 3)", "statement": "We consider an infinite horizon discounted Markov decision process (MDP) $\\mathcal{M}$ defined by the tuple $\\left(\\mathcal{S},\\mathcal{A},P,r^{1},r^{2},\\cdot\\cdot\\cdot,r^{K},\\rho_{0}\\right)$ . $\\mathcal{S}$ denotes a finite set of state space of size S, and $\\mathcal{A}$ denotes a finite set of A actions. $P:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]^{S}$ denotes the probability transition distribution. K denotes the number of objectives and $[K]=\\{1,2,\\cdots,K\\}$ is the set of K objectives. Let $r^{k}:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ be the bounded reward function for objective $k \\in [ K ]$. Lastly, $\\rho_{0}:\\mathcal{S}\\to [0,1]$ is the distribution of initial state. We motivate our choice of bounds on rewards from the fact that many problems in practice require explicit reward shaping. Hence, the controller or the learner is aware of the bounds on the rewards. We consider the bounds to be [0, 1] for our case which is easy to satisfy by reward shaping.\n\nWe use a stochastic policy $\\pi:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ which returns the probability of selecting action $a \\in{\\mathcal{A}}$ for any given state $s \\in{\\mathcal{S}}$. Following policy π on the MDP, the agent observes a sequence of random variables $\\{S_{t}, A_{t}\\}_{t}$ where $S_{t}$ denotes the state of the agent at time t and $A_{t}$ denotes the action taken by the agent at time t. The expected discounted long term reward and expected per step reward of the objective k are given by $J_{\\pi}^{P,k}$ and $\\lambda_{\\pi}^{P,k}$ , respectively, when the joint policy π is followed. Formally, for discount factor $\\gamma\\in(0,1)$ , $J_{\\pi}^{P,k}$ and $\\lambda_{\\pi}^{P,k}$ are defined as\n\n$$\nJ_{\\pi}^{P,k}=\\mathbb{E}_{S_{0},A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\sum_{t=0}^{\\tau}\\gamma^{t}r^{k}(S_{t},A_{t})]\\tag{1}\n$$\n\n$$\nS_{0}\\sim\\rho_{0}(S_{0}),\\ A_{t}\\sim\\pi(A_{t}|S_{t}),\\ S_{t+1}\\sim P(\\cdot|S_{t},A_{t})\n$$\n\n$$\n\\lambda_{\\pi}^{P,k}=\\mathbb{E}_{S_{0},A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\frac{1}{\\tau}\\sum_{t=0}^{\\tau}r^{k}(S_{t},A_{t})]\\tag{2}\n$$\n\n$$\n=\\operatorname*{lim}_{\\gamma\\to 1}(1-\\gamma)J_{\\pi}^{P,k}\\tag{3}\n$$"}, {"name": "Bias Bellman equation, eq. (4)", "statement": "Equation (3) follows from the Laurent series expansion of $J_{\\pi}^{k}$ (Puterman, 1994). For brevity, in the rest of the paper $\\mathbb{E}_{S_{t}, A_{t}, S_{t + 1}; t \\geq 0}[ \\cdot ]$ will be denoted as $\\mathbb{E}_{\\rho,\\pi,P}[\\cdot]$ , where $S_{0}\\sim \\rho_{0}, \\ A_{t}\\sim$ $\\pi(\\cdot|S_{t}),\\ S_{t+1}\\sim P(\\cdot|S_{t},A_{t})$ . The expected per step reward satisfies the following Bellman equation\n\n$$\nh_{\\pi}^{P,k}(s)+\\lambda_{\\pi}^{P,k}=\\mathbb{E}_{a\\sim\\pi}\\left[r^{k}(s,a)\\right]+\\mathbb{E}_{a\\sim\\pi}\\left[\\sum_{s^{\\prime}}P(s^{\\prime}|s,a)h_{\\pi}^{P,k}(s^{\\prime})\\right]\\tag{4}\n$$\n\nwhere $h_{\\pi}^{k}(s)$ is the bias of policy π for state s."}, {"name": "Induced Markov chain and Definition 1 (Diameter), eq. (8)", "statement": "Note that each policy induces a Markov Chain on the states $\\mathcal{S}$ with transition probabilities $\\begin{array}{r}{P_{\\pi,s}(s^{\\prime})=\\sum_{a\\in\\mathcal{A}}\\pi(a|s)P(s^{\\prime}|s,a)}\\end{array}$ . After defining a policy, we can now define the diameter of the MDP $\\mathcal{M}$ as:\n\nDefinition 1 (Diameter) Consider the Markov Chain induced by the policy π on the MDP $\\mathcal{M}$. Let $T(s^{\\prime}|\\mathcal{M},\\pi,s)$ be a random variable that denotes the first time step when this Markov Chain enters state s' starting from state s. Then, the diameter of the MDP $\\mathcal{M}$ is defined as:\n\n$$\nD(\\mathcal{M})=\\operatorname*{max}_{\\pi}\\operatorname*{max}_{s^{\\prime}\\neq s}\\mathbb{E}\\left[T(s^{\\prime}|\\mathcal{M},\\pi,s)\\right]\\tag{8}\n$$"}, {"name": "Assumption 1 (ergodic MDP), eq. (9)", "statement": "Further, starting from an arbitrary initial state distribution, the state distribution may take a while to converge to the steady state distribution. For any policy π, let $P_{\\pi,s}^{t}=(P_{\\pi,s})^{t}$ be the t-step probability distribution of the states when policy π is applied to MDP with transition probabilities P starting from state s.\n\nWe are now ready to state our first assumption on the MDP. We assume that the Markov Decision Process is ergodic. This implies that: 1. for any policy all states, $s \\in{\\mathcal{S}}$ communicate with each other; 2. for any policy, the process converges to the steady state distribution exponentially fast. Formally, we have\n\nAssumption 1 The Markov Decision Process, $\\mathcal{M}$, is ergodic. Then, we have,\n\n1. The diameter, D, of $\\mathcal{M}$ is finite.\n\n2. For any policy π, for some $C > 0$ and $0 \\leq \\rho < 1$ , we have\n\n$$\n\\|P_{\\pi,s}^{t}-d_{\\pi}\\|_{TV}\\leq C\\rho^{t}\\tag{9}\n$$\n\nwhere $d_{\\pi}$ is the steady state distribution induced by policy π on the MDP."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nvariable {S : Type*} {A : Type*}\n\nnoncomputable def bellmanError (Pt P : S → A → S → ℝ) (h : S → ℝ)\n    [Fintype S] (s : S) (a : A) : ℝ :=\n  ∑ s' : S, (Pt s a s' - P s a s') * h s'\n\nnoncomputable def transL1 (Pt P : S → A → S → ℝ)\n    [Fintype S] (s : S) (a : A) : ℝ :=\n  ∑ s' : S, |Pt s a s' - P s a s'|\n\nnoncomputable def biasSupNorm [Fintype S] [Nonempty S] (h : S → ℝ) : ℝ :=\n  Finset.univ.sup' Finset.univ_nonempty (fun s' => |h s'|)\n\nnoncomputable def hitTime {S Ω : Type*} (X : ℕ → Ω → S) (s' : S) (ω : Ω) : ℕ :=\n  sInf {t : ℕ | 1 ≤ t ∧ X t ω = s'}\n\ndef IsDiameterBoundOfChain {S Ω : Type*} [Fintype S] [MeasurableSpace Ω]\n    (μ : Measure Ω) (X : S → ℕ → Ω → S) (D : ℝ) : Prop :=\n  ∀ s s' : S, s ≠ s' →\n    (∀ᵐ ω ∂μ, ∃ t : ℕ, 1 ≤ t ∧ X s t ω = s') ∧\n    Integrable (fun ω => (hitTime (X s) s' ω : ℝ)) μ ∧\n    ∫ ω, (hitTime (X s) s' ω : ℝ) ∂μ ≤ D\n\ndef IsDiameterUpperBound (Ω : Type*) {S A : Type*} [Fintype S] [MeasurableSpace S]\n    [Fintype A] [MeasurableSpace Ω] (P : S → A → S → ℝ) (D : ℝ) : Prop :=\n  ∀ (μ₂ : Measure Ω), IsProbabilityMeasure μ₂ →\n  ∀ π₂ : S → A → ℝ, (∀ s, (∀ a, 0 ≤ π₂ s a) ∧ ∑ a, π₂ s a = 1) →\n  ∀ (X₂ : S → ℕ → Ω → S) (F₂ : ℕ → MeasurableSpace Ω),\n    (∀ s₀ t, Measurable (X₂ s₀ t)) →\n    (∀ s₀ : S, μ₂ {ω | X₂ s₀ 1 ω = s₀} = 1) →\n    Monotone F₂ → (∀ t, F₂ t ≤ (inferInstance : MeasurableSpace Ω)) →\n    (∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F₂ t'] (X₂ s₀ t)) →\n    (∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F₂ t] H →\n        ∀ x y : S, μ₂ (H ∩ {ω | X₂ s₀ t ω = x}) ≠ 0 →\n        μ₂ (H ∩ {ω | X₂ s₀ t ω = x} ∩ {ω | X₂ s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (∑ a : A, π₂ x a * P x a y)\n              * μ₂ (H ∩ {ω | X₂ s₀ t ω = x})) →\n    IsDiameterBoundOfChain μ₂ X₂ D\n\ndef IsDiameterBound (Ω : Type*) {S A : Type*} [Fintype S] [MeasurableSpace S]\n    [Fintype A] [MeasurableSpace Ω] (P : S → A → S → ℝ) (D : ℝ) : Prop :=\n  IsDiameterUpperBound Ω P D ∧ ∀ D' : ℝ, IsDiameterUpperBound Ω P D' → D ≤ D'\n\n\ntheorem lemma6 {Ω : Type*} [MeasurableSpace Ω] [Fintype S] [Nonempty S] [DecidableEq S] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    (Pt P : S → A → S → ℝ) (h : S → ℝ) (s : S) (a : A) (Dt ε : ℝ)\n    (hD : 0 ≤ Dt)\n    (hP_nonneg : ∀ s', 0 ≤ P s a s') (hP_sum : ∑ s', P s a s' = 1)\n    (hPt_nonneg : ∀ s', 0 ≤ Pt s a s') (hPt_sum : ∑ s', Pt s a s' = 1)\n    (hP_prob : ∀ x b, (∀ y, 0 ≤ P x b y) ∧ ∑ y, P x b y = 1)\n    (hPt_prob : ∀ x b, (∀ y, 0 ≤ Pt x b y) ∧ ∑ y, Pt x b y = 1)\n    (hε : transL1 Pt P s a ≤ ε)\n    (hspan : ∀ x y : S, h x - h y ≤ Dt)\n    (hDiam : IsDiameterUpperBound Ω Pt Dt)\n    (π : S → A → ℝ) (hπ_prob : ∀ x, (∀ a', 0 ≤ π x a') ∧ ∑ a', π x a' = 1)\n    (r : S → A → ℝ) (hr_bound : ∀ x b, r x b ∈ Set.Icc (0:ℝ) 1)\n    (Pπ : S → S → ℝ) (hPπ_def : ∀ x y, Pπ x y = ∑ a' : A, π x a' * Pt x a' y)\n    (hPπ_prob : ∀ x, (∀ y, 0 ≤ Pπ x y) ∧ ∑ y, Pπ x y = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ x, rπ x = ∑ a' : A, π x a' * r x a')\n    (lam : ℝ)\n    (hBellman : ∀ x, h x = rπ x - lam + ∑ y, Pπ x y * h y)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y) :\n    bellmanError Pt P h s a ≤ transL1 Pt P s a * biasSupNorm h\n      ∧ |bellmanError Pt P h s a| ≤ Dt * min ε 2 := by\n  sorry\n\ntheorem lemma11 {S A Ω : Type*} [Fintype S] [Nonempty S] [DecidableEq S]\n    [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (r : S → A → ℝ) (hr_bound : ∀ s a, r s a ∈ Set.Icc (0 : ℝ) 1)\n    (P : S → A → S → ℝ) (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (π : S → A → ℝ)\n    (hπ_prob : ∀ s, (∀ a, 0 ≤ π s a) ∧ ∑ a, π s a = 1)\n    (hPπ : S → S → ℝ) (hPπ_def : ∀ s s', hPπ s s' = ∑ a : A, π s a * P s a s')\n    (hPπ_prob : ∀ s, (∀ s', 0 ≤ hPπ s s') ∧ ∑ s', hPπ s s' = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ s, rπ s = ∑ a : A, π s a * r s a)\n    (lamPi : ℝ) (h : S → ℝ)\n    (hBellman : ∀ s, h s = rπ s - lamPi + ∑ s', hPπ s s' * h s')\n    (D : ℝ) (hD0 : 0 ≤ D)\n    (X : S → ℕ → Ω → S) (hXmeas : ∀ s₀ t, Measurable (X s₀ t))\n    (hXstart : ∀ s₀ : S, μ {ω | X s₀ 1 ω = s₀} = 1)\n    (F : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone F) (hFle : ∀ t, F t ≤ (inferInstance : MeasurableSpace Ω))\n    (hXAdapted : ∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F t'] (X s₀ t))\n    (hMarkov : ∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F t] H →\n        ∀ x y : S, μ (H ∩ {ω | X s₀ t ω = x}) ≠ 0 →\n        μ (H ∩ {ω | X s₀ t ω = x} ∩ {ω | X s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (hPπ x y) * μ (H ∩ {ω | X s₀ t ω = x}))\n    (hD : IsDiameterUpperBound Ω P D)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (s s' : S) :\n    h s - h s' ≤ D := by\n  sorry\n\nnoncomputable def visitCount {S A Ω : Type*} [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (t : ℕ) (s : S) (a : A) (ω : Ω) : ℕ :=\n  ∑ t' ∈ Finset.Icc 1 t, if Sp t' ω = s ∧ Ap t' ω = a then 1 else 0\n\nnoncomputable def empTrans {S A Ω : Type*} [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (t : ℕ) (s : S) (a : A) (s' : S) (ω : Ω) : ℝ :=\n  (∑ t' ∈ Finset.Icc 1 t, if Sp t' ω = s ∧ Ap t' ω = a ∧ Sp (t' + 1) ω = s'\n      then (1 : ℝ) else 0) / (visitCount Sp Ap t s a ω : ℝ)\n\ndef goodEvent {S A Ω : Type*} [Fintype S] [Fintype A] [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (P : S → A → S → ℝ) (t T : ℕ) (ω : Ω) : Prop :=\n  ∀ s : S, ∀ a : A,\n    transL1 (fun s a s' => empTrans Sp Ap t s a s' ω) P s a\n      ≤ Real.sqrt (14 * (Fintype.card S : ℝ) * Real.log (2 * (Fintype.card A : ℝ) * T)\n          / max 1 (visitCount Sp Ap t s a ω : ℝ))\n\ntheorem lemma7 {S A Ω : Type*} [Fintype S] [Nonempty S] [Fintype A] [Nonempty A]\n    [DecidableEq S] [DecidableEq A] [MeasurableSpace S] [MeasurableSpace A]\n    [MeasurableSingletonClass S] [MeasurableSingletonClass A]\n    [MeasurableSpace Ω]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A)\n    (hSpmeas : ∀ t', Measurable (Sp t')) (hApmeas : ∀ t', Measurable (Ap t'))\n    (ℱ : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone ℱ) (hFle : ∀ t', ℱ t' ≤ (inferInstance : MeasurableSpace Ω))\n    (hSpAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Sp t'))\n    (hApAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Ap t'))\n    (P : S → A → S → ℝ)\n    (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y)\n    (hMarkov : ∀ t' : ℕ, ∀ H : Set Ω, MeasurableSet[ℱ t'] H →\n        ∀ x : S, ∀ y1 : A, ∀ y2 : S,\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}) ≠ 0 →\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1} ∩ {ω | Sp (t' + 1) ω = y2})\n          = ENNReal.ofReal (P x y1 y2) * μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}))\n    (t T : ℕ) (ht : 1 ≤ t) (hT : t ≤ T) :\n    μ {ω | ¬ goodEvent Sp Ap P t T ω} ≤ ENNReal.ofReal (1 / (20 * (t : ℝ) ^ 6)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma6", "lemma11", "lemma7"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "holes": [{"start": 3668, "end": 5314, "gold": "{S A Ω : Type*} [Fintype S] [Nonempty S] [DecidableEq S]\n    [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (r : S → A → ℝ) (hr_bound : ∀ s a, r s a ∈ Set.Icc (0 : ℝ) 1)\n    (P : S → A → S → ℝ) (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (π : S → A → ℝ)\n    (hπ_prob : ∀ s, (∀ a, 0 ≤ π s a) ∧ ∑ a, π s a = 1)\n    (hPπ : S → S → ℝ) (hPπ_def : ∀ s s', hPπ s s' = ∑ a : A, π s a * P s a s')\n    (hPπ_prob : ∀ s, (∀ s', 0 ≤ hPπ s s') ∧ ∑ s', hPπ s s' = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ s, rπ s = ∑ a : A, π s a * r s a)\n    (lamPi : ℝ) (h : S → ℝ)\n    (hBellman : ∀ s, h s = rπ s - lamPi + ∑ s', hPπ s s' * h s')\n    (D : ℝ) (hD0 : 0 ≤ D)\n    (X : S → ℕ → Ω → S) (hXmeas : ∀ s₀ t, Measurable (X s₀ t))\n    (hXstart : ∀ s₀ : S, μ {ω | X s₀ 1 ω = s₀} = 1)\n    (F : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone F) (hFle : ∀ t, F t ≤ (inferInstance : MeasurableSpace Ω))\n    (hXAdapted : ∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F t'] (X s₀ t))\n    (hMarkov : ∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F t] H →\n        ∀ x y : S, μ (H ∩ {ω | X s₀ t ω = x}) ≠ 0 →\n        μ (H ∩ {ω | X s₀ t ω = x} ∩ {ω | X s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (hPπ x y) * μ (H ∩ {ω | X s₀ t ω = x}))\n    (hD : IsDiameterUpperBound Ω P D)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (s s' : S) :\n    h s - h s' ≤ D"}]}
{"item_id": "v24-19-980::lemma7::A-stmt", "paper_id": "v24-19-980", "target_name": "lemma7", "source": "Research-level", "nl_statement": "Lemma 7 For any $1 \\leq t \\leq T$ , the probability that the event $\\mathcal{E}_{t}$ fails to occur us upper bounded by $\\frac{1}{20t^{6}}$", "context": [{"name": "Setup: MDP, rewards, policy and average reward (Section 3)", "statement": "We consider an infinite horizon discounted Markov decision process (MDP) $\\mathcal{M}$ defined by the tuple $\\left(\\mathcal{S},\\mathcal{A},P,r^{1},r^{2},\\cdot\\cdot\\cdot,r^{K},\\rho_{0}\\right)$ . $\\mathcal{S}$ denotes a finite set of state space of size S, and $\\mathcal{A}$ denotes a finite set of A actions. $P:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]^{S}$ denotes the probability transition distribution. K denotes the number of objectives and $[K]=\\{1,2,\\cdots,K\\}$ is the set of K objectives. Let $r^{k}:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ be the bounded reward function for objective $k \\in [ K ]$. Lastly, $\\rho_{0}:\\mathcal{S}\\to [0,1]$ is the distribution of initial state. We motivate our choice of bounds on rewards from the fact that many problems in practice require explicit reward shaping. Hence, the controller or the learner is aware of the bounds on the rewards. We consider the bounds to be [0, 1] for our case which is easy to satisfy by reward shaping.\n\nWe use a stochastic policy $\\pi:\\mathcal{S}\\times\\mathcal{A}\\to [0,1]$ which returns the probability of selecting action $a \\in{\\mathcal{A}}$ for any given state $s \\in{\\mathcal{S}}$. Following policy π on the MDP, the agent observes a sequence of random variables $\\{S_{t}, A_{t}\\}_{t}$ where $S_{t}$ denotes the state of the agent at time t and $A_{t}$ denotes the action taken by the agent at time t. The expected discounted long term reward and expected per step reward of the objective k are given by $J_{\\pi}^{P,k}$ and $\\lambda_{\\pi}^{P,k}$ , respectively, when the joint policy π is followed. Formally, for discount factor $\\gamma\\in(0,1)$ , $J_{\\pi}^{P,k}$ and $\\lambda_{\\pi}^{P,k}$ are defined as\n\n$$\nJ_{\\pi}^{P,k}=\\mathbb{E}_{S_{0},A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\sum_{t=0}^{\\tau}\\gamma^{t}r^{k}(S_{t},A_{t})]\\tag{1}\n$$\n\n$$\nS_{0}\\sim\\rho_{0}(S_{0}),\\ A_{t}\\sim\\pi(A_{t}|S_{t}),\\ S_{t+1}\\sim P(\\cdot|S_{t},A_{t})\n$$\n\n$$\n\\lambda_{\\pi}^{P,k}=\\mathbb{E}_{S_{0},A_{0},S_{1},A_{1},\\cdots}[\\operatorname*{lim}_{\\tau\\to\\infty}\\frac{1}{\\tau}\\sum_{t=0}^{\\tau}r^{k}(S_{t},A_{t})]\\tag{2}\n$$\n\n$$\n=\\operatorname*{lim}_{\\gamma\\to 1}(1-\\gamma)J_{\\pi}^{P,k}\\tag{3}\n$$"}, {"name": "Good event E_t, eq. (29), and empirical estimates", "statement": "A major part of the regret analysis is how well the learned model estimates the true system model. For that, bound the deviation of the estimates of the estimated transition probabilities of the Markov Decision Processes $\\mathcal{M}$. For that we use $\\ell_{1}$ deviation bounds from (Weissman et al., 2003). Consider, the following event,\n\n$$\n\\mathcal{E}_{t}=\\left\\{\\|\\bar{P}_{t}(\\cdot|s,a)-P(\\cdot|s,a)\\|_{1}\\leq\\sqrt{\\frac{14S\\log(2AT)}{\\operatorname*{max}\\{1,n_{t}(s,a)\\}}}\\forall(s,a)\\in\\mathcal{S}\\times\\mathcal{A}\\right\\}\\tag{29}\n$$\n\nwhere $\\begin{array}{r}{n_{t}(s,a)=\\sum_{t^{\\prime}=1}^{t}\\mathbf{1}_{\\{S_{t^{\\prime}}=s,a_{t^{\\prime}}=a\\}}}\\end{array}$ and $\\begin{array}{r}{\\bar{P}(s^{\\prime}|s,a)=\\left(\\sum_{t^{\\prime}=1}^{t}\\mathbf{1}_{\\{S_{t^{\\prime}}=s,a_{t^{\\prime}}=a,S_{t^{\\prime}+1}=s^{\\prime}\\}}\\right)/n_{t}(s,a)}\\end{array}$ of the empirical estimate of the transition probabilities."}, {"name": "Algorithm 1 (Model-Based Joint Decision Making) and its description (Section 5.1)", "statement": "Algorithm 1 Model-Based Joint Decision Making Algorithm   \n1: procedure Model Based Online($\\mathcal{S},\\mathcal{A},[K],f,r$)   \n2: Initialize $N(s,a,s^{\\prime})=1\\forall(s,a,s^{\\prime})\\in\\mathcal{S}\\times\\mathcal{A}\\times\\mathcal{S}$   \n3: Initialize $\\nu_{0}(s,a)=0~\\forall~(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$   \n4: Initialize $\\textstyle\\pi_{1}(a|s)={\\frac{1}{|{\\mathcal{A}}|}}\\forall(a,s)\\in{\\mathcal{A}}\\times{\\mathcal{S}}$   \n5: Initialize $e = 1$   \n6: for time steps $t=1,2,\\cdots$ do   \n7: Observe current state $S_{t}$   \n8: Sample action to play $A_{t}\\sim\\pi_{e}(\\cdot|S_{t})$   \n9: Play $A_{t},$ obtain reward $r_{t}\\in [0,1]^{K}$ and observe next state $S_{t + 1}$   \n10: Update $N(S_{t},A_{t},S_{t+1})\\gets N(S_{t},A_{t},S_{t+1})+1,\\nu_{e}(S_{t},A_{t})\\gets\\nu_{e}(S_{t},A_{t})+1$   \n11: if $\\begin{array}{r}{\\nu_{e}(S_{t},A_{t})\\ge\\operatorname*{max}\\{1,\\sum_{e^{\\prime}<e}\\nu_{e^{\\prime}}(S_{t},A_{t})\\}}\\end{array}$ then   \n12: Update epoch counter $e \\gets e + 1$   \n13: Initialize $\\nu_{e}(s,a)=0~\\forall~(s,a)\\in\\mathcal{S}\\times\\mathcal{A}$   \n14: $P_{e}(s^{\\prime}|s,a)\\sim Dir(N(s,a,\\cdot))\\ \\forall\\ (s,a)$   \n15: Solve steady state distribution $d(s,a)$ as the solution of the optimization   \nproblem in Equations (20-23)   \n16: Obtain optimal policy $\\pi_{e}$ as   \n$\\pi_{e}(a|s)={\\frac{d(s,a)}{\\sum_{a\\in{\\mathcal{A}}}d(s,a)}}$   \n17: end if   \n18: end for   \n19: end procedure\n\nAlgorithm 1 describes the overall procedure that estimates the transition probabilities and the reward functions. The algorithm takes as input the state space $\\mathcal{S}$, action space ${\\mathcal{A}},$ set of agents [K], the reward structure $r$, and the objective function $f .$ It initializes the next state visit count for each state-action pair $N(s,a,s^{\\prime})$ by one for Dirichlet sampling. For initial exploration, the policy uses a uniform distribution over all the actions. The algorithm proceeds in epochs e with $t_{e}$ denoting the time step t at which epoch e starts. Also, $\\nu_{e}(s,a)$ stores the number of times a state-action pair is visited in epoch e. We assume that the controller is optimizing for infinite horizon and thus there is no stopping condition for epoch loop in Line 6. For each time index $t \\in [ t_{e}, t_{e + 1})$ in epoch $e,$ the controller observes the state, samples and plays the action according to $\\pi_{e}(\\cdot|s)$ , and observes the rewards for each agent and next state. It then updates the state visit count for the observed state and played action pair. We break the current epoch e if the total number of visitations of any state action pair in current epoch exceeds total visitations in previous epoch. This epoch breaking strategy ensures that each state-action triggers epoch switches exponentially slower. This increasing epoch duration bounds the policy switching cost (Jaksch et al., 2010). After breaking the epoch, we sample transition probabilities from the updated posterior and find a new optimal policy for the sampled MDP by solving the optimization framework described in Equation(20)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nvariable {S : Type*} {A : Type*}\n\nnoncomputable def bellmanError (Pt P : S → A → S → ℝ) (h : S → ℝ)\n    [Fintype S] (s : S) (a : A) : ℝ :=\n  ∑ s' : S, (Pt s a s' - P s a s') * h s'\n\nnoncomputable def transL1 (Pt P : S → A → S → ℝ)\n    [Fintype S] (s : S) (a : A) : ℝ :=\n  ∑ s' : S, |Pt s a s' - P s a s'|\n\nnoncomputable def biasSupNorm [Fintype S] [Nonempty S] (h : S → ℝ) : ℝ :=\n  Finset.univ.sup' Finset.univ_nonempty (fun s' => |h s'|)\n\nnoncomputable def hitTime {S Ω : Type*} (X : ℕ → Ω → S) (s' : S) (ω : Ω) : ℕ :=\n  sInf {t : ℕ | 1 ≤ t ∧ X t ω = s'}\n\ndef IsDiameterBoundOfChain {S Ω : Type*} [Fintype S] [MeasurableSpace Ω]\n    (μ : Measure Ω) (X : S → ℕ → Ω → S) (D : ℝ) : Prop :=\n  ∀ s s' : S, s ≠ s' →\n    (∀ᵐ ω ∂μ, ∃ t : ℕ, 1 ≤ t ∧ X s t ω = s') ∧\n    Integrable (fun ω => (hitTime (X s) s' ω : ℝ)) μ ∧\n    ∫ ω, (hitTime (X s) s' ω : ℝ) ∂μ ≤ D\n\ndef IsDiameterUpperBound (Ω : Type*) {S A : Type*} [Fintype S] [MeasurableSpace S]\n    [Fintype A] [MeasurableSpace Ω] (P : S → A → S → ℝ) (D : ℝ) : Prop :=\n  ∀ (μ₂ : Measure Ω), IsProbabilityMeasure μ₂ →\n  ∀ π₂ : S → A → ℝ, (∀ s, (∀ a, 0 ≤ π₂ s a) ∧ ∑ a, π₂ s a = 1) →\n  ∀ (X₂ : S → ℕ → Ω → S) (F₂ : ℕ → MeasurableSpace Ω),\n    (∀ s₀ t, Measurable (X₂ s₀ t)) →\n    (∀ s₀ : S, μ₂ {ω | X₂ s₀ 1 ω = s₀} = 1) →\n    Monotone F₂ → (∀ t, F₂ t ≤ (inferInstance : MeasurableSpace Ω)) →\n    (∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F₂ t'] (X₂ s₀ t)) →\n    (∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F₂ t] H →\n        ∀ x y : S, μ₂ (H ∩ {ω | X₂ s₀ t ω = x}) ≠ 0 →\n        μ₂ (H ∩ {ω | X₂ s₀ t ω = x} ∩ {ω | X₂ s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (∑ a : A, π₂ x a * P x a y)\n              * μ₂ (H ∩ {ω | X₂ s₀ t ω = x})) →\n    IsDiameterBoundOfChain μ₂ X₂ D\n\ndef IsDiameterBound (Ω : Type*) {S A : Type*} [Fintype S] [MeasurableSpace S]\n    [Fintype A] [MeasurableSpace Ω] (P : S → A → S → ℝ) (D : ℝ) : Prop :=\n  IsDiameterUpperBound Ω P D ∧ ∀ D' : ℝ, IsDiameterUpperBound Ω P D' → D ≤ D'\n\n\ntheorem lemma6 {Ω : Type*} [MeasurableSpace Ω] [Fintype S] [Nonempty S] [DecidableEq S] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    (Pt P : S → A → S → ℝ) (h : S → ℝ) (s : S) (a : A) (Dt ε : ℝ)\n    (hD : 0 ≤ Dt)\n    (hP_nonneg : ∀ s', 0 ≤ P s a s') (hP_sum : ∑ s', P s a s' = 1)\n    (hPt_nonneg : ∀ s', 0 ≤ Pt s a s') (hPt_sum : ∑ s', Pt s a s' = 1)\n    (hP_prob : ∀ x b, (∀ y, 0 ≤ P x b y) ∧ ∑ y, P x b y = 1)\n    (hPt_prob : ∀ x b, (∀ y, 0 ≤ Pt x b y) ∧ ∑ y, Pt x b y = 1)\n    (hε : transL1 Pt P s a ≤ ε)\n    (hspan : ∀ x y : S, h x - h y ≤ Dt)\n    (hDiam : IsDiameterUpperBound Ω Pt Dt)\n    (π : S → A → ℝ) (hπ_prob : ∀ x, (∀ a', 0 ≤ π x a') ∧ ∑ a', π x a' = 1)\n    (r : S → A → ℝ) (hr_bound : ∀ x b, r x b ∈ Set.Icc (0:ℝ) 1)\n    (Pπ : S → S → ℝ) (hPπ_def : ∀ x y, Pπ x y = ∑ a' : A, π x a' * Pt x a' y)\n    (hPπ_prob : ∀ x, (∀ y, 0 ≤ Pπ x y) ∧ ∑ y, Pπ x y = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ x, rπ x = ∑ a' : A, π x a' * r x a')\n    (lam : ℝ)\n    (hBellman : ∀ x, h x = rπ x - lam + ∑ y, Pπ x y * h y)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y) :\n    bellmanError Pt P h s a ≤ transL1 Pt P s a * biasSupNorm h\n      ∧ |bellmanError Pt P h s a| ≤ Dt * min ε 2 := by\n  sorry\n\ntheorem lemma11 {S A Ω : Type*} [Fintype S] [Nonempty S] [DecidableEq S]\n    [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype A]\n    [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (r : S → A → ℝ) (hr_bound : ∀ s a, r s a ∈ Set.Icc (0 : ℝ) 1)\n    (P : S → A → S → ℝ) (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (π : S → A → ℝ)\n    (hπ_prob : ∀ s, (∀ a, 0 ≤ π s a) ∧ ∑ a, π s a = 1)\n    (hPπ : S → S → ℝ) (hPπ_def : ∀ s s', hPπ s s' = ∑ a : A, π s a * P s a s')\n    (hPπ_prob : ∀ s, (∀ s', 0 ≤ hPπ s s') ∧ ∑ s', hPπ s s' = 1)\n    (rπ : S → ℝ) (hrπ_def : ∀ s, rπ s = ∑ a : A, π s a * r s a)\n    (lamPi : ℝ) (h : S → ℝ)\n    (hBellman : ∀ s, h s = rπ s - lamPi + ∑ s', hPπ s s' * h s')\n    (D : ℝ) (hD0 : 0 ≤ D)\n    (X : S → ℕ → Ω → S) (hXmeas : ∀ s₀ t, Measurable (X s₀ t))\n    (hXstart : ∀ s₀ : S, μ {ω | X s₀ 1 ω = s₀} = 1)\n    (F : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone F) (hFle : ∀ t, F t ≤ (inferInstance : MeasurableSpace Ω))\n    (hXAdapted : ∀ s₀ : S, ∀ t t' : ℕ, t ≤ t' → Measurable[F t'] (X s₀ t))\n    (hMarkov : ∀ s₀ : S, ∀ t : ℕ, 1 ≤ t → ∀ H : Set Ω, MeasurableSet[F t] H →\n        ∀ x y : S, μ (H ∩ {ω | X s₀ t ω = x}) ≠ 0 →\n        μ (H ∩ {ω | X s₀ t ω = x} ∩ {ω | X s₀ (t + 1) ω = y})\n          = ENNReal.ofReal (hPπ x y) * μ (H ∩ {ω | X s₀ t ω = x}))\n    (hD : IsDiameterUpperBound Ω P D)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (s s' : S) :\n    h s - h s' ≤ D := by\n  sorry\n\nnoncomputable def visitCount {S A Ω : Type*} [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (t : ℕ) (s : S) (a : A) (ω : Ω) : ℕ :=\n  ∑ t' ∈ Finset.Icc 1 t, if Sp t' ω = s ∧ Ap t' ω = a then 1 else 0\n\nnoncomputable def empTrans {S A Ω : Type*} [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (t : ℕ) (s : S) (a : A) (s' : S) (ω : Ω) : ℝ :=\n  (∑ t' ∈ Finset.Icc 1 t, if Sp t' ω = s ∧ Ap t' ω = a ∧ Sp (t' + 1) ω = s'\n      then (1 : ℝ) else 0) / (visitCount Sp Ap t s a ω : ℝ)\n\ndef goodEvent {S A Ω : Type*} [Fintype S] [Fintype A] [DecidableEq S] [DecidableEq A]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A) (P : S → A → S → ℝ) (t T : ℕ) (ω : Ω) : Prop :=\n  ∀ s : S, ∀ a : A,\n    transL1 (fun s a s' => empTrans Sp Ap t s a s' ω) P s a\n      ≤ Real.sqrt (14 * (Fintype.card S : ℝ) * Real.log (2 * (Fintype.card A : ℝ) * T)\n          / max 1 (visitCount Sp Ap t s a ω : ℝ))\n\ntheorem lemma7 {S A Ω : Type*} [Fintype S] [Nonempty S] [Fintype A] [Nonempty A]\n    [DecidableEq S] [DecidableEq A] [MeasurableSpace S] [MeasurableSpace A]\n    [MeasurableSingletonClass S] [MeasurableSingletonClass A]\n    [MeasurableSpace Ω]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A)\n    (hSpmeas : ∀ t', Measurable (Sp t')) (hApmeas : ∀ t', Measurable (Ap t'))\n    (ℱ : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone ℱ) (hFle : ∀ t', ℱ t' ≤ (inferInstance : MeasurableSpace Ω))\n    (hSpAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Sp t'))\n    (hApAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Ap t'))\n    (P : S → A → S → ℝ)\n    (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y)\n    (hMarkov : ∀ t' : ℕ, ∀ H : Set Ω, MeasurableSet[ℱ t'] H →\n        ∀ x : S, ∀ y1 : A, ∀ y2 : S,\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}) ≠ 0 →\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1} ∩ {ω | Sp (t' + 1) ω = y2})\n          = ENNReal.ofReal (P x y1 y2) * μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}))\n    (t T : ℕ) (ht : 1 ≤ t) (hT : t ≤ T) :\n    μ {ω | ¬ goodEvent Sp Ap P t T ω} ≤ ENNReal.ofReal (1 / (20 * (t : ℝ) ^ 6)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma6", "lemma11", "lemma7"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "holes": [{"start": 6256, "end": 7894, "gold": "{S A Ω : Type*} [Fintype S] [Nonempty S] [Fintype A] [Nonempty A]\n    [DecidableEq S] [DecidableEq A] [MeasurableSpace S] [MeasurableSpace A]\n    [MeasurableSingletonClass S] [MeasurableSingletonClass A]\n    [MeasurableSpace Ω]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (Sp : ℕ → Ω → S) (Ap : ℕ → Ω → A)\n    (hSpmeas : ∀ t', Measurable (Sp t')) (hApmeas : ∀ t', Measurable (Ap t'))\n    (ℱ : ℕ → MeasurableSpace Ω)\n    (hFmono : Monotone ℱ) (hFle : ∀ t', ℱ t' ≤ (inferInstance : MeasurableSpace Ω))\n    (hSpAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Sp t'))\n    (hApAdapted : ∀ t' t'' : ℕ, t' ≤ t'' → Measurable[ℱ t''] (Ap t'))\n    (P : S → A → S → ℝ)\n    (hP_prob : ∀ s a, (∀ s', 0 ≤ P s a s') ∧ ∑ s', P s a s' = 1)\n    (hErgodic : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∃ (d : S → ℝ) (C ρ : ℝ), 0 ≤ C ∧ 0 ≤ ρ ∧ ρ < 1 ∧\n          ∀ t : ℕ, ∀ x : S,\n            ∑ y, |((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y - d y|\n              ≤ C * ρ ^ t)\n    (hCommunicate : ∀ π₂ : S → A → ℝ, (∀ x, (∀ a, 0 ≤ π₂ x a) ∧ ∑ a, π₂ x a = 1) →\n        ∀ x y : S, ∃ t : ℕ,\n          0 < ((Matrix.of fun u v => ∑ a, π₂ u a * P u a v) ^ t) x y)\n    (hMarkov : ∀ t' : ℕ, ∀ H : Set Ω, MeasurableSet[ℱ t'] H →\n        ∀ x : S, ∀ y1 : A, ∀ y2 : S,\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}) ≠ 0 →\n        μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1} ∩ {ω | Sp (t' + 1) ω = y2})\n          = ENNReal.ofReal (P x y1 y2) * μ (H ∩ {ω | Sp t' ω = x} ∩ {ω | Ap t' ω = y1}))\n    (t T : ℕ) (ht : 1 ≤ t) (hT : t ≤ T) :\n    μ {ω | ¬ goodEvent Sp Ap P t T ω} ≤ ENNReal.ofReal (1 / (20 * (t : ℝ) ^ 6))"}]}
{"item_id": "v24-22-0881::theorem7::A-stmt", "paper_id": "v24-22-0881", "target_name": "theorem7", "source": "Research-level", "nl_statement": "Theorem 7 For any $\\mathcal{H}$ with $vc(\\mathcal{H}) = d$ and $\\mathcal{U}(P)=\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\},\\rho>0,\\forall\\varepsilon,\\delta\\in(0,1)$ (the constant hidden in $O(\\cdot)$ may depend on $\\rho$)\n\n$$\n\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})=O\\left(\\frac{d}{\\varepsilon^{4}}\\log\\left(\\frac{d}{\\varepsilon^{4}}\\right)+\\frac{1}{\\varepsilon^{4}}\\Big(\\frac{1}{c_{2}(\\rho)-1}\\vee 1\\Big)+\\frac{d}{\\varepsilon^{4}}\\log\\left(\\frac{e}{d}\\right)+\\frac{\\log(2/\\delta)}{\\varepsilon^{4}}\\right).\\tag{3}\n$$", "context": [{"name": "Setup: robust risk Eq. (1) and DRERM (Section 1)", "statement": "Concretely, let $\\mathcal{X}$ be the instance space, while $\\mathcal{Y}=\\{+1,-1\\}$ denotes the label space. We formalize the distributional shift that we would like to protect against as an uncertainty set $\\mathcal{U}(P)$ containing distributions with certain constraints, such as moment condition (Delage and Ye, 2010), Wasserstein distance (Gao, 2020) and f-divergence (Duchi and Namkoong, 2021). For a distribution P over $\\mathcal{X}\\times\\mathcal{Y}=\\{(x,y):x\\in\\mathcal{X},y\\in\\mathcal{Y}\\}$ , we observe m i.i.d. samples $S \\sim P^{m}$ , and distributionally robust learning attempts to learn a predictor $h : \\mathcal{X}\\to \\mathcal{Y}$ having small distributionally robust risk,\n\n$$\nR_{\\mathcal{U}}(h;P): =\\operatorname*{sup}_{Q\\in\\mathcal{U}(P)}\\mathbb{E}_{(x,y)\\sim Q}[\\mathbb{1}[h(x)\\neq y]].\\tag{1}\n$$\n\nThe common approach to distributionally robust learning involves selecting a hypothesis class $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ and learning a predictor $\\hat{h}: \\mathcal{X}\\to \\mathcal{Y}$ from $\\mathcal{H}$ through Distributionally Robust Empirical Risk Minimization:\n\n$$\n\\begin{array}{r}{\\hat{h}\\in\\mathrm{DRERM}_{\\mathcal{H}}(S): =\\underset{h\\in\\mathcal{H}}{\\mathrm{argmin}}\\hat{R}_{\\mathcal{U}}(h;S),}\\end{array}\n$$\n\nwhere $\\begin{array}{r}{\\hat{R}_{\\mathcal{U}}(h;S): =\\operatorname*{sup}_{Q\\in\\mathcal{U}(P_{m})}\\mathbb{E}_{(x,y)\\sim Q}[\\mathbb{1}\\left[h(x)\\neq y\\right]]}\\end{array}$ and $P_{m}$ denotes the empirical distribution over samples S."}, {"name": "Uncertainty set and chi^2-divergence D_2 (Section 1)", "statement": "Namely, we consider the distributional shift as follows:\n\n$$\n\\mathcal{U}(P)=\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\},\n$$\n\nwhere $\\begin{array}{r}{D_{2}(Q\\|P): =\\frac{1}{2}\\int\\left(\\frac{dQ}{dP}-1\\right)^{2}dP}\\end{array}$ and $Q \\ll P$ indicates that distribution Q is absolutely continuous with respect to $P .$ The $\\chi^{2}$-divergence is a commonly explored concept in the distributionally robust optimization (DRO) literature (Duchi and Namkoong, 2019). Moreover, it is also a crucial concept in a variety of fields such as information theory, statistics, learning, signal processing, and various branches of mathematics (Park et al., 2011; Saraswat, 2014; Nishiyama and Sason, 2020). The $\\chi^{2}$-divergence plays a fundamental role in problems related to source and channel coding, combinatorics, large deviation theory, goodness-of-fit, and independence tests in statistics, as demonstrated by Csiszár et al. (2004). Additionally, it is widely recognized for its computational feasibility and statistical properties, as noted in Tsybakov (2009)."}, {"name": "Problem setup (Section 2)", "statement": "Given a hypothesis class $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ , our goal is to design a learning rule $\\mathcal{A}:(\\mathcal{X}\\times\\mathcal{Y})^{*}\\mapsto\\mathcal{Y}^{\\mathcal{X}}$ such that for any distribution P over $\\mathcal{X}\\times \\mathcal{Y}$ , the rule $\\mathcal{A}$ will find a predictor that can compete with the best predictor $h^{\\ast}\\in \\mathcal{H}$ in terms of the distributionally robust risk using a number of samples that is independent of the distribution P. In this paper, we use $(\\mathcal{X}\\times\\mathcal{Y})^{*}$ to denote the set of all sequences in the space $\\mathcal{X}\\times \\mathcal{Y}$ . The following definitions formalize the notion of distributionally robust PAC learning under the realizable case and agnostic settings."}, {"name": "Definition 1 (Agnostic Distributionally Robust PAC Learnability)", "statement": "Definition 1 (Agnostic Distributionally Robust PAC Learnability) For any $\\varepsilon,\\delta\\in$ (0, 1), the sample complexity of agnostic distributionally robust $(\\varepsilon,\\delta)$-PAC learning of $\\mathcal{H}$ with respect to the distributional shift $\\mathcal{U}$, denoted by $\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})$ , is defined as the smallest $m \\in \\mathbb{N}\\cup \\{0 \\}$ for which there exists a learning rule $\\mathcal{A}:(\\mathcal{X}\\times\\mathcal{Y})^{*}\\mapsto\\mathcal{Y}^{\\mathcal{X}}$ such that, for every data distribution P over $\\mathcal{X}\\times \\mathcal{Y}$ , with probability of at least $1 - \\delta$ over $S \\sim P^{m}$ ,\n\n$$\nR_{\\mathcal{U}}(\\mathcal{A}(S);P)\\leq\\operatorname*{inf}_{h\\in\\mathcal{H}}R_{\\mathcal{U}}(h;P)+\\varepsilon .\n$$\n\nIf no such m exists, define $\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})=\\infty$ . We say that $\\mathcal{H}$ is distributionally robustly PAC-learnable in the agnostic setting with respect to the distributional shift $\\mathcal{U}$ if $\\forall\\varepsilon,\\delta\\in(0,1)$ , $\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})$ scales polynomially with $1 / \\varepsilon$ and $1 / \\delta$"}, {"name": "Definition 3 (VC dimension)", "statement": "Definition 3 (VC dimension) We say that a sequence $\\{x_{1},\\ldots,x_{k}\\}\\subseteq{\\mathcal{X}}$ is shattered by $\\mathcal{H}$ if $\\forall y_{1},\\ldots,y_{k}\\in\\mathcal{Y},\\exists h\\in\\mathcal{H}$ such that $\\forall i\\in [k],h(x_{i})=y_{i}$ . The VC dimension of $\\mathcal{H}$ (denoted as $vc(\\mathcal{H}))$ is then defined as the largest integer k for which there exists $\\{x_{1},\\ldots,x_{k}\\}\\subseteq{\\mathcal{X}}$ that is shattered by $\\mathcal{H}$. If no such k exists, then $vc(\\mathcal{H})$ is said to be infinite."}, {"name": "Loss class L_H", "statement": "Denote the loss class of $\\mathcal{H}$ by $\\mathcal{L}_{\\mathcal{H}}$ , where\n\n$$\n\\mathcal{L}_{\\mathcal{H}}=\\Big\\{(x,y)\\mapsto\\mathbb{1}[h(x)\\neq y]:h\\in\\mathcal{H}\\Big\\}.\n$$"}, {"name": "Notation R_2(h;P) (Section 3)", "statement": "We use $R_{2}(h;P)$ to denote the distributionally robust risk under distributional shift\n\n$$\n\\begin{array}{r}{\\mathcal{U}(P)=\\left\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\right\\}.}\\end{array}\n$$"}, {"name": "Proposition 4 (Duality Formulation), Eq. (2) (Shapiro 2017)", "statement": "We recall the following duality formulation (Shapiro, 2017, Section 3.2) for distributionally robust risk, which is essential in our derivation.\n\nProposition 4 (Duality Formulation) For any probability P on $\\mathcal{X}\\times \\mathcal{Y}$ , any $\\rho > 0$ , and $c_{2}(\\rho): =\\sqrt{1+2\\rho},$ for all $h \\in \\mathcal H$ , we have\n\n$$\nR_{2}(h;P)=\\operatorname*{inf}_{\\eta\\in\\mathbb{R}}\\Big\\{c_{2}(\\rho)\\mathbb{E}_{P}\\Big [(\\mathbb{1}[h(x)\\neq y]-\\eta)_{+}^{2}\\Big]^{1/2}+\\eta\\Big\\},\\tag{2}\n$$\n\nwhere $a_{+}: =\\operatorname*{max}\\left(a,0\\right)$"}, {"name": "g_2 and Lemma 5 (compact range for eta), Remark 6", "statement": "To bring the above proposition into effect, we need the following lemma. In the interests of simplicity, for any fixed $h \\in \\mathcal H$ , let\n\n$$\n\\begin{array}{r}{g_{2}(\\eta,P): =c_{2}(\\rho)\\mathbb{E}_{P}\\left[(\\mathbb{1}\\left[h(x)\\neq y\\right]-\\eta)_{+}^{2}\\right]^{1/2}+\\eta .}\\end{array}\n$$\n\nLemma 5 For any distribution $P$ ,\n\n$$\n\\operatorname*{inf}_{\\eta\\in\\mathbb{R}}g_{2}(\\eta,P)=\\operatorname*{inf}_{\\eta}\\left\\{g_{2}(\\eta,P):\\eta\\in\\left[-\\frac{1}{c_{2}(\\rho)-1},1\\right]\\right\\}.\n$$\n\nRemark 6 The above lemma restricts the domain of η to a compact set, which is crucial to our uniform convergence result."}, {"name": "Remark 8 (dependence on 1/(c_2(rho)-1))", "statement": "Remark 8 The dependence on $\\frac{1}{c_{2}(\\rho)-1}$ is due to the lower bound for η in Lemma 5. While more advanced techniques could potentially yield a better dependence on $\\rho,$ it is worth noting that the upper bound shows that the finite VC dimension is sufficient for distributionally robust PAC learnability for finite non-zero values of ρ."}, {"name": "Notation phi_{eta,h} (Appendix A.1)", "statement": "We use $\\varphi_{\\eta,h}(x,y)$ to denote $\\mathbb{1}[h(x)\\neq y]-\\eta$ and $c_{2}$ as the shorthand of $c_{2}(\\rho)$ when there is no ambiguity."}, {"name": "Definition 16 (Rademacher Complexity)", "statement": "Moreover, we introduce the definition of Rademacher Complexity and Growth Function in order to bound the excess risk in terms of VC dimension.\n\nDefinition 16 (Rademacher Complexity) We define the empirical Rademacher Complexity of a hypothesis class $\\mathcal{F}$ for a given sample $z_{i}=(x_{i},y_{i}),i=1,\\dots,m$ as follows :\n\n$$\n\\hat{\\mathcal{R}}_{m}(\\mathcal{F}): =\\mathbb{E}_{\\sigma}\\left[\\operatorname*{sup}_{f\\in\\mathcal{F}}\\frac{1}{m}\\sum_{i=1}^{m}\\sigma_{i}f(z_{i})\\right],\n$$\n\nwhere $\\sigma=(\\sigma_{1},\\ldots,\\sigma_{m}) $ is a vector of i.i.d. Rademacher variables. The Rademacher Complexity is defined as the expectation of this quantity:\n\n$$\n\\mathcal{R}_{m}(\\mathcal{F}): =\\mathbb{E}_{(z_{1},\\ldots,z_{m})\\sim P^{m}}\\left[\\hat{\\mathcal{R}}_{m}(\\mathcal{F})\\right].\n$$"}, {"name": "Lemma 20 (Hoeffding's Inequality; Shalev-Shwartz & Ben-David Lemma 4.5)", "statement": "The following Hoeffding's lemma can be found in (Shalev-Shwartz and Ben-David, 2014, Lemma 4.5).\n\nLemma 20 (Hoeffding's Inequality) Let $\\theta_{1},\\ldots,\\theta_{m}$ be a sequence of i.i.d. random variables, and assume that for all $i,\\mathbb{E}[\\theta_{i}]=\\mu$ and $\\mathbb{P}[ a \\leq \\theta_{i}\\leq b ] = 1$ . Then, for any $\\varepsilon > 0$ ,\n\n$$\n\\mathbb{P}\\left[\\left|\\frac{1}{m}\\sum_{i=1}^{m}\\theta_{i}-\\mu\\right| >\\varepsilon\\right]\\leq 2\\exp{(-2m\\varepsilon^{2}/(b-a)^{2})}.\n$$"}, {"name": "Lemma 21 (Talagrand's Lemma; Mohri et al. Lemma 4.2)", "statement": "We next recall an important lemma that is useful for bounding the Rademacher complexity and can be found in (Mohri et al., 2012, Lemma 4.2).\n\nLemma 21 (Talagrand's Lemma) Let ψ be a ρ-Lipschitz function. For any function class $\\mathcal{H}$, we have:\n\n$$\n\\mathcal{R}_{m}(\\psi\\circ\\mathcal{H})\\leq\\rho\\mathcal{R}_{m}(\\mathcal{H}).\n$$"}, {"name": "Lemma 22 (Shalev-Shwartz & Ben-David Lemma A.1)", "statement": "The following lemma is fundamental and can be found in (Shalev-Shwartz and Ben-David, 2014, Lemma A.1).\n\nLemma 22 Let $a > 1$ and $b > 0$ Then: $x\\geq 4a\\log(2a)+2b\\implies x\\geq a\\log(x)+b.$"}], "nl_notes": "### Theorem 7\nNOTE (Theorems 7, 9 and 12) — In the Lean the robust risk is defined directly by the dual formula of Proposition 4, with η restricted as in Lemma 5: it is the infimum over η of c₂(ρ)·√(E[(1[h(x) ≠ y] − η)₊²]) + η, with η ∈ [−1/(c₂(ρ) − 1), 1] when ρ > 0 and η ≤ 1 when ρ = 0. The paper's own definition, eq. (1), a supremum over Q ≪ P with D₂(Q‖P) ≤ ρ, does not appear, so any step of the paper's proofs that uses the supremum form has to be redone with the dual formula. Theorem 12 also covers ρ = 0, for which Proposition 4 is not stated; at ρ = 0 the formula equals the ordinary error P(h(x) ≠ y).\n\nNOTE (Theorem 7) — Read with a constant independent of ρ, as eq. (3) displays it, the bound is false: a two-point family needs at least on the order of ρ·log(1/δ) samples, while the right-hand side of (3) stays bounded as ρ grows; the paper's own proof (eqs. (17)–(21)) carries c₂(ρ)-dependent factors into its constant. The statement therefore fixes ρ > 0 and lets the constant hidden in O(·) depend on ρ; only the dependence on ε, δ and d is claimed.\n\nNOTE (Theorems 7 and 12) — The paper's proofs need a measurability condition on H, which the paper leaves implicit: Theorem 7's proof bounds a supremum over the class by \"a standard symmetrization argument\" (whose high-probability step is McDiarmid's inequality), and Theorem 12's proof applies Fubini (Claim 1, eq. (23)) to events defined by ∃h ∈ H. Several standard choices exist (a countable class, an image-admissible Suslin class, a pointwise-measurable class); the Lean uses the pointwise-measurable one: H has a countable subclass H₀ such that every h ∈ H is the pointwise limit of a sequence in H₀, which for Bool-valued h means eventually equal to h at each x (`∃ H₀ ⊆ H, H₀.Countable ∧ …`).", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen MeasureTheory\n\nstructure Predictor (𝒳 𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] where\n\n  f : 𝒳 → 𝒴'\n\n  hf : Measurable f\n\nabbrev HypothesisClass (𝒳 𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] :=\n  Set (Predictor 𝒳 𝒴')\n\nvariable {𝒳 : Type*} [MeasurableSpace 𝒳]\n\ndef Shatters (H : HypothesisClass 𝒳 Bool) (S : Finset 𝒳) : Prop :=\n  ∀ b : S → Bool, ∃ h ∈ H, ∀ x : S, h.f x.val = b x\n\nnoncomputable def VCDim (H : HypothesisClass 𝒳 Bool) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : Shatters H S), (S.card : ℕ∞)\n\nnoncomputable def c2 (ρ : ℝ) : ℝ := Real.sqrt (1 + 2 * ρ)\n\nnoncomputable def etaDomain (ρ : ℝ) : Set ℝ :=\n  if ρ = 0 then Set.Iic 1 else Set.Icc (-(1 / (c2 ρ - 1))) 1\n\nnoncomputable def robustRisk {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (h : Predictor 𝒳 Bool) (P : Measure (𝒳 × Bool)) (ρ : ℝ) : ℝ :=\n  ⨅ η : etaDomain ρ,\n    c2 ρ * Real.sqrt\n        ((P {z : 𝒳 × Bool | h.f z.1 ≠ z.2}).toReal * (max (1 - (η : ℝ)) 0) ^ 2\n          + (P {z : 𝒳 × Bool | h.f z.1 = z.2}).toReal * (max (0 - (η : ℝ)) 0) ^ 2)\n      + (η : ℝ)\n\ndef AgnosticPACAtSize {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) (n : ℕ) : Prop :=\n  ∃ A : (Fin n → 𝒳 × Bool) → Predictor 𝒳 Bool,\n    ∀ (P : Measure (𝒳 × Bool)) [IsProbabilityMeasure P],\n      (Measure.pi fun _ : Fin n => P)\n          {S : Fin n → 𝒳 × Bool |\n            (⨅ h : ↥H, robustRisk (h : Predictor 𝒳 Bool) P ρ) + ε < robustRisk (A S) P ρ}\n        ≤ ENNReal.ofReal δ\n\nnoncomputable def agnosticSampleComplexity {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) : ℕ∞ :=\n  sInf {n : ℕ∞ | ∃ m : ℕ, n = (m : ℕ∞) ∧ AgnosticPACAtSize H ρ ε δ m}\n\ndef RealizablePACAtSize {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) (n : ℕ) : Prop :=\n  ∃ A : (Fin n → 𝒳 × Bool) → Predictor 𝒳 Bool,\n    ∀ (P : Measure (𝒳 × Bool)) [IsProbabilityMeasure P],\n      (∃ hstar ∈ H, robustRisk hstar P ρ = 0) →\n      (Measure.pi fun _ : Fin n => P)\n          {S : Fin n → 𝒳 × Bool | ε < robustRisk (A S) P ρ}\n        ≤ ENNReal.ofReal δ\n\nnoncomputable def realizableSampleComplexity {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) : ℕ∞ :=\n  sInf {n : ℕ∞ | ∃ m : ℕ, n = (m : ℕ∞) ∧ RealizablePACAtSize H ρ ε δ m}\n\ntheorem theorem7 :\n    ∀ ρ : ℝ, 0 < ρ →\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) 1,\n      ∃ n : ℕ, agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((d / ε ^ 4) * Real.log (d / ε ^ 4)\n                        + (1 / ε ^ 4) * max (1 / (c2 ρ - 1)) 1\n                        + (d / ε ^ 4) * Real.log (Real.exp 1 / d)\n                        + Real.log (2 / δ) / ε ^ 4) := by\n  sorry\n\ntheorem theorem9 :\n    ∃ c : ℝ, 0 < c ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty → ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ρ : ℝ, 0 < ρ → ρ < (3 - 2 * Real.sqrt 2) / 2 →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) (1 / (8 * Real.sqrt 2)), ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 8), ∀ n : ℕ,\n      agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) →\n      c * (((1 / 2 - (Real.sqrt 2 / 16) * Real.sqrt ρ) ^ 2 * d + Real.log (1 / δ)) / ε ^ 2)\n        ≤ (n : ℝ) := by\n  sorry\n\ntheorem theorem12 :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d → ∀ ρ : ℝ, 0 ≤ ρ →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 4),\n      ∃ n : ℕ, realizableSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((16 * (1 + 2 * ρ) * d / ε ^ 2) * Real.log (8 * (1 + 2 * ρ) * d / ε ^ 2)\n                        + (8 * (1 + 2 * ρ) / ε ^ 2)\n                            * (d * Real.log (2 * Real.exp 1 / d) + Real.log (2 / δ))) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem7", "theorem9", "theorem12"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 2306, "end": 2993, "gold": ":\n    ∀ ρ : ℝ, 0 < ρ →\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) 1,\n      ∃ n : ℕ, agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((d / ε ^ 4) * Real.log (d / ε ^ 4)\n                        + (1 / ε ^ 4) * max (1 / (c2 ρ - 1)) 1\n                        + (d / ε ^ 4) * Real.log (Real.exp 1 / d)\n                        + Real.log (2 / δ) / ε ^ 4)"}]}
{"item_id": "v24-22-0881::theorem9::A-stmt", "paper_id": "v24-22-0881", "target_name": "theorem9", "source": "Research-level", "nl_statement": "Theorem 9 For any $\\mathcal{H}$ with $vc(\\mathcal{H})=d$ and $\\mathcal{U}(P)=\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\}$ with $\\rho \\in$ $\\begin{array}{r}{\\left(0,\\frac{3-2\\sqrt{2}}{2}\\right),\\forall\\varepsilon\\in(0,\\frac{1}{8\\sqrt{2}}),\\forall\\delta\\in(0,\\frac{1}{8})}\\end{array}$\n\n$$\n\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})=\\Omega\\Bigg(\\frac{\\left(\\frac{1}{2}-\\frac{\\sqrt{2}}{16}\\rho^{1/2}\\right)^{2}d+\\log(1/\\delta)}{\\varepsilon^{2}}\\Bigg).\\tag{4}\n$$", "context": [{"name": "Setup: robust risk Eq. (1) and DRERM (Section 1)", "statement": "Concretely, let $\\mathcal{X}$ be the instance space, while $\\mathcal{Y}=\\{+1,-1\\}$ denotes the label space. We formalize the distributional shift that we would like to protect against as an uncertainty set $\\mathcal{U}(P)$ containing distributions with certain constraints, such as moment condition (Delage and Ye, 2010), Wasserstein distance (Gao, 2020) and f-divergence (Duchi and Namkoong, 2021). For a distribution P over $\\mathcal{X}\\times\\mathcal{Y}=\\{(x,y):x\\in\\mathcal{X},y\\in\\mathcal{Y}\\}$ , we observe m i.i.d. samples $S \\sim P^{m}$ , and distributionally robust learning attempts to learn a predictor $h : \\mathcal{X}\\to \\mathcal{Y}$ having small distributionally robust risk,\n\n$$\nR_{\\mathcal{U}}(h;P): =\\operatorname*{sup}_{Q\\in\\mathcal{U}(P)}\\mathbb{E}_{(x,y)\\sim Q}[\\mathbb{1}[h(x)\\neq y]].\\tag{1}\n$$\n\nThe common approach to distributionally robust learning involves selecting a hypothesis class $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ and learning a predictor $\\hat{h}: \\mathcal{X}\\to \\mathcal{Y}$ from $\\mathcal{H}$ through Distributionally Robust Empirical Risk Minimization:\n\n$$\n\\begin{array}{r}{\\hat{h}\\in\\mathrm{DRERM}_{\\mathcal{H}}(S): =\\underset{h\\in\\mathcal{H}}{\\mathrm{argmin}}\\hat{R}_{\\mathcal{U}}(h;S),}\\end{array}\n$$\n\nwhere $\\begin{array}{r}{\\hat{R}_{\\mathcal{U}}(h;S): =\\operatorname*{sup}_{Q\\in\\mathcal{U}(P_{m})}\\mathbb{E}_{(x,y)\\sim Q}[\\mathbb{1}\\left[h(x)\\neq y\\right]]}\\end{array}$ and $P_{m}$ denotes the empirical distribution over samples S."}, {"name": "Uncertainty set and chi^2-divergence D_2 (Section 1)", "statement": "Namely, we consider the distributional shift as follows:\n\n$$\n\\mathcal{U}(P)=\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\},\n$$\n\nwhere $\\begin{array}{r}{D_{2}(Q\\|P): =\\frac{1}{2}\\int\\left(\\frac{dQ}{dP}-1\\right)^{2}dP}\\end{array}$ and $Q \\ll P$ indicates that distribution Q is absolutely continuous with respect to $P .$ The $\\chi^{2}$-divergence is a commonly explored concept in the distributionally robust optimization (DRO) literature (Duchi and Namkoong, 2019). Moreover, it is also a crucial concept in a variety of fields such as information theory, statistics, learning, signal processing, and various branches of mathematics (Park et al., 2011; Saraswat, 2014; Nishiyama and Sason, 2020). The $\\chi^{2}$-divergence plays a fundamental role in problems related to source and channel coding, combinatorics, large deviation theory, goodness-of-fit, and independence tests in statistics, as demonstrated by Csiszár et al. (2004). Additionally, it is widely recognized for its computational feasibility and statistical properties, as noted in Tsybakov (2009)."}, {"name": "Problem setup (Section 2)", "statement": "Given a hypothesis class $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ , our goal is to design a learning rule $\\mathcal{A}:(\\mathcal{X}\\times\\mathcal{Y})^{*}\\mapsto\\mathcal{Y}^{\\mathcal{X}}$ such that for any distribution P over $\\mathcal{X}\\times \\mathcal{Y}$ , the rule $\\mathcal{A}$ will find a predictor that can compete with the best predictor $h^{\\ast}\\in \\mathcal{H}$ in terms of the distributionally robust risk using a number of samples that is independent of the distribution P. In this paper, we use $(\\mathcal{X}\\times\\mathcal{Y})^{*}$ to denote the set of all sequences in the space $\\mathcal{X}\\times \\mathcal{Y}$ . The following definitions formalize the notion of distributionally robust PAC learning under the realizable case and agnostic settings."}, {"name": "Definition 1 (Agnostic Distributionally Robust PAC Learnability)", "statement": "Definition 1 (Agnostic Distributionally Robust PAC Learnability) For any $\\varepsilon,\\delta\\in$ (0, 1), the sample complexity of agnostic distributionally robust $(\\varepsilon,\\delta)$-PAC learning of $\\mathcal{H}$ with respect to the distributional shift $\\mathcal{U}$, denoted by $\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})$ , is defined as the smallest $m \\in \\mathbb{N}\\cup \\{0 \\}$ for which there exists a learning rule $\\mathcal{A}:(\\mathcal{X}\\times\\mathcal{Y})^{*}\\mapsto\\mathcal{Y}^{\\mathcal{X}}$ such that, for every data distribution P over $\\mathcal{X}\\times \\mathcal{Y}$ , with probability of at least $1 - \\delta$ over $S \\sim P^{m}$ ,\n\n$$\nR_{\\mathcal{U}}(\\mathcal{A}(S);P)\\leq\\operatorname*{inf}_{h\\in\\mathcal{H}}R_{\\mathcal{U}}(h;P)+\\varepsilon .\n$$\n\nIf no such m exists, define $\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})=\\infty$ . We say that $\\mathcal{H}$ is distributionally robustly PAC-learnable in the agnostic setting with respect to the distributional shift $\\mathcal{U}$ if $\\forall\\varepsilon,\\delta\\in(0,1)$ , $\\mathcal{M}_{AG}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})$ scales polynomially with $1 / \\varepsilon$ and $1 / \\delta$"}, {"name": "Notation: error rate er(h;P) and empirical error", "statement": "We also denote $\\mathrm{er}(h;P)=P(h(x)\\neq y)$ , the (non-robust) error rate under 0-1 loss, and $\\begin{array}{r}{\\hat{\\mathrm{er}}(h;S)=\\frac{1}{|S|}\\sum_{(x,y)\\in S}\\mathbb{1}[h(x)\\neq y]}\\end{array}$ , the empirical error rate; here, |S| denotes the cardinality of set S, while $\\mathbb{1}[\\cdot]$ is the indicator function that takes 1 when the statement in the square brackets is true and 0 otherwise. The definition of Vapnik-Chervonenkis dimension (VC dimension) is provided below."}, {"name": "Definition 3 (VC dimension)", "statement": "Definition 3 (VC dimension) We say that a sequence $\\{x_{1},\\ldots,x_{k}\\}\\subseteq{\\mathcal{X}}$ is shattered by $\\mathcal{H}$ if $\\forall y_{1},\\ldots,y_{k}\\in\\mathcal{Y},\\exists h\\in\\mathcal{H}$ such that $\\forall i\\in [k],h(x_{i})=y_{i}$ . The VC dimension of $\\mathcal{H}$ (denoted as $vc(\\mathcal{H}))$ is then defined as the largest integer k for which there exists $\\{x_{1},\\ldots,x_{k}\\}\\subseteq{\\mathcal{X}}$ that is shattered by $\\mathcal{H}$. If no such k exists, then $vc(\\mathcal{H})$ is said to be infinite."}, {"name": "Notation R_2(h;P) (Section 3)", "statement": "We use $R_{2}(h;P)$ to denote the distributionally robust risk under distributional shift\n\n$$\n\\begin{array}{r}{\\mathcal{U}(P)=\\left\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\right\\}.}\\end{array}\n$$"}, {"name": "Proposition 4 (Duality Formulation), Eq. (2) (Shapiro 2017)", "statement": "We recall the following duality formulation (Shapiro, 2017, Section 3.2) for distributionally robust risk, which is essential in our derivation.\n\nProposition 4 (Duality Formulation) For any probability P on $\\mathcal{X}\\times \\mathcal{Y}$ , any $\\rho > 0$ , and $c_{2}(\\rho): =\\sqrt{1+2\\rho},$ for all $h \\in \\mathcal H$ , we have\n\n$$\nR_{2}(h;P)=\\operatorname*{inf}_{\\eta\\in\\mathbb{R}}\\Big\\{c_{2}(\\rho)\\mathbb{E}_{P}\\Big [(\\mathbb{1}[h(x)\\neq y]-\\eta)_{+}^{2}\\Big]^{1/2}+\\eta\\Big\\},\\tag{2}\n$$\n\nwhere $a_{+}: =\\operatorname*{max}\\left(a,0\\right)$"}, {"name": "Remark 10 (range of rho in Theorem 9)", "statement": "Remark 10 We derive the result under the assumption that the perturbation size $\\rho \\in$ $\\textstyle\\left(0,{\\frac{3-2{\\sqrt{2}}}{2}}\\right)$ . Intuitively, if the perturbation size is prohibitively large, the learning problem can become \"easier\", since the benchmark in $\\inf_{h\\in\\mathcal{H}}R_{2}(h;P)$ may increase too much. When $\\rho$ approaches 0, the lower bound recovers that of the classical statistical learning (Mohri et al., 2012; Shalev-Shwartz and Ben-David, 2014)."}, {"name": "Lemma 23 (closed form of R_2 on two-point distributions)", "statement": "Lemma 23 For $P_{b}(x,y)=\\begin{cases}\\frac{1+yb\\varepsilon}{2},&\\text{if } x=c,\\\\ 0,&\\text{otherwise.}\\end{cases}$ and $\\rho\\in\\left(0,{\\frac{3-2{\\sqrt{2}}}{2}}\\right)$ , we have $R_{2}(h;P_{b})=$ $\\left[1-h(c)b\\varepsilon+\\sqrt{2\\rho(1-\\varepsilon^{2})}\\right]/2.$"}, {"name": "Lemma 24 (Slud's Inequality; Slud 1977)", "statement": "We frequently employ the following estimate on the binomial random variable probability (Slud, 1977):\n\nLemma 24 (Slud's Inequality) Let X be $a\\ (m,p)$ binomial random variable and assume that $p=(1-\\varepsilon)/2$ . Then,\n\n$$\n\\mathbb{P}[X\\ge m/2]\\ge\\frac{1}{2}\\left(1-\\sqrt{1-\\exp(-m\\varepsilon^{2}/(1-\\varepsilon^{2}))}\\right).\n$$"}, {"name": "Lemma 25 (Maximum-Likelihood rule minimizes (14))", "statement": "Lemma 25 Among all algorithms, (14) is minimized for $\\mathcal{A}$ being the Maximum-Likelihood algorithm, $\\mathcal{A}_{M L}$ , defined as\n\n$$\n\\forall i,\\quad\\mathcal{A}_{ML}(S)(c_{i})=sign\\left(\\sum_{k:x_{k}=c_{i}}y_{k}\\right).\n$$"}, {"name": "Lemma 26 (range of Delta)", "statement": "Lemma 26 Let $\\Delta=\\frac{1}{r}\\left(R_{2}(\\mathcal{A}(S);P)-\\operatorname*{inf}_{h\\in\\mathcal{H}}R_{2}(h;P)\\right)$ , where r is defined in the proof of Theorem 9. We have $\\Delta\\in [0,1+\\sqrt{2}\\rho^{1/2}/4]$"}, {"name": "Lemma 27 (reverse Markov inequality)", "statement": "Lemma 27 Let Z be a random variable that takes values in $[0,c],c>1$ . Assume that $\\mathbb{E}[ Z ] = \\mu$ . Then, for any $a\\in(0,c)$ ,\n\n$$\n\\mathbb{P}[ Z > a ] \\geq \\frac{\\mu - a}{c - a}.\n$$"}], "nl_notes": "### Theorem 9\nNOTE (Theorems 7, 9 and 12) — In the Lean the robust risk is defined directly by the dual formula of Proposition 4, with η restricted as in Lemma 5: it is the infimum over η of c₂(ρ)·√(E[(1[h(x) ≠ y] − η)₊²]) + η, with η ∈ [−1/(c₂(ρ) − 1), 1] when ρ > 0 and η ≤ 1 when ρ = 0. The paper's own definition, eq. (1), a supremum over Q ≪ P with D₂(Q‖P) ≤ ρ, does not appear, so any step of the paper's proofs that uses the supremum form has to be redone with the dual formula. Theorem 12 also covers ρ = 0, for which Proposition 4 is not stated; at ρ = 0 the formula equals the ordinary error P(h(x) ≠ y).\n\nNOTE (Theorem 9) — The paper states (4) for all ε, δ ∈ (0, 1), which is false on that range (for the class of the two constant classifiers, which has d = 1, and any δ ≥ 1/2, a single sample suffices for every ε, while the right-hand side grows without bound as ε → 0). Its proof in Section 6 obtains the d-dependent part only for ε < 1/(8√2) and δ ≤ 1/8, and the statement claims the lower bound only for ε ∈ (0, 1/(8√2)) and δ ∈ (0, 1/8).", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen MeasureTheory\n\nstructure Predictor (𝒳 𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] where\n\n  f : 𝒳 → 𝒴'\n\n  hf : Measurable f\n\nabbrev HypothesisClass (𝒳 𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] :=\n  Set (Predictor 𝒳 𝒴')\n\nvariable {𝒳 : Type*} [MeasurableSpace 𝒳]\n\ndef Shatters (H : HypothesisClass 𝒳 Bool) (S : Finset 𝒳) : Prop :=\n  ∀ b : S → Bool, ∃ h ∈ H, ∀ x : S, h.f x.val = b x\n\nnoncomputable def VCDim (H : HypothesisClass 𝒳 Bool) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : Shatters H S), (S.card : ℕ∞)\n\nnoncomputable def c2 (ρ : ℝ) : ℝ := Real.sqrt (1 + 2 * ρ)\n\nnoncomputable def etaDomain (ρ : ℝ) : Set ℝ :=\n  if ρ = 0 then Set.Iic 1 else Set.Icc (-(1 / (c2 ρ - 1))) 1\n\nnoncomputable def robustRisk {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (h : Predictor 𝒳 Bool) (P : Measure (𝒳 × Bool)) (ρ : ℝ) : ℝ :=\n  ⨅ η : etaDomain ρ,\n    c2 ρ * Real.sqrt\n        ((P {z : 𝒳 × Bool | h.f z.1 ≠ z.2}).toReal * (max (1 - (η : ℝ)) 0) ^ 2\n          + (P {z : 𝒳 × Bool | h.f z.1 = z.2}).toReal * (max (0 - (η : ℝ)) 0) ^ 2)\n      + (η : ℝ)\n\ndef AgnosticPACAtSize {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) (n : ℕ) : Prop :=\n  ∃ A : (Fin n → 𝒳 × Bool) → Predictor 𝒳 Bool,\n    ∀ (P : Measure (𝒳 × Bool)) [IsProbabilityMeasure P],\n      (Measure.pi fun _ : Fin n => P)\n          {S : Fin n → 𝒳 × Bool |\n            (⨅ h : ↥H, robustRisk (h : Predictor 𝒳 Bool) P ρ) + ε < robustRisk (A S) P ρ}\n        ≤ ENNReal.ofReal δ\n\nnoncomputable def agnosticSampleComplexity {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) : ℕ∞ :=\n  sInf {n : ℕ∞ | ∃ m : ℕ, n = (m : ℕ∞) ∧ AgnosticPACAtSize H ρ ε δ m}\n\ndef RealizablePACAtSize {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) (n : ℕ) : Prop :=\n  ∃ A : (Fin n → 𝒳 × Bool) → Predictor 𝒳 Bool,\n    ∀ (P : Measure (𝒳 × Bool)) [IsProbabilityMeasure P],\n      (∃ hstar ∈ H, robustRisk hstar P ρ = 0) →\n      (Measure.pi fun _ : Fin n => P)\n          {S : Fin n → 𝒳 × Bool | ε < robustRisk (A S) P ρ}\n        ≤ ENNReal.ofReal δ\n\nnoncomputable def realizableSampleComplexity {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) : ℕ∞ :=\n  sInf {n : ℕ∞ | ∃ m : ℕ, n = (m : ℕ∞) ∧ RealizablePACAtSize H ρ ε δ m}\n\ntheorem theorem7 :\n    ∀ ρ : ℝ, 0 < ρ →\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) 1,\n      ∃ n : ℕ, agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((d / ε ^ 4) * Real.log (d / ε ^ 4)\n                        + (1 / ε ^ 4) * max (1 / (c2 ρ - 1)) 1\n                        + (d / ε ^ 4) * Real.log (Real.exp 1 / d)\n                        + Real.log (2 / δ) / ε ^ 4) := by\n  sorry\n\ntheorem theorem9 :\n    ∃ c : ℝ, 0 < c ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty → ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ρ : ℝ, 0 < ρ → ρ < (3 - 2 * Real.sqrt 2) / 2 →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) (1 / (8 * Real.sqrt 2)), ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 8), ∀ n : ℕ,\n      agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) →\n      c * (((1 / 2 - (Real.sqrt 2 / 16) * Real.sqrt ρ) ^ 2 * d + Real.log (1 / δ)) / ε ^ 2)\n        ≤ (n : ℝ) := by\n  sorry\n\ntheorem theorem12 :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d → ∀ ρ : ℝ, 0 ≤ ρ →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 4),\n      ∃ n : ℕ, realizableSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((16 * (1 + 2 * ρ) * d / ε ^ 2) * Real.log (8 * (1 + 2 * ρ) * d / ε ^ 2)\n                        + (8 * (1 + 2 * ρ) / ε ^ 2)\n                            * (d * Real.log (2 * Real.exp 1 / d) + Real.log (2 / δ))) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem7", "theorem9", "theorem12"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "holes": [{"start": 3026, "end": 3491, "gold": ":\n    ∃ c : ℝ, 0 < c ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty → ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ρ : ℝ, 0 < ρ → ρ < (3 - 2 * Real.sqrt 2) / 2 →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) (1 / (8 * Real.sqrt 2)), ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 8), ∀ n : ℕ,\n      agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) →\n      c * (((1 / 2 - (Real.sqrt 2 / 16) * Real.sqrt ρ) ^ 2 * d + Real.log (1 / δ)) / ε ^ 2)\n        ≤ (n : ℝ)"}]}
{"item_id": "v24-22-0881::theorem12::A-stmt", "paper_id": "v24-22-0881", "target_name": "theorem12", "source": "Research-level", "nl_statement": "Theorem 12 Let $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ with $vc(\\mathcal{H})=d$ and $\\mathcal{U}(P)=\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\}$ . $\\forall\\varepsilon\\in(0,1),\\forall\\delta\\in(0,1/4)$ , we have\n\n$$\n\\begin{array}{r l}&{\\mathcal{M}_{RE}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})=O\\Bigg(\\frac{16(1+2\\rho)d}{\\varepsilon^{2}}\\log\\left(\\frac{8(1+2\\rho)d}{\\varepsilon^{2}}\\right)}\\\\ &{\\qquad+\\frac{8(1+2\\rho)}{\\varepsilon^{2}}\\left(d\\log\\left(\\frac{2e}{d}\\right)+\\log\\left(\\frac{2}{\\delta}\\right)\\right)\\Bigg).}\\end{array}\\tag{6}\n$$", "context": [{"name": "Setup: robust risk Eq. (1) and DRERM (Section 1)", "statement": "Concretely, let $\\mathcal{X}$ be the instance space, while $\\mathcal{Y}=\\{+1,-1\\}$ denotes the label space. We formalize the distributional shift that we would like to protect against as an uncertainty set $\\mathcal{U}(P)$ containing distributions with certain constraints, such as moment condition (Delage and Ye, 2010), Wasserstein distance (Gao, 2020) and f-divergence (Duchi and Namkoong, 2021). For a distribution P over $\\mathcal{X}\\times\\mathcal{Y}=\\{(x,y):x\\in\\mathcal{X},y\\in\\mathcal{Y}\\}$ , we observe m i.i.d. samples $S \\sim P^{m}$ , and distributionally robust learning attempts to learn a predictor $h : \\mathcal{X}\\to \\mathcal{Y}$ having small distributionally robust risk,\n\n$$\nR_{\\mathcal{U}}(h;P): =\\operatorname*{sup}_{Q\\in\\mathcal{U}(P)}\\mathbb{E}_{(x,y)\\sim Q}[\\mathbb{1}[h(x)\\neq y]].\\tag{1}\n$$\n\nThe common approach to distributionally robust learning involves selecting a hypothesis class $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ and learning a predictor $\\hat{h}: \\mathcal{X}\\to \\mathcal{Y}$ from $\\mathcal{H}$ through Distributionally Robust Empirical Risk Minimization:\n\n$$\n\\begin{array}{r}{\\hat{h}\\in\\mathrm{DRERM}_{\\mathcal{H}}(S): =\\underset{h\\in\\mathcal{H}}{\\mathrm{argmin}}\\hat{R}_{\\mathcal{U}}(h;S),}\\end{array}\n$$\n\nwhere $\\begin{array}{r}{\\hat{R}_{\\mathcal{U}}(h;S): =\\operatorname*{sup}_{Q\\in\\mathcal{U}(P_{m})}\\mathbb{E}_{(x,y)\\sim Q}[\\mathbb{1}\\left[h(x)\\neq y\\right]]}\\end{array}$ and $P_{m}$ denotes the empirical distribution over samples S."}, {"name": "Uncertainty set and chi^2-divergence D_2 (Section 1)", "statement": "Namely, we consider the distributional shift as follows:\n\n$$\n\\mathcal{U}(P)=\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\},\n$$\n\nwhere $\\begin{array}{r}{D_{2}(Q\\|P): =\\frac{1}{2}\\int\\left(\\frac{dQ}{dP}-1\\right)^{2}dP}\\end{array}$ and $Q \\ll P$ indicates that distribution Q is absolutely continuous with respect to $P .$ The $\\chi^{2}$-divergence is a commonly explored concept in the distributionally robust optimization (DRO) literature (Duchi and Namkoong, 2019). Moreover, it is also a crucial concept in a variety of fields such as information theory, statistics, learning, signal processing, and various branches of mathematics (Park et al., 2011; Saraswat, 2014; Nishiyama and Sason, 2020). The $\\chi^{2}$-divergence plays a fundamental role in problems related to source and channel coding, combinatorics, large deviation theory, goodness-of-fit, and independence tests in statistics, as demonstrated by Csiszár et al. (2004). Additionally, it is widely recognized for its computational feasibility and statistical properties, as noted in Tsybakov (2009)."}, {"name": "Problem setup (Section 2)", "statement": "Given a hypothesis class $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ , our goal is to design a learning rule $\\mathcal{A}:(\\mathcal{X}\\times\\mathcal{Y})^{*}\\mapsto\\mathcal{Y}^{\\mathcal{X}}$ such that for any distribution P over $\\mathcal{X}\\times \\mathcal{Y}$ , the rule $\\mathcal{A}$ will find a predictor that can compete with the best predictor $h^{\\ast}\\in \\mathcal{H}$ in terms of the distributionally robust risk using a number of samples that is independent of the distribution P. In this paper, we use $(\\mathcal{X}\\times\\mathcal{Y})^{*}$ to denote the set of all sequences in the space $\\mathcal{X}\\times \\mathcal{Y}$ . The following definitions formalize the notion of distributionally robust PAC learning under the realizable case and agnostic settings."}, {"name": "Definition 2 (Realizable Distributionally Robust PAC Learnability)", "statement": "Definition 2 (Realizable Distributionally Robust PAC Learnability) For any $\\varepsilon,\\delta\\in(0,1)$, the sample complexity of realizable distributionally robust $(\\varepsilon,\\delta)$-PAC learning of $\\mathcal{H}$ with respect to the distributional shift $\\mathcal{U}$, denoted by $\\mathcal{M}_{RE}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})$ , is defined as the smallest $m \\in \\mathbb{N}\\cup \\{0 \\}$ for which there exists a learning rule $\\mathcal{A}:(\\mathcal{X}\\times\\mathcal{Y})^{*}\\mapsto\\mathcal{Y}^{\\mathcal{X}}$ such that, for every data distribution P over $\\mathcal{X}\\times \\mathcal{Y}$ where there exists a predictor $h^{\\ast}\\in \\mathcal{H}$ with zero distributionally robust risk, $R_{{\\mathcal U}}(h^{*};P)=0$ , with probability of at least $1 - \\delta$ over $S \\sim P^{m}$ ,\n\n$$\nR_{\\mathcal{U}}(\\mathcal{A}(S);P)\\leq\\varepsilon .\n$$\n\nIf no such m exists, define $\\mathcal{M}_{RE}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})=\\infty$ . We say that $\\mathcal{H}$ is distributionally robustly PAC-learnable in the realizable setting with respect to the distributional shift $\\mathcal{U}$ if $\\forall\\varepsilon,\\delta\\in(0,1)$ , $\\mathcal{M}_{RE}(\\varepsilon,\\delta;\\mathcal{H},\\mathcal{U})$ scales polynomially with $1 / \\varepsilon$ and $1 / \\delta$"}, {"name": "Notation: error rate er(h;P) and empirical error", "statement": "We also denote $\\mathrm{er}(h;P)=P(h(x)\\neq y)$ , the (non-robust) error rate under 0-1 loss, and $\\begin{array}{r}{\\hat{\\mathrm{er}}(h;S)=\\frac{1}{|S|}\\sum_{(x,y)\\in S}\\mathbb{1}[h(x)\\neq y]}\\end{array}$ , the empirical error rate; here, |S| denotes the cardinality of set S, while $\\mathbb{1}[\\cdot]$ is the indicator function that takes 1 when the statement in the square brackets is true and 0 otherwise. The definition of Vapnik-Chervonenkis dimension (VC dimension) is provided below."}, {"name": "Definition 3 (VC dimension)", "statement": "Definition 3 (VC dimension) We say that a sequence $\\{x_{1},\\ldots,x_{k}\\}\\subseteq{\\mathcal{X}}$ is shattered by $\\mathcal{H}$ if $\\forall y_{1},\\ldots,y_{k}\\in\\mathcal{Y},\\exists h\\in\\mathcal{H}$ such that $\\forall i\\in [k],h(x_{i})=y_{i}$ . The VC dimension of $\\mathcal{H}$ (denoted as $vc(\\mathcal{H}))$ is then defined as the largest integer k for which there exists $\\{x_{1},\\ldots,x_{k}\\}\\subseteq{\\mathcal{X}}$ that is shattered by $\\mathcal{H}$. If no such k exists, then $vc(\\mathcal{H})$ is said to be infinite."}, {"name": "Notation R_2(h;P) (Section 3)", "statement": "We use $R_{2}(h;P)$ to denote the distributionally robust risk under distributional shift\n\n$$\n\\begin{array}{r}{\\mathcal{U}(P)=\\left\\{Q\\ll P:D_{2}(Q\\|P)\\leq\\rho\\right\\}.}\\end{array}\n$$"}, {"name": "Proposition 4 (Duality Formulation), Eq. (2) (Shapiro 2017)", "statement": "We recall the following duality formulation (Shapiro, 2017, Section 3.2) for distributionally robust risk, which is essential in our derivation.\n\nProposition 4 (Duality Formulation) For any probability P on $\\mathcal{X}\\times \\mathcal{Y}$ , any $\\rho > 0$ , and $c_{2}(\\rho): =\\sqrt{1+2\\rho},$ for all $h \\in \\mathcal H$ , we have\n\n$$\nR_{2}(h;P)=\\operatorname*{inf}_{\\eta\\in\\mathbb{R}}\\Big\\{c_{2}(\\rho)\\mathbb{E}_{P}\\Big [(\\mathbb{1}[h(x)\\neq y]-\\eta)_{+}^{2}\\Big]^{1/2}+\\eta\\Big\\},\\tag{2}\n$$\n\nwhere $a_{+}: =\\operatorname*{max}\\left(a,0\\right)$"}, {"name": "Realizable case setup: C_h and R_2 in primal form (Section 4)", "statement": "To study the upper bound of the sample complexity under the realizable case, it is necessary to introduce the definition of Distributionally Robust ε-net, which is similar to the definition in (Shalev-Shwartz and Ben-David, 2014, Definition 28.2). In the realizable case, we have a target hypothesis $h^{*}$ that generates the label. We will frequently refer $\\mathcal{C}_{h}$ to the set $\\{x\\in\\mathcal{X}:h(x)\\neq h^{*}(x)\\}$ , where h is a predictor in hypothesis class $\\mathcal{H}$. The distributionally robust risk has the following form:\n\n$$\nR_{2}(h;P)=\\operatorname*{sup}_{Q\\ll P}\\left\\{\\mathbb{E}_{x\\sim Q}[\\mathbb{1}[h(x)\\neq h^{\\ast}(x)]]:D_{2}(Q\\|P)\\leq\\rho\\right\\}.\n$$"}, {"name": "Definition 11 (Distributionally Robust eps-net), Eq. (5)", "statement": "Definition 11 (Distributionally Robust ε-net) Let $\\mathcal{X}$ be a domain. $S \\subseteq{\\mathcal{X}}$ is a Distributionally Robust ε-net for $\\mathcal{H}\\subseteq \\mathcal{Y}^{\\mathcal{X}}$ with respect to a distribution P over $\\mathcal{X}$ if:\n\n$$\n\\forall h\\in{\\mathcal{H}}:R_{2}(h;P)\\geq\\varepsilon\\implies\\mathcal{C}_{h}\\cap S\\neq\\emptyset .\\tag{5}\n$$"}, {"name": "Definition 17 (Growth Function)", "statement": "Definition 17 (Growth Function) Let $\\mathcal{H}$ be a hypothesis class. The growth function of $\\mathcal{H}$ ,denoted as $\\tau_{\\mathcal{H}}: \\mathbb{N}\\to \\mathbb{N}$ , is defined as follows:\n\n$$\n\\tau_{\\mathcal{H}}(m): =\\underset{C\\subset\\mathcal{X}: |C| =m}{max}\\left|\\mathcal{H}_{C}\\right|,\n$$\n\nwhere $\\mathcal{H}_{C}: =\\left\\{(h(x_{1}),\\cdot\\cdot\\cdot,h(x_{|C|})):h\\in\\mathcal{H}\\right\\}$ for $C = \\{x_{i}: 1 \\leq i \\leq | C | \\}$"}, {"name": "Lemma 22 (Shalev-Shwartz & Ben-David Lemma A.1)", "statement": "The following lemma is fundamental and can be found in (Shalev-Shwartz and Ben-David, 2014, Lemma A.1).\n\nLemma 22 Let $a > 1$ and $b > 0$ Then: $x\\geq 4a\\log(2a)+2b\\implies x\\geq a\\log(x)+b.$"}, {"name": "Lemma 28 (R_2 <= c_2(rho) er^{1/2})", "statement": "Lemma 28 For any probability distribution P and predictor $h \\in \\mathcal H$ , we have\n\n$$\nR_{2}(h;P)\\leq c_{2}(\\rho)er(h;P)^{1/2}.\n$$"}], "nl_notes": "### Theorem 12\nNOTE (Theorems 7, 9 and 12) — In the Lean the robust risk is defined directly by the dual formula of Proposition 4, with η restricted as in Lemma 5: it is the infimum over η of c₂(ρ)·√(E[(1[h(x) ≠ y] − η)₊²]) + η, with η ∈ [−1/(c₂(ρ) − 1), 1] when ρ > 0 and η ≤ 1 when ρ = 0. The paper's own definition, eq. (1), a supremum over Q ≪ P with D₂(Q‖P) ≤ ρ, does not appear, so any step of the paper's proofs that uses the supremum form has to be redone with the dual formula. Theorem 12 also covers ρ = 0, for which Proposition 4 is not stated; at ρ = 0 the formula equals the ordinary error P(h(x) ≠ y).\n\nNOTE (Theorems 7 and 12) — The paper's proofs need a measurability condition on H, which the paper leaves implicit: Theorem 7's proof bounds a supremum over the class by \"a standard symmetrization argument\" (whose high-probability step is McDiarmid's inequality), and Theorem 12's proof applies Fubini (Claim 1, eq. (23)) to events defined by ∃h ∈ H. Several standard choices exist (a countable class, an image-admissible Suslin class, a pointwise-measurable class); the Lean uses the pointwise-measurable one: H has a countable subclass H₀ such that every h ∈ H is the pointwise limit of a sequence in H₀, which for Bool-valued h means eventually equal to h at each x (`∃ H₀ ⊆ H, H₀.Countable ∧ …`).", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen MeasureTheory\n\nstructure Predictor (𝒳 𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] where\n\n  f : 𝒳 → 𝒴'\n\n  hf : Measurable f\n\nabbrev HypothesisClass (𝒳 𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] :=\n  Set (Predictor 𝒳 𝒴')\n\nvariable {𝒳 : Type*} [MeasurableSpace 𝒳]\n\ndef Shatters (H : HypothesisClass 𝒳 Bool) (S : Finset 𝒳) : Prop :=\n  ∀ b : S → Bool, ∃ h ∈ H, ∀ x : S, h.f x.val = b x\n\nnoncomputable def VCDim (H : HypothesisClass 𝒳 Bool) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : Shatters H S), (S.card : ℕ∞)\n\nnoncomputable def c2 (ρ : ℝ) : ℝ := Real.sqrt (1 + 2 * ρ)\n\nnoncomputable def etaDomain (ρ : ℝ) : Set ℝ :=\n  if ρ = 0 then Set.Iic 1 else Set.Icc (-(1 / (c2 ρ - 1))) 1\n\nnoncomputable def robustRisk {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (h : Predictor 𝒳 Bool) (P : Measure (𝒳 × Bool)) (ρ : ℝ) : ℝ :=\n  ⨅ η : etaDomain ρ,\n    c2 ρ * Real.sqrt\n        ((P {z : 𝒳 × Bool | h.f z.1 ≠ z.2}).toReal * (max (1 - (η : ℝ)) 0) ^ 2\n          + (P {z : 𝒳 × Bool | h.f z.1 = z.2}).toReal * (max (0 - (η : ℝ)) 0) ^ 2)\n      + (η : ℝ)\n\ndef AgnosticPACAtSize {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) (n : ℕ) : Prop :=\n  ∃ A : (Fin n → 𝒳 × Bool) → Predictor 𝒳 Bool,\n    ∀ (P : Measure (𝒳 × Bool)) [IsProbabilityMeasure P],\n      (Measure.pi fun _ : Fin n => P)\n          {S : Fin n → 𝒳 × Bool |\n            (⨅ h : ↥H, robustRisk (h : Predictor 𝒳 Bool) P ρ) + ε < robustRisk (A S) P ρ}\n        ≤ ENNReal.ofReal δ\n\nnoncomputable def agnosticSampleComplexity {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) : ℕ∞ :=\n  sInf {n : ℕ∞ | ∃ m : ℕ, n = (m : ℕ∞) ∧ AgnosticPACAtSize H ρ ε δ m}\n\ndef RealizablePACAtSize {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) (n : ℕ) : Prop :=\n  ∃ A : (Fin n → 𝒳 × Bool) → Predictor 𝒳 Bool,\n    ∀ (P : Measure (𝒳 × Bool)) [IsProbabilityMeasure P],\n      (∃ hstar ∈ H, robustRisk hstar P ρ = 0) →\n      (Measure.pi fun _ : Fin n => P)\n          {S : Fin n → 𝒳 × Bool | ε < robustRisk (A S) P ρ}\n        ≤ ENNReal.ofReal δ\n\nnoncomputable def realizableSampleComplexity {𝒳 : Type*} [MeasurableSpace 𝒳]\n    (H : HypothesisClass 𝒳 Bool) (ρ ε δ : ℝ) : ℕ∞ :=\n  sInf {n : ℕ∞ | ∃ m : ℕ, n = (m : ℕ∞) ∧ RealizablePACAtSize H ρ ε δ m}\n\ntheorem theorem7 :\n    ∀ ρ : ℝ, 0 < ρ →\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) 1,\n      ∃ n : ℕ, agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((d / ε ^ 4) * Real.log (d / ε ^ 4)\n                        + (1 / ε ^ 4) * max (1 / (c2 ρ - 1)) 1\n                        + (d / ε ^ 4) * Real.log (Real.exp 1 / d)\n                        + Real.log (2 / δ) / ε ^ 4) := by\n  sorry\n\ntheorem theorem9 :\n    ∃ c : ℝ, 0 < c ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty → ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d →\n        ∀ ρ : ℝ, 0 < ρ → ρ < (3 - 2 * Real.sqrt 2) / 2 →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) (1 / (8 * Real.sqrt 2)), ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 8), ∀ n : ℕ,\n      agnosticSampleComplexity H ρ ε δ ≤ (n : ℕ∞) →\n      c * (((1 / 2 - (Real.sqrt 2 / 16) * Real.sqrt ρ) ^ 2 * d + Real.log (1 / δ)) / ε ^ 2)\n        ≤ (n : ℝ) := by\n  sorry\n\ntheorem theorem12 :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d → ∀ ρ : ℝ, 0 ≤ ρ →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 4),\n      ∃ n : ℕ, realizableSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((16 * (1 + 2 * ρ) * d / ε ^ 2) * Real.log (8 * (1 + 2 * ρ) * d / ε ^ 2)\n                        + (8 * (1 + 2 * ρ) / ε ^ 2)\n                            * (d * Real.log (2 * Real.exp 1 / d) + Real.log (2 / δ))) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem7", "theorem9", "theorem12"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 3525, "end": 4210, "gold": ":\n    ∃ C : ℝ, 0 < C ∧\n      ∀ {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool),\n        H.Nonempty →\n        (∃ H₀ ⊆ H, H₀.Countable ∧ ∀ h ∈ H, ∃ s : ℕ → Predictor 𝒳 Bool,\n            (∀ i, s i ∈ H₀) ∧ ∀ x, ∀ᶠ i in Filter.atTop, (s i).f x = h.f x) →\n        ∀ d : ℕ, VCDim H = (d : ℕ∞) → 0 < d → ∀ ρ : ℝ, 0 ≤ ρ →\n        ∀ ε ∈ Set.Ioo (0 : ℝ) 1, ∀ δ ∈ Set.Ioo (0 : ℝ) (1 / 4),\n      ∃ n : ℕ, realizableSampleComplexity H ρ ε δ ≤ (n : ℕ∞) ∧\n        (n : ℝ) ≤ C * ((16 * (1 + 2 * ρ) * d / ε ^ 2) * Real.log (8 * (1 + 2 * ρ) * d / ε ^ 2)\n                        + (8 * (1 + 2 * ρ) / ε ^ 2)\n                            * (d * Real.log (2 * Real.exp 1 / d) + Real.log (2 / δ)))"}]}
{"item_id": "v24-22-1210::lemma11::A-stmt", "paper_id": "v24-22-1210", "target_name": "lemma11", "source": "Research-level", "nl_statement": "Lemma 11 Letting $\\mathbf{x}_{t}, \\mathbf{y}_{t}, \\mathbf{s}_{t}$ be generated by Algorithm 3 with $\\eta=1/(2L)$ , it holds that\n\n$$\nV_{t+1}\\leq(1-\\alpha)V_{t}+\\frac{13\\eta}{m}\\left\\|\\mathbf{s}_{t}-\\mathbf{1}\\bar{s}_{t}\\right\\|^{2}+\\frac{20M^{2}\\eta+10\\eta^{-1}+(1+8/\\alpha^{3})M^{2}/L}{m}\\left\\|\\mathbf{y}_{t}-\\mathbf{1}\\bar{y}_{t}\\right\\|^{2}.\\tag{25}\n$$", "context": [{"name": "Setup: decentralized composite problem (Section 1, eq. (1))", "statement": "In this paper, we consider the decentralized optimization problem, where the objective function is composed of m local functions $f_{i}(x)$ that are located on m different agents. The agents form a connected and undirected network and each of them only accesses its local function and communicates\n\nwith its neighbors. All of the agents target to cooperatively solve the convex optimization problem\n\n$$\n\\operatorname*{min}_{x\\in\\mathbb{R}^{d}}h(x)\\triangleq f(x)+r(x)\\quad\\mathrm{with}\\quad f(x)\\triangleq\\frac{1}{m}\\sum_{i=1}^{m}f_{i}(x),\\tag{1}\n$$\n\nwhere $f(x)$ is L-smooth and $\\mu$-strongly convex, $r(x)$ is convex but may be non-differentiable."}, {"name": "Notation: aggregate variables, averages, prox and generalized gradients (Section 3, eqs. (2)-(6))", "statement": "## 3. Preliminaries\n\nWe let $x_{i}\\in \\mathbb{R}^{d}$ be the local copy of the variable of x for agent i and we introduce the aggregate variable $\\mathbf{x}\\in \\mathbb{R}^{m \\times d}$ , aggregate objective function $F(\\mathbf{x})$ and aggregate gradient $\\nabla F(\\mathbf{x})\\in\\mathbb{R}^{m\\times d}$ as\n\n$$\n\\mathbf{x}=\\left[\\begin{array}{l}{x_{1}^{\\top}}\\\\ {\\vdots}\\\\ {x_{m}^{\\top}}\\end{array}\\right],\\quad F(\\mathbf{x})=\\frac{1}{m}\\sum_{i=1}^{m}f_{i}(x_{i})\\quad\\mathrm{and}\\quad\\nabla F(\\mathbf{x})=\\left[\\begin{array}{l}{\\nabla f_{1}(x_{1})^{\\top}}\\\\ {\\vdots}\\\\ {\\nabla f_{m}(x_{m})^{\\top}}\\end{array}\\right].\\tag{2}\n$$\n\nWe denote that\n\n$$\n\\bar{\\boldsymbol{x}}_{t}=\\frac{1}{m}\\sum_{i=0}^{m}\\mathbf{x}_{t}^{(i)},\\quad\\bar{y}_{t}=\\frac{1}{m}\\sum_{i=0}^{m}\\mathbf{y}_{t}^{(i)}\\quad\\mathrm{and}\\quad\\bar{g}_{t}=\\frac{1}{m}\\sum_{i=0}^{m}\\nabla f_{i}(\\mathbf{y}_{t}^{(i)}),\\tag{3}\n$$\n\nwhere $\\mathbf{x}^{(i)}$ means the i-th row of matrix x. Moreover, we use $\\lVert \\cdot \\rVert$ to denote the Frobenius norm of vector or matrix and use $\\langle x,y\\rangle$ to denote the inner product of vectors x and y.\n\nFurthermore, we denote\n\n$$\nR(\\mathbf{x})=\\frac{1}{m}\\sum_{i=1}^{m}r(x_{i}).\n$$\n\nAccordingly, we introduce the proximal operator and aggregated proximal operator with respect to $r(\\cdot)$ and $R(\\cdot)$ as\n\n$$\n\\mathbf{prox}_{\\eta,r}(x)=\\operatorname*{argmin}_{z\\in\\mathbb{R}^{d}}\\left(r(z)+\\frac{1}{2\\eta}\\|z-x\\|^{2}\\right)\\mathrm{~a n d~}\\mathbf{prox}_{m\\eta,R}(\\mathbf{x})=\\operatorname*{argmin}_{z\\in\\mathbb{R}^{m\\times d}}\\left(R(\\mathbf{z})+\\frac{1}{2 m\\eta}\\|\\mathbf{z}-\\mathbf{x}\\|^{2}\\right).\\tag{4}\n$$\n\nUsing above notations, we define the (aggregated) generalized gradients as\n\n$$\nG_{t}=\\eta^{-1}\\left(\\mathbf{y}_{t}-\\mathbf{prox}_{\\eta m,R}(\\mathbf{y}_{t}-\\eta\\mathbf{s}_{t})\\right)\\ \\mathrm{~a n d~}\\ G_{t}^{(i)}=\\eta^{-1}\\left(\\mathbf{y}_{t}^{(i)}-\\mathbf{prox}_{\\eta,r}(\\mathbf{y}_{t}^{(i)}-\\eta\\mathbf{s}_{t}^{(i)})\\right)\\tag{5}\n$$\n\nWe also denote\n\n$$\n\\bar{G}_{t}=\\frac{1}{m}\\sum_{i=1}^{m}G_{t}^{(i)}.\\tag{6}\n$$"}, {"name": "Definitions: L-smoothness, strong convexity, local smoothness, condition numbers (eqs. (7)-(9))", "statement": "Then we introduce the following definitions that will be used in the whole paper:\n\nWe say $f(x)$ is L-smooth if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf(y)\\leq f(x)+\\langle\\nabla f(x),y-x\\rangle+{\\frac{L}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f(x)$ is µ-strongly convex, if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf(y)\\geq f(x)+\\langle\\nabla f(x),y-x\\rangle+{\\frac{\\mu}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f_{i}(x)$ is locally Mi-smooth if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf_{i}(y)\\leq f_{i}(x)+\\langle\\nabla f_{i}(x),y-x\\rangle+{\\frac{M_{i}}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f_{i}(x)$ is locally νi-strongly convex if for all $x,y\\in\\mathbb{R}^{d}$ ,it holds that\n\n$$\nf_{i}(y)\\geq f_{i}(x)+\\langle\\nabla f_{i}(x),y-x\\rangle+{\\frac{\\nu_{i}}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nBased on the smoothness and strong convexity, we can define global and local condition numbers of the objective function as\n\n$$\n\\kappa_{g}=\\frac{L}{\\mu},\\qquad\\hat{\\kappa}_{g}=\\frac{M}{\\mu}\\qquad\\mathrm{and}\\qquad\\kappa_{\\ell}=\\frac{M}{\\nu},\\tag{7}\n$$\n\nwhere\n\n$$\nM=\\operatorname*{max}_{i\\in\\{1,...,m\\}}M_{i}\\mathrm{~a n d~}\\nu=\\operatorname*{min}_{i\\in\\{1,...,m\\}}\\nu_{i}.\\tag{8}\n$$\n\nIt is easy to verify that\n\n$$\nL\\leq M\\qquad\\mathrm{and}\\qquad\\kappa_{g}\\leq\\hat{\\kappa}_{g}\\leq\\kappa_{\\ell}.\\tag{9}\n$$"}, {"name": "Assumption: properties of the weight matrix W", "statement": "For the topology of the network, we let W be the weight matrix associated with the network, indicating how agents are connected to each other. We assume that the weight matrix W has the following properties:\n\n1. W is symmetric with $W_{i,j}\\neq 0$ if and if only agents i and j are connected or $i = j ;$\n\n$$\n2 .\\ \\mathbf{0}\\preceq W\\preceq I,W\\mathbf{1}=\\mathbf{1},\\mathrm{null}(I-W)=\\mathrm{span}(\\mathbf{1});\n$$\n\nwhere we use I to denote the $m \\times m$ identity matrix and $\\mathbf{1}=[1,\\ldots,1]^{\\top}\\in\\mathbb{R}^{m}$ denotes the vector with all ones. The weight matrix has an important property that $\\begin{array}{r}{W^{\\infty}= \\frac{1}{m}\\mathbf{1}\\mathbf{1}^{\\top}}\\end{array}$ (Xiao and Boyd, 2004). Thus, one can achieve the effect of averaging local $x_{i}$ on different agents by using Wx for iterations. Instead of directly multiplying W, Liu and Morse (2011) proposed a more efficient way to achieve averaging described in Algorithm 2, which has the following important proposition."}, {"name": "Proposition 1 (FastMix averaging and contraction)", "statement": "Proposition 1 Let $\\mathbf{x}^{K}$ be the output of Algorithm 2 with $\\eta_{w}=1/(1+\\sqrt{1-\\lambda_{2}^{2}(W)})$ and we denote $\\begin{array}{r}{\\bar{\\boldsymbol{x}}= \\frac{1}{m}\\mathbf{1}^{\\top}\\mathbf{x}^{0}}\\end{array}$ Then it holds that\n\n$$\n\\bar{x}=\\frac{1}{m}\\mathbf{1}^{\\mathsf{T}}\\mathbf{x}^{K}\\quad and\\quad\\|\\mathbf{x}_{K}-\\mathbf{1}\\bar{x}\\|\\leq\\sqrt{14}\\left(1-\\left(1-\\frac{1}{\\sqrt{2}}\\right)\\sqrt{1-\\lambda_{2}(W)}\\right)^{K}\\|\\mathbf{x}_{0}-\\mathbf{1}\\bar{x}\\|,\n$$\n\nwhere $\\lambda_{2}(W)$ is the second largest eigenvalue of W."}, {"name": "Algorithm 2 (FastMix)", "statement": "Algorithm 2 FastMix   \n1: Input: $\\mathbf{x}^{0},K,W$ and $\\eta_{w}$   \n2: $\\mathbf{x}^{- 1}= \\mathbf{x}^{0}$   \n3: for $k=0,\\ldots,K$ do   \n4: $\\mathbf{x}^{k+1}=(1+\\eta_{w})W\\mathbf{x}^{k}-\\eta_{w}\\mathbf{x}^{k-1}$   \n5: end for   \n6: Output: $\\mathbf{x}^{K}$"}, {"name": "ProxMudag updates (eqs. (10)-(12)) and Algorithm 3", "statement": "Our algorithms are based on the multi-consensus, gradient-tracking and Nesterov's acceleration technique. We first introduce ProxMudag (Algorithm 3) for solving the problem with $r(x)\\neq 0$ It has the following algorithmic procedure:\n\n$$\n\\begin{array}{r}{{\\bf x}_{t+1}={\\bf prox}_{\\eta m,R}({\\bf y}_{t}-\\eta{\\bf s}_{t}),}\\end{array}\\tag{10}\n$$\n\n$$\n\\mathbf{y}_{t+1}=\\mathrm{FastMix}\\left(\\mathbf{x}_{t+1}+\\frac{1-\\alpha}{1+\\alpha}(\\mathbf{x}_{t+1}-\\mathbf{x}_{t}),K\\right),\\tag{11}\n$$\n\n$$\n\\begin{array}{r}{\\mathbf{s}_{t+1}=\\mathrm{FastMix}(\\mathbf{s}_{t}+\\nabla F(\\mathbf{y}_{t+1})-\\nabla F(\\mathbf{y}_{t}),\\ K),}\\end{array}\\tag{12}\n$$\n\nwhere η is the step size and K is the step number in multi-consensus. We can observe that Eqs. (10) and (11) belong to the algorithmic framework of accelerated proximal gradient descent since $\\mathbf{s}_{t}$ can approximate the average gradient. In Eq. (12), we introduce $\\mathbf{s}_{t}$ to track the gradient by using history information and the gradient difference. Thus, $\\mathbf{s}_{t}$ can well approximate the average gradient ${\\bf 1}\\bar{g}_{t}$ (defined in Eq. (3)). Furthermore, the variable $\\mathbf{y}_{t}$ can also approximate $\\mathbf{1}\\bar{y}_{t}$ well by the \"FastMix\" operator. Since both $\\bar{y}_{t}$ and $\\bar{g}_{t}$ well approximate the averages, then we can obtain that $\\bar{g}_{t}\\approx\\nabla f(\\bar{y}_{t})$ Thus, the convergence properties of our algorithm are similar to the centralized accelerated proximal gradient descent, which is the main idea behind our approach to the decentralized optimization. In other words, we combine multi-consensus with gradient-tracking to approximate the centralized accelerated proximal gradient descent. As we will show, this seemingly simple idea leads to establishing a near optimal algorithm for the decentralized optimization. Note that Algorithm 3 only takes two multi-consensus steps at each iteration. In contrast, the algorithm in conference version (Ye et al., 2020, Algorithm 1) of this paper requires three multi-consensus steps at each round.\n\nAlgorithm 3 ProxMudag   \n1: Input: $x_{0},\\eta,\\alpha,K$   \n2: Initialization: $\\mathbf{x}_{0}=\\mathbf{1}x_{0},\\mathbf{y}_{0}=\\mathbf{x}_{0},\\mathbf{s}_{0}=\\nabla F(\\mathbf{x}_{0})$   \n3: for $t=0,\\ldots,T$ do   \n4: $\\mathbf{x}_{t+1}=\\mathbf{prox}_{\\eta m,R}(\\mathbf{y}_{t}-\\eta\\mathbf{s}_{t})$   \n5: $\\begin{array}{r}{\\mathbf{y}_{t+1}=\\mathrm{FastMix}\\left(\\mathbf{x}_{t+1}+\\frac{1-\\alpha}{1+\\alpha}(\\mathbf{x}_{t+1}-\\mathbf{x}_{t}),K\\right)}\\end{array}$   \n6: $\\mathbf{s}_{t+1}=\\mathrm{FastMix}(\\mathbf{s}_{t}+\\nabla F(\\mathbf{y}_{t+1})-\\nabla F(\\mathbf{y}_{t}),\\ K)$   \n7: end for   \n8: Output: $\\mathbf{x}_{T}$"}, {"name": "Theorem 3 (parameter setting for Algorithm 3)", "statement": "Theorem 3 Let $f(x)$ be L-smooth and µ-strongly convex. Assume each $f_{i}(x)$ is M-smooth. We set $\\eta=1/(2L)$ and $\\alpha = \\sqrt{\\mu \\eta}$ in Algorithm 3. Letting K in Algorithm 3 satisfy that\n\n$$\nK=\\frac{\\sqrt{2}}{\\sqrt{2}-1}\\sqrt{\\frac{1}{1-\\lambda_{2}(W)}}\\log{\\left(\\frac{\\sqrt{14}}{\\rho}\\right)}\\qquad with\\qquad\\rho\\leq\\frac{1}{5.5\\cdot 10^{8}}\\cdot\\left(\\frac{L}{M}\\right)^{6}\\kappa_{g}^{-3/2},\n$$\n\nthen sequence $\\{\\bar{x}_{t}\\}$ generated by Algorithm 3 satisfies that\n\n$$\nh(\\bar{x}_{T})-h(x^{*})\\leq\\left(1-\\frac{\\alpha}{2}\\right)^{T}\\left(h(\\bar{x}_{0})-h(x^{*})+\\frac{\\mu\\|\\bar{x}_{0}-x^{*}\\|^{2}}{2}+\\frac{52L}{m}\\sum_{i}^{m}\\|\\nabla f_{i}(\\bar{x}_{0})-\\nabla f(\\bar{x}_{0})\\|^{2}\\right),\n$$\n\nwhere $x^{*}$ is the global minimum of $h(x)$ . To achieve $\\mathbf{x}_{T}$ such that $\\|\\mathbf{x}_{T}-\\mathbf{1}x^{*}\\|^{2}={\\mathcal{O}}(m\\epsilon/\\mu)$ and $h(\\bar{x}_{T})-h(x^{*})\\leq\\epsilon,$ , the computation and communication complexities of Algorithm 1 are at most\n\n$$\nT=\\mathcal{O}\\left(\\sqrt{\\kappa_{g}}\\log\\left(\\frac{1}{\\epsilon}\\right)\\right)\\qquad and\\qquad Q=\\mathcal{O}\\left(\\sqrt{\\frac{\\kappa_{g}}{1-\\lambda_{2}(W)}}\\log\\left(\\frac{M\\kappa_{g}}{L}\\right)\\log\\left(\\frac{1}{\\epsilon}\\right)\\right)\n$$\n\nrespectively."}, {"name": "Lyapunov function V_t and v_bar_t (eqs. (17)-(18))", "statement": "We first introduce the Lyapunov function as follows\n\n$$\nV_{t}=h(\\bar{x}_{t})-h(x^{*})+\\frac{\\mu}{2}\\left\\|\\bar{v}_{t}-x^{*}\\right\\|^{2},\\tag{17}\n$$\n\nwhere $\\bar{v}_{t}$ is defined as\n\n$$\n\\bar{v}_{t}=\\bar{x}_{t-1}+\\frac{1}{\\alpha}\\big(\\bar{x}_{t}-\\bar{x}_{t-1}\\big)\\quad\\mathrm{with}\\qquad\\alpha=\\sqrt{\\mu\\eta}.\\tag{18}\n$$"}, {"name": "Lemma 8", "statement": "Lemma 8 Let $\\bar{x}_{t}, \\bar{y}_{t}, \\bar{g}_{t}$ and $\\bar{G}_{t}$ (defined in Eqs. (3) and (6)) be generated by Algorithm 3. By setting $\\begin{array}{r}{\\bar{s}_{t}= \\frac{1}{m}\\mathbf{1}^{\\top}\\mathbf{s}_{t}}\\end{array}$ with $\\mathbf{s}_{t}$ defined in Eq. (12), it satisfies:\n\n$$\n\\bar{x}_{t + 1}= \\bar{y}_{t}- \\eta \\bar{G}_{t},\\tag{19}\n$$\n\n$$\n\\bar{y}_{t+1}=\\bar{x}_{t+1}+\\frac{1-\\alpha}{1+\\alpha}(\\bar{x}_{t+1}-\\bar{x}_{t}),\\tag{20}\n$$\n\n$$\n\\bar{s}_{t + 1}= \\bar{s}_{t}+ \\bar{g}_{t + 1}- \\bar{g}_{t}= \\bar{g}_{t + 1}.\\tag{21}\n$$"}, {"name": "Lemma 17 (row-wise prox)", "statement": "Lemma 17 Letting $\\mathbf{prox}_{\\eta m,R}^{(i)}(\\mathbf{x})$ denote the i-th row of the matrix $\\mathbf{prox}_{\\eta m,R}(\\mathbf{x})$ (defined in Eqn. (4)), we have the following equation\n\n$$\n\\begin{array}{r}{\\mathbf{prox}_{\\eta m,R}^{(i)}(\\mathbf{x})=\\mathbf{prox}_{\\eta,r}(\\mathbf{x}^{(i)}),}\\end{array}\n$$\n\nwhich implies that $G_{t}^{(i)}$ equals to the i-th row of $G_{t}$ defined in Eq. (5)."}, {"name": "Lemma 19", "statement": "Lemma 19 Letting $\\mathbf{s}_{t}^{(i)}$ be the i-th row of $\\mathbf{s}_{t}$ and $G_{t}^{(i)},\\bar{G}_{t}$ (defined in Eqs. (5), (6)) generated by Algorithm 3, we have\n\n$$\n\\sum_{i=1}^{m}\\left\\|\\mathbf{s}_{t}^{(i)}-\\nabla f(\\mathbf{y}_{t}^{(i)})\\right\\|^{2}\\leq 2\\left\\|\\mathbf{s}_{t}-\\mathbf{1}\\bar{s}_{t}\\right\\|^{2}+8M^{2}\\left\\|\\mathbf{y}_{t}-\\mathbf{1}\\bar{y}_{t}\\right\\|^{2}\\tag{44}\n$$\n\nand\n\n$$\n\\eta^{2}\\sum_{i=1}^{m}\\left\\|G_{t}^{(i)}-\\bar{G}_{t}\\right\\|^{2}\\leq 18\\left\\|\\mathbf{y}_{t}-\\mathbf{1}\\bar{y}_{t}\\right\\|^{2}+16\\eta^{2}\\left\\|\\mathbf{s}_{t}-\\mathbf{1}\\bar{s}_{t}\\right\\|^{2}.\\tag{45}\n$$"}, {"name": "Lemma 14", "statement": "Lemma 14 For $\\textstyle{\\bar{x}}_{t},{\\bar{y}}_{t}$ and $\\bar{v}_{t}$ defined in Eqs. (3) and (18), then we can obtain that\n\n$$\n\\bar{y}_{t}-\\bar{x}_{t}=\\alpha(\\bar{v}_{t}-\\bar{y}_{t}),\\tag{36}\n$$\n\n$$\n\\bar{y}_{t + 1}= \\frac{\\bar{x}_{t + 1}+ \\alpha \\bar{v}_{t + 1}}{1 + \\alpha},\\tag{37}\n$$\n\nand\n\n$$\n\\bar{v}_{t+1}=\\left\\{\\begin{array}{l l}{(1-\\alpha)\\bar{v}_{t}+\\alpha\\bar{y}_{t}-\\displaystyle\\frac{\\eta}{\\alpha}\\bar{G}_{t}\\quad}&{r(x),isconvex,}\\\\ {(1-\\alpha)\\bar{v}_{t}+\\alpha\\bar{y}_{t}-\\displaystyle\\frac{\\eta}{\\alpha}\\bar{g}_{t},\\quad}&{r(x)=0.}\\end{array}\\right.\\tag{38}\n$$"}, {"name": "Lemma 13", "statement": "Lemma 13 We have\n\n$$\n\\|\\nabla F(\\mathbf{y})-\\nabla F(\\mathbf{x})\\|\\leq M\\|\\mathbf{y}-\\mathbf{x}\\|\\tag{33}\n$$\n\nand\n\n$$\n\\left\\|{\\bar{g}}_{t}- \\nabla f ({\\bar{y}}_{t}) \\right\\| \\leq{\\frac{M}{\\sqrt{m}}}\\left\\| \\mathbf{y}_{t}- \\mathbf{1}{\\bar{y}}_{t}\\right\\| .\\tag{34}\n$$\n\nFurthermore, we have the $\\left(L+2/\\eta\\right)$ -smooth property for the generalized gradient $\\tilde{\\nabla}h(\\cdot)$ (defined in Eq. (42)), i.e.,\n\n$$\n\\left\\|\\tilde{\\nabla}h(x)-\\tilde{\\nabla}h(y)\\right\\|\\leq\\left(\\frac{2}{\\eta}+L\\right)\\|x-y\\|.\\tag{35}\n$$"}, {"name": "Lemma 15", "statement": "Lemma 15 Let $f(x)$ be µ-strongly convex. For $\\bar{y}_{t},$ and $V_{t}$ defined in Eqs. (3) and (17) and $x^{*}$ being the optimum, we have the following inequality,\n\n$$\n\\|{\\bar{y}}_{t}- x^{*}\\| \\leq \\sqrt{\\frac{2 V_{t}}{\\mu}}.\\tag{39}\n$$"}, {"name": "Lemma 16 (generalized gradient, eq. (42))", "statement": "Lemma 16 (Nesterov (2018)) Letting $\\tilde{\\nabla}h(x)$ the generalized gradient of $h(x)$ (refer to Eq. (1)) be defined as\n\n$$\n\\tilde{\\nabla}h(x)\\triangleq\\frac{x-\\mathbf{prox}_{\\eta,r}(x-\\eta\\nabla f(x))}{\\eta},w i t h\\eta b e i n g t h e s t e p s i z e,\\tag{42}\n$$\n\nthen it holds that $\\tilde{\\nabla}h(x^{*})=0$ if $x^{*}$ minimizes $h(x)$"}], "nl_notes": "### Lemma 11\nREPAIRED STATEMENT (Lemma 11) — The paper's Lemma 11 is false as printed: eq. (25) gives the y-consensus coefficient (20M²η + 10η⁻¹)/m, independent of α, but the proof of the paper's Lemma 21 drops the residual s̄_t − (1/m)Σ_i ∇f(y_t^i), which is not zero (eq. (21) only gives s̄_t = (1/m)Σ_i ∇F_i(y_t^i), each agent's local gradient at its own point). The target's coefficient therefore carries the extra term (1 + 8/α³)M²/L, the same factor the paper's Lemma 27 (eq. (72)) carries; with it the bound is true. Prove the bound exactly as stated in the Lean file, including that term.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped BigOperators\nopen Polynomial\n\nnoncomputable def rowAvg {m d : ℕ} (x : Fin m → EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) :=\n  (1 / (m : ℝ)) • ∑ i, x i\n\nnoncomputable def Wapply {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ)\n    (x : Fin m → EuclideanSpace ℝ (Fin d)) : Fin m → EuclideanSpace ℝ (Fin d) :=\n  fun i => ∑ j, W i j • x j\n\nnoncomputable def fastMixAux {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (x0 : Fin m → EuclideanSpace ℝ (Fin d)) :\n    ℕ → (Fin m → EuclideanSpace ℝ (Fin d)) × (Fin m → EuclideanSpace ℝ (Fin d))\n  | 0 => (x0, x0)\n  | (k+1) =>\n      ((fun i => (1 + ηw) • Wapply W (fastMixAux W ηw x0 k).1 i\n                  - ηw • (fastMixAux W ηw x0 k).2 i),\n       (fastMixAux W ηw x0 k).1)\n\nnoncomputable def FastMix {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) : Fin m → EuclideanSpace ℝ (Fin d) :=\n  (fastMixAux W ηw x0 K).1\n\ntheorem Wapply_sum {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (x : Fin m → EuclideanSpace ℝ (Fin d)) :\n    ∑ i, Wapply W x i = ∑ i, x i := by\n  simp only [Wapply]\n  rw [Finset.sum_comm]\n  refine Finset.sum_congr rfl (fun j _ => ?_)\n  rw [← Finset.sum_smul, hW j, one_smul]\n\ntheorem fastMixAux_sum {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (hW : ∀ j, ∑ i, W i j = 1) (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) :\n    (∑ i, (fastMixAux W ηw x0 K).1 i = ∑ i, x0 i) ∧\n    (∑ i, (fastMixAux W ηw x0 K).2 i = ∑ i, x0 i) := by\n  induction K with\n  | zero => simp [fastMixAux]\n  | succ k ih =>\n    obtain ⟨ih1, ih2⟩ := ih\n    constructor\n    · simp only [fastMixAux]\n      rw [Finset.sum_sub_distrib, ← Finset.smul_sum, ← Finset.smul_sum,\n          Wapply_sum W hW, ih1, ih2, ← sub_smul,\n          show (1 + ηw) - ηw = (1 : ℝ) from by ring, one_smul]\n    · simpa only [fastMixAux] using ih1\n\ntheorem FastMix_rowAvg {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (hW : ∀ j, ∑ i, W i j = 1) (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) :\n    rowAvg (FastMix W ηw x0 K) = rowAvg x0 := by\n  unfold rowAvg FastMix\n  rw [(fastMixAux_sum W ηw hW x0 K).1]\n\ndef IsSmooth {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (L : ℝ) (f : E → ℝ) (g : E → E) : Prop :=\n  ∀ x y, f y ≤ f x + (inner (𝕜 := ℝ) (g x) (y - x)) + (L / 2) * ‖y - x‖ ^ 2\n\ndef IsProxStep {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (c : ℝ) (R : E → ℝ) (a xplus : E) : Prop :=\n  ∀ u, R xplus + (1 / (2 * c)) * ‖xplus - a‖ ^ 2 ≤ R u + (1 / (2 * c)) * ‖u - a‖ ^ 2\n\ntheorem lemma11 {d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / (2 * L))\n    (hα : α = Real.sqrt (μ * η)) (hα1 : α ≤ 1)\n    (R : EuclideanSpace ℝ (Fin d) → ℝ) (hRconv : ConvexOn ℝ Set.univ R)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar + R xstar ≤ f x + R x)\n    (xtprev xt xt1 yt yt1 st st1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = FastMix W ηw (fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i)) K)\n    (hxt1 : ∀ i, IsProxStep η R (yt i - η • st i) (xt1 i))\n    (hyt1 : yt1 = FastMix W ηw (fun i => xt1 i + ((1 - α) / (1 + α)) • (xt1 i - xt i)) K)\n    (hst1 : st1 = FastMix W ηw (fun i => st i + gradFi i (yt1 i) - gradFi i (yt i)) K)\n    (hgtrack : rowAvg st = (1 / (m : ℝ)) • ∑ i, gradFi i (yt i))\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) + R (rowAvg xt) - (f xstar + R xstar))\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) + R (rowAvg xt1) - (f xstar + R xstar))\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - α) * Vt + (13 * η / m) * (∑ i, ‖st i - rowAvg st‖ ^ 2)\n        + ((20 * M ^ 2 * η + 10 * η⁻¹ + (1 + 8 / α ^ 3) * M ^ 2 / L) / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2) := by\n  sorry\n\ntheorem lemma27 {d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / L)\n    (hα : α = Real.sqrt (μ * η))\n    (hα1 : α ≤ 1)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar ≤ f x)\n    (xtprev ytprev xt yt xt1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i))\n    (hprevavg : rowAvg xt = rowAvg ytprev\n        - η • ((1 / (m : ℝ)) • ∑ i, gradFi i (ytprev i)))\n    (hxt1 : xt1 = FastMix W ηw (fun i => yt i + (xt i - ytprev i)\n                - η • (gradFi i (yt i) - gradFi i (ytprev i))) K)\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) - f xstar)\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) - f xstar)\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - (3 / 4) * α) * Vt\n        + (1 + 8 / α ^ 3) * (M ^ 2 / L) * (1 / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2) := by\n  sorry\n\ntheorem lemma10 (L M ρ : ℝ) (hL : 0 < L) (η : ℝ) (hηeq : η = 1 / (2 * L)) (hM : 0 < M) (hLM : L ≤ M) (hρpos : 0 < ρ)\n    (hρ : ρ ≤ L ^ 3 / (1280 * M ^ 3))\n    (A : Matrix (Fin 3) (Fin 3) ℝ)\n    (hA : A = !![0, 2, 2; 2 * ρ, 4 * ρ, 4 * ρ; ρ * M / L, 8 * ρ * M ^ 2 / L ^ 2, 5 * ρ * M / L]) :\n    ∃ l1 l2 l3 : ℝ, A.charpoly = (X - C l1) * (X - C l2) * (X - C l3) ∧\n      0 < l1 ∧ |l3| ≤ |l2| ∧ |l2| < l1 ∧ l1 ≤ 1 / 2 ∧\n      ∃ v : Fin 3 → ℝ, A.mulVec v = l1 • v ∧\n        0 < v 0 ∧ v 0 ≤ 8 * v 2 / Real.sqrt ρ ∧\n        0 < v 1 ∧ v 1 ≤ 3 * v 2 ∧\n        0 < v 2 := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma11", "lemma27", "lemma10"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 2592, "end": 4947, "gold": "{d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / (2 * L))\n    (hα : α = Real.sqrt (μ * η)) (hα1 : α ≤ 1)\n    (R : EuclideanSpace ℝ (Fin d) → ℝ) (hRconv : ConvexOn ℝ Set.univ R)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar + R xstar ≤ f x + R x)\n    (xtprev xt xt1 yt yt1 st st1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = FastMix W ηw (fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i)) K)\n    (hxt1 : ∀ i, IsProxStep η R (yt i - η • st i) (xt1 i))\n    (hyt1 : yt1 = FastMix W ηw (fun i => xt1 i + ((1 - α) / (1 + α)) • (xt1 i - xt i)) K)\n    (hst1 : st1 = FastMix W ηw (fun i => st i + gradFi i (yt1 i) - gradFi i (yt i)) K)\n    (hgtrack : rowAvg st = (1 / (m : ℝ)) • ∑ i, gradFi i (yt i))\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) + R (rowAvg xt) - (f xstar + R xstar))\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) + R (rowAvg xt1) - (f xstar + R xstar))\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - α) * Vt + (13 * η / m) * (∑ i, ‖st i - rowAvg st‖ ^ 2)\n        + ((20 * M ^ 2 * η + 10 * η⁻¹ + (1 + 8 / α ^ 3) * M ^ 2 / L) / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2)"}]}
{"item_id": "v24-22-1210::lemma27::A-stmt", "paper_id": "v24-22-1210", "target_name": "lemma27", "source": "Research-level", "nl_statement": "Lemma 27 Letting $V_{t}$ be the Lyapunov function defined in $E q .$ (17) associated to Algorithm 1 with $\\eta = 1 / L$ , then it satisfies the following property\n\n$$\nV_{t+1}\\leq\\left(1-\\frac{3}{4}\\alpha\\right)V_{t}+\\left(1+\\frac{8}{\\alpha^{3}}\\right)\\cdot\\frac{M^{2}}{L}\\cdot\\frac{1}{m}\\left\\|\\mathbf{y}_{t}-\\mathbf{1}\\bar{y}_{t}\\right\\|^{2}.\\tag{72}\n$$", "context": [{"name": "Setup: decentralized composite problem (Section 1, eq. (1))", "statement": "In this paper, we consider the decentralized optimization problem, where the objective function is composed of m local functions $f_{i}(x)$ that are located on m different agents. The agents form a connected and undirected network and each of them only accesses its local function and communicates\n\nwith its neighbors. All of the agents target to cooperatively solve the convex optimization problem\n\n$$\n\\operatorname*{min}_{x\\in\\mathbb{R}^{d}}h(x)\\triangleq f(x)+r(x)\\quad\\mathrm{with}\\quad f(x)\\triangleq\\frac{1}{m}\\sum_{i=1}^{m}f_{i}(x),\\tag{1}\n$$\n\nwhere $f(x)$ is L-smooth and $\\mu$-strongly convex, $r(x)$ is convex but may be non-differentiable."}, {"name": "Notation: aggregate variables, averages, prox and generalized gradients (Section 3, eqs. (2)-(6))", "statement": "## 3. Preliminaries\n\nWe let $x_{i}\\in \\mathbb{R}^{d}$ be the local copy of the variable of x for agent i and we introduce the aggregate variable $\\mathbf{x}\\in \\mathbb{R}^{m \\times d}$ , aggregate objective function $F(\\mathbf{x})$ and aggregate gradient $\\nabla F(\\mathbf{x})\\in\\mathbb{R}^{m\\times d}$ as\n\n$$\n\\mathbf{x}=\\left[\\begin{array}{l}{x_{1}^{\\top}}\\\\ {\\vdots}\\\\ {x_{m}^{\\top}}\\end{array}\\right],\\quad F(\\mathbf{x})=\\frac{1}{m}\\sum_{i=1}^{m}f_{i}(x_{i})\\quad\\mathrm{and}\\quad\\nabla F(\\mathbf{x})=\\left[\\begin{array}{l}{\\nabla f_{1}(x_{1})^{\\top}}\\\\ {\\vdots}\\\\ {\\nabla f_{m}(x_{m})^{\\top}}\\end{array}\\right].\\tag{2}\n$$\n\nWe denote that\n\n$$\n\\bar{\\boldsymbol{x}}_{t}=\\frac{1}{m}\\sum_{i=0}^{m}\\mathbf{x}_{t}^{(i)},\\quad\\bar{y}_{t}=\\frac{1}{m}\\sum_{i=0}^{m}\\mathbf{y}_{t}^{(i)}\\quad\\mathrm{and}\\quad\\bar{g}_{t}=\\frac{1}{m}\\sum_{i=0}^{m}\\nabla f_{i}(\\mathbf{y}_{t}^{(i)}),\\tag{3}\n$$\n\nwhere $\\mathbf{x}^{(i)}$ means the i-th row of matrix x. Moreover, we use $\\lVert \\cdot \\rVert$ to denote the Frobenius norm of vector or matrix and use $\\langle x,y\\rangle$ to denote the inner product of vectors x and y.\n\nFurthermore, we denote\n\n$$\nR(\\mathbf{x})=\\frac{1}{m}\\sum_{i=1}^{m}r(x_{i}).\n$$\n\nAccordingly, we introduce the proximal operator and aggregated proximal operator with respect to $r(\\cdot)$ and $R(\\cdot)$ as\n\n$$\n\\mathbf{prox}_{\\eta,r}(x)=\\operatorname*{argmin}_{z\\in\\mathbb{R}^{d}}\\left(r(z)+\\frac{1}{2\\eta}\\|z-x\\|^{2}\\right)\\mathrm{~a n d~}\\mathbf{prox}_{m\\eta,R}(\\mathbf{x})=\\operatorname*{argmin}_{z\\in\\mathbb{R}^{m\\times d}}\\left(R(\\mathbf{z})+\\frac{1}{2 m\\eta}\\|\\mathbf{z}-\\mathbf{x}\\|^{2}\\right).\\tag{4}\n$$\n\nUsing above notations, we define the (aggregated) generalized gradients as\n\n$$\nG_{t}=\\eta^{-1}\\left(\\mathbf{y}_{t}-\\mathbf{prox}_{\\eta m,R}(\\mathbf{y}_{t}-\\eta\\mathbf{s}_{t})\\right)\\ \\mathrm{~a n d~}\\ G_{t}^{(i)}=\\eta^{-1}\\left(\\mathbf{y}_{t}^{(i)}-\\mathbf{prox}_{\\eta,r}(\\mathbf{y}_{t}^{(i)}-\\eta\\mathbf{s}_{t}^{(i)})\\right)\\tag{5}\n$$\n\nWe also denote\n\n$$\n\\bar{G}_{t}=\\frac{1}{m}\\sum_{i=1}^{m}G_{t}^{(i)}.\\tag{6}\n$$"}, {"name": "Definitions: L-smoothness, strong convexity, local smoothness, condition numbers (eqs. (7)-(9))", "statement": "Then we introduce the following definitions that will be used in the whole paper:\n\nWe say $f(x)$ is L-smooth if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf(y)\\leq f(x)+\\langle\\nabla f(x),y-x\\rangle+{\\frac{L}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f(x)$ is µ-strongly convex, if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf(y)\\geq f(x)+\\langle\\nabla f(x),y-x\\rangle+{\\frac{\\mu}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f_{i}(x)$ is locally Mi-smooth if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf_{i}(y)\\leq f_{i}(x)+\\langle\\nabla f_{i}(x),y-x\\rangle+{\\frac{M_{i}}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f_{i}(x)$ is locally νi-strongly convex if for all $x,y\\in\\mathbb{R}^{d}$ ,it holds that\n\n$$\nf_{i}(y)\\geq f_{i}(x)+\\langle\\nabla f_{i}(x),y-x\\rangle+{\\frac{\\nu_{i}}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nBased on the smoothness and strong convexity, we can define global and local condition numbers of the objective function as\n\n$$\n\\kappa_{g}=\\frac{L}{\\mu},\\qquad\\hat{\\kappa}_{g}=\\frac{M}{\\mu}\\qquad\\mathrm{and}\\qquad\\kappa_{\\ell}=\\frac{M}{\\nu},\\tag{7}\n$$\n\nwhere\n\n$$\nM=\\operatorname*{max}_{i\\in\\{1,...,m\\}}M_{i}\\mathrm{~a n d~}\\nu=\\operatorname*{min}_{i\\in\\{1,...,m\\}}\\nu_{i}.\\tag{8}\n$$\n\nIt is easy to verify that\n\n$$\nL\\leq M\\qquad\\mathrm{and}\\qquad\\kappa_{g}\\leq\\hat{\\kappa}_{g}\\leq\\kappa_{\\ell}.\\tag{9}\n$$"}, {"name": "Assumption: properties of the weight matrix W", "statement": "For the topology of the network, we let W be the weight matrix associated with the network, indicating how agents are connected to each other. We assume that the weight matrix W has the following properties:\n\n1. W is symmetric with $W_{i,j}\\neq 0$ if and if only agents i and j are connected or $i = j ;$\n\n$$\n2 .\\ \\mathbf{0}\\preceq W\\preceq I,W\\mathbf{1}=\\mathbf{1},\\mathrm{null}(I-W)=\\mathrm{span}(\\mathbf{1});\n$$\n\nwhere we use I to denote the $m \\times m$ identity matrix and $\\mathbf{1}=[1,\\ldots,1]^{\\top}\\in\\mathbb{R}^{m}$ denotes the vector with all ones. The weight matrix has an important property that $\\begin{array}{r}{W^{\\infty}= \\frac{1}{m}\\mathbf{1}\\mathbf{1}^{\\top}}\\end{array}$ (Xiao and Boyd, 2004). Thus, one can achieve the effect of averaging local $x_{i}$ on different agents by using Wx for iterations. Instead of directly multiplying W, Liu and Morse (2011) proposed a more efficient way to achieve averaging described in Algorithm 2, which has the following important proposition."}, {"name": "Proposition 1 (FastMix averaging and contraction)", "statement": "Proposition 1 Let $\\mathbf{x}^{K}$ be the output of Algorithm 2 with $\\eta_{w}=1/(1+\\sqrt{1-\\lambda_{2}^{2}(W)})$ and we denote $\\begin{array}{r}{\\bar{\\boldsymbol{x}}= \\frac{1}{m}\\mathbf{1}^{\\top}\\mathbf{x}^{0}}\\end{array}$ Then it holds that\n\n$$\n\\bar{x}=\\frac{1}{m}\\mathbf{1}^{\\mathsf{T}}\\mathbf{x}^{K}\\quad and\\quad\\|\\mathbf{x}_{K}-\\mathbf{1}\\bar{x}\\|\\leq\\sqrt{14}\\left(1-\\left(1-\\frac{1}{\\sqrt{2}}\\right)\\sqrt{1-\\lambda_{2}(W)}\\right)^{K}\\|\\mathbf{x}_{0}-\\mathbf{1}\\bar{x}\\|,\n$$\n\nwhere $\\lambda_{2}(W)$ is the second largest eigenvalue of W."}, {"name": "Algorithm 1 (Mudag)", "statement": "Algorithm 1 Mudag   \n1: Input: $x_{0},\\eta,\\alpha$ , and K   \n2: Initialization: $\\mathbf{x}_{0}= \\mathbf{1}x_{0}, \\mathbf{y}_{0}= \\mathbf{x}_{0}$   \n3: $\\mathbf{x}_{1}=\\mathrm{FastMix}(\\mathbf{y}_{0}-\\eta\\nabla F(\\mathbf{y}_{0}))$   \n4: $\\begin{array}{r}{\\mathbf{y}_{1}=\\mathbf{x}_{1}+\\frac{1-\\alpha}{1+\\alpha}(\\mathbf{x}_{1}-\\mathbf{x}_{0})}\\end{array}$   \n5: for $t=1,\\dots,T$ do   \n6: $\\mathbf{x}_{t+1}={\\mathrm{FastMix}}(\\mathbf{y}_{t}+(\\mathbf{x}_{t}-\\mathbf{y}_{t-1})-\\eta(\\nabla F(\\mathbf{y}_{t})-\\nabla F(\\mathbf{y}_{t-1})),\\ K)$   \n7: $\\begin{array}{r}{\\mathbf{y}_{t+1}=\\mathbf{x}_{t+1}+\\frac{1-\\alpha}{1+\\alpha}\\left(\\mathbf{x}_{t+1}-\\mathbf{x}_{t}\\right)}\\end{array}$   \n8: end for   \n9: Output: $\\mathbf{x}_{T}$"}, {"name": "Algorithm 2 (FastMix)", "statement": "Algorithm 2 FastMix   \n1: Input: $\\mathbf{x}^{0},K,W$ and $\\eta_{w}$   \n2: $\\mathbf{x}^{- 1}= \\mathbf{x}^{0}$   \n3: for $k=0,\\ldots,K$ do   \n4: $\\mathbf{x}^{k+1}=(1+\\eta_{w})W\\mathbf{x}^{k}-\\eta_{w}\\mathbf{x}^{k-1}$   \n5: end for   \n6: Output: $\\mathbf{x}^{K}$"}, {"name": "Mudag updates (eqs. (13)-(16))", "statement": "In the case of $r(x)=0$ , we propose Mudag (Algorithm 1) that only needs one multi-consensus step for each iteration. The Mudag has the following algorithmic procedure:\n\n$$\n\\begin{array}{r l}&{\\mathbf{x}_{t+1}=\\mathrm{FastMix}(\\mathbf{y}_{t}+(\\mathbf{x}_{t}-\\mathbf{y}_{t-1})-\\eta(\\nabla F(\\mathbf{y}_{t})-\\nabla F(\\mathbf{y}_{t-1})),\\ K),}\\\\ &{\\mathbf{y}_{t+1}=\\mathbf{x}_{t+1}+\\displaystyle\\frac{1-\\alpha}{1+\\alpha}\\left(\\mathbf{x}_{t+1}-\\mathbf{x}_{t}\\right).}\\end{array}\\tag{13}\n$$\n\nTo understand Mudag from perspective of gradient tracking, we can reformulate the above procedure in a form similar to Eqs. (10) to (12) as follows (The reformulation is proved in Lemma 23)\n\n$$\n\\mathbf{x}_{t+1}=\\mathrm{FastMix}\\left(\\mathbf{y}_{t}-\\eta\\mathbf{s}_{t},K\\right),\\tag{14}\n$$\n\n$$\n{\\bf y}_{t+1}={\\bf x}_{t+1}+\\frac{1-\\alpha}{1+\\alpha}({\\bf x}_{t+1}-{\\bf x}_{t}),\\tag{15}\n$$\n\n$$\n\\mathbf{s}_{t+1}=\\mathrm{FastMix}(\\mathbf{s}_{t},K)+(\\nabla F(\\mathbf{y}_{t+1})-\\nabla F(\\mathbf{y}_{t}))-\\eta^{-1}(\\mathrm{FastMix}(\\mathbf{y}_{t},K)-\\mathbf{y}_{t}).\\tag{16}\n$$"}, {"name": "Theorem 2 (parameter setting for Algorithm 1)", "statement": "Theorem 2 Let $f(x)$ be L-smooth and µ-strongly convex. Assume each $f_{i}(x)$ is M-smooth. We set $\\eta = 1 / L$ and $\\alpha = \\sqrt{\\mu \\eta}$ in Algorithm 1. Letting K in Algorithm 1 satisfy that\n\n$$\nK=\\frac{\\sqrt{2}}{\\sqrt{2}-1}\\sqrt{\\frac{1}{1-\\lambda_{2}(W)}}\\log\\left(\\frac{\\sqrt{14}}{\\rho}\\right)\\qquad with\\qquad\\rho\\leq\\frac{1}{4^{3}\\cdot 9\\cdot 288}\\cdot\\left(\\frac{L}{M}\\right)^{4}\\kappa_{g}^{-3},\n$$\n\nthen the sequence $\\left\\{\\bar{x}_{t}\\right\\}$ satisfies that\n\n$$\nf(\\bar{x}_{T})-f(x^{*})\\leq\\left(1-\\frac{\\alpha}{2}\\right)^{T}\\left(f(\\bar{x}_{0})-f(x^{*})+\\frac{\\mu\\|\\bar{x}_{0}-x^{*}\\|^{2}}{2}+\\frac{\\mu}{288m}\\sum_{i=1}^{m}\\|\\nabla f_{i}(\\bar{x}_{0})-\\nabla f(\\bar{x}_{0})\\|^{2}\\right),\n$$\n\nwhere $x^{*}$ is the global minimum of $f(x)$ To achieve $\\mathbf{x}_{T}$ such that $\\|\\mathbf{x}_{T}-\\mathbf{1}x^{*}\\|^{2}={\\mathcal{O}}(m\\epsilon/\\mu)$ and $f({\\bar{x}}_{T})-f(x^{*})\\leq\\epsilon,$ the computation and communication complexities of Algorithm 1 are at most\n\n$$\nT=\\mathcal{O}\\left(\\sqrt{\\kappa_{g}}\\log\\left(\\frac{1}{\\epsilon}\\right)\\right)\\qquad and\\qquad Q=\\mathcal{O}\\left(\\sqrt{\\frac{\\kappa_{g}}{1-\\lambda_{2}(W)}}\\log\\left(\\frac{M\\kappa_{g}}{L}\\right)\\log\\left(\\frac{1}{\\epsilon}\\right)\\right)\n$$\n\nrespectively."}, {"name": "Lyapunov function V_t and v_bar_t (eqs. (17)-(18))", "statement": "We first introduce the Lyapunov function as follows\n\n$$\nV_{t}=h(\\bar{x}_{t})-h(x^{*})+\\frac{\\mu}{2}\\left\\|\\bar{v}_{t}-x^{*}\\right\\|^{2},\\tag{17}\n$$\n\nwhere $\\bar{v}_{t}$ is defined as\n\n$$\n\\bar{v}_{t}=\\bar{x}_{t-1}+\\frac{1}{\\alpha}\\big(\\bar{x}_{t}-\\bar{x}_{t-1}\\big)\\quad\\mathrm{with}\\qquad\\alpha=\\sqrt{\\mu\\eta}.\\tag{18}\n$$"}, {"name": "Lemma 14", "statement": "Lemma 14 For $\\textstyle{\\bar{x}}_{t},{\\bar{y}}_{t}$ and $\\bar{v}_{t}$ defined in Eqs. (3) and (18), then we can obtain that\n\n$$\n\\bar{y}_{t}-\\bar{x}_{t}=\\alpha(\\bar{v}_{t}-\\bar{y}_{t}),\\tag{36}\n$$\n\n$$\n\\bar{y}_{t + 1}= \\frac{\\bar{x}_{t + 1}+ \\alpha \\bar{v}_{t + 1}}{1 + \\alpha},\\tag{37}\n$$\n\nand\n\n$$\n\\bar{v}_{t+1}=\\left\\{\\begin{array}{l l}{(1-\\alpha)\\bar{v}_{t}+\\alpha\\bar{y}_{t}-\\displaystyle\\frac{\\eta}{\\alpha}\\bar{G}_{t}\\quad}&{r(x),isconvex,}\\\\ {(1-\\alpha)\\bar{v}_{t}+\\alpha\\bar{y}_{t}-\\displaystyle\\frac{\\eta}{\\alpha}\\bar{g}_{t},\\quad}&{r(x)=0.}\\end{array}\\right.\\tag{38}\n$$"}, {"name": "Lemma 13", "statement": "Lemma 13 We have\n\n$$\n\\|\\nabla F(\\mathbf{y})-\\nabla F(\\mathbf{x})\\|\\leq M\\|\\mathbf{y}-\\mathbf{x}\\|\\tag{33}\n$$\n\nand\n\n$$\n\\left\\|{\\bar{g}}_{t}- \\nabla f ({\\bar{y}}_{t}) \\right\\| \\leq{\\frac{M}{\\sqrt{m}}}\\left\\| \\mathbf{y}_{t}- \\mathbf{1}{\\bar{y}}_{t}\\right\\| .\\tag{34}\n$$\n\nFurthermore, we have the $\\left(L+2/\\eta\\right)$ -smooth property for the generalized gradient $\\tilde{\\nabla}h(\\cdot)$ (defined in Eq. (42)), i.e.,\n\n$$\n\\left\\|\\tilde{\\nabla}h(x)-\\tilde{\\nabla}h(y)\\right\\|\\leq\\left(\\frac{2}{\\eta}+L\\right)\\|x-y\\|.\\tag{35}\n$$"}, {"name": "Lemma 15", "statement": "Lemma 15 Let $f(x)$ be µ-strongly convex. For $\\bar{y}_{t},$ and $V_{t}$ defined in Eqs. (3) and (17) and $x^{*}$ being the optimum, we have the following inequality,\n\n$$\n\\|{\\bar{y}}_{t}- x^{*}\\| \\leq \\sqrt{\\frac{2 V_{t}}{\\mu}}.\\tag{39}\n$$"}, {"name": "Lemma 23 (reformulation of Algorithm 1)", "statement": "Lemma 23 The update procedure of Algorithm 1 can be represented as\n\n$$\n\\mathbf{x}_{t+1}=\\mathrm{FastMix}\\left(\\mathbf{y}_{t}-\\eta\\mathbf{s}_{t},K\\right),\\tag{64}\n$$\n\n$$\n{\\bf y}_{t+1}={\\bf x}_{t+1}+\\frac{1-\\alpha}{1+\\alpha}({\\bf x}_{t+1}-{\\bf x}_{t}),\\tag{65}\n$$\n\n$$\n\\mathbf{s}_{t+1}=\\mathrm{FastMix}(\\mathbf{s}_{t},K)+(\\nabla F(\\mathbf{y}_{t+1})-\\nabla F(\\mathbf{y}_{t}))-\\eta^{-1}(\\mathrm{FastMix}(\\mathbf{y}_{t},K)-\\mathbf{y}_{t}),\\tag{66}\n$$\n\nwith ${\\bf s}_{0}= \\nabla F ({\\bf y}_{0})$"}, {"name": "Lemma 24", "statement": "Lemma 24 Let $\\bar{x}_{t}, \\bar{y}_{t}, \\bar{g}_{t}$ (defined in Eq. (3)) be generated by Algorithm 1. Then they satisfy the following equalities:\n\n$$\n\\begin{array}{l l}{\\bar{x}_{t+1}=\\bar{y}_{t}-\\eta\\bar{g}_{t},}\\\\ {\\bar{y}_{t+1}=\\bar{x}_{t+1}+\\displaystyle\\frac{1-\\alpha}{1+\\alpha}(\\bar{x}_{t+1}-\\bar{x}_{t}),}\\\\ {\\bar{s}_{t+1}=\\bar{s}_{t}+\\bar{g}_{t+1}-\\bar{g}_{t}=\\bar{g}_{t+1}.}\\end{array}\\tag{68}\n$$"}], "nl_notes": "### Lemma 27\nNOTE (Lemma 27) — Model a single step t → t+1 of Algorithm 1, not a full run. Take the iterates x_{t−1}, y_{t−1}, x_t as given, with y_t and x_{t+1} produced from them by the algorithm's updates. Assume the first identity of Lemma 24 at the previous step, x̄_t = ȳ_{t−1} − η·(1/m)Σ_i ∇f_i(y_{t−1}^i), instead of deriving it from the initialization.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped BigOperators\nopen Polynomial\n\nnoncomputable def rowAvg {m d : ℕ} (x : Fin m → EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) :=\n  (1 / (m : ℝ)) • ∑ i, x i\n\nnoncomputable def Wapply {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ)\n    (x : Fin m → EuclideanSpace ℝ (Fin d)) : Fin m → EuclideanSpace ℝ (Fin d) :=\n  fun i => ∑ j, W i j • x j\n\nnoncomputable def fastMixAux {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (x0 : Fin m → EuclideanSpace ℝ (Fin d)) :\n    ℕ → (Fin m → EuclideanSpace ℝ (Fin d)) × (Fin m → EuclideanSpace ℝ (Fin d))\n  | 0 => (x0, x0)\n  | (k+1) =>\n      ((fun i => (1 + ηw) • Wapply W (fastMixAux W ηw x0 k).1 i\n                  - ηw • (fastMixAux W ηw x0 k).2 i),\n       (fastMixAux W ηw x0 k).1)\n\nnoncomputable def FastMix {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) : Fin m → EuclideanSpace ℝ (Fin d) :=\n  (fastMixAux W ηw x0 K).1\n\ntheorem Wapply_sum {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (x : Fin m → EuclideanSpace ℝ (Fin d)) :\n    ∑ i, Wapply W x i = ∑ i, x i := by\n  simp only [Wapply]\n  rw [Finset.sum_comm]\n  refine Finset.sum_congr rfl (fun j _ => ?_)\n  rw [← Finset.sum_smul, hW j, one_smul]\n\ntheorem fastMixAux_sum {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (hW : ∀ j, ∑ i, W i j = 1) (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) :\n    (∑ i, (fastMixAux W ηw x0 K).1 i = ∑ i, x0 i) ∧\n    (∑ i, (fastMixAux W ηw x0 K).2 i = ∑ i, x0 i) := by\n  induction K with\n  | zero => simp [fastMixAux]\n  | succ k ih =>\n    obtain ⟨ih1, ih2⟩ := ih\n    constructor\n    · simp only [fastMixAux]\n      rw [Finset.sum_sub_distrib, ← Finset.smul_sum, ← Finset.smul_sum,\n          Wapply_sum W hW, ih1, ih2, ← sub_smul,\n          show (1 + ηw) - ηw = (1 : ℝ) from by ring, one_smul]\n    · simpa only [fastMixAux] using ih1\n\ntheorem FastMix_rowAvg {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (hW : ∀ j, ∑ i, W i j = 1) (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) :\n    rowAvg (FastMix W ηw x0 K) = rowAvg x0 := by\n  unfold rowAvg FastMix\n  rw [(fastMixAux_sum W ηw hW x0 K).1]\n\ndef IsSmooth {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (L : ℝ) (f : E → ℝ) (g : E → E) : Prop :=\n  ∀ x y, f y ≤ f x + (inner (𝕜 := ℝ) (g x) (y - x)) + (L / 2) * ‖y - x‖ ^ 2\n\ndef IsProxStep {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (c : ℝ) (R : E → ℝ) (a xplus : E) : Prop :=\n  ∀ u, R xplus + (1 / (2 * c)) * ‖xplus - a‖ ^ 2 ≤ R u + (1 / (2 * c)) * ‖u - a‖ ^ 2\n\ntheorem lemma11 {d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / (2 * L))\n    (hα : α = Real.sqrt (μ * η)) (hα1 : α ≤ 1)\n    (R : EuclideanSpace ℝ (Fin d) → ℝ) (hRconv : ConvexOn ℝ Set.univ R)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar + R xstar ≤ f x + R x)\n    (xtprev xt xt1 yt yt1 st st1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = FastMix W ηw (fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i)) K)\n    (hxt1 : ∀ i, IsProxStep η R (yt i - η • st i) (xt1 i))\n    (hyt1 : yt1 = FastMix W ηw (fun i => xt1 i + ((1 - α) / (1 + α)) • (xt1 i - xt i)) K)\n    (hst1 : st1 = FastMix W ηw (fun i => st i + gradFi i (yt1 i) - gradFi i (yt i)) K)\n    (hgtrack : rowAvg st = (1 / (m : ℝ)) • ∑ i, gradFi i (yt i))\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) + R (rowAvg xt) - (f xstar + R xstar))\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) + R (rowAvg xt1) - (f xstar + R xstar))\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - α) * Vt + (13 * η / m) * (∑ i, ‖st i - rowAvg st‖ ^ 2)\n        + ((20 * M ^ 2 * η + 10 * η⁻¹ + (1 + 8 / α ^ 3) * M ^ 2 / L) / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2) := by\n  sorry\n\ntheorem lemma27 {d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / L)\n    (hα : α = Real.sqrt (μ * η))\n    (hα1 : α ≤ 1)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar ≤ f x)\n    (xtprev ytprev xt yt xt1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i))\n    (hprevavg : rowAvg xt = rowAvg ytprev\n        - η • ((1 / (m : ℝ)) • ∑ i, gradFi i (ytprev i)))\n    (hxt1 : xt1 = FastMix W ηw (fun i => yt i + (xt i - ytprev i)\n                - η • (gradFi i (yt i) - gradFi i (ytprev i))) K)\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) - f xstar)\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) - f xstar)\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - (3 / 4) * α) * Vt\n        + (1 + 8 / α ^ 3) * (M ^ 2 / L) * (1 / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2) := by\n  sorry\n\ntheorem lemma10 (L M ρ : ℝ) (hL : 0 < L) (η : ℝ) (hηeq : η = 1 / (2 * L)) (hM : 0 < M) (hLM : L ≤ M) (hρpos : 0 < ρ)\n    (hρ : ρ ≤ L ^ 3 / (1280 * M ^ 3))\n    (A : Matrix (Fin 3) (Fin 3) ℝ)\n    (hA : A = !![0, 2, 2; 2 * ρ, 4 * ρ, 4 * ρ; ρ * M / L, 8 * ρ * M ^ 2 / L ^ 2, 5 * ρ * M / L]) :\n    ∃ l1 l2 l3 : ℝ, A.charpoly = (X - C l1) * (X - C l2) * (X - C l3) ∧\n      0 < l1 ∧ |l3| ≤ |l2| ∧ |l2| < l1 ∧ l1 ≤ 1 / 2 ∧\n      ∃ v : Fin 3 → ℝ, A.mulVec v = l1 • v ∧\n        0 < v 0 ∧ v 0 ≤ 8 * v 2 / Real.sqrt ρ ∧\n        0 < v 1 ∧ v 1 ≤ 3 * v 2 ∧\n        0 < v 2 := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma11", "lemma27", "lemma10"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 4979, "end": 7036, "gold": "{d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / L)\n    (hα : α = Real.sqrt (μ * η))\n    (hα1 : α ≤ 1)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar ≤ f x)\n    (xtprev ytprev xt yt xt1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i))\n    (hprevavg : rowAvg xt = rowAvg ytprev\n        - η • ((1 / (m : ℝ)) • ∑ i, gradFi i (ytprev i)))\n    (hxt1 : xt1 = FastMix W ηw (fun i => yt i + (xt i - ytprev i)\n                - η • (gradFi i (yt i) - gradFi i (ytprev i))) K)\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) - f xstar)\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) - f xstar)\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - (3 / 4) * α) * Vt\n        + (1 + 8 / α ^ 3) * (M ^ 2 / L) * (1 / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2)"}]}
{"item_id": "v24-22-1210::lemma10::A-stmt", "paper_id": "v24-22-1210", "target_name": "lemma10", "source": "Research-level", "nl_statement": "Lemma 10 Matrix A defined in $E q .$ (23) satisfies that\n\n$$\n0<\\lambda_{1}(\\mathbf{A})\\qquad and\\qquad|\\lambda_{3}(\\mathbf{A})|\\leq|\\lambda_{2}(\\mathbf{A})| <\\lambda_{1}(\\mathbf{A})\n$$\n\nwith $\\lambda_{i}(\\mathbf{A})$ being the i-th largest eigenvalue of A.Letting $\\eta=1/(2L)$ and $\\rho\\le L^{3}/(1280M^{3})$ , then it holds that\n\n$$\n\\lambda_{1}(\\mathbf{A})\\leq\\frac{1}{2},\n$$\n\nand the eigenvector v associated with $\\lambda_{1}(\\mathbf{A})$ is positive and its entries satisfy\n\n$$\n0<{\\bf v}(1)\\leq\\frac{8{\\bf v}(3)}{\\sqrt{\\rho}},\\qquad 0<{\\bf v}(2)\\leq 3{\\bf v}(3)\\qquad and\\quad 0<{\\bf v}(3),\\tag{24}\n$$\n\nwhere $\\mathbf{v}(i)$ is i-th entry of v.", "context": [{"name": "Setup: decentralized composite problem (Section 1, eq. (1))", "statement": "In this paper, we consider the decentralized optimization problem, where the objective function is composed of m local functions $f_{i}(x)$ that are located on m different agents. The agents form a connected and undirected network and each of them only accesses its local function and communicates\n\nwith its neighbors. All of the agents target to cooperatively solve the convex optimization problem\n\n$$\n\\operatorname*{min}_{x\\in\\mathbb{R}^{d}}h(x)\\triangleq f(x)+r(x)\\quad\\mathrm{with}\\quad f(x)\\triangleq\\frac{1}{m}\\sum_{i=1}^{m}f_{i}(x),\\tag{1}\n$$\n\nwhere $f(x)$ is L-smooth and $\\mu$-strongly convex, $r(x)$ is convex but may be non-differentiable."}, {"name": "Definitions: L-smoothness, strong convexity, local smoothness, condition numbers (eqs. (7)-(9))", "statement": "Then we introduce the following definitions that will be used in the whole paper:\n\nWe say $f(x)$ is L-smooth if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf(y)\\leq f(x)+\\langle\\nabla f(x),y-x\\rangle+{\\frac{L}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f(x)$ is µ-strongly convex, if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf(y)\\geq f(x)+\\langle\\nabla f(x),y-x\\rangle+{\\frac{\\mu}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f_{i}(x)$ is locally Mi-smooth if for all $x,y\\in\\mathbb{R}^{d}$ , it holds that\n\n$$\nf_{i}(y)\\leq f_{i}(x)+\\langle\\nabla f_{i}(x),y-x\\rangle+{\\frac{M_{i}}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nWe say $f_{i}(x)$ is locally νi-strongly convex if for all $x,y\\in\\mathbb{R}^{d}$ ,it holds that\n\n$$\nf_{i}(y)\\geq f_{i}(x)+\\langle\\nabla f_{i}(x),y-x\\rangle+{\\frac{\\nu_{i}}{2}}\\left\\|y-x\\right\\|^{2}.\n$$\n\nBased on the smoothness and strong convexity, we can define global and local condition numbers of the objective function as\n\n$$\n\\kappa_{g}=\\frac{L}{\\mu},\\qquad\\hat{\\kappa}_{g}=\\frac{M}{\\mu}\\qquad\\mathrm{and}\\qquad\\kappa_{\\ell}=\\frac{M}{\\nu},\\tag{7}\n$$\n\nwhere\n\n$$\nM=\\operatorname*{max}_{i\\in\\{1,...,m\\}}M_{i}\\mathrm{~a n d~}\\nu=\\operatorname*{min}_{i\\in\\{1,...,m\\}}\\nu_{i}.\\tag{8}\n$$\n\nIt is easy to verify that\n\n$$\nL\\leq M\\qquad\\mathrm{and}\\qquad\\kappa_{g}\\leq\\hat{\\kappa}_{g}\\leq\\kappa_{\\ell}.\\tag{9}\n$$"}, {"name": "Assumption: properties of the weight matrix W", "statement": "For the topology of the network, we let W be the weight matrix associated with the network, indicating how agents are connected to each other. We assume that the weight matrix W has the following properties:\n\n1. W is symmetric with $W_{i,j}\\neq 0$ if and if only agents i and j are connected or $i = j ;$\n\n$$\n2 .\\ \\mathbf{0}\\preceq W\\preceq I,W\\mathbf{1}=\\mathbf{1},\\mathrm{null}(I-W)=\\mathrm{span}(\\mathbf{1});\n$$\n\nwhere we use I to denote the $m \\times m$ identity matrix and $\\mathbf{1}=[1,\\ldots,1]^{\\top}\\in\\mathbb{R}^{m}$ denotes the vector with all ones. The weight matrix has an important property that $\\begin{array}{r}{W^{\\infty}= \\frac{1}{m}\\mathbf{1}\\mathbf{1}^{\\top}}\\end{array}$ (Xiao and Boyd, 2004). Thus, one can achieve the effect of averaging local $x_{i}$ on different agents by using Wx for iterations. Instead of directly multiplying W, Liu and Morse (2011) proposed a more efficient way to achieve averaging described in Algorithm 2, which has the following important proposition."}, {"name": "Lemma 9 and the matrix A (eqs. (22)-(23))", "statement": "Lemma 9 Let $\\mathbf{z}_{t}\\ =\\ \\left[\\left\\|\\mathbf{x}_{t}-\\mathbf{1}\\bar{x}_{t}\\right\\|,\\left\\|\\mathbf{y}_{t}-\\mathbf{1}\\bar{y}_{t}\\right\\|,\\eta\\left\\|\\mathbf{s}_{t}-\\mathbf{1}\\bar{s}_{t}\\right\\|\\right]^{\\top}$ with $\\mathbf x_{t}, \\ \\mathbf y_{t}$ and st generated by Algorithm 3, then it holds that\n\n$$\n\\mathbf{z}_{t+1}\\leq\\mathbf{A}\\mathbf{z}_{t}+\\frac{4\\rho\\sqrt{m}M}{L}\\left[0,0,\\sqrt{\\frac{2V_{t}}{\\mu}}\\right]^{\\top},\\tag{22}\n$$\n\nwhere $\\rho$ and $\\mathbf{A}$ are defined as\n\n$$\n\\rho=\\left(1-\\sqrt{1-\\lambda_{2}(W)}\\right)^{K}\\quad and\\quad\\mathbf{A}=\\left[\\begin{array}{c c c}{0}&{2}&{2}\\\\ {2\\rho}&{4\\rho}&{4\\rho}\\\\ {\\rho M/L}&{8\\rho M^{2}/L^{2}}&{5\\rho M/L}\\end{array}\\right].\\tag{23}\n$$"}], "nl_notes": "### Lemma 10\nNOTE (Lemma 10) — The statement presupposes that the eigenvalues of A are real numbers, ordered by size. The Lean makes this part of the claim: the characteristic polynomial of A factors as (X − λ₁)(X − λ₂)(X − λ₃) with real λ₁, λ₂, λ₃. The paper's proof does not show that all three eigenvalues are real (Perron–Frobenius gives only λ₁), so this has to be proved for the stated range of ρ.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nopen scoped BigOperators\nopen Polynomial\n\nnoncomputable def rowAvg {m d : ℕ} (x : Fin m → EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) :=\n  (1 / (m : ℝ)) • ∑ i, x i\n\nnoncomputable def Wapply {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ)\n    (x : Fin m → EuclideanSpace ℝ (Fin d)) : Fin m → EuclideanSpace ℝ (Fin d) :=\n  fun i => ∑ j, W i j • x j\n\nnoncomputable def fastMixAux {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (x0 : Fin m → EuclideanSpace ℝ (Fin d)) :\n    ℕ → (Fin m → EuclideanSpace ℝ (Fin d)) × (Fin m → EuclideanSpace ℝ (Fin d))\n  | 0 => (x0, x0)\n  | (k+1) =>\n      ((fun i => (1 + ηw) • Wapply W (fastMixAux W ηw x0 k).1 i\n                  - ηw • (fastMixAux W ηw x0 k).2 i),\n       (fastMixAux W ηw x0 k).1)\n\nnoncomputable def FastMix {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) : Fin m → EuclideanSpace ℝ (Fin d) :=\n  (fastMixAux W ηw x0 K).1\n\ntheorem Wapply_sum {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (x : Fin m → EuclideanSpace ℝ (Fin d)) :\n    ∑ i, Wapply W x i = ∑ i, x i := by\n  simp only [Wapply]\n  rw [Finset.sum_comm]\n  refine Finset.sum_congr rfl (fun j _ => ?_)\n  rw [← Finset.sum_smul, hW j, one_smul]\n\ntheorem fastMixAux_sum {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (hW : ∀ j, ∑ i, W i j = 1) (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) :\n    (∑ i, (fastMixAux W ηw x0 K).1 i = ∑ i, x0 i) ∧\n    (∑ i, (fastMixAux W ηw x0 K).2 i = ∑ i, x0 i) := by\n  induction K with\n  | zero => simp [fastMixAux]\n  | succ k ih =>\n    obtain ⟨ih1, ih2⟩ := ih\n    constructor\n    · simp only [fastMixAux]\n      rw [Finset.sum_sub_distrib, ← Finset.smul_sum, ← Finset.smul_sum,\n          Wapply_sum W hW, ih1, ih2, ← sub_smul,\n          show (1 + ηw) - ηw = (1 : ℝ) from by ring, one_smul]\n    · simpa only [fastMixAux] using ih1\n\ntheorem FastMix_rowAvg {m d : ℕ} (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ)\n    (hW : ∀ j, ∑ i, W i j = 1) (x0 : Fin m → EuclideanSpace ℝ (Fin d)) (K : ℕ) :\n    rowAvg (FastMix W ηw x0 K) = rowAvg x0 := by\n  unfold rowAvg FastMix\n  rw [(fastMixAux_sum W ηw hW x0 K).1]\n\ndef IsSmooth {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (L : ℝ) (f : E → ℝ) (g : E → E) : Prop :=\n  ∀ x y, f y ≤ f x + (inner (𝕜 := ℝ) (g x) (y - x)) + (L / 2) * ‖y - x‖ ^ 2\n\ndef IsProxStep {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (c : ℝ) (R : E → ℝ) (a xplus : E) : Prop :=\n  ∀ u, R xplus + (1 / (2 * c)) * ‖xplus - a‖ ^ 2 ≤ R u + (1 / (2 * c)) * ‖u - a‖ ^ 2\n\ntheorem lemma11 {d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / (2 * L))\n    (hα : α = Real.sqrt (μ * η)) (hα1 : α ≤ 1)\n    (R : EuclideanSpace ℝ (Fin d) → ℝ) (hRconv : ConvexOn ℝ Set.univ R)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar + R xstar ≤ f x + R x)\n    (xtprev xt xt1 yt yt1 st st1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = FastMix W ηw (fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i)) K)\n    (hxt1 : ∀ i, IsProxStep η R (yt i - η • st i) (xt1 i))\n    (hyt1 : yt1 = FastMix W ηw (fun i => xt1 i + ((1 - α) / (1 + α)) • (xt1 i - xt i)) K)\n    (hst1 : st1 = FastMix W ηw (fun i => st i + gradFi i (yt1 i) - gradFi i (yt i)) K)\n    (hgtrack : rowAvg st = (1 / (m : ℝ)) • ∑ i, gradFi i (yt i))\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) + R (rowAvg xt) - (f xstar + R xstar))\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) + R (rowAvg xt1) - (f xstar + R xstar))\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - α) * Vt + (13 * η / m) * (∑ i, ‖st i - rowAvg st‖ ^ 2)\n        + ((20 * M ^ 2 * η + 10 * η⁻¹ + (1 + 8 / α ^ 3) * M ^ 2 / L) / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2) := by\n  sorry\n\ntheorem lemma27 {d m : ℕ} (hm : 0 < m) (μ L M η α : ℝ)\n    (hμ : 0 < μ) (hL : 0 < L) (hM : 0 < M)\n    (hLM : L ≤ M) (hη : 0 < η) (hηeq : η = 1 / L)\n    (hα : α = Real.sqrt (μ * η))\n    (hα1 : α ≤ 1)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hfsmooth : IsSmooth L f gradf) (hfconv : StrongConvexOn Set.univ μ f)\n    (Fi : Fin m → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradFi : Fin m → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hFismooth : ∀ i, IsSmooth M (Fi i) (gradFi i))\n\n    (hFiLip : ∀ i x y, ‖gradFi i x - gradFi i y‖ ≤ M * ‖x - y‖)\n    (hfavg : f = fun x => (1 / (m : ℝ)) * ∑ i, Fi i x)\n    (hgradf_avg : gradf = fun x => (1 / (m : ℝ)) • ∑ i, gradFi i x)\n    (xstar : EuclideanSpace ℝ (Fin d)) (hxstar : ∀ x, f xstar ≤ f x)\n    (xtprev ytprev xt yt xt1 : Fin m → EuclideanSpace ℝ (Fin d))\n    (K : ℕ)\n    (W : Matrix (Fin m) (Fin m) ℝ) (ηw : ℝ) (hW : ∀ j, ∑ i, W i j = 1)\n    (adj : Fin m → Fin m → Prop) (hWsupp : ∀ i j, W i j ≠ 0 ↔ adj i j ∨ i = j)\n    (hWsymm : W.IsSymm) (hWpsd : W.PosSemidef) (hWle : (1 - W).PosSemidef)\n    (hWnull : ∀ v : Fin m → ℝ, (1 - W).mulVec v = 0 ↔ ∃ c : ℝ, v = fun _ => c)\n    (lam2 : ℝ)\n    (hlam2 : IsGreatest {μ : ℝ | ∃ v : Fin m → ℝ, v ≠ 0 ∧ (∑ i, v i = 0) ∧ W.mulVec v = μ • v} lam2)\n    (hηw : ηw = 1 / (1 + Real.sqrt (1 - lam2 ^ 2)))\n    (hyt : yt = fun i => xt i + ((1 - α) / (1 + α)) • (xt i - xtprev i))\n    (hprevavg : rowAvg xt = rowAvg ytprev\n        - η • ((1 / (m : ℝ)) • ∑ i, gradFi i (ytprev i)))\n    (hxt1 : xt1 = FastMix W ηw (fun i => yt i + (xt i - ytprev i)\n                - η • (gradFi i (yt i) - gradFi i (ytprev i))) K)\n    (Vt Vt1 : ℝ)\n    (hVt : Vt = (f (rowAvg xt) - f xstar)\n              + (μ / 2) * ‖(rowAvg xtprev + (1 / α) • (rowAvg xt - rowAvg xtprev)) - xstar‖ ^ 2)\n    (hVt1 : Vt1 = (f (rowAvg xt1) - f xstar)\n               + (μ / 2) * ‖(rowAvg xt + (1 / α) • (rowAvg xt1 - rowAvg xt)) - xstar‖ ^ 2) :\n    Vt1 ≤ (1 - (3 / 4) * α) * Vt\n        + (1 + 8 / α ^ 3) * (M ^ 2 / L) * (1 / m) * (∑ i, ‖yt i - rowAvg yt‖ ^ 2) := by\n  sorry\n\ntheorem lemma10 (L M ρ : ℝ) (hL : 0 < L) (η : ℝ) (hηeq : η = 1 / (2 * L)) (hM : 0 < M) (hLM : L ≤ M) (hρpos : 0 < ρ)\n    (hρ : ρ ≤ L ^ 3 / (1280 * M ^ 3))\n    (A : Matrix (Fin 3) (Fin 3) ℝ)\n    (hA : A = !![0, 2, 2; 2 * ρ, 4 * ρ, 4 * ρ; ρ * M / L, 8 * ρ * M ^ 2 / L ^ 2, 5 * ρ * M / L]) :\n    ∃ l1 l2 l3 : ℝ, A.charpoly = (X - C l1) * (X - C l2) * (X - C l3) ∧\n      0 < l1 ∧ |l3| ≤ |l2| ∧ |l2| < l1 ∧ l1 ≤ 1 / 2 ∧\n      ∃ v : Fin 3 → ℝ, A.mulVec v = l1 • v ∧\n        0 < v 0 ∧ v 0 ≤ 8 * v 2 / Real.sqrt ρ ∧\n        0 < v 1 ∧ v 1 ≤ 3 * v 2 ∧\n        0 < v 2 := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma11", "lemma27", "lemma10"], "topic_domain": "Others", "topic_subtopic": "Matrix & Tensor Analysis", "holes": [{"start": 7068, "end": 7609, "gold": "(L M ρ : ℝ) (hL : 0 < L) (η : ℝ) (hηeq : η = 1 / (2 * L)) (hM : 0 < M) (hLM : L ≤ M) (hρpos : 0 < ρ)\n    (hρ : ρ ≤ L ^ 3 / (1280 * M ^ 3))\n    (A : Matrix (Fin 3) (Fin 3) ℝ)\n    (hA : A = !![0, 2, 2; 2 * ρ, 4 * ρ, 4 * ρ; ρ * M / L, 8 * ρ * M ^ 2 / L ^ 2, 5 * ρ * M / L]) :\n    ∃ l1 l2 l3 : ℝ, A.charpoly = (X - C l1) * (X - C l2) * (X - C l3) ∧\n      0 < l1 ∧ |l3| ≤ |l2| ∧ |l2| < l1 ∧ l1 ≤ 1 / 2 ∧\n      ∃ v : Fin 3 → ℝ, A.mulVec v = l1 • v ∧\n        0 < v 0 ∧ v 0 ≤ 8 * v 2 / Real.sqrt ρ ∧\n        0 < v 1 ∧ v 1 ≤ 3 * v 2 ∧\n        0 < v 2"}]}
{"item_id": "v7-evendar06a::naive_PAC::A-stmt", "paper_id": "v7-evendar06a", "target_name": "naive_PAC", "source": "Research-level", "nl_statement": "Theorem 6 Naive(ε,δ) is an (ε, δ)-PAC algorithm, and its arm sample complexity is $O\\left((n/\\varepsilon^{2})\\log(n/\\delta)\\right)$.", "context": [{"name": "Setup: multi-armed bandit model (Section 2.1)", "statement": "The model consists of a set A of arms, where $n = \\left| A \\right|$. Sampling an arm $a \\in A$ yields a reward given by a random variable $R(a)$. Rewards are assumed binary: for each arm $a \\in A$, $R(a)\\in\\{0,1\\}$ (every result holds unchanged when the reward is bounded in [0, 1], and, with suitable modifications, more generally whenever the reward is bounded). We write $a_{1},\\cdots,a_{n}$ for the arms and set $p_{i}=\\mathbb{E}[R(a_{i})]$. To simplify notation, the arms are indexed in order of expected reward, $p_{1}> p_{2}> \\ldots > p_{n}$.\n\nThe best arm, denoted $a^{*}$, is an arm whose expected reward is highest; its expected reward $r^{*}$ is called the optimal reward. A non-best arm is an arm with expected reward strictly smaller than $r^{*}$, the expected reward of the best arm. An arm a is ε-optimal if its expected reward is within ε of the optimal reward, that is, $\\mathbb{E}[R(a)]\\geq r^{*}-\\varepsilon$.\n\nAt every time step t, an algorithm for the MAB problem samples some arm $a_{t}$ and obtains a reward $r_{t}$ (whose distribution is $R(a_{t})$). Its choice may depend on the history (that is, the actions taken and rewards received) up to time t − 1. In the end, the algorithm has to commit to one arm and select it."}, {"name": "Definition 1 ((ε,δ)-PAC algorithm) and Remark 2 (worst-case sample complexity)", "statement": "Definition 1 An algorithm for the multi armed bandit is an $(\\varepsilon,\\delta)$-PAC algorithm with sample complexity T if, upon termination, it outputs an ε-optimal arm a' with probability at least 1 − δ, and the number of time steps it executes before terminating is at most T.\n\nRemark 2 The MAB algorithm is allowed to terminate before T steps have elapsed. The sample complexity used here is that of the worst trajectory. Mannor and Tsitsiklis (2004) studied the expected sample complexity (with expectation over both the model and the algorithm), which behaves as $\\Omega((n+\\log(1/\\delta))/\\varepsilon^{2})$; this differs from the complexity established below in Theorem 10. Note also that the algorithm of Mannor and Tsitsiklis (2004) does not have a worst-case bound on its running time."}, {"name": "Lemma 4 (Hoeffding, 1963)", "statement": "Large deviation bounds are used repeatedly in this paper. Since boundedness is assumed, Hoeffding's inequality is available.\n\nLemma 4 (Hoeffding, 1963) Let X be a set, let D be a probability distribution on X, and let $f_{1},\\ldots,f_{m}$ be real-valued functions on X with $f_{i}: X \\longrightarrow [ a_{i}, b_{i}]$ for $i=1,\\ldots,m$, where $a_{i}$ and $b_{i}$ are real numbers with $a_{i}< b_{i}$. Let $x_{1},\\ldots,x_{m}$ be samples drawn independently and identically from D. Then\n\n$$\n\\begin{array}{r l r}{\\mathbf{P}\\left[\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}f_{i}(x_{i})-\\left(\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}\\int_{a_{i}}^{b_{i}}f_{i}(x)D(x)\\right)\\ge\\varepsilon\\right]}&{\\le}&{e^{-\\frac{2\\varepsilon^{2}m^{2}}{\\sum_{i=1}^{m}(b_{i}-a_{i})^{2}}}}\\\\ {\\mathbf{P}\\left[\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}f_{i}(x_{i})-\\left(\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}\\int_{a_{i}}^{b_{i}}f_{i}(x)D(x)\\right)\\le-\\varepsilon\\right]}&{\\le}&{e^{-\\frac{2\\varepsilon^{2}m^{2}}{\\sum_{i=1}^{m}(b_{i}-a_{i})^{2}}}.}\\end{array}\n$$"}, {"name": "Naive algorithm: Section 3 overview and Algorithm 1", "statement": "This section studies (ε, δ)-PAC algorithms for the MAB problem, i.e., algorithms that must, with probability 1 − δ, output an ε-optimal arm. The starting point is a naive approach: sample every arm $1/(\\varepsilon/2)^{2}\\ln(2n/\\delta)$ times and return the arm whose empirical reward is largest. This naive algorithm has sample complexity $O(n/\\varepsilon^{2}\\log(n/\\delta))$, and it is given as Algorithm 1. Section 3.1 treats an algorithm that removes arms one at a time, and Section 3.2 ends with the Median Elimination algorithm, which attains worst-case optimal sample complexity.\n\nAlgorithm 1 (Naive Algorithm).\nGiven: accuracy $\\varepsilon > 0$ and confidence $\\delta > 0$. Returns: one arm.\n1. For each arm $a \\in A$: draw $\\ell = \\frac{4}{\\varepsilon^{2}}\\ln (\\frac{2 n}{\\delta})$ samples of $a$, and record their mean reward as $\\hat{p}_{a}$.\n2. Return $a^{\\prime}= \\arg \\max_{a \\in A}\\{\\hat{p}_{a}\\}$."}], "nl_notes": "### Theorem 6\nNOTE (Theorem 6) — (1) The paper prints ℓ = (4/ε²) ln(2n/δ) in Algorithm 1 but uses ℓ = (2/ε²) ln(2n/δ) in the proof of Theorem 6 (the smallest ℓ with 2·exp(−2(ε/2)²ℓ) ≤ δ/n). The Lean follows the proof: ℓ ∈ [(2/ε²) log(2n/δ), (2/ε²) log(2n/δ) + 1] (`hℓ`, `hℓ_le`; the +1 is integer rounding). (2) The printed bound O((n/ε²) log(n/δ)) has no uniform constant (at n = 1, ε = 1, δ → 1, log(n/δ) → 0 while ℓ ≥ 2), so the conclusion is ∃ C > 0, n·ℓ ≤ C·((n/ε²) log(2n/δ) + n), with C chosen before all other quantities. (3) δ < 1 is assumed (`hδ1`).", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ResearchLevel\n\ndef epsOptimal {n : ℕ} (p : Fin n → ℝ) (pmax ε : ℝ) (a : Fin n) : Prop :=\n  pmax - p a ≤ ε\n\nnoncomputable def empMean {n ℓ : ℕ} {Ω : Type*} (X : Fin n → Fin ℓ → Ω → ℝ)\n    (ω : Ω) (a : Fin n) : ℝ :=\n  (∑ k, X a k ω) / ℓ\n\nopen Classical in\n\nnoncomputable def naiveOutput {n ℓ : ℕ} [NeZero n] {Ω : Type*}\n    (X : Fin n → Fin ℓ → Ω → ℝ) (ω : Ω) : Fin n :=\n  (Finite.exists_max (empMean X ω)).choose\n\nnoncomputable def empMeanT {n : ℕ} {Ω : Type*} (X : Fin n → ℕ → Ω → ℝ)\n    (a : Fin n) (t : ℕ) (ω : Ω) : ℝ :=\n  (∑ k ∈ Finset.range t, X a k ω) / t\n\nnoncomputable def confRadiusSE (c : ℝ) (n t : ℕ) (δ : ℝ) : ℝ :=\n  Real.sqrt (Real.log (c * n * (t : ℝ) ^ 2 / δ) / t)\n\nnoncomputable def phaseEps (ε : ℝ) : ℕ → ℝ\n  | 0 => ε / 4\n  | (ℓ + 1) => (3 / 4) * phaseEps ε ℓ\n\nnoncomputable def phaseDelta (δ : ℝ) : ℕ → ℝ\n  | 0 => δ / 2\n  | (ℓ + 1) => phaseDelta δ ℓ / 2\n\nnoncomputable def empMeanPhase {n : ℕ} {Ω : Type*} (X : Fin n → ℕ → ℕ → Ω → ℝ)\n    (a : Fin n) (ℓ m : ℕ) (ω : Ω) : ℝ :=\n  (∑ k ∈ Finset.range m, X a ℓ k ω) / m\n\ntheorem naive_PAC :\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (ℓ : ℕ) (hℓ : (2 / ε ^ 2) * Real.log (2 * n / δ) ≤ ℓ)\n      (hℓ_le : (ℓ : ℝ) ≤ (2 / ε ^ 2) * Real.log (2 * n / δ) + 1)\n      (X : Fin n → Fin ℓ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × Fin ℓ => X q.1 q.2) P)\n      (hX_ident : ∀ a (k k' : Fin ℓ), ProbabilityTheory.IdentDistrib (X a k) (X a k') P P),\n    (1 - δ ≤ P.real {ω | epsOptimal p (p astar) ε (naiveOutput X ω)}) ∧\n    ((n : ℝ) * ℓ ≤ C * ((n / ε ^ 2) * Real.log (2 * n / δ) + n)) := by\n  sorry\n\ntheorem successive_elimination_PAC :\n    ∀ (c : ℝ), 4 < c →\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (hgap : ∀ a, a ≠ astar → p a < p astar)\n      (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (X : Fin n → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ => X q.1 q.2) P)\n      (hX_ident : ∀ a k, IdentDistrib (X a k) (X a 0) P P)\n      (S : ℕ → Ω → Finset (Fin n))\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSsub : ∀ t ω, S (t + 1) ω ⊆ S t ω)\n      (hSrule : ∀ t ω a, a ∈ S t ω →\n          (a ∈ S (t + 1) ω ↔\n            ∀ b ∈ S t ω, empMeanT X b (t + 1) ω - empMeanT X a (t + 1) ω <\n              2 * confRadiusSE c n (t + 1) δ)),\n      1 - δ ≤ P.real {ω | ∃ T : ℕ,\n        (∑ t ∈ Finset.range T, (S t ω).card : ℝ) ≤\n          C * ∑ a ∈ Finset.univ.erase astar,\n            Real.log (n / (δ * (p astar - p a))) / (p astar - p a) ^ 2\n        ∧ ∀ t, T ≤ t → S t ω = {astar}} := by\n  sorry\n\ntheorem median_elimination_PAC :\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} (hn : 0 < n) (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (L : ℕ) (hL : L = Nat.clog 2 n)\n      (m : ℕ → ℕ)\n      (hm : ∀ ℓ, ℓ < L →\n        (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) ≤ m ℓ)\n      (hm_le : ∀ ℓ, ℓ < L →\n        (m ℓ : ℝ) ≤ (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) + 1)\n      (X : Fin n → ℕ → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a ℓ k, Measurable (X a ℓ k))\n      (hX_bound : ∀ a ℓ k, ∀ᵐ ω ∂P, X a ℓ k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a ℓ k, ∫ ω, X a ℓ k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ × ℕ => X q.1 q.2.1 q.2.2) P)\n    (hX_ident : ∀ a ℓ k, IdentDistrib (X a ℓ k) (X a 0 0) P P)\n      (S : ℕ → Ω → Finset (Fin n)) (hS_meas : ∀ ℓ s, MeasurableSet {ω | S ℓ ω = s})\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSnonempty : ∀ ℓ ω, (S ℓ ω).Nonempty)\n      (hScard : ∀ ℓ ω, 2 * (S (ℓ + 1) ω).card ≤ (S ℓ ω).card + 1)\n      (hScard_lo : ∀ ℓ ω, (S ℓ ω).card ≤ 2 * (S (ℓ + 1) ω).card)\n      (median : ℕ → Ω → ℝ) (hmedian_meas : ∀ ℓ, Measurable (median ℓ))\n      (hmedian_ge : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω)).card ≥ (S ℓ ω).card)\n      (hmedian_le : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => empMeanPhase X a ℓ (m ℓ) ω ≤ median ℓ ω)).card ≥ (S ℓ ω).card)\n      (hSstep_lo : ∀ ℓ ω, (S ℓ ω).filter\n          (fun a => median ℓ ω < empMeanPhase X a ℓ (m ℓ) ω) ⊆ S (ℓ + 1) ω)\n      (hSstep_hi : ∀ ℓ ω, S (ℓ + 1) ω ⊆ (S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω))\n      (hS_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → S (ℓ + 1) ω = S (ℓ + 1) ω')\n      (hmedian_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → median ℓ ω = median ℓ ω'),\n    (1 - δ ≤ P.real {ω | ∀ a ∈ S L ω, epsOptimal p (p astar) ε a}) ∧\n    (∀ ω, (∑ ℓ ∈ Finset.range L, (S ℓ ω).card * m ℓ : ℝ) ≤\n        C * ((n / ε ^ 2) * Real.log (3 / δ) + n)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["naive_PAC", "successive_elimination_PAC", "median_elimination_PAC"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 1115, "end": 1973, "gold": ":\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (ℓ : ℕ) (hℓ : (2 / ε ^ 2) * Real.log (2 * n / δ) ≤ ℓ)\n      (hℓ_le : (ℓ : ℝ) ≤ (2 / ε ^ 2) * Real.log (2 * n / δ) + 1)\n      (X : Fin n → Fin ℓ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × Fin ℓ => X q.1 q.2) P)\n      (hX_ident : ∀ a (k k' : Fin ℓ), ProbabilityTheory.IdentDistrib (X a k) (X a k') P P),\n    (1 - δ ≤ P.real {ω | epsOptimal p (p astar) ε (naiveOutput X ω)}) ∧\n    ((n : ℝ) * ℓ ≤ C * ((n / ε ^ 2) * Real.log (2 * n / δ) + n))"}]}
{"item_id": "v7-evendar06a::successive_elimination_PAC::A-stmt", "paper_id": "v7-evendar06a", "target_name": "successive_elimination_PAC", "source": "Research-level", "nl_statement": "Theorem 8 Assume $\\Delta_{i} > 0$ for every $i = 2, 3, \\ldots, n$. Let the constant $c$ in Algorithm 3 be any value with $c > 4$, and let $0 < \\delta < 1$. Then the Successive Elimination algorithm (Algorithm 3) is (0, δ)-PAC in the following sense: with probability no less than $1 - \\delta$, there exists a time from which on the set of surviving arms consists exactly of the best arm, and the number of samples drawn before that time is at most (where the constant hidden in the O depends on c alone)\n\n$$\nO\\left(\\sum_{i=2}^{n}\\frac{\\ln(\\frac{n}{\\delta\\Delta_{i}})}{\\Delta_{i}^{2}}\\right).\n$$", "context": [{"name": "Setup: multi-armed bandit model (Section 2.1)", "statement": "The model consists of a set A of arms, where $n = \\left| A \\right|$. Sampling an arm $a \\in A$ yields a reward given by a random variable $R(a)$. Rewards are assumed binary: for each arm $a \\in A$, $R(a)\\in\\{0,1\\}$ (every result holds unchanged when the reward is bounded in [0, 1], and, with suitable modifications, more generally whenever the reward is bounded). We write $a_{1},\\cdots,a_{n}$ for the arms and set $p_{i}=\\mathbb{E}[R(a_{i})]$. To simplify notation, the arms are indexed in order of expected reward, $p_{1}> p_{2}> \\ldots > p_{n}$.\n\nThe best arm, denoted $a^{*}$, is an arm whose expected reward is highest; its expected reward $r^{*}$ is called the optimal reward. A non-best arm is an arm with expected reward strictly smaller than $r^{*}$, the expected reward of the best arm. An arm a is ε-optimal if its expected reward is within ε of the optimal reward, that is, $\\mathbb{E}[R(a)]\\geq r^{*}-\\varepsilon$.\n\nAt every time step t, an algorithm for the MAB problem samples some arm $a_{t}$ and obtains a reward $r_{t}$ (whose distribution is $R(a_{t})$). Its choice may depend on the history (that is, the actions taken and rewards received) up to time t − 1. In the end, the algorithm has to commit to one arm and select it."}, {"name": "Definition 1 ((ε,δ)-PAC algorithm) and Remark 2 (worst-case sample complexity)", "statement": "Definition 1 An algorithm for the multi armed bandit is an $(\\varepsilon,\\delta)$-PAC algorithm with sample complexity T if, upon termination, it outputs an ε-optimal arm a' with probability at least 1 − δ, and the number of time steps it executes before terminating is at most T.\n\nRemark 2 The MAB algorithm is allowed to terminate before T steps have elapsed. The sample complexity used here is that of the worst trajectory. Mannor and Tsitsiklis (2004) studied the expected sample complexity (with expectation over both the model and the algorithm), which behaves as $\\Omega((n+\\log(1/\\delta))/\\varepsilon^{2})$; this differs from the complexity established below in Theorem 10. Note also that the algorithm of Mannor and Tsitsiklis (2004) does not have a worst-case bound on its running time."}, {"name": "Lemma 4 (Hoeffding, 1963)", "statement": "Large deviation bounds are used repeatedly in this paper. Since boundedness is assumed, Hoeffding's inequality is available.\n\nLemma 4 (Hoeffding, 1963) Let X be a set, let D be a probability distribution on X, and let $f_{1},\\ldots,f_{m}$ be real-valued functions on X with $f_{i}: X \\longrightarrow [ a_{i}, b_{i}]$ for $i=1,\\ldots,m$, where $a_{i}$ and $b_{i}$ are real numbers with $a_{i}< b_{i}$. Let $x_{1},\\ldots,x_{m}$ be samples drawn independently and identically from D. Then\n\n$$\n\\begin{array}{r l r}{\\mathbf{P}\\left[\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}f_{i}(x_{i})-\\left(\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}\\int_{a_{i}}^{b_{i}}f_{i}(x)D(x)\\right)\\ge\\varepsilon\\right]}&{\\le}&{e^{-\\frac{2\\varepsilon^{2}m^{2}}{\\sum_{i=1}^{m}(b_{i}-a_{i})^{2}}}}\\\\ {\\mathbf{P}\\left[\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}f_{i}(x_{i})-\\left(\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}\\int_{a_{i}}^{b_{i}}f_{i}(x)D(x)\\right)\\le-\\varepsilon\\right]}&{\\le}&{e^{-\\frac{2\\varepsilon^{2}m^{2}}{\\sum_{i=1}^{m}(b_{i}-a_{i})^{2}}}.}\\end{array}\n$$"}, {"name": "Gap Δ_i (Section 3.1)", "statement": "Define $\\Delta_{i}= p_{1}- p_{i}> 0$."}, {"name": "Algorithm 3: Successive elimination with unknown biases", "statement": "We now drop the assumption that the arms' expected rewards are known beforehand, and present the Successive Elimination algorithm, which handles an arbitrary set of biases. Algorithm 3 below identifies the best arm (not merely an ε-best one) with high probability; Remark 9 later describes how to turn it into an (ε, δ)-PAC algorithm.\n\nAlgorithm 3 (Successive elimination with unknown biases).\nGiven: $\\delta > 0$. Returns: one arm.\n1. Initialise $t = 1$ and $S = A$, and put $\\hat{p}_{a}^{1}= 0$ for each arm a.\n2. Draw one sample from each arm $a \\in S$; write $\\hat{p}_{a}^{t}$ for the mean reward of arm a up to time $t$.\n3. Loop body:\n   (a) compute $\\hat{p}_{max}^{t}= \\max_{a \\in S}\\hat{p}_{a}^{t}$ and $\\alpha_{t}=\\sqrt{\\ln(cnt^{2}/\\delta)/t}$, with c a constant;\n   (b) remove from $S$ (i.e. $S = S \\backslash \\{a \\}$) every arm $a \\in S$ for which $\\hat{p}_{max}^{t}- \\hat{p}_{a}^{t}\\geq 2 \\alpha_{t}$;\n   (c) increase $t$ by one ($t = t + 1$).\n4. Repeat step 3 until $| S | > 1$ (termination condition exactly as printed in the paper)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ResearchLevel\n\ndef epsOptimal {n : ℕ} (p : Fin n → ℝ) (pmax ε : ℝ) (a : Fin n) : Prop :=\n  pmax - p a ≤ ε\n\nnoncomputable def empMean {n ℓ : ℕ} {Ω : Type*} (X : Fin n → Fin ℓ → Ω → ℝ)\n    (ω : Ω) (a : Fin n) : ℝ :=\n  (∑ k, X a k ω) / ℓ\n\nopen Classical in\n\nnoncomputable def naiveOutput {n ℓ : ℕ} [NeZero n] {Ω : Type*}\n    (X : Fin n → Fin ℓ → Ω → ℝ) (ω : Ω) : Fin n :=\n  (Finite.exists_max (empMean X ω)).choose\n\nnoncomputable def empMeanT {n : ℕ} {Ω : Type*} (X : Fin n → ℕ → Ω → ℝ)\n    (a : Fin n) (t : ℕ) (ω : Ω) : ℝ :=\n  (∑ k ∈ Finset.range t, X a k ω) / t\n\nnoncomputable def confRadiusSE (c : ℝ) (n t : ℕ) (δ : ℝ) : ℝ :=\n  Real.sqrt (Real.log (c * n * (t : ℝ) ^ 2 / δ) / t)\n\nnoncomputable def phaseEps (ε : ℝ) : ℕ → ℝ\n  | 0 => ε / 4\n  | (ℓ + 1) => (3 / 4) * phaseEps ε ℓ\n\nnoncomputable def phaseDelta (δ : ℝ) : ℕ → ℝ\n  | 0 => δ / 2\n  | (ℓ + 1) => phaseDelta δ ℓ / 2\n\nnoncomputable def empMeanPhase {n : ℕ} {Ω : Type*} (X : Fin n → ℕ → ℕ → Ω → ℝ)\n    (a : Fin n) (ℓ m : ℕ) (ω : Ω) : ℝ :=\n  (∑ k ∈ Finset.range m, X a ℓ k ω) / m\n\ntheorem naive_PAC :\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (ℓ : ℕ) (hℓ : (2 / ε ^ 2) * Real.log (2 * n / δ) ≤ ℓ)\n      (hℓ_le : (ℓ : ℝ) ≤ (2 / ε ^ 2) * Real.log (2 * n / δ) + 1)\n      (X : Fin n → Fin ℓ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × Fin ℓ => X q.1 q.2) P)\n      (hX_ident : ∀ a (k k' : Fin ℓ), ProbabilityTheory.IdentDistrib (X a k) (X a k') P P),\n    (1 - δ ≤ P.real {ω | epsOptimal p (p astar) ε (naiveOutput X ω)}) ∧\n    ((n : ℝ) * ℓ ≤ C * ((n / ε ^ 2) * Real.log (2 * n / δ) + n)) := by\n  sorry\n\ntheorem successive_elimination_PAC :\n    ∀ (c : ℝ), 4 < c →\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (hgap : ∀ a, a ≠ astar → p a < p astar)\n      (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (X : Fin n → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ => X q.1 q.2) P)\n      (hX_ident : ∀ a k, IdentDistrib (X a k) (X a 0) P P)\n      (S : ℕ → Ω → Finset (Fin n))\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSsub : ∀ t ω, S (t + 1) ω ⊆ S t ω)\n      (hSrule : ∀ t ω a, a ∈ S t ω →\n          (a ∈ S (t + 1) ω ↔\n            ∀ b ∈ S t ω, empMeanT X b (t + 1) ω - empMeanT X a (t + 1) ω <\n              2 * confRadiusSE c n (t + 1) δ)),\n      1 - δ ≤ P.real {ω | ∃ T : ℕ,\n        (∑ t ∈ Finset.range T, (S t ω).card : ℝ) ≤\n          C * ∑ a ∈ Finset.univ.erase astar,\n            Real.log (n / (δ * (p astar - p a))) / (p astar - p a) ^ 2\n        ∧ ∀ t, T ≤ t → S t ω = {astar}} := by\n  sorry\n\ntheorem median_elimination_PAC :\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} (hn : 0 < n) (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (L : ℕ) (hL : L = Nat.clog 2 n)\n      (m : ℕ → ℕ)\n      (hm : ∀ ℓ, ℓ < L →\n        (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) ≤ m ℓ)\n      (hm_le : ∀ ℓ, ℓ < L →\n        (m ℓ : ℝ) ≤ (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) + 1)\n      (X : Fin n → ℕ → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a ℓ k, Measurable (X a ℓ k))\n      (hX_bound : ∀ a ℓ k, ∀ᵐ ω ∂P, X a ℓ k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a ℓ k, ∫ ω, X a ℓ k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ × ℕ => X q.1 q.2.1 q.2.2) P)\n    (hX_ident : ∀ a ℓ k, IdentDistrib (X a ℓ k) (X a 0 0) P P)\n      (S : ℕ → Ω → Finset (Fin n)) (hS_meas : ∀ ℓ s, MeasurableSet {ω | S ℓ ω = s})\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSnonempty : ∀ ℓ ω, (S ℓ ω).Nonempty)\n      (hScard : ∀ ℓ ω, 2 * (S (ℓ + 1) ω).card ≤ (S ℓ ω).card + 1)\n      (hScard_lo : ∀ ℓ ω, (S ℓ ω).card ≤ 2 * (S (ℓ + 1) ω).card)\n      (median : ℕ → Ω → ℝ) (hmedian_meas : ∀ ℓ, Measurable (median ℓ))\n      (hmedian_ge : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω)).card ≥ (S ℓ ω).card)\n      (hmedian_le : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => empMeanPhase X a ℓ (m ℓ) ω ≤ median ℓ ω)).card ≥ (S ℓ ω).card)\n      (hSstep_lo : ∀ ℓ ω, (S ℓ ω).filter\n          (fun a => median ℓ ω < empMeanPhase X a ℓ (m ℓ) ω) ⊆ S (ℓ + 1) ω)\n      (hSstep_hi : ∀ ℓ ω, S (ℓ + 1) ω ⊆ (S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω))\n      (hS_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → S (ℓ + 1) ω = S (ℓ + 1) ω')\n      (hmedian_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → median ℓ ω = median ℓ ω'),\n    (1 - δ ≤ P.real {ω | ∀ a ∈ S L ω, epsOptimal p (p astar) ε a}) ∧\n    (∀ ω, (∑ ℓ ∈ Finset.range L, (S ℓ ω).card * m ℓ : ℝ) ≤\n        C * ((n / ε ^ 2) * Real.log (3 / δ) + n)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["naive_PAC", "successive_elimination_PAC", "median_elimination_PAC"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 2024, "end": 3181, "gold": ":\n    ∀ (c : ℝ), 4 < c →\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (hgap : ∀ a, a ≠ astar → p a < p astar)\n      (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (X : Fin n → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ => X q.1 q.2) P)\n      (hX_ident : ∀ a k, IdentDistrib (X a k) (X a 0) P P)\n      (S : ℕ → Ω → Finset (Fin n))\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSsub : ∀ t ω, S (t + 1) ω ⊆ S t ω)\n      (hSrule : ∀ t ω a, a ∈ S t ω →\n          (a ∈ S (t + 1) ω ↔\n            ∀ b ∈ S t ω, empMeanT X b (t + 1) ω - empMeanT X a (t + 1) ω <\n              2 * confRadiusSE c n (t + 1) δ)),\n      1 - δ ≤ P.real {ω | ∃ T : ℕ,\n        (∑ t ∈ Finset.range T, (S t ω).card : ℝ) ≤\n          C * ∑ a ∈ Finset.univ.erase astar,\n            Real.log (n / (δ * (p astar - p a))) / (p astar - p a) ^ 2\n        ∧ ∀ t, T ≤ t → S t ω = {astar}}"}]}
{"item_id": "v7-evendar06a::median_elimination_PAC::A-stmt", "paper_id": "v7-evendar06a", "target_name": "median_elimination_PAC", "source": "Research-level", "nl_statement": "Theorem 10 Median Elimination(ε,δ) is an (ε, δ)-PAC algorithm, with sample complexity\n\n$$\nO\\left(\\frac{n}{\\varepsilon^{2}}\\log\\left(\\frac{1}{\\delta}\\right)\\right).\n$$", "context": [{"name": "Setup: multi-armed bandit model (Section 2.1)", "statement": "The model consists of a set A of arms, where $n = \\left| A \\right|$. Sampling an arm $a \\in A$ yields a reward given by a random variable $R(a)$. Rewards are assumed binary: for each arm $a \\in A$, $R(a)\\in\\{0,1\\}$ (every result holds unchanged when the reward is bounded in [0, 1], and, with suitable modifications, more generally whenever the reward is bounded). We write $a_{1},\\cdots,a_{n}$ for the arms and set $p_{i}=\\mathbb{E}[R(a_{i})]$. To simplify notation, the arms are indexed in order of expected reward, $p_{1}> p_{2}> \\ldots > p_{n}$.\n\nThe best arm, denoted $a^{*}$, is an arm whose expected reward is highest; its expected reward $r^{*}$ is called the optimal reward. A non-best arm is an arm with expected reward strictly smaller than $r^{*}$, the expected reward of the best arm. An arm a is ε-optimal if its expected reward is within ε of the optimal reward, that is, $\\mathbb{E}[R(a)]\\geq r^{*}-\\varepsilon$.\n\nAt every time step t, an algorithm for the MAB problem samples some arm $a_{t}$ and obtains a reward $r_{t}$ (whose distribution is $R(a_{t})$). Its choice may depend on the history (that is, the actions taken and rewards received) up to time t − 1. In the end, the algorithm has to commit to one arm and select it."}, {"name": "Definition 1 ((ε,δ)-PAC algorithm) and Remark 2 (worst-case sample complexity)", "statement": "Definition 1 An algorithm for the multi armed bandit is an $(\\varepsilon,\\delta)$-PAC algorithm with sample complexity T if, upon termination, it outputs an ε-optimal arm a' with probability at least 1 − δ, and the number of time steps it executes before terminating is at most T.\n\nRemark 2 The MAB algorithm is allowed to terminate before T steps have elapsed. The sample complexity used here is that of the worst trajectory. Mannor and Tsitsiklis (2004) studied the expected sample complexity (with expectation over both the model and the algorithm), which behaves as $\\Omega((n+\\log(1/\\delta))/\\varepsilon^{2})$; this differs from the complexity established below in Theorem 10. Note also that the algorithm of Mannor and Tsitsiklis (2004) does not have a worst-case bound on its running time."}, {"name": "Lemma 4 (Hoeffding, 1963)", "statement": "Large deviation bounds are used repeatedly in this paper. Since boundedness is assumed, Hoeffding's inequality is available.\n\nLemma 4 (Hoeffding, 1963) Let X be a set, let D be a probability distribution on X, and let $f_{1},\\ldots,f_{m}$ be real-valued functions on X with $f_{i}: X \\longrightarrow [ a_{i}, b_{i}]$ for $i=1,\\ldots,m$, where $a_{i}$ and $b_{i}$ are real numbers with $a_{i}< b_{i}$. Let $x_{1},\\ldots,x_{m}$ be samples drawn independently and identically from D. Then\n\n$$\n\\begin{array}{r l r}{\\mathbf{P}\\left[\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}f_{i}(x_{i})-\\left(\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}\\int_{a_{i}}^{b_{i}}f_{i}(x)D(x)\\right)\\ge\\varepsilon\\right]}&{\\le}&{e^{-\\frac{2\\varepsilon^{2}m^{2}}{\\sum_{i=1}^{m}(b_{i}-a_{i})^{2}}}}\\\\ {\\mathbf{P}\\left[\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}f_{i}(x_{i})-\\left(\\frac{1}{m}\\displaystyle\\sum_{i=1}^{m}\\int_{a_{i}}^{b_{i}}f_{i}(x)D(x)\\right)\\le-\\varepsilon\\right]}&{\\le}&{e^{-\\frac{2\\varepsilon^{2}m^{2}}{\\sum_{i=1}^{m}(b_{i}-a_{i})^{2}}}.}\\end{array}\n$$"}, {"name": "Algorithm 4: Median Elimination", "statement": "The algorithm below replaces the ${\\cal O}(\\log(n/\\delta))$ term of the naive bound by ${\\cal O}(\\log(1/\\delta))$. Its idea is to discard the worse half of the arms in every iteration. The best arm need not come out empirically \"the best\"; all that is required is that some ε-optimal arm lies above the median.\n\nAlgorithm 4 (Median Elimination).\nGiven: $\\varepsilon > 0$ and $\\delta > 0$. Returns: one arm.\n1. Initialise $S_{1}= A$, $\\varepsilon_{1}= \\varepsilon/ 4$, $\\delta_{1}= \\delta / 2$, $\\ell = 1$.\n2. Loop body:\n   (a) draw $1/(\\varepsilon_{\\ell}/2)^{2}\\log(3/\\delta_{\\ell})$ samples from each arm $a \\in S_{\\ell}$, writing $\\hat{p}_{a}^{\\ell}$ for its empirical value;\n   (b) let $m_{\\ell}$ be the median of the values $\\hat{p}_{a}^{\\ell}$;\n   (c) set $S_{\\ell + 1}= S_{\\ell}\\backslash \\{a : \\hat{p}_{a}^{\\ell}< m_{\\ell}\\}$;\n   (d) update $\\varepsilon_{\\ell + 1}= \\frac{3}{4}\\varepsilon_{\\ell}$ and $\\delta_{\\ell + 1}= \\delta_{\\ell}/ 2$, then $\\ell = \\ell + 1$.\n3. Repeat step 2 until $| S_{\\ell}| = 1$."}], "nl_notes": "### Theorem 10\nREPAIRED STATEMENT (Theorem 10) — Theorem 10 as published is FALSE. Algorithm 4 as printed keeps S_{ℓ+1} = S_ℓ \\ {a : p̂_a < m_ℓ}, which does not halve the candidate set under ties; with binary rewards the algorithm can then run forever (e.g. n = 2 with both arms' rewards ≡ 1). The printed O((n/ε²)log(1/δ)) sample bound also fails for large δ. This target is the repaired version: (i) the next candidate set keeps every arm strictly above the median, keeps only arms at least the median, and satisfies |S_ℓ| ≤ 2|S_{ℓ+1}| and 2|S_{ℓ+1}| ≤ |S_ℓ| + 1 (`hSstep_lo`, `hSstep_hi`, `hScard_lo`, `hScard`); (ii) δ < 1 (`hδ1`); (iii) the sample bound is C((n/ε²) log(3/δ) + n) for a universal constant C. Prove the statement exactly as given in the Lean file.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ResearchLevel\n\ndef epsOptimal {n : ℕ} (p : Fin n → ℝ) (pmax ε : ℝ) (a : Fin n) : Prop :=\n  pmax - p a ≤ ε\n\nnoncomputable def empMean {n ℓ : ℕ} {Ω : Type*} (X : Fin n → Fin ℓ → Ω → ℝ)\n    (ω : Ω) (a : Fin n) : ℝ :=\n  (∑ k, X a k ω) / ℓ\n\nopen Classical in\n\nnoncomputable def naiveOutput {n ℓ : ℕ} [NeZero n] {Ω : Type*}\n    (X : Fin n → Fin ℓ → Ω → ℝ) (ω : Ω) : Fin n :=\n  (Finite.exists_max (empMean X ω)).choose\n\nnoncomputable def empMeanT {n : ℕ} {Ω : Type*} (X : Fin n → ℕ → Ω → ℝ)\n    (a : Fin n) (t : ℕ) (ω : Ω) : ℝ :=\n  (∑ k ∈ Finset.range t, X a k ω) / t\n\nnoncomputable def confRadiusSE (c : ℝ) (n t : ℕ) (δ : ℝ) : ℝ :=\n  Real.sqrt (Real.log (c * n * (t : ℝ) ^ 2 / δ) / t)\n\nnoncomputable def phaseEps (ε : ℝ) : ℕ → ℝ\n  | 0 => ε / 4\n  | (ℓ + 1) => (3 / 4) * phaseEps ε ℓ\n\nnoncomputable def phaseDelta (δ : ℝ) : ℕ → ℝ\n  | 0 => δ / 2\n  | (ℓ + 1) => phaseDelta δ ℓ / 2\n\nnoncomputable def empMeanPhase {n : ℕ} {Ω : Type*} (X : Fin n → ℕ → ℕ → Ω → ℝ)\n    (a : Fin n) (ℓ m : ℕ) (ω : Ω) : ℝ :=\n  (∑ k ∈ Finset.range m, X a ℓ k ω) / m\n\ntheorem naive_PAC :\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (ℓ : ℕ) (hℓ : (2 / ε ^ 2) * Real.log (2 * n / δ) ≤ ℓ)\n      (hℓ_le : (ℓ : ℝ) ≤ (2 / ε ^ 2) * Real.log (2 * n / δ) + 1)\n      (X : Fin n → Fin ℓ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × Fin ℓ => X q.1 q.2) P)\n      (hX_ident : ∀ a (k k' : Fin ℓ), ProbabilityTheory.IdentDistrib (X a k) (X a k') P P),\n    (1 - δ ≤ P.real {ω | epsOptimal p (p astar) ε (naiveOutput X ω)}) ∧\n    ((n : ℝ) * ℓ ≤ C * ((n / ε ^ 2) * Real.log (2 * n / δ) + n)) := by\n  sorry\n\ntheorem successive_elimination_PAC :\n    ∀ (c : ℝ), 4 < c →\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} [NeZero n] (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (hgap : ∀ a, a ≠ astar → p a < p astar)\n      (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (X : Fin n → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a k, Measurable (X a k))\n      (hX_bound : ∀ a k, ∀ᵐ ω ∂P, X a k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a k, ∫ ω, X a k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ => X q.1 q.2) P)\n      (hX_ident : ∀ a k, IdentDistrib (X a k) (X a 0) P P)\n      (S : ℕ → Ω → Finset (Fin n))\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSsub : ∀ t ω, S (t + 1) ω ⊆ S t ω)\n      (hSrule : ∀ t ω a, a ∈ S t ω →\n          (a ∈ S (t + 1) ω ↔\n            ∀ b ∈ S t ω, empMeanT X b (t + 1) ω - empMeanT X a (t + 1) ω <\n              2 * confRadiusSE c n (t + 1) δ)),\n      1 - δ ≤ P.real {ω | ∃ T : ℕ,\n        (∑ t ∈ Finset.range T, (S t ω).card : ℝ) ≤\n          C * ∑ a ∈ Finset.univ.erase astar,\n            Real.log (n / (δ * (p astar - p a))) / (p astar - p a) ^ 2\n        ∧ ∀ t, T ≤ t → S t ω = {astar}} := by\n  sorry\n\ntheorem median_elimination_PAC :\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} (hn : 0 < n) (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (L : ℕ) (hL : L = Nat.clog 2 n)\n      (m : ℕ → ℕ)\n      (hm : ∀ ℓ, ℓ < L →\n        (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) ≤ m ℓ)\n      (hm_le : ∀ ℓ, ℓ < L →\n        (m ℓ : ℝ) ≤ (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) + 1)\n      (X : Fin n → ℕ → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a ℓ k, Measurable (X a ℓ k))\n      (hX_bound : ∀ a ℓ k, ∀ᵐ ω ∂P, X a ℓ k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a ℓ k, ∫ ω, X a ℓ k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ × ℕ => X q.1 q.2.1 q.2.2) P)\n    (hX_ident : ∀ a ℓ k, IdentDistrib (X a ℓ k) (X a 0 0) P P)\n      (S : ℕ → Ω → Finset (Fin n)) (hS_meas : ∀ ℓ s, MeasurableSet {ω | S ℓ ω = s})\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSnonempty : ∀ ℓ ω, (S ℓ ω).Nonempty)\n      (hScard : ∀ ℓ ω, 2 * (S (ℓ + 1) ω).card ≤ (S ℓ ω).card + 1)\n      (hScard_lo : ∀ ℓ ω, (S ℓ ω).card ≤ 2 * (S (ℓ + 1) ω).card)\n      (median : ℕ → Ω → ℝ) (hmedian_meas : ∀ ℓ, Measurable (median ℓ))\n      (hmedian_ge : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω)).card ≥ (S ℓ ω).card)\n      (hmedian_le : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => empMeanPhase X a ℓ (m ℓ) ω ≤ median ℓ ω)).card ≥ (S ℓ ω).card)\n      (hSstep_lo : ∀ ℓ ω, (S ℓ ω).filter\n          (fun a => median ℓ ω < empMeanPhase X a ℓ (m ℓ) ω) ⊆ S (ℓ + 1) ω)\n      (hSstep_hi : ∀ ℓ ω, S (ℓ + 1) ω ⊆ (S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω))\n      (hS_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → S (ℓ + 1) ω = S (ℓ + 1) ω')\n      (hmedian_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → median ℓ ω = median ℓ ω'),\n    (1 - δ ≤ P.real {ω | ∀ a ∈ S L ω, epsOptimal p (p astar) ε a}) ∧\n    (∀ ω, (∑ ℓ ∈ Finset.range L, (S ℓ ω).card * m ℓ : ℝ) ≤\n        C * ((n / ε ^ 2) * Real.log (3 / δ) + n)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["naive_PAC", "successive_elimination_PAC", "median_elimination_PAC"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 3228, "end": 5358, "gold": ":\n    ∃ C : ℝ, 0 < C ∧\n    ∀ {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n      {n : ℕ} (hn : 0 < n) (p : Fin n → ℝ)\n      (astar : Fin n) (hopt : ∀ a, p a ≤ p astar)\n      (ε δ : ℝ) (hε : 0 < ε) (hδ : 0 < δ) (hδ1 : δ < 1)\n      (L : ℕ) (hL : L = Nat.clog 2 n)\n      (m : ℕ → ℕ)\n      (hm : ∀ ℓ, ℓ < L →\n        (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) ≤ m ℓ)\n      (hm_le : ∀ ℓ, ℓ < L →\n        (m ℓ : ℝ) ≤ (1 / (phaseEps ε ℓ / 2) ^ 2) * Real.log (3 / phaseDelta δ ℓ) + 1)\n      (X : Fin n → ℕ → ℕ → Ω → ℝ)\n      (hX_meas : ∀ a ℓ k, Measurable (X a ℓ k))\n      (hX_bound : ∀ a ℓ k, ∀ᵐ ω ∂P, X a ℓ k ω ∈ Set.Icc (0 : ℝ) 1)\n      (hX_mean : ∀ a ℓ k, ∫ ω, X a ℓ k ω ∂P = p a)\n      (hX_indep : iIndepFun (fun q : Fin n × ℕ × ℕ => X q.1 q.2.1 q.2.2) P)\n    (hX_ident : ∀ a ℓ k, IdentDistrib (X a ℓ k) (X a 0 0) P P)\n      (S : ℕ → Ω → Finset (Fin n)) (hS_meas : ∀ ℓ s, MeasurableSet {ω | S ℓ ω = s})\n      (hS0 : ∀ ω, S 0 ω = Finset.univ)\n      (hSnonempty : ∀ ℓ ω, (S ℓ ω).Nonempty)\n      (hScard : ∀ ℓ ω, 2 * (S (ℓ + 1) ω).card ≤ (S ℓ ω).card + 1)\n      (hScard_lo : ∀ ℓ ω, (S ℓ ω).card ≤ 2 * (S (ℓ + 1) ω).card)\n      (median : ℕ → Ω → ℝ) (hmedian_meas : ∀ ℓ, Measurable (median ℓ))\n      (hmedian_ge : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω)).card ≥ (S ℓ ω).card)\n      (hmedian_le : ∀ ℓ ω, 2 * ((S ℓ ω).filter\n          (fun a => empMeanPhase X a ℓ (m ℓ) ω ≤ median ℓ ω)).card ≥ (S ℓ ω).card)\n      (hSstep_lo : ∀ ℓ ω, (S ℓ ω).filter\n          (fun a => median ℓ ω < empMeanPhase X a ℓ (m ℓ) ω) ⊆ S (ℓ + 1) ω)\n      (hSstep_hi : ∀ ℓ ω, S (ℓ + 1) ω ⊆ (S ℓ ω).filter\n          (fun a => median ℓ ω ≤ empMeanPhase X a ℓ (m ℓ) ω))\n      (hS_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → S (ℓ + 1) ω = S (ℓ + 1) ω')\n      (hmedian_adapted : ∀ ℓ ω ω',\n          (∀ a j k, j ≤ ℓ → X a j k ω = X a j k ω') → median ℓ ω = median ℓ ω'),\n    (1 - δ ≤ P.real {ω | ∀ a ∈ S L ω, epsOptimal p (p astar) ε a}) ∧\n    (∀ ω, (∑ ℓ ∈ Finset.range L, (S ℓ ω).card * m ℓ : ℝ) ≤\n        C * ((n / ε ^ 2) * Real.log (3 / δ) + n))"}]}
{"item_id": "v25-23-0422::lemma3_1::A-stmt", "paper_id": "v25-23-0422", "target_name": "lemma3_1", "source": "Research-level", "nl_statement": "Lemma 3.1 (Key self-boundedness properties). Consider the setup of Section 2 and let Assumptions 1-2 hold. Further assume the loss is self-bounded as per Assumption 4. Then, the objective satisfies the following self-boundedness properties for its Gradient and Hessian:\n\n1. Self-bounded gradient: $\\begin{array}{r}{\\left\\|\\nabla\\widehat{F}_{i}(w)\\right\\|\\leq\\ell R\\widehat{F}_{i}(w),\\forall i\\in [n].}\\end{array}$\n\n2. Self-bounded weak convexity: $\\begin{array}{r}{\\lambda_{\\operatorname*{min}}\\left(\\nabla^{2}\\widehat{F}(w)\\right)\\geq-\\frac{L R^{2}}{\\sqrt{m}}\\widehat{F}(w)}\\end{array}$", "context": [{"name": "Notation", "statement": "## Notation\n\nWe define $[n]: =\\{1,2,\\cdots,n\\}$. We use the standard notation $O(\\cdot),\\Omega(\\cdot)$ and use $\\tilde{O}(\\cdot),\\tilde{\\Omega}(\\cdot)$ to hide polylogarithmic factors. Occasionally we use $\\lesssim$ to hide numerical constants. The Gradient and Hessian of a function $\\Phi : \\mathbb{R}^{d_{1}\\times d_{2}}\\to \\mathbb{R}$ with respect to the ith input $(i=1,2)$ are denoted by $\\nabla_{i}\\Phi$ and $\\nabla_{i}^{2}\\Phi$ , respectively. All logarithms are in base e. We use $\\| \\cdot \\|$ for the $\\ell_{2}$ norm of vectors and the operator norm of matrices. We denote $[ w_{1}, w_{2}] : = \\{w \\ : \\ w =$ $\\alpha w_{1}+(1-\\alpha)w_{2},\\alpha\\in [0,1]\\}$ the line segment between w1, $w_{2}\\in \\mathbb{R}^{d^{\\prime}}$ ."}, {"name": "Setup: empirical risk objective (Section 2, eqs. (1)-(2))", "statement": "## 2. Problem Setup\n\nGiven n i.i.d. samples $(x_{i},y_{i})\\sim\\mathcal{D},i\\in [n]$ from data distribution D, we study unconstrained empirical risk minimization with objective $\\widehat{F}: \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$:\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d^{\\prime}}}\\Big\\{\\widehat{F}(w): =\\frac{1}{n}\\sum_{i=1}^{n}\\widehat{F}_{i}(w)=\\frac{1}{n}\\sum_{i=1}^{n}f\\left(y_{i}\\Phi\\left(w,x_{i}\\right)\\right)\\Big\\}.\\tag{1}\n$$\n\nThis serves as a proxy for minimizing the test loss $F : \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$:\n\n$$\nF(w): =\\mathbb{E}_{(x,y)\\sim\\mathcal{D}}\\left[f\\left(y\\Phi(w,x)\\right)\\right].\\tag{2}\n$$\n\nWe introduce our assumptions on the data $(x,y)$ , the model $\\Phi(\\cdot,x)$ , and the loss function $f(\\cdot)$ , below. We start by imposing the following mild assumption on the data distribution."}, {"name": "Assumption 1 (Bounded features)", "statement": "Assumption 1 (Bounded features). Assume any $(x,y)\\sim\\mathcal{D}$ has almost surely bounded features, i.e. $\\| x \\| \\leq R$ , and binary label $y \\in \\{\\pm 1 \\}$."}, {"name": "Setup: two-layer network model (eq. (3)) and second-layer weights", "statement": "The model $\\Phi:\\mathbb{R}^{d^{\\prime}}\\times\\mathbb{R}^{d}\\to\\mathbb{R}$ is parameterized by trainable weights $w\\in\\mathbb{R}^{d^{\\prime}}$ and takes input $x\\in\\mathbb{R}^{d}$ . For our main results, we assume $\\Phi$ is a one-hidden layer neural-net of m neurons, i.e.\n\n$$\n\\Phi(w,x): =\\frac{1}{\\sqrt{m}}\\sum_{j=1}^{m}a_{j}\\sigma(\\langle w_{j},x\\rangle),\\tag{3}\n$$\n\nwhere $\\sigma:\\mathbb{R}\\to\\mathbb{R}$ is the activation function, $w_{j}\\in\\mathbb{R}^{d}$ denotes the weight vector of the jth hidden neuron and $\\textstyle{\\frac{a_{j}}{\\sqrt{m}}},j\\in [m]$ are the second-layer weights. For the second layer weights, we assume that they are fixed during training taking values $a_{j}\\in\\{\\pm 1\\}$ . We assume that for half of second layer weights $a_{j}=1$ and for the other half $a_{j}= -1$ . On the other hand, all the first-layer weights are updated during training. Thus, the total number of trainable parameters is $d^{\\prime}=m d$ and we denote $w=[w_{1};w_{2};\\ldots;w_{m}]\\in\\mathbb{R}^{d^{\\prime}}$ the vector of trainable weights. Throughout, we make the following assumptions on the activation function."}, {"name": "Assumption 2 (Lipschitz and smooth activation)", "statement": "Assumption 2 (Lipschitz and smooth activation). The activation function $\\sigma : \\mathbb{R}\\to \\mathbb{R}$ satisfies the following for non-negative constants $\\ell,L$:\n\n$$\n|\\sigma^{\\prime}(u)|\\leq\\ell,\\quad|\\sigma^{\\prime\\prime}(u)|\\leq L,\\qquad\\forall u\\in\\mathbb{R}.\n$$"}, {"name": "Loss class and Assumption 3 (Lipschitz and smooth loss)", "statement": "Next, we discuss conditions on the loss function. Of primal interest is the commonly used logistic loss function $f(u)=\\log(1+e^{-u})$ . However, our results hold for a broader class of convex, non-negative and monotonically decreasing functions $(\\operatorname*{lim}_{u\\to\\infty}f(u)=0)$ that satisfy the following:\n\nAssumption 3 (Lipschitz and smooth loss). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R_{+}}$ satisfies for all $u \\in \\mathbb{R}$\n\n3.A: Lipschitzness: $|f^{\\prime}(u)|\\leq G_{f}$\n\n3.B: Smoothness: $f^{\\prime\\prime}(u)\\leq L_{f}$"}, {"name": "Assumption 4 (Self-bounded loss) and normalization G_f = L_f = beta_f = 1", "statement": "Assumption 4 (Self-bounded loss). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R}_{+}$ is self-bounded with some constant $\\beta_{f}>0,\\ i.e.,\\ |f^{\\prime}(u)|\\leq\\beta_{f}f(u),\\forall u\\in\\mathbb{R}$\n\nThe self-boundedness Assumption 4 is the key property of the loss that drives our analysis and justifies the polylogarithmic width requirement, as will become evident. Note that the logistic loss naturally satisfies Assumptions 3.A and 3.B (with $G_{f}=1,L_{f}=1/4)$ , as well as, Assumption 4 with $\\beta_{f}= 1$ . Other interesting examples of loss functions satisfying those assumptions include polynomial losses, with the tail behavior $f(u)=1/u^{\\beta}$ for $\\beta > 0$, which we discuss in Remark 2. To lighten the notation and without loss of generality, we set $G_{f}= L_{f}= \\beta_{f}= 1$ for the rest of the paper. We remark that our training-loss results also hold for the exponential loss $e^{- u}$ . The exponential loss is self-bounded and while it is not Lipschitz or smooth it satisfies a second-order self-bounded property $f^{\\prime\\prime}(u)\\leq f(u)$ , which we can leverage instead; see Appendix A for details."}, {"name": "Definition 1 (Self-bounded weak convexity, eq. (4))", "statement": "Definition 1 (Self-bounded weak convexity). We say a function $\\widehat{F}: \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$ is self-bounded weakly convex if there exists constant $\\kappa > 0$ such that for all w,\n\n$$\n\\lambda_{\\mathrm{min}}\\left(\\nabla^{2}{\\widehat F}(w)\\right)\\geq-\\kappa{\\widehat F}(w).\\tag{4}\n$$\n\nRecall a function $G : \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$ is weakly convex if $\\exists \\kappa \\geq 0$ such that uniformly over all $w\\in\\mathbb{R}^{d^{\\prime}},\\lambda_{\\operatorname*{min}}\\left(\\nabla^{2}G(w)\\right)\\geq-\\kappa$ . If $\\kappa = 0$ , the function is convex. Instead, property (4) lower bounds the curvature by $-\\kappa G(w)$ that changes proportionally with the function value $G(w)$. We explain below how this is exploited in our setting."}, {"name": "Definitions for the gradient/Hessian calculations (Appendix D.1, eqs. (50)-(52b))", "statement": "## D.1 Definitions\n\nAssume IID data $(x,y)\\sim\\mathcal{D},x\\in\\mathbb{R}^{d},y\\in\\{\\pm 1\\}$ . Denote for convenience $z : = y x$ . Suppose two-layer neural network model\n\n$$\n\\Phi(w,x_{i})=\\frac{1}{\\sqrt{m}}\\sum_{j\\in [m]}a_{j}\\sigma(\\langle w_{j},x\\rangle)\\tag{50}\n$$\n\n$a_{j}\\in \\{\\pm 1 \\}, j \\in [ m ]$ and first-layer weights trained by GD on\n\n$$\n\\widehat F(w)=\\frac{1}{n}\\sum_{i\\in [n]}f(y_{i}\\Phi(w,x_{i}))= :\\frac{1}{n}\\sum_{i\\in [n]}f(w,z_{i}).\\tag{51}\n$$\n\nfor loss function $f : \\mathbb{R}\\to \\mathbb{R}$\n\nFor convenience define\n\n$$\n\\widehat{F}^{\\prime}(w)=\\frac{1}{n}\\sum_{i\\in [n]}\\vert f^{\\prime}(y_{i}\\Phi(w,x_{i}))\\vert\\tag{52a}\n$$\n\n$$\n\\widehat{F}^{\\prime\\prime}(w)=\\frac{1}{n}\\sum_{i\\in [n]}|f^{\\prime\\prime}\\left(y_{i}\\Phi(w,x_{i})\\right)|\\tag{52b}\n$$"}, {"name": "Corollary D.2.1 (Self-boundedness of Objective)", "statement": "Corollary D.2.1 (Self-boundedness of Objective). Let Assumption 2 hold. If the loss satisfies Assumptions 4 (with $\\beta_{f}=1)$ and 8, then\n\n1. $\\lVert\\nabla\\widehat{F}(w)\\rVert\\leq\\ell R\\widehat{F}(w)$\n\n$$\n\\begin{array}{r}{2.\\|\\nabla^{2}\\widehat{F}(w)\\|{\\le}\\left(\\ell^{2}R^{2}+\\frac{LR^{2}}{\\sqrt{m}}\\right)\\widehat{F}(w).}\\end{array}\n$$\n\n3. $\\begin{array}{r}{\\lambda_{\\operatorname*{min}}\\left(\\nabla^{2}\\widehat{F}(w)\\right)\\geq-\\frac{L R^{2}}{\\sqrt{m}}\\widehat{F}(w).}\\end{array}$\n\nIf in addition the loss satisfies Assumptions 3.A and 3.B with $L_{f}= G_{f}= 1$ , then\n\n6. $\\lVert\\nabla\\widehat{F}(w)\\rVert\\leq\\ell R.$\n\n$$\n\\begin{array}{r}{7.\\ \\|\\nabla^{2}\\widehat{F}(w)\\|\\le\\ell^{2}R^{2}+\\frac{LR^{2}}{\\sqrt{m}}.}\\end{array}\n$$"}, {"name": "Assumption 8 (2nd order self-boundedness)", "statement": "Assumption 8 (2nd order self-boundedness). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R}_{+}$ satisfies the 2nd order self-boundedness property, i.e.\n\n$$\nf^{\\prime\\prime}(u)\\leq f(u),\\forall u\\in\\mathbb{R}.\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nabbrev Params (m d : ℕ) := PiLp 2 (fun _ : Fin m => EuclideanSpace ℝ (Fin d))\n\nnoncomputable def phi {m d : ℕ} (a : Fin m → ℝ) (σ : ℝ → ℝ)\n    (w : Params m d) (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  (1 / Real.sqrt (m : ℝ)) * ∑ j : Fin m, a j * σ (inner ℝ (w j) x)\n\nnoncomputable def Fhat {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (w : Params m d) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i : Fin n, f (ys i * phi a σ w (xs i))\n\nnoncomputable def FhatI {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (i : Fin n) (w : Params m d) : ℝ :=\n  f (ys i * phi a σ w (xs i))\n\nnoncomputable def gdSeq {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (η : ℝ) (w0 : Params m d) : ℕ → Params m d\n  | 0 => w0\n  | (t + 1) =>\n      gdSeq a σ f xs ys η w0 t - η • gradient (Fhat a σ f xs ys) (gdSeq a σ f xs ys η w0 t)\n\ntheorem lemma3_1 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n)\n    (R ℓ L : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n\n    (hfconv : ConvexOn ℝ Set.univ f)\n    (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w : Params m d) :\n    (∀ i, ‖gradient (FhatI a σ f xs ys i) w‖ ≤ ℓ * R * FhatI a σ f xs ys i w)\n    ∧ (∀ v : Params m d,\n        -(L * R ^ 2 / Real.sqrt (m : ℝ)) * Fhat a σ f xs ys w * ‖v‖ ^ 2\n          ≤ iteratedFDeriv ℝ 2 (Fhat a σ f xs ys) w (fun _ => v)) := by\n  sorry\n\ntheorem thm3_2 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η) (hη : η ≤ 1 / LF)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 w : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (hnorm : ‖w - w0‖ ^ 2 ≥ max (η * T * Fhat a σ f xs ys w) (η * Fhat a σ f xs ys w0))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * ‖w - w0‖ ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n    ∧ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n        ≤ 2 * Fhat a σ f xs ys w + 5 * ‖w - w0‖ ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * ‖w - w0‖ := by\n  sorry\n\ntheorem thm3_3 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (g : ℝ → ℝ) (hgpos : ∀ ε, 0 < ε → 0 < g ε)\n    (hgdec : ∀ ε₁ ε₂, 0 < ε₁ → ε₁ ≤ ε₂ → g ε₂ ≤ g ε₁)\n\n    (ε₀ : ℝ) (hε₀ : 0 < ε₀)\n    (hreal : ∀ ε, 0 < ε → ε ≤ ε₀ → ∃ wε : Params m d,\n        Fhat a σ f xs ys wε ≤ ε ∧ ‖wε - w0‖ = g ε)\n    (hTsmall : 1 / (T : ℝ) ≤ ε₀)\n    (hFw0 : 0 < Fhat a σ f xs ys w0)\n    (hη : η ≤ min (1 / LF) (min (g 1 ^ 2) (g 1 ^ 2 / Fhat a σ f xs ys w0)))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * g (1 / (T : ℝ)) ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ 2 / (T : ℝ) + 5 * g (1 / (T : ℝ)) ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * g (1 / (T : ℝ)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma3_1", "thm3_2", "thm3_3"], "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "holes": [{"start": 1078, "end": 2028, "gold": "{m d n : ℕ} (hm : 0 < m) (hn : 0 < n)\n    (R ℓ L : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n\n    (hfconv : ConvexOn ℝ Set.univ f)\n    (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w : Params m d) :\n    (∀ i, ‖gradient (FhatI a σ f xs ys i) w‖ ≤ ℓ * R * FhatI a σ f xs ys i w)\n    ∧ (∀ v : Params m d,\n        -(L * R ^ 2 / Real.sqrt (m : ℝ)) * Fhat a σ f xs ys w * ‖v‖ ^ 2\n          ≤ iteratedFDeriv ℝ 2 (Fhat a σ f xs ys) w (fun _ => v))"}]}
{"item_id": "v25-23-0422::thm3_2::A-stmt", "paper_id": "v25-23-0422", "target_name": "thm3_2", "source": "Research-level", "nl_statement": "Theorem 3.2 (Training loss – General bound). Suppose Assumptions $1 \\mathrm{-}\\mathit{4}$ hold. Fix any training horizon $T \\geq 0$ and any step-size $\\eta \\leq 1 / L_{\\widehat{F}}$ where $L_{\\widehat{F}}$ is the objective's smoothness parameter. Assume any w $\\in \\mathbb{R}^{d^{\\prime}}$ and hidden-layer width m such that $\\| w - w_{0}\\|^{2}\\ge$ max $\\{\\eta T\\widehat{F}(w),\\eta\\widehat{F}(w_{0})\\}$ and $m\\geq 18^{2}L^{2}R^{4}\\Vert w-w_{0}\\Vert^{4}$. Then, the training loss and the parameters' norm satisfy\n\n$$\n\\begin{array}{r l}&{\\widehat{F}(w_{T})\\ \\leq\\ \\displaystyle\\frac{1}{T}\\sum_{t=1}^{T}\\widehat{F}(w_{t})\\ \\leq\\ 2\\widehat{F}(w)+\\frac{5\\|w-w_{0}\\|^{2}}{2\\eta T},}\\\\ &{\\forall t\\in [T]\\ :\\ \\|w_{t}-w_{0}\\|\\ \\leq\\ 4\\|w-w_{0}\\|.}\\end{array}\\tag{5}\n$$", "context": [{"name": "Notation", "statement": "## Notation\n\nWe define $[n]: =\\{1,2,\\cdots,n\\}$. We use the standard notation $O(\\cdot),\\Omega(\\cdot)$ and use $\\tilde{O}(\\cdot),\\tilde{\\Omega}(\\cdot)$ to hide polylogarithmic factors. Occasionally we use $\\lesssim$ to hide numerical constants. The Gradient and Hessian of a function $\\Phi : \\mathbb{R}^{d_{1}\\times d_{2}}\\to \\mathbb{R}$ with respect to the ith input $(i=1,2)$ are denoted by $\\nabla_{i}\\Phi$ and $\\nabla_{i}^{2}\\Phi$ , respectively. All logarithms are in base e. We use $\\| \\cdot \\|$ for the $\\ell_{2}$ norm of vectors and the operator norm of matrices. We denote $[ w_{1}, w_{2}] : = \\{w \\ : \\ w =$ $\\alpha w_{1}+(1-\\alpha)w_{2},\\alpha\\in [0,1]\\}$ the line segment between w1, $w_{2}\\in \\mathbb{R}^{d^{\\prime}}$ ."}, {"name": "Setup: empirical risk objective (Section 2, eqs. (1)-(2))", "statement": "## 2. Problem Setup\n\nGiven n i.i.d. samples $(x_{i},y_{i})\\sim\\mathcal{D},i\\in [n]$ from data distribution D, we study unconstrained empirical risk minimization with objective $\\widehat{F}: \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$:\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d^{\\prime}}}\\Big\\{\\widehat{F}(w): =\\frac{1}{n}\\sum_{i=1}^{n}\\widehat{F}_{i}(w)=\\frac{1}{n}\\sum_{i=1}^{n}f\\left(y_{i}\\Phi\\left(w,x_{i}\\right)\\right)\\Big\\}.\\tag{1}\n$$\n\nThis serves as a proxy for minimizing the test loss $F : \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$:\n\n$$\nF(w): =\\mathbb{E}_{(x,y)\\sim\\mathcal{D}}\\left[f\\left(y\\Phi(w,x)\\right)\\right].\\tag{2}\n$$\n\nWe introduce our assumptions on the data $(x,y)$ , the model $\\Phi(\\cdot,x)$ , and the loss function $f(\\cdot)$ , below. We start by imposing the following mild assumption on the data distribution."}, {"name": "Assumption 1 (Bounded features)", "statement": "Assumption 1 (Bounded features). Assume any $(x,y)\\sim\\mathcal{D}$ has almost surely bounded features, i.e. $\\| x \\| \\leq R$ , and binary label $y \\in \\{\\pm 1 \\}$."}, {"name": "Setup: two-layer network model (eq. (3)) and second-layer weights", "statement": "The model $\\Phi:\\mathbb{R}^{d^{\\prime}}\\times\\mathbb{R}^{d}\\to\\mathbb{R}$ is parameterized by trainable weights $w\\in\\mathbb{R}^{d^{\\prime}}$ and takes input $x\\in\\mathbb{R}^{d}$ . For our main results, we assume $\\Phi$ is a one-hidden layer neural-net of m neurons, i.e.\n\n$$\n\\Phi(w,x): =\\frac{1}{\\sqrt{m}}\\sum_{j=1}^{m}a_{j}\\sigma(\\langle w_{j},x\\rangle),\\tag{3}\n$$\n\nwhere $\\sigma:\\mathbb{R}\\to\\mathbb{R}$ is the activation function, $w_{j}\\in\\mathbb{R}^{d}$ denotes the weight vector of the jth hidden neuron and $\\textstyle{\\frac{a_{j}}{\\sqrt{m}}},j\\in [m]$ are the second-layer weights. For the second layer weights, we assume that they are fixed during training taking values $a_{j}\\in\\{\\pm 1\\}$ . We assume that for half of second layer weights $a_{j}=1$ and for the other half $a_{j}= -1$ . On the other hand, all the first-layer weights are updated during training. Thus, the total number of trainable parameters is $d^{\\prime}=m d$ and we denote $w=[w_{1};w_{2};\\ldots;w_{m}]\\in\\mathbb{R}^{d^{\\prime}}$ the vector of trainable weights. Throughout, we make the following assumptions on the activation function."}, {"name": "Assumption 2 (Lipschitz and smooth activation)", "statement": "Assumption 2 (Lipschitz and smooth activation). The activation function $\\sigma : \\mathbb{R}\\to \\mathbb{R}$ satisfies the following for non-negative constants $\\ell,L$:\n\n$$\n|\\sigma^{\\prime}(u)|\\leq\\ell,\\quad|\\sigma^{\\prime\\prime}(u)|\\leq L,\\qquad\\forall u\\in\\mathbb{R}.\n$$"}, {"name": "Loss class and Assumption 3 (Lipschitz and smooth loss)", "statement": "Next, we discuss conditions on the loss function. Of primal interest is the commonly used logistic loss function $f(u)=\\log(1+e^{-u})$ . However, our results hold for a broader class of convex, non-negative and monotonically decreasing functions $(\\operatorname*{lim}_{u\\to\\infty}f(u)=0)$ that satisfy the following:\n\nAssumption 3 (Lipschitz and smooth loss). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R_{+}}$ satisfies for all $u \\in \\mathbb{R}$\n\n3.A: Lipschitzness: $|f^{\\prime}(u)|\\leq G_{f}$\n\n3.B: Smoothness: $f^{\\prime\\prime}(u)\\leq L_{f}$"}, {"name": "Assumption 4 (Self-bounded loss) and normalization G_f = L_f = beta_f = 1", "statement": "Assumption 4 (Self-bounded loss). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R}_{+}$ is self-bounded with some constant $\\beta_{f}>0,\\ i.e.,\\ |f^{\\prime}(u)|\\leq\\beta_{f}f(u),\\forall u\\in\\mathbb{R}$\n\nThe self-boundedness Assumption 4 is the key property of the loss that drives our analysis and justifies the polylogarithmic width requirement, as will become evident. Note that the logistic loss naturally satisfies Assumptions 3.A and 3.B (with $G_{f}=1,L_{f}=1/4)$ , as well as, Assumption 4 with $\\beta_{f}= 1$ . Other interesting examples of loss functions satisfying those assumptions include polynomial losses, with the tail behavior $f(u)=1/u^{\\beta}$ for $\\beta > 0$, which we discuss in Remark 2. To lighten the notation and without loss of generality, we set $G_{f}= L_{f}= \\beta_{f}= 1$ for the rest of the paper. We remark that our training-loss results also hold for the exponential loss $e^{- u}$ . The exponential loss is self-bounded and while it is not Lipschitz or smooth it satisfies a second-order self-bounded property $f^{\\prime\\prime}(u)\\leq f(u)$ , which we can leverage instead; see Appendix A for details."}, {"name": "Gradient descent update", "statement": "We present bounds on the train loss and generalization gap of gradient-descent (GD) under the setting of Section 2. Formally, GD with step-size $\\eta > 0$ optimizes (1) by performing the following updates starting from an initialization w0:\n\n$$\n\\forall t\\geq 0:w_{t+1}=w_{t}-\\eta\\nabla\\widehat{F}(w_{t}).\n$$"}, {"name": "Definition 1 (Self-bounded weak convexity, eq. (4))", "statement": "Definition 1 (Self-bounded weak convexity). We say a function $\\widehat{F}: \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$ is self-bounded weakly convex if there exists constant $\\kappa > 0$ such that for all w,\n\n$$\n\\lambda_{\\mathrm{min}}\\left(\\nabla^{2}{\\widehat F}(w)\\right)\\geq-\\kappa{\\widehat F}(w).\\tag{4}\n$$\n\nRecall a function $G : \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$ is weakly convex if $\\exists \\kappa \\geq 0$ such that uniformly over all $w\\in\\mathbb{R}^{d^{\\prime}},\\lambda_{\\operatorname*{min}}\\left(\\nabla^{2}G(w)\\right)\\geq-\\kappa$ . If $\\kappa = 0$ , the function is convex. Instead, property (4) lower bounds the curvature by $-\\kappa G(w)$ that changes proportionally with the function value $G(w)$. We explain below how this is exploited in our setting."}, {"name": "Corollary D.2.1 (Self-boundedness of Objective)", "statement": "Corollary D.2.1 (Self-boundedness of Objective). Let Assumption 2 hold. If the loss satisfies Assumptions 4 (with $\\beta_{f}=1)$ and 8, then\n\n1. $\\lVert\\nabla\\widehat{F}(w)\\rVert\\leq\\ell R\\widehat{F}(w)$\n\n$$\n\\begin{array}{r}{2.\\|\\nabla^{2}\\widehat{F}(w)\\|{\\le}\\left(\\ell^{2}R^{2}+\\frac{LR^{2}}{\\sqrt{m}}\\right)\\widehat{F}(w).}\\end{array}\n$$\n\n3. $\\begin{array}{r}{\\lambda_{\\operatorname*{min}}\\left(\\nabla^{2}\\widehat{F}(w)\\right)\\geq-\\frac{L R^{2}}{\\sqrt{m}}\\widehat{F}(w).}\\end{array}$\n\nIf in addition the loss satisfies Assumptions 3.A and 3.B with $L_{f}= G_{f}= 1$ , then\n\n6. $\\lVert\\nabla\\widehat{F}(w)\\rVert\\leq\\ell R.$\n\n$$\n\\begin{array}{r}{7.\\ \\|\\nabla^{2}\\widehat{F}(w)\\|\\le\\ell^{2}R^{2}+\\frac{LR^{2}}{\\sqrt{m}}.}\\end{array}\n$$"}, {"name": "Assumption 8 (2nd order self-boundedness)", "statement": "Assumption 8 (2nd order self-boundedness). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R}_{+}$ satisfies the 2nd order self-boundedness property, i.e.\n\n$$\nf^{\\prime\\prime}(u)\\leq f(u),\\forall u\\in\\mathbb{R}.\n$$"}, {"name": "Theorem A.1 (General statement of Theorem 3.2)", "statement": "Theorem A.1 (General statement of Theorem 3.2). Let Assumptions 1-2 hold. Assume the loss function satisfies self-bounded Assumption 4. Moreover, suppose either Assumption 3 or Assumption 8 hold. Fix any $T \\geq 0$ . Let the step-size satisfy the assumptions of the descent lemma (Lemma A.2). Assume any w and m such that $\\| w - w_{0}\\|^{2}\\ge$ max $\\left\\{\\eta T\\widehat{F}(w),\\eta\\widehat{F}(w_{0})\\right\\}$ and $m\\geq 18^{2}L^{2}R^{4}\\|w-w_{0}\\|^{4}$ . Then, the training loss and the parameters' norm satisfy\n\n$$\n\\begin{array}{r l r}{{\\frac{1}{T}\\sum_{t=1}^{T}\\widehat{F}(w_{t})\\leq 2\\widehat{F}(w)+\\frac{5\\|w-w_{0}\\|^{2}}{2\\eta T},}}\\\\ &{}&{\\forall t\\in [T]\\ :\\ \\|w_{t}-w_{0}\\|\\leq 4\\|w-w_{0}\\|.}\\end{array}\\tag{16}\n$$"}, {"name": "Lemma A.2 (Descent lemma)", "statement": "Lemma A.2 (Descent lemma). Let Assumptions 1-2 hold. Assume the loss function satisfies self-boundedness Assumptions 4, 8. Then, for any $\\begin{array}{r}{\\eta<\\frac{1}{R^{2}\\widehat{F}(w_{t})}\\operatorname*{min}\\{\\frac{1}{\\ell^{2}+L},\\frac{1}{\\sqrt{L}\\ell}\\}}\\end{array}$ the descent property holds, i.e.,\n\n$$\n\\widehat{F}(w_{t+1})\\leq\\widehat{F}(w_{t})-\\frac{\\eta}{2}\\|\\nabla\\widehat{F}(w_{t})\\|^{2}.\n$$\n\nMoreover, if f satisfies Assumption 3 then the descent property holds for any $\\eta \\leq 1 / L_{\\widehat{F}}$ where $\\begin{array}{r}{L_{\\widehat{F}}: = \\ell^{2}R^{2}+ \\frac{L R^{2}}{\\sqrt{m}}}\\end{array}$ is the smoothness parameter of the training objective."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nabbrev Params (m d : ℕ) := PiLp 2 (fun _ : Fin m => EuclideanSpace ℝ (Fin d))\n\nnoncomputable def phi {m d : ℕ} (a : Fin m → ℝ) (σ : ℝ → ℝ)\n    (w : Params m d) (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  (1 / Real.sqrt (m : ℝ)) * ∑ j : Fin m, a j * σ (inner ℝ (w j) x)\n\nnoncomputable def Fhat {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (w : Params m d) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i : Fin n, f (ys i * phi a σ w (xs i))\n\nnoncomputable def FhatI {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (i : Fin n) (w : Params m d) : ℝ :=\n  f (ys i * phi a σ w (xs i))\n\nnoncomputable def gdSeq {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (η : ℝ) (w0 : Params m d) : ℕ → Params m d\n  | 0 => w0\n  | (t + 1) =>\n      gdSeq a σ f xs ys η w0 t - η • gradient (Fhat a σ f xs ys) (gdSeq a σ f xs ys η w0 t)\n\ntheorem lemma3_1 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n)\n    (R ℓ L : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n\n    (hfconv : ConvexOn ℝ Set.univ f)\n    (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w : Params m d) :\n    (∀ i, ‖gradient (FhatI a σ f xs ys i) w‖ ≤ ℓ * R * FhatI a σ f xs ys i w)\n    ∧ (∀ v : Params m d,\n        -(L * R ^ 2 / Real.sqrt (m : ℝ)) * Fhat a σ f xs ys w * ‖v‖ ^ 2\n          ≤ iteratedFDeriv ℝ 2 (Fhat a σ f xs ys) w (fun _ => v)) := by\n  sorry\n\ntheorem thm3_2 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η) (hη : η ≤ 1 / LF)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 w : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (hnorm : ‖w - w0‖ ^ 2 ≥ max (η * T * Fhat a σ f xs ys w) (η * Fhat a σ f xs ys w0))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * ‖w - w0‖ ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n    ∧ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n        ≤ 2 * Fhat a σ f xs ys w + 5 * ‖w - w0‖ ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * ‖w - w0‖ := by\n  sorry\n\ntheorem thm3_3 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (g : ℝ → ℝ) (hgpos : ∀ ε, 0 < ε → 0 < g ε)\n    (hgdec : ∀ ε₁ ε₂, 0 < ε₁ → ε₁ ≤ ε₂ → g ε₂ ≤ g ε₁)\n\n    (ε₀ : ℝ) (hε₀ : 0 < ε₀)\n    (hreal : ∀ ε, 0 < ε → ε ≤ ε₀ → ∃ wε : Params m d,\n        Fhat a σ f xs ys wε ≤ ε ∧ ‖wε - w0‖ = g ε)\n    (hTsmall : 1 / (T : ℝ) ≤ ε₀)\n    (hFw0 : 0 < Fhat a σ f xs ys w0)\n    (hη : η ≤ min (1 / LF) (min (g 1 ^ 2) (g 1 ^ 2 / Fhat a σ f xs ys w0)))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * g (1 / (T : ℝ)) ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ 2 / (T : ℝ) + 5 * g (1 / (T : ℝ)) ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * g (1 / (T : ℝ)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma3_1", "thm3_2", "thm3_3"], "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "holes": [{"start": 2059, "end": 3504, "gold": "{m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η) (hη : η ≤ 1 / LF)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 w : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (hnorm : ‖w - w0‖ ^ 2 ≥ max (η * T * Fhat a σ f xs ys w) (η * Fhat a σ f xs ys w0))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * ‖w - w0‖ ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n    ∧ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n        ≤ 2 * Fhat a σ f xs ys w + 5 * ‖w - w0‖ ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * ‖w - w0‖"}]}
{"item_id": "v25-23-0422::thm3_3::A-stmt", "paper_id": "v25-23-0422", "target_name": "thm3_3", "source": "Research-level", "nl_statement": "Theorem 3.3 (Training loss under interpolation). Let Assumptions 1-5 hold. Let $\\eta \\leq$ min $\\begin{array}{r}{\\{\\frac{1}{L_{\\widehat{F}}},g(1)^{2},\\frac{g(1)^{2}}{\\widehat{F}(w_{0})}\\}}\\end{array}$ and assume the width satisfies $m\\geq 18^{2}L^{2}R^{4}g(\\frac{1}{T})^{4}$ for a fixed training horizon T such that $\\varepsilon = 1 / T$ is small enough for Assumption 5 to apply. Then,\n\n$$\n\\widehat{F}(w_{T})\\leq\\frac{2}{T}+\\frac{5g(\\frac{1}{T})^{2}}{2\\eta T},\\tag{6}\n$$\n\n$$\n\\forall t\\in [T]\\ :\\ \\|w_{t}-w_{0}\\|\\ \\leq\\ 4g(\\frac{1}{T}).\n$$", "context": [{"name": "Notation", "statement": "## Notation\n\nWe define $[n]: =\\{1,2,\\cdots,n\\}$. We use the standard notation $O(\\cdot),\\Omega(\\cdot)$ and use $\\tilde{O}(\\cdot),\\tilde{\\Omega}(\\cdot)$ to hide polylogarithmic factors. Occasionally we use $\\lesssim$ to hide numerical constants. The Gradient and Hessian of a function $\\Phi : \\mathbb{R}^{d_{1}\\times d_{2}}\\to \\mathbb{R}$ with respect to the ith input $(i=1,2)$ are denoted by $\\nabla_{i}\\Phi$ and $\\nabla_{i}^{2}\\Phi$ , respectively. All logarithms are in base e. We use $\\| \\cdot \\|$ for the $\\ell_{2}$ norm of vectors and the operator norm of matrices. We denote $[ w_{1}, w_{2}] : = \\{w \\ : \\ w =$ $\\alpha w_{1}+(1-\\alpha)w_{2},\\alpha\\in [0,1]\\}$ the line segment between w1, $w_{2}\\in \\mathbb{R}^{d^{\\prime}}$ ."}, {"name": "Setup: empirical risk objective (Section 2, eqs. (1)-(2))", "statement": "## 2. Problem Setup\n\nGiven n i.i.d. samples $(x_{i},y_{i})\\sim\\mathcal{D},i\\in [n]$ from data distribution D, we study unconstrained empirical risk minimization with objective $\\widehat{F}: \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$:\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d^{\\prime}}}\\Big\\{\\widehat{F}(w): =\\frac{1}{n}\\sum_{i=1}^{n}\\widehat{F}_{i}(w)=\\frac{1}{n}\\sum_{i=1}^{n}f\\left(y_{i}\\Phi\\left(w,x_{i}\\right)\\right)\\Big\\}.\\tag{1}\n$$\n\nThis serves as a proxy for minimizing the test loss $F : \\mathbb{R}^{d^{\\prime}}\\to \\mathbb{R}$:\n\n$$\nF(w): =\\mathbb{E}_{(x,y)\\sim\\mathcal{D}}\\left[f\\left(y\\Phi(w,x)\\right)\\right].\\tag{2}\n$$\n\nWe introduce our assumptions on the data $(x,y)$ , the model $\\Phi(\\cdot,x)$ , and the loss function $f(\\cdot)$ , below. We start by imposing the following mild assumption on the data distribution."}, {"name": "Assumption 1 (Bounded features)", "statement": "Assumption 1 (Bounded features). Assume any $(x,y)\\sim\\mathcal{D}$ has almost surely bounded features, i.e. $\\| x \\| \\leq R$ , and binary label $y \\in \\{\\pm 1 \\}$."}, {"name": "Setup: two-layer network model (eq. (3)) and second-layer weights", "statement": "The model $\\Phi:\\mathbb{R}^{d^{\\prime}}\\times\\mathbb{R}^{d}\\to\\mathbb{R}$ is parameterized by trainable weights $w\\in\\mathbb{R}^{d^{\\prime}}$ and takes input $x\\in\\mathbb{R}^{d}$ . For our main results, we assume $\\Phi$ is a one-hidden layer neural-net of m neurons, i.e.\n\n$$\n\\Phi(w,x): =\\frac{1}{\\sqrt{m}}\\sum_{j=1}^{m}a_{j}\\sigma(\\langle w_{j},x\\rangle),\\tag{3}\n$$\n\nwhere $\\sigma:\\mathbb{R}\\to\\mathbb{R}$ is the activation function, $w_{j}\\in\\mathbb{R}^{d}$ denotes the weight vector of the jth hidden neuron and $\\textstyle{\\frac{a_{j}}{\\sqrt{m}}},j\\in [m]$ are the second-layer weights. For the second layer weights, we assume that they are fixed during training taking values $a_{j}\\in\\{\\pm 1\\}$ . We assume that for half of second layer weights $a_{j}=1$ and for the other half $a_{j}= -1$ . On the other hand, all the first-layer weights are updated during training. Thus, the total number of trainable parameters is $d^{\\prime}=m d$ and we denote $w=[w_{1};w_{2};\\ldots;w_{m}]\\in\\mathbb{R}^{d^{\\prime}}$ the vector of trainable weights. Throughout, we make the following assumptions on the activation function."}, {"name": "Assumption 2 (Lipschitz and smooth activation)", "statement": "Assumption 2 (Lipschitz and smooth activation). The activation function $\\sigma : \\mathbb{R}\\to \\mathbb{R}$ satisfies the following for non-negative constants $\\ell,L$:\n\n$$\n|\\sigma^{\\prime}(u)|\\leq\\ell,\\quad|\\sigma^{\\prime\\prime}(u)|\\leq L,\\qquad\\forall u\\in\\mathbb{R}.\n$$"}, {"name": "Loss class and Assumption 3 (Lipschitz and smooth loss)", "statement": "Next, we discuss conditions on the loss function. Of primal interest is the commonly used logistic loss function $f(u)=\\log(1+e^{-u})$ . However, our results hold for a broader class of convex, non-negative and monotonically decreasing functions $(\\operatorname*{lim}_{u\\to\\infty}f(u)=0)$ that satisfy the following:\n\nAssumption 3 (Lipschitz and smooth loss). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R_{+}}$ satisfies for all $u \\in \\mathbb{R}$\n\n3.A: Lipschitzness: $|f^{\\prime}(u)|\\leq G_{f}$\n\n3.B: Smoothness: $f^{\\prime\\prime}(u)\\leq L_{f}$"}, {"name": "Assumption 4 (Self-bounded loss) and normalization G_f = L_f = beta_f = 1", "statement": "Assumption 4 (Self-bounded loss). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R}_{+}$ is self-bounded with some constant $\\beta_{f}>0,\\ i.e.,\\ |f^{\\prime}(u)|\\leq\\beta_{f}f(u),\\forall u\\in\\mathbb{R}$\n\nThe self-boundedness Assumption 4 is the key property of the loss that drives our analysis and justifies the polylogarithmic width requirement, as will become evident. Note that the logistic loss naturally satisfies Assumptions 3.A and 3.B (with $G_{f}=1,L_{f}=1/4)$ , as well as, Assumption 4 with $\\beta_{f}= 1$ . Other interesting examples of loss functions satisfying those assumptions include polynomial losses, with the tail behavior $f(u)=1/u^{\\beta}$ for $\\beta > 0$, which we discuss in Remark 2. To lighten the notation and without loss of generality, we set $G_{f}= L_{f}= \\beta_{f}= 1$ for the rest of the paper. We remark that our training-loss results also hold for the exponential loss $e^{- u}$ . The exponential loss is self-bounded and while it is not Lipschitz or smooth it satisfies a second-order self-bounded property $f^{\\prime\\prime}(u)\\leq f(u)$ , which we can leverage instead; see Appendix A for details."}, {"name": "Gradient descent update", "statement": "We present bounds on the train loss and generalization gap of gradient-descent (GD) under the setting of Section 2. Formally, GD with step-size $\\eta > 0$ optimizes (1) by performing the following updates starting from an initialization w0:\n\n$$\n\\forall t\\geq 0:w_{t+1}=w_{t}-\\eta\\nabla\\widehat{F}(w_{t}).\n$$"}, {"name": "Corollary D.2.1 (Self-boundedness of Objective)", "statement": "Corollary D.2.1 (Self-boundedness of Objective). Let Assumption 2 hold. If the loss satisfies Assumptions 4 (with $\\beta_{f}=1)$ and 8, then\n\n1. $\\lVert\\nabla\\widehat{F}(w)\\rVert\\leq\\ell R\\widehat{F}(w)$\n\n$$\n\\begin{array}{r}{2.\\|\\nabla^{2}\\widehat{F}(w)\\|{\\le}\\left(\\ell^{2}R^{2}+\\frac{LR^{2}}{\\sqrt{m}}\\right)\\widehat{F}(w).}\\end{array}\n$$\n\n3. $\\begin{array}{r}{\\lambda_{\\operatorname*{min}}\\left(\\nabla^{2}\\widehat{F}(w)\\right)\\geq-\\frac{L R^{2}}{\\sqrt{m}}\\widehat{F}(w).}\\end{array}$\n\nIf in addition the loss satisfies Assumptions 3.A and 3.B with $L_{f}= G_{f}= 1$ , then\n\n6. $\\lVert\\nabla\\widehat{F}(w)\\rVert\\leq\\ell R.$\n\n$$\n\\begin{array}{r}{7.\\ \\|\\nabla^{2}\\widehat{F}(w)\\|\\le\\ell^{2}R^{2}+\\frac{LR^{2}}{\\sqrt{m}}.}\\end{array}\n$$"}, {"name": "Assumption 5 (NN-Realizability)", "statement": "Assumption 5 (NN-Realizability). There exists a decreasing function $g : \\mathbb{R}_{+}\\to \\mathbb{R}_{+}$ which measures the norm of deviations from initialization of models that achieve arbitrarily small training error.\n\nFormally, for almost surely all n training samples and for any sufficiently small $\\varepsilon > 0$ there exists $\\boldsymbol{w}^{(\\varepsilon)}\\in\\mathbb{R}^{d^{\\prime}}$ such that\n\n$$\n{\\widehat F}(w^{(\\varepsilon)})\\leq\\varepsilon,\\ and\\ g(\\varepsilon)=\\Big\\|w^{(\\varepsilon)}-w_{0}\\Big\\|.\n$$\n\nSince Assumption 5 holds for arbitrarily small ε, it guarantees that the model has enough capacity to interpolate the data, i.e., attain train error that is arbitrarily small (ε). Additionally, this is accomplished for model weights whose distance from initialization is managed by the function $g(\\varepsilon)$ . By using these model weights to select w in Theorem 3.2 we obtain train loss bounds for interpolating models."}, {"name": "Assumption 8 (2nd order self-boundedness)", "statement": "Assumption 8 (2nd order self-boundedness). The convex loss function $f : \\mathbb{R}\\to \\mathbb{R}_{+}$ satisfies the 2nd order self-boundedness property, i.e.\n\n$$\nf^{\\prime\\prime}(u)\\leq f(u),\\forall u\\in\\mathbb{R}.\n$$"}, {"name": "Theorem A.1 (General statement of Theorem 3.2)", "statement": "Theorem A.1 (General statement of Theorem 3.2). Let Assumptions 1-2 hold. Assume the loss function satisfies self-bounded Assumption 4. Moreover, suppose either Assumption 3 or Assumption 8 hold. Fix any $T \\geq 0$ . Let the step-size satisfy the assumptions of the descent lemma (Lemma A.2). Assume any w and m such that $\\| w - w_{0}\\|^{2}\\ge$ max $\\left\\{\\eta T\\widehat{F}(w),\\eta\\widehat{F}(w_{0})\\right\\}$ and $m\\geq 18^{2}L^{2}R^{4}\\|w-w_{0}\\|^{4}$ . Then, the training loss and the parameters' norm satisfy\n\n$$\n\\begin{array}{r l r}{{\\frac{1}{T}\\sum_{t=1}^{T}\\widehat{F}(w_{t})\\leq 2\\widehat{F}(w)+\\frac{5\\|w-w_{0}\\|^{2}}{2\\eta T},}}\\\\ &{}&{\\forall t\\in [T]\\ :\\ \\|w_{t}-w_{0}\\|\\leq 4\\|w-w_{0}\\|.}\\end{array}\\tag{16}\n$$"}, {"name": "Lemma A.2 (Descent lemma)", "statement": "Lemma A.2 (Descent lemma). Let Assumptions 1-2 hold. Assume the loss function satisfies self-boundedness Assumptions 4, 8. Then, for any $\\begin{array}{r}{\\eta<\\frac{1}{R^{2}\\widehat{F}(w_{t})}\\operatorname*{min}\\{\\frac{1}{\\ell^{2}+L},\\frac{1}{\\sqrt{L}\\ell}\\}}\\end{array}$ the descent property holds, i.e.,\n\n$$\n\\widehat{F}(w_{t+1})\\leq\\widehat{F}(w_{t})-\\frac{\\eta}{2}\\|\\nabla\\widehat{F}(w_{t})\\|^{2}.\n$$\n\nMoreover, if f satisfies Assumption 3 then the descent property holds for any $\\eta \\leq 1 / L_{\\widehat{F}}$ where $\\begin{array}{r}{L_{\\widehat{F}}: = \\ell^{2}R^{2}+ \\frac{L R^{2}}{\\sqrt{m}}}\\end{array}$ is the smoothness parameter of the training objective."}, {"name": "Remark 1 (Example: linearly-separable data)", "statement": "Remark 1 (Example: Linearly-separable data). Consider logistic-loss minimization, tanh activation $\\begin{array}{r}{\\sigma(u)=\\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}\\end{array}$ and data distribution that is linearly separable with margin γ, $i . e .$ , for almost surely all n samples there exists unit-norm vector $v^{\\star}\\in \\mathbb{R}^{d}$ such that $\\mathrm{min}_{i\\in [n]}y_{i}\\langle v^{\\star},x_{i}\\rangle=\\gamma$. We initialize the weights to zero, i.e. $w_{0}~ = ~ 0$ and show that the realizability Assumption 5 naturally holds in this setting. To see this, for any fixed $\\varepsilon > 0$ , set $\\alpha=2\\log(1/\\varepsilon)/\\gamma\\sqrt{m}$ and assume m $\\geq 4\\log^{2}(1/\\varepsilon)$ . With this choice, select weights $\\begin{array}{r}{w_{j}^{(\\varepsilon)}: =\\alpha v^{\\star},a_{j}=\\frac{1}{\\sqrt{m}}\\ for\\ j\\in [1,\\cdot\\cdot\\cdot,\\frac{m}{2}]}\\end{array}$ and $\\begin{array}{r}{w_{j}^{(\\varepsilon)}: = -\\alpha v^{\\star},a_{j}=\\frac{-1}{\\sqrt{m}}\\ for\\ j\\in\\{\\frac{m}{2}+1,\\cdot\\cdot\\cdot,m\\}}\\end{array}$ Then, the model output for any sample $(x_{i},y_{i})$ satisfies\n\n$$\n\\begin{array}{r l}&{y_{i}\\Phi(w^{(\\varepsilon)},x_{i})=\\displaystyle\\frac{y_{i}\\sqrt{m}}{2}\\left(\\sigma(\\alpha\\langle v^{\\star},x_{i}\\rangle)-\\sigma(-\\alpha\\langle v^{\\star},x_{i}\\rangle)\\right)}\\\\ &{\\quad\\quad\\quad=y_{i}\\sqrt{m}\\sigma(\\alpha\\langle v^{\\star},x_{i}\\rangle)\\ge\\sqrt{m}\\sigma(\\alpha\\gamma)\\ge\\displaystyle\\frac{\\sqrt{m}}{2}\\alpha\\gamma=\\log(1/\\varepsilon),}\\end{array}\n$$\n\nwhere the second equality uses the fact that tanh is odd, the first inequality follows by the increasing nature of tanh and data separability, and the last inequality follows since $\\alpha \\gamma \\leq 1$ and $\\sigma(u)\\geq u/2$ for all $u\\in [0,1]$ . Thus, the loss satisfies $\\widehat{F}(w^{(\\varepsilon)})\\leq\\varepsilon$ since for the logistic function log $\\left(1+e^{u}\\right)\\leq e^{u}$ . Moreover, our choice of α implies $g(\\varepsilon)=\\|w^{(\\varepsilon)}-w_{0}\\|=$ $\\|w^{(\\varepsilon)}\\|{=}\\alpha\\sqrt{m}=2\\log(1/\\varepsilon)/\\gamma$ . To conclude, the NN-Realizability Assumption 5 holds with $g(\\varepsilon)=2\\log(1/\\varepsilon)/\\gamma$ and thus applying Theorems 3.3 and 3.5 shows that with $m=\\Omega(\\log^{4}(T))$ neurons, the training loss and generalization gap are respectively bounded by $\\begin{array}{r}{{\\tilde{O}}(\\frac{1}{\\gamma^{2}T})}\\end{array}$ and $\\begin{array}{r}{{\\cal\\tilde{O}}(\\frac{1}{\\gamma^{2}n})}\\end{array}$ , which are known to be optimal under this data assumption. We note that the same conclusion as above holds for other smooth activations such as Softmax or GELU."}, {"name": "Section 4: NTK separability, initialization bound, Proposition 4.1 (eq. (9))", "statement": "In this section, we interpret our results for NTK-separable data by showing that our realizability condition holds for this class. We recall the definition of NTK-separability below Nitanda et al. (2019); Chen et al. (2020); Cao and Gu (2020).\n\nAssumption 6 (Separability by NTK). For almost surely all n training samples from the data distribution there exists $w^{\\star}\\in \\mathbb{R}^{d^{\\prime}}$ and $\\gamma > 0$ such that $\\| w^{\\star}\\| = 1$ and for all $i \\in [ n ]$ ,\n\n$$\n\\begin{array}{r}{y_{i}\\Bigl\\langle\\nabla_{1}\\Phi(w_{0},x_{i}),w^{\\star}\\Bigr\\rangle\\geq\\gamma .}\\end{array}\\tag{8}\n$$\n\nWe also assume a bound on the model's output at initialization. Similar assumptions, but for the value of the loss, also appear in prior works that study generalization using the algorithmic stability framework Richards and Kuzborskij (2021); Lei et al. (2022).\n\nAssumption 7 (Initialization bound). There exists parameter C such that $\\forall i \\in [ n ] \\ :$ $|\\Phi(w_{0},x_{i})|\\leq C,$ for almost surely all n training samples from the data distribution\n\nThe next proposition relates the NTK-separability assumption to our realizability assumption. The proofs for this section are given in Appendix C.\n\nProposition 4.1 (Realizability of NTK-separable data). Let Assumptions $1-2,6-7$ hold. Assume f (·) to be the logistic loss. Fix $\\varepsilon > 0$ and let $\\begin{array}{r}{m\\geq\\frac{L^{2}R^{4}}{4\\gamma^{4}C^{2}}(2C+\\log(1/\\varepsilon))^{4}}\\end{array}$ . Then the realizability Assumption 5 holds with $\\begin{array}{r}{g(\\varepsilon)=\\frac{1}{\\gamma}(2C+\\log(1/\\varepsilon))}\\end{array}$ . In other words, there exists $w^{(\\varepsilon)}$ such that\n\n$$\n\\widehat{F}(w^{(\\varepsilon)})\\leq\\varepsilon,\\quad and\\quad\\left\\|w^{(\\varepsilon)}-w_{0}\\right\\|=\\frac{1}{\\gamma}\\left(2C+\\log(1/\\varepsilon)\\right).\\tag{9}\n$$"}, {"name": "Theorem 3.2 (another result of the same paper)", "statement": "Theorem 3.2 (Training loss – General bound). Suppose Assumptions $1 \\mathrm{-}\\mathit{4}$ hold. Fix any training horizon $T \\geq 0$ and any step-size $\\eta \\leq 1 / L_{\\widehat{F}}$ where $L_{\\widehat{F}}$ is the objective's smoothness parameter. Assume any w $\\in \\mathbb{R}^{d^{\\prime}}$ and hidden-layer width m such that $\\| w - w_{0}\\|^{2}\\ge$ max $\\{\\eta T\\widehat{F}(w),\\eta\\widehat{F}(w_{0})\\}$ and $m\\geq 18^{2}L^{2}R^{4}\\Vert w-w_{0}\\Vert^{4}$. Then, the training loss and the parameters' norm satisfy\n\n$$\n\\begin{array}{r l}&{\\widehat{F}(w_{T})\\ \\leq\\ \\displaystyle\\frac{1}{T}\\sum_{t=1}^{T}\\widehat{F}(w_{t})\\ \\leq\\ 2\\widehat{F}(w)+\\frac{5\\|w-w_{0}\\|^{2}}{2\\eta T},}\\\\ &{\\forall t\\in [T]\\ :\\ \\|w_{t}-w_{0}\\|\\ \\leq\\ 4\\|w-w_{0}\\|.}\\end{array}\\tag{5}\n$$"}], "nl_notes": "### Theorem 3.3\nNOTE (Theorem 3.3) — The Lean assumes g(ε) > 0 for every ε > 0 (`hgpos`) and uses g only through `hreal` (on 0 < ε ≤ ε₀), its monotonicity (`hgdec`), g(1) and g(1/T). The paper's Remark 1 example g(ε) = 2 log(1/ε)/γ has g(1) = 0, which would force η ≤ 0 in the step-size condition; the intended shape is the one in Section 4, g(ε) = (2C + log(1/ε))/γ, with g(1) > 0.", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace BigOperators\n\nnamespace ResearchLevel\n\nabbrev Params (m d : ℕ) := PiLp 2 (fun _ : Fin m => EuclideanSpace ℝ (Fin d))\n\nnoncomputable def phi {m d : ℕ} (a : Fin m → ℝ) (σ : ℝ → ℝ)\n    (w : Params m d) (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  (1 / Real.sqrt (m : ℝ)) * ∑ j : Fin m, a j * σ (inner ℝ (w j) x)\n\nnoncomputable def Fhat {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (w : Params m d) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i : Fin n, f (ys i * phi a σ w (xs i))\n\nnoncomputable def FhatI {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (i : Fin n) (w : Params m d) : ℝ :=\n  f (ys i * phi a σ w (xs i))\n\nnoncomputable def gdSeq {m d n : ℕ} (a : Fin m → ℝ) (σ f : ℝ → ℝ)\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (η : ℝ) (w0 : Params m d) : ℕ → Params m d\n  | 0 => w0\n  | (t + 1) =>\n      gdSeq a σ f xs ys η w0 t - η • gradient (Fhat a σ f xs ys) (gdSeq a σ f xs ys η w0 t)\n\ntheorem lemma3_1 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n)\n    (R ℓ L : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n\n    (hfconv : ConvexOn ℝ Set.univ f)\n    (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w : Params m d) :\n    (∀ i, ‖gradient (FhatI a σ f xs ys i) w‖ ≤ ℓ * R * FhatI a σ f xs ys i w)\n    ∧ (∀ v : Params m d,\n        -(L * R ^ 2 / Real.sqrt (m : ℝ)) * Fhat a σ f xs ys w * ‖v‖ ^ 2\n          ≤ iteratedFDeriv ℝ 2 (Fhat a σ f xs ys) w (fun _ => v)) := by\n  sorry\n\ntheorem thm3_2 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η) (hη : η ≤ 1 / LF)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 w : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (hnorm : ‖w - w0‖ ^ 2 ≥ max (η * T * Fhat a σ f xs ys w) (η * Fhat a σ f xs ys w0))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * ‖w - w0‖ ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n    ∧ (1 / (T : ℝ)) * ∑ t ∈ Finset.Icc 1 T, Fhat a σ f xs ys (ws t)\n        ≤ 2 * Fhat a σ f xs ys w + 5 * ‖w - w0‖ ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * ‖w - w0‖ := by\n  sorry\n\ntheorem thm3_3 {m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (g : ℝ → ℝ) (hgpos : ∀ ε, 0 < ε → 0 < g ε)\n    (hgdec : ∀ ε₁ ε₂, 0 < ε₁ → ε₁ ≤ ε₂ → g ε₂ ≤ g ε₁)\n\n    (ε₀ : ℝ) (hε₀ : 0 < ε₀)\n    (hreal : ∀ ε, 0 < ε → ε ≤ ε₀ → ∃ wε : Params m d,\n        Fhat a σ f xs ys wε ≤ ε ∧ ‖wε - w0‖ = g ε)\n    (hTsmall : 1 / (T : ℝ) ≤ ε₀)\n    (hFw0 : 0 < Fhat a σ f xs ys w0)\n    (hη : η ≤ min (1 / LF) (min (g 1 ^ 2) (g 1 ^ 2 / Fhat a σ f xs ys w0)))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * g (1 / (T : ℝ)) ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ 2 / (T : ℝ) + 5 * g (1 / (T : ℝ)) ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * g (1 / (T : ℝ)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma3_1", "thm3_2", "thm3_3"], "topic_domain": "Deep Learning Theory", "topic_subtopic": "Training Dynamics & Implicit Bias", "holes": [{"start": 3535, "end": 5123, "gold": "{m d n : ℕ} (hm : 0 < m) (hn : 0 < n) (T : ℕ) (hT : 0 < T)\n    (R ℓ L LF η : ℝ) (hR : 0 ≤ R) (hℓ : 0 ≤ ℓ) (hL : 0 ≤ L)\n    (hLF : 0 < LF) (hη0 : 0 < η)\n    (a : Fin m → ℝ) (ha : ∀ j, a j = 1 ∨ a j = -1)\n    (hbal : (∑ j, a j) = 0)\n    (σ f : ℝ → ℝ)\n    (hσ : Differentiable ℝ σ) (hσd : Differentiable ℝ (deriv σ)) (hσ' : ∀ u, |deriv σ u| ≤ ℓ)\n    (hσ'' : ∀ u, |deriv (deriv σ) u| ≤ L)\n    (hf : Differentiable ℝ f) (hfd : Differentiable ℝ (deriv f)) (hfpos : ∀ u, 0 ≤ f u)\n    (hfconv : ConvexOn ℝ Set.univ f) (hf' : ∀ u, |deriv f u| ≤ 1)\n    (hf'' : ∀ u, deriv (deriv f) u ≤ 1) (hfsb : ∀ u, |deriv f u| ≤ f u)\n    (hfdec : Antitone f)\n    (hflim : Filter.Tendsto f Filter.atTop (nhds 0))\n    (xs : Fin n → EuclideanSpace ℝ (Fin d)) (ys : Fin n → ℝ)\n    (hx : ∀ i, ‖xs i‖ ≤ R) (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (w0 : Params m d)\n    (hsmooth : ∀ u v, ‖gradient (Fhat a σ f xs ys) u - gradient (Fhat a σ f xs ys) v‖\n        ≤ LF * ‖u - v‖)\n    (g : ℝ → ℝ) (hgpos : ∀ ε, 0 < ε → 0 < g ε)\n    (hgdec : ∀ ε₁ ε₂, 0 < ε₁ → ε₁ ≤ ε₂ → g ε₂ ≤ g ε₁)\n\n    (ε₀ : ℝ) (hε₀ : 0 < ε₀)\n    (hreal : ∀ ε, 0 < ε → ε ≤ ε₀ → ∃ wε : Params m d,\n        Fhat a σ f xs ys wε ≤ ε ∧ ‖wε - w0‖ = g ε)\n    (hTsmall : 1 / (T : ℝ) ≤ ε₀)\n    (hFw0 : 0 < Fhat a σ f xs ys w0)\n    (hη : η ≤ min (1 / LF) (min (g 1 ^ 2) (g 1 ^ 2 / Fhat a σ f xs ys w0)))\n    (hwidth : (m : ℝ) ≥ 18 ^ 2 * L ^ 2 * R ^ 4 * g (1 / (T : ℝ)) ^ 4) :\n    let ws := gdSeq a σ f xs ys η w0\n    Fhat a σ f xs ys (ws T)\n        ≤ 2 / (T : ℝ) + 5 * g (1 / (T : ℝ)) ^ 2 / (2 * η * (T : ℝ))\n    ∧ ∀ t, 1 ≤ t → t ≤ T → ‖ws t - w0‖ ≤ 4 * g (1 / (T : ℝ))"}]}
{"item_id": "v24-22-0339::lemma11_strongDuality::A-stmt", "paper_id": "v24-22-0339", "target_name": "lemma11_strongDuality", "source": "Research-level", "nl_statement": "Lemma 11 Let U be an uncertainty set given by (4) that satisfies R1 or R2. For any h $\\in T(\\mathcal{X},\\mathcal{Y})$ , we have that\n\n$$\n\\operatorname*{sup}_{\\mathbf{p}\\in\\mathcal{U}}\\ell(\\mathbf{h},\\mathbf{p})=\\operatorname*{min}_{\\mu\\in\\mathbb{R}^{m}}1-\\tau^{T}\\pmb{\\mu}+\\operatorname*{sup}_{x\\in\\mathcal{X},y\\in\\mathcal{Y}}\\left\\{\\Phi(x,y)^{T}\\pmb{\\mu}-\\mathrm{h}(y|x)\\right\\}+\\lambda^{T}|\\mu|\\tag{37}\n$$", "context": [{"name": "Notation (Section 1)", "statement": "Notation: calligraphic upper case letters denote sets, $\\mathrm{e . g .}, \\mathcal{Z}; | \\mathcal{Z}|$ denotes de cardinality of set $\\mathcal{Z};$ vectors and matrices are denoted by bold lower and upper case letters, respectively, e.g., v and M; for a vector v, $\\|\\mathbf{v}\\|,\\|\\mathbf{v}\\|_{1}$ , and $\\| \\mathbf{v}\\|_{\\infty}$ denote its L2-, L1-, and L-infinity norms, respectively, $\\mathbf{v}^{\\mathrm{T}}$ denotes its transpose, $v^{(i)}$ denotes its i-th component, $| \\mathbf{v}|$ and $\\mathbf{v}_{+}$ denote the vector given by the component-wise absolute value and positive part of $\\mathbf{v},$ respectively, and diag(v) denotes the diagonal matrix with diagonal given by v; probability distributions and classification rules are denoted by upright fonts, e.g., p and $\\mathrm{h}; \\mathbb{E}_{z \\sim \\mathrm{p}}$ or simply $\\mathbb{E}_{\\mathrm{p}}$ denotes the expectation w.r.t. probability distribution p of random variable $z ;$ for a probability distribution p over $\\mathcal{X}\\times \\mathcal{Y}$ , we denote by ${\\mathrm{p}}_{x},{\\mathrm{p}}_{y}$ its corresponding marginals over $\\mathcal{X}$ and $\\mathcal{Y}$, respectively, and by $\\mathrm{p}_{x \\mid y}$ the corresponding conditional distribution given $y \\in \\mathcal{Y}; \\preceq$ and $\\succeq$ denote vector (component-wise) inequalities; $\\mathbf{1}$ denotes a vector with all components equal to 1 and $\\mathbb{I}\\{\\cdot \\}$ denotes the indicator function; finally, for a function of two variables $f:\\mathcal{X}\\times\\mathcal{Y}\\to\\mathcal{Z},f(x,\\cdot)$ denotes the function $f(x,\\cdot):\\mathcal{Y}\\to\\mathcal{Z}$ obtained by fixing $x \\in \\mathcal{X}$"}, {"name": "Setup: instances, labels, distributions, classification rules (Section 2.1)", "statement": "Classification techniques assign instances in a set $\\mathcal{X}$ to labels in a set $\\mathcal{Y}$, where $\\mathcal{X}$ is a Borel subset of $\\mathbb{R}^{d}$ and $\\mathcal{Y}$ is a finite set represented by $\\{1,2,\\ldots,|\\mathcal{Y}|\\}$ . We denote by $\\Delta(\\mathcal{X}\\times\\mathcal{Y})$ the set of Borel probability measures p on $\\mathcal{X}\\times \\mathcal{Y}$ , also referred to as probability distributions. Both randomized and deterministic classification rules are given by functions from instances to probability distributions on labels (Markov transitions). We denote the set of all classification rules by $\\mathrm{T}(\\mathcal{X},\\mathcal{Y})$ , and for $\\mathrm{~h~}\\in\\mathrm{~T~}(\\mathcal{X},\\mathcal{Y})$ we denote by $\\mathrm{h}(y|x)$ the probability with which instance $x \\in \\mathcal{X}$ is classified by label $y\\in\\mathcal{Y}\\ (\\mathrm{h}(y|x)\\in\\{0,1\\}$ if labels are deterministically assigned to instances)."}, {"name": "0-1 loss and expected loss l(h,p) (Section 2.1)", "statement": "Supervised classification techniques use n training instance-label pairs $(x_{1},y_{1}),(x_{2},y_{2}),\\ldots,(x_{n},y_{n})$ from the underlying distribution $\\mathrm{p}^{\\ast}$ to find classification rules that assign labels to instances with small error probability. If $\\mathrm{p}^{*}\\in\\Delta(\\mathcal{X}\\times\\mathcal{Y})$ is the true underlying distribution of instance-label pairs, the error probability of a classification rule $\\mathrm{h}\\in\\mathrm{T}(\\mathcal{X},\\mathcal{Y})$ is its expected 0-1 loss denoted by $R\\mathrm{(h)}$ , that is\n\n$$\nR(\\mathrm{h})=\\mathbb{E}_{\\mathrm{p}^{*}}\\{\\ell(\\mathrm{h},(x,y))\\}\n$$\n\nwith $\\ell(\\mathrm{h},(x,y))=1-\\mathrm{h}(y|x)$ the 0-1 loss of rule h at instance-label pair $(x,y)$ . In the following, we overload the notation for the loss function and denote the expected loss of h with respect to probability distribution p as $\\ell(\\mathrm{h},\\mathrm{p}): =\\mathbb{E}_{\\mathrm{p}}\\{\\ell(\\mathrm{h},(x,y))\\}$"}, {"name": "Feature mapping: bounded Borel measurable (Section 2.2)", "statement": "The proposed MRCs minimize the worst-case expected 0-1 loss with respect to distributions in uncertainty sets that can contain the true underlying distribution with a tunable confidence. These uncertainty sets are given by constraints on the expectations of a vector-valued bounded and Borel measurable function $\\Phi : \\mathcal{X}\\times \\mathcal{Y}\\to \\mathbb{R}^{m}$ referred to as feature mapping, e.g., multiple polynomials on x and y or one-hot encodings of the last layers in an NN. Such mappings are commonly used in machine learning to represent instance-label pairs as real vectors (see e.g., Mohri et al. (2018); Bengio et al. (2013) and next Section 3.1)."}, {"name": "Uncertainty set (4) and regularity conditions R1, R2 (Section 2.2)", "statement": "In the following we consider uncertainty sets\n\n$$\n\\mathcal{U}=\\left\\{{\\mathrm p}\\in\\Delta(\\mathcal{X},\\mathcal{Y}):~|\\mathbb{E}_{\\mathrm{p}}\\{\\Phi\\}-\\pmb{\\tau}|\\preceq\\lambda\\right\\}\\tag{4}\n$$\n\ngiven by a feature mapping $\\Phi$ together with mean and confidence vectors $\\tau \\in \\mathbb{R}^{m}$ and $\\pmb{\\lambda}\\in \\mathbb{R}^{m}$ , and satisfying one of the following regularity conditions.\n\nR1 The set $\\mathcal{X}$ is finite and there exists a probability measure p in $\\mathcal{U}$\n\nR2 There exists a probability measure p in $\\mathcal{U}$ such that $|\\mathbb{E}_{\\mathrm{p}}\\{\\Phi(x,y)^{(i)}\\}-\\pmb{\\tau}^{(i)}| <\\lambda^{(i)}$ for any $i\\in\\{1,2,\\dots,m\\}$ with $\\lambda^{(i)}>0$ and\n\nR2.1 $\\lambda^{(i)}>0$ for all $i\\in\\{1,2,\\dots,m\\}$ or\n\nR2.2 the support of the r.v. $\\Phi(x,y)$ for $(x,y)\\sim\\mathrm{p}$ is not contained in a proper affine subspace of $\\mathbb{R}^{m}$"}, {"name": "Role of the regularity conditions (Section 2.2)", "statement": "These regularity conditions are utilized to ensure strong duality holds for the inner maximization in the minimax problems, and they are satisfied with wide generality."}, {"name": "Lemma 11, display (36) (weak duality bound, Appendix A)", "statement": "Lemma 11 Let U be an uncertainty set given by (4). For any h $\\in T(\\mathcal{X},\\mathcal{Y})$ , we have that\n\n$$\n\\operatorname*{sup}_{\\mathbf{p}\\in\\mathcal{U}}\\ell(\\mathbf{h},\\mathbf{p})\\le\\operatorname*{inf}_{\\mu\\in\\mathbb{R}^{m}}1-\\tau^{T}\\pmb{\\mu}+\\operatorname*{sup}_{x\\in\\mathcal{X},y\\in\\mathcal{Y}}\\left\\{\\Phi(x,y)^{T}\\pmb{\\mu}-\\mathrm{h}(y|x)\\right\\}+\\lambda^{T}|\\mu|\\tag{36}\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def lambdaDelta (C δ : ℝ) (m n : ℕ) : ℝ :=\n  C * Real.sqrt (2 * Real.log (2 * (m : ℝ) / δ) / (n : ℝ))\n\nnoncomputable def featExpectM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (p : Measure (𝒳 × 𝒴)) (i : Fin m) : ℝ :=\n  ∫ z, Φ z.1 z.2 i ∂p\n\ndef inUncertaintySetM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (p : Measure (𝒳 × 𝒴)) : Prop :=\n  IsProbabilityMeasure p ∧ (∀ i, |featExpectM Φ p i - τ i| ≤ lam i)\n\nnoncomputable def expectedLossM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴)) : ℝ :=\n  ∫ z, (1 - h z.1 z.2) ∂p\n\nnoncomputable def RunderM {𝒳 𝒴 : Type*} {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  ⨆ μ : Fin m → ℝ, (1 - (∑ i, τ i * μ i)\n    + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n    - (∑ i, lam i * |μ i|))\n\nnoncomputable def RoverM {𝒳 𝒴 : Type*} {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  ⨅ μ : Fin m → ℝ, (1 - (∑ i, τ i * μ i)\n    + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n    + (∑ i, lam i * |μ i|))\n\nnoncomputable def worstCaseRiskM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  sSup {r : ℝ | ∃ p, inUncertaintySetM Φ τ lam p ∧ r = expectedLossM h p}\n\ndef R1 {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*} [MeasurableSpace 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) : Prop :=\n  Finite 𝒳 ∧ ∃ p : Measure (𝒳 × 𝒴), inUncertaintySetM Φ τ lam p\n\ndef R2 {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*} [MeasurableSpace 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) : Prop :=\n  ∃ p : Measure (𝒳 × 𝒴), inUncertaintySetM Φ τ lam p ∧\n    (∀ i, 0 < lam i → |featExpectM Φ p i - τ i| < lam i) ∧\n    ((∀ i, 0 < lam i) ∨\n      ¬ ∃ (α : Fin m → ℝ) (c : ℝ), α ≠ 0 ∧ ∀ᵐ z ∂p, (∑ i, α i * Φ z.1 z.2 i) = c)\n\ntheorem theorem5 {𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳] {𝒴 : Type*} [Fintype 𝒴]\n    [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴))\n\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hp : inUncertaintySetM Φ τ lam p)\n    (hba : BddAbove (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) - (∑ i, lam i * |μ i|))))\n    (hbb : BddBelow (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) + (∑ i, lam i * |μ i|)))) :\n    (RunderM Φ τ lam h ≤ expectedLossM h p ∧ expectedLossM h p ≤ RoverM Φ τ lam h)\n      ∧ ((R1 Φ τ lam ∨ R2 Φ τ lam) →\n          (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              - (∑ i, lam i * |μ i|) = RunderM Φ τ lam h)\n        ∧ (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              + (∑ i, lam i * |μ i|) = RoverM Φ τ lam h)) := by\n  sorry\n\ntheorem lemma11_strongDuality {𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳]\n    {𝒴 : Type*} [Fintype 𝒴] [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴]\n    [Nonempty 𝒳] [Nonempty 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ)\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hlam : ∀ i, 0 ≤ lam i)\n    (hreg : R1 Φ τ lam ∨ R2 Φ τ lam) :\n    worstCaseRiskM Φ τ lam h = RoverM Φ τ lam h\n      ∧ ∃ μ : Fin m → ℝ, RoverM Φ τ lam h = 1 - (∑ i, τ i * μ i)\n          + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n          + (∑ i, lam i * |μ i|) := by\n  sorry\n\ntheorem theorem4_hoeffding {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)\n    [IsProbabilityMeasure P] (m n : ℕ) (C δ : ℝ)\n    (τ : Fin m → ℝ) (Z : Fin n → Ω → Fin m → ℝ)\n    (hm : 0 < m) (hn : 0 < n) (hC : 0 ≤ C) (hδ : 0 < δ) (hδ1 : δ < 1)\n    (nY : ℕ) (hnY : 0 < nY) (labelOf : Fin m → Fin nY)\n    (D : ℕ) (hD : 0 < D)\n    (hblocks : ∀ y : Fin nY, (Finset.univ.filter fun i => labelOf i = y).card = D)\n    (hlabelOnto : Function.Surjective labelOf)\n    (honehot : ∀ k ω, ∃ y : Fin nY, ∀ i, labelOf i ≠ y → Z k ω i = 0)\n    (hbdd : ∀ k i ω, |Z k ω i| ≤ C)\n    (hmeas : ∀ k, Measurable (Z k))\n    (hindep : ProbabilityTheory.iIndepFun Z P)\n    (hident : ∀ k l, ProbabilityTheory.IdentDistrib (Z k) (Z l) P P)\n    (hmean : ∀ k i, ∫ ω, Z k ω i ∂P = τ i) :\n    ENNReal.ofReal (1 - δ)\n      ≤ P {ω | ∀ i : Fin m, |τ i - (∑ k, Z k ω i) / (n : ℝ)| ≤ lambdaDelta C δ m n} := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma11_strongDuality", "theorem5", "theorem4_hoeffding"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 3471, "end": 4218, "gold": "{𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳]\n    {𝒴 : Type*} [Fintype 𝒴] [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴]\n    [Nonempty 𝒳] [Nonempty 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ)\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hlam : ∀ i, 0 ≤ lam i)\n    (hreg : R1 Φ τ lam ∨ R2 Φ τ lam) :\n    worstCaseRiskM Φ τ lam h = RoverM Φ τ lam h\n      ∧ ∃ μ : Fin m → ℝ, RoverM Φ τ lam h = 1 - (∑ i, τ i * μ i)\n          + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n          + (∑ i, lam i * |μ i|)"}]}
{"item_id": "v24-22-0339::theorem5::A-stmt", "paper_id": "v24-22-0339", "target_name": "theorem5", "source": "Research-level", "nl_statement": "Theorem 5 Let U be an uncertainty set given by (4), h be any classification rule, and $\\underline{{R}}(\\mathcal{U},\\mathrm{h})$ and $\\overline{{R}}(\\mathcal{U},\\mathrm{h})$ be given by\n\n$$\n\\underline{{R}}(\\mathcal{U},\\mathrm{h})=\\operatorname*{sup}_{\\mu}\\ 1-{\\tau^{T}}\\mu+\\operatorname*{inf}_{x\\in\\mathcal{X},y\\in\\mathcal{Y}}\\{\\Phi(x,y)^{T}\\mu-\\mathrm{h}(y|x)\\}-{\\lambda^{T}}|\\mu|\\tag{20}\n$$\n\n$$\n\\overline{{R}}(\\mathcal{U},\\mathrm{h})=\\operatorname*{inf}_{\\mu}\\ 1-\\tau^{T}\\pmb{\\mu}+\\operatorname*{sup}_{x\\in\\mathcal{X},y\\in\\mathcal{Y}}\\{\\Phi(x,y)^{T}\\pmb{\\mu}-\\mathrm{h}(y|x)\\}+\\pmb{\\lambda}^{T}|\\pmb{\\mu}|.\\tag{21}\n$$\n\nThen, for any ${\\mathrm{p}}\\in \\mathcal{U}$ we have that $\\underline{{R}}(\\mathcal{U},\\mathrm{h})\\le\\ell(\\mathrm{h},\\mathrm{p})\\le\\overline{{R}}(\\mathcal{U},\\mathrm{h})$ . In addition, if U satisfies R1 or R2 the optimal values in (20) and (21) are attained.", "context": [{"name": "Notation (Section 1)", "statement": "Notation: calligraphic upper case letters denote sets, $\\mathrm{e . g .}, \\mathcal{Z}; | \\mathcal{Z}|$ denotes de cardinality of set $\\mathcal{Z};$ vectors and matrices are denoted by bold lower and upper case letters, respectively, e.g., v and M; for a vector v, $\\|\\mathbf{v}\\|,\\|\\mathbf{v}\\|_{1}$ , and $\\| \\mathbf{v}\\|_{\\infty}$ denote its L2-, L1-, and L-infinity norms, respectively, $\\mathbf{v}^{\\mathrm{T}}$ denotes its transpose, $v^{(i)}$ denotes its i-th component, $| \\mathbf{v}|$ and $\\mathbf{v}_{+}$ denote the vector given by the component-wise absolute value and positive part of $\\mathbf{v},$ respectively, and diag(v) denotes the diagonal matrix with diagonal given by v; probability distributions and classification rules are denoted by upright fonts, e.g., p and $\\mathrm{h}; \\mathbb{E}_{z \\sim \\mathrm{p}}$ or simply $\\mathbb{E}_{\\mathrm{p}}$ denotes the expectation w.r.t. probability distribution p of random variable $z ;$ for a probability distribution p over $\\mathcal{X}\\times \\mathcal{Y}$ , we denote by ${\\mathrm{p}}_{x},{\\mathrm{p}}_{y}$ its corresponding marginals over $\\mathcal{X}$ and $\\mathcal{Y}$, respectively, and by $\\mathrm{p}_{x \\mid y}$ the corresponding conditional distribution given $y \\in \\mathcal{Y}; \\preceq$ and $\\succeq$ denote vector (component-wise) inequalities; $\\mathbf{1}$ denotes a vector with all components equal to 1 and $\\mathbb{I}\\{\\cdot \\}$ denotes the indicator function; finally, for a function of two variables $f:\\mathcal{X}\\times\\mathcal{Y}\\to\\mathcal{Z},f(x,\\cdot)$ denotes the function $f(x,\\cdot):\\mathcal{Y}\\to\\mathcal{Z}$ obtained by fixing $x \\in \\mathcal{X}$"}, {"name": "Setup: instances, labels, distributions, classification rules (Section 2.1)", "statement": "Classification techniques assign instances in a set $\\mathcal{X}$ to labels in a set $\\mathcal{Y}$, where $\\mathcal{X}$ is a Borel subset of $\\mathbb{R}^{d}$ and $\\mathcal{Y}$ is a finite set represented by $\\{1,2,\\ldots,|\\mathcal{Y}|\\}$ . We denote by $\\Delta(\\mathcal{X}\\times\\mathcal{Y})$ the set of Borel probability measures p on $\\mathcal{X}\\times \\mathcal{Y}$ , also referred to as probability distributions. Both randomized and deterministic classification rules are given by functions from instances to probability distributions on labels (Markov transitions). We denote the set of all classification rules by $\\mathrm{T}(\\mathcal{X},\\mathcal{Y})$ , and for $\\mathrm{~h~}\\in\\mathrm{~T~}(\\mathcal{X},\\mathcal{Y})$ we denote by $\\mathrm{h}(y|x)$ the probability with which instance $x \\in \\mathcal{X}$ is classified by label $y\\in\\mathcal{Y}\\ (\\mathrm{h}(y|x)\\in\\{0,1\\}$ if labels are deterministically assigned to instances)."}, {"name": "0-1 loss and expected loss l(h,p) (Section 2.1)", "statement": "Supervised classification techniques use n training instance-label pairs $(x_{1},y_{1}),(x_{2},y_{2}),\\ldots,(x_{n},y_{n})$ from the underlying distribution $\\mathrm{p}^{\\ast}$ to find classification rules that assign labels to instances with small error probability. If $\\mathrm{p}^{*}\\in\\Delta(\\mathcal{X}\\times\\mathcal{Y})$ is the true underlying distribution of instance-label pairs, the error probability of a classification rule $\\mathrm{h}\\in\\mathrm{T}(\\mathcal{X},\\mathcal{Y})$ is its expected 0-1 loss denoted by $R\\mathrm{(h)}$ , that is\n\n$$\nR(\\mathrm{h})=\\mathbb{E}_{\\mathrm{p}^{*}}\\{\\ell(\\mathrm{h},(x,y))\\}\n$$\n\nwith $\\ell(\\mathrm{h},(x,y))=1-\\mathrm{h}(y|x)$ the 0-1 loss of rule h at instance-label pair $(x,y)$ . In the following, we overload the notation for the loss function and denote the expected loss of h with respect to probability distribution p as $\\ell(\\mathrm{h},\\mathrm{p}): =\\mathbb{E}_{\\mathrm{p}}\\{\\ell(\\mathrm{h},(x,y))\\}$"}, {"name": "Feature mapping: bounded Borel measurable (Section 2.2)", "statement": "The proposed MRCs minimize the worst-case expected 0-1 loss with respect to distributions in uncertainty sets that can contain the true underlying distribution with a tunable confidence. These uncertainty sets are given by constraints on the expectations of a vector-valued bounded and Borel measurable function $\\Phi : \\mathcal{X}\\times \\mathcal{Y}\\to \\mathbb{R}^{m}$ referred to as feature mapping, e.g., multiple polynomials on x and y or one-hot encodings of the last layers in an NN. Such mappings are commonly used in machine learning to represent instance-label pairs as real vectors (see e.g., Mohri et al. (2018); Bengio et al. (2013) and next Section 3.1)."}, {"name": "Uncertainty set (4) and regularity conditions R1, R2 (Section 2.2)", "statement": "In the following we consider uncertainty sets\n\n$$\n\\mathcal{U}=\\left\\{{\\mathrm p}\\in\\Delta(\\mathcal{X},\\mathcal{Y}):~|\\mathbb{E}_{\\mathrm{p}}\\{\\Phi\\}-\\pmb{\\tau}|\\preceq\\lambda\\right\\}\\tag{4}\n$$\n\ngiven by a feature mapping $\\Phi$ together with mean and confidence vectors $\\tau \\in \\mathbb{R}^{m}$ and $\\pmb{\\lambda}\\in \\mathbb{R}^{m}$ , and satisfying one of the following regularity conditions.\n\nR1 The set $\\mathcal{X}$ is finite and there exists a probability measure p in $\\mathcal{U}$\n\nR2 There exists a probability measure p in $\\mathcal{U}$ such that $|\\mathbb{E}_{\\mathrm{p}}\\{\\Phi(x,y)^{(i)}\\}-\\pmb{\\tau}^{(i)}| <\\lambda^{(i)}$ for any $i\\in\\{1,2,\\dots,m\\}$ with $\\lambda^{(i)}>0$ and\n\nR2.1 $\\lambda^{(i)}>0$ for all $i\\in\\{1,2,\\dots,m\\}$ or\n\nR2.2 the support of the r.v. $\\Phi(x,y)$ for $(x,y)\\sim\\mathrm{p}$ is not contained in a proper affine subspace of $\\mathbb{R}^{m}$"}, {"name": "Role of the regularity conditions (Section 2.2)", "statement": "These regularity conditions are utilized to ensure strong duality holds for the inner maximization in the minimax problems, and they are satisfied with wide generality."}, {"name": "Lemma 11, display (36) (weak duality bound, Appendix A)", "statement": "Lemma 11 Let U be an uncertainty set given by (4). For any h $\\in T(\\mathcal{X},\\mathcal{Y})$ , we have that\n\n$$\n\\operatorname*{sup}_{\\mathbf{p}\\in\\mathcal{U}}\\ell(\\mathbf{h},\\mathbf{p})\\le\\operatorname*{inf}_{\\mu\\in\\mathbb{R}^{m}}1-\\tau^{T}\\pmb{\\mu}+\\operatorname*{sup}_{x\\in\\mathcal{X},y\\in\\mathcal{Y}}\\left\\{\\Phi(x,y)^{T}\\pmb{\\mu}-\\mathrm{h}(y|x)\\right\\}+\\lambda^{T}|\\mu|\\tag{36}\n$$"}, {"name": "Lemma 11 (minimax strong duality) (another result of the same paper)", "statement": "Lemma 11 Let U be an uncertainty set given by (4) that satisfies R1 or R2. For any h $\\in T(\\mathcal{X},\\mathcal{Y})$ , we have that\n\n$$\n\\operatorname*{sup}_{\\mathbf{p}\\in\\mathcal{U}}\\ell(\\mathbf{h},\\mathbf{p})=\\operatorname*{min}_{\\mu\\in\\mathbb{R}^{m}}1-\\tau^{T}\\pmb{\\mu}+\\operatorname*{sup}_{x\\in\\mathcal{X},y\\in\\mathcal{Y}}\\left\\{\\Phi(x,y)^{T}\\pmb{\\mu}-\\mathrm{h}(y|x)\\right\\}+\\lambda^{T}|\\mu|\\tag{37}\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def lambdaDelta (C δ : ℝ) (m n : ℕ) : ℝ :=\n  C * Real.sqrt (2 * Real.log (2 * (m : ℝ) / δ) / (n : ℝ))\n\nnoncomputable def featExpectM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (p : Measure (𝒳 × 𝒴)) (i : Fin m) : ℝ :=\n  ∫ z, Φ z.1 z.2 i ∂p\n\ndef inUncertaintySetM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (p : Measure (𝒳 × 𝒴)) : Prop :=\n  IsProbabilityMeasure p ∧ (∀ i, |featExpectM Φ p i - τ i| ≤ lam i)\n\nnoncomputable def expectedLossM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴)) : ℝ :=\n  ∫ z, (1 - h z.1 z.2) ∂p\n\nnoncomputable def RunderM {𝒳 𝒴 : Type*} {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  ⨆ μ : Fin m → ℝ, (1 - (∑ i, τ i * μ i)\n    + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n    - (∑ i, lam i * |μ i|))\n\nnoncomputable def RoverM {𝒳 𝒴 : Type*} {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  ⨅ μ : Fin m → ℝ, (1 - (∑ i, τ i * μ i)\n    + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n    + (∑ i, lam i * |μ i|))\n\nnoncomputable def worstCaseRiskM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  sSup {r : ℝ | ∃ p, inUncertaintySetM Φ τ lam p ∧ r = expectedLossM h p}\n\ndef R1 {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*} [MeasurableSpace 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) : Prop :=\n  Finite 𝒳 ∧ ∃ p : Measure (𝒳 × 𝒴), inUncertaintySetM Φ τ lam p\n\ndef R2 {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*} [MeasurableSpace 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) : Prop :=\n  ∃ p : Measure (𝒳 × 𝒴), inUncertaintySetM Φ τ lam p ∧\n    (∀ i, 0 < lam i → |featExpectM Φ p i - τ i| < lam i) ∧\n    ((∀ i, 0 < lam i) ∨\n      ¬ ∃ (α : Fin m → ℝ) (c : ℝ), α ≠ 0 ∧ ∀ᵐ z ∂p, (∑ i, α i * Φ z.1 z.2 i) = c)\n\ntheorem theorem5 {𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳] {𝒴 : Type*} [Fintype 𝒴]\n    [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴))\n\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hp : inUncertaintySetM Φ τ lam p)\n    (hba : BddAbove (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) - (∑ i, lam i * |μ i|))))\n    (hbb : BddBelow (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) + (∑ i, lam i * |μ i|)))) :\n    (RunderM Φ τ lam h ≤ expectedLossM h p ∧ expectedLossM h p ≤ RoverM Φ τ lam h)\n      ∧ ((R1 Φ τ lam ∨ R2 Φ τ lam) →\n          (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              - (∑ i, lam i * |μ i|) = RunderM Φ τ lam h)\n        ∧ (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              + (∑ i, lam i * |μ i|) = RoverM Φ τ lam h)) := by\n  sorry\n\ntheorem lemma11_strongDuality {𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳]\n    {𝒴 : Type*} [Fintype 𝒴] [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴]\n    [Nonempty 𝒳] [Nonempty 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ)\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hlam : ∀ i, 0 ≤ lam i)\n    (hreg : R1 Φ τ lam ∨ R2 Φ τ lam) :\n    worstCaseRiskM Φ τ lam h = RoverM Φ τ lam h\n      ∧ ∃ μ : Fin m → ℝ, RoverM Φ τ lam h = 1 - (∑ i, τ i * μ i)\n          + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n          + (∑ i, lam i * |μ i|) := by\n  sorry\n\ntheorem theorem4_hoeffding {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)\n    [IsProbabilityMeasure P] (m n : ℕ) (C δ : ℝ)\n    (τ : Fin m → ℝ) (Z : Fin n → Ω → Fin m → ℝ)\n    (hm : 0 < m) (hn : 0 < n) (hC : 0 ≤ C) (hδ : 0 < δ) (hδ1 : δ < 1)\n    (nY : ℕ) (hnY : 0 < nY) (labelOf : Fin m → Fin nY)\n    (D : ℕ) (hD : 0 < D)\n    (hblocks : ∀ y : Fin nY, (Finset.univ.filter fun i => labelOf i = y).card = D)\n    (hlabelOnto : Function.Surjective labelOf)\n    (honehot : ∀ k ω, ∃ y : Fin nY, ∀ i, labelOf i ≠ y → Z k ω i = 0)\n    (hbdd : ∀ k i ω, |Z k ω i| ≤ C)\n    (hmeas : ∀ k, Measurable (Z k))\n    (hindep : ProbabilityTheory.iIndepFun Z P)\n    (hident : ∀ k l, ProbabilityTheory.IdentDistrib (Z k) (Z l) P P)\n    (hmean : ∀ k i, ∫ ω, Z k ω i ∂P = τ i) :\n    ENNReal.ofReal (1 - δ)\n      ≤ P {ω | ∀ i : Fin m, |τ i - (∑ k, Z k ω i) / (n : ℝ)| ≤ lambdaDelta C δ m n} := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma11_strongDuality", "theorem5", "theorem4_hoeffding"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 2124, "end": 3425, "gold": "{𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳] {𝒴 : Type*} [Fintype 𝒴]\n    [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴))\n\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hp : inUncertaintySetM Φ τ lam p)\n    (hba : BddAbove (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) - (∑ i, lam i * |μ i|))))\n    (hbb : BddBelow (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) + (∑ i, lam i * |μ i|)))) :\n    (RunderM Φ τ lam h ≤ expectedLossM h p ∧ expectedLossM h p ≤ RoverM Φ τ lam h)\n      ∧ ((R1 Φ τ lam ∨ R2 Φ τ lam) →\n          (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              - (∑ i, lam i * |μ i|) = RunderM Φ τ lam h)\n        ∧ (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              + (∑ i, lam i * |μ i|) = RoverM Φ τ lam h))"}]}
{"item_id": "v24-22-0339::theorem4_hoeffding::A-stmt", "paper_id": "v24-22-0339", "target_name": "theorem4_hoeffding", "source": "Research-level", "nl_statement": "Theorem 4 Let the mean vector be given by $\\tau = \\tau_{n}$ as in (5) for n training samples, and $\\mathcal{F}$ be a family of bounded scalar features, that is $|\\psi(x)|\\leq C$ for all $\\psi \\in{\\mathcal{F}}$ and $x \\in \\mathcal{X}$\n\nIf $\\left| \\mathcal{F}\\right| < \\infty$ and $\\Phi$ is given by (16) using all $D = |\\mathcal{F}|$ scalar features of $\\mathcal{F}$ (so that $m = |\\mathcal{F}||\\mathcal{Y}|$), $\\lambda_{\\delta}^{(i)}$ for $i=1,2,\\ldots,m$ can be taken as\n\n$$\n\\lambda_{\\delta}^{(i)}=C\\sqrt{\\frac{2\\log 2\\vert\\mathcal{F}\\vert\\vert\\mathcal{Y}\\vert/\\delta}{n}}.\\tag{17}\n$$", "context": [{"name": "Notation (Section 1)", "statement": "Notation: calligraphic upper case letters denote sets, $\\mathrm{e . g .}, \\mathcal{Z}; | \\mathcal{Z}|$ denotes de cardinality of set $\\mathcal{Z};$ vectors and matrices are denoted by bold lower and upper case letters, respectively, e.g., v and M; for a vector v, $\\|\\mathbf{v}\\|,\\|\\mathbf{v}\\|_{1}$ , and $\\| \\mathbf{v}\\|_{\\infty}$ denote its L2-, L1-, and L-infinity norms, respectively, $\\mathbf{v}^{\\mathrm{T}}$ denotes its transpose, $v^{(i)}$ denotes its i-th component, $| \\mathbf{v}|$ and $\\mathbf{v}_{+}$ denote the vector given by the component-wise absolute value and positive part of $\\mathbf{v},$ respectively, and diag(v) denotes the diagonal matrix with diagonal given by v; probability distributions and classification rules are denoted by upright fonts, e.g., p and $\\mathrm{h}; \\mathbb{E}_{z \\sim \\mathrm{p}}$ or simply $\\mathbb{E}_{\\mathrm{p}}$ denotes the expectation w.r.t. probability distribution p of random variable $z ;$ for a probability distribution p over $\\mathcal{X}\\times \\mathcal{Y}$ , we denote by ${\\mathrm{p}}_{x},{\\mathrm{p}}_{y}$ its corresponding marginals over $\\mathcal{X}$ and $\\mathcal{Y}$, respectively, and by $\\mathrm{p}_{x \\mid y}$ the corresponding conditional distribution given $y \\in \\mathcal{Y}; \\preceq$ and $\\succeq$ denote vector (component-wise) inequalities; $\\mathbf{1}$ denotes a vector with all components equal to 1 and $\\mathbb{I}\\{\\cdot \\}$ denotes the indicator function; finally, for a function of two variables $f:\\mathcal{X}\\times\\mathcal{Y}\\to\\mathcal{Z},f(x,\\cdot)$ denotes the function $f(x,\\cdot):\\mathcal{Y}\\to\\mathcal{Z}$ obtained by fixing $x \\in \\mathcal{X}$"}, {"name": "Setup: instances, labels, distributions, classification rules (Section 2.1)", "statement": "Classification techniques assign instances in a set $\\mathcal{X}$ to labels in a set $\\mathcal{Y}$, where $\\mathcal{X}$ is a Borel subset of $\\mathbb{R}^{d}$ and $\\mathcal{Y}$ is a finite set represented by $\\{1,2,\\ldots,|\\mathcal{Y}|\\}$ . We denote by $\\Delta(\\mathcal{X}\\times\\mathcal{Y})$ the set of Borel probability measures p on $\\mathcal{X}\\times \\mathcal{Y}$ , also referred to as probability distributions. Both randomized and deterministic classification rules are given by functions from instances to probability distributions on labels (Markov transitions). We denote the set of all classification rules by $\\mathrm{T}(\\mathcal{X},\\mathcal{Y})$ , and for $\\mathrm{~h~}\\in\\mathrm{~T~}(\\mathcal{X},\\mathcal{Y})$ we denote by $\\mathrm{h}(y|x)$ the probability with which instance $x \\in \\mathcal{X}$ is classified by label $y\\in\\mathcal{Y}\\ (\\mathrm{h}(y|x)\\in\\{0,1\\}$ if labels are deterministically assigned to instances)."}, {"name": "Feature mapping: bounded Borel measurable (Section 2.2)", "statement": "The proposed MRCs minimize the worst-case expected 0-1 loss with respect to distributions in uncertainty sets that can contain the true underlying distribution with a tunable confidence. These uncertainty sets are given by constraints on the expectations of a vector-valued bounded and Borel measurable function $\\Phi : \\mathcal{X}\\times \\mathcal{Y}\\to \\mathbb{R}^{m}$ referred to as feature mapping, e.g., multiple polynomials on x and y or one-hot encodings of the last layers in an NN. Such mappings are commonly used in machine learning to represent instance-label pairs as real vectors (see e.g., Mohri et al. (2018); Bengio et al. (2013) and next Section 3.1)."}, {"name": "Uncertainty set (4) and regularity conditions R1, R2 (Section 2.2)", "statement": "In the following we consider uncertainty sets\n\n$$\n\\mathcal{U}=\\left\\{{\\mathrm p}\\in\\Delta(\\mathcal{X},\\mathcal{Y}):~|\\mathbb{E}_{\\mathrm{p}}\\{\\Phi\\}-\\pmb{\\tau}|\\preceq\\lambda\\right\\}\\tag{4}\n$$\n\ngiven by a feature mapping $\\Phi$ together with mean and confidence vectors $\\tau \\in \\mathbb{R}^{m}$ and $\\pmb{\\lambda}\\in \\mathbb{R}^{m}$ , and satisfying one of the following regularity conditions.\n\nR1 The set $\\mathcal{X}$ is finite and there exists a probability measure p in $\\mathcal{U}$\n\nR2 There exists a probability measure p in $\\mathcal{U}$ such that $|\\mathbb{E}_{\\mathrm{p}}\\{\\Phi(x,y)^{(i)}\\}-\\pmb{\\tau}^{(i)}| <\\lambda^{(i)}$ for any $i\\in\\{1,2,\\dots,m\\}$ with $\\lambda^{(i)}>0$ and\n\nR2.1 $\\lambda^{(i)}>0$ for all $i\\in\\{1,2,\\dots,m\\}$ or\n\nR2.2 the support of the r.v. $\\Phi(x,y)$ for $(x,y)\\sim\\mathrm{p}$ is not contained in a proper affine subspace of $\\mathbb{R}^{m}$"}, {"name": "Mean vector estimate tau_n, eq. (5) (Section 2.2)", "statement": "The mean vector $\\tau \\in \\mathbb{R}^{m}$ in (4) is an estimate of the feature mapping expectation $\\mathbb{E}_{\\mathrm{p}^{*}}\\{\\Phi \\}$ with respect to the underlying distribution. In this paper, we consider expectation estimates obtained as the sample average\n\n$$\n\\tau_{n}=\\frac{1}{n}\\sum_{i=1}^{n}\\Phi(x_{i},y_{i})\\tag{5}\n$$\n\nobtained from the n training samples $(x_{1},y_{1}),(x_{2},y_{2}),\\dotsc,(x_{n},y_{n})$ that are assumed to be independent samples from the underlying distribution $\\mathrm{p}^{\\ast}$ . Nevertheless, it is important to note that most of the results presented in the paper as well as the general methodology proposed can be utilized with general types of expectation estimates. Alternative estimators for the expectation can be preferred in several practical scenarios including cases with different distributions at training and test (Mohri and Medina, 2012; Mazuelas and Perez, 2020; Álvarez et al., 2022) and cases where the distribution of features has heavy tails (Lugosi and Mendelson, 2019; Hsu and Sabato, 2016)."}, {"name": "Confidence vector lambda (Section 2.2)", "statement": "The confidence vector $\\pmb{\\lambda}\\in \\mathbb{R}^{m}$ in (4) is an estimate of the mean vector component-wise accuracy $\\left| \\mathbb{E}_{\\mathrm{p}^{*}}\\{\\Phi \\}- \\tau \\right|$ , and controls the size of the uncertainty set considered. In particular, if λ corresponds to the length of confidence intervals at level $1 - \\delta$ centered at $\\tau$, then the true underlying distribution is included in the uncertainty set (4) with probability at least $1 - \\delta$. Section 3.2 describes how to choose such confidence vectors for different feature mappings."}, {"name": "Feature mapping (16): one-hot label blocks of scalar features (Section 3.1)", "statement": "Most of the results presented in the paper are valid for general feature mappings $\\Phi : \\mathcal{X}\\times \\mathcal{Y}\\to \\mathbb{R}^{m}$. In this section we describe feature mappings that are common in supervised classification and will be used later in the paper. Such feature mappings are given by real-valued functions over the set of instances $\\psi : \\mathcal{X}\\to \\mathbb{R}$ , referred to as scalar features. The most common and simple way to define feature mappings over $\\mathcal{X}$ and $\\mathcal{Y}$ is to use multiple scalar features over $\\mathcal{X}$ together with a one-hot encoding of the elements of $\\mathcal{Y}$ (Tsochantaridis et al., 2005; Crammer et al., 2006; Mohri et al., 2018) as follows\n\n$$\n\\Phi(x,y)=[\\mathbb{I}\\{y=1\\}\\Psi(x)^{\\mathrm{T}},\\mathbb{I}\\{y=2\\}\\Psi(x)^{\\mathrm{T}},\\ldots,\\mathbb{I}\\{y= |\\mathcal{Y}|\\}\\Psi(x)^{\\mathrm{T}}]^{\\mathrm{T}}=\\mathbf{e}_{y}\\otimes\\Psi(x)\\tag{16}\n$$\n\nwhere $\\Psi(x)=[\\psi_{1}(x),\\psi_{2}(x),\\ldots,\\psi_{D}(x)]^{\\mathrm{T}}$ for D scalar features $\\psi_{1},\\psi_{2},\\ldots,\\psi_{D},\\mathbf{e}_{y}$ is the $y \\mathrm{- t h}$ vector in the standard basis of $\\mathbb{R}^{| \\mathcal{Y}|}$ , and $\\otimes$ denotes the Kronecker product. The feature mapping $\\Phi$ represents each instance-label pair $(x,y)$ by an m-dimensional real vector with $m = | \\mathcal{Y}| D$ , so that $\\Phi(x,y)$ is composed by $|\\mathcal{Y}|$ $D$-dimensional blocks with values $\\Psi(x)$ in the block corresponding to y and zero otherwise. Other feature mappings can exploit relationships among classes as described for instance in Bakir et al. (2007); Caponnetto et al. (2008)."}, {"name": "Section 3.2 setup and coverage definition of lambda_delta", "statement": "In this section, we describe conditions for confidence vectors λ that lead to uncertainty sets given by (4) that contain the true underlying distribution with high probability. Specifically, we consider uncertainty sets U given by feature mappings defined by (16) using scalar features in a family $\\mathcal{F}$ together with mean vectors obtained as in (5) using the sample average of n independent samples from the underlying distribution $\\mathrm{p}^{\\ast}$\n\nWe denote by $\\lambda_{\\delta}$ any confidence vector that has coverage probability at least $1-\\delta,$ $\\begin{array}{r}{\\mathrm{i.e.,}\\operatorname{\\mathbb{P}}\\{|{\\mathbb{E}}_{\\mathrm{p}^{*}}\\{\\Phi\\}-\\pmb{\\tau}|\\preceq\\lambda_{\\delta}\\}\\geq 1-\\delta,}\\end{array}$ so that the underlying distribution $\\mathrm{p}^{\\ast}$ is included with probability at least $1 - \\delta$ in the uncertainty set given by $\\lambda_{\\delta}$ . The following result describes such confidence vectors in terms of the cardinality $|\\mathcal{F}|$, the empirical variance of the feature mapping, and the Rademacher complexity $\\mathcal{R}_{n}(\\mathcal{F})$ . In most of the results in the paper, we utilize feature mappings given by scalar features chosen before the training samples are observed. In such cases, appropriate confidence vectors are given in terms of the cardinality $| \\mathcal F |$ , i.e., the number of scalar features used. However, the methods proposed can also be used with data-dependent feature mappings. In such cases, appropriate confidence vectors are given in terms of the Rademacher complexity $\\mathcal{R}_{n}(\\mathcal{F})$ where $\\mathcal{F}$ is the family of candidate scalar features. For instance, $\\mathcal{F}$ can be family of all decision stumps and the feature mapping can be defined using the decision stumps found by one-dimensional decision trees learned using the training samples, as in Mazuelas et al. (2020)."}], "nl_notes": "### Theorem 4 (Hoeffding case, eq. 17)\nNOTE (Theorem 4, eq. 17) — Do not model F, Ψ, Φ or p*. Model the n training samples directly as random vectors Z_1, …, Z_n in ℝ^m standing for Φ(x_k, y_k): independent, identically distributed, measurable, with every coordinate bounded by C in absolute value. Let τ be their common mean (the paper's E_{p*}Φ) and compare it with their sample average. Model the one-hot block structure of (16) only by a labelling of the m coordinates by the labels, each label owning D coordinates, with every sample vector zero outside the block of one label.", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def lambdaDelta (C δ : ℝ) (m n : ℕ) : ℝ :=\n  C * Real.sqrt (2 * Real.log (2 * (m : ℝ) / δ) / (n : ℝ))\n\nnoncomputable def featExpectM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (p : Measure (𝒳 × 𝒴)) (i : Fin m) : ℝ :=\n  ∫ z, Φ z.1 z.2 i ∂p\n\ndef inUncertaintySetM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (p : Measure (𝒳 × 𝒴)) : Prop :=\n  IsProbabilityMeasure p ∧ (∀ i, |featExpectM Φ p i - τ i| ≤ lam i)\n\nnoncomputable def expectedLossM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴)) : ℝ :=\n  ∫ z, (1 - h z.1 z.2) ∂p\n\nnoncomputable def RunderM {𝒳 𝒴 : Type*} {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  ⨆ μ : Fin m → ℝ, (1 - (∑ i, τ i * μ i)\n    + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n    - (∑ i, lam i * |μ i|))\n\nnoncomputable def RoverM {𝒳 𝒴 : Type*} {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ)\n    (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  ⨅ μ : Fin m → ℝ, (1 - (∑ i, τ i * μ i)\n    + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n    + (∑ i, lam i * |μ i|))\n\nnoncomputable def worstCaseRiskM {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*}\n    [MeasurableSpace 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) : ℝ :=\n  sSup {r : ℝ | ∃ p, inUncertaintySetM Φ τ lam p ∧ r = expectedLossM h p}\n\ndef R1 {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*} [MeasurableSpace 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) : Prop :=\n  Finite 𝒳 ∧ ∃ p : Measure (𝒳 × 𝒴), inUncertaintySetM Φ τ lam p\n\ndef R2 {𝒳 : Type*} [MeasurableSpace 𝒳] {𝒴 : Type*} [MeasurableSpace 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) : Prop :=\n  ∃ p : Measure (𝒳 × 𝒴), inUncertaintySetM Φ τ lam p ∧\n    (∀ i, 0 < lam i → |featExpectM Φ p i - τ i| < lam i) ∧\n    ((∀ i, 0 < lam i) ∨\n      ¬ ∃ (α : Fin m → ℝ) (c : ℝ), α ≠ 0 ∧ ∀ᵐ z ∂p, (∑ i, α i * Φ z.1 z.2 i) = c)\n\ntheorem theorem5 {𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳] {𝒴 : Type*} [Fintype 𝒴]\n    [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴] {m : ℕ} (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ)\n    (h : 𝒳 → 𝒴 → ℝ) (p : Measure (𝒳 × 𝒴))\n\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hp : inUncertaintySetM Φ τ lam p)\n    (hba : BddAbove (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) - (∑ i, lam i * |μ i|))))\n    (hbb : BddBelow (Set.range (fun μ : Fin m → ℝ => 1 - (∑ i, τ i * μ i)\n      + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2)) + (∑ i, lam i * |μ i|)))) :\n    (RunderM Φ τ lam h ≤ expectedLossM h p ∧ expectedLossM h p ≤ RoverM Φ τ lam h)\n      ∧ ((R1 Φ τ lam ∨ R2 Φ τ lam) →\n          (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨅ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              - (∑ i, lam i * |μ i|) = RunderM Φ τ lam h)\n        ∧ (∃ μ : Fin m → ℝ, 1 - (∑ i, τ i * μ i)\n              + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n              + (∑ i, lam i * |μ i|) = RoverM Φ τ lam h)) := by\n  sorry\n\ntheorem lemma11_strongDuality {𝒳 : Type*} [MeasurableSpace 𝒳] [StandardBorelSpace 𝒳]\n    {𝒴 : Type*} [Fintype 𝒴] [MeasurableSpace 𝒴] [MeasurableSingletonClass 𝒴]\n    [Nonempty 𝒳] [Nonempty 𝒴] {m : ℕ}\n    (Φ : 𝒳 → 𝒴 → Fin m → ℝ) (τ lam : Fin m → ℝ) (h : 𝒳 → 𝒴 → ℝ)\n    (hΦmeas : ∀ i, Measurable (fun z : 𝒳 × 𝒴 => Φ z.1 z.2 i))\n    (hΦbdd : ∃ B : ℝ, ∀ x y i, |Φ x y i| ≤ B)\n    (hhmeas : Measurable (fun z : 𝒳 × 𝒴 => h z.1 z.2))\n    (hrule_nonneg : ∀ x y, 0 ≤ h x y) (hrule_sum : ∀ x, ∑ y, h x y = 1)\n    (hlam : ∀ i, 0 ≤ lam i)\n    (hreg : R1 Φ τ lam ∨ R2 Φ τ lam) :\n    worstCaseRiskM Φ τ lam h = RoverM Φ τ lam h\n      ∧ ∃ μ : Fin m → ℝ, RoverM Φ τ lam h = 1 - (∑ i, τ i * μ i)\n          + (⨆ q : 𝒳 × 𝒴, ((∑ i, Φ q.1 q.2 i * μ i) - h q.1 q.2))\n          + (∑ i, lam i * |μ i|) := by\n  sorry\n\ntheorem theorem4_hoeffding {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)\n    [IsProbabilityMeasure P] (m n : ℕ) (C δ : ℝ)\n    (τ : Fin m → ℝ) (Z : Fin n → Ω → Fin m → ℝ)\n    (hm : 0 < m) (hn : 0 < n) (hC : 0 ≤ C) (hδ : 0 < δ) (hδ1 : δ < 1)\n    (nY : ℕ) (hnY : 0 < nY) (labelOf : Fin m → Fin nY)\n    (D : ℕ) (hD : 0 < D)\n    (hblocks : ∀ y : Fin nY, (Finset.univ.filter fun i => labelOf i = y).card = D)\n    (hlabelOnto : Function.Surjective labelOf)\n    (honehot : ∀ k ω, ∃ y : Fin nY, ∀ i, labelOf i ≠ y → Z k ω i = 0)\n    (hbdd : ∀ k i ω, |Z k ω i| ≤ C)\n    (hmeas : ∀ k, Measurable (Z k))\n    (hindep : ProbabilityTheory.iIndepFun Z P)\n    (hident : ∀ k l, ProbabilityTheory.IdentDistrib (Z k) (Z l) P P)\n    (hmean : ∀ k i, ∫ ω, Z k ω i ∂P = τ i) :\n    ENNReal.ofReal (1 - δ)\n      ≤ P {ω | ∀ i : Fin m, |τ i - (∑ k, Z k ω i) / (n : ℝ)| ≤ lambdaDelta C δ m n} := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma11_strongDuality", "theorem5", "theorem4_hoeffding"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 4261, "end": 5099, "gold": "{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)\n    [IsProbabilityMeasure P] (m n : ℕ) (C δ : ℝ)\n    (τ : Fin m → ℝ) (Z : Fin n → Ω → Fin m → ℝ)\n    (hm : 0 < m) (hn : 0 < n) (hC : 0 ≤ C) (hδ : 0 < δ) (hδ1 : δ < 1)\n    (nY : ℕ) (hnY : 0 < nY) (labelOf : Fin m → Fin nY)\n    (D : ℕ) (hD : 0 < D)\n    (hblocks : ∀ y : Fin nY, (Finset.univ.filter fun i => labelOf i = y).card = D)\n    (hlabelOnto : Function.Surjective labelOf)\n    (honehot : ∀ k ω, ∃ y : Fin nY, ∀ i, labelOf i ≠ y → Z k ω i = 0)\n    (hbdd : ∀ k i ω, |Z k ω i| ≤ C)\n    (hmeas : ∀ k, Measurable (Z k))\n    (hindep : ProbabilityTheory.iIndepFun Z P)\n    (hident : ∀ k l, ProbabilityTheory.IdentDistrib (Z k) (Z l) P P)\n    (hmean : ∀ k i, ∫ ω, Z k ω i ∂P = τ i) :\n    ENNReal.ofReal (1 - δ)\n      ≤ P {ω | ∀ i : Fin m, |τ i - (∑ k, Z k ω i) / (n : ℝ)| ≤ lambdaDelta C δ m n}"}]}
{"item_id": "v23-20-1353::lemma16::A-stmt", "paper_id": "v23-20-1353", "target_name": "lemma16", "source": "Research-level", "nl_statement": "Lemma 16 Let $G:\\{-1,1\\}^{k}\\to\\{-1,1\\}$ and let ${\\mathcal{F}}_{1},\\ldots,{\\mathcal{F}}_{k}\\subseteq\\{-1,1\\}^{\\mathcal{X}}$ be hypothesis classes with $\\operatorname{VC}(\\mathcal{F}_{j})~=~d_{j}$ Denote $\\begin{array}{r}{\\bar{d}: = \\frac{1}{k}\\sum_{i = 1}^{k}d_{j}}\\end{array}$ . Define the function class $G\\left({\\mathcal{F}}_{1},\\ldots,{\\mathcal{F}}_{k}\\right)= :$ $\\{\\mathcal{X}\\ni x\\mapsto G\\left(f_{1}(x),\\ldots,f_{k}(x)\\right):f_{i}\\in\\mathcal{F}_{i}\\}$ . Then,\n\n$$\n\\mathrm{VC}\\left(G\\left(\\mathcal{F}_{1},\\ldots,\\mathcal{F}_{k}\\right)\\right)\\leq 2k\\log_2(3k)\\bar{d}\n$$", "context": [{"name": "Notation (Section 1.3)", "statement": "Notation. d denotes the VC dimension, $d^{*}$ denote the dual dual-VC dimension, $\\mathrm{fat}_{\\gamma}(\\cdot)$ is the γ—fat shattering dimension, and k is the size of possible corruptions for each input. $\\tilde{\\mathcal{O}}(\\cdot)$ stands for for omitting poly-logarithmic factors of $d,d^{*},1/\\epsilon,1/\\delta$"}, {"name": "Growth function and VC-dimension (Section 3.1)", "statement": "Growth Function. The growth function $\\Pi_{\\mathcal{H}}: \\mathbb{N}\\mapsto \\mathbb{N}$ for a binary function class $\\mathcal{H}:\\mathcal{X}\\mapsto\\{0,1\\}$ is defined by\n\n$$\n\\forall m\\in\\mathbb{N},\\ \\Pi_{\\mathcal{H}}(m)=\\operatorname*{max}_{\\{x_{1},\\ldots,x_{m}\\}\\subseteq\\mathcal{X}}|\\left\\{(h(x_{1}),\\ldots,h(x_{m})):h\\in\\mathcal{H}\\right\\}|\n$$\n\nAnd the VC-dimension of $\\mathcal{H}$ is defined by\n\n$$\n\\mathrm{VC}(\\mathcal{H})=\\operatorname*{max}\\left\\{m:\\Pi_{\\mathcal{H}}(m)=2^{m}\\right\\}.\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nvariable {𝒳 : Type*}\n\ndef ShattersF (H : Set (𝒳 → Bool)) (S : Finset 𝒳) : Prop :=\n  ∀ b : S → Bool, ∃ h ∈ H, ∀ x : S, h x.val = b x\n\nnoncomputable def VCDimF (H : Set (𝒳 → Bool)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : ShattersF H S), (S.card : ℕ∞)\n\ndef combinedClass {k : ℕ} (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) : Set (𝒳 → Bool) :=\n  { g | ∃ f : Fin k → (𝒳 → Bool), (∀ j, f j ∈ F j) ∧ ∀ x, g x = G (fun j => f j x) }\n\ndef robustLossClass (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳) :\n    Set (𝒳 × Bool → Bool) :=\n  { g | ∃ h ∈ H, ∀ xy : 𝒳 × Bool,\n      g xy = (ρ xy.1).sup (fun z => decide (h z ≠ xy.2)) }\n\ndef GammaShatters (γ : ℝ) (F : Set (𝒳 → ℝ)) (S : Finset 𝒳) : Prop :=\n  ∃ r : 𝒳 → ℝ, ∀ b : 𝒳 → Bool, ∃ f ∈ F, ∀ x ∈ S,\n    (b x = true → f x ≥ r x + γ) ∧ (b x = false → f x ≤ r x - γ)\n\ndef GammaShattersZero (γ : ℝ) (F : Set (𝒳 → ℝ)) (S : Finset 𝒳) : Prop :=\n  ∀ b : 𝒳 → Bool, ∃ f ∈ F, ∀ x ∈ S,\n    (b x = true → f x ≥ γ) ∧ (b x = false → f x ≤ - γ)\n\nnoncomputable def fatDim (γ : ℝ) (F : Set (𝒳 → ℝ)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : GammaShatters γ F S), (S.card : ℕ∞)\n\nnoncomputable def fatDimZero (γ : ℝ) (F : Set (𝒳 → ℝ)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : GammaShattersZero γ F S), (S.card : ℕ∞)\n\ndef shiftClass (F : Set (𝒳 → ℝ)) (r : 𝒳 → ℝ) : Set (𝒳 → ℝ) :=\n  { g | ∃ f ∈ F, g = fun x => f x - r x }\n\ntheorem lemma16 {k : ℕ} (hk : 1 ≤ k) (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) (d : Fin k → ℕ)\n    (hd : ∀ j, VCDimF (F j) = (d j : ℕ∞)) :\n    ∃ D : ℕ, VCDimF (combinedClass G F) = (D : ℕ∞) ∧\n      (D : ℝ) ≤ 2 * (k : ℝ) * Real.logb 2 (3 * k) * ((∑ j, (d j : ℝ)) / k) := by\n  sorry\n\ntheorem lemma1 :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (k d : ℕ) (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳),\n        2 ≤ k → (∀ x, (ρ x).Nonempty) → (∀ x, (ρ x).card ≤ k) → VCDimF H = (d : ℕ∞) →\n          ∃ D : ℕ, VCDimF (robustLossClass H ρ) = (D : ℕ∞) ∧\n            (D : ℝ) ≤ C * (d : ℝ) * Real.log (k : ℝ) := by\n  sorry\n\ntheorem lemma14 {γ : ℝ} (hγ : 0 < γ) (F : Set (𝒳 → ℝ)) :\n    ∃ r₀ : 𝒳 → ℝ, fatDim γ F = fatDimZero γ (shiftClass F r₀) ∧\n      ∀ r : 𝒳 → ℝ, fatDimZero γ (shiftClass F r) ≤ fatDim γ F := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma16", "lemma1", "lemma14"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 1379, "end": 1648, "gold": "{k : ℕ} (hk : 1 ≤ k) (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) (d : Fin k → ℕ)\n    (hd : ∀ j, VCDimF (F j) = (d j : ℕ∞)) :\n    ∃ D : ℕ, VCDimF (combinedClass G F) = (D : ℕ∞) ∧\n      (D : ℝ) ≤ 2 * (k : ℝ) * Real.logb 2 (3 * k) * ((∑ j, (d j : ℝ)) / k)"}]}
{"item_id": "v23-20-1353::lemma1::A-stmt", "paper_id": "v23-20-1353", "target_name": "lemma1", "source": "Research-level", "nl_statement": "Lemma 1 For any class $\\mathcal{H}$ of VC dimension d, and any adversary $\\rho : \\mathcal{X}\\to 2^{\\mathcal{X}}$ such that $|\\rho(x)|\\leq$ $k,$ the VC-dimension of the zero-one robust loss class $L_{\\mathcal{H}}^{\\rho}=\\left\\{(x,y)\\mapsto\\operatorname*{max}_{z\\in\\rho(x)}\\mathbb{I}\\left[h(z)\\neq y\\right]:h\\in\\mathcal{H}\\right\\}$ is at most ${\\mathcal{O}}(d\\log k)$", "context": [{"name": "Model: domain, labels, adversary rho with |rho(x)| <= k (Section 2)", "statement": "There is an unknown distribution D over some domain $\\mathcal{X}$ of uncorrupted examples and a finite domain of corrupted examples $\\mathcal{Z}$ , possibly the same as $\\mathcal{X}$. Our setting is the agnostic PAC-learning framework in a deterministic scenario. The label of each input is uniquely determined by an arbitrary unknown target function $c : \\mathcal{X}\\to \\mathcal{Y}$ . The function c maps each uncorrupted input $x \\in \\mathcal{X}$ to a label $c(x)=y$ , where the set of labels $\\mathcal{Y}$ can be $\\{1,\\ldots,l\\}$ or $\\mathbb{R}$.\n\nThe adversary is able to corrupt an input by mapping an uncorrupted input $x \\in \\mathcal{X}$ to a corrupted one $z \\in{\\mathcal{Z}}$ . There is a mapping $\\rho$ which for every $x \\in \\mathcal{X}$ defines a set $\\rho(x)\\subseteq{\\mathcal{Z}}$ ,such that $|\\rho(x)|\\le k$ The adversary can map an uncorrupted input x to any corrupted input $z\\in\\rho(x)$ . We assume that the learner has an access to $\\rho(\\cdot)$ during the training phase."}, {"name": "Zero-one robust loss, eq. (2) (Section 1.1)", "statement": "Montasser et al. (2019, 2020b) defined a loss function $\\ell:\\mathcal{H}\\times\\mathcal{X}\\times\\mathcal{Y}\\to\\{0,1\\}$ as follows\n\n$$\n\\ell_{2}(h,x,y)=\\operatorname*{max}_{z\\in\\rho(x)}\\mathbb{I}\\left[h(z)\\neq y\\right],\\tag{2}\n$$\n\nwhich we refer to as the zero-one robust loss."}, {"name": "Notation (Section 1.3)", "statement": "Notation. d denotes the VC dimension, $d^{*}$ denote the dual dual-VC dimension, $\\mathrm{fat}_{\\gamma}(\\cdot)$ is the γ—fat shattering dimension, and k is the size of possible corruptions for each input. $\\tilde{\\mathcal{O}}(\\cdot)$ stands for for omitting poly-logarithmic factors of $d,d^{*},1/\\epsilon,1/\\delta$"}, {"name": "Growth function and VC-dimension (Section 3.1)", "statement": "Growth Function. The growth function $\\Pi_{\\mathcal{H}}: \\mathbb{N}\\mapsto \\mathbb{N}$ for a binary function class $\\mathcal{H}:\\mathcal{X}\\mapsto\\{0,1\\}$ is defined by\n\n$$\n\\forall m\\in\\mathbb{N},\\ \\Pi_{\\mathcal{H}}(m)=\\operatorname*{max}_{\\{x_{1},\\ldots,x_{m}\\}\\subseteq\\mathcal{X}}|\\left\\{(h(x_{1}),\\ldots,h(x_{m})):h\\in\\mathcal{H}\\right\\}|\n$$\n\nAnd the VC-dimension of $\\mathcal{H}$ is defined by\n\n$$\n\\mathrm{VC}(\\mathcal{H})=\\operatorname*{max}\\left\\{m:\\Pi_{\\mathcal{H}}(m)=2^{m}\\right\\}.\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nvariable {𝒳 : Type*}\n\ndef ShattersF (H : Set (𝒳 → Bool)) (S : Finset 𝒳) : Prop :=\n  ∀ b : S → Bool, ∃ h ∈ H, ∀ x : S, h x.val = b x\n\nnoncomputable def VCDimF (H : Set (𝒳 → Bool)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : ShattersF H S), (S.card : ℕ∞)\n\ndef combinedClass {k : ℕ} (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) : Set (𝒳 → Bool) :=\n  { g | ∃ f : Fin k → (𝒳 → Bool), (∀ j, f j ∈ F j) ∧ ∀ x, g x = G (fun j => f j x) }\n\ndef robustLossClass (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳) :\n    Set (𝒳 × Bool → Bool) :=\n  { g | ∃ h ∈ H, ∀ xy : 𝒳 × Bool,\n      g xy = (ρ xy.1).sup (fun z => decide (h z ≠ xy.2)) }\n\ndef GammaShatters (γ : ℝ) (F : Set (𝒳 → ℝ)) (S : Finset 𝒳) : Prop :=\n  ∃ r : 𝒳 → ℝ, ∀ b : 𝒳 → Bool, ∃ f ∈ F, ∀ x ∈ S,\n    (b x = true → f x ≥ r x + γ) ∧ (b x = false → f x ≤ r x - γ)\n\ndef GammaShattersZero (γ : ℝ) (F : Set (𝒳 → ℝ)) (S : Finset 𝒳) : Prop :=\n  ∀ b : 𝒳 → Bool, ∃ f ∈ F, ∀ x ∈ S,\n    (b x = true → f x ≥ γ) ∧ (b x = false → f x ≤ - γ)\n\nnoncomputable def fatDim (γ : ℝ) (F : Set (𝒳 → ℝ)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : GammaShatters γ F S), (S.card : ℕ∞)\n\nnoncomputable def fatDimZero (γ : ℝ) (F : Set (𝒳 → ℝ)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : GammaShattersZero γ F S), (S.card : ℕ∞)\n\ndef shiftClass (F : Set (𝒳 → ℝ)) (r : 𝒳 → ℝ) : Set (𝒳 → ℝ) :=\n  { g | ∃ f ∈ F, g = fun x => f x - r x }\n\ntheorem lemma16 {k : ℕ} (hk : 1 ≤ k) (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) (d : Fin k → ℕ)\n    (hd : ∀ j, VCDimF (F j) = (d j : ℕ∞)) :\n    ∃ D : ℕ, VCDimF (combinedClass G F) = (D : ℕ∞) ∧\n      (D : ℝ) ≤ 2 * (k : ℝ) * Real.logb 2 (3 * k) * ((∑ j, (d j : ℝ)) / k) := by\n  sorry\n\ntheorem lemma1 :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (k d : ℕ) (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳),\n        2 ≤ k → (∀ x, (ρ x).Nonempty) → (∀ x, (ρ x).card ≤ k) → VCDimF H = (d : ℕ∞) →\n          ∃ D : ℕ, VCDimF (robustLossClass H ρ) = (D : ℕ∞) ∧\n            (D : ℝ) ≤ C * (d : ℝ) * Real.log (k : ℝ) := by\n  sorry\n\ntheorem lemma14 {γ : ℝ} (hγ : 0 < γ) (F : Set (𝒳 → ℝ)) :\n    ∃ r₀ : 𝒳 → ℝ, fatDim γ F = fatDimZero γ (shiftClass F r₀) ∧\n      ∀ r : 𝒳 → ℝ, fatDimZero γ (shiftClass F r) ≤ fatDim γ F := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma16", "lemma1", "lemma14"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 1679, "end": 1960, "gold": ":\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (k d : ℕ) (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳),\n        2 ≤ k → (∀ x, (ρ x).Nonempty) → (∀ x, (ρ x).card ≤ k) → VCDimF H = (d : ℕ∞) →\n          ∃ D : ℕ, VCDimF (robustLossClass H ρ) = (D : ℕ∞) ∧\n            (D : ℝ) ≤ C * (d : ℝ) * Real.log (k : ℝ)"}]}
{"item_id": "v23-20-1353::lemma14::A-stmt", "paper_id": "v23-20-1353", "target_name": "lemma14", "source": "Research-level", "nl_statement": "Lemma 14 For all $\\mathcal{F}\\subseteq \\mathbb{R}^{\\mathcal{X}}$ and $\\gamma > 0$ ,we have\n\n$$\n\\mathrm{fat}_{\\gamma}(\\mathcal{F})=\\operatorname*{max}_{r\\in\\mathbb{R}^{\\mathcal{X}}}\\mathrm{fat}_{\\gamma}^{0}(\\mathcal{F}-r),\\tag{7}\n$$\n\nwhere ${\\mathcal{F}}- r ~ = ~ \\{f - r : f \\in{\\mathcal{F}}\\}$ is the r-shifted class; in particular, the maximum is always achieved.", "context": [{"name": "Notation (Section 1.3)", "statement": "Notation. d denotes the VC dimension, $d^{*}$ denote the dual dual-VC dimension, $\\mathrm{fat}_{\\gamma}(\\cdot)$ is the γ—fat shattering dimension, and k is the size of possible corruptions for each input. $\\tilde{\\mathcal{O}}(\\cdot)$ stands for for omitting poly-logarithmic factors of $d,d^{*},1/\\epsilon,1/\\delta$"}, {"name": "Fat-shattering dimension and shift (Section 3.1)", "statement": "Fat-Shattering Dimension. For $\\mathcal{F}\\subset \\mathbb{R}^{\\mathcal{X}}$ and $\\gamma ~ > ~ 0$ , we say that $\\mathcal{F}$ $\\gamma$-shatters a set $S \\ =$ $\\{x_{1},\\ldots,x_{m}\\}\\subset\\mathcal{X}$ if there exists an $r=(r_{1},\\ldots,r_{m})\\in\\mathbb{R}^{m}$ such that for each $b\\in\\{-1,1\\}^{m}$ there is a function $f_{b}\\in \\mathcal{F}$ such that\n\n$$\n\\forall i\\in [m]:\\begin{cases}f_{b}(x_{i})\\geq r_{i}+\\gamma&\\text{if } b_{i}=1\\\\ f_{b}(x_{i})\\leq r_{i}-\\gamma&\\text{if } b_{i}= -1\\end{cases}.\n$$\n\nWe refer to r as the shift. The γ-fat-shattering dimension, denoted by $\\mathrm{fat}_{\\gamma}(\\mathcal{F})$ , is the size of the largest γ-shattered set (possibly ∞)."}, {"name": "Shattering at zero, fat^0_gamma (Section 6.1)", "statement": "Before presenting the proof, we introduce some auxiliary notions. We say that $\\mathcal{F}$ “$\\gamma$-shatters a set $S$ at zero” if the shift $r$ is constrained to be 0 in the the usual γ-shattering definition (has appeared previously in Gottlieb et al. (2014)). The analogous dimension will be denoted by $\\mathrm{fat}_{\\gamma}^{0}(\\mathcal{F})$."}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nvariable {𝒳 : Type*}\n\ndef ShattersF (H : Set (𝒳 → Bool)) (S : Finset 𝒳) : Prop :=\n  ∀ b : S → Bool, ∃ h ∈ H, ∀ x : S, h x.val = b x\n\nnoncomputable def VCDimF (H : Set (𝒳 → Bool)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : ShattersF H S), (S.card : ℕ∞)\n\ndef combinedClass {k : ℕ} (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) : Set (𝒳 → Bool) :=\n  { g | ∃ f : Fin k → (𝒳 → Bool), (∀ j, f j ∈ F j) ∧ ∀ x, g x = G (fun j => f j x) }\n\ndef robustLossClass (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳) :\n    Set (𝒳 × Bool → Bool) :=\n  { g | ∃ h ∈ H, ∀ xy : 𝒳 × Bool,\n      g xy = (ρ xy.1).sup (fun z => decide (h z ≠ xy.2)) }\n\ndef GammaShatters (γ : ℝ) (F : Set (𝒳 → ℝ)) (S : Finset 𝒳) : Prop :=\n  ∃ r : 𝒳 → ℝ, ∀ b : 𝒳 → Bool, ∃ f ∈ F, ∀ x ∈ S,\n    (b x = true → f x ≥ r x + γ) ∧ (b x = false → f x ≤ r x - γ)\n\ndef GammaShattersZero (γ : ℝ) (F : Set (𝒳 → ℝ)) (S : Finset 𝒳) : Prop :=\n  ∀ b : 𝒳 → Bool, ∃ f ∈ F, ∀ x ∈ S,\n    (b x = true → f x ≥ γ) ∧ (b x = false → f x ≤ - γ)\n\nnoncomputable def fatDim (γ : ℝ) (F : Set (𝒳 → ℝ)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : GammaShatters γ F S), (S.card : ℕ∞)\n\nnoncomputable def fatDimZero (γ : ℝ) (F : Set (𝒳 → ℝ)) : ℕ∞ :=\n  ⨆ (S : Finset 𝒳) (_ : GammaShattersZero γ F S), (S.card : ℕ∞)\n\ndef shiftClass (F : Set (𝒳 → ℝ)) (r : 𝒳 → ℝ) : Set (𝒳 → ℝ) :=\n  { g | ∃ f ∈ F, g = fun x => f x - r x }\n\ntheorem lemma16 {k : ℕ} (hk : 1 ≤ k) (G : (Fin k → Bool) → Bool)\n    (F : Fin k → Set (𝒳 → Bool)) (d : Fin k → ℕ)\n    (hd : ∀ j, VCDimF (F j) = (d j : ℕ∞)) :\n    ∃ D : ℕ, VCDimF (combinedClass G F) = (D : ℕ∞) ∧\n      (D : ℝ) ≤ 2 * (k : ℝ) * Real.logb 2 (3 * k) * ((∑ j, (d j : ℝ)) / k) := by\n  sorry\n\ntheorem lemma1 :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (k d : ℕ) (H : Set (𝒳 → Bool)) (ρ : 𝒳 → Finset 𝒳),\n        2 ≤ k → (∀ x, (ρ x).Nonempty) → (∀ x, (ρ x).card ≤ k) → VCDimF H = (d : ℕ∞) →\n          ∃ D : ℕ, VCDimF (robustLossClass H ρ) = (D : ℕ∞) ∧\n            (D : ℝ) ≤ C * (d : ℝ) * Real.log (k : ℝ) := by\n  sorry\n\ntheorem lemma14 {γ : ℝ} (hγ : 0 < γ) (F : Set (𝒳 → ℝ)) :\n    ∃ r₀ : 𝒳 → ℝ, fatDim γ F = fatDimZero γ (shiftClass F r₀) ∧\n      ∀ r : 𝒳 → ℝ, fatDimZero γ (shiftClass F r) ≤ fatDim γ F := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma16", "lemma1", "lemma14"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 1992, "end": 2158, "gold": "{γ : ℝ} (hγ : 0 < γ) (F : Set (𝒳 → ℝ)) :\n    ∃ r₀ : 𝒳 → ℝ, fatDim γ F = fatDimZero γ (shiftClass F r₀) ∧\n      ∀ r : 𝒳 → ℝ, fatDimZero γ (shiftClass F r) ≤ fatDim γ F"}]}
{"item_id": "v23-21-0152::asymptotic_bias_one_node::A-stmt", "paper_id": "v23-21-0152", "target_name": "asymptotic_bias_one_node", "source": "Research-level", "nl_statement": "Theorem 1 (Asymptotic Bias of the Posterior Mode of 1-Node BNN+LV) Fix any $W \\in \\mathbb{R}$ and any bounded prior $p_{W}(W)$ on W. Suppose that inputs $\\{x_{1},\\ldots,x_{N}\\}$ are sampled i.i.d from $p_{x}$ with finite first and second moments (mean $\\mu_{x}$ and variance $\\sigma_{x}^{2}$ ) , and that $\\{z_{1},\\dots,z_{N}\\}$ are sampled i.i.d from $\\mathcal{N}(0,\\sigma_{z}^{2})$ , where $\\mu_{x}^{2}+ \\sigma_{x}^{2}> \\sigma_{z}^{2}$ . Suppose $p_{W}(W)>0$ . For every $C$ with $\\frac{\\sigma_{x}^{2}+ \\mu_{x}^{2}- \\sigma_{z}^{2}}{\\sigma_{x}^{2}+ \\mu_{x}^{2}+ \\sigma_{z}^{2}}< C < 1$ such that $p_{W}(\\widehat{W}^{(C)})>0$ , the probability that the scaled values $(\\widehat{W}^{(C)},\\{\\widehat{z}_{n}^{(C)}\\}_{n=1}^{N})$ are more likely than $\\left(W,\\{z_{n}\\}_{n=1}^{N}\\right)$ under the posterior approaches 1 as $N \\to \\infty$", "context": [{"name": "Setup: data, BNN and BNN+LV model, joint posterior (Section 3, Eqs. 1-3)", "statement": "Let $\\mathcal{D}=\\{(x_{1},y_{1}),...(x_{N},y_{N})\\}$ be a data-set of N observations from the true data distribution $p(y|x)p(x)$ , in which each input $\\boldsymbol{x}_{n}\\in \\mathbb{R}^{D}$ is a D-dimensional vector and each output $y_{n}\\in \\mathbb{R}^{L}$ is L-dimensional. Let I denote the identity matrix and let capital letters denote sets of variables, e.g. $X = \\{x_{n}\\}_{n = 1}^{N}, Y = \\{y_{n}\\}_{n = 1}^{N}, Z = \\{z_{n}\\}_{n = 1}^{N}$ . See Appendix A for a summary of the notation.\n\nA Bayesian Neural Network (BNN). A BNN assumes a predictor of the form $y =$ $f(x;W)+\\epsilon$ , where f is a neural network parametrized by W and $\\epsilon\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2}\\cdot I)$ is a normally distributed noise variable. It places a prior p(W ) on the network parameters; given a data-set $\\mathcal{D},$ we can apply Bayesian inference to compute the posterior distribution $p(W|\\mathcal{D})$ over W or the posterior predictive distribution $p(\\boldsymbol{y}^{*}|\\boldsymbol{x}^{*},\\mathcal{D})$ of over outputs $y^{*}$ given a new input $x^{*}$ . As commonly done, we assume $p(W)=\\mathcal{N}(0,\\sigma_{w}^{2}\\cdot I)$\n\nA Bayesian Neural Network with Latent Variables (BNN+LV). A BNN+LV enables more flexible noise distributions for BNNs by introducing a latent variable $z_{n}\\sim\\mathcal{N}(0,\\sigma_{z}^{2}\\cdot I)$ for each observation $(x_{n},y_{n})$ (Depeweg et al., 2018). It assumes the following data generation process (Figure 1):\n\n$$\n\\begin{array}{l}{{W\\sim p(W),\\quad z_{n}\\sim p(z),\\quad\\epsilon_{n}\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2}\\cdot I),}}\\\\ {{y_{n}=f(x_{n},z_{n};W)+\\epsilon_{n},\\quad n=1,\\ldots,N.}}\\end{array}\\tag{1}\n$$\n\nIn this model, z is independent from $x,$ and can represent either white noise or meaningful latent explanatory variables. When $f$ is non-linear, BNN+LV is able to model heteroscedastic noise by transforming $z .$ Inference for BNN+LVs involves approximating the posterior distribution,\n\n$$\np(W,Z|\\mathcal D)\\propto p(W)\\cdot\\prod_{n}p(y_{n}|x_{n},z_{n},W)p(z_{n}),\\tag{2}\n$$\n\n(where $Z=\\{z_{1},\\ldots,z_{N}\\})$ , over both network weights W and the latent input $z_{n}$ for each observation $x_{n}$. When Z represents meaningful latent variables, we may infer the latent information $z_{n}$ for each input $x_{n}$ by marginalizing $p(W,Z|\\mathcal{D})$ over W. For a new input $x^{*}$ the posterior predictive is given by the expected likelihood under the posterior of $W$, and the prior of $z$ (Depeweg et al., 2018):\n\n$$\n\\begin{array}{l}{{p(y^{*}|x^{*},\\mathcal{D})=\\displaystyle\\int p(y^{*}|x^{*},W)p(W|\\mathcal{D})dW}}\\\\ {{\\displaystyle\\qquad=\\int\\int p(y^{*}|x^{*},z^{*},W)p(z^{*})dz^{*}p(W|\\mathcal{D})dW.}}\\end{array}\\tag{3}\n$$"}, {"name": "Notation (Appendix A)", "statement": "<table><tr><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=1>Definition</td></tr><tr><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>Number of observations (or training points)</td></tr><tr><td rowspan=1 colspan=1> $(x_{n},y_{n})$ </td><td rowspan=1 colspan=1>The nth observed input and output, where $x_{n}\\in$  $\\mathbb{R}^{D}, y_{n}\\in \\mathbb{R}^{L}$ </td></tr><tr><td rowspan=1 colspan=1> $(x^{*},y^{*})$ </td><td rowspan=1 colspan=1>A point from the test-set.</td></tr><tr><td rowspan=1 colspan=1> $z_{n}$ </td><td rowspan=1 colspan=1>The latent code corresponding to the nth observation. Since the latent variables are sampled i.i.d from the prior, $p(z_{n})=p(z)$. </td></tr><tr><td rowspan=1 colspan=1> $z^{*}$ </td><td rowspan=1 colspan=1>The latent code corresponding to $x^{*}$ . Again, $p(z^{*})=$  $p(z)$ </td></tr><tr><td rowspan=1 colspan=1> $\\mathcal{D}$ </td><td rowspan=1 colspan=1> $\\{(x_{n},y_{n})\\}_{n=1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $X$ </td><td rowspan=1 colspan=1> $\\{x_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $Y$ </td><td rowspan=1 colspan=1> $\\{y_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $Z$ </td><td rowspan=1 colspan=1> $\\{z_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $p(y|x)p(x)$ or $p(y|x;W^{\\mathrm{true}})p(x)$ </td><td rowspan=1 colspan=1>The ground-truth data-generating distribution (that generated $\\mathcal{D}$).</td></tr><tr><td rowspan=1 colspan=1> $f(\\cdot,\\cdot;W)$ </td><td rowspan=1 colspan=1>A neural network $f$ parameterized by weights $W .$ </td></tr><tr><td rowspan=1 colspan=1> $W$ </td><td rowspan=1 colspan=1>The set of all neural network weights $w_{i}$ </td></tr><tr><td rowspan=1 colspan=1> $p(W,Z|\\mathcal{D})$ </td><td rowspan=1 colspan=1>The BNN+LV joint posterior (Equation 2).</td></tr><tr><td rowspan=1 colspan=1> $p(W|\\mathcal{D})$ </td><td rowspan=1 colspan=1>The marginal posterior of the weights, $\\begin{array}{r}{\\int_{Z}p(W,Z|\\mathcal{D})dZ}\\end{array}$ (intractable).</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(Z,W|\\mathcal D)=q_{\\phi}(Z|\\mathcal D)q_{\\phi}(W|\\mathcal D)$ </td><td rowspan=1 colspan=1>The mean-field variational approximation of the joint posterior (Equation 4), parameterized by φ.</td></tr><tr><td rowspan=1 colspan=1> $Z^{\\mathrm{MAP}},W^{\\mathrm{MAP}}$ </td><td rowspan=1 colspan=1>The MAP of the true joint posterior: $Z^{\\mathrm{MAP}},W^{\\mathrm{MAP}}=$  $\\operatorname{argmax}_{Z,W}p(Z,W|\\mathcal{D})$ </td></tr><tr><td rowspan=1 colspan=1> $Z^{\\mathrm{true}},W^{\\mathrm{true}}$ </td><td rowspan=1 colspan=1>The ground-truth data-generating weights $W^{\\mathrm{true}}$ and latent codes $Z^{\\mathrm{true}}=\\{z_{n}^{\\mathrm{true}}\\}_{n=1}^{N}$ that produced the observed data.</td></tr><tr><td rowspan=1 colspan=1> $\\widehat{Z}, \\widehat{W}$ </td><td rowspan=1 colspan=1>The alternative set of latent variables $\\widehat{Z}\\ = \\ \\{\\widehat{z}_{n}\\}_{n = 1}^{N}$ and weights, defined in Equation 8, that have a higher probability under the joint posterior.</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(z)$ </td><td rowspan=1 colspan=1>The aggregated posterior (Equation 11).</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(z|x)$ </td><td rowspan=1 colspan=1>The approximate posterior marginalized over y: $\\begin{array}{r}{\\int_{y}q_{\\phi}(z|x,y)p(y|x)dy}\\end{array}$ . Since we only observe one y for every x, we cannot obtain unbiased estimates of this distribution in practice.</td></tr></table>"}, {"name": "Section 4 preamble: well-specified model", "statement": "For intuition and clarity, we begin by characterizing likelihood non-identifiability and posterior bias for a single node of a BNN+LV and then generalize these results to a 1-layer $\\mathrm{B N N + L V}$ . We assume the model is well-specified."}, {"name": "1-node non-identifiability: W/C and (C-1)x+Cz (Section 4.1)", "statement": "Non-Identifiability. Consider univariate output generated by a single hidden-node neural network with LeakyReLU activation. For simplicity, we study a case with zero network biases, unit output weights, and additive input noise:\n\n$$\nf(x,z;W)=\\operatorname*{max}\\left\\{W(x+z),\\alpha W(x+z)\\right\\},\n$$\n\nwhere $0 < \\alpha < 1$ . For any non-zero constant $C,$ the pair $\\widehat W^{(C)}=W/C,\\widehat z^{(C)}=(C-1)x+Cz$ reconstructs the observed data equally well:\n\n$$\n\\operatorname*{max}\\left\\{W(x+z),\\alpha W(x+z)\\right\\}=\\operatorname*{max}\\left\\{\\widehat{W}^{(C)}\\left(x+\\widehat{z}^{(C)}\\right),\\alpha\\widehat{W}^{(C)}\\left(x+\\widehat{z}^{(C)}\\right)\\right\\}.\n$$\n\nNow, suppose that the output is observed with Gaussian noise: $\\displaystyle y\\sim\\mathcal{N}(f(x,z;W),\\sigma_{\\epsilon}^{2})$ . Then the true values of the parameter W and the latent input noise z are equally likely as $\\widehat{W}^{(C)}$ and $\\widehat{z}^{(C)}$ under the likelihood: $p(y|f(x,z,W),\\sigma_{\\epsilon}^{2})=p(y|f(x,\\widehat{z}^{(C)},\\widehat{W}^{(C)}),\\sigma_{\\epsilon}^{2})$ . We show in Theorem 1 that for this model, the posterior over the model parameter and latent inputs W, Z is biased away from the ground-truth towards parameters that generalize poorly as the sample size grows, regardless of the choice of W prior."}, {"name": "1-node model (Appendix B.1, Eq. 15)", "statement": "Consider univariate output generated by a single hidden-node neural network with LeakyReLU activation:\n\n$$\n\\begin{array}{l}{z\\sim\\mathcal{N}(0,\\sigma_{z}^{2})}\\\\ {\\epsilon\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2})}\\\\ {y=\\operatorname*{max}\\left\\{W(x+z),\\alpha W(x+z)\\right\\}+\\epsilon}\\end{array}\\tag{15}\n$$\n\nwhere α is a fixed constant in (0, 1)."}, {"name": "Generative modeling assumptions 1-2 (Section 4.3)", "statement": "Generally, when assuming a probabilistic model, we hope that in the limit of infinite data, inference recovers model parameters that satisfy our generative modeling assumptions. For example, when assuming a BNN, we hope that in the limit of infinite data, the MLE and MAP over the weights parameterize the true function that generated the data. For the BNN+LV model, however, the parameters given by the MAP of the joint posterior $W^{\\mathrm{MAP}},Z^{\\mathrm{MAP}}$ do not satisfy our generative modeling assumptions. To illustrate this, we compare $W^{\\mathrm{MAP}},Z^{\\mathrm{MAP}}$ with the ground-truth data generating parameters, $W^{\\mathrm{true}},Z^{\\mathrm{true}}$ ; under our ground-truth data-generating process from Equation 1, $Z^{\\mathrm{true}}$ satisfies two modeling assumptions:\n\nAssumption 1: $z^{\\mathrm{true}}$ is independent of x—that is, $p(z,x)$ factorizes.\n\nAssumption 2: $z^{\\mathrm{true}}$ is distributed like the prior $p(z)$"}], "nl_notes": "### Theorem 1\nNOTE (Theorem 1) — The paper assumes only a bounded prior and asserts that, for some non-zero 𝒞, the conclusion holds for every C ∈ (𝒞, 1). That fails when the prior vanishes at the scaled parameter Ŵ^(C) = W/C: if p_W(Ŵ^(C)) = 0 < p_W(W), the scaled values have posterior 0 (for instance, a uniform prior on (W − 1, W] with W > 0 gives p_W(Ŵ^(C)) = 0 for every C ∈ (0, 1)). The statement therefore assumes p_W(W) > 0 and restricts to C with p_W(Ŵ^(C)) > 0.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Filter Topology\n\nnamespace ResearchLevel\n\nnoncomputable def logPriorRatioZ (σz C x z : ℝ) : ℝ :=\n  (z ^ 2 - ((C - 1) * x + C * z) ^ 2) / (2 * σz ^ 2)\n\nnoncomputable def Dseq1 {Ω : Type*} (Δw σz C : ℝ) (X Z : ℕ → Ω → ℝ)\n    (N : ℕ) (ω : Ω) : ℝ :=\n  Δw + ∑ n ∈ Finset.range N, logPriorRatioZ σz C (X n ω) (Z n ω)\n\nnoncomputable def logPriorRatioZ2 {D : ℕ} (σz : ℝ) (t x z : Fin D → ℝ) : ℝ :=\n  (∑ d, ((z d) ^ 2 - (t d * (z d - x d)) ^ 2)) / (2 * σz ^ 2)\n\nnoncomputable def Dseq2 {Ω : Type*} {D : ℕ} (Δw σz : ℝ) (t : Fin D → ℝ)\n    (X Z : ℕ → Ω → (Fin D → ℝ)) (N : ℕ) (ω : Ω) : ℝ :=\n  Δw + ∑ n ∈ Finset.range N, logPriorRatioZ2 σz t (X n ω) (Z n ω)\n\ntheorem asymptotic_bias_one_node\n    {Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    (X Z : ℕ → Ω → ℝ)\n    (μx σx σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint1 : Integrable (X 0) P) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) P)\n    (hμx : ∫ ω, X 0 ω ∂P = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂P = σx ^ 2 + μx ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZgauss : ∀ n, Measure.map (Z n) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n\n    (Δw : ℝ)\n\n    (W : ℝ) (pW : ℝ → ℝ) (hpWbdd : ∃ M, ∀ w, pW w ≤ M) (hpWposW : 0 < pW W)\n    (C : ℝ)\n    (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n    (hChi : C < 1)\n    (hpWposWC : 0 < pW (W / C)) (hΔw : Δw = Real.log (pW (W / C)) - Real.log (pW W)) :\n    Tendsto (fun N => P {ω | 0 < Dseq1 Δw σz C X Z N ω}) atTop (𝓝 1) := by\n  sorry\n\ntheorem asymptotic_bias_one_layer\n    {Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {D : ℕ} (hD : 0 < D)\n    (X Z : ℕ → Ω → (Fin D → ℝ))\n    (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint : ∀ d, Integrable (fun ω => X 0 ω d) P ∧\n                  Integrable (fun ω => (X 0 ω d) ^ 2) P)\n    (hμx : ∀ d, ∫ ω, X 0 ω d ∂P = μx d)\n    (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂P = (σx d) ^ 2 + (μx d) ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) P)\n    (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (pW : (Fin D → ℝ) → ℝ) (hpWbdd2 : ∃ M, ∀ v, pW v ≤ M) (W : Fin D → ℝ) (hpWposW : 0 < pW W) :\n    ∃ (t : Fin D → ℝ) (Ŵ : Fin D → ℝ),\n      (∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))) ∧\n      0 < pW Ŵ ∧\n      Tendsto (fun N => P {ω | 0 < Dseq2 (Real.log (pW Ŵ) - Real.log (pW W)) σz t X Z N ω}) atTop (𝓝 1) := by\n  sorry\n\nnoncomputable def RavgZ1 {Ω : Type*} (σz C : ℝ) (X Z : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ :=\n  (1 / (N : ℝ)) * ∑ n ∈ Finset.range N, logPriorRatioZ σz C (X n ω) (Z n ω)\n\nnoncomputable def RavgZ2 {Ω : Type*} {D : ℕ} (σz : ℝ) (t : Fin D → ℝ)\n    (X Z : ℕ → Ω → (Fin D → ℝ)) (N : ℕ) (ω : Ω) : ℝ :=\n  (1 / (N : ℝ)) * ∑ n ∈ Finset.range N, logPriorRatioZ2 σz t (X n ω) (Z n ω)\n\ntheorem bias_concentration_cantelli\n    {Ω : Type*} [MeasureSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] :\n    (∀ (X Z : ℕ → Ω → ℝ)\n      (μx σx σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint1 : Integrable (X 0) μ) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) μ)\n      (hμx : ∫ ω, X 0 ω ∂μ = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂μ = σx ^ 2 + μx ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZgauss : ∀ n, Measure.map (Z n) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n      (C : ℝ)\n      (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n      (hChi : C < 1),\n      Tendsto (fun N => μ {ω | 0 < RavgZ1 σz C X Z N ω}) atTop (𝓝 1)) ∧\n    (∀ {D : ℕ} (hD : 0 < D)\n      (X Z : ℕ → Ω → (Fin D → ℝ))\n      (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint : ∀ d, Integrable (fun ω => X 0 ω d) μ ∧\n                    Integrable (fun ω => (X 0 ω d) ^ 2) μ)\n      (hμx : ∀ d, ∫ ω, X 0 ω d ∂μ = μx d)\n      (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂μ = (σx d) ^ 2 + (μx d) ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) μ)\n      (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (t : Fin D → ℝ)\n      (ht : ∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))),\n      Tendsto (fun N => μ {ω | 0 < RavgZ2 σz t X Z N ω}) atTop (𝓝 1)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["asymptotic_bias_one_node", "asymptotic_bias_one_layer", "bias_concentration_cantelli"], "topic_domain": "Deep Learning Theory", "topic_subtopic": "Bayesian Neural Networks", "holes": [{"start": 737, "end": 1662, "gold": "{Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    (X Z : ℕ → Ω → ℝ)\n    (μx σx σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint1 : Integrable (X 0) P) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) P)\n    (hμx : ∫ ω, X 0 ω ∂P = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂P = σx ^ 2 + μx ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZgauss : ∀ n, Measure.map (Z n) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n\n    (Δw : ℝ)\n\n    (W : ℝ) (pW : ℝ → ℝ) (hpWbdd : ∃ M, ∀ w, pW w ≤ M) (hpWposW : 0 < pW W)\n    (C : ℝ)\n    (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n    (hChi : C < 1)\n    (hpWposWC : 0 < pW (W / C)) (hΔw : Δw = Real.log (pW (W / C)) - Real.log (pW W)) :\n    Tendsto (fun N => P {ω | 0 < Dseq1 Δw σz C X Z N ω}) atTop (𝓝 1)"}]}
{"item_id": "v23-21-0152::asymptotic_bias_one_layer::A-stmt", "paper_id": "v23-21-0152", "target_name": "asymptotic_bias_one_layer", "source": "Research-level", "nl_statement": "Theorem 2 (Asymptotic Bias of the Posterior Mode of 1-Layer BNN+LV) Fix any set of parameters W and any bounded prior $p_{W}(W)$ on W with $p_{W}(W)>0$ . Suppose that $\\{x_{1},\\ldots,x_{N}\\}$ is sampled i.i.d from $p_{x}$ , with finite first and second moments (the dth marginal having mean $\\mu_{x_{d}}$ and variance $\\sigma_{x_{d}}^{2}$ ), and that $\\{z_{1},\\dots,z_{N}\\}$ are sampled i.i.d from $\\mathcal{N}(0,\\sigma_{z}^{2}\\cdot I)$ . There exists an alternate set of parameters $\\left(\\widehat{W},\\{\\widehat{z}_{n}\\}_{n=1}^{N}\\right)$ with $p_{W}(\\widehat{W})>0$ and $\\widehat{z}_{n}= T z_{n}- T x_{n}$ , where $T=\\mathrm{diag}(t)$ with $0 < | t_{d}| < \\sqrt{\\frac{\\sigma_{z}^{2}}{\\mu_{x_{d}}^{2}+ \\sigma_{x_{d}}^{2}+ \\sigma_{z}^{2}}}$ for every $d$ , such that the probability that these alternative parameters are valued as more likely than $\\left(W,\\{z_{n}\\}_{n=1}^{N}\\right)$ under the posterior approaches 1 as $N \\to \\infty$", "context": [{"name": "Setup: data, BNN and BNN+LV model, joint posterior (Section 3, Eqs. 1-3)", "statement": "Let $\\mathcal{D}=\\{(x_{1},y_{1}),...(x_{N},y_{N})\\}$ be a data-set of N observations from the true data distribution $p(y|x)p(x)$ , in which each input $\\boldsymbol{x}_{n}\\in \\mathbb{R}^{D}$ is a D-dimensional vector and each output $y_{n}\\in \\mathbb{R}^{L}$ is L-dimensional. Let I denote the identity matrix and let capital letters denote sets of variables, e.g. $X = \\{x_{n}\\}_{n = 1}^{N}, Y = \\{y_{n}\\}_{n = 1}^{N}, Z = \\{z_{n}\\}_{n = 1}^{N}$ . See Appendix A for a summary of the notation.\n\nA Bayesian Neural Network (BNN). A BNN assumes a predictor of the form $y =$ $f(x;W)+\\epsilon$ , where f is a neural network parametrized by W and $\\epsilon\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2}\\cdot I)$ is a normally distributed noise variable. It places a prior p(W ) on the network parameters; given a data-set $\\mathcal{D},$ we can apply Bayesian inference to compute the posterior distribution $p(W|\\mathcal{D})$ over W or the posterior predictive distribution $p(\\boldsymbol{y}^{*}|\\boldsymbol{x}^{*},\\mathcal{D})$ of over outputs $y^{*}$ given a new input $x^{*}$ . As commonly done, we assume $p(W)=\\mathcal{N}(0,\\sigma_{w}^{2}\\cdot I)$\n\nA Bayesian Neural Network with Latent Variables (BNN+LV). A BNN+LV enables more flexible noise distributions for BNNs by introducing a latent variable $z_{n}\\sim\\mathcal{N}(0,\\sigma_{z}^{2}\\cdot I)$ for each observation $(x_{n},y_{n})$ (Depeweg et al., 2018). It assumes the following data generation process (Figure 1):\n\n$$\n\\begin{array}{l}{{W\\sim p(W),\\quad z_{n}\\sim p(z),\\quad\\epsilon_{n}\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2}\\cdot I),}}\\\\ {{y_{n}=f(x_{n},z_{n};W)+\\epsilon_{n},\\quad n=1,\\ldots,N.}}\\end{array}\\tag{1}\n$$\n\nIn this model, z is independent from $x,$ and can represent either white noise or meaningful latent explanatory variables. When $f$ is non-linear, BNN+LV is able to model heteroscedastic noise by transforming $z .$ Inference for BNN+LVs involves approximating the posterior distribution,\n\n$$\np(W,Z|\\mathcal D)\\propto p(W)\\cdot\\prod_{n}p(y_{n}|x_{n},z_{n},W)p(z_{n}),\\tag{2}\n$$\n\n(where $Z=\\{z_{1},\\ldots,z_{N}\\})$ , over both network weights W and the latent input $z_{n}$ for each observation $x_{n}$. When Z represents meaningful latent variables, we may infer the latent information $z_{n}$ for each input $x_{n}$ by marginalizing $p(W,Z|\\mathcal{D})$ over W. For a new input $x^{*}$ the posterior predictive is given by the expected likelihood under the posterior of $W$, and the prior of $z$ (Depeweg et al., 2018):\n\n$$\n\\begin{array}{l}{{p(y^{*}|x^{*},\\mathcal{D})=\\displaystyle\\int p(y^{*}|x^{*},W)p(W|\\mathcal{D})dW}}\\\\ {{\\displaystyle\\qquad=\\int\\int p(y^{*}|x^{*},z^{*},W)p(z^{*})dz^{*}p(W|\\mathcal{D})dW.}}\\end{array}\\tag{3}\n$$"}, {"name": "Notation (Appendix A)", "statement": "<table><tr><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=1>Definition</td></tr><tr><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>Number of observations (or training points)</td></tr><tr><td rowspan=1 colspan=1> $(x_{n},y_{n})$ </td><td rowspan=1 colspan=1>The nth observed input and output, where $x_{n}\\in$  $\\mathbb{R}^{D}, y_{n}\\in \\mathbb{R}^{L}$ </td></tr><tr><td rowspan=1 colspan=1> $(x^{*},y^{*})$ </td><td rowspan=1 colspan=1>A point from the test-set.</td></tr><tr><td rowspan=1 colspan=1> $z_{n}$ </td><td rowspan=1 colspan=1>The latent code corresponding to the nth observation. Since the latent variables are sampled i.i.d from the prior, $p(z_{n})=p(z)$. </td></tr><tr><td rowspan=1 colspan=1> $z^{*}$ </td><td rowspan=1 colspan=1>The latent code corresponding to $x^{*}$ . Again, $p(z^{*})=$  $p(z)$ </td></tr><tr><td rowspan=1 colspan=1> $\\mathcal{D}$ </td><td rowspan=1 colspan=1> $\\{(x_{n},y_{n})\\}_{n=1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $X$ </td><td rowspan=1 colspan=1> $\\{x_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $Y$ </td><td rowspan=1 colspan=1> $\\{y_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $Z$ </td><td rowspan=1 colspan=1> $\\{z_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $p(y|x)p(x)$ or $p(y|x;W^{\\mathrm{true}})p(x)$ </td><td rowspan=1 colspan=1>The ground-truth data-generating distribution (that generated $\\mathcal{D}$).</td></tr><tr><td rowspan=1 colspan=1> $f(\\cdot,\\cdot;W)$ </td><td rowspan=1 colspan=1>A neural network $f$ parameterized by weights $W .$ </td></tr><tr><td rowspan=1 colspan=1> $W$ </td><td rowspan=1 colspan=1>The set of all neural network weights $w_{i}$ </td></tr><tr><td rowspan=1 colspan=1> $p(W,Z|\\mathcal{D})$ </td><td rowspan=1 colspan=1>The BNN+LV joint posterior (Equation 2).</td></tr><tr><td rowspan=1 colspan=1> $p(W|\\mathcal{D})$ </td><td rowspan=1 colspan=1>The marginal posterior of the weights, $\\begin{array}{r}{\\int_{Z}p(W,Z|\\mathcal{D})dZ}\\end{array}$ (intractable).</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(Z,W|\\mathcal D)=q_{\\phi}(Z|\\mathcal D)q_{\\phi}(W|\\mathcal D)$ </td><td rowspan=1 colspan=1>The mean-field variational approximation of the joint posterior (Equation 4), parameterized by φ.</td></tr><tr><td rowspan=1 colspan=1> $Z^{\\mathrm{MAP}},W^{\\mathrm{MAP}}$ </td><td rowspan=1 colspan=1>The MAP of the true joint posterior: $Z^{\\mathrm{MAP}},W^{\\mathrm{MAP}}=$  $\\operatorname{argmax}_{Z,W}p(Z,W|\\mathcal{D})$ </td></tr><tr><td rowspan=1 colspan=1> $Z^{\\mathrm{true}},W^{\\mathrm{true}}$ </td><td rowspan=1 colspan=1>The ground-truth data-generating weights $W^{\\mathrm{true}}$ and latent codes $Z^{\\mathrm{true}}=\\{z_{n}^{\\mathrm{true}}\\}_{n=1}^{N}$ that produced the observed data.</td></tr><tr><td rowspan=1 colspan=1> $\\widehat{Z}, \\widehat{W}$ </td><td rowspan=1 colspan=1>The alternative set of latent variables $\\widehat{Z}\\ = \\ \\{\\widehat{z}_{n}\\}_{n = 1}^{N}$ and weights, defined in Equation 8, that have a higher probability under the joint posterior.</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(z)$ </td><td rowspan=1 colspan=1>The aggregated posterior (Equation 11).</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(z|x)$ </td><td rowspan=1 colspan=1>The approximate posterior marginalized over y: $\\begin{array}{r}{\\int_{y}q_{\\phi}(z|x,y)p(y|x)dy}\\end{array}$ . Since we only observe one y for every x, we cannot obtain unbiased estimates of this distribution in practice.</td></tr></table>"}, {"name": "Section 4 preamble: well-specified model", "statement": "For intuition and clarity, we begin by characterizing likelihood non-identifiability and posterior bias for a single node of a BNN+LV and then generalize these results to a 1-layer $\\mathrm{B N N + L V}$ . We assume the model is well-specified."}, {"name": "1-layer non-identifiability (Section 4.2, Eq. 8)", "statement": "Non-Identifiability. Sources of non-identifiability increase when f is a neural network. Consider a single-output neural network, f, that takes as input x and z (represented as a concatenated vector in $\\mathbb{R}^{2 D})$ and has a single hidden layer containing H hidden nodes. At the output node, we fix the activation to be the identity. Thus, the activation of the output node is computed as\n\n$$\na^{\\mathrm{out}}=(a^{\\mathrm{hidden}})^{\\top}W^{\\mathrm{out}}+b^{\\mathrm{out}},\n$$\n\nwhere $W^{\\mathrm{out}}$ is a H-dimensional weight vector, $b^{\\mathrm{out}}$ is the bias and $a^{\\mathrm{hidden}}$ is the H-dimensional vector of activations of the hidden nodes. We can further expand $a^{\\mathrm{hidden}}$ as\n\n$$\na^{\\mathrm{hidden}}=g\\left(W^{x}x+W^{z}z+b^{\\mathrm{hidden}}\\right),\n$$\n\nwhere $W^{x}$ and $W^{z}$ are weight matrices in $\\mathbb{R}^{H \\times D}$ , $b^{\\mathrm{hidden}}$ is a H-dimensional bias vector and $g$ is the activation function, applied element-wise. We characterize ways that the models parameters and the latent input variables z are non-identifiable given a set of observed data. For any choice of diagonal matrix $S \\in \\mathbb{R}^{D \\times D}$ , vector $U \\in \\mathbb{R}^{D}$ , and any factorization $W^{z}= R T$ where T is in $\\mathbb{R}^{D \\times D}$ , we can express $a^{\\mathrm{hidden}}$ in two equivalent ways:\n\n$$\na^{\\mathrm{hidden}}=g\\left(W^{x}x+W^{z}z+b^{\\mathrm{hidden}}\\right)=g\\left(\\widehat{W}^{x}x+\\widehat{W}^{z}\\widehat{z}+\\widehat{b}^{\\mathrm{hidden}}\\right)\n$$\n\nby setting:\n\n$$\n\\widehat{W}^{x}=W^{x}+W^{z}S,\\widehat{W}^{z}=R,\\widehat{z}=T z-T S x-U,\\widehat{b}^{\\mathrm{hidden}}=b+R U .\\tag{8}\n$$\n\nSuppose that the output is observed with Gaussian noise: $\\displaystyle y\\sim\\mathcal{N}(f(x,z;W),\\sigma_{\\epsilon}^{2})$ , where W denotes the set of weights $(W^{\\mathrm{out}},W^{x},W^{z},b^{\\mathrm{out}},b^{\\mathrm{hidden}})$ . Then, by only observing outputs generated by this network given observed input, one cannot identify ground-truth parameter and latent variable values under the likelihood: $p(y|f(x,z,W),\\sigma_{\\epsilon}^{2})=p(y|f(x,\\widehat{z},\\widehat{W}),\\sigma_{\\epsilon}^{2})$ . Just like in the case of a network with a single node, we show in Theorem 2 that the joint posterior is biased away from the ground-truth parameters, regardless of the choice of weight prior."}, {"name": "1-layer model (Appendix B.2, Eq. 16)", "statement": "Let W be the set of network weights $\\{W^{x},W^{z},W^{\\mathrm{out}},b^{\\mathrm{hidden}},b^{\\mathrm{out}}\\}$ . Consider a data-set generated from the following model:\n\n$$\n\\begin{array}{r}{\\begin{array}{c}{W\\sim p(W)}\\\\ {z\\sim\\mathcal{N}(0,\\sigma_{z}^{2}\\cdot I)}\\\\ {\\epsilon\\sim N(0,\\sigma_{\\epsilon}^{2})}\\\\ {a^{\\mathrm{hidden}}=g\\left(W^{x}x+W^{z}z+b^{\\mathrm{hidden}}\\right),}\\\\ {y=(a^{\\mathrm{hidden}})^{\\top}W^{\\mathrm{out}}+b^{\\mathrm{out}}+\\epsilon}\\end{array}}\\end{array}\\tag{16}\n$$"}, {"name": "Generative modeling assumptions 1-2 (Section 4.3)", "statement": "Generally, when assuming a probabilistic model, we hope that in the limit of infinite data, inference recovers model parameters that satisfy our generative modeling assumptions. For example, when assuming a BNN, we hope that in the limit of infinite data, the MLE and MAP over the weights parameterize the true function that generated the data. For the BNN+LV model, however, the parameters given by the MAP of the joint posterior $W^{\\mathrm{MAP}},Z^{\\mathrm{MAP}}$ do not satisfy our generative modeling assumptions. To illustrate this, we compare $W^{\\mathrm{MAP}},Z^{\\mathrm{MAP}}$ with the ground-truth data generating parameters, $W^{\\mathrm{true}},Z^{\\mathrm{true}}$ ; under our ground-truth data-generating process from Equation 1, $Z^{\\mathrm{true}}$ satisfies two modeling assumptions:\n\nAssumption 1: $z^{\\mathrm{true}}$ is independent of x—that is, $p(z,x)$ factorizes.\n\nAssumption 2: $z^{\\mathrm{true}}$ is distributed like the prior $p(z)$"}], "nl_notes": "### Theorem 2\nNOTE (Theorem 2) — Do not model the network of Eq. 16 or the likelihood. Take the proof's likelihood step as given, and compare the two parameter sets by their log-priors only, through `Dseq2`. Treat the weights W as a single vector in ℝ^D. The alternate weights Ŵ are another such vector. Ŵ enters only through the offset Δw = log p_W(Ŵ) − log p_W(W), and is not tied to t.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Filter Topology\n\nnamespace ResearchLevel\n\nnoncomputable def logPriorRatioZ (σz C x z : ℝ) : ℝ :=\n  (z ^ 2 - ((C - 1) * x + C * z) ^ 2) / (2 * σz ^ 2)\n\nnoncomputable def Dseq1 {Ω : Type*} (Δw σz C : ℝ) (X Z : ℕ → Ω → ℝ)\n    (N : ℕ) (ω : Ω) : ℝ :=\n  Δw + ∑ n ∈ Finset.range N, logPriorRatioZ σz C (X n ω) (Z n ω)\n\nnoncomputable def logPriorRatioZ2 {D : ℕ} (σz : ℝ) (t x z : Fin D → ℝ) : ℝ :=\n  (∑ d, ((z d) ^ 2 - (t d * (z d - x d)) ^ 2)) / (2 * σz ^ 2)\n\nnoncomputable def Dseq2 {Ω : Type*} {D : ℕ} (Δw σz : ℝ) (t : Fin D → ℝ)\n    (X Z : ℕ → Ω → (Fin D → ℝ)) (N : ℕ) (ω : Ω) : ℝ :=\n  Δw + ∑ n ∈ Finset.range N, logPriorRatioZ2 σz t (X n ω) (Z n ω)\n\ntheorem asymptotic_bias_one_node\n    {Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    (X Z : ℕ → Ω → ℝ)\n    (μx σx σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint1 : Integrable (X 0) P) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) P)\n    (hμx : ∫ ω, X 0 ω ∂P = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂P = σx ^ 2 + μx ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZgauss : ∀ n, Measure.map (Z n) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n\n    (Δw : ℝ)\n\n    (W : ℝ) (pW : ℝ → ℝ) (hpWbdd : ∃ M, ∀ w, pW w ≤ M) (hpWposW : 0 < pW W)\n    (C : ℝ)\n    (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n    (hChi : C < 1)\n    (hpWposWC : 0 < pW (W / C)) (hΔw : Δw = Real.log (pW (W / C)) - Real.log (pW W)) :\n    Tendsto (fun N => P {ω | 0 < Dseq1 Δw σz C X Z N ω}) atTop (𝓝 1) := by\n  sorry\n\ntheorem asymptotic_bias_one_layer\n    {Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {D : ℕ} (hD : 0 < D)\n    (X Z : ℕ → Ω → (Fin D → ℝ))\n    (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint : ∀ d, Integrable (fun ω => X 0 ω d) P ∧\n                  Integrable (fun ω => (X 0 ω d) ^ 2) P)\n    (hμx : ∀ d, ∫ ω, X 0 ω d ∂P = μx d)\n    (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂P = (σx d) ^ 2 + (μx d) ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) P)\n    (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (pW : (Fin D → ℝ) → ℝ) (hpWbdd2 : ∃ M, ∀ v, pW v ≤ M) (W : Fin D → ℝ) (hpWposW : 0 < pW W) :\n    ∃ (t : Fin D → ℝ) (Ŵ : Fin D → ℝ),\n      (∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))) ∧\n      0 < pW Ŵ ∧\n      Tendsto (fun N => P {ω | 0 < Dseq2 (Real.log (pW Ŵ) - Real.log (pW W)) σz t X Z N ω}) atTop (𝓝 1) := by\n  sorry\n\nnoncomputable def RavgZ1 {Ω : Type*} (σz C : ℝ) (X Z : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ :=\n  (1 / (N : ℝ)) * ∑ n ∈ Finset.range N, logPriorRatioZ σz C (X n ω) (Z n ω)\n\nnoncomputable def RavgZ2 {Ω : Type*} {D : ℕ} (σz : ℝ) (t : Fin D → ℝ)\n    (X Z : ℕ → Ω → (Fin D → ℝ)) (N : ℕ) (ω : Ω) : ℝ :=\n  (1 / (N : ℝ)) * ∑ n ∈ Finset.range N, logPriorRatioZ2 σz t (X n ω) (Z n ω)\n\ntheorem bias_concentration_cantelli\n    {Ω : Type*} [MeasureSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] :\n    (∀ (X Z : ℕ → Ω → ℝ)\n      (μx σx σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint1 : Integrable (X 0) μ) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) μ)\n      (hμx : ∫ ω, X 0 ω ∂μ = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂μ = σx ^ 2 + μx ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZgauss : ∀ n, Measure.map (Z n) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n      (C : ℝ)\n      (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n      (hChi : C < 1),\n      Tendsto (fun N => μ {ω | 0 < RavgZ1 σz C X Z N ω}) atTop (𝓝 1)) ∧\n    (∀ {D : ℕ} (hD : 0 < D)\n      (X Z : ℕ → Ω → (Fin D → ℝ))\n      (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint : ∀ d, Integrable (fun ω => X 0 ω d) μ ∧\n                    Integrable (fun ω => (X 0 ω d) ^ 2) μ)\n      (hμx : ∀ d, ∫ ω, X 0 ω d ∂μ = μx d)\n      (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂μ = (σx d) ^ 2 + (μx d) ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) μ)\n      (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (t : Fin D → ℝ)\n      (ht : ∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))),\n      Tendsto (fun N => μ {ω | 0 < RavgZ2 σz t X Z N ω}) atTop (𝓝 1)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["asymptotic_bias_one_node", "asymptotic_bias_one_layer", "bias_concentration_cantelli"], "topic_domain": "Deep Learning Theory", "topic_subtopic": "Bayesian Neural Networks", "holes": [{"start": 1716, "end": 2773, "gold": "{Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {D : ℕ} (hD : 0 < D)\n    (X Z : ℕ → Ω → (Fin D → ℝ))\n    (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint : ∀ d, Integrable (fun ω => X 0 ω d) P ∧\n                  Integrable (fun ω => (X 0 ω d) ^ 2) P)\n    (hμx : ∀ d, ∫ ω, X 0 ω d ∂P = μx d)\n    (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂P = (σx d) ^ 2 + (μx d) ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) P)\n    (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (pW : (Fin D → ℝ) → ℝ) (hpWbdd2 : ∃ M, ∀ v, pW v ≤ M) (W : Fin D → ℝ) (hpWposW : 0 < pW W) :\n    ∃ (t : Fin D → ℝ) (Ŵ : Fin D → ℝ),\n      (∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))) ∧\n      0 < pW Ŵ ∧\n      Tendsto (fun N => P {ω | 0 < Dseq2 (Real.log (pW Ŵ) - Real.log (pW W)) σz t X Z N ω}) atTop (𝓝 1)"}]}
{"item_id": "v23-21-0152::bias_concentration_cantelli::A-stmt", "paper_id": "v23-21-0152", "target_name": "bias_concentration_cantelli", "source": "Research-level", "nl_statement": "Concentration crux of Theorems 1 and 2 (Appendix B.1/B.2, the step where Cantelli's inequality is applied), for the concrete per-sample log-prior-ratio average $R_N$. Throughout, $p_z$ is the Gaussian latent prior and $z$ is independent of $x$. (Theorem 1, 1-node model, Appendix B.1.) Suppose that inputs $x_1, x_2, \\ldots$ are sampled i.i.d. from $p_x$ with finite first and second moments $\\mu_x, \\sigma_x^2$, and that $z_1, z_2, \\ldots$ are sampled i.i.d. from $\\mathcal{N}(0, \\sigma_z^2)$, where $\\mu_x^2 + \\sigma_x^2 > \\sigma_z^2$. Let $C$ satisfy $0 < \\frac{\\sigma_x^2 + \\mu_x^2 - \\sigma_z^2}{\\sigma_x^2 + \\mu_x^2 + \\sigma_z^2} < C < 1$, let $\\widehat{z}_n^{(C)} = (C-1)x_n + C z_n$, and let $R_N^{(C)} = \\frac{1}{N} \\sum_{n=1}^{N} \\big(\\log p_z(\\widehat{z}_n^{(C)}) - \\log p_z(z_n)\\big)$. Then $\\Pr[R_N^{(C)} > 0] \\to 1$ as $N \\to \\infty$. (Theorem 2, 1-layer model, Appendix B.2.) Suppose that $x_1, x_2, \\ldots \\in \\mathbb{R}^D$ are sampled i.i.d. from $p_x$ with finite first and second moments, the $d$th marginal of $x$ having mean $\\mu_{x_d}$ and variance $\\sigma_{x_d}^2$, and that $z_1, z_2, \\ldots$ are sampled i.i.d. from $\\mathcal{N}(0, \\sigma_z^2 \\cdot I)$. Let $T = \\mathrm{diag}(t)$ with $0 < |t_d| < \\sqrt{\\frac{\\sigma_z^2}{\\mu_{x_d}^2 + \\sigma_{x_d}^2 + \\sigma_z^2}}$ for every $d$, let $\\widehat{z}_n = T z_n - T x_n$, and let $R_N = \\frac{1}{N} \\sum_{n=1}^{N} \\big(\\log p_z(\\widehat{z}_n) - \\log p_z(z_n)\\big)$. Then $\\Pr[R_N > 0] \\to 1$ as $N \\to \\infty$. (In the paper, the condition on $C$, resp. on $t_d$, is what makes $\\mathbb{E}[R_N] > 0$, after which Cantelli's inequality gives $\\Pr[R_N < 0] \\to 0$.)", "context": [{"name": "Setup: data, BNN and BNN+LV model, joint posterior (Section 3, Eqs. 1-3)", "statement": "Let $\\mathcal{D}=\\{(x_{1},y_{1}),...(x_{N},y_{N})\\}$ be a data-set of N observations from the true data distribution $p(y|x)p(x)$ , in which each input $\\boldsymbol{x}_{n}\\in \\mathbb{R}^{D}$ is a D-dimensional vector and each output $y_{n}\\in \\mathbb{R}^{L}$ is L-dimensional. Let I denote the identity matrix and let capital letters denote sets of variables, e.g. $X = \\{x_{n}\\}_{n = 1}^{N}, Y = \\{y_{n}\\}_{n = 1}^{N}, Z = \\{z_{n}\\}_{n = 1}^{N}$ . See Appendix A for a summary of the notation.\n\nA Bayesian Neural Network (BNN). A BNN assumes a predictor of the form $y =$ $f(x;W)+\\epsilon$ , where f is a neural network parametrized by W and $\\epsilon\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2}\\cdot I)$ is a normally distributed noise variable. It places a prior p(W ) on the network parameters; given a data-set $\\mathcal{D},$ we can apply Bayesian inference to compute the posterior distribution $p(W|\\mathcal{D})$ over W or the posterior predictive distribution $p(\\boldsymbol{y}^{*}|\\boldsymbol{x}^{*},\\mathcal{D})$ of over outputs $y^{*}$ given a new input $x^{*}$ . As commonly done, we assume $p(W)=\\mathcal{N}(0,\\sigma_{w}^{2}\\cdot I)$\n\nA Bayesian Neural Network with Latent Variables (BNN+LV). A BNN+LV enables more flexible noise distributions for BNNs by introducing a latent variable $z_{n}\\sim\\mathcal{N}(0,\\sigma_{z}^{2}\\cdot I)$ for each observation $(x_{n},y_{n})$ (Depeweg et al., 2018). It assumes the following data generation process (Figure 1):\n\n$$\n\\begin{array}{l}{{W\\sim p(W),\\quad z_{n}\\sim p(z),\\quad\\epsilon_{n}\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2}\\cdot I),}}\\\\ {{y_{n}=f(x_{n},z_{n};W)+\\epsilon_{n},\\quad n=1,\\ldots,N.}}\\end{array}\\tag{1}\n$$\n\nIn this model, z is independent from $x,$ and can represent either white noise or meaningful latent explanatory variables. When $f$ is non-linear, BNN+LV is able to model heteroscedastic noise by transforming $z .$ Inference for BNN+LVs involves approximating the posterior distribution,\n\n$$\np(W,Z|\\mathcal D)\\propto p(W)\\cdot\\prod_{n}p(y_{n}|x_{n},z_{n},W)p(z_{n}),\\tag{2}\n$$\n\n(where $Z=\\{z_{1},\\ldots,z_{N}\\})$ , over both network weights W and the latent input $z_{n}$ for each observation $x_{n}$. When Z represents meaningful latent variables, we may infer the latent information $z_{n}$ for each input $x_{n}$ by marginalizing $p(W,Z|\\mathcal{D})$ over W. For a new input $x^{*}$ the posterior predictive is given by the expected likelihood under the posterior of $W$, and the prior of $z$ (Depeweg et al., 2018):\n\n$$\n\\begin{array}{l}{{p(y^{*}|x^{*},\\mathcal{D})=\\displaystyle\\int p(y^{*}|x^{*},W)p(W|\\mathcal{D})dW}}\\\\ {{\\displaystyle\\qquad=\\int\\int p(y^{*}|x^{*},z^{*},W)p(z^{*})dz^{*}p(W|\\mathcal{D})dW.}}\\end{array}\\tag{3}\n$$"}, {"name": "Notation (Appendix A)", "statement": "<table><tr><td rowspan=1 colspan=1>Symbol</td><td rowspan=1 colspan=1>Definition</td></tr><tr><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>Number of observations (or training points)</td></tr><tr><td rowspan=1 colspan=1> $(x_{n},y_{n})$ </td><td rowspan=1 colspan=1>The nth observed input and output, where $x_{n}\\in$  $\\mathbb{R}^{D}, y_{n}\\in \\mathbb{R}^{L}$ </td></tr><tr><td rowspan=1 colspan=1> $(x^{*},y^{*})$ </td><td rowspan=1 colspan=1>A point from the test-set.</td></tr><tr><td rowspan=1 colspan=1> $z_{n}$ </td><td rowspan=1 colspan=1>The latent code corresponding to the nth observation. Since the latent variables are sampled i.i.d from the prior, $p(z_{n})=p(z)$. </td></tr><tr><td rowspan=1 colspan=1> $z^{*}$ </td><td rowspan=1 colspan=1>The latent code corresponding to $x^{*}$ . Again, $p(z^{*})=$  $p(z)$ </td></tr><tr><td rowspan=1 colspan=1> $\\mathcal{D}$ </td><td rowspan=1 colspan=1> $\\{(x_{n},y_{n})\\}_{n=1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $X$ </td><td rowspan=1 colspan=1> $\\{x_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $Y$ </td><td rowspan=1 colspan=1> $\\{y_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $Z$ </td><td rowspan=1 colspan=1> $\\{z_{n}\\}_{n = 1}^{N}$ </td></tr><tr><td rowspan=1 colspan=1> $p(y|x)p(x)$ or $p(y|x;W^{\\mathrm{true}})p(x)$ </td><td rowspan=1 colspan=1>The ground-truth data-generating distribution (that generated $\\mathcal{D}$).</td></tr><tr><td rowspan=1 colspan=1> $f(\\cdot,\\cdot;W)$ </td><td rowspan=1 colspan=1>A neural network $f$ parameterized by weights $W .$ </td></tr><tr><td rowspan=1 colspan=1> $W$ </td><td rowspan=1 colspan=1>The set of all neural network weights $w_{i}$ </td></tr><tr><td rowspan=1 colspan=1> $p(W,Z|\\mathcal{D})$ </td><td rowspan=1 colspan=1>The BNN+LV joint posterior (Equation 2).</td></tr><tr><td rowspan=1 colspan=1> $p(W|\\mathcal{D})$ </td><td rowspan=1 colspan=1>The marginal posterior of the weights, $\\begin{array}{r}{\\int_{Z}p(W,Z|\\mathcal{D})dZ}\\end{array}$ (intractable).</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(Z,W|\\mathcal D)=q_{\\phi}(Z|\\mathcal D)q_{\\phi}(W|\\mathcal D)$ </td><td rowspan=1 colspan=1>The mean-field variational approximation of the joint posterior (Equation 4), parameterized by φ.</td></tr><tr><td rowspan=1 colspan=1> $Z^{\\mathrm{MAP}},W^{\\mathrm{MAP}}$ </td><td rowspan=1 colspan=1>The MAP of the true joint posterior: $Z^{\\mathrm{MAP}},W^{\\mathrm{MAP}}=$  $\\operatorname{argmax}_{Z,W}p(Z,W|\\mathcal{D})$ </td></tr><tr><td rowspan=1 colspan=1> $Z^{\\mathrm{true}},W^{\\mathrm{true}}$ </td><td rowspan=1 colspan=1>The ground-truth data-generating weights $W^{\\mathrm{true}}$ and latent codes $Z^{\\mathrm{true}}=\\{z_{n}^{\\mathrm{true}}\\}_{n=1}^{N}$ that produced the observed data.</td></tr><tr><td rowspan=1 colspan=1> $\\widehat{Z}, \\widehat{W}$ </td><td rowspan=1 colspan=1>The alternative set of latent variables $\\widehat{Z}\\ = \\ \\{\\widehat{z}_{n}\\}_{n = 1}^{N}$ and weights, defined in Equation 8, that have a higher probability under the joint posterior.</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(z)$ </td><td rowspan=1 colspan=1>The aggregated posterior (Equation 11).</td></tr><tr><td rowspan=1 colspan=1> $q_{\\phi}(z|x)$ </td><td rowspan=1 colspan=1>The approximate posterior marginalized over y: $\\begin{array}{r}{\\int_{y}q_{\\phi}(z|x,y)p(y|x)dy}\\end{array}$ . Since we only observe one y for every x, we cannot obtain unbiased estimates of this distribution in practice.</td></tr></table>"}, {"name": "Section 4 preamble: well-specified model", "statement": "For intuition and clarity, we begin by characterizing likelihood non-identifiability and posterior bias for a single node of a BNN+LV and then generalize these results to a 1-layer $\\mathrm{B N N + L V}$ . We assume the model is well-specified."}, {"name": "1-node non-identifiability: W/C and (C-1)x+Cz (Section 4.1)", "statement": "Non-Identifiability. Consider univariate output generated by a single hidden-node neural network with LeakyReLU activation. For simplicity, we study a case with zero network biases, unit output weights, and additive input noise:\n\n$$\nf(x,z;W)=\\operatorname*{max}\\left\\{W(x+z),\\alpha W(x+z)\\right\\},\n$$\n\nwhere $0 < \\alpha < 1$ . For any non-zero constant $C,$ the pair $\\widehat W^{(C)}=W/C,\\widehat z^{(C)}=(C-1)x+Cz$ reconstructs the observed data equally well:\n\n$$\n\\operatorname*{max}\\left\\{W(x+z),\\alpha W(x+z)\\right\\}=\\operatorname*{max}\\left\\{\\widehat{W}^{(C)}\\left(x+\\widehat{z}^{(C)}\\right),\\alpha\\widehat{W}^{(C)}\\left(x+\\widehat{z}^{(C)}\\right)\\right\\}.\n$$\n\nNow, suppose that the output is observed with Gaussian noise: $\\displaystyle y\\sim\\mathcal{N}(f(x,z;W),\\sigma_{\\epsilon}^{2})$ . Then the true values of the parameter W and the latent input noise z are equally likely as $\\widehat{W}^{(C)}$ and $\\widehat{z}^{(C)}$ under the likelihood: $p(y|f(x,z,W),\\sigma_{\\epsilon}^{2})=p(y|f(x,\\widehat{z}^{(C)},\\widehat{W}^{(C)}),\\sigma_{\\epsilon}^{2})$ . We show in Theorem 1 that for this model, the posterior over the model parameter and latent inputs W, Z is biased away from the ground-truth towards parameters that generalize poorly as the sample size grows, regardless of the choice of W prior."}, {"name": "1-node model (Appendix B.1, Eq. 15)", "statement": "Consider univariate output generated by a single hidden-node neural network with LeakyReLU activation:\n\n$$\n\\begin{array}{l}{z\\sim\\mathcal{N}(0,\\sigma_{z}^{2})}\\\\ {\\epsilon\\sim\\mathcal{N}(0,\\sigma_{\\epsilon}^{2})}\\\\ {y=\\operatorname*{max}\\left\\{W(x+z),\\alpha W(x+z)\\right\\}+\\epsilon}\\end{array}\\tag{15}\n$$\n\nwhere α is a fixed constant in (0, 1)."}, {"name": "1-layer non-identifiability (Section 4.2, Eq. 8)", "statement": "Non-Identifiability. Sources of non-identifiability increase when f is a neural network. Consider a single-output neural network, f, that takes as input x and z (represented as a concatenated vector in $\\mathbb{R}^{2 D})$ and has a single hidden layer containing H hidden nodes. At the output node, we fix the activation to be the identity. Thus, the activation of the output node is computed as\n\n$$\na^{\\mathrm{out}}=(a^{\\mathrm{hidden}})^{\\top}W^{\\mathrm{out}}+b^{\\mathrm{out}},\n$$\n\nwhere $W^{\\mathrm{out}}$ is a H-dimensional weight vector, $b^{\\mathrm{out}}$ is the bias and $a^{\\mathrm{hidden}}$ is the H-dimensional vector of activations of the hidden nodes. We can further expand $a^{\\mathrm{hidden}}$ as\n\n$$\na^{\\mathrm{hidden}}=g\\left(W^{x}x+W^{z}z+b^{\\mathrm{hidden}}\\right),\n$$\n\nwhere $W^{x}$ and $W^{z}$ are weight matrices in $\\mathbb{R}^{H \\times D}$ , $b^{\\mathrm{hidden}}$ is a H-dimensional bias vector and $g$ is the activation function, applied element-wise. We characterize ways that the models parameters and the latent input variables z are non-identifiable given a set of observed data. For any choice of diagonal matrix $S \\in \\mathbb{R}^{D \\times D}$ , vector $U \\in \\mathbb{R}^{D}$ , and any factorization $W^{z}= R T$ where T is in $\\mathbb{R}^{D \\times D}$ , we can express $a^{\\mathrm{hidden}}$ in two equivalent ways:\n\n$$\na^{\\mathrm{hidden}}=g\\left(W^{x}x+W^{z}z+b^{\\mathrm{hidden}}\\right)=g\\left(\\widehat{W}^{x}x+\\widehat{W}^{z}\\widehat{z}+\\widehat{b}^{\\mathrm{hidden}}\\right)\n$$\n\nby setting:\n\n$$\n\\widehat{W}^{x}=W^{x}+W^{z}S,\\widehat{W}^{z}=R,\\widehat{z}=T z-T S x-U,\\widehat{b}^{\\mathrm{hidden}}=b+R U .\\tag{8}\n$$\n\nSuppose that the output is observed with Gaussian noise: $\\displaystyle y\\sim\\mathcal{N}(f(x,z;W),\\sigma_{\\epsilon}^{2})$ , where W denotes the set of weights $(W^{\\mathrm{out}},W^{x},W^{z},b^{\\mathrm{out}},b^{\\mathrm{hidden}})$ . Then, by only observing outputs generated by this network given observed input, one cannot identify ground-truth parameter and latent variable values under the likelihood: $p(y|f(x,z,W),\\sigma_{\\epsilon}^{2})=p(y|f(x,\\widehat{z},\\widehat{W}),\\sigma_{\\epsilon}^{2})$ . Just like in the case of a network with a single node, we show in Theorem 2 that the joint posterior is biased away from the ground-truth parameters, regardless of the choice of weight prior."}, {"name": "1-layer model (Appendix B.2, Eq. 16)", "statement": "Let W be the set of network weights $\\{W^{x},W^{z},W^{\\mathrm{out}},b^{\\mathrm{hidden}},b^{\\mathrm{out}}\\}$ . Consider a data-set generated from the following model:\n\n$$\n\\begin{array}{r}{\\begin{array}{c}{W\\sim p(W)}\\\\ {z\\sim\\mathcal{N}(0,\\sigma_{z}^{2}\\cdot I)}\\\\ {\\epsilon\\sim N(0,\\sigma_{\\epsilon}^{2})}\\\\ {a^{\\mathrm{hidden}}=g\\left(W^{x}x+W^{z}z+b^{\\mathrm{hidden}}\\right),}\\\\ {y=(a^{\\mathrm{hidden}})^{\\top}W^{\\mathrm{out}}+b^{\\mathrm{out}}+\\epsilon}\\end{array}}\\end{array}\\tag{16}\n$$"}, {"name": "Generative modeling assumptions 1-2 (Section 4.3)", "statement": "Generally, when assuming a probabilistic model, we hope that in the limit of infinite data, inference recovers model parameters that satisfy our generative modeling assumptions. For example, when assuming a BNN, we hope that in the limit of infinite data, the MLE and MAP over the weights parameterize the true function that generated the data. For the BNN+LV model, however, the parameters given by the MAP of the joint posterior $W^{\\mathrm{MAP}},Z^{\\mathrm{MAP}}$ do not satisfy our generative modeling assumptions. To illustrate this, we compare $W^{\\mathrm{MAP}},Z^{\\mathrm{MAP}}$ with the ground-truth data generating parameters, $W^{\\mathrm{true}},Z^{\\mathrm{true}}$ ; under our ground-truth data-generating process from Equation 1, $Z^{\\mathrm{true}}$ satisfies two modeling assumptions:\n\nAssumption 1: $z^{\\mathrm{true}}$ is independent of x—that is, $p(z,x)$ factorizes.\n\nAssumption 2: $z^{\\mathrm{true}}$ is distributed like the prior $p(z)$"}, {"name": "From the proof of Theorem 1 (the priors p_W, p_z and the quantity D_N^(C))", "statement": "We assume the model in Equation 15. We denote the prior on $W$ as $p_{W}$ and suppose that it is bounded, we denote the prior on $z$ as $p_{z}$ , and we suppose that $p_{x}$, the distribution over the input $x$, has bounded first and second moments $\\mu_{x}$ and $\\sigma_{x}^{2}.$ For any non-zero constant $0 < C < 1$ , we define\n\n$$\n\\widehat W^{(C)}=W/C,~\\widehat z_{n}^{(C)}=(C-1)x_{n}+Cz_{n},\n$$\n\nand we define $D_{N}^{(C)}$ to be the difference between the scaled parameters $(\\widehat{W}^{(C)},\\widehat{z}_{1}^{(C)},\\dots\\widehat{z}_{N}^{(C)})$ and the ground-truth parameters $(W,z_{1},\\dots z_{N})$ under the joint posterior log $p(W,Z|\\mathcal{D})$\n\nFor any set of parameters $(W,z_{1},\\dots z_{N})$ , we show that we can find alternate parameters $(\\widehat{W}^{(C)},\\widehat{z}_{1}^{(C)},\\ldots\\widehat{z}_{N}^{(C)})$ , that are scored as more likely under the log-posterior log $p(W,\\{z_{n}\\}|\\mathcal{D})$ To do this, we first show that the alternative parameters are scored as equally likely under the log-likelihood, and then we show that the alternate parameters are scored as more likely under the log-prior.\n\nSince we have that\n\n$$\n\\operatorname*{max}\\left\\{W^{\\mathrm{true}}(x+z_{n}^{\\mathrm{true}}),\\alpha W^{\\mathrm{true}}(x+z_{n}^{\\mathrm{true}})\\right\\}=\\operatorname*{max}\\left\\{\\widehat{W}^{(C)}(x+\\widehat{z}_{n}^{(C)}),\\alpha\\widehat{W}^{(C)}(x+\\widehat{z}_{n}^{(C)})\\right\\},\n$$\n\nwe see that the alternate parameters are as likely as the original parameters under the likelihood; that is,\n\n$$\n\\prod_{n}p(y_{n}|x_{n},z_{n}^{\\mathrm{true}},W^{\\mathrm{true}})=\\prod_{n}p\\left(y_{n}|x_{n},\\widehat{z}_{n}^{(C)},\\widehat{W}^{(C)}\\right).\n$$\n\nSince the likelihood under both sets of parameters is equal, the difference between the log-posterior, $D_{N}^{(C)}$ , is simply the difference of the two sets of parameters under the log-prior:\n\n$$\nD_{N}^{(C)}=\\log p_{W}(\\widehat{W}^{(C)})-\\log p_{W}(W)+\\sum_{n=1}^{N}\\log p_{z}(\\widehat{z}_{n}^{(C)})-\\log p_{z}(z_{n}).\n$$"}], "nl_notes": "### Concentration crux (Cantelli step, Appendix B.1/B.2)\nNOTE (Concentration crux) — The paper bounds Pr[R_N < 0] with Cantelli's inequality and V[R_N] = V[R_1]/N, which needs E[x⁴] < ∞, but Theorems 1–2 assume only finite first and second moments. The Lean carries exactly those moment hypotheses (`hXint1`/`hXint2`, resp. `hXint`, with `hμx`, `hσx`) and no fourth-moment or finite-variance hypothesis, so the Cantelli route does not apply as written. The claim still holds by the strong law of large numbers, since E[R_1] > 0 under `hClo`/`hChi` (resp. `ht`). The goal is Pr[R_N > 0] → 1, the form the paper sets as its aim; it agrees with Pr[R_N < 0] → 0 because R_1 has an atomless law.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Filter Topology\n\nnamespace ResearchLevel\n\nnoncomputable def logPriorRatioZ (σz C x z : ℝ) : ℝ :=\n  (z ^ 2 - ((C - 1) * x + C * z) ^ 2) / (2 * σz ^ 2)\n\nnoncomputable def Dseq1 {Ω : Type*} (Δw σz C : ℝ) (X Z : ℕ → Ω → ℝ)\n    (N : ℕ) (ω : Ω) : ℝ :=\n  Δw + ∑ n ∈ Finset.range N, logPriorRatioZ σz C (X n ω) (Z n ω)\n\nnoncomputable def logPriorRatioZ2 {D : ℕ} (σz : ℝ) (t x z : Fin D → ℝ) : ℝ :=\n  (∑ d, ((z d) ^ 2 - (t d * (z d - x d)) ^ 2)) / (2 * σz ^ 2)\n\nnoncomputable def Dseq2 {Ω : Type*} {D : ℕ} (Δw σz : ℝ) (t : Fin D → ℝ)\n    (X Z : ℕ → Ω → (Fin D → ℝ)) (N : ℕ) (ω : Ω) : ℝ :=\n  Δw + ∑ n ∈ Finset.range N, logPriorRatioZ2 σz t (X n ω) (Z n ω)\n\ntheorem asymptotic_bias_one_node\n    {Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    (X Z : ℕ → Ω → ℝ)\n    (μx σx σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint1 : Integrable (X 0) P) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) P)\n    (hμx : ∫ ω, X 0 ω ∂P = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂P = σx ^ 2 + μx ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZgauss : ∀ n, Measure.map (Z n) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n\n    (Δw : ℝ)\n\n    (W : ℝ) (pW : ℝ → ℝ) (hpWbdd : ∃ M, ∀ w, pW w ≤ M) (hpWposW : 0 < pW W)\n    (C : ℝ)\n    (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n    (hChi : C < 1)\n    (hpWposWC : 0 < pW (W / C)) (hΔw : Δw = Real.log (pW (W / C)) - Real.log (pW W)) :\n    Tendsto (fun N => P {ω | 0 < Dseq1 Δw σz C X Z N ω}) atTop (𝓝 1) := by\n  sorry\n\ntheorem asymptotic_bias_one_layer\n    {Ω : Type*} [MeasureSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {D : ℕ} (hD : 0 < D)\n    (X Z : ℕ → Ω → (Fin D → ℝ))\n    (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n    (hXindep : iIndepFun X P) (hXid : ∀ n, IdentDistrib (X n) (X 0) P P)\n\n    (hXint : ∀ d, Integrable (fun ω => X 0 ω d) P ∧\n                  Integrable (fun ω => (X 0 ω d) ^ 2) P)\n    (hμx : ∀ d, ∫ ω, X 0 ω d ∂P = μx d)\n    (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂P = (σx d) ^ 2 + (μx d) ^ 2)\n\n    (hZindep : iIndepFun Z P)\n    (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) P)\n    (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) P = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n    (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) P)\n\n    (pW : (Fin D → ℝ) → ℝ) (hpWbdd2 : ∃ M, ∀ v, pW v ≤ M) (W : Fin D → ℝ) (hpWposW : 0 < pW W) :\n    ∃ (t : Fin D → ℝ) (Ŵ : Fin D → ℝ),\n      (∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))) ∧\n      0 < pW Ŵ ∧\n      Tendsto (fun N => P {ω | 0 < Dseq2 (Real.log (pW Ŵ) - Real.log (pW W)) σz t X Z N ω}) atTop (𝓝 1) := by\n  sorry\n\nnoncomputable def RavgZ1 {Ω : Type*} (σz C : ℝ) (X Z : ℕ → Ω → ℝ) (N : ℕ) (ω : Ω) : ℝ :=\n  (1 / (N : ℝ)) * ∑ n ∈ Finset.range N, logPriorRatioZ σz C (X n ω) (Z n ω)\n\nnoncomputable def RavgZ2 {Ω : Type*} {D : ℕ} (σz : ℝ) (t : Fin D → ℝ)\n    (X Z : ℕ → Ω → (Fin D → ℝ)) (N : ℕ) (ω : Ω) : ℝ :=\n  (1 / (N : ℝ)) * ∑ n ∈ Finset.range N, logPriorRatioZ2 σz t (X n ω) (Z n ω)\n\ntheorem bias_concentration_cantelli\n    {Ω : Type*} [MeasureSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] :\n    (∀ (X Z : ℕ → Ω → ℝ)\n      (μx σx σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint1 : Integrable (X 0) μ) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) μ)\n      (hμx : ∫ ω, X 0 ω ∂μ = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂μ = σx ^ 2 + μx ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZgauss : ∀ n, Measure.map (Z n) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n      (C : ℝ)\n      (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n      (hChi : C < 1),\n      Tendsto (fun N => μ {ω | 0 < RavgZ1 σz C X Z N ω}) atTop (𝓝 1)) ∧\n    (∀ {D : ℕ} (hD : 0 < D)\n      (X Z : ℕ → Ω → (Fin D → ℝ))\n      (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint : ∀ d, Integrable (fun ω => X 0 ω d) μ ∧\n                    Integrable (fun ω => (X 0 ω d) ^ 2) μ)\n      (hμx : ∀ d, ∫ ω, X 0 ω d ∂μ = μx d)\n      (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂μ = (σx d) ^ 2 + (μx d) ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) μ)\n      (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (t : Fin D → ℝ)\n      (ht : ∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))),\n      Tendsto (fun N => μ {ω | 0 < RavgZ2 σz t X Z N ω}) atTop (𝓝 1)) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["asymptotic_bias_one_node", "asymptotic_bias_one_layer", "bias_concentration_cantelli"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 3198, "end": 4828, "gold": "{Ω : Type*} [MeasureSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] :\n    (∀ (X Z : ℕ → Ω → ℝ)\n      (μx σx σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint1 : Integrable (X 0) μ) (hXint2 : Integrable (fun ω => (X 0 ω) ^ 2) μ)\n      (hμx : ∫ ω, X 0 ω ∂μ = μx) (hσx : ∫ ω, (X 0 ω) ^ 2 ∂μ = σx ^ 2 + μx ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZgauss : ∀ n, Measure.map (Z n) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (hcond : σz ^ 2 < μx ^ 2 + σx ^ 2)\n      (C : ℝ)\n      (hClo : (σx ^ 2 + μx ^ 2 - σz ^ 2) / (σx ^ 2 + μx ^ 2 + σz ^ 2) < C)\n      (hChi : C < 1),\n      Tendsto (fun N => μ {ω | 0 < RavgZ1 σz C X Z N ω}) atTop (𝓝 1)) ∧\n    (∀ {D : ℕ} (hD : 0 < D)\n      (X Z : ℕ → Ω → (Fin D → ℝ))\n      (μx σx : Fin D → ℝ) (σz : ℝ) (hσz : 0 < σz)\n\n      (hXindep : iIndepFun X μ) (hXid : ∀ n, IdentDistrib (X n) (X 0) μ μ)\n\n      (hXint : ∀ d, Integrable (fun ω => X 0 ω d) μ ∧\n                    Integrable (fun ω => (X 0 ω d) ^ 2) μ)\n      (hμx : ∀ d, ∫ ω, X 0 ω d ∂μ = μx d)\n      (hσx : ∀ d, ∫ ω, (X 0 ω d) ^ 2 ∂μ = (σx d) ^ 2 + (μx d) ^ 2)\n\n      (hZindep : iIndepFun Z μ)\n      (hZcoord : ∀ n, iIndepFun (fun d ω => Z n ω d) μ)\n      (hZgauss : ∀ n d, Measure.map (fun ω => Z n ω d) μ = gaussianReal 0 (Real.toNNReal (σz ^ 2)))\n\n      (hXZ : IndepFun (fun ω n => X n ω) (fun ω n => Z n ω) μ)\n\n      (t : Fin D → ℝ)\n      (ht : ∀ d, 0 < |t d| ∧ |t d| < Real.sqrt (σz ^ 2 / ((μx d) ^ 2 + (σx d) ^ 2 + σz ^ 2))),\n      Tendsto (fun N => μ {ω | 0 < RavgZ2 σz t X Z N ω}) atTop (𝓝 1))"}]}
{"item_id": "v24-22-1104::theorem1::A-stmt", "paper_id": "v24-22-1104", "target_name": "theorem1", "source": "Research-level", "nl_statement": "Theorem 1 (Classification-calibration) Let L be the LHS loss. For any x such that $p(\\mathbf{x})\\in$ (0, 1) with $p(\\mathbf{x})\\neq 1/2$, a global minimizer of $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))\\mid\\mathbf{X}=\\mathbf{x}]$ uniquely exists and is\n\n$$\n\\bar{f}({\\bf x})=\\left\\{\\begin{array}{l l}{-(1-p({\\bf x}))(p({\\bf x}))^{-1},\\quad}&{ifp({\\bf x})<1/2,}\\\\ {+p({\\bf x})(1-p({\\bf x}))^{-1},\\quad}&{ifp({\\bf x})>1/2.}\\end{array}\\right.\n$$\n\nAlso, sgn $\\{{\\bar{f}}({\\bf x}) \\}= f^{*}({\\bf x})$ , where $f^{*}(\\mathbf{x})=\\mathrm{sgn}\\{p(\\mathbf{x})-1/2\\}$ is the Bayes rule.", "context": [{"name": "Setup: binary classification, p(x), Bayes rule (Section 1)", "statement": "This paper concerns binary classification, where the task is to predict an unobserved binary output value $y\\in\\{-1,1\\}$ based on an observed input vector $\\mathbf{x}\\in \\mathbb{R}^{p}$ . The classifier is a mapping from the input space $\\mathcal{X}$ to {−1, 1} via a classification function $\\hat{f}$, and the predicted y value is $\\mathrm{sgn}\\{\\hat{f}(\\mathbf{x})\\}$ . The decision boundary is the set $\\{\\mathbf{x}:{\\hat{f}}(\\mathbf{x})=0\\}$ . Suppose that data are generated from some underlying distribution $\\mathrm{pr}(\\mathbf{X},\\mathbf{Y})$ , and let $p(\\mathbf{x})=\\mathrm{pr}(\\mathbf{Y}=1\\mid$ $\\mathbf{X}= \\mathbf{x})$ Under the standard 0-1 loss, the optimal classification rule is sgn $[p(\\mathbf{x})-1/2]$ (a.k.a. Bayes rule). Throughout the paper, we assume that $p(\\mathbf{X})\\neq 1/2$ almost surely. The optimal decision boundary is $\\{\\mathbf{x}:p(\\mathbf{x})=1/2\\}$ Given training data, $\\{({\\bf x}_{i},y_{i}),i=$ $1,\\ldots,n\\}$ , one aims to develop a classifier that mimics the Bayes rule as closely as possible."}, {"name": "LHS loss, eq. (1)", "statement": "In this paper, we provide an affirmative answer and the new function called the leaky hockey stick loss (LHS loss). The expression of the LHS loss is given in (1) and the picture of this loss function is displayed in Figure 1. We use the word 'hockey stick' because our loss function resembles it and is continuously differentiable. We use the term 'leaky' because the right side diverges to negative infinity motivated by the name of leaky ReLU.\n\n$$\nL(yf)=\\left\\{{\\begin{array}{l l}{-\\log yf,}&{yf>1,}\\\\ {1-yf,}&{yf\\leq 1,}\\end{array}}\\right.\\tag{1}\n$$"}, {"name": "Classification-calibration and Fisher consistency (Section 2)", "statement": "Lin (2004) proposed Fisher consistency as a necessary condition on a loss function. It is defined to be that any global minimizer $\\check{f}$ (if it exists) of $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))]$ , generally referred to as a population minimizer, has the same sign function as the Bayes rule almost surely. Later, Bartlett et al. (2006) defined classification-calibration. A loss function L is called classification-calibrated if any global minimizer $\\bar{f}$ (if it exists) of $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))\\mid\\mathbf{X}=\\mathbf{x}]$ has the same sign as the Bayes rule almost surely. For example, the hinge loss is classification-calibrated, which can be directly shown by Theorem 2 of Bartlett et al. (2006).\n\nAn unspoken assumption in those two definitions is the existence of a global minimizer. When $\\bar{f}$ exists (which can be easily checked) and $|\\mathrm{E}[L({\\bf Y}\\bar{f}({\\bf X}))]| <\\infty .$ then a population minimizer $\\check{f}$ exists. We further see that loss function L is classification-calibrated if and only if it is Fisher consistent. The 'if' statement follows from the fact that any $\\bar{f}$ is also a population minimizer and the definition of Fisher consistency. The 'only if' statement comes from the fact that $\\operatorname{E}[L(\\mathbf{Y}{\\bar{f}}(\\mathbf{X}))\\mid\\mathbf{X}]=\\operatorname{E}[L(\\mathbf{Y}{\\check{f}}(\\mathbf{X}))\\mid\\mathbf{X}]$ almost surely and that for any $\\delta_{\\mathbf{X}}$ such that $\\operatorname{sgn}(\\delta_{\\mathbf{X}})\\neq\\operatorname{sgn}(p(\\mathbf{X})-1/2),\\operatorname{E}[L(\\mathbf{Y}{\\bar{f}}(\\mathbf{X}))\\mid\\mathbf{X}]<\\operatorname{E}[L(\\mathbf{Y}\\delta_{\\mathbf{X}})\\mid\\mathbf{X}]$ almost surely by the definition of classification-calibration.\n\nFor a non-negative loss function L, $\\operatorname{E}[L(\\mathbf{Y}\\bar{f}(\\mathbf{X}))]$ is always finite. However, the LHS loss does not guarantee the finiteness of $\\operatorname{E}[L(\\mathbf{Y}\\bar{f}(\\mathbf{X}))]$ (see Theorem 4 for further information), thereby not ensuring the existence of a global minimizer that Fisher consistency implicitly assumes. Instead, we verify that the LHS loss is classification-calibrated. The LHS loss is the first negatively divergent loss function proven to satisfy the classification-calibration property."}, {"name": "Remark 1 (why p(x) in (0,1))", "statement": "Remark 1 In general, a global minimizer of $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))\\mid\\mathbf{X}=\\mathbf{x}]$ is allowed to take values $\\pm \\infty$ because what matters is only the sign of the minimizer $(Lin,~2004)$ If we allow the minimum objective to take values $\\pm \\infty$ , Theorem 1 can be extended to include $p(\\mathbf{x})$ equals to 0 and 1. It is because if $p(\\mathbf{x})=0$, then $\\operatorname{E}[L(\\mathbf{Y}\\alpha)\\mid\\mathbf{X}=\\mathbf{x}]=L(-\\alpha)\\to-\\infty$ as $\\alpha \\to - \\infty$. If $p ({\\bf x}) = 1$ , then $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))\\mid\\mathbf{X}=\\mathbf{x}]=L(\\alpha)\\to-\\infty$ as $\\alpha \\to \\infty$."}, {"name": "Remark 2 (uniqueness of the linear LHS solution)", "statement": "Remark 2 (Uniqueness) The minimizer $\\hat{\\boldsymbol{\\beta}}$ is uniquely determined because the LHS loss is convex and the $\\ell_{2}$ regularizer is strictly convex. However, $\\hat{\\beta}_{0}$ may not. Here is an illustrative example. Let the data be\n\n$$\n\\begin{array}{r l r l r l}{y_{1}= -1,}&{{}y_{2}= -1,}&{}&{{}y_{3}=1,}&{}&{{}y_{4}=1,}\\\\ {x_{1}=1,}&{{}x_{2}= -1,}&{}&{{}x_{3}=1,}&{}&{{}x_{4}= -1.}\\end{array}\n$$\n\nThen both $(0,0)$ and $(1,0)$ are global minimizers. Non-uniqueness of the intercept term can also occur in the linear SVM."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace ResearchLevel\n\nnoncomputable def lhs (u : ℝ) : ℝ := if 1 < u then -Real.log u else 1 - u\n\nnoncomputable def condRisk (p α : ℝ) : ℝ := p * lhs α + (1 - p) * lhs (-α)\n\nnoncomputable def fbar (p : ℝ) : ℝ := if p < 1 / 2 then -(1 - p) / p else p / (1 - p)\n\nnoncomputable def objective {n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (lam : ℝ) (b0 : ℝ) (b : Fin p → ℝ) : ℝ :=\n  (∑ i, lhs (ys i * (b0 + ∑ k, xs i k * b k))) / n + lam * (∑ k, b k * b k)\n\nnoncomputable def lhsR (r u : ℝ) : ℝ := if 1 < u then r * (1 - u ^ (1 / r)) else 1 - u\n\nnoncomputable def condRiskR (r p α : ℝ) : ℝ := p * lhsR r α + (1 - p) * lhsR r (-α)\n\nnoncomputable def fbarR (r p : ℝ) : ℝ :=\n  if p < 1 / 2 then -(((1 - p) / p) ^ (r / (r - 1))) else (p / (1 - p)) ^ (r / (r - 1))\n\ntheorem theorem1 {p : ℝ} (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRisk p (fbar p) ≤ condRisk p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRisk p α ≤ condRisk p β) → α = fbar p) ∧\n    Real.sign (fbar p) = Real.sign (p - 1 / 2) := by\n  sorry\n\ntheorem theorem2 {n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (hpos : ∃ i, ys i = 1) (hneg : ∃ j, ys j = -1)\n    {lam : ℝ} (hlam : 0 < lam) :\n    ∃ b0 : ℝ, ∃ b : Fin p → ℝ, ∀ c0 : ℝ, ∀ c : Fin p → ℝ,\n      objective xs ys lam b0 b ≤ objective xs ys lam c0 c := by\n  sorry\n\ntheorem theorem6 {r p : ℝ} (hr : 1 < r) (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRiskR r p (fbarR r p) ≤ condRiskR r p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRiskR r p α ≤ condRiskR r p β) → α = fbarR r p) ∧\n    Real.sign (fbarR r p) = Real.sign (p - 1 / 2) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem1", "theorem2", "theorem6"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 830, "end": 1053, "gold": "{p : ℝ} (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRisk p (fbar p) ≤ condRisk p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRisk p α ≤ condRisk p β) → α = fbar p) ∧\n    Real.sign (fbar p) = Real.sign (p - 1 / 2)"}]}
{"item_id": "v24-22-1104::theorem2::A-stmt", "paper_id": "v24-22-1104", "target_name": "theorem2", "source": "Research-level", "nl_statement": "Theorem 2 (Existence of global solution) Let $(\\mathbf{x}_{i},y_{i})\\in\\mathbb{R}^{p}\\times\\{-1,+1\\}$ for $i=1,\\ldots,n$ be training data. Suppose there exist $i,j$ with $y_{i}= + 1$ and $y_{j}= - 1$ . Then, there exists a global solution to\n\n$$\n\\operatorname*{min}\\mathcal{L}(\\beta_{0},\\beta)=\\operatorname*{min}_{\\beta_{0},\\beta}\\left[\\sum_{i=1}^{n}L\\left(y_{i}(\\beta_{0}+\\mathbf{x}_{i}^{T}\\beta)\\right)/n+\\lambda\\beta^{T}\\beta\\right],\\tag{2}\n$$\n\nwhere L is the LHS loss and $\\lambda > 0$ is a tuning parameter.", "context": [{"name": "Setup: binary classification, p(x), Bayes rule (Section 1)", "statement": "This paper concerns binary classification, where the task is to predict an unobserved binary output value $y\\in\\{-1,1\\}$ based on an observed input vector $\\mathbf{x}\\in \\mathbb{R}^{p}$ . The classifier is a mapping from the input space $\\mathcal{X}$ to {−1, 1} via a classification function $\\hat{f}$, and the predicted y value is $\\mathrm{sgn}\\{\\hat{f}(\\mathbf{x})\\}$ . The decision boundary is the set $\\{\\mathbf{x}:{\\hat{f}}(\\mathbf{x})=0\\}$ . Suppose that data are generated from some underlying distribution $\\mathrm{pr}(\\mathbf{X},\\mathbf{Y})$ , and let $p(\\mathbf{x})=\\mathrm{pr}(\\mathbf{Y}=1\\mid$ $\\mathbf{X}= \\mathbf{x})$ Under the standard 0-1 loss, the optimal classification rule is sgn $[p(\\mathbf{x})-1/2]$ (a.k.a. Bayes rule). Throughout the paper, we assume that $p(\\mathbf{X})\\neq 1/2$ almost surely. The optimal decision boundary is $\\{\\mathbf{x}:p(\\mathbf{x})=1/2\\}$ Given training data, $\\{({\\bf x}_{i},y_{i}),i=$ $1,\\ldots,n\\}$ , one aims to develop a classifier that mimics the Bayes rule as closely as possible."}, {"name": "LHS loss, eq. (1)", "statement": "In this paper, we provide an affirmative answer and the new function called the leaky hockey stick loss (LHS loss). The expression of the LHS loss is given in (1) and the picture of this loss function is displayed in Figure 1. We use the word 'hockey stick' because our loss function resembles it and is continuously differentiable. We use the term 'leaky' because the right side diverges to negative infinity motivated by the name of leaky ReLU.\n\n$$\nL(yf)=\\left\\{{\\begin{array}{l l}{-\\log yf,}&{yf>1,}\\\\ {1-yf,}&{yf\\leq 1,}\\end{array}}\\right.\\tag{1}\n$$"}, {"name": "Existence approach for the linear LHS classifier (Section 3.1)", "statement": "When L is a non-negative continuous loss function, there exists a global minimizer to the regularized ERM problem. Proving existence is considered well-understood. The sublevel set of the objective function L of the regularized ERM problem, $\\{\\beta:{\\mathcal{L}}(\\beta)\\leq c\\}$ , is compact for any $c \\in \\mathbb{R}$ , and the global minimizer of the continuous objective function on a compact set must exist by the extreme value theorem.\n\nIn contrast, it is not trivial to show the existence of a global minimizer of the linear LHS classifier. The loss term in the regularized ERM can diverge to negative infinity even though the LHS loss is convex and training data have finite samples. To show the existence, we find a specific compact set D of $\\beta,$ so that the objective function of the linear LHS classifier can have a global minimizer on D. Then, we show that the objective value at the global minimizer on D is always smaller than any objective value on $D^{c}$"}, {"name": "Remark 2 (uniqueness of the linear LHS solution)", "statement": "Remark 2 (Uniqueness) The minimizer $\\hat{\\boldsymbol{\\beta}}$ is uniquely determined because the LHS loss is convex and the $\\ell_{2}$ regularizer is strictly convex. However, $\\hat{\\beta}_{0}$ may not. Here is an illustrative example. Let the data be\n\n$$\n\\begin{array}{r l r l r l}{y_{1}= -1,}&{{}y_{2}= -1,}&{}&{{}y_{3}=1,}&{}&{{}y_{4}=1,}\\\\ {x_{1}=1,}&{{}x_{2}= -1,}&{}&{{}x_{3}=1,}&{}&{{}x_{4}= -1.}\\end{array}\n$$\n\nThen both $(0,0)$ and $(1,0)$ are global minimizers. Non-uniqueness of the intercept term can also occur in the linear SVM."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace ResearchLevel\n\nnoncomputable def lhs (u : ℝ) : ℝ := if 1 < u then -Real.log u else 1 - u\n\nnoncomputable def condRisk (p α : ℝ) : ℝ := p * lhs α + (1 - p) * lhs (-α)\n\nnoncomputable def fbar (p : ℝ) : ℝ := if p < 1 / 2 then -(1 - p) / p else p / (1 - p)\n\nnoncomputable def objective {n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (lam : ℝ) (b0 : ℝ) (b : Fin p → ℝ) : ℝ :=\n  (∑ i, lhs (ys i * (b0 + ∑ k, xs i k * b k))) / n + lam * (∑ k, b k * b k)\n\nnoncomputable def lhsR (r u : ℝ) : ℝ := if 1 < u then r * (1 - u ^ (1 / r)) else 1 - u\n\nnoncomputable def condRiskR (r p α : ℝ) : ℝ := p * lhsR r α + (1 - p) * lhsR r (-α)\n\nnoncomputable def fbarR (r p : ℝ) : ℝ :=\n  if p < 1 / 2 then -(((1 - p) / p) ^ (r / (r - 1))) else (p / (1 - p)) ^ (r / (r - 1))\n\ntheorem theorem1 {p : ℝ} (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRisk p (fbar p) ≤ condRisk p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRisk p α ≤ condRisk p β) → α = fbar p) ∧\n    Real.sign (fbar p) = Real.sign (p - 1 / 2) := by\n  sorry\n\ntheorem theorem2 {n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (hpos : ∃ i, ys i = 1) (hneg : ∃ j, ys j = -1)\n    {lam : ℝ} (hlam : 0 < lam) :\n    ∃ b0 : ℝ, ∃ b : Fin p → ℝ, ∀ c0 : ℝ, ∀ c : Fin p → ℝ,\n      objective xs ys lam b0 b ≤ objective xs ys lam c0 c := by\n  sorry\n\ntheorem theorem6 {r p : ℝ} (hr : 1 < r) (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRiskR r p (fbarR r p) ≤ condRiskR r p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRiskR r p α ≤ condRiskR r p β) → α = fbarR r p) ∧\n    Real.sign (fbarR r p) = Real.sign (p - 1 / 2) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem1", "theorem2", "theorem6"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 1086, "end": 1376, "gold": "{n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (hpos : ∃ i, ys i = 1) (hneg : ∃ j, ys j = -1)\n    {lam : ℝ} (hlam : 0 < lam) :\n    ∃ b0 : ℝ, ∃ b : Fin p → ℝ, ∀ c0 : ℝ, ∀ c : Fin p → ℝ,\n      objective xs ys lam b0 b ≤ objective xs ys lam c0 c"}]}
{"item_id": "v24-22-1104::theorem6::A-stmt", "paper_id": "v24-22-1104", "target_name": "theorem6", "source": "Research-level", "nl_statement": "Theorem 6 (Classification-calibration) For any x such that $p(\\mathbf{x})\\in(0,1)$ with $p(\\mathbf{x})\\neq 1/2$ , a global minimizer of $\\operatorname{E}[L_{r}(\\mathbf{Y}f(\\mathbf{X}))\\mid\\mathbf{X}=\\mathbf{x}]$ uniquely exists and is\n\n$$\n\\begin{array}{r l}{-\\left\\{(1-p({\\bf x}))/p({\\bf x})\\right\\}^{r/(r-1)},}&{{}\\quad if~p({\\bf x})<1/2,}\\\\ {and+\\left\\{p({\\bf x})/(1-p({\\bf x}))\\right\\}^{r/(r-1)},}&{{}\\quad if~p({\\bf x})>1/2.}\\end{array}\n$$\n\nNote that sgn $\\{{\\bar{f}}({\\bf x}) \\}= f^{*}({\\bf x})$ , where $f^{*}$ is the Bayes rule.", "context": [{"name": "Setup: binary classification, p(x), Bayes rule (Section 1)", "statement": "This paper concerns binary classification, where the task is to predict an unobserved binary output value $y\\in\\{-1,1\\}$ based on an observed input vector $\\mathbf{x}\\in \\mathbb{R}^{p}$ . The classifier is a mapping from the input space $\\mathcal{X}$ to {−1, 1} via a classification function $\\hat{f}$, and the predicted y value is $\\mathrm{sgn}\\{\\hat{f}(\\mathbf{x})\\}$ . The decision boundary is the set $\\{\\mathbf{x}:{\\hat{f}}(\\mathbf{x})=0\\}$ . Suppose that data are generated from some underlying distribution $\\mathrm{pr}(\\mathbf{X},\\mathbf{Y})$ , and let $p(\\mathbf{x})=\\mathrm{pr}(\\mathbf{Y}=1\\mid$ $\\mathbf{X}= \\mathbf{x})$ Under the standard 0-1 loss, the optimal classification rule is sgn $[p(\\mathbf{x})-1/2]$ (a.k.a. Bayes rule). Throughout the paper, we assume that $p(\\mathbf{X})\\neq 1/2$ almost surely. The optimal decision boundary is $\\{\\mathbf{x}:p(\\mathbf{x})=1/2\\}$ Given training data, $\\{({\\bf x}_{i},y_{i}),i=$ $1,\\ldots,n\\}$ , one aims to develop a classifier that mimics the Bayes rule as closely as possible."}, {"name": "Classification-calibration and Fisher consistency (Section 2)", "statement": "Lin (2004) proposed Fisher consistency as a necessary condition on a loss function. It is defined to be that any global minimizer $\\check{f}$ (if it exists) of $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))]$ , generally referred to as a population minimizer, has the same sign function as the Bayes rule almost surely. Later, Bartlett et al. (2006) defined classification-calibration. A loss function L is called classification-calibrated if any global minimizer $\\bar{f}$ (if it exists) of $\\operatorname{E}[L(\\mathbf{Y}f(\\mathbf{X}))\\mid\\mathbf{X}=\\mathbf{x}]$ has the same sign as the Bayes rule almost surely. For example, the hinge loss is classification-calibrated, which can be directly shown by Theorem 2 of Bartlett et al. (2006).\n\nAn unspoken assumption in those two definitions is the existence of a global minimizer. When $\\bar{f}$ exists (which can be easily checked) and $|\\mathrm{E}[L({\\bf Y}\\bar{f}({\\bf X}))]| <\\infty .$ then a population minimizer $\\check{f}$ exists. We further see that loss function L is classification-calibrated if and only if it is Fisher consistent. The 'if' statement follows from the fact that any $\\bar{f}$ is also a population minimizer and the definition of Fisher consistency. The 'only if' statement comes from the fact that $\\operatorname{E}[L(\\mathbf{Y}{\\bar{f}}(\\mathbf{X}))\\mid\\mathbf{X}]=\\operatorname{E}[L(\\mathbf{Y}{\\check{f}}(\\mathbf{X}))\\mid\\mathbf{X}]$ almost surely and that for any $\\delta_{\\mathbf{X}}$ such that $\\operatorname{sgn}(\\delta_{\\mathbf{X}})\\neq\\operatorname{sgn}(p(\\mathbf{X})-1/2),\\operatorname{E}[L(\\mathbf{Y}{\\bar{f}}(\\mathbf{X}))\\mid\\mathbf{X}]<\\operatorname{E}[L(\\mathbf{Y}\\delta_{\\mathbf{X}})\\mid\\mathbf{X}]$ almost surely by the definition of classification-calibration.\n\nFor a non-negative loss function L, $\\operatorname{E}[L(\\mathbf{Y}\\bar{f}(\\mathbf{X}))]$ is always finite. However, the LHS loss does not guarantee the finiteness of $\\operatorname{E}[L(\\mathbf{Y}\\bar{f}(\\mathbf{X}))]$ (see Theorem 4 for further information), thereby not ensuring the existence of a global minimizer that Fisher consistency implicitly assumes. Instead, we verify that the LHS loss is classification-calibrated. The LHS loss is the first negatively divergent loss function proven to satisfy the classification-calibration property."}, {"name": "Loss family L_r, eq. (15) (Section 7.2)", "statement": "We define a negatively divergent loss functions $L_{r}$ as follows: with $r\\in(1,\\infty)$,\n\n$$\nL_{r}(yf)={\\left\\{\\begin{array}{l l}{r\\left(1-(yf)^{1/r}\\right),}&{yf>1,}\\\\ {1-yf,}&{yf\\leq 1.}\\end{array}\\right.}\\tag{15}\n$$\n\nFigure 3 illustrates (15) for various r values. Note the LHS loss is (15) with r diverging to ∞."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace ResearchLevel\n\nnoncomputable def lhs (u : ℝ) : ℝ := if 1 < u then -Real.log u else 1 - u\n\nnoncomputable def condRisk (p α : ℝ) : ℝ := p * lhs α + (1 - p) * lhs (-α)\n\nnoncomputable def fbar (p : ℝ) : ℝ := if p < 1 / 2 then -(1 - p) / p else p / (1 - p)\n\nnoncomputable def objective {n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (lam : ℝ) (b0 : ℝ) (b : Fin p → ℝ) : ℝ :=\n  (∑ i, lhs (ys i * (b0 + ∑ k, xs i k * b k))) / n + lam * (∑ k, b k * b k)\n\nnoncomputable def lhsR (r u : ℝ) : ℝ := if 1 < u then r * (1 - u ^ (1 / r)) else 1 - u\n\nnoncomputable def condRiskR (r p α : ℝ) : ℝ := p * lhsR r α + (1 - p) * lhsR r (-α)\n\nnoncomputable def fbarR (r p : ℝ) : ℝ :=\n  if p < 1 / 2 then -(((1 - p) / p) ^ (r / (r - 1))) else (p / (1 - p)) ^ (r / (r - 1))\n\ntheorem theorem1 {p : ℝ} (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRisk p (fbar p) ≤ condRisk p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRisk p α ≤ condRisk p β) → α = fbar p) ∧\n    Real.sign (fbar p) = Real.sign (p - 1 / 2) := by\n  sorry\n\ntheorem theorem2 {n p : ℕ} (xs : Fin n → (Fin p → ℝ)) (ys : Fin n → ℝ)\n    (hy : ∀ i, ys i = 1 ∨ ys i = -1)\n    (hpos : ∃ i, ys i = 1) (hneg : ∃ j, ys j = -1)\n    {lam : ℝ} (hlam : 0 < lam) :\n    ∃ b0 : ℝ, ∃ b : Fin p → ℝ, ∀ c0 : ℝ, ∀ c : Fin p → ℝ,\n      objective xs ys lam b0 b ≤ objective xs ys lam c0 c := by\n  sorry\n\ntheorem theorem6 {r p : ℝ} (hr : 1 < r) (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRiskR r p (fbarR r p) ≤ condRiskR r p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRiskR r p α ≤ condRiskR r p β) → α = fbarR r p) ∧\n    Real.sign (fbarR r p) = Real.sign (p - 1 / 2) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem1", "theorem2", "theorem6"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1409, "end": 1668, "gold": "{r p : ℝ} (hr : 1 < r) (hp0 : 0 < p) (hp1 : p < 1) (hne : p ≠ 1 / 2) :\n    (∀ α : ℝ, condRiskR r p (fbarR r p) ≤ condRiskR r p α) ∧\n    (∀ α : ℝ, (∀ β : ℝ, condRiskR r p α ≤ condRiskR r p β) → α = fbarR r p) ∧\n    Real.sign (fbarR r p) = Real.sign (p - 1 / 2)"}]}
{"item_id": "v24-22-1149::powerball_lower::A-stmt", "paper_id": "v24-22-1149", "target_name": "powerball_lower", "source": "Research-level", "nl_statement": "**Eqs. (14)–(15)**.\n$(\\langle x, \\sigma_\\gamma(x)\\rangle)^2 / \\|\\sigma_\\gamma(x)\\|^2 \\ge \\|x\\|_{1+\\gamma}^2 / \\|\\mathbb{1}\\|_p$, with $p = (1+\\gamma)/(1-\\gamma)$,\nfor $x \\in \\mathbb{R}^d$ with $0 < d$, $0 \\le \\gamma < 1$ (the paper's standing range for the power\ncoefficient, eq. (4)), and $\\sigma_\\gamma(x) \\neq 0$ so that the left denominator is nonzero.", "context": [{"name": "Setup: finite-sum problem (1) (Section 1)", "statement": "Large numbers of applications in machine learning (Le Thi et al., 2022), computer vision (Xu et al., 2021) and numerical optimization (Fehrman et al., 2020) are often implemented to be a finite-sum structure of the form\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d}}f(w)=\\frac{1}{n}\\sum_{i=1}^{n}f_{i}(w),\\tag{1}\n$$\n\nwhere n denotes the total number of the instances, w defines the parameter to learn and $f_{i}(\\cdot):\\mathbb{R}^{d}\\to\\mathbb{R}$ represents a loss function corresponding to the i-th instance with d dimensions. In this work, we are mainly interested in each $f_{i}(w)$ and $f(w)$ that are L-smooth but may be non-convex. More concretely and vividly, the cases Problem (1) referred into logistic regression, eigenvector computation, multi-kernel learning, neural networks, to name a few."}, {"name": "Notation (Section 1.2)", "statement": "Throughout this work, we take $\\mathbb{R}^{d}$ to define the set of d-dimensional real number vectors. For vectors, the mark w · v is used to denote their inner product. Additionally, we use [n] to define the set $\\{1,2,\\ldots,\\mathtt{n}\\}$ . The symbol $\\| \\cdot \\|$ and $\\| \\cdot \\|_{p}$ are chosen to represent the Euclidean norm and p-norm on $\\mathbb{R}^{d}$ respectively. Moreover, Denote by $\\nabla f(\\boldsymbol{w})$ the gradient of the loss function $f(w)$ . We adopt $\\mathbb{E}[ \\cdot ]$ to represent the expectation with respect to the underlying probability space. We use $(w)_{i}$ to represent the i-th element of vector $\\boldsymbol{w}\\in \\mathbb{R}^{d}$"}, {"name": "SVRG update (3) and Powerball update (4); definition of σ_γ (Section 3)", "statement": "Since this work will delve into the theoretical and numerical characteristics of SVRGE in PSO algorithms, we offer some discussions about the updating scheme of the original SVRG algorithm, originally developed in (Johnson and Zhang, 2013). Comparing with the conventional SGD algorithm that worked with the update scheme:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\nabla f_{i}(w_{k}),\\tag{2}\n$$\n\nfor solving Problem (1), the update scheme of SVRG usually took the form:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\big(\\nabla f_{i}\\big(w_{k}\\big)-\\nabla f_{i}\\big(\\tilde{w}\\big)+\\nabla f\\big(\\tilde{w}\\big)\\big),\\tag{3}\n$$\n\nwhere $\\tilde{w}$ was a snapshot point and $\\eta_{k}> 0$ was the learning rate. In general, there were two loops in the SVRG-like approach, where the iterative rule (3) located in the inner loop. The outer loop was used to store the historical information, then transferred it into the inner loop.\n\nThe iterative scheme (3), utilizing historical information $(\\mathrm{i}.\\mathrm{e}.,\\nabla f_{i}(\\tilde{w})$ and $\\nabla f(\\tilde{w}))$ , made the stochastic optimization algorithms enjoy a lower variance than classical SGD, and resulted in a faster convergence rate and a better generalization in practice.\n\nIn contrast, when dealing with Problem (1), the classical iterative scheme of the PSO algorithm was:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\sigma_{\\gamma}(\\nabla f_{i}(w_{k})),\\tag{4}\n$$\n\nwhere the power coefficient $\\gamma\\in [0,1),\\sigma_{\\gamma}(\\nabla f_{i}(w_{k}))=\\mathrm{sign}(\\nabla f_{i}(w_{k}))|\\nabla f_{i}(w_{k})|^{\\gamma}$ Here $\\mathrm{sign}(x)$ backed to the sign of $x$, or 0 if $x = 0$. Also, it was noted that the operations sign(w) and $\\sigma_{\\gamma}(w)$ was applied to each element if the parameter $y \\in \\mathbb{R}^{d}$ was a vector. For example, $\\sigma_{\\gamma}(y)=(\\sigma_{\\gamma}((y)_{1}),\\sigma_{\\gamma}((y)_{2}),\\dots,\\sigma_{\\gamma}((y)_{d}))$"}, {"name": "Algorithm 1: PB-SVRGE", "statement": "Naturally, it was asked whether SVRGE created a positive effect on PSO algorithms? In order to respond to this question, we first proposed the PB-SVRGE algorithm (Algorithm 1) and analyzed its theoretical and empirical properties in the upcoming sections.\n\nAlgorithm 1 PB-SVRGE   \nInput: $\\widetilde{w}^{0}= w_{K}^{0}= w^{0}$ , epoch length $K,$ outer loop size ${\\mathfrak{S}}= T / K$ , mini-batch size $b$, learning rate   \n$\\{\\eta_{j}\\}_{j = 0}^{K - 1}$ , power coefficient $\\gamma .$   \nfor $s=0,\\ldots,{\\mathfrak{S}}-1$ do   \n$w_{0}^{s + 1}= w_{K}^{s}$   \n$g^{s+1}=\\nabla f(\\tilde{w}^{s})$   \nfor $k=0,\\ldots,K-1$ do   \nRandomly choose $S \\subseteq [ n ]$ with $|S| =b,$ compute a new stochastic estimate $\\nabla f(w_{k}^{s})$   \n$\\nu_{k}^{s+1}=\\nabla f_{S}(w_{k}^{s+1})-\\nabla f_{S}(\\tilde{w}^{s})+g^{s+1}$   \n$w_{k+1}^{s+1}=w_{k}^{s+1}-\\eta_{k}\\sigma(\\nu_{k}^{s+1})$   \nend for   \nSet $\\widetilde{w}^{s + 1}= w_{K}^{s + 1}$   \nend for"}, {"name": "Remark 1(i): mini-batch estimators ∇f_S", "statement": "Remark 1: For PB-SVRGE, the following remarks were necessary to be demonstrated:\n\n(i) PR-SVRGE utilized the similar algorithm framework as SVRG-like algorithms. The main difference between these two approaches is that the introduction of the Powerball function in PB-SVRGE. Specifically, we can view the SVRG-like algorithm as an extension of PB-SVRGE when taking $\\gamma = 1$ . Moreover, we considered PSO algorithms in the mini-batching setting, where $\\begin{array}{r}{\\nu_{k}^{s+1}=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(w_{k}^{s+1})-\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{S}(\\tilde{w}^{s})+\\nabla f(\\tilde{w}^{s})\\ (S\\subseteq [n]}\\end{array}$ with b batch samples). Here, $\\begin{array}{r}{\\nabla f_{S}(w_{k}^{s+1})=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(w_{k}^{s+1})}\\end{array}$ and $\\begin{array}{r}{\\nabla f_{S}(\\tilde{w}^{s})=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(\\tilde{w}^{s})}\\end{array}$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def sigmaGamma {d : ℕ} (γ : ℝ) (x : EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) :=\n  (WithLp.equiv 2 (Fin d → ℝ)).symm (fun i => Real.sign (x i) * |x i| ^ γ)\n\nnoncomputable def lpNorm {d : ℕ} (r : ℝ) (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  (∑ i, |x i| ^ r) ^ (1 / r)\n\nnoncomputable def onesNorm (p : ℝ) (d : ℕ) : ℝ := (d : ℝ) ^ (1 / p)\n\nlemma powerball_lower {d : ℕ} (γ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1)\n    (p : ℝ) (hp : p = (1 + γ) / (1 - γ)) (hd : 0 < d)\n    (x : EuclideanSpace ℝ (Fin d)) (hx : sigmaGamma γ x ≠ 0) :\n    (⟪x, sigmaGamma γ x⟫) ^ 2 / ‖sigmaGamma γ x‖ ^ 2\n      ≥ (lpNorm (1 + γ) x) ^ 2 / onesNorm p d := by\n  sorry\n\ntheorem theorem1 {d n K Sout T b : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbn : b ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hθ1 : θ < 1) (hσ : 0 ≤ σ2)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (heta : ∀ k s ω, eta k s ω\n      = ⟪nu k s ω, sigmaGamma γ (nu k s ω)⟫ / (L * ‖sigmaGamma γ (nu k s ω)‖ ^ 2))\n    (hsigmaNu : ∀ k < K, ∀ s < Sout, ∀ ω, sigmaGamma γ (nu k s ω) ≠ 0)\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ (4 * L * onesP) / ((T : ℝ) * (1 - θ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * onesP * σ2 / ((b : ℝ) * θ * (1 - θ)) := by\n  sorry\n\ntheorem theorem2 {d n K Sout T b bH : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbHpos : 0 < bH) (hbn : b ≤ n) (hbHn : bH ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 ζ τ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hσ : 0 ≤ σ2)\n    (hτ : 0 < τ) (hζ : 0 < ζ)\n    (hbH2 : 0 < 2 * (bH : ℝ) - θ * (bH : ℝ) - 1)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n)) (SidxH : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hindepSH : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => SidxH ks.1 ks.2 ω) μ)\n    (gSHa gSHb : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hSHcard : ∀ k s ω, (SidxH k s ω).card = bH)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hgSHa : ∀ k s ω, gSHa k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W (k + 1) (s + 1) ω))\n    (hgSHb : ∀ k s ω, gSHb k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W k (s + 1) ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (eta0 : ℝ) (heta0 : ∀ s ω, eta 0 s ω = eta0)\n    (heta : ∀ k s ω, eta (k + 1) s ω = (ζ / bH)\n      * (‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2\n        / (⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n              sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫\n            + τ * ‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2)))\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hΔW : ∀ k < K, ∀ s < Sout, ∀ ω, W (k + 1) (s + 1) ω - W k (s + 1) ω ≠ 0)\n\n    (hbb_ip : ∀ k s ω, 0 ≤ ⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n        sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫)\n    (hLem1 : ∀ k s ω, eta k s ω ≤ ζ / ((bH : ℝ) * τ))\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ)\n    (hintFgap : ∀ k s,\n      Integrable (fun ω => f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) μ)\n    (hdescent : ∀ k < K, ∀ s < Sout,\n      (∫ ω, (f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) ∂μ)\n        ≤ -((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) / (4 * L * (bH : ℝ) ^ 2 * onesP))\n              * (∫ ω, (lpNorm (1 + γ) (gS k s ω)) ^ 2 ∂μ)\n          + 2 * σ2 / (L * (bH : ℝ) * (b : ℝ) * θ)) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ 4 * (bH : ℝ) ^ 2 * L * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (T : ℝ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * σ2 * (bH : ℝ) * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (b : ℝ) * θ) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["powerball_lower", "theorem1", "theorem2"], "topic_domain": "Others", "topic_subtopic": "Analysis & Geometry", "holes": [{"start": 481, "end": 740, "gold": "{d : ℕ} (γ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1)\n    (p : ℝ) (hp : p = (1 + γ) / (1 - γ)) (hd : 0 < d)\n    (x : EuclideanSpace ℝ (Fin d)) (hx : sigmaGamma γ x ≠ 0) :\n    (⟪x, sigmaGamma γ x⟫) ^ 2 / ‖sigmaGamma γ x‖ ^ 2\n      ≥ (lpNorm (1 + γ) x) ^ 2 / onesNorm p d"}]}
{"item_id": "v24-22-1149::theorem1::A-stmt", "paper_id": "v24-22-1149", "target_name": "theorem1", "source": "Research-level", "nl_statement": "Theorem 1. Set $w^{*}=\\arg\\operatorname*{min}_{w}f(w)$ , and choose $S \\subseteq [ n ]$ with $| S | = b .$ Let T denote the number of total iterations, then, under Assumption 1 and Assumption 2 (used in the mini-batch form $\\mathbb{E}[\\nabla f_{S}(w)]=\\mathbb{E}[\\nabla f(w)]$ and $\\mathbb{E}[\\|\\nabla f_{S}(w)-\\nabla f(w)\\|^{2}]\\le\\hat{\\sigma}^{2}/b$ at $w = w_{k}^{s + 1}$ and at $w = \\tilde{w}^{s}$), for any $T \\geq 1$ , PB-SVRGE (Algorithm 1) with learning rate $\\eta_{k}=\\frac{\\nu_{k}^{s+1}\\cdot\\sigma_{\\gamma}(\\nu_{k}^{s+1})}{L\\|\\sigma_{\\gamma}(\\nu_{k}^{s+1})\\|^{2}}$ (where $\\sigma_{\\gamma}(\\nu_{k}^{s+1})\\neq 0$) can lead to, for every $\\theta\\in(0,1)$ and with $p = \\frac{1 + \\gamma}{1 - \\gamma}$,\n\n$$\n\\mathbb{E}\\left[\\frac{1}{T}\\sum_{s=0}^{\\mathfrak{S}-1}\\sum_{k=0}^{K-1}\\lVert\\nabla f_{S}(w_{k}^{s+1})\\rVert_{1+\\gamma}^{2}\\right]\\leq\\frac{4L\\lVert\\mathbf{1}\\rVert_{p}}{T(1-\\theta)}\\mathbb{E}[f(\\tilde{w}^{0})-f(w^{*})]+\\frac{8\\lVert\\mathbf{1}\\rVert_{p}\\hat{\\sigma}^{2}}{b\\theta(1-\\theta)}.\\tag{8}\n$$", "context": [{"name": "Setup: finite-sum problem (1) (Section 1)", "statement": "Large numbers of applications in machine learning (Le Thi et al., 2022), computer vision (Xu et al., 2021) and numerical optimization (Fehrman et al., 2020) are often implemented to be a finite-sum structure of the form\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d}}f(w)=\\frac{1}{n}\\sum_{i=1}^{n}f_{i}(w),\\tag{1}\n$$\n\nwhere n denotes the total number of the instances, w defines the parameter to learn and $f_{i}(\\cdot):\\mathbb{R}^{d}\\to\\mathbb{R}$ represents a loss function corresponding to the i-th instance with d dimensions. In this work, we are mainly interested in each $f_{i}(w)$ and $f(w)$ that are L-smooth but may be non-convex. More concretely and vividly, the cases Problem (1) referred into logistic regression, eigenvector computation, multi-kernel learning, neural networks, to name a few."}, {"name": "Notation (Section 1.2)", "statement": "Throughout this work, we take $\\mathbb{R}^{d}$ to define the set of d-dimensional real number vectors. For vectors, the mark w · v is used to denote their inner product. Additionally, we use [n] to define the set $\\{1,2,\\ldots,\\mathtt{n}\\}$ . The symbol $\\| \\cdot \\|$ and $\\| \\cdot \\|_{p}$ are chosen to represent the Euclidean norm and p-norm on $\\mathbb{R}^{d}$ respectively. Moreover, Denote by $\\nabla f(\\boldsymbol{w})$ the gradient of the loss function $f(w)$ . We adopt $\\mathbb{E}[ \\cdot ]$ to represent the expectation with respect to the underlying probability space. We use $(w)_{i}$ to represent the i-th element of vector $\\boldsymbol{w}\\in \\mathbb{R}^{d}$"}, {"name": "SVRG update (3) and Powerball update (4); definition of σ_γ (Section 3)", "statement": "Since this work will delve into the theoretical and numerical characteristics of SVRGE in PSO algorithms, we offer some discussions about the updating scheme of the original SVRG algorithm, originally developed in (Johnson and Zhang, 2013). Comparing with the conventional SGD algorithm that worked with the update scheme:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\nabla f_{i}(w_{k}),\\tag{2}\n$$\n\nfor solving Problem (1), the update scheme of SVRG usually took the form:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\big(\\nabla f_{i}\\big(w_{k}\\big)-\\nabla f_{i}\\big(\\tilde{w}\\big)+\\nabla f\\big(\\tilde{w}\\big)\\big),\\tag{3}\n$$\n\nwhere $\\tilde{w}$ was a snapshot point and $\\eta_{k}> 0$ was the learning rate. In general, there were two loops in the SVRG-like approach, where the iterative rule (3) located in the inner loop. The outer loop was used to store the historical information, then transferred it into the inner loop.\n\nThe iterative scheme (3), utilizing historical information $(\\mathrm{i}.\\mathrm{e}.,\\nabla f_{i}(\\tilde{w})$ and $\\nabla f(\\tilde{w}))$ , made the stochastic optimization algorithms enjoy a lower variance than classical SGD, and resulted in a faster convergence rate and a better generalization in practice.\n\nIn contrast, when dealing with Problem (1), the classical iterative scheme of the PSO algorithm was:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\sigma_{\\gamma}(\\nabla f_{i}(w_{k})),\\tag{4}\n$$\n\nwhere the power coefficient $\\gamma\\in [0,1),\\sigma_{\\gamma}(\\nabla f_{i}(w_{k}))=\\mathrm{sign}(\\nabla f_{i}(w_{k}))|\\nabla f_{i}(w_{k})|^{\\gamma}$ Here $\\mathrm{sign}(x)$ backed to the sign of $x$, or 0 if $x = 0$. Also, it was noted that the operations sign(w) and $\\sigma_{\\gamma}(w)$ was applied to each element if the parameter $y \\in \\mathbb{R}^{d}$ was a vector. For example, $\\sigma_{\\gamma}(y)=(\\sigma_{\\gamma}((y)_{1}),\\sigma_{\\gamma}((y)_{2}),\\dots,\\sigma_{\\gamma}((y)_{d}))$"}, {"name": "Algorithm 1: PB-SVRGE", "statement": "Naturally, it was asked whether SVRGE created a positive effect on PSO algorithms? In order to respond to this question, we first proposed the PB-SVRGE algorithm (Algorithm 1) and analyzed its theoretical and empirical properties in the upcoming sections.\n\nAlgorithm 1 PB-SVRGE   \nInput: $\\widetilde{w}^{0}= w_{K}^{0}= w^{0}$ , epoch length $K,$ outer loop size ${\\mathfrak{S}}= T / K$ , mini-batch size $b$, learning rate   \n$\\{\\eta_{j}\\}_{j = 0}^{K - 1}$ , power coefficient $\\gamma .$   \nfor $s=0,\\ldots,{\\mathfrak{S}}-1$ do   \n$w_{0}^{s + 1}= w_{K}^{s}$   \n$g^{s+1}=\\nabla f(\\tilde{w}^{s})$   \nfor $k=0,\\ldots,K-1$ do   \nRandomly choose $S \\subseteq [ n ]$ with $|S| =b,$ compute a new stochastic estimate $\\nabla f(w_{k}^{s})$   \n$\\nu_{k}^{s+1}=\\nabla f_{S}(w_{k}^{s+1})-\\nabla f_{S}(\\tilde{w}^{s})+g^{s+1}$   \n$w_{k+1}^{s+1}=w_{k}^{s+1}-\\eta_{k}\\sigma(\\nu_{k}^{s+1})$   \nend for   \nSet $\\widetilde{w}^{s + 1}= w_{K}^{s + 1}$   \nend for"}, {"name": "Remark 1(i): mini-batch estimators ∇f_S", "statement": "Remark 1: For PB-SVRGE, the following remarks were necessary to be demonstrated:\n\n(i) PR-SVRGE utilized the similar algorithm framework as SVRG-like algorithms. The main difference between these two approaches is that the introduction of the Powerball function in PB-SVRGE. Specifically, we can view the SVRG-like algorithm as an extension of PB-SVRGE when taking $\\gamma = 1$ . Moreover, we considered PSO algorithms in the mini-batching setting, where $\\begin{array}{r}{\\nu_{k}^{s+1}=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(w_{k}^{s+1})-\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{S}(\\tilde{w}^{s})+\\nabla f(\\tilde{w}^{s})\\ (S\\subseteq [n]}\\end{array}$ with b batch samples). Here, $\\begin{array}{r}{\\nabla f_{S}(w_{k}^{s+1})=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(w_{k}^{s+1})}\\end{array}$ and $\\begin{array}{r}{\\nabla f_{S}(\\tilde{w}^{s})=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(\\tilde{w}^{s})}\\end{array}$"}, {"name": "Assumption 1 (L-smoothness) and inequality (6)", "statement": "This work considers the theoretical properties of PB-SVRGE on non-convex optimization problems.   \nTo proceed the analysis, we make the following standard smoothness assumptions.\n\nAssumption 1. Each cost function $f_{i}(w)$ in Problem (1) is differential and has a L-Lipschitz gradient, that is,\n\n$$\n\\forall w,\\nu\\in\\mathbb{R}^{d},\\quad\\|\\nabla f_{i}(w)-\\nabla f_{i}(\\nu)\\|\\le L\\|w-\\nu\\|.\\tag{5}\n$$\n\nThe conclusion in Assumption 1 indicates that the average function $f(w)$ , presented in Problem (1), has a L-Lipschitz gradient as well.\n\nClearly, from the smooth property of the cost function, we have the following result:\n\n$$\nf(w)\\leq f(\\nu)+\\nabla f(\\nu)^{T}(w-\\nu)+\\frac{L}{2}\\parallel w-\\nu\\parallel^{2}.\\tag{6}\n$$"}, {"name": "Assumption 2 (unbiased oracle, bounded variance) and the mini-batch variance bound σ̂²/b", "statement": "Assumption 2. The stochastic gradient oracle is an independent and unbiased estimator of the gradient and satisfies\n\n$$\n\\begin{array}{r l}&{\\mathbb{E}_{\\xi_{i}}[\\nabla f(w,\\xi_{i})]=\\nabla f(w),\\forall w\\in\\mathbb{R}^{d},}\\\\ &{\\mathbb{E}_{\\xi_{i}}[\\|\\nabla f(w,\\xi_{i})-\\nabla f(w)\\|^{2}]\\le\\hat{\\sigma}^{2},\\forall w\\in\\mathbb{R}^{d},}\\end{array}\\tag{7}\n$$\n\nwhere $\\nabla f_{i}(w)=\\nabla f(w,\\xi_{i})$ and $\\xi_{i}$ denotes a random variable.\n\nTwo assumptions mentioned above were very common in the most existing studies for finding a first-order stationary point of non-convex or convex loss functions (Ghadimi and Lan, 2013; Wei et al., 2021; Zhou et al., 2018). Observe that, by (5), the gradient of the loss function does not change arbitrary rapid, and by (7), the variance of the random variable $\\mathbb{E}[\\|\\nabla f_{i}(w)-\\nabla f(w)\\|^{2}]$ has an upper bound. Notice that, when taking b samples in vanilla SGD at each iterative step, we have $\\begin{array}{r}{\\mathbb{E}[\\|\\nabla f_{S}(w)-\\nabla f(w)\\|^{2}]\\le\\frac{\\hat{\\sigma}^{2}}{b}}\\end{array}$"}, {"name": "Eqs. (14)-(15) Hölder lower bound (another result of the same paper)", "statement": "**Eqs. (14)–(15)**.\n$(\\langle x, \\sigma_\\gamma(x)\\rangle)^2 / \\|\\sigma_\\gamma(x)\\|^2 \\ge \\|x\\|_{1+\\gamma}^2 / \\|\\mathbb{1}\\|_p$, with $p = (1+\\gamma)/(1-\\gamma)$,\nfor $x \\in \\mathbb{R}^d$ with $0 < d$, $0 \\le \\gamma < 1$ (the paper's standing range for the power\ncoefficient, eq. (4)), and $\\sigma_\\gamma(x) \\neq 0$ so that the left denominator is nonzero."}], "nl_notes": "### Theorem 1 (PB-SVRGE, eq. 8)\nPROOF NOTE (Theorem 1) — The stated bound (8) is true, but the paper's route to it has an invalid step: from (19) the paper continues with (20), which is (19) with the nonnegative term (1 − θ)/(2L‖𝟙‖_p) · ‖∇f_S(w̃^s) − ∇f(w̃^s)‖²_{1+γ} deleted. (20) does not follow from (19), and repairing that route only gives the constant 8/(θ(1 − θ)) + 2. Do not copy the step (19) → (20); the stated constant 8/(θ(1 − θ)) needs a different per-step argument.\n\nNOTE (Theorem 1) — Assumption 2 is stated for the single-sample oracle, and the paper gives the mini-batch variance bound σ̂²/b only as a remark on vanilla SGD. The proof applies both to the mini-batch gradient ∇f_S at the inner iterate w_k^{s+1} and at the snapshot w̃^s, so the statement names that use explicitly: E[∇f_S(w)] = E[∇f(w)] and E[‖∇f_S(w) − ∇f(w)‖²] ≤ σ̂²/b at those two points. It also requires σ_γ(ν_k^{s+1}) ≠ 0, without which the learning rate η_k is undefined.", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def sigmaGamma {d : ℕ} (γ : ℝ) (x : EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) :=\n  (WithLp.equiv 2 (Fin d → ℝ)).symm (fun i => Real.sign (x i) * |x i| ^ γ)\n\nnoncomputable def lpNorm {d : ℕ} (r : ℝ) (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  (∑ i, |x i| ^ r) ^ (1 / r)\n\nnoncomputable def onesNorm (p : ℝ) (d : ℕ) : ℝ := (d : ℝ) ^ (1 / p)\n\nlemma powerball_lower {d : ℕ} (γ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1)\n    (p : ℝ) (hp : p = (1 + γ) / (1 - γ)) (hd : 0 < d)\n    (x : EuclideanSpace ℝ (Fin d)) (hx : sigmaGamma γ x ≠ 0) :\n    (⟪x, sigmaGamma γ x⟫) ^ 2 / ‖sigmaGamma γ x‖ ^ 2\n      ≥ (lpNorm (1 + γ) x) ^ 2 / onesNorm p d := by\n  sorry\n\ntheorem theorem1 {d n K Sout T b : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbn : b ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hθ1 : θ < 1) (hσ : 0 ≤ σ2)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (heta : ∀ k s ω, eta k s ω\n      = ⟪nu k s ω, sigmaGamma γ (nu k s ω)⟫ / (L * ‖sigmaGamma γ (nu k s ω)‖ ^ 2))\n    (hsigmaNu : ∀ k < K, ∀ s < Sout, ∀ ω, sigmaGamma γ (nu k s ω) ≠ 0)\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ (4 * L * onesP) / ((T : ℝ) * (1 - θ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * onesP * σ2 / ((b : ℝ) * θ * (1 - θ)) := by\n  sorry\n\ntheorem theorem2 {d n K Sout T b bH : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbHpos : 0 < bH) (hbn : b ≤ n) (hbHn : bH ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 ζ τ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hσ : 0 ≤ σ2)\n    (hτ : 0 < τ) (hζ : 0 < ζ)\n    (hbH2 : 0 < 2 * (bH : ℝ) - θ * (bH : ℝ) - 1)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n)) (SidxH : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hindepSH : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => SidxH ks.1 ks.2 ω) μ)\n    (gSHa gSHb : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hSHcard : ∀ k s ω, (SidxH k s ω).card = bH)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hgSHa : ∀ k s ω, gSHa k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W (k + 1) (s + 1) ω))\n    (hgSHb : ∀ k s ω, gSHb k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W k (s + 1) ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (eta0 : ℝ) (heta0 : ∀ s ω, eta 0 s ω = eta0)\n    (heta : ∀ k s ω, eta (k + 1) s ω = (ζ / bH)\n      * (‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2\n        / (⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n              sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫\n            + τ * ‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2)))\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hΔW : ∀ k < K, ∀ s < Sout, ∀ ω, W (k + 1) (s + 1) ω - W k (s + 1) ω ≠ 0)\n\n    (hbb_ip : ∀ k s ω, 0 ≤ ⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n        sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫)\n    (hLem1 : ∀ k s ω, eta k s ω ≤ ζ / ((bH : ℝ) * τ))\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ)\n    (hintFgap : ∀ k s,\n      Integrable (fun ω => f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) μ)\n    (hdescent : ∀ k < K, ∀ s < Sout,\n      (∫ ω, (f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) ∂μ)\n        ≤ -((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) / (4 * L * (bH : ℝ) ^ 2 * onesP))\n              * (∫ ω, (lpNorm (1 + γ) (gS k s ω)) ^ 2 ∂μ)\n          + 2 * σ2 / (L * (bH : ℝ) * (b : ℝ) * θ)) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ 4 * (bH : ℝ) ^ 2 * L * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (T : ℝ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * σ2 * (bH : ℝ) * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (b : ℝ) * θ) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["powerball_lower", "theorem1", "theorem2"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 773, "end": 4147, "gold": "{d n K Sout T b : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbn : b ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hθ1 : θ < 1) (hσ : 0 ≤ σ2)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (heta : ∀ k s ω, eta k s ω\n      = ⟪nu k s ω, sigmaGamma γ (nu k s ω)⟫ / (L * ‖sigmaGamma γ (nu k s ω)‖ ^ 2))\n    (hsigmaNu : ∀ k < K, ∀ s < Sout, ∀ ω, sigmaGamma γ (nu k s ω) ≠ 0)\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ (4 * L * onesP) / ((T : ℝ) * (1 - θ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * onesP * σ2 / ((b : ℝ) * θ * (1 - θ))"}]}
{"item_id": "v24-22-1149::theorem2::A-stmt", "paper_id": "v24-22-1149", "target_name": "theorem2", "source": "Research-level", "nl_statement": "Theorem 2. Set $w^{*}=\\arg\\operatorname*{min}_{w}f(w)$ , and choose $S \\subseteq [ n ]$ and $S_{H}\\subseteq [ n ]$ with $| S | = b$ and $| S_{H}| = b_{H}$ respectively. Let T denote the number of total iterations, then, under Assumption 1, Assumption 2 (used in the mini-batch form $\\mathbb{E}[\\nabla f_{S}(w)]=\\mathbb{E}[\\nabla f(w)]$ and $\\mathbb{E}[\\|\\nabla f_{S}(w)-\\nabla f(w)\\|^{2}]\\le\\hat{\\sigma}^{2}/b$ at $w = w_{k}^{s + 1}$ and at $w = \\tilde{w}^{s}$) and Lemma 1, for any $T \\geq 1$ , PB-SVRGE-RSBB (Algorithm 2), for any $\\theta > 0$ with $2 b_{H}- \\theta b_{H}- 1 > 0$, with $p = \\frac{1 + \\gamma}{1 - \\gamma}$, and assuming that for all $0 \\le k \\le K - 1$ and $0 \\le s \\le \\mathfrak{S}- 1$ the per-step bound $\\mathbb{E}[f(w_{k+1}^{s+1})-f(w_{k}^{s+1})]\\leq-\\frac{2b_{H}-\\theta b_{H}-1}{4Lb_{H}^{2}\\|\\mathbf{1}\\|_{p}}\\mathbb{E}[\\|\\nabla f_{S}(w_{k}^{s+1})\\|_{1+\\gamma}^{2}]+\\frac{2\\hat{\\sigma}^{2}}{Lb_{H}b\\theta}$ (eq. (34)) holds, leads to\n\n$$\n\\mathbb{E}\\left[\\frac{1}{T}\\sum_{s=0}^{\\mathfrak{S}-1}\\sum_{k=0}^{K-1}\\|\\nabla f_{S}(w_{k}^{s+1})\\|_{1+\\gamma}^{2}\\right]\\leq\\frac{4b_{H}^{2}L\\|\\mathbf{1}\\|_{p}}{(2b_{H}-\\theta b_{H}-1)T}\\mathbb{E}[f(\\tilde{w}^{0})-f(w^{*})]+\\frac{8\\hat{\\sigma}^{2}b_{H}\\|\\mathbf{1}\\|_{p}}{(2b_{H}-\\theta b_{H}-1)b\\theta}.\\tag{27}\n$$", "context": [{"name": "Setup: finite-sum problem (1) (Section 1)", "statement": "Large numbers of applications in machine learning (Le Thi et al., 2022), computer vision (Xu et al., 2021) and numerical optimization (Fehrman et al., 2020) are often implemented to be a finite-sum structure of the form\n\n$$\n\\operatorname*{min}_{w\\in\\mathbb{R}^{d}}f(w)=\\frac{1}{n}\\sum_{i=1}^{n}f_{i}(w),\\tag{1}\n$$\n\nwhere n denotes the total number of the instances, w defines the parameter to learn and $f_{i}(\\cdot):\\mathbb{R}^{d}\\to\\mathbb{R}$ represents a loss function corresponding to the i-th instance with d dimensions. In this work, we are mainly interested in each $f_{i}(w)$ and $f(w)$ that are L-smooth but may be non-convex. More concretely and vividly, the cases Problem (1) referred into logistic regression, eigenvector computation, multi-kernel learning, neural networks, to name a few."}, {"name": "Notation (Section 1.2)", "statement": "Throughout this work, we take $\\mathbb{R}^{d}$ to define the set of d-dimensional real number vectors. For vectors, the mark w · v is used to denote their inner product. Additionally, we use [n] to define the set $\\{1,2,\\ldots,\\mathtt{n}\\}$ . The symbol $\\| \\cdot \\|$ and $\\| \\cdot \\|_{p}$ are chosen to represent the Euclidean norm and p-norm on $\\mathbb{R}^{d}$ respectively. Moreover, Denote by $\\nabla f(\\boldsymbol{w})$ the gradient of the loss function $f(w)$ . We adopt $\\mathbb{E}[ \\cdot ]$ to represent the expectation with respect to the underlying probability space. We use $(w)_{i}$ to represent the i-th element of vector $\\boldsymbol{w}\\in \\mathbb{R}^{d}$"}, {"name": "SVRG update (3) and Powerball update (4); definition of σ_γ (Section 3)", "statement": "Since this work will delve into the theoretical and numerical characteristics of SVRGE in PSO algorithms, we offer some discussions about the updating scheme of the original SVRG algorithm, originally developed in (Johnson and Zhang, 2013). Comparing with the conventional SGD algorithm that worked with the update scheme:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\nabla f_{i}(w_{k}),\\tag{2}\n$$\n\nfor solving Problem (1), the update scheme of SVRG usually took the form:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\big(\\nabla f_{i}\\big(w_{k}\\big)-\\nabla f_{i}\\big(\\tilde{w}\\big)+\\nabla f\\big(\\tilde{w}\\big)\\big),\\tag{3}\n$$\n\nwhere $\\tilde{w}$ was a snapshot point and $\\eta_{k}> 0$ was the learning rate. In general, there were two loops in the SVRG-like approach, where the iterative rule (3) located in the inner loop. The outer loop was used to store the historical information, then transferred it into the inner loop.\n\nThe iterative scheme (3), utilizing historical information $(\\mathrm{i}.\\mathrm{e}.,\\nabla f_{i}(\\tilde{w})$ and $\\nabla f(\\tilde{w}))$ , made the stochastic optimization algorithms enjoy a lower variance than classical SGD, and resulted in a faster convergence rate and a better generalization in practice.\n\nIn contrast, when dealing with Problem (1), the classical iterative scheme of the PSO algorithm was:\n\n$$\nw_{k+1}=w_{k}-\\eta_{k}\\sigma_{\\gamma}(\\nabla f_{i}(w_{k})),\\tag{4}\n$$\n\nwhere the power coefficient $\\gamma\\in [0,1),\\sigma_{\\gamma}(\\nabla f_{i}(w_{k}))=\\mathrm{sign}(\\nabla f_{i}(w_{k}))|\\nabla f_{i}(w_{k})|^{\\gamma}$ Here $\\mathrm{sign}(x)$ backed to the sign of $x$, or 0 if $x = 0$. Also, it was noted that the operations sign(w) and $\\sigma_{\\gamma}(w)$ was applied to each element if the parameter $y \\in \\mathbb{R}^{d}$ was a vector. For example, $\\sigma_{\\gamma}(y)=(\\sigma_{\\gamma}((y)_{1}),\\sigma_{\\gamma}((y)_{2}),\\dots,\\sigma_{\\gamma}((y)_{d}))$"}, {"name": "Algorithm 1: PB-SVRGE", "statement": "Naturally, it was asked whether SVRGE created a positive effect on PSO algorithms? In order to respond to this question, we first proposed the PB-SVRGE algorithm (Algorithm 1) and analyzed its theoretical and empirical properties in the upcoming sections.\n\nAlgorithm 1 PB-SVRGE   \nInput: $\\widetilde{w}^{0}= w_{K}^{0}= w^{0}$ , epoch length $K,$ outer loop size ${\\mathfrak{S}}= T / K$ , mini-batch size $b$, learning rate   \n$\\{\\eta_{j}\\}_{j = 0}^{K - 1}$ , power coefficient $\\gamma .$   \nfor $s=0,\\ldots,{\\mathfrak{S}}-1$ do   \n$w_{0}^{s + 1}= w_{K}^{s}$   \n$g^{s+1}=\\nabla f(\\tilde{w}^{s})$   \nfor $k=0,\\ldots,K-1$ do   \nRandomly choose $S \\subseteq [ n ]$ with $|S| =b,$ compute a new stochastic estimate $\\nabla f(w_{k}^{s})$   \n$\\nu_{k}^{s+1}=\\nabla f_{S}(w_{k}^{s+1})-\\nabla f_{S}(\\tilde{w}^{s})+g^{s+1}$   \n$w_{k+1}^{s+1}=w_{k}^{s+1}-\\eta_{k}\\sigma(\\nu_{k}^{s+1})$   \nend for   \nSet $\\widetilde{w}^{s + 1}= w_{K}^{s + 1}$   \nend for"}, {"name": "Remark 1(i): mini-batch estimators ∇f_S", "statement": "Remark 1: For PB-SVRGE, the following remarks were necessary to be demonstrated:\n\n(i) PR-SVRGE utilized the similar algorithm framework as SVRG-like algorithms. The main difference between these two approaches is that the introduction of the Powerball function in PB-SVRGE. Specifically, we can view the SVRG-like algorithm as an extension of PB-SVRGE when taking $\\gamma = 1$ . Moreover, we considered PSO algorithms in the mini-batching setting, where $\\begin{array}{r}{\\nu_{k}^{s+1}=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(w_{k}^{s+1})-\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{S}(\\tilde{w}^{s})+\\nabla f(\\tilde{w}^{s})\\ (S\\subseteq [n]}\\end{array}$ with b batch samples). Here, $\\begin{array}{r}{\\nabla f_{S}(w_{k}^{s+1})=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(w_{k}^{s+1})}\\end{array}$ and $\\begin{array}{r}{\\nabla f_{S}(\\tilde{w}^{s})=\\frac{1}{b}\\sum_{i\\in S}\\nabla f_{i}(\\tilde{w}^{s})}\\end{array}$"}, {"name": "Assumption 1 (L-smoothness) and inequality (6)", "statement": "This work considers the theoretical properties of PB-SVRGE on non-convex optimization problems.   \nTo proceed the analysis, we make the following standard smoothness assumptions.\n\nAssumption 1. Each cost function $f_{i}(w)$ in Problem (1) is differential and has a L-Lipschitz gradient, that is,\n\n$$\n\\forall w,\\nu\\in\\mathbb{R}^{d},\\quad\\|\\nabla f_{i}(w)-\\nabla f_{i}(\\nu)\\|\\le L\\|w-\\nu\\|.\\tag{5}\n$$\n\nThe conclusion in Assumption 1 indicates that the average function $f(w)$ , presented in Problem (1), has a L-Lipschitz gradient as well.\n\nClearly, from the smooth property of the cost function, we have the following result:\n\n$$\nf(w)\\leq f(\\nu)+\\nabla f(\\nu)^{T}(w-\\nu)+\\frac{L}{2}\\parallel w-\\nu\\parallel^{2}.\\tag{6}\n$$"}, {"name": "Assumption 2 (unbiased oracle, bounded variance) and the mini-batch variance bound σ̂²/b", "statement": "Assumption 2. The stochastic gradient oracle is an independent and unbiased estimator of the gradient and satisfies\n\n$$\n\\begin{array}{r l}&{\\mathbb{E}_{\\xi_{i}}[\\nabla f(w,\\xi_{i})]=\\nabla f(w),\\forall w\\in\\mathbb{R}^{d},}\\\\ &{\\mathbb{E}_{\\xi_{i}}[\\|\\nabla f(w,\\xi_{i})-\\nabla f(w)\\|^{2}]\\le\\hat{\\sigma}^{2},\\forall w\\in\\mathbb{R}^{d},}\\end{array}\\tag{7}\n$$\n\nwhere $\\nabla f_{i}(w)=\\nabla f(w,\\xi_{i})$ and $\\xi_{i}$ denotes a random variable.\n\nTwo assumptions mentioned above were very common in the most existing studies for finding a first-order stationary point of non-convex or convex loss functions (Ghadimi and Lan, 2013; Wei et al., 2021; Zhou et al., 2018). Observe that, by (5), the gradient of the loss function does not change arbitrary rapid, and by (7), the variance of the random variable $\\mathbb{E}[\\|\\nabla f_{i}(w)-\\nabla f(w)\\|^{2}]$ has an upper bound. Notice that, when taking b samples in vanilla SGD at each iterative step, we have $\\begin{array}{r}{\\mathbb{E}[\\|\\nabla f_{S}(w)-\\nabla f(w)\\|^{2}]\\le\\frac{\\hat{\\sigma}^{2}}{b}}\\end{array}$"}, {"name": "RSBB learning rate (25) (Section 4)", "statement": "We here briefly discussed the RSBB method. In RSBB, it also employed the similar updating rule to SGD in (2), i.e., $w_{k+1}=w_{k}-\\eta_{k}\\nabla f_{i}\\big(w_{k}\\big)$ , but the learning rate\n\n$$\n\\eta_{k}= \\frac{\\zeta}{b_{H}}\\cdot \\frac{\\| s_{k}\\|^{2}}{s_{k}\\cdot y_{k}+ \\tau \\| s_{k}\\|^{2}},\\tag{25}\n$$\n\nwhere $s_{k}= w_{k}- w_{k - 1}$ and $y_{k}=\\nabla f_{S_{H}}(w_{k})-\\nabla f_{S_{H}}(w_{k-1})$ for $k \\geq 1$ . Note that, here, $S_{H}\\subseteq [ n ]$ with $b_{H}$ samples $(\\mathrm{i}.\\mathrm{e}.,|S_{H}| =b_{H})$ and $\\begin{array}{r}{\\nabla f_{S_{H}}(w_{k})=\\frac{1}{b_{H}}\\sum_{i\\in S_{H}}\\nabla f_{i}(w_{k})}\\end{array}$ . In addition, $\\zeta$ and $\\tau$ were two positive parameters. Specifically, $\\zeta$ was employed to control the convergence speed. The term $\\tau \\| s_{k}\\|^{2}$ was used to avoid the denominator of the original BB step size being zero. For the BB-type technique, it enjoyed a high-order information, resorting employing a simple but a low computational cost way to approximate the Hessian matrix unlike Newton or quasi-Newton type methods."}, {"name": "Algorithm 2: PB-SVRGE-RSBB", "statement": "Algorithm 2 PB-SVRGE-RSBB\nInput: $\\widetilde{w}^{0}= w_{K}^{0}= w^{0},$ epoch length $K,$ outer loop size ${\\mathfrak{S}}= T / K$ ,mini-batch size $b$, initial\nlearning rate $\\eta_{0}$, power coefficient $\\gamma,$ positive parameters $\\zeta$ and $\\tau$.\nfor $s=0,\\ldots,\\mathfrak{S}-1$ do\n$w_{0}^{s + 1}= w_{K}^{s}$\n$\\boldsymbol{g}^{s+1}=\\nabla f(\\tilde{w}^{s})$\nfor $k=0,\\ldots,K-1$ do\nRandomly choose $S \\subseteq [ n ]$ with $|S| =b,$ compute a new stochastic estimate $\\nabla f(w_{k}^{s+1})$\n$\\nu_{k}^{s+1}=\\nabla f_{S}(w_{k}^{s+1})-\\nabla f_{S}(\\tilde{w}^{s})+g^{s+1}$\n$w_{k+1}^{s+1}=w_{k}^{s+1}-\\eta_{k}\\sigma_{\\gamma}(\\nu_{k}^{s+1})$\nRandomly choose $S_{H}\\subseteq [ n ]$ with $| S_{H}| = b_{H}$ and compute the learning rate\n$\\eta_{k+1}=\\frac{\\zeta}{b_{H}}\\cdot\\frac{\\|w_{k+1}^{s+1}-w_{k}^{s+1}\\|^{2}}{(w_{k+1}^{s+1}-w_{k}^{s+1})\\cdot(\\sigma_{\\gamma}(\\nabla f_{S_{H}}(w_{k+1}^{s+1}))-\\sigma_{\\gamma}(\\nabla f_{S_{H}}(w_{k}^{s+1})))+\\tau\\|w_{k+1}^{s+1}-w_{k}^{s+1}\\|^{2}}$\nend for\nSet $\\widetilde{w}^{s + 1}= w_{K}^{s + 1}$\nend for"}, {"name": "Lemma 1 (step-size bound η_k ≤ ζ/(b_H τ))", "statement": "The boundary of the learning rate $\\eta_{k}$ was first established in the following lemma.\n\nLemma 1. From the definition of $\\eta_{k}$ in PB-SVRGE-RSBB (Algorithm 2), we have the following conclusion\n\n$$\n\\eta_{k}\\leq \\frac{\\zeta}{b_{H}\\tau}.\\tag{26}\n$$"}, {"name": "Remark after Lemma 1: positivity of s_k·y_k", "statement": "Note that the original BB technique (Barzilai and Borwein, 1988) that used the iterative scheme $\\begin{array}{r}{\\eta_{k}= \\frac{\\| s_{k}\\|^{2}}{s_{k}^{T}y_{k}}}\\end{array}$ to update the learning rate usually kept such the learning rate positive under deterministic optimization background, which meant that the inner product of the vectors $s_{k}$ and $y_{k}$ was positive. Due to the introduction of the mini-batching and variance reduced techniques under stochastic optimization setting, it, here, still almost kept the inner product of the vectors $s_{k}$ and $y_{k}$ positive. This is indeed the case and can be easily confirmed through numerical experiments."}, {"name": "Inequalities (17) and (18), from the proof of Theorem 1", "statement": "Now, we consider the second term and the third term of the right hand side of (16), respectively. First, considering $\\| \\nu_{k}^{s + 1}\\|_{1 + \\gamma}^{2},$ defining in PB-SVRGE (Algorithm 1), we have\n\n$$\n\\begin{array}{r l}&{\\|\\boldsymbol{\\nu}_{k}^{s+1}\\|_{1+\\gamma}^{2}=\\|\\nabla f_{S}(\\boldsymbol{w}_{k}^{s+1})-\\nabla f_{S}(\\tilde{\\boldsymbol{w}}^{s})+\\nabla f(\\tilde{\\boldsymbol{w}}^{s})\\|_{1+r}^{2}}\\\\ &{\\qquad\\ge\\displaystyle\\frac{1}{2}\\|\\nabla f_{S}(\\boldsymbol{w}_{k}^{s+1})\\|_{1+\\gamma}^{2}-\\|\\nabla f_{S}(\\tilde{\\boldsymbol{w}}^{s})-\\nabla f(\\tilde{\\boldsymbol{w}}^{s})\\|_{1+r}^{2},}\\end{array}\\tag{17}\n$$\n\nwhere the first inequality holds due to the fact $\\begin{array}{r}{\\| a \\|_{1 + \\gamma}^{2}\\geq \\frac{1}{2}\\| b \\|_{1 + \\gamma}^{2}- \\| b - a \\|_{1 + \\gamma}^{2}.}\\end{array}$\n\nSecond, considering $\\|\\nu_{k}^{s+1}-\\nabla f(w_{k}^{s+1})\\|^{2}$ , we further obtain\n\n$$\n\\begin{array}{r l}&{\\|\\nu_{k}^{s+1}-\\nabla f(w_{k}^{s+1})\\|^{2}=\\|\\nabla f_{S}(w_{k}^{s+1})-\\nabla f_{S}(\\tilde{w}^{s})+\\nabla f(\\tilde{w}^{s})-\\nabla f(w_{k}^{s+1})\\|^{2}}\\\\ &{\\qquad\\le 2\\|\\nabla f_{S}(w_{k}^{s+1})-\\nabla f(w_{k}^{s+1})\\|^{2}+2\\|\\nabla f_{S}(\\tilde{w}^{s})-\\nabla f(\\tilde{w}^{s})\\|^{2}}\\end{array}\\tag{18}\n$$\n\nwhere the first inequality can be satisfied since the fact $\\| a + b \\|^{2}\\leq 2 \\| a \\|^{2}+ 2 \\| b \\|^{2}$ holds."}, {"name": "Eqs. (14)-(15) Hölder lower bound (another result of the same paper)", "statement": "**Eqs. (14)–(15)**.\n$(\\langle x, \\sigma_\\gamma(x)\\rangle)^2 / \\|\\sigma_\\gamma(x)\\|^2 \\ge \\|x\\|_{1+\\gamma}^2 / \\|\\mathbb{1}\\|_p$, with $p = (1+\\gamma)/(1-\\gamma)$,\nfor $x \\in \\mathbb{R}^d$ with $0 < d$, $0 \\le \\gamma < 1$ (the paper's standing range for the power\ncoefficient, eq. (4)), and $\\sigma_\\gamma(x) \\neq 0$ so that the left denominator is nonzero."}], "nl_notes": "### Theorem 2 (PB-SVRGE-RSBB, eq. 27)\nFOR THE SOLVER: use `hdescent` directly; you do not need to derive eq. (34).\n\nNOTE (Theorem 2) — Under the assumptions the paper names, (27) is false. Its proof replaces η_k in (28) by the upper bound of Lemma 1 where a lower bound is needed, sets ζ = τν_k^{s+1}·σ_γ(ν_k^{s+1})/(L‖σ_γ(ν_k^{s+1})‖²), which changes with the iterate, although ζ is a fixed positive parameter of Algorithm 2, and passes from (32) to (33) by deleting a nonnegative term. The statement therefore assumes the per-step descent estimate (34) instead of deriving it.", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def sigmaGamma {d : ℕ} (γ : ℝ) (x : EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) :=\n  (WithLp.equiv 2 (Fin d → ℝ)).symm (fun i => Real.sign (x i) * |x i| ^ γ)\n\nnoncomputable def lpNorm {d : ℕ} (r : ℝ) (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  (∑ i, |x i| ^ r) ^ (1 / r)\n\nnoncomputable def onesNorm (p : ℝ) (d : ℕ) : ℝ := (d : ℝ) ^ (1 / p)\n\nlemma powerball_lower {d : ℕ} (γ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1)\n    (p : ℝ) (hp : p = (1 + γ) / (1 - γ)) (hd : 0 < d)\n    (x : EuclideanSpace ℝ (Fin d)) (hx : sigmaGamma γ x ≠ 0) :\n    (⟪x, sigmaGamma γ x⟫) ^ 2 / ‖sigmaGamma γ x‖ ^ 2\n      ≥ (lpNorm (1 + γ) x) ^ 2 / onesNorm p d := by\n  sorry\n\ntheorem theorem1 {d n K Sout T b : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbn : b ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hθ1 : θ < 1) (hσ : 0 ≤ σ2)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (heta : ∀ k s ω, eta k s ω\n      = ⟪nu k s ω, sigmaGamma γ (nu k s ω)⟫ / (L * ‖sigmaGamma γ (nu k s ω)‖ ^ 2))\n    (hsigmaNu : ∀ k < K, ∀ s < Sout, ∀ ω, sigmaGamma γ (nu k s ω) ≠ 0)\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ (4 * L * onesP) / ((T : ℝ) * (1 - θ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * onesP * σ2 / ((b : ℝ) * θ * (1 - θ)) := by\n  sorry\n\ntheorem theorem2 {d n K Sout T b bH : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbHpos : 0 < bH) (hbn : b ≤ n) (hbHn : bH ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 ζ τ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hσ : 0 ≤ σ2)\n    (hτ : 0 < τ) (hζ : 0 < ζ)\n    (hbH2 : 0 < 2 * (bH : ℝ) - θ * (bH : ℝ) - 1)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n)) (SidxH : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hindepSH : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => SidxH ks.1 ks.2 ω) μ)\n    (gSHa gSHb : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hSHcard : ∀ k s ω, (SidxH k s ω).card = bH)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hgSHa : ∀ k s ω, gSHa k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W (k + 1) (s + 1) ω))\n    (hgSHb : ∀ k s ω, gSHb k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W k (s + 1) ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (eta0 : ℝ) (heta0 : ∀ s ω, eta 0 s ω = eta0)\n    (heta : ∀ k s ω, eta (k + 1) s ω = (ζ / bH)\n      * (‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2\n        / (⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n              sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫\n            + τ * ‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2)))\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hΔW : ∀ k < K, ∀ s < Sout, ∀ ω, W (k + 1) (s + 1) ω - W k (s + 1) ω ≠ 0)\n\n    (hbb_ip : ∀ k s ω, 0 ≤ ⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n        sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫)\n    (hLem1 : ∀ k s ω, eta k s ω ≤ ζ / ((bH : ℝ) * τ))\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ)\n    (hintFgap : ∀ k s,\n      Integrable (fun ω => f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) μ)\n    (hdescent : ∀ k < K, ∀ s < Sout,\n      (∫ ω, (f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) ∂μ)\n        ≤ -((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) / (4 * L * (bH : ℝ) ^ 2 * onesP))\n              * (∫ ω, (lpNorm (1 + γ) (gS k s ω)) ^ 2 ∂μ)\n          + 2 * σ2 / (L * (bH : ℝ) * (b : ℝ) * θ)) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ 4 * (bH : ℝ) ^ 2 * L * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (T : ℝ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * σ2 * (bH : ℝ) * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (b : ℝ) * θ) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["powerball_lower", "theorem1", "theorem2"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 4180, "end": 9053, "gold": "{d n K Sout T b bH : ℕ}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (hT : T = K * Sout) (hT1 : 1 ≤ T) (hK : 0 < K) (hSout : 0 < Sout)\n    (hb : 0 < b) (hbHpos : 0 < bH) (hbn : b ≤ n) (hbHn : bH ≤ n) (hd : 0 < d) (hn : 0 < n)\n    (γ p L θ onesP σ2 ζ τ : ℝ) (hγ0 : 0 ≤ γ) (hγ1 : γ < 1) (hp : p = (1 + γ) / (1 - γ))\n    (hL : 0 < L) (hθ0 : 0 < θ) (hσ : 0 ≤ σ2)\n    (hτ : 0 < τ) (hζ : 0 < ζ)\n    (hbH2 : 0 < 2 * (bH : ℝ) - θ * (bH : ℝ) - 1)\n    (honesP : onesP = onesNorm p d)\n    (fi : Fin n → EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradfi : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hsmoothi : ∀ i, ∀ x y : EuclideanSpace ℝ (Fin d),\n      ‖gradfi i x - gradfi i y‖ ≤ L * ‖x - y‖)\n    (hdiff : ∀ i, ∀ x : EuclideanSpace ℝ (Fin d), HasGradientAt (fi i) (gradfi i x) x)\n    (f : EuclideanSpace ℝ (Fin d) → ℝ)\n    (gradf : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d))\n    (hf : f = fun w => (1 / (n : ℝ)) * ∑ i, fi i w)\n    (hgradf : gradf = fun w => (1 / (n : ℝ)) • ∑ i, gradfi i w)\n    (hsmooth : ∀ x y : EuclideanSpace ℝ (Fin d),\n      f x ≤ f y + ⟪gradf y, x - y⟫ + L / 2 * ‖x - y‖ ^ 2)\n    (wstar : EuclideanSpace ℝ (Fin d)) (hmin : ∀ x, f wstar ≤ f x)\n    (W gS : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (gSt : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (nu : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d)) (eta : ℕ → ℕ → Ω → ℝ)\n    (Sidx : ℕ → ℕ → Ω → Finset (Fin n)) (SidxH : ℕ → ℕ → Ω → Finset (Fin n))\n    (hindepS : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => Sidx ks.1 ks.2 ω) μ)\n    (hindepSH : ProbabilityTheory.iIndepFun (ι := ℕ × ℕ)\n        (m := fun _ => (⊤ : MeasurableSpace (Finset (Fin n))))\n        (fun ks ω => SidxH ks.1 ks.2 ω) μ)\n    (gSHa gSHb : ℕ → ℕ → Ω → EuclideanSpace ℝ (Fin d))\n    (hScard : ∀ k s ω, (Sidx k s ω).card = b)\n    (hSHcard : ∀ k s ω, (SidxH k s ω).card = bH)\n    (hgS : ∀ k s ω, gS k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W k (s + 1) ω))\n    (hgSt : ∀ k s ω, gSt k s ω = (1 / (b : ℝ)) • ∑ i ∈ Sidx k s ω, gradfi i (W K s ω))\n    (hgSHa : ∀ k s ω, gSHa k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W (k + 1) (s + 1) ω))\n    (hgSHb : ∀ k s ω, gSHb k s ω = (1 / (bH : ℝ)) • ∑ i ∈ SidxH k s ω, gradfi i (W k (s + 1) ω))\n    (hnu : ∀ k s ω, nu k s ω = gS k s ω - gSt k s ω + gradf (W K s ω))\n    (eta0 : ℝ) (heta0 : ∀ s ω, eta 0 s ω = eta0)\n    (heta : ∀ k s ω, eta (k + 1) s ω = (ζ / bH)\n      * (‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2\n        / (⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n              sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫\n            + τ * ‖W (k + 1) (s + 1) ω - W k (s + 1) ω‖ ^ 2)))\n    (hupd : ∀ k s ω,\n      W (k + 1) (s + 1) ω = W k (s + 1) ω - eta k s ω • sigmaGamma γ (nu k s ω))\n    (hlink : ∀ s ω, W 0 (s + 1) ω = W K s ω)\n\n    (hΔW : ∀ k < K, ∀ s < Sout, ∀ ω, W (k + 1) (s + 1) ω - W k (s + 1) ω ≠ 0)\n\n    (hbb_ip : ∀ k s ω, 0 ≤ ⟪W (k + 1) (s + 1) ω - W k (s + 1) ω,\n        sigmaGamma γ (gSHa k s ω) - sigmaGamma γ (gSHb k s ω)⟫)\n    (hLem1 : ∀ k s ω, eta k s ω ≤ ζ / ((bH : ℝ) * τ))\n\n    (hintGS : ∀ k s, Integrable (fun ω => gS k s ω) μ)\n    (hintGSt : ∀ k s, Integrable (fun ω => gSt k s ω) μ)\n    (hintGW : ∀ k s, Integrable (fun ω => gradf (W k (s + 1) ω)) μ)\n    (hintGWt : ∀ s, Integrable (fun ω => gradf (W K s ω)) μ)\n    (hunb1 : ∀ k s, ∫ ω, gS k s ω ∂μ = ∫ ω, gradf (W k (s + 1) ω) ∂μ)\n    (hunb2 : ∀ k s, ∫ ω, gSt k s ω ∂μ = ∫ ω, gradf (W K s ω) ∂μ)\n\n    (hintVar1 : ∀ k s, Integrable (fun ω => ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2) μ)\n    (hintVar2 : ∀ k s, Integrable (fun ω => ‖gSt k s ω - gradf (W K s ω)‖ ^ 2) μ)\n    (hvar1 : ∀ k s, ∫ ω, ‖gS k s ω - gradf (W k (s + 1) ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvar2 : ∀ k s, ∫ ω, ‖gSt k s ω - gradf (W K s ω)‖ ^ 2 ∂μ ≤ σ2 / (b : ℝ))\n    (hvarPrim : ∀ w : EuclideanSpace ℝ (Fin d),\n      (1 / (n : ℝ)) * ∑ i, ‖gradfi i w - gradf w‖ ^ 2 ≤ σ2)\n    (hintSum : Integrable (fun ω => ∑ s ∈ Finset.range Sout,\n      ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) μ)\n    (hintGap : Integrable (fun ω => f (W K 0 ω) - f wstar) μ)\n    (hintFgap : ∀ k s,\n      Integrable (fun ω => f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) μ)\n    (hdescent : ∀ k < K, ∀ s < Sout,\n      (∫ ω, (f (W (k + 1) (s + 1) ω) - f (W k (s + 1) ω)) ∂μ)\n        ≤ -((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) / (4 * L * (bH : ℝ) ^ 2 * onesP))\n              * (∫ ω, (lpNorm (1 + γ) (gS k s ω)) ^ 2 ∂μ)\n          + 2 * σ2 / (L * (bH : ℝ) * (b : ℝ) * θ)) :\n    (1 / (T : ℝ)) * (∫ ω, (∑ s ∈ Finset.range Sout,\n        ∑ k ∈ Finset.range K, (lpNorm (1 + γ) (gS k s ω)) ^ 2) ∂μ)\n      ≤ 4 * (bH : ℝ) ^ 2 * L * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (T : ℝ))\n          * (∫ ω, (f (W K 0 ω) - f wstar) ∂μ)\n        + 8 * σ2 * (bH : ℝ) * onesP\n          / ((2 * (bH : ℝ) - θ * (bH : ℝ) - 1) * (b : ℝ) * θ)"}]}
{"item_id": "v3-bartlett02a::thm8::A-stmt", "paper_id": "v3-bartlett02a", "target_name": "thm8", "source": "Research-level", "nl_statement": "Theorem 8 Let $\\mathcal{L}:\\mathcal{Y}\\times\\mathcal{A}\\longrightarrow[0,1]$ be a loss function, and let $\\phi:\\mathcal{Y}\\times\\mathcal{A}\\longrightarrow[0,1]$ be a cost function that dominates it. Take $F$ to be a class of functions from $\\mathcal{X}$ into $\\mathcal{A}$, and suppose the pairs $(X_{i},Y_{i})_{i=1}^{n}$ are drawn independently from the probability measure $P$. Then for every integer $n$ and every $0 < \\delta < 1$, the following holds with probability no less than $1 - \\delta$ over samples of size $n$: for all $f$ in $F$,\n\n$$\n\\mathbf{E}\\mathcal{L}(Y,f(X))\\leq\\hat{\\mathbf{E}}_{n}\\phi(Y,f(X))+R_{n}(\\tilde{\\phi}\\circ F)+\\sqrt{\\frac{8\\ln(2/\\delta)}{n}},\n$$\n\nwith $\\tilde{\\phi}\\circ F=\\{(x,y)\\mapsto\\phi(y,f(x))-\\phi(y,0):f\\in F\\}$", "context": [{"name": "Setup: decision-theoretic learning setting (Section 1)", "statement": "The following section presents a bound of this type that holds in a broader, decision-theoretic framework. In that framework there are an input space $\\mathcal{X}$, an action space $\\mathcal{A}$ and an output space $\\mathcal{Y}$. The $n$ training examples $(X_{1},Y_{1}),\\dots,(X_{n},Y_{n})$ are drawn independently from a probability measure $P$ on $\\mathcal{X}\\times \\mathcal{Y}$. A loss function $\\mathcal{L}:\\mathcal{Y}\\times\\mathcal{A}\\longrightarrow[0,1]$ is given, where $\\mathcal{L}(y,a)$ measures the cost incurred by choosing action $a \\in{\\mathcal{A}}$ when the outcome turns out to be $y \\in \\mathcal{Y}$. Learning seeks a function $f$ from $\\mathcal{X}$ to ${\\mathcal{A}}$ whose expected loss $\\mathbf{E}{\\mathcal{L}}(Y,f(X))$ is as small as possible."}, {"name": "Definition 2 (maximum discrepancy, Rademacher and Gaussian complexities)", "statement": "Definition 2 Suppose $\\mu$ is a probability distribution on a set $\\mathcal{X}$, and $X_{1},\\ldots,X_{n}$ are independent samples drawn from $\\mu$. Let $F$ be a class of functions from $\\mathcal{X}$ to $\\mathbb{R}$. The maximum discrepancy of $F$ is the random variable\n\n$$\n\\hat{D}_{n}(F)=\\operatorname*{sup}_{f\\in F}\\left(\\frac{2}{n}\\sum_{i=1}^{n/2}f(X_{i})-\\frac{2}{n}\\sum_{i=n/2+1}^{n}f(X_{i})\\right).\n$$\n\nIts expectation, the expected maximum discrepancy of $F$, is written $D_{n}(F)=\\mathbf{E}\\hat{D}_{n}(F)$\n\nNext, let\n\n$$\n{\\hat{R}}_{n}(F)=\\mathbf{E}[\\operatorname*{sup}_{f\\in F}|{\\frac{2}{n}}\\sum_{i=1}^{n}\\sigma_{i}f(X_{i})| |X_{1},\\ldots,X_{n}]\\ ,\n$$\n\nwith $\\sigma_{1},\\ldots,\\sigma_{n}$ independent random variables, each uniform on $\\{\\pm 1 \\}$. The Rademacher complexity of $F$ is then $R_{n}(F)=\\mathbf{E}\\hat{R}_{n}(F)$. In the same way, let\n\n$$\n\\hat{G}_{n}(F)=\\mathbf{E}[\\operatorname*{sup}_{f\\in F}|{\\frac{2}{n}}\\sum_{i=1}^{n}g_{i}f(X_{i})| |X_{1},\\ldots,X_{n}]\\ ,\n$$\n\nwith $g_{1},\\ldots,g_{n}$ independent Gaussian $N(0,1)$ random variables. The Gaussian complexity of $F$ is $G_{n}(F)=\\mathbf{E}\\hat{G}_{n}(F)$"}, {"name": "Notation: empirical average hat E_n h", "statement": "Given a training sample $(X_{1},Y_{1}),\\dots,(X_{n},Y_{n})$ and a function $h : \\mathcal{X}\\times \\mathcal{Y}\\longrightarrow \\mathbb{R}$, we use the notation\n\n$$\n\\hat{\\mathbf{E}}_{n}h(X,Y)=(1/n)\\sum_{i=1}^{n}h(X_{i},Y_{i}).\n$$"}, {"name": "Notation: dominating cost function, phi o F, function-class operations (Section 2)", "statement": "First, some notation. For an independent sample $(X_{i},Y_{i})_{i=1}^{n}$ with the distribution of $(X,Y)$, $P_{n}$ is the empirical measure supported on the sample, and $\\mu_{n}$ is the empirical measure supported on $(X_{i})_{i=1}^{n}$. A function $\\phi : \\mathcal{Y}\\times \\mathcal{A}\\to \\mathbb{R}$ is said to dominate a loss function $\\mathcal{L}$ when $\\phi(y,a)\\geq{\\mathcal{L}}(y,a)$ holds for every $y \\in \\mathcal{Y}$ and every $a \\in{\\mathcal{A}}$. Given a class of functions $F$: conv $F$ denotes the set of convex combinations of members of $F$; $- F = \\{- f : f \\in F \\}$; absconv $F$ denotes the set of convex combinations of members of $F \\cup{- F}$; and $c F = \\{c f : f \\in F \\}$. When $\\phi$ is defined on the range of the functions in $F$, we put $\\phi \\circ F = \\{\\phi \\circ f | f \\in F \\}$. The characteristic function of a set $A$ is written ${\\bf 1}_{A}$ or $\\mathbf{1}(A)$. Lastly, $C$ and $c$ denote constants whose values may differ from one line to the next, or even inside a single line."}, {"name": "Theorem 9 (McDiarmid's Inequality)", "statement": "The argument relies on McDiarmid's inequality (McDiarmid, 1989).\n\nTheorem 9 (McDiarmid's Inequality) Suppose $X_{1},...,X_{n}$ are independent random variables with values in a set $A$, and $f : A^{n}\\to \\mathbb{R}$ is such that\n\n$$\n\\operatorname*{sup}_{x_{1},\\ldots,x_{n},x_{i}^{\\prime}\\in A}\\left|f(x_{1},...,x_{n})-f(x_{1},...,x_{i-1},x_{i}^{\\prime},x_{i+1},... x_{n})\\right|\\leq c_{i}\n$$\n\nholds for each $1 \\leq i \\leq n$. Then every $t > 0$ satisfies\n\n$$\nP\\left\\{f(X_{1},...,X_{n})-{\\bf E}f(X_{1},...,X_{n})\\geq t\\right\\}\\leq e^{-2t^{2}/\\sum_{i=1}^{n}c_{i}^{2}}.\n$$"}], "nl_notes": "### Theorem 8\nNOTE (Theorem 8) — The Lean assumes the index set of the function class is countable (`[Countable ι]`); the paper states no such condition. Without it the theorem is not provable in ZFC: under CH, a class indexed by a well-order of [0,1] of type ω₁ satisfies every other hypothesis while the empirical and expected quantities separate maximally at the supremum of the sample points. This is the usual countability / permissibility convention for suprema over a function class.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nnoncomputable section\n\nuniverse u v w\n\nopen MeasureTheory ProbabilityTheory Real\n\ndef Signs (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n : ℕ} : Fintype (Signs n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\ndef paperEmpRad {ι : Type v} {𝒮 : Type w} (n : ℕ) (f : ι → 𝒮 → ℝ) (S : Fin n → 𝒮) : ℝ :=\n  (Fintype.card (Signs n) : ℝ)⁻¹ *\n    ∑ σ : Signs n, ⨆ i, |(2 / (n : ℝ)) * ∑ k : Fin n, (σ k : ℝ) * f i (S k)|\n\ndef paperRad {Ω : Type u} {ι : Type v} {𝒮 : Type w} [MeasurableSpace Ω]\n    (n : ℕ) (f : ι → 𝒮 → ℝ) (μ : Measure Ω) (X : Ω → 𝒮) : ℝ :=\n  ∫ ω : Fin n → Ω, paperEmpRad n f (fun k => X (ω k)) ∂(Measure.pi (fun _ => μ))\n\ndef centeredClass {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v} [Zero 𝒜]\n    (φ : 𝒱 → 𝒜 → ℝ) (F : ι → 𝒳 → 𝒜) : ι → (𝒳 × 𝒱) → ℝ :=\n  fun i p => φ p.2 (F i p.1) - φ p.2 0\n\ndef expectedLoss {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} [MeasurableSpace Ω]\n    (μ : Measure Ω) (X : Ω → 𝒳) (Y : Ω → 𝒱) (L : 𝒱 → 𝒜 → ℝ) (f : 𝒳 → 𝒜) : ℝ :=\n  ∫ ω, L (Y ω) (f (X ω)) ∂μ\n\ndef empiricalCost {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*}\n    (n : ℕ) (X : Ω → 𝒳) (Y : Ω → 𝒱) (L : 𝒱 → 𝒜 → ℝ) (f : 𝒳 → 𝒜)\n    (ω : Fin n → Ω) : ℝ :=\n  (n : ℝ)⁻¹ * ∑ k : Fin n, L (Y (ω k)) (f (X (ω k)))\n\ndef zeroOneLoss (y a : ℝ) : ℝ := by\n  classical exact if y = a then 0 else 1\n\ntheorem thm8 {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Zero 𝒜] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → 𝒱)\n    (L φ : 𝒱 → 𝒜 → ℝ)\n    (hL01 : ∀ y a, L y a ∈ Set.Icc (0 : ℝ) 1)\n    (hφ01 : ∀ y a, φ y a ∈ Set.Icc (0 : ℝ) 1)\n    (hdom : ∀ y a, L y a ≤ φ y a)\n    (F : ι → 𝒳 → 𝒜)\n    (hLmeas : ∀ i, Measurable (fun ω => L (Y ω) (F i (X ω))))\n    (hφmeas : ∀ i, Measurable (fun ω => φ (Y ω) (F i (X ω))))\n    (hφ0meas : Measurable (fun ω => φ (Y ω) 0))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω =>\n        paperEmpRad n (centeredClass φ F) (fun k => (X (ω k), Y (ω k))))\n      (Measure.pi (fun _ => μ)))\n    (hDevMeas : Measurable\n      (fun ω : Fin n → Ω =>\n        ⨆ i : ι, ((∫ ω' : Ω, centeredClass φ F i (X ω', Y ω') ∂μ)\n          - ((n : ℝ)⁻¹ * ∑ k : Fin n, centeredClass φ F i (X (ω k), Y (ω k))))))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y L (F i) ≤\n            empiricalCost n X Y φ (F i) ω\n            + paperRad n (centeredClass φ F) μ (fun ω' => (X ω', Y ω'))\n            + Real.sqrt (8 * Real.log (2 / δ) / n)} := by\n  sorry\n\ntheorem theorem5b_rademacher_bound {Ω : Type u} {𝒳 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → ℝ)\n    (hY : ∀ ω, Y ω ∈ ({-1, 1} : Set ℝ))\n    (F : ι → 𝒳 → ℝ) (hF : ∀ i x, F i x ∈ ({-1, 1} : Set ℝ))\n    (hLmeas : ∀ i, Measurable (fun ω => zeroOneLoss (Y ω) (F i (X ω))))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (hEvt : MeasurableSet\n      {ω : Fin n → Ω |\n        ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n          empiricalCost n X Y zeroOneLoss (F i) ω\n          + paperRad n F μ X / 2\n          + Real.sqrt (Real.log (1 / δ) / (2 * n))}) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n            empiricalCost n X Y zeroOneLoss (F i) ω\n            + paperRad n F μ X / 2\n            + Real.sqrt (Real.log (1 / δ) / (2 * n))} := by\n  sorry\n\ntheorem thm11_rademacher_concentration {Ω : Type u} {𝒮 : Type w} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒮)\n    (F : ι → 𝒮 → ℝ) (hF : ∀ i s, F i s ∈ Set.Icc (-1 : ℝ) 1)\n    (n : ℕ) (hn : 0 < n)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hMcD : ∀ (g : (Fin n → Ω) → ℝ) (c : ℝ), 0 ≤ c →\n        Integrable g (Measure.pi (fun _ => μ)) →\n        (∀ (ω ω' : Fin n → Ω) (j : Fin n),\n            (∀ k, k ≠ j → ω k = ω' k) → |g ω - g ω'| ≤ c) →\n        ∀ t : ℝ, 0 < t →\n          (Measure.pi (fun _ : Fin n => μ)).real\n              {ω | g ω - ∫ ω', g ω' ∂(Measure.pi (fun _ => μ)) ≥ t}\n            ≤ Real.exp (-2 * t ^ 2 / (n * c ^ 2)))\n    (ε : ℝ) (hε : 0 < ε) :\n    (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          |paperRad n F μ X - paperEmpRad n F (fun k => X (ω k))| ≥ ε}\n      ≤ 2 * Real.exp (-(ε ^ 2 * n) / 8) := by\n  sorry\n\nend\n\nend ResearchLevel\n", "target_names": ["thm8", "theorem5b_rademacher_bound", "thm11_rademacher_concentration"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1370, "end": 2625, "gold": "{Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Zero 𝒜] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → 𝒱)\n    (L φ : 𝒱 → 𝒜 → ℝ)\n    (hL01 : ∀ y a, L y a ∈ Set.Icc (0 : ℝ) 1)\n    (hφ01 : ∀ y a, φ y a ∈ Set.Icc (0 : ℝ) 1)\n    (hdom : ∀ y a, L y a ≤ φ y a)\n    (F : ι → 𝒳 → 𝒜)\n    (hLmeas : ∀ i, Measurable (fun ω => L (Y ω) (F i (X ω))))\n    (hφmeas : ∀ i, Measurable (fun ω => φ (Y ω) (F i (X ω))))\n    (hφ0meas : Measurable (fun ω => φ (Y ω) 0))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω =>\n        paperEmpRad n (centeredClass φ F) (fun k => (X (ω k), Y (ω k))))\n      (Measure.pi (fun _ => μ)))\n    (hDevMeas : Measurable\n      (fun ω : Fin n → Ω =>\n        ⨆ i : ι, ((∫ ω' : Ω, centeredClass φ F i (X ω', Y ω') ∂μ)\n          - ((n : ℝ)⁻¹ * ∑ k : Fin n, centeredClass φ F i (X (ω k), Y (ω k))))))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y L (F i) ≤\n            empiricalCost n X Y φ (F i) ω\n            + paperRad n (centeredClass φ F) μ (fun ω' => (X ω', Y ω'))\n            + Real.sqrt (8 * Real.log (2 / δ) / n)}"}]}
{"item_id": "v3-bartlett02a::theorem5b_rademacher_bound::A-stmt", "paper_id": "v3-bartlett02a", "target_name": "theorem5b_rademacher_bound", "source": "Research-level", "nl_statement": "Theorem 5 Suppose $P$ is a probability distribution on $\\mathcal{X}\\times \\left\\{\\pm 1 \\right\\}$, $F$ is a collection of functions on $\\mathcal{X}$ taking values in $\\{\\pm 1 \\}$, and the training sample $(X_{i},Y_{i})_{i=1}^{n}$ is distributed according to $P^{n}$\n\n(b) With probability no less than $1 - \\delta$, the following inequality holds simultaneously for all $f$ in $F$:\n\n$$\nP(Y\\neq f(X))\\leq\\hat{P}_{n}(Y\\neq f(X))+\\frac{R_{n}(F)}{2}+\\sqrt{\\frac{\\ln(1/\\delta)}{2n}}.\n$$", "context": [{"name": "Setup: decision-theoretic learning setting (Section 1)", "statement": "The following section presents a bound of this type that holds in a broader, decision-theoretic framework. In that framework there are an input space $\\mathcal{X}$, an action space $\\mathcal{A}$ and an output space $\\mathcal{Y}$. The $n$ training examples $(X_{1},Y_{1}),\\dots,(X_{n},Y_{n})$ are drawn independently from a probability measure $P$ on $\\mathcal{X}\\times \\mathcal{Y}$. A loss function $\\mathcal{L}:\\mathcal{Y}\\times\\mathcal{A}\\longrightarrow[0,1]$ is given, where $\\mathcal{L}(y,a)$ measures the cost incurred by choosing action $a \\in{\\mathcal{A}}$ when the outcome turns out to be $y \\in \\mathcal{Y}$. Learning seeks a function $f$ from $\\mathcal{X}$ to ${\\mathcal{A}}$ whose expected loss $\\mathbf{E}{\\mathcal{L}}(Y,f(X))$ is as small as possible."}, {"name": "Notation: empirical frequency hat P_n (Theorem 1)", "statement": "here VCdim(F) stands for the Vapnik-Chervonenkis dimension of $F$,\n\n$$\n{\\hat{P}}_{n}(S)={\\frac{1}{n}}\\sum_{i=1}^{n}\\mathbf{1}_{S}(X_{i},Y_{i}),\n$$\n\nand ${\\bf 1}_{S}$ denotes the indicator function of $S$."}, {"name": "Definition 2 (maximum discrepancy, Rademacher and Gaussian complexities)", "statement": "Definition 2 Suppose $\\mu$ is a probability distribution on a set $\\mathcal{X}$, and $X_{1},\\ldots,X_{n}$ are independent samples drawn from $\\mu$. Let $F$ be a class of functions from $\\mathcal{X}$ to $\\mathbb{R}$. The maximum discrepancy of $F$ is the random variable\n\n$$\n\\hat{D}_{n}(F)=\\operatorname*{sup}_{f\\in F}\\left(\\frac{2}{n}\\sum_{i=1}^{n/2}f(X_{i})-\\frac{2}{n}\\sum_{i=n/2+1}^{n}f(X_{i})\\right).\n$$\n\nIts expectation, the expected maximum discrepancy of $F$, is written $D_{n}(F)=\\mathbf{E}\\hat{D}_{n}(F)$\n\nNext, let\n\n$$\n{\\hat{R}}_{n}(F)=\\mathbf{E}[\\operatorname*{sup}_{f\\in F}|{\\frac{2}{n}}\\sum_{i=1}^{n}\\sigma_{i}f(X_{i})| |X_{1},\\ldots,X_{n}]\\ ,\n$$\n\nwith $\\sigma_{1},\\ldots,\\sigma_{n}$ independent random variables, each uniform on $\\{\\pm 1 \\}$. The Rademacher complexity of $F$ is then $R_{n}(F)=\\mathbf{E}\\hat{R}_{n}(F)$. In the same way, let\n\n$$\n\\hat{G}_{n}(F)=\\mathbf{E}[\\operatorname*{sup}_{f\\in F}|{\\frac{2}{n}}\\sum_{i=1}^{n}g_{i}f(X_{i})| |X_{1},\\ldots,X_{n}]\\ ,\n$$\n\nwith $g_{1},\\ldots,g_{n}$ independent Gaussian $N(0,1)$ random variables. The Gaussian complexity of $F$ is $G_{n}(F)=\\mathbf{E}\\hat{G}_{n}(F)$"}, {"name": "Remark: Theorem 5(b) is proved via the argument of Theorem 8", "statement": "Part (b) is proved by a small sharpening of the proof of a more general statement, Theorem 8 below; the argument is given in Appendix B."}, {"name": "Notation: empirical average hat E_n h", "statement": "Given a training sample $(X_{1},Y_{1}),\\dots,(X_{n},Y_{n})$ and a function $h : \\mathcal{X}\\times \\mathcal{Y}\\longrightarrow \\mathbb{R}$, we use the notation\n\n$$\n\\hat{\\mathbf{E}}_{n}h(X,Y)=(1/n)\\sum_{i=1}^{n}h(X_{i},Y_{i}).\n$$"}, {"name": "Notation: dominating cost function, phi o F, function-class operations (Section 2)", "statement": "First, some notation. For an independent sample $(X_{i},Y_{i})_{i=1}^{n}$ with the distribution of $(X,Y)$, $P_{n}$ is the empirical measure supported on the sample, and $\\mu_{n}$ is the empirical measure supported on $(X_{i})_{i=1}^{n}$. A function $\\phi : \\mathcal{Y}\\times \\mathcal{A}\\to \\mathbb{R}$ is said to dominate a loss function $\\mathcal{L}$ when $\\phi(y,a)\\geq{\\mathcal{L}}(y,a)$ holds for every $y \\in \\mathcal{Y}$ and every $a \\in{\\mathcal{A}}$. Given a class of functions $F$: conv $F$ denotes the set of convex combinations of members of $F$; $- F = \\{- f : f \\in F \\}$; absconv $F$ denotes the set of convex combinations of members of $F \\cup{- F}$; and $c F = \\{c f : f \\in F \\}$. When $\\phi$ is defined on the range of the functions in $F$, we put $\\phi \\circ F = \\{\\phi \\circ f | f \\in F \\}$. The characteristic function of a set $A$ is written ${\\bf 1}_{A}$ or $\\mathbf{1}(A)$. Lastly, $C$ and $c$ denote constants whose values may differ from one line to the next, or even inside a single line."}, {"name": "Theorem 9 (McDiarmid's Inequality)", "statement": "The argument relies on McDiarmid's inequality (McDiarmid, 1989).\n\nTheorem 9 (McDiarmid's Inequality) Suppose $X_{1},...,X_{n}$ are independent random variables with values in a set $A$, and $f : A^{n}\\to \\mathbb{R}$ is such that\n\n$$\n\\operatorname*{sup}_{x_{1},\\ldots,x_{n},x_{i}^{\\prime}\\in A}\\left|f(x_{1},...,x_{n})-f(x_{1},...,x_{i-1},x_{i}^{\\prime},x_{i+1},... x_{n})\\right|\\leq c_{i}\n$$\n\nholds for each $1 \\leq i \\leq n$. Then every $t > 0$ satisfies\n\n$$\nP\\left\\{f(X_{1},...,X_{n})-{\\bf E}f(X_{1},...,X_{n})\\geq t\\right\\}\\leq e^{-2t^{2}/\\sum_{i=1}^{n}c_{i}^{2}}.\n$$"}, {"name": "From the proof of Theorem 8 (the argument this proof repeats)", "statement": "What is left is to bound the second term on the right by $R_{n}(\\tilde{\\phi}\\circ F)$. Let $(X_{1}^{\\prime},Y_{1}^{\\prime}),\\dots,(X_{n}^{\\prime},Y_{n}^{\\prime})$ be independent random variables, each distributed as $(X,Y)$. Then\n\n$$\n\\begin{array}{r c l}{\\mathbf{E}\\underset{h\\in\\tilde{\\phi}\\circ F}{\\operatorname*{sup}}\\left(\\mathbf{E}h-\\hat{\\mathbf{E}}_{n}h\\right)}&{=}&{\\mathbf{E}\\underset{h\\in\\tilde{\\phi}\\circ F}{\\operatorname*{sup}}\\mathbf{E}\\left[\\frac{1}{n}\\displaystyle\\sum_{i=1}^{n}h(X_{i}^{\\prime},Y_{i}^{\\prime})-\\hat{\\mathbf{E}}_{n}h\\,\\middle|\\,(X_{i},Y_{i})\\right]}\\\\ &{\\leq}&{\\mathbf{E}\\underset{h\\in\\tilde{\\phi}\\circ F}{\\operatorname*{sup}}\\left(\\frac{1}{n}\\displaystyle\\sum_{i=1}^{n}h(X_{i}^{\\prime},Y_{i}^{\\prime})-\\hat{\\mathbf{E}}_{n}h\\right)}\\\\ &{=}&{\\mathbf{E}\\underset{h\\in\\tilde{\\phi}\\circ F}{\\operatorname*{sup}}\\frac{1}{n}\\displaystyle\\sum_{i=1}^{n}\\sigma_{i}\\left(h(X_{i}^{\\prime},Y_{i}^{\\prime})-h(X_{i},Y_{i})\\right)}\\\\ &{\\leq}&{2\\mathbf{E}\\underset{h\\in\\tilde{\\phi}\\circ F}{\\operatorname*{sup}}\\frac{1}{n}\\displaystyle\\sum_{i=1}^{n}\\sigma_{i}h(X_{i},Y_{i})}\\\\ &{\\leq}&{R_{n}(\\tilde{\\phi}\\circ F).}\\end{array}\n$$"}, {"name": "Theorem 8 (another result of the same paper)", "statement": "Theorem 8 Let $\\mathcal{L}:\\mathcal{Y}\\times\\mathcal{A}\\longrightarrow[0,1]$ be a loss function, and let $\\phi:\\mathcal{Y}\\times\\mathcal{A}\\longrightarrow[0,1]$ be a cost function that dominates it. Take $F$ to be a class of functions from $\\mathcal{X}$ into $\\mathcal{A}$, and suppose the pairs $(X_{i},Y_{i})_{i=1}^{n}$ are drawn independently from the probability measure $P$. Then for every integer $n$ and every $0 < \\delta < 1$, the following holds with probability no less than $1 - \\delta$ over samples of size $n$: for all $f$ in $F$,\n\n$$\n\\mathbf{E}\\mathcal{L}(Y,f(X))\\leq\\hat{\\mathbf{E}}_{n}\\phi(Y,f(X))+R_{n}(\\tilde{\\phi}\\circ F)+\\sqrt{\\frac{8\\ln(2/\\delta)}{n}},\n$$\n\nwith $\\tilde{\\phi}\\circ F=\\{(x,y)\\mapsto\\phi(y,f(x))-\\phi(y,0):f\\in F\\}$"}], "nl_notes": "### Theorem 5(b)\nNOTE (Theorem 5(b)) — The Lean assumes the index set of the function class is countable (`[Countable ι]`); the paper states no such condition. Without it the theorem is not provable in ZFC: under CH, take a well-order of [0,1] of type ω₁ and F_t(x) = 1 if x precedes or equals t, else −1; every other hypothesis holds, yet at t = the supremum of the sample points the empirical risk is 0 and the expected risk is 1. This is the usual countability / permissibility convention for suprema over a function class.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nnoncomputable section\n\nuniverse u v w\n\nopen MeasureTheory ProbabilityTheory Real\n\ndef Signs (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n : ℕ} : Fintype (Signs n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\ndef paperEmpRad {ι : Type v} {𝒮 : Type w} (n : ℕ) (f : ι → 𝒮 → ℝ) (S : Fin n → 𝒮) : ℝ :=\n  (Fintype.card (Signs n) : ℝ)⁻¹ *\n    ∑ σ : Signs n, ⨆ i, |(2 / (n : ℝ)) * ∑ k : Fin n, (σ k : ℝ) * f i (S k)|\n\ndef paperRad {Ω : Type u} {ι : Type v} {𝒮 : Type w} [MeasurableSpace Ω]\n    (n : ℕ) (f : ι → 𝒮 → ℝ) (μ : Measure Ω) (X : Ω → 𝒮) : ℝ :=\n  ∫ ω : Fin n → Ω, paperEmpRad n f (fun k => X (ω k)) ∂(Measure.pi (fun _ => μ))\n\ndef centeredClass {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v} [Zero 𝒜]\n    (φ : 𝒱 → 𝒜 → ℝ) (F : ι → 𝒳 → 𝒜) : ι → (𝒳 × 𝒱) → ℝ :=\n  fun i p => φ p.2 (F i p.1) - φ p.2 0\n\ndef expectedLoss {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} [MeasurableSpace Ω]\n    (μ : Measure Ω) (X : Ω → 𝒳) (Y : Ω → 𝒱) (L : 𝒱 → 𝒜 → ℝ) (f : 𝒳 → 𝒜) : ℝ :=\n  ∫ ω, L (Y ω) (f (X ω)) ∂μ\n\ndef empiricalCost {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*}\n    (n : ℕ) (X : Ω → 𝒳) (Y : Ω → 𝒱) (L : 𝒱 → 𝒜 → ℝ) (f : 𝒳 → 𝒜)\n    (ω : Fin n → Ω) : ℝ :=\n  (n : ℝ)⁻¹ * ∑ k : Fin n, L (Y (ω k)) (f (X (ω k)))\n\ndef zeroOneLoss (y a : ℝ) : ℝ := by\n  classical exact if y = a then 0 else 1\n\ntheorem thm8 {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Zero 𝒜] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → 𝒱)\n    (L φ : 𝒱 → 𝒜 → ℝ)\n    (hL01 : ∀ y a, L y a ∈ Set.Icc (0 : ℝ) 1)\n    (hφ01 : ∀ y a, φ y a ∈ Set.Icc (0 : ℝ) 1)\n    (hdom : ∀ y a, L y a ≤ φ y a)\n    (F : ι → 𝒳 → 𝒜)\n    (hLmeas : ∀ i, Measurable (fun ω => L (Y ω) (F i (X ω))))\n    (hφmeas : ∀ i, Measurable (fun ω => φ (Y ω) (F i (X ω))))\n    (hφ0meas : Measurable (fun ω => φ (Y ω) 0))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω =>\n        paperEmpRad n (centeredClass φ F) (fun k => (X (ω k), Y (ω k))))\n      (Measure.pi (fun _ => μ)))\n    (hDevMeas : Measurable\n      (fun ω : Fin n → Ω =>\n        ⨆ i : ι, ((∫ ω' : Ω, centeredClass φ F i (X ω', Y ω') ∂μ)\n          - ((n : ℝ)⁻¹ * ∑ k : Fin n, centeredClass φ F i (X (ω k), Y (ω k))))))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y L (F i) ≤\n            empiricalCost n X Y φ (F i) ω\n            + paperRad n (centeredClass φ F) μ (fun ω' => (X ω', Y ω'))\n            + Real.sqrt (8 * Real.log (2 / δ) / n)} := by\n  sorry\n\ntheorem theorem5b_rademacher_bound {Ω : Type u} {𝒳 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → ℝ)\n    (hY : ∀ ω, Y ω ∈ ({-1, 1} : Set ℝ))\n    (F : ι → 𝒳 → ℝ) (hF : ∀ i x, F i x ∈ ({-1, 1} : Set ℝ))\n    (hLmeas : ∀ i, Measurable (fun ω => zeroOneLoss (Y ω) (F i (X ω))))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (hEvt : MeasurableSet\n      {ω : Fin n → Ω |\n        ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n          empiricalCost n X Y zeroOneLoss (F i) ω\n          + paperRad n F μ X / 2\n          + Real.sqrt (Real.log (1 / δ) / (2 * n))}) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n            empiricalCost n X Y zeroOneLoss (F i) ω\n            + paperRad n F μ X / 2\n            + Real.sqrt (Real.log (1 / δ) / (2 * n))} := by\n  sorry\n\ntheorem thm11_rademacher_concentration {Ω : Type u} {𝒮 : Type w} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒮)\n    (F : ι → 𝒮 → ℝ) (hF : ∀ i s, F i s ∈ Set.Icc (-1 : ℝ) 1)\n    (n : ℕ) (hn : 0 < n)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hMcD : ∀ (g : (Fin n → Ω) → ℝ) (c : ℝ), 0 ≤ c →\n        Integrable g (Measure.pi (fun _ => μ)) →\n        (∀ (ω ω' : Fin n → Ω) (j : Fin n),\n            (∀ k, k ≠ j → ω k = ω' k) → |g ω - g ω'| ≤ c) →\n        ∀ t : ℝ, 0 < t →\n          (Measure.pi (fun _ : Fin n => μ)).real\n              {ω | g ω - ∫ ω', g ω' ∂(Measure.pi (fun _ => μ)) ≥ t}\n            ≤ Real.exp (-2 * t ^ 2 / (n * c ^ 2)))\n    (ε : ℝ) (hε : 0 < ε) :\n    (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          |paperRad n F μ X - paperEmpRad n F (fun k => X (ω k))| ≥ ε}\n      ≤ 2 * Real.exp (-(ε ^ 2 * n) / 8) := by\n  sorry\n\nend\n\nend ResearchLevel\n", "target_names": ["thm8", "theorem5b_rademacher_bound", "thm11_rademacher_concentration"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 2676, "end": 3713, "gold": "{Ω : Type u} {𝒳 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → ℝ)\n    (hY : ∀ ω, Y ω ∈ ({-1, 1} : Set ℝ))\n    (F : ι → 𝒳 → ℝ) (hF : ∀ i x, F i x ∈ ({-1, 1} : Set ℝ))\n    (hLmeas : ∀ i, Measurable (fun ω => zeroOneLoss (Y ω) (F i (X ω))))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (hEvt : MeasurableSet\n      {ω : Fin n → Ω |\n        ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n          empiricalCost n X Y zeroOneLoss (F i) ω\n          + paperRad n F μ X / 2\n          + Real.sqrt (Real.log (1 / δ) / (2 * n))}) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n            empiricalCost n X Y zeroOneLoss (F i) ω\n            + paperRad n F μ X / 2\n            + Real.sqrt (Real.log (1 / δ) / (2 * n))}"}]}
{"item_id": "v3-bartlett02a::thm11_rademacher_concentration::A-stmt", "paper_id": "v3-bartlett02a", "target_name": "thm11_rademacher_concentration", "source": "Research-level", "nl_statement": "Theorem 11 Suppose $F$ is a class of functions taking values in $[-1, 1]$. Then for every integer $n$,\n\n$$\nP\\left\\{\\Big|R_{n}(F)-\\hat{R}_{n}(F)\\Big|\\geq\\epsilon\\right\\}\\leq 2\\exp\\left(\\frac{-\\epsilon^{2}n}{8}\\right),\n$$", "context": [{"name": "Definition 2 (maximum discrepancy, Rademacher and Gaussian complexities)", "statement": "Definition 2 Suppose $\\mu$ is a probability distribution on a set $\\mathcal{X}$, and $X_{1},\\ldots,X_{n}$ are independent samples drawn from $\\mu$. Let $F$ be a class of functions from $\\mathcal{X}$ to $\\mathbb{R}$. The maximum discrepancy of $F$ is the random variable\n\n$$\n\\hat{D}_{n}(F)=\\operatorname*{sup}_{f\\in F}\\left(\\frac{2}{n}\\sum_{i=1}^{n/2}f(X_{i})-\\frac{2}{n}\\sum_{i=n/2+1}^{n}f(X_{i})\\right).\n$$\n\nIts expectation, the expected maximum discrepancy of $F$, is written $D_{n}(F)=\\mathbf{E}\\hat{D}_{n}(F)$\n\nNext, let\n\n$$\n{\\hat{R}}_{n}(F)=\\mathbf{E}[\\operatorname*{sup}_{f\\in F}|{\\frac{2}{n}}\\sum_{i=1}^{n}\\sigma_{i}f(X_{i})| |X_{1},\\ldots,X_{n}]\\ ,\n$$\n\nwith $\\sigma_{1},\\ldots,\\sigma_{n}$ independent random variables, each uniform on $\\{\\pm 1 \\}$. The Rademacher complexity of $F$ is then $R_{n}(F)=\\mathbf{E}\\hat{R}_{n}(F)$. In the same way, let\n\n$$\n\\hat{G}_{n}(F)=\\mathbf{E}[\\operatorname*{sup}_{f\\in F}|{\\frac{2}{n}}\\sum_{i=1}^{n}g_{i}f(X_{i})| |X_{1},\\ldots,X_{n}]\\ ,\n$$\n\nwith $g_{1},\\ldots,g_{n}$ independent Gaussian $N(0,1)$ random variables. The Gaussian complexity of $F$ is $G_{n}(F)=\\mathbf{E}\\hat{G}_{n}(F)$"}, {"name": "Notation: dominating cost function, phi o F, function-class operations (Section 2)", "statement": "First, some notation. For an independent sample $(X_{i},Y_{i})_{i=1}^{n}$ with the distribution of $(X,Y)$, $P_{n}$ is the empirical measure supported on the sample, and $\\mu_{n}$ is the empirical measure supported on $(X_{i})_{i=1}^{n}$. A function $\\phi : \\mathcal{Y}\\times \\mathcal{A}\\to \\mathbb{R}$ is said to dominate a loss function $\\mathcal{L}$ when $\\phi(y,a)\\geq{\\mathcal{L}}(y,a)$ holds for every $y \\in \\mathcal{Y}$ and every $a \\in{\\mathcal{A}}$. Given a class of functions $F$: conv $F$ denotes the set of convex combinations of members of $F$; $- F = \\{- f : f \\in F \\}$; absconv $F$ denotes the set of convex combinations of members of $F \\cup{- F}$; and $c F = \\{c f : f \\in F \\}$. When $\\phi$ is defined on the range of the functions in $F$, we put $\\phi \\circ F = \\{\\phi \\circ f | f \\in F \\}$. The characteristic function of a set $A$ is written ${\\bf 1}_{A}$ or $\\mathbf{1}(A)$. Lastly, $C$ and $c$ denote constants whose values may differ from one line to the next, or even inside a single line."}, {"name": "Theorem 9 (McDiarmid's Inequality)", "statement": "The argument relies on McDiarmid's inequality (McDiarmid, 1989).\n\nTheorem 9 (McDiarmid's Inequality) Suppose $X_{1},...,X_{n}$ are independent random variables with values in a set $A$, and $f : A^{n}\\to \\mathbb{R}$ is such that\n\n$$\n\\operatorname*{sup}_{x_{1},\\ldots,x_{n},x_{i}^{\\prime}\\in A}\\left|f(x_{1},...,x_{n})-f(x_{1},...,x_{i-1},x_{i}^{\\prime},x_{i+1},... x_{n})\\right|\\leq c_{i}\n$$\n\nholds for each $1 \\leq i \\leq n$. Then every $t > 0$ satisfies\n\n$$\nP\\left\\{f(X_{1},...,X_{n})-{\\bf E}f(X_{1},...,X_{n})\\geq t\\right\\}\\leq e^{-2t^{2}/\\sum_{i=1}^{n}c_{i}^{2}}.\n$$"}, {"name": "Remark: Theorem 11 follows from McDiarmid's inequality", "statement": "The next result is a consequence of McDiarmid's inequality. An analogous statement holds for the Gaussian complexity."}], "nl_notes": "### Theorem 11 (second bound)\nHYPOTHESIS NOTE (Theorem 11) — `hMcD` is McDiarmid's bounded-differences inequality (McDiarmid 1989), an external result the paper cites without proof. Use it directly; you do not need to prove it.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\nnoncomputable section\n\nuniverse u v w\n\nopen MeasureTheory ProbabilityTheory Real\n\ndef Signs (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n : ℕ} : Fintype (Signs n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\ndef paperEmpRad {ι : Type v} {𝒮 : Type w} (n : ℕ) (f : ι → 𝒮 → ℝ) (S : Fin n → 𝒮) : ℝ :=\n  (Fintype.card (Signs n) : ℝ)⁻¹ *\n    ∑ σ : Signs n, ⨆ i, |(2 / (n : ℝ)) * ∑ k : Fin n, (σ k : ℝ) * f i (S k)|\n\ndef paperRad {Ω : Type u} {ι : Type v} {𝒮 : Type w} [MeasurableSpace Ω]\n    (n : ℕ) (f : ι → 𝒮 → ℝ) (μ : Measure Ω) (X : Ω → 𝒮) : ℝ :=\n  ∫ ω : Fin n → Ω, paperEmpRad n f (fun k => X (ω k)) ∂(Measure.pi (fun _ => μ))\n\ndef centeredClass {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v} [Zero 𝒜]\n    (φ : 𝒱 → 𝒜 → ℝ) (F : ι → 𝒳 → 𝒜) : ι → (𝒳 × 𝒱) → ℝ :=\n  fun i p => φ p.2 (F i p.1) - φ p.2 0\n\ndef expectedLoss {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} [MeasurableSpace Ω]\n    (μ : Measure Ω) (X : Ω → 𝒳) (Y : Ω → 𝒱) (L : 𝒱 → 𝒜 → ℝ) (f : 𝒳 → 𝒜) : ℝ :=\n  ∫ ω, L (Y ω) (f (X ω)) ∂μ\n\ndef empiricalCost {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*}\n    (n : ℕ) (X : Ω → 𝒳) (Y : Ω → 𝒱) (L : 𝒱 → 𝒜 → ℝ) (f : 𝒳 → 𝒜)\n    (ω : Fin n → Ω) : ℝ :=\n  (n : ℝ)⁻¹ * ∑ k : Fin n, L (Y (ω k)) (f (X (ω k)))\n\ndef zeroOneLoss (y a : ℝ) : ℝ := by\n  classical exact if y = a then 0 else 1\n\ntheorem thm8 {Ω : Type u} {𝒳 : Type*} {𝒱 : Type*} {𝒜 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Zero 𝒜] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → 𝒱)\n    (L φ : 𝒱 → 𝒜 → ℝ)\n    (hL01 : ∀ y a, L y a ∈ Set.Icc (0 : ℝ) 1)\n    (hφ01 : ∀ y a, φ y a ∈ Set.Icc (0 : ℝ) 1)\n    (hdom : ∀ y a, L y a ≤ φ y a)\n    (F : ι → 𝒳 → 𝒜)\n    (hLmeas : ∀ i, Measurable (fun ω => L (Y ω) (F i (X ω))))\n    (hφmeas : ∀ i, Measurable (fun ω => φ (Y ω) (F i (X ω))))\n    (hφ0meas : Measurable (fun ω => φ (Y ω) 0))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω =>\n        paperEmpRad n (centeredClass φ F) (fun k => (X (ω k), Y (ω k))))\n      (Measure.pi (fun _ => μ)))\n    (hDevMeas : Measurable\n      (fun ω : Fin n → Ω =>\n        ⨆ i : ι, ((∫ ω' : Ω, centeredClass φ F i (X ω', Y ω') ∂μ)\n          - ((n : ℝ)⁻¹ * ∑ k : Fin n, centeredClass φ F i (X (ω k), Y (ω k))))))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y L (F i) ≤\n            empiricalCost n X Y φ (F i) ω\n            + paperRad n (centeredClass φ F) μ (fun ω' => (X ω', Y ω'))\n            + Real.sqrt (8 * Real.log (2 / δ) / n)} := by\n  sorry\n\ntheorem theorem5b_rademacher_bound {Ω : Type u} {𝒳 : Type*} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι] [Countable ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒳) (Y : Ω → ℝ)\n    (hY : ∀ ω, Y ω ∈ ({-1, 1} : Set ℝ))\n    (F : ι → 𝒳 → ℝ) (hF : ∀ i x, F i x ∈ ({-1, 1} : Set ℝ))\n    (hLmeas : ∀ i, Measurable (fun ω => zeroOneLoss (Y ω) (F i (X ω))))\n    (n : ℕ)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hn : 0 < n) (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (hEvt : MeasurableSet\n      {ω : Fin n → Ω |\n        ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n          empiricalCost n X Y zeroOneLoss (F i) ω\n          + paperRad n F μ X / 2\n          + Real.sqrt (Real.log (1 / δ) / (2 * n))}) :\n    1 - δ ≤\n      (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          ∀ i, expectedLoss μ X Y zeroOneLoss (F i) ≤\n            empiricalCost n X Y zeroOneLoss (F i) ω\n            + paperRad n F μ X / 2\n            + Real.sqrt (Real.log (1 / δ) / (2 * n))} := by\n  sorry\n\ntheorem thm11_rademacher_concentration {Ω : Type u} {𝒮 : Type w} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒮)\n    (F : ι → 𝒮 → ℝ) (hF : ∀ i s, F i s ∈ Set.Icc (-1 : ℝ) 1)\n    (n : ℕ) (hn : 0 < n)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hMcD : ∀ (g : (Fin n → Ω) → ℝ) (c : ℝ), 0 ≤ c →\n        Integrable g (Measure.pi (fun _ => μ)) →\n        (∀ (ω ω' : Fin n → Ω) (j : Fin n),\n            (∀ k, k ≠ j → ω k = ω' k) → |g ω - g ω'| ≤ c) →\n        ∀ t : ℝ, 0 < t →\n          (Measure.pi (fun _ : Fin n => μ)).real\n              {ω | g ω - ∫ ω', g ω' ∂(Measure.pi (fun _ => μ)) ≥ t}\n            ≤ Real.exp (-2 * t ^ 2 / (n * c ^ 2)))\n    (ε : ℝ) (hε : 0 < ε) :\n    (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          |paperRad n F μ X - paperEmpRad n F (fun k => X (ω k))| ≥ ε}\n      ≤ 2 * Real.exp (-(ε ^ 2 * n) / 8) := by\n  sorry\n\nend\n\nend ResearchLevel\n", "target_names": ["thm8", "theorem5b_rademacher_bound", "thm11_rademacher_concentration"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 3768, "end": 4717, "gold": "{Ω : Type u} {𝒮 : Type w} {ι : Type v}\n    [MeasurableSpace Ω] [Nonempty ι]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (X : Ω → 𝒮)\n    (F : ι → 𝒮 → ℝ) (hF : ∀ i s, F i s ∈ Set.Icc (-1 : ℝ) 1)\n    (n : ℕ) (hn : 0 < n)\n    (hRadInt : Integrable\n      (fun ω : Fin n → Ω => paperEmpRad n F (fun k => X (ω k)))\n      (Measure.pi (fun _ => μ)))\n    (hMcD : ∀ (g : (Fin n → Ω) → ℝ) (c : ℝ), 0 ≤ c →\n        Integrable g (Measure.pi (fun _ => μ)) →\n        (∀ (ω ω' : Fin n → Ω) (j : Fin n),\n            (∀ k, k ≠ j → ω k = ω' k) → |g ω - g ω'| ≤ c) →\n        ∀ t : ℝ, 0 < t →\n          (Measure.pi (fun _ : Fin n => μ)).real\n              {ω | g ω - ∫ ω', g ω' ∂(Measure.pi (fun _ => μ)) ≥ t}\n            ≤ Real.exp (-2 * t ^ 2 / (n * c ^ 2)))\n    (ε : ℝ) (hε : 0 < ε) :\n    (Measure.pi (fun _ : Fin n => μ)).real\n        {ω : Fin n → Ω |\n          |paperRad n F μ X - paperEmpRad n F (fun k => X (ω k))| ≥ ε}\n      ≤ 2 * Real.exp (-(ε ^ 2 * n) / 8)"}]}
{"item_id": "v25-21-0748::theorem1::A-stmt", "paper_id": "v25-21-0748", "target_name": "theorem1", "source": "Research-level", "nl_statement": "Theorem 1. Suppose that the regularizer ψ $: \\mathcal{X}\\mapsto \\mathbb{R}$ is 1-strongly convex with respect to the norm $\\| \\cdot \\|$ , and let $\\| \\cdot \\|_{*}$ be the dual norm of $\\| \\cdot \\|$ . The dynamic regret of OpTIMISTIC OMD whose update rule is specified in (6) is bounded as follows:\n\n$$\n\\begin{array}{r}{\\displaystyle\\sum_{t=1}^{T}f_{t}(\\mathbf{x}_{t})-\\displaystyle\\sum_{t=1}^{T}f_{t}(\\mathbf{u}_{t})\\leq\\displaystyle\\sum_{t=1}^{T}\\eta_{t}\\|\\nabla f_{t}(\\mathbf{x}_{t})-M_{t}\\|_{*}^{2}+\\displaystyle\\sum_{t=1}^{T}\\frac{1}{\\eta_{t}}\\Big(\\mathcal{D}_{\\psi}(\\mathbf{u}_{t},\\widehat{\\mathbf{x}}_{t})-\\mathcal{D}_{\\psi}(\\mathbf{u}_{t},\\widehat{\\mathbf{x}}_{t+1})\\Big)}\\\\ {-\\displaystyle\\sum_{t=1}^{T}\\frac{1}{\\eta_{t}}\\Big(\\mathcal{D}_{\\psi}(\\widehat{\\mathbf{x}}_{t+1},\\mathbf{x}_{t})+\\mathcal{D}_{\\psi}(\\mathbf{x}_{t},\\widehat{\\mathbf{x}}_{t})\\Big),}\\end{array}\\tag{7}\n$$\n\nwhich holds for any comparator sequence $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$", "context": [{"name": "Setup: online convex optimization protocol (Introduction)", "statement": "At iteration t, the player chooses a decision vector $\\mathbf{x}_{t}$ from a convex set $\\boldsymbol{\\mathcal{X}}\\subseteq \\mathbb{R}^{d}$ . Subsequently, the adversary discloses a convex function $f_{t}: \\mathcal{X}\\mapsto \\mathbb{R}$ ,and the player incurs a loss denoted by $f_{t}(\\mathbf{x}_{t})$"}, {"name": "Setup: problem setup and feedback models (Section 3.1)", "statement": "Online Convex Optimization (OCO) can be modeled as an iterated game between the player and the environments. At iteration $t \\in [ T ]$ , the player first chooses the decision $\\mathbf{x}_{t}$ from a convex feasible set $\\boldsymbol{\\mathcal{X}}\\subseteq \\mathbb{R}^{d}$ , then the environments reveal the loss function $f_{t}: \\mathcal{X}\\mapsto \\mathbb{R}$ and the player suffers the loss $f_{t}(\\mathbf{x}_{t})$ and observes a certain information about the function $f_{t}(\\cdot).$ 2 According to the revealed information, the online learning problems are typically classified into full-information online learning and partial-information online learning (or sometimes called bandit online learning). In this paper, we focus on the full-information one, which can be further categorized into the following two setups:\n\n(i) multi-gradient feedback: the player can access the entire gradient function $\\nabla f_{t}(\\cdot)$ and thus can evaluate the gradient multiple times at each round;\n\n(ii) one-gradient feedback: the player can observe the gradient information $\\nabla f_{t}(\\mathbf{x}_{t})$ after submitting the decision $\\mathbf{x}_{t}$ at each round.\n\nIn Section 4.2, we develop the Sword algorithm, which achieves the gradient-variation dynamic regret under the multi-gradient feedback model. In Section 4.3, we present an improved algorithm called Sword++ that can achieve the same dynamic regret guarantee (up to constants) under the more challenging one-gradient feedback model.\n\nTo handle non-stationary environments, we focus on the dynamic regret measure, which compares the online algorithm to a sequence of time-varying comparators $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$ as defined in (2). An upper bound of dynamic regret should be a function of comparators, and typically the bound depends on some regularities that measure the fluctuation of the comparator sequence, such as the path length $\\begin{array}{r}{P_{T}= \\sum_{t = 2}^{T}\\lVert \\mathbf{u}_{t}- \\mathbf{u}_{t - 1}\\rVert_{2}}\\end{array}$ Throughout the paper, we focus on the Euclidean norm for simplicity, and it is straightforward to extend our results to general primal-dual norms.\n\nIn addition to the regret measure, we further consider the gradient query complexity. Note that algorithms designed for the multi-gradient feedback model may query the gradients for multiple times at each round. However, most algorithms designed for the static regret minimization only require one gradient per iteration, namely, using $\\nabla f_{t}(\\mathbf{x}_{t})$ for the next update only. Therefore, it is more desirable to achieve the favorable regret guarantees under the one-gradient feedback model. In other words, our aim is to develop first-order methods for dynamic regret minimization that require only one gradient query per iteration."}, {"name": "Dynamic regret, eq. (2)", "statement": "To address this limitation, dynamic regret is proposed to compete with changing comparators $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$\n\n$$\n\\mathrm{D-Regret}_{T}(\\mathbf{u}_{1},\\ldots,\\mathbf{u}_{T})=\\sum_{t=1}^{T}f_{t}(\\mathbf{x}_{t})-\\sum_{t=1}^{T}f_{t}(\\mathbf{u}_{t}),\\tag{2}\n$$"}, {"name": "Optimistic OMD: update rule (6), Bregman divergence, strong convexity (Section 3.2)", "statement": "We employ the algorithmic framework of Optimistic Online Mirror Descent (OpTIMISTIC OMD) (Chiang et al., 2012; Rakhlin and Sridharan, 2013) as a general building block for designing algorithms for non-stationary online learning. Optimistic OMD is an algorithmic realization of optimistic online learning. Compared to the standard online learning setup, the player receives an additional element at each round: an optimistic vector $M_{t}\\in \\mathbb{R}^{d}$ This vector acts as a predictive hint or an optimistic estimate of the upcoming gradient, thereby called \"optimistic vector\" or simply \"optimism\". Optimistic OMD starts from the initial point $\\widehat{\\mathbf{x}}_{1}\\in \\mathcal{X}$ and performs the following two-step updates at each round:\n\n$$\n\\begin{array}{r l}&{\\mathbf{x}_{t}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}~\\big\\{\\eta_{t}\\langle M_{t},\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\widehat{\\mathbf{x}}_{t})\\big\\},}\\\\ &{\\widehat{\\mathbf{x}}_{t+1}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}~\\big\\{\\eta_{t}\\langle\\nabla f_{t}(\\mathbf{x}_{t}),\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\widehat{\\mathbf{x}}_{t})\\big\\},}\\end{array}\\tag{6}\n$$\n\nwhich firstly updates by the optimistic vector $M_{t}$ and then updates by the received gradient $\\nabla f_{t}(\\mathbf{x}_{t})$ . In above, $\\eta_{t}> 0$ is a (potentially) time-varying step size, and $\\mathcal{D}_{\\psi}(\\cdot,\\cdot)$ denotes the Bregman divergence associated with the regularizer $\\psi$ defined as $\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{y})=\\psi(\\mathbf{x})-\\psi(\\mathbf{y})-$ $\\langle\\nabla\\psi(\\mathbf{y}),\\mathbf{x}-\\mathbf{y}\\rangle$ . We may assume the regularizer to be σ-strongly convex with respect to the norm $\\begin{array}{r}{\\|\\cdot\\|,\\operatorname{i.e.,}\\psi(\\mathbf{y})\\geq\\psi(\\mathbf{x})+\\langle\\nabla\\psi(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle+\\frac{\\sigma}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2}}\\end{array}$ holds for any $\\mathbf{x}, \\mathbf{y}\\in{\\mathcal{X}}$ . We have the following general result regarding dynamic regret of optimistic OMD."}, {"name": "Lemma 5 (stability lemma, Chiang et al. 2012, Proposition 7)", "statement": "Lemma 5 (stability lemma (Chiang et al., 2012, Proposition 7)). Consider the following two updates: $\\begin{array}{r}{(i)\\mathbf{x}_{*}=\\arg\\operatorname*{min}_{\\mathbf{x}\\in\\mathcal{X}}\\langle\\mathbf{a},\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{c})}\\end{array}$ , and $(i i)\\mathbf{x}_{*}^{\\prime}=\\arg\\operatorname*{min}_{\\mathbf{x}\\in\\mathcal{X}}\\langle\\mathbf{a}^{\\prime},\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{c})$ When the regularizer $\\psi : \\mathcal{X}\\mapsto \\mathbb{R}$ is a 1-strongly convex function with respect to the norm $\\| \\cdot \\|$ , we have $\\| \\mathbf{x}_{*}- \\mathbf{x}_{*}^{\\prime}\\| \\leq \\| \\mathbf{a}- \\mathbf{a}^{\\prime}\\|_{*}$"}, {"name": "Remark: Theorem 1 does not need smoothness", "statement": "We have the following remarks regarding the assumptions. The general dynamic regret analysis of Optimistic OMD (Theorem 1) does not require the smoothness assumption. Nevertheless, this assumption is crucial for achieving problem-dependent dynamic regret bounds."}, {"name": "Lemma 4 (another result of the same paper)", "statement": "Lemma 4 (Bregman proximal inequality (Chen and Teboulle, 1993, Lemma 3.2)). Let X be a convex set in a Banach space. Let $f : \\mathcal{X}\\mapsto \\mathbb{R}$ be a closed proper convex function on X. Given a convex regularizer $\\psi : \\mathcal{X}\\mapsto \\mathbb{R}$ , we denote its induced Bregman divergence by $\\mathcal{D}_{\\psi}(\\cdot,\\cdot)$ . Then, any update of the form\n\n$$\n\\mathbf{x}_{k}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}\\{f(\\mathbf{x})+\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{x}_{k-1})\\}\n$$\n\nsatisfies the following inequality for any $\\mathbf{u}\\in \\mathcal{X}$\n\n$$\n\\begin{array}{r}{f ({\\mathbf x}_{k}) - f ({\\mathbf u}) \\le \\mathcal{D}_{\\psi}({\\mathbf u},{\\mathbf x}_{k - 1}) - \\mathcal{D}_{\\psi}({\\mathbf u},{\\mathbf x}_{k}) - \\mathcal{D}_{\\psi}({\\mathbf x}_{k},{\\mathbf x}_{k - 1}) .}\\end{array}\\tag{60}\n$$"}], "nl_notes": "### Theorem 1\nNOTE (Theorem 1) — Theorem 1 does not say which feedback model it assumes, and its proof evaluates ∇f_t only at the played point x_t. The Lean takes this one-gradient premise: each f_t is convex on X (`hfconv`) and differentiable at x_t with gradient `gradf t (x t)` (`hfgrad`); no subgradient inequality at other points is assumed. The first-order inequality at x_t that the proof needs follows from `hfconv` and `hfgrad`.", "lean": "import Mathlib\n\nnoncomputable section\n\nopen scoped BigOperators RealInnerProductSpace\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]\n\ndef bregman (ψ : E → ℝ) (gradψ : E → E) (x y : E) : ℝ :=\n  ψ x - ψ y - ⟪gradψ y, x - y⟫\n\ndef bregmanDual {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ)) (x y : F) : ℝ :=\n  ψ x - ψ y - gradψ y (x - y)\n\ndef pathLength (u : ℕ → E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 2 T, ‖u t - u (t - 1)‖\n\ndef gradVarSup (X : Set E) (gradf : ℕ → E → E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 2 T, ⨆ x : X, ‖gradf t (x : E) - gradf (t - 1) (x : E)‖ ^ 2\n\ntheorem theorem1\n    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]\n    {T : ℕ} (X : Set F)\n    (f : ℕ → F → ℝ) (gradf : ℕ → F → (F →L[ℝ] ℝ))\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (x xhat u : ℕ → F) (M : ℕ → (F →L[ℝ] ℝ)) (η : ℕ → ℝ)\n\n    (hXconv : Convex ℝ X)\n\n    (hη : ∀ t ∈ Finset.Icc 1 T, 0 < η t)\n\n    (hsc : ∀ a ∈ X, ∀ b ∈ X, ψ a + gradψ a (b - a) + (1 / 2) * ‖b - a‖ ^ 2 ≤ ψ b)\n\n    (hψgrad : ∀ x ∈ X, HasFDerivAt ψ (gradψ x) x)\n\n    (hfconv : ∀ t ∈ Finset.Icc 1 T, ConvexOn ℝ X (f t))\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, HasFDerivAt (f t) (gradf t (x t)) (x t))\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * M t (x t) + bregmanDual ψ gradψ (x t) (xhat t)\n        ≤ η t * M t z + bregmanDual ψ gradψ z (xhat t))\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * gradf t (x t) (xhat (t + 1)) + bregmanDual ψ gradψ (xhat (t + 1)) (xhat t)\n        ≤ η t * gradf t (x t) z + bregmanDual ψ gradψ z (xhat t)) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ (∑ t ∈ Finset.Icc 1 T, η t * ‖gradf t (x t) - M t‖ ^ 2)\n        + (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (u t) (xhat t)\n              - bregmanDual ψ gradψ (u t) (xhat (t + 1))))\n        - (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (xhat (t + 1)) (x t)\n              + bregmanDual ψ gradψ (x t) (xhat t))) := by\n  sorry\n\ntheorem lemma4_bregmanProximal\n    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]\n    (X : Set F) (f ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (xk xkm1 u : F)\n\n    (hXconv : Convex ℝ X)\n    (hxkX : xk ∈ X) (hxkm1X : xkm1 ∈ X) (huX : u ∈ X)\n\n    (hfconv : ConvexOn ℝ X f)\n    (hflsc : LowerSemicontinuousOn f X)\n\n    (hψconv : ConvexOn ℝ X ψ)\n\n    (hψgrad : HasFDerivAt ψ (gradψ xk) xk)\n    (hψgrad' : HasFDerivAt ψ (gradψ xkm1) xkm1)\n\n    (hprox : ∀ z ∈ X,\n      f xk + bregmanDual ψ gradψ xk xkm1 ≤ f z + bregmanDual ψ gradψ z xkm1) :\n    f xk - f u\n      ≤ bregmanDual ψ gradψ u xkm1 - bregmanDual ψ gradψ u xk\n          - bregmanDual ψ gradψ xk xkm1 := by\n  sorry\n\ntheorem lemma1_OEGD\n    {T : ℕ} (X : Set E)\n    (f : ℕ → E → ℝ) (gradf : ℕ → E → E)\n    (x xhat u : ℕ → E) (η G D L : ℝ)\n    (hη : 0 < η) (hL : 0 < L) (hηL : η ≤ 1 / (4 * L))\n    (hXne : X.Nonempty)\n\n    (hXconv : Convex ℝ X)\n\n    (hG : ∀ t, ∀ a ∈ X, ‖gradf t a‖ ≤ G)\n    (hgrad0 : ∀ a, gradf 0 a = 0)\n\n    (h0X : (0 : E) ∈ X) (hD : ∀ a ∈ X, ∀ b ∈ X, ‖a - b‖ ≤ D)\n\n    (hsmooth : ∀ t, ∀ a b, ‖gradf t a - gradf t b‖ ≤ L * ‖a - b‖)\n\n    (hconv : ∀ t ∈ Finset.Icc 1 T, ∀ a ∈ X, ∀ b ∈ X,\n      f t a + ⟪gradf t a, b - a⟫ ≤ f t b)\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, ∀ a, HasGradientAt (f t) (gradf t a) a)\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖x t - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), x t⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), z⟫)\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖xhat (t + 1) - xhat t‖ ^ 2 + η * ⟪gradf t (x t), xhat (t + 1)⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf t (x t), z⟫) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ η * (G ^ 2 + 2 * gradVarSup X gradf T)\n        + (1 / (2 * η)) * (D ^ 2 + 2 * D * pathLength u T) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem1", "lemma4_bregmanProximal", "lemma1_OEGD"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 719, "end": 2269, "gold": "{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]\n    {T : ℕ} (X : Set F)\n    (f : ℕ → F → ℝ) (gradf : ℕ → F → (F →L[ℝ] ℝ))\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (x xhat u : ℕ → F) (M : ℕ → (F →L[ℝ] ℝ)) (η : ℕ → ℝ)\n\n    (hXconv : Convex ℝ X)\n\n    (hη : ∀ t ∈ Finset.Icc 1 T, 0 < η t)\n\n    (hsc : ∀ a ∈ X, ∀ b ∈ X, ψ a + gradψ a (b - a) + (1 / 2) * ‖b - a‖ ^ 2 ≤ ψ b)\n\n    (hψgrad : ∀ x ∈ X, HasFDerivAt ψ (gradψ x) x)\n\n    (hfconv : ∀ t ∈ Finset.Icc 1 T, ConvexOn ℝ X (f t))\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, HasFDerivAt (f t) (gradf t (x t)) (x t))\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * M t (x t) + bregmanDual ψ gradψ (x t) (xhat t)\n        ≤ η t * M t z + bregmanDual ψ gradψ z (xhat t))\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * gradf t (x t) (xhat (t + 1)) + bregmanDual ψ gradψ (xhat (t + 1)) (xhat t)\n        ≤ η t * gradf t (x t) z + bregmanDual ψ gradψ z (xhat t)) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ (∑ t ∈ Finset.Icc 1 T, η t * ‖gradf t (x t) - M t‖ ^ 2)\n        + (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (u t) (xhat t)\n              - bregmanDual ψ gradψ (u t) (xhat (t + 1))))\n        - (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (xhat (t + 1)) (x t)\n              + bregmanDual ψ gradψ (x t) (xhat t)))"}]}
{"item_id": "v25-21-0748::lemma4_bregmanProximal::A-stmt", "paper_id": "v25-21-0748", "target_name": "lemma4_bregmanProximal", "source": "Research-level", "nl_statement": "Lemma 4 (Bregman proximal inequality (Chen and Teboulle, 1993, Lemma 3.2)). Let X be a convex set in a Banach space. Let $f : \\mathcal{X}\\mapsto \\mathbb{R}$ be a closed proper convex function on X. Given a convex regularizer $\\psi : \\mathcal{X}\\mapsto \\mathbb{R}$ , we denote its induced Bregman divergence by $\\mathcal{D}_{\\psi}(\\cdot,\\cdot)$ . Then, any update of the form\n\n$$\n\\mathbf{x}_{k}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}\\{f(\\mathbf{x})+\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{x}_{k-1})\\}\n$$\n\nsatisfies the following inequality for any $\\mathbf{u}\\in \\mathcal{X}$\n\n$$\n\\begin{array}{r}{f ({\\mathbf x}_{k}) - f ({\\mathbf u}) \\le \\mathcal{D}_{\\psi}({\\mathbf u},{\\mathbf x}_{k - 1}) - \\mathcal{D}_{\\psi}({\\mathbf u},{\\mathbf x}_{k}) - \\mathcal{D}_{\\psi}({\\mathbf x}_{k},{\\mathbf x}_{k - 1}) .}\\end{array}\\tag{60}\n$$", "context": [{"name": "Optimistic OMD: update rule (6), Bregman divergence, strong convexity (Section 3.2)", "statement": "We employ the algorithmic framework of Optimistic Online Mirror Descent (OpTIMISTIC OMD) (Chiang et al., 2012; Rakhlin and Sridharan, 2013) as a general building block for designing algorithms for non-stationary online learning. Optimistic OMD is an algorithmic realization of optimistic online learning. Compared to the standard online learning setup, the player receives an additional element at each round: an optimistic vector $M_{t}\\in \\mathbb{R}^{d}$ This vector acts as a predictive hint or an optimistic estimate of the upcoming gradient, thereby called \"optimistic vector\" or simply \"optimism\". Optimistic OMD starts from the initial point $\\widehat{\\mathbf{x}}_{1}\\in \\mathcal{X}$ and performs the following two-step updates at each round:\n\n$$\n\\begin{array}{r l}&{\\mathbf{x}_{t}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}~\\big\\{\\eta_{t}\\langle M_{t},\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\widehat{\\mathbf{x}}_{t})\\big\\},}\\\\ &{\\widehat{\\mathbf{x}}_{t+1}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}~\\big\\{\\eta_{t}\\langle\\nabla f_{t}(\\mathbf{x}_{t}),\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\widehat{\\mathbf{x}}_{t})\\big\\},}\\end{array}\\tag{6}\n$$\n\nwhich firstly updates by the optimistic vector $M_{t}$ and then updates by the received gradient $\\nabla f_{t}(\\mathbf{x}_{t})$ . In above, $\\eta_{t}> 0$ is a (potentially) time-varying step size, and $\\mathcal{D}_{\\psi}(\\cdot,\\cdot)$ denotes the Bregman divergence associated with the regularizer $\\psi$ defined as $\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{y})=\\psi(\\mathbf{x})-\\psi(\\mathbf{y})-$ $\\langle\\nabla\\psi(\\mathbf{y}),\\mathbf{x}-\\mathbf{y}\\rangle$ . We may assume the regularizer to be σ-strongly convex with respect to the norm $\\begin{array}{r}{\\|\\cdot\\|,\\operatorname{i.e.,}\\psi(\\mathbf{y})\\geq\\psi(\\mathbf{x})+\\langle\\nabla\\psi(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle+\\frac{\\sigma}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2}}\\end{array}$ holds for any $\\mathbf{x}, \\mathbf{y}\\in{\\mathcal{X}}$ . We have the following general result regarding dynamic regret of optimistic OMD."}], "nl_notes": "", "lean": "import Mathlib\n\nnoncomputable section\n\nopen scoped BigOperators RealInnerProductSpace\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]\n\ndef bregman (ψ : E → ℝ) (gradψ : E → E) (x y : E) : ℝ :=\n  ψ x - ψ y - ⟪gradψ y, x - y⟫\n\ndef bregmanDual {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ)) (x y : F) : ℝ :=\n  ψ x - ψ y - gradψ y (x - y)\n\ndef pathLength (u : ℕ → E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 2 T, ‖u t - u (t - 1)‖\n\ndef gradVarSup (X : Set E) (gradf : ℕ → E → E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 2 T, ⨆ x : X, ‖gradf t (x : E) - gradf (t - 1) (x : E)‖ ^ 2\n\ntheorem theorem1\n    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]\n    {T : ℕ} (X : Set F)\n    (f : ℕ → F → ℝ) (gradf : ℕ → F → (F →L[ℝ] ℝ))\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (x xhat u : ℕ → F) (M : ℕ → (F →L[ℝ] ℝ)) (η : ℕ → ℝ)\n\n    (hXconv : Convex ℝ X)\n\n    (hη : ∀ t ∈ Finset.Icc 1 T, 0 < η t)\n\n    (hsc : ∀ a ∈ X, ∀ b ∈ X, ψ a + gradψ a (b - a) + (1 / 2) * ‖b - a‖ ^ 2 ≤ ψ b)\n\n    (hψgrad : ∀ x ∈ X, HasFDerivAt ψ (gradψ x) x)\n\n    (hfconv : ∀ t ∈ Finset.Icc 1 T, ConvexOn ℝ X (f t))\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, HasFDerivAt (f t) (gradf t (x t)) (x t))\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * M t (x t) + bregmanDual ψ gradψ (x t) (xhat t)\n        ≤ η t * M t z + bregmanDual ψ gradψ z (xhat t))\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * gradf t (x t) (xhat (t + 1)) + bregmanDual ψ gradψ (xhat (t + 1)) (xhat t)\n        ≤ η t * gradf t (x t) z + bregmanDual ψ gradψ z (xhat t)) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ (∑ t ∈ Finset.Icc 1 T, η t * ‖gradf t (x t) - M t‖ ^ 2)\n        + (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (u t) (xhat t)\n              - bregmanDual ψ gradψ (u t) (xhat (t + 1))))\n        - (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (xhat (t + 1)) (x t)\n              + bregmanDual ψ gradψ (x t) (xhat t))) := by\n  sorry\n\ntheorem lemma4_bregmanProximal\n    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]\n    (X : Set F) (f ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (xk xkm1 u : F)\n\n    (hXconv : Convex ℝ X)\n    (hxkX : xk ∈ X) (hxkm1X : xkm1 ∈ X) (huX : u ∈ X)\n\n    (hfconv : ConvexOn ℝ X f)\n    (hflsc : LowerSemicontinuousOn f X)\n\n    (hψconv : ConvexOn ℝ X ψ)\n\n    (hψgrad : HasFDerivAt ψ (gradψ xk) xk)\n    (hψgrad' : HasFDerivAt ψ (gradψ xkm1) xkm1)\n\n    (hprox : ∀ z ∈ X,\n      f xk + bregmanDual ψ gradψ xk xkm1 ≤ f z + bregmanDual ψ gradψ z xkm1) :\n    f xk - f u\n      ≤ bregmanDual ψ gradψ u xkm1 - bregmanDual ψ gradψ u xk\n          - bregmanDual ψ gradψ xk xkm1 := by\n  sorry\n\ntheorem lemma1_OEGD\n    {T : ℕ} (X : Set E)\n    (f : ℕ → E → ℝ) (gradf : ℕ → E → E)\n    (x xhat u : ℕ → E) (η G D L : ℝ)\n    (hη : 0 < η) (hL : 0 < L) (hηL : η ≤ 1 / (4 * L))\n    (hXne : X.Nonempty)\n\n    (hXconv : Convex ℝ X)\n\n    (hG : ∀ t, ∀ a ∈ X, ‖gradf t a‖ ≤ G)\n    (hgrad0 : ∀ a, gradf 0 a = 0)\n\n    (h0X : (0 : E) ∈ X) (hD : ∀ a ∈ X, ∀ b ∈ X, ‖a - b‖ ≤ D)\n\n    (hsmooth : ∀ t, ∀ a b, ‖gradf t a - gradf t b‖ ≤ L * ‖a - b‖)\n\n    (hconv : ∀ t ∈ Finset.Icc 1 T, ∀ a ∈ X, ∀ b ∈ X,\n      f t a + ⟪gradf t a, b - a⟫ ≤ f t b)\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, ∀ a, HasGradientAt (f t) (gradf t a) a)\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖x t - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), x t⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), z⟫)\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖xhat (t + 1) - xhat t‖ ^ 2 + η * ⟪gradf t (x t), xhat (t + 1)⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf t (x t), z⟫) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ η * (G ^ 2 + 2 * gradVarSup X gradf T)\n        + (1 / (2 * η)) * (D ^ 2 + 2 * D * pathLength u T) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem1", "lemma4_bregmanProximal", "lemma1_OEGD"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 2320, "end": 2960, "gold": "{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]\n    (X : Set F) (f ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (xk xkm1 u : F)\n\n    (hXconv : Convex ℝ X)\n    (hxkX : xk ∈ X) (hxkm1X : xkm1 ∈ X) (huX : u ∈ X)\n\n    (hfconv : ConvexOn ℝ X f)\n    (hflsc : LowerSemicontinuousOn f X)\n\n    (hψconv : ConvexOn ℝ X ψ)\n\n    (hψgrad : HasFDerivAt ψ (gradψ xk) xk)\n    (hψgrad' : HasFDerivAt ψ (gradψ xkm1) xkm1)\n\n    (hprox : ∀ z ∈ X,\n      f xk + bregmanDual ψ gradψ xk xkm1 ≤ f z + bregmanDual ψ gradψ z xkm1) :\n    f xk - f u\n      ≤ bregmanDual ψ gradψ u xkm1 - bregmanDual ψ gradψ u xk\n          - bregmanDual ψ gradψ xk xkm1"}]}
{"item_id": "v25-21-0748::lemma1_OEGD::A-stmt", "paper_id": "v25-21-0748", "target_name": "lemma1_OEGD", "source": "Research-level", "nl_statement": "Lemma 1. Under Assumptions 1, 2, and 3, by choosing $\\begin{array}{r}{\\eta \\le \\frac{1}{4 L}}\\end{array}$ , dynamic regret of OEGD (namely, OPTIMISTIC OMD with $\\begin{array}{r}{\\psi(\\mathbf{x})=\\frac{1}{2}\\|\\mathbf{x}\\|_{2}^{2}}\\end{array}$ and $M_{t}=\\nabla f_{t-1}(\\mathbf x_{t-1}))$ satisfies\n\n$$\n\\sum_{t=1}^{T}f_{t}(\\mathbf x_{t})-\\sum_{t=1}^{T}f_{t}(\\mathbf u_{t})\\leq\\eta(G^{2}+2V_{T})+\\frac{1}{2\\eta}(D^{2}+2DP_{T})\\tag{9}\n$$\n\nfor any comparator sequence $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$", "context": [{"name": "Setup: online convex optimization protocol (Introduction)", "statement": "At iteration t, the player chooses a decision vector $\\mathbf{x}_{t}$ from a convex set $\\boldsymbol{\\mathcal{X}}\\subseteq \\mathbb{R}^{d}$ . Subsequently, the adversary discloses a convex function $f_{t}: \\mathcal{X}\\mapsto \\mathbb{R}$ ,and the player incurs a loss denoted by $f_{t}(\\mathbf{x}_{t})$"}, {"name": "Setup: problem setup and feedback models (Section 3.1)", "statement": "Online Convex Optimization (OCO) can be modeled as an iterated game between the player and the environments. At iteration $t \\in [ T ]$ , the player first chooses the decision $\\mathbf{x}_{t}$ from a convex feasible set $\\boldsymbol{\\mathcal{X}}\\subseteq \\mathbb{R}^{d}$ , then the environments reveal the loss function $f_{t}: \\mathcal{X}\\mapsto \\mathbb{R}$ and the player suffers the loss $f_{t}(\\mathbf{x}_{t})$ and observes a certain information about the function $f_{t}(\\cdot).$ 2 According to the revealed information, the online learning problems are typically classified into full-information online learning and partial-information online learning (or sometimes called bandit online learning). In this paper, we focus on the full-information one, which can be further categorized into the following two setups:\n\n(i) multi-gradient feedback: the player can access the entire gradient function $\\nabla f_{t}(\\cdot)$ and thus can evaluate the gradient multiple times at each round;\n\n(ii) one-gradient feedback: the player can observe the gradient information $\\nabla f_{t}(\\mathbf{x}_{t})$ after submitting the decision $\\mathbf{x}_{t}$ at each round.\n\nIn Section 4.2, we develop the Sword algorithm, which achieves the gradient-variation dynamic regret under the multi-gradient feedback model. In Section 4.3, we present an improved algorithm called Sword++ that can achieve the same dynamic regret guarantee (up to constants) under the more challenging one-gradient feedback model.\n\nTo handle non-stationary environments, we focus on the dynamic regret measure, which compares the online algorithm to a sequence of time-varying comparators $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$ as defined in (2). An upper bound of dynamic regret should be a function of comparators, and typically the bound depends on some regularities that measure the fluctuation of the comparator sequence, such as the path length $\\begin{array}{r}{P_{T}= \\sum_{t = 2}^{T}\\lVert \\mathbf{u}_{t}- \\mathbf{u}_{t - 1}\\rVert_{2}}\\end{array}$ Throughout the paper, we focus on the Euclidean norm for simplicity, and it is straightforward to extend our results to general primal-dual norms.\n\nIn addition to the regret measure, we further consider the gradient query complexity. Note that algorithms designed for the multi-gradient feedback model may query the gradients for multiple times at each round. However, most algorithms designed for the static regret minimization only require one gradient per iteration, namely, using $\\nabla f_{t}(\\mathbf{x}_{t})$ for the next update only. Therefore, it is more desirable to achieve the favorable regret guarantees under the one-gradient feedback model. In other words, our aim is to develop first-order methods for dynamic regret minimization that require only one gradient query per iteration."}, {"name": "Dynamic regret, eq. (2)", "statement": "To address this limitation, dynamic regret is proposed to compete with changing comparators $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$\n\n$$\n\\mathrm{D-Regret}_{T}(\\mathbf{u}_{1},\\ldots,\\mathbf{u}_{T})=\\sum_{t=1}^{T}f_{t}(\\mathbf{x}_{t})-\\sum_{t=1}^{T}f_{t}(\\mathbf{u}_{t}),\\tag{2}\n$$"}, {"name": "Optimistic OMD: update rule (6), Bregman divergence, strong convexity (Section 3.2)", "statement": "We employ the algorithmic framework of Optimistic Online Mirror Descent (OpTIMISTIC OMD) (Chiang et al., 2012; Rakhlin and Sridharan, 2013) as a general building block for designing algorithms for non-stationary online learning. Optimistic OMD is an algorithmic realization of optimistic online learning. Compared to the standard online learning setup, the player receives an additional element at each round: an optimistic vector $M_{t}\\in \\mathbb{R}^{d}$ This vector acts as a predictive hint or an optimistic estimate of the upcoming gradient, thereby called \"optimistic vector\" or simply \"optimism\". Optimistic OMD starts from the initial point $\\widehat{\\mathbf{x}}_{1}\\in \\mathcal{X}$ and performs the following two-step updates at each round:\n\n$$\n\\begin{array}{r l}&{\\mathbf{x}_{t}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}~\\big\\{\\eta_{t}\\langle M_{t},\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\widehat{\\mathbf{x}}_{t})\\big\\},}\\\\ &{\\widehat{\\mathbf{x}}_{t+1}=\\underset{\\mathbf{x}\\in\\mathcal{X}}{\\arg\\operatorname*{min}}~\\big\\{\\eta_{t}\\langle\\nabla f_{t}(\\mathbf{x}_{t}),\\mathbf{x}\\rangle+\\mathcal{D}_{\\psi}(\\mathbf{x},\\widehat{\\mathbf{x}}_{t})\\big\\},}\\end{array}\\tag{6}\n$$\n\nwhich firstly updates by the optimistic vector $M_{t}$ and then updates by the received gradient $\\nabla f_{t}(\\mathbf{x}_{t})$ . In above, $\\eta_{t}> 0$ is a (potentially) time-varying step size, and $\\mathcal{D}_{\\psi}(\\cdot,\\cdot)$ denotes the Bregman divergence associated with the regularizer $\\psi$ defined as $\\mathcal{D}_{\\psi}(\\mathbf{x},\\mathbf{y})=\\psi(\\mathbf{x})-\\psi(\\mathbf{y})-$ $\\langle\\nabla\\psi(\\mathbf{y}),\\mathbf{x}-\\mathbf{y}\\rangle$ . We may assume the regularizer to be σ-strongly convex with respect to the norm $\\begin{array}{r}{\\|\\cdot\\|,\\operatorname{i.e.,}\\psi(\\mathbf{y})\\geq\\psi(\\mathbf{x})+\\langle\\nabla\\psi(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle+\\frac{\\sigma}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2}}\\end{array}$ holds for any $\\mathbf{x}, \\mathbf{y}\\in{\\mathcal{X}}$ . We have the following general result regarding dynamic regret of optimistic OMD."}, {"name": "Assumption 1 (bounded gradients)", "statement": "Assumption 1. The norm of the gradients of online functions over the domain $\\mathcal{X}$ is bounded by G, i.e., $\\|\\nabla f_{t}(\\mathbf{x})\\|_{2}\\leq G$ , for all $\\mathbf{x}\\in \\mathcal{X}$ and $t \\in [ T ]$ ."}, {"name": "Assumption 2 (domain contains origin, diameter D)", "statement": "Assumption 2. The domain $\\mathcal{X}\\subseteq \\mathbb{R}^{d}$ contains the origin $\\mathbf{0}$, and the diameter of the domain $\\mathcal{X}$ is at most D, i.e., $\\| \\mathbf{x}- \\mathbf{x}^{\\prime}\\|_{2}\\leq D$ for any $\\mathbf{x}, \\mathbf{x}^{\\prime}\\in \\mathcal{X}$"}, {"name": "Assumption 3 (L-smoothness)", "statement": "Assumption 3. All the online functions are L-smooth, i.e., $\\|\\nabla f_{t}(\\mathbf{x})-\\nabla f_{t}(\\mathbf{x}^{\\prime})\\|_{2}\\leq L\\|\\mathbf{x}-\\mathbf{x}^{\\prime}\\|_{2}$ for any $\\mathbf{x}, \\mathbf{x}^{\\prime}\\in \\mathbb{R}^{d}$ and $t \\in [ T ]$"}, {"name": "Remark: Theorem 1 does not need smoothness", "statement": "We have the following remarks regarding the assumptions. The general dynamic regret analysis of Optimistic OMD (Theorem 1) does not require the smoothness assumption. Nevertheless, this assumption is crucial for achieving problem-dependent dynamic regret bounds."}, {"name": "OEGD update (8) and its reduction to Optimistic OMD (Section 4.1)", "statement": "In the study of static regret, Chiang et al. (2012) propose the online extra-gradient descent (OEGD) algorithm and prove that the algorithm enjoys gradient-variation static regret. Specifically, OEGD starts from $\\widehat{\\mathbf{x}}_{1}\\in \\mathcal{X}$ and then updates by\n\n$$\n\\begin{array}{r}{\\mathbf{x}_{t}=\\Pi_{\\mathcal{X}}\\left[\\widehat{\\mathbf{x}}_{t}-\\eta\\nabla f_{t-1}(\\mathbf{x}_{t-1})\\right],\\widehat{\\mathbf{x}}_{t+1}=\\Pi_{\\mathcal{X}}\\left[\\widehat{\\mathbf{x}}_{t}-\\eta\\nabla f_{t}(\\mathbf{x}_{t})\\right],}\\end{array}\\tag{8}\n$$\n\nwhere we define $f_{0}(\\mathbf{x}_{0})=\\mathbf{0}$ and $\\Pi_{\\mathcal{X}}[ \\cdot ]$ the Euclidean projection onto the nearest point in $\\mathcal{X}.$ We consider a fixed step size $\\eta > 0$ for simplicity. For convex and smooth functions, Chiang et al. (2012) prove that OEGD enjoys an $\\mathcal{O}(\\sqrt{1+V_{T}})$ gradient-variation static regret.\n\nActually, OEGD can be viewed as a specialization of Optimistic OMD (6) presented in Section 3.2, by choosing the regularizer $\\begin{array}{r}{\\psi(\\mathbf{x})=\\frac{1}{2}\\|\\mathbf{x}\\|_{2}^{2}}\\end{array}$ and the optimistic vector $M_{t}=$ $\\nabla f_{t-1}\\big(\\mathbf{x}_{t-1}\\big)$ as well as a fixed step size $\\eta > 0$ Therefore, Theorem 1 directly implies the following dynamic regret upper bound for OEGD, with proof in Appendix A."}, {"name": "Path length P_T, eq. (4)", "statement": "$$\nP_{T}= \\sum_{t = 2}^{T}\\lVert \\mathbf{u}_{t}- \\mathbf{u}_{t - 1}\\rVert_{2}\\tag{4}\n$$\n\nis the path length of comparators $\\mathbf{u}_{1},\\ldots,\\mathbf{u}_{T}$ and thus reflects the non-stationarity of environments."}, {"name": "Gradient variation V_T, eq. (5)", "statement": "Specifically, we focus on the following two problem-dependent quantities: the gradient variation of online functions $V_{T}$ , and the cumulative loss of the comparator sequence $F_{T}$ , defined as\n\n$$\nV_{T}=\\sum_{t=2}^{T}\\operatorname*{sup}_{\\mathbf x\\in\\mathcal X}\\|\\nabla f_{t}(\\mathbf x)-\\nabla f_{t-1}(\\mathbf x)\\|_{2}^{2},~\\mathrm{and}~F_{T}=\\sum_{t=1}^{T}f_{t}(\\mathbf u_{t}).\\tag{5}\n$$"}, {"name": "Theorem 1 (another result of the same paper)", "statement": "Theorem 1. Suppose that the regularizer ψ $: \\mathcal{X}\\mapsto \\mathbb{R}$ is 1-strongly convex with respect to the norm $\\| \\cdot \\|$ , and let $\\| \\cdot \\|_{*}$ be the dual norm of $\\| \\cdot \\|$ . The dynamic regret of OpTIMISTIC OMD whose update rule is specified in (6) is bounded as follows:\n\n$$\n\\begin{array}{r}{\\displaystyle\\sum_{t=1}^{T}f_{t}(\\mathbf{x}_{t})-\\displaystyle\\sum_{t=1}^{T}f_{t}(\\mathbf{u}_{t})\\leq\\displaystyle\\sum_{t=1}^{T}\\eta_{t}\\|\\nabla f_{t}(\\mathbf{x}_{t})-M_{t}\\|_{*}^{2}+\\displaystyle\\sum_{t=1}^{T}\\frac{1}{\\eta_{t}}\\Big(\\mathcal{D}_{\\psi}(\\mathbf{u}_{t},\\widehat{\\mathbf{x}}_{t})-\\mathcal{D}_{\\psi}(\\mathbf{u}_{t},\\widehat{\\mathbf{x}}_{t+1})\\Big)}\\\\ {-\\displaystyle\\sum_{t=1}^{T}\\frac{1}{\\eta_{t}}\\Big(\\mathcal{D}_{\\psi}(\\widehat{\\mathbf{x}}_{t+1},\\mathbf{x}_{t})+\\mathcal{D}_{\\psi}(\\mathbf{x}_{t},\\widehat{\\mathbf{x}}_{t})\\Big),}\\end{array}\\tag{7}\n$$\n\nwhich holds for any comparator sequence $\\mathbf{u}_{1},\\dots,\\mathbf{u}_{T}\\in\\mathcal{X}$"}], "nl_notes": "", "lean": "import Mathlib\n\nnoncomputable section\n\nopen scoped BigOperators RealInnerProductSpace\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] [FiniteDimensional ℝ E]\n\ndef bregman (ψ : E → ℝ) (gradψ : E → E) (x y : E) : ℝ :=\n  ψ x - ψ y - ⟪gradψ y, x - y⟫\n\ndef bregmanDual {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ)) (x y : F) : ℝ :=\n  ψ x - ψ y - gradψ y (x - y)\n\ndef pathLength (u : ℕ → E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 2 T, ‖u t - u (t - 1)‖\n\ndef gradVarSup (X : Set E) (gradf : ℕ → E → E) (T : ℕ) : ℝ :=\n  ∑ t ∈ Finset.Icc 2 T, ⨆ x : X, ‖gradf t (x : E) - gradf (t - 1) (x : E)‖ ^ 2\n\ntheorem theorem1\n    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]\n    {T : ℕ} (X : Set F)\n    (f : ℕ → F → ℝ) (gradf : ℕ → F → (F →L[ℝ] ℝ))\n    (ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (x xhat u : ℕ → F) (M : ℕ → (F →L[ℝ] ℝ)) (η : ℕ → ℝ)\n\n    (hXconv : Convex ℝ X)\n\n    (hη : ∀ t ∈ Finset.Icc 1 T, 0 < η t)\n\n    (hsc : ∀ a ∈ X, ∀ b ∈ X, ψ a + gradψ a (b - a) + (1 / 2) * ‖b - a‖ ^ 2 ≤ ψ b)\n\n    (hψgrad : ∀ x ∈ X, HasFDerivAt ψ (gradψ x) x)\n\n    (hfconv : ∀ t ∈ Finset.Icc 1 T, ConvexOn ℝ X (f t))\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, HasFDerivAt (f t) (gradf t (x t)) (x t))\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * M t (x t) + bregmanDual ψ gradψ (x t) (xhat t)\n        ≤ η t * M t z + bregmanDual ψ gradψ z (xhat t))\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      η t * gradf t (x t) (xhat (t + 1)) + bregmanDual ψ gradψ (xhat (t + 1)) (xhat t)\n        ≤ η t * gradf t (x t) z + bregmanDual ψ gradψ z (xhat t)) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ (∑ t ∈ Finset.Icc 1 T, η t * ‖gradf t (x t) - M t‖ ^ 2)\n        + (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (u t) (xhat t)\n              - bregmanDual ψ gradψ (u t) (xhat (t + 1))))\n        - (∑ t ∈ Finset.Icc 1 T,\n            (1 / η t) * (bregmanDual ψ gradψ (xhat (t + 1)) (x t)\n              + bregmanDual ψ gradψ (x t) (xhat t))) := by\n  sorry\n\ntheorem lemma4_bregmanProximal\n    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]\n    (X : Set F) (f ψ : F → ℝ) (gradψ : F → (F →L[ℝ] ℝ))\n    (xk xkm1 u : F)\n\n    (hXconv : Convex ℝ X)\n    (hxkX : xk ∈ X) (hxkm1X : xkm1 ∈ X) (huX : u ∈ X)\n\n    (hfconv : ConvexOn ℝ X f)\n    (hflsc : LowerSemicontinuousOn f X)\n\n    (hψconv : ConvexOn ℝ X ψ)\n\n    (hψgrad : HasFDerivAt ψ (gradψ xk) xk)\n    (hψgrad' : HasFDerivAt ψ (gradψ xkm1) xkm1)\n\n    (hprox : ∀ z ∈ X,\n      f xk + bregmanDual ψ gradψ xk xkm1 ≤ f z + bregmanDual ψ gradψ z xkm1) :\n    f xk - f u\n      ≤ bregmanDual ψ gradψ u xkm1 - bregmanDual ψ gradψ u xk\n          - bregmanDual ψ gradψ xk xkm1 := by\n  sorry\n\ntheorem lemma1_OEGD\n    {T : ℕ} (X : Set E)\n    (f : ℕ → E → ℝ) (gradf : ℕ → E → E)\n    (x xhat u : ℕ → E) (η G D L : ℝ)\n    (hη : 0 < η) (hL : 0 < L) (hηL : η ≤ 1 / (4 * L))\n    (hXne : X.Nonempty)\n\n    (hXconv : Convex ℝ X)\n\n    (hG : ∀ t, ∀ a ∈ X, ‖gradf t a‖ ≤ G)\n    (hgrad0 : ∀ a, gradf 0 a = 0)\n\n    (h0X : (0 : E) ∈ X) (hD : ∀ a ∈ X, ∀ b ∈ X, ‖a - b‖ ≤ D)\n\n    (hsmooth : ∀ t, ∀ a b, ‖gradf t a - gradf t b‖ ≤ L * ‖a - b‖)\n\n    (hconv : ∀ t ∈ Finset.Icc 1 T, ∀ a ∈ X, ∀ b ∈ X,\n      f t a + ⟪gradf t a, b - a⟫ ≤ f t b)\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, ∀ a, HasGradientAt (f t) (gradf t a) a)\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖x t - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), x t⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), z⟫)\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖xhat (t + 1) - xhat t‖ ^ 2 + η * ⟪gradf t (x t), xhat (t + 1)⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf t (x t), z⟫) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ η * (G ^ 2 + 2 * gradVarSup X gradf T)\n        + (1 / (2 * η)) * (D ^ 2 + 2 * D * pathLength u T) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["theorem1", "lemma4_bregmanProximal", "lemma1_OEGD"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 3000, "end": 4305, "gold": "{T : ℕ} (X : Set E)\n    (f : ℕ → E → ℝ) (gradf : ℕ → E → E)\n    (x xhat u : ℕ → E) (η G D L : ℝ)\n    (hη : 0 < η) (hL : 0 < L) (hηL : η ≤ 1 / (4 * L))\n    (hXne : X.Nonempty)\n\n    (hXconv : Convex ℝ X)\n\n    (hG : ∀ t, ∀ a ∈ X, ‖gradf t a‖ ≤ G)\n    (hgrad0 : ∀ a, gradf 0 a = 0)\n\n    (h0X : (0 : E) ∈ X) (hD : ∀ a ∈ X, ∀ b ∈ X, ‖a - b‖ ≤ D)\n\n    (hsmooth : ∀ t, ∀ a b, ‖gradf t a - gradf t b‖ ≤ L * ‖a - b‖)\n\n    (hconv : ∀ t ∈ Finset.Icc 1 T, ∀ a ∈ X, ∀ b ∈ X,\n      f t a + ⟪gradf t a, b - a⟫ ≤ f t b)\n\n    (hfgrad : ∀ t ∈ Finset.Icc 1 T, ∀ a, HasGradientAt (f t) (gradf t a) a)\n\n    (hu : ∀ t ∈ Finset.Icc 1 T, u t ∈ X)\n\n    (hxX : ∀ t ∈ Finset.Icc 1 T, x t ∈ X)\n    (hxhat1 : xhat 1 ∈ X)\n    (hxhatX : ∀ t ∈ Finset.Icc 1 T, xhat (t + 1) ∈ X)\n\n    (hx : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖x t - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), x t⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf (t - 1) (x (t - 1)), z⟫)\n\n    (hxhat : ∀ t ∈ Finset.Icc 1 T, ∀ z ∈ X,\n      (1 / 2) * ‖xhat (t + 1) - xhat t‖ ^ 2 + η * ⟪gradf t (x t), xhat (t + 1)⟫\n        ≤ (1 / 2) * ‖z - xhat t‖ ^ 2 + η * ⟪gradf t (x t), z⟫) :\n    (∑ t ∈ Finset.Icc 1 T, f t (x t)) - (∑ t ∈ Finset.Icc 1 T, f t (u t))\n      ≤ η * (G ^ 2 + 2 * gradVarSup X gradf T)\n        + (1 / (2 * η)) * (D ^ 2 + 2 * D * pathLength u T)"}]}
{"item_id": "v24-21-1250::strategic_agnostic_PAC::A-stmt", "paper_id": "v24-21-1250", "target_name": "strategic_agnostic_PAC", "source": "Research-level", "nl_statement": "Theorem 4 Any strategic classification instance $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ is agnostic PAC learnable with sample complexity $m_{\\mathcal{H},R,c}(\\delta,\\epsilon)\\leq C\\epsilon^{-2}[d+\\log(\\frac{1}{\\delta})]$ by the SERM in Eq. (4), where $d=SVC(\\mathcal{H},R,c)$ is the strategic VC-dimension and $C$ is an absolute constant.", "context": [{"name": "Setup: data points, preferences, classifiers (Section 2)", "statement": "Basic Setup. We consider binary classification, where each data point is characterized by a tuple $(x,y,r)$ . Like classic classification setups, $\\mathbf{\\boldsymbol{x}}\\in \\mathcal{X}$ is the feature vector and $y \\in$ $\\{+1,-1\\}$ is its label. The only difference of our setup from classic classification problems is the additional $r \\in R \\subseteq \\mathbb{R}$ , which is the data point's (positive or negative) preference/reward of being labeled as +1. The data point's reward for label −1 is, without loss of generality, normalized to be 0. A classifier is a mapping $h:\\mathcal{X}\\to\\{+1,-1\\}$ . Our model is essentially the same as that of (Hardt et al., 2016; Miller et al., 2019), except that the r in our model can be any real value from set R whereas the aforementioned works assume $r = 1$ for all data points. Notably, we also allow r to be negative, which means some data points prefer to be classified as label $-1$. This generalization is natural and very useful because it allows much richer agent preferences. For instance, it casts the adversarial/robust classification problem as a special case of our model as well (see discussions later). Intuitively, the set R captures the richness of agents' preferences. As we will prove, how rich it is will affect both the statistical learnability and computational tractability of the learning problem."}, {"name": "Setup: strategic manipulation and cost functions (Section 2)", "statement": "The Strategic Manipulation of Test Data. We consider strategic behaviors during the test phase and assume that the training data is unaltered/uncontaminated. An illustration of the setup can be found in Figure 1. A generic test data point is denoted as $(x,y,r)$ . The test data point is strategic and may shift its feature to vector z with cost $c(z;\\pmb{x})$ where $c : \\mathcal{X}\\times \\mathcal{X}\\to \\mathbb{R}_{\\ge 0}$ . In general, function c can be an arbitrary non-negative cost function. In our study of strategic linear classification, we assume the cost functions are induced by seminorms. We will consider the following two types of cost functions, with increasing generality.\n\n1. Instance-invariant cost functions: A cost function c is instance-invariant if there is a common function l such that $c(z;\\pmb{x})=l(\\pmb{z}-\\pmb{x})$ for any point $(x,y,r)$ .\n\n2. Instance-wise cost functions: With instance-wise cost functions, each data point is allowed to possess its own loss function. To capture this situation, we need to augment each data point's representation from $(x,y,r)$ to $({\\pmb x},y,r,l)$ , where l is a function depending on x such that $c ({\\boldsymbol{z}};{\\pmb x}) = l ({\\boldsymbol{z}}-{\\pmb x})$ for some semi-norm l. For notational convenience, we often refer to function l associated with this data point as $l_{x}$ though we emphasize that this is a slight abuse of notation because $l_{x}$ is determined not just by the data point's feature x but rather by the data point itself.4\n\nBoth cases have been considered in previous works. For instance, the separable cost function studied in (Hardt et al., 2016) is instance-wise, and the cost function induced by a seminorm as assumed by the main theorem of (Cullina et al., 2018) is instance-invariant. We shall prove later that the choice among these two types of cost functions will largely affect the efficient learnability of the problem."}, {"name": "Eq. (1): data point best response and tie-breaking", "statement": "Given a classifier h, the strategic test data point $(x,y,r)$ may shift its feature vector to z and would like to pick the best such z by solving the following optimization problem:\n\nData Point Best Response:\n\n$$\n\\Delta_{c}(\\pmb{x},r;h)=\\arg\\operatorname*{max}_{\\pmb{\\mathrm{z}}\\in\\mathcal{X}}\\big [\\mathbb{I}(h(\\pmb{z})=1)\\cdot r-c(\\pmb{z};\\pmb{x})\\big].\\tag{1}\n$$\n\nwhere $\\mathbb{I}(S)$ is the indicator function and $[\\mathbb{I}(h(z)=1)\\cdot r-c(z;\\pmb{x})]$ is the quasi-linear utility function of data point $(x,y,r)$ . We call $\\Delta_{c}({\\pmb x},r;h)$ the manipulated feature. When there are multiple best responses, we assume the data point may choose any best response and thus will adopt the standard worst-case analysis. Note that the test data's strategic behaviors do not change its true label. Such strategic gaming behaviors differ from strategic improvements (see (Miller et al., 2019) for more discussions on their differences). We also point out that the preference r is assumed to be known for any training data (though no need to be known for testing data). While this may appear a strong assumption at the first glance, we believe this parameter can be estimated in real applications (e.g., how much it matters to someone for being admitted to a university, for receiving a loan or for being tested COVID positive). An interesting future direction may be to understand how robust the strategic classifier is to the estimation error of r in the training data."}, {"name": "StraC problem and Eq. (2): strategic 0-1 loss (Section 2.1)", "statement": "A STRAC problem is described by a hypothesis class H, the set of preferences R and a manipulation cost function c. We thus denote it as $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ . Adopting the standard statistical learning framework, the input to our learning task is n uncontaminated training data points $({\\pmb x}_{1},y_{1},r_{1}),\\dots,({\\pmb x}_{n},y_{n},r_{n})$ drawn independently and identically (i.i.d.) from distribution D. Given these training data, we look to learn a classifier $h \\in \\mathcal H$ which minimizes the basic 0-1 loss, defined as follows:\n\nStrategic 0-1 Loss of classifier h :\n\n$$\nL_{c}(h;\\mathcal{D})=\\operatorname*{Pr}_{(\\mathbf{x},y,r)\\sim\\mathcal{D}}\\big [h(\\Delta_{c}(\\mathbf{x},r;h))\\neq y\\big].\\tag{2}\n$$\n\nNotably, the classifier h in the above loss definition takes the manipulated feature $\\textstyle\\Delta_{c}({\\pmb x},r;h)\\big)$ as input and, nevertheless, looks to correctly predict the true label y. For notational convenience, we sometimes omit c when it is clear from the context and simply write $\\Delta({\\pmb x},r;h)$ and $L(h;\\mathcal{D})$ ."}, {"name": "Definition 1 (Strategic Shattering Coefficients)", "statement": "In this section, we introduce the notion of strategic VC-dimension (SVC) and show that it properly captures the behaviors of a hypothesis class in the strategic setup introduced above. We then show the connection of SVC with previous studies on both strategic and adversarial learning. Before formally introducing SVC, we first define the shattering coefficients in strategic setups.\n\nDefinition 1 (Strategic Shattering Coefficients) The n-th shattering coefficient of any strategic classification problem $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ is defined as\n\n$$\n\\sigma_{n}(\\mathcal{H},R,c)=\\operatorname*{max}_{(\\boldsymbol{x},\\boldsymbol{r})\\in\\mathcal{X}^{n}\\times R^{n}}~|\\{(h(\\Delta_{c}(\\boldsymbol{x}_{1},\\boldsymbol{r}_{1};h)),\\cdots,h(\\Delta_{c}(\\boldsymbol{x}_{n},\\boldsymbol{r}_{n};h)):h\\in\\mathcal{H}\\}|,\n$$\n\nwhere $\\Delta_{c}(\\pmb{x}_{i},r_{i};h)$ defined in Eq. (1) is a best response of data point $({\\pmb x}_{i}, y_{i}, r_{i})$ to classifier h under cost function c.\n\nThat is, $\\sigma_{n}(\\mathcal{H},R,c)$ captures the maximum number of classification behaviors/outcomes (among all choices of data points) that classifiers in H can possibly induce by using manipulated features as input. Like classic definition of shattering coefficient, the $\\sigma_{n}(\\mathcal{H},R,c)$ here does not involve the labels of the data points at all. In contrast, in the shattering coefficient definition for adversarial VC-dimension of (Cullina et al., 2018), the \"max\" is allowed to be over data labels as well. This is an important difference compared to our setting. Given the definition of the strategic shattering coefficients, the definition of strategic VC-dimension is standard."}, {"name": "Definition 2 (Strategic VC-dimension), Eq. (3)", "statement": "Definition 2 The Strategic VC-dimension (SVC) for strategic classification problem $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ is defined as\n\n$$\nS V C(\\mathcal{H},R,c)=\\operatorname*{sup}\\{n\\in\\mathbb{N}:\\sigma_{n}(\\mathcal{H},R,c)=2^{n}\\}.\\tag{3}\n$$"}, {"name": "SERM, Eq. (4)", "statement": "We show that the SVC defined above correctly characterizes the learnability of any strategic classification problem $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ . We consider the standard Empirical risk minimization (ERM) paradigm for strategic classification, but take into account training data's manipulation behaviors. Specifically, given any cost function c, any n uncontaminated training data points $(\\mathbf{x}_{1},y_{1},r_{1}),\\cdot\\cdot\\cdot,(\\mathbf{x}_{n},y_{n},r_{n})$ drawn independently and identically (i.i.d.) from the same distribution $\\mathcal{D}.$ , the strategic empirical risk minimization (SERM) problem computes a classifier $h \\in \\mathcal H$ that minimizes the empirical strategic 0-1 loss in Eq. (2). Formally, the SERM for $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ is defined as follows:\n\n$$\n\\mathrm{SERM}:\\quad\\mathrm{argmin}_{h\\in\\mathcal{H}}\\ L_{c}(h,\\{(x_{i},y_{i},r_{i})\\}_{i=1}^{n})=\\sum_{i=1}^{n}\\mathbb{I}\\big [h(\\Delta_{c}(\\mathbf{x}_{i},r_{i};h))\\neq y_{i}\\big]\\tag{4}\n$$\n\nwhere $L_{c}(h,\\{(x_{i},y_{i},r_{i})\\}_{i=1}^{n})$ is the empirical loss (compared to the expected loss $L_{c}(h,\\mathcal{D})$ defined in Eq. (2)). Unlike the standard (non-strategic) ERM problem and similar in spirit to the \"incentive-aware ERM\" in (Zhang and Conitzer, 2021), classifiers in the SERM problem take each data point's strategic response $\\Delta_{c}~({\\bf x}_{i},r_{i};h)$ as input, while not the original feature vector $\\pmb{x}_{i}$"}, {"name": "Definition 3 (PAC-Learnability)", "statement": "Given the definition of strategic VC-dimension and the SERM framework, we state the sample complexity result for PAC-learning in our strategic setup:\n\nDefinition 3 (PAC-Learnability) In a strategic classification problem $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ the hypothesis class $\\mathcal{H}\\subseteq(\\mathcal{X}\\to\\{+1,-1\\})$ is Probably Approximately Correctly (PAC) learnable by an algorithm A if there is a function $m_{\\mathcal{H},R,c}:(0,1)^{2}\\to\\mathbb{N}$ such that $\\forall(\\delta,\\epsilon)\\in$ $(0,1)^{2}$ , for any $n\\ge m_{\\mathcal{H},R,c}(\\delta,\\epsilon)$ and any distribution D for $({\\pmb x},y,r)$ , with at least probability $1 - \\delta$ , we have $L_{c}(h^{\\ast},\\mathcal{D})\\leq\\epsilon$ where $h^{*}$ is the output of the algorithm A with n i.i.d. samples from D as input. The problem is agnostic PAC learnable if $\\begin{array}{r}{L_{c}(h^{*},\\mathcal{D})-\\operatorname*{inf}_{h\\in\\mathcal{H}}L_{c}(h,\\mathcal{D})\\leq\\epsilon}\\end{array}$"}], "nl_notes": "### Theorem 4\nNOTE (Theorem 4) — The Lean assumes the hypothesis class H is countable (`_hH_ctble`); the paper states no such condition (H ⊆ (X → {±1}) is otherwise arbitrary). Without it the theorem is not provable in ZFC: under CH, a class built from the initial segments of a well-order of [0,1] of type ω₁ satisfies every other hypothesis, yet with probability → 1 every SERM output has excess risk 0.75. This is the usual countability / admissibility convention for uniform convergence over a hypothesis class.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\ndef stratUtil {X : Type*} (c : X → X → ℝ) (h : X → Bool) (x : X) (r : ℝ) (z : X) : ℝ :=\n  (if h z then r else 0) - c z x\n\ndef IsBestResponse {X : Type*} (c : X → X → ℝ) (h : X → Bool) (x : X) (r : ℝ) (z : X) :\n    Prop :=\n  ∀ z' : X, stratUtil c h x r z' ≤ stratUtil c h x r z\n\ndef StrategicShatters {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) (n : ℕ) : Prop :=\n  ∃ x : Fin n → X, ∃ r : Fin n → ℝ, (∀ i, r i ∈ R) ∧\n    ∀ v : Fin n → Bool, ∃ h ∈ H, ∀ i, h (Δ h (x i) (r i)) = v i\n\nnoncomputable def SVC {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) : ℕ∞ :=\n  ⨆ (n : ℕ) (_ : StrategicShatters H R c Δ n), (n : ℕ∞)\n\nnoncomputable def strategicRisk {X : Type*} [MeasurableSpace X] (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) (h : X → Bool)\n    (D : MeasureTheory.Measure (X × Bool × ℝ)) : ℝ :=\n  (D {p : X × Bool × ℝ | h (Δ h p.1 p.2.2) ≠ p.2.1}).toReal\n\ndef empStrategicLoss {X : Type*} (c : X → X → ℝ) (Δ : (X → Bool) → X → ℝ → X)\n    (h : X → Bool) {n : ℕ} (s : Fin n → X × Bool × ℝ) : ℕ :=\n  (Finset.univ.filter (fun i => h (Δ h (s i).1 (s i).2.2) ≠ (s i).2.1)).card\n\ndef IsSERM {X : Type*} (H : Set (X → Bool)) (c : X → X → ℝ) (Δ : (X → Bool) → X → ℝ → X)\n    (hstar : X → Bool) {n : ℕ} (s : Fin n → X × Bool × ℝ) : Prop :=\n  hstar ∈ H ∧ ∀ h ∈ H, empStrategicLoss c Δ hstar s ≤ empStrategicLoss c Δ h s\n\ndef SeparableCost {X : Type*} (c : X → X → ℝ) : Prop :=\n  ∃ c₁ c₂ : X → ℝ, ∀ z x : X, c z x = max (c₂ z - c₁ x) 0\n\ndef dotp {d : ℕ} (w u : Fin d → ℝ) : ℝ := ∑ i, w i * u i\n\ntheorem strategic_agnostic_PAC :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (X : Type*) [MeasurableSpace X]\n        (H : Set (X → Bool)) (_hH : H.Nonempty) (_hH_ctble : H.Countable)\n        (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (_hc0 : ∀ x : X, c x x = 0)\n        (Δ : (X → Bool) → X → ℝ → X)\n        (_hΔ : ∀ h ∈ H, ∀ x, ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n        (_hmeas : ∀ h ∈ H, Measurable (fun p : X × Bool × ℝ => h (Δ h p.1 p.2.2)))\n        (d : ℕ) (_hd : SVC H R c Δ = (d : ℕ∞)),\n      ∃ m : ℝ → ℝ → ℕ,\n        (∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n            (m δ ε : ℝ) ≤ C * (ε ^ 2)⁻¹ * ((d : ℝ) + Real.log (1 / δ))) ∧\n        ∃ A : (N : ℕ) → (Fin N → X × Bool × ℝ) → (X → Bool),\n          (∀ N s, IsSERM H c Δ (A N s) s) ∧\n          ∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n          ∀ n : ℕ, m δ ε ≤ n →\n          ∀ D : MeasureTheory.Measure (X × Bool × ℝ), MeasureTheory.IsProbabilityMeasure D →\n\n          D {p : X × Bool × ℝ | p.2.2 ∉ R} = 0 →\n            1 - δ ≤\n              ((MeasureTheory.Measure.pi (fun _ : Fin n => D))\n                {s : Fin n → X × Bool × ℝ |\n                  strategicRisk c Δ (A n s) D - ⨅ h : ↥H, strategicRisk c Δ (h : X → Bool) D\n                    ≤ ε}).toReal := by\n  sorry\n\ntheorem separable_SVC_le_two\n    {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (Δ : (X → Bool) → X → ℝ → X)\n    (hΔ : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n\n    (htie : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R,\n      (∃ z, IsBestResponse c h x r z ∧ h z = decide (0 < r)) →\n      h (Δ h x r) = decide (0 < r))\n    (hR : (0 : ℝ) ∉ R) (hsep : SeparableCost c) (hc0 : ∀ x : X, c x x = 0) :\n    SVC H R c Δ ≤ 2 := by\n  sorry\n\ntheorem min_manipulation_cost_lower_bound\n    {d : ℕ} (l : (Fin d → ℝ) → ℝ) (w : Fin d → ℝ) (b : ℝ) (x : Fin d → ℝ)\n    (lstar : EReal) (hlstar : 0 < lstar)\n    (hnn  : ∀ u : Fin d → ℝ, 0 ≤ l u)\n    (hhom : ∀ (a : ℝ) (u : Fin d → ℝ), l (fun i => a * u i) = |a| * l u)\n    (htri : ∀ u v : Fin d → ℝ, l (fun i => u i + v i) ≤ l u + l v)\n    (hlub : IsLUB {t : EReal | ∃ z : Fin d → ℝ, l z ≤ 1 ∧ t = ((dotp w z : ℝ) : EReal)} lstar) :\n    IsLeast {t : EReal | ∃ x' : Fin d → ℝ, dotp w x' + b = 0 ∧\n        t = ((l (fun i => x' i - x i) : ℝ) : EReal)}\n      (((|dotp w x + b| : ℝ) : EReal) / lstar) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["strategic_agnostic_PAC", "separable_SVC_le_two", "min_manipulation_cost_lower_bound"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1653, "end": 2860, "gold": ":\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (X : Type*) [MeasurableSpace X]\n        (H : Set (X → Bool)) (_hH : H.Nonempty) (_hH_ctble : H.Countable)\n        (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (_hc0 : ∀ x : X, c x x = 0)\n        (Δ : (X → Bool) → X → ℝ → X)\n        (_hΔ : ∀ h ∈ H, ∀ x, ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n        (_hmeas : ∀ h ∈ H, Measurable (fun p : X × Bool × ℝ => h (Δ h p.1 p.2.2)))\n        (d : ℕ) (_hd : SVC H R c Δ = (d : ℕ∞)),\n      ∃ m : ℝ → ℝ → ℕ,\n        (∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n            (m δ ε : ℝ) ≤ C * (ε ^ 2)⁻¹ * ((d : ℝ) + Real.log (1 / δ))) ∧\n        ∃ A : (N : ℕ) → (Fin N → X × Bool × ℝ) → (X → Bool),\n          (∀ N s, IsSERM H c Δ (A N s) s) ∧\n          ∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n          ∀ n : ℕ, m δ ε ≤ n →\n          ∀ D : MeasureTheory.Measure (X × Bool × ℝ), MeasureTheory.IsProbabilityMeasure D →\n\n          D {p : X × Bool × ℝ | p.2.2 ∉ R} = 0 →\n            1 - δ ≤\n              ((MeasureTheory.Measure.pi (fun _ : Fin n => D))\n                {s : Fin n → X × Bool × ℝ |\n                  strategicRisk c Δ (A n s) D - ⨅ h : ↥H, strategicRisk c Δ (h : X → Bool) D\n                    ≤ ε}).toReal"}]}
{"item_id": "v24-21-1250::separable_SVC_le_two::A-stmt", "paper_id": "v24-21-1250", "target_name": "separable_SVC_le_two", "source": "Research-level", "nl_statement": "Proposition 9 For any hypothesis class H, any preference set R satisfying $0\\notin R,$ and any separable cost function $c(z;\\pmb{x})$ , we have $SVC(\\mathcal{H},R,c)\\leq 2$", "context": [{"name": "Setup: data points, preferences, classifiers (Section 2)", "statement": "Basic Setup. We consider binary classification, where each data point is characterized by a tuple $(x,y,r)$ . Like classic classification setups, $\\mathbf{\\boldsymbol{x}}\\in \\mathcal{X}$ is the feature vector and $y \\in$ $\\{+1,-1\\}$ is its label. The only difference of our setup from classic classification problems is the additional $r \\in R \\subseteq \\mathbb{R}$ , which is the data point's (positive or negative) preference/reward of being labeled as +1. The data point's reward for label −1 is, without loss of generality, normalized to be 0. A classifier is a mapping $h:\\mathcal{X}\\to\\{+1,-1\\}$ . Our model is essentially the same as that of (Hardt et al., 2016; Miller et al., 2019), except that the r in our model can be any real value from set R whereas the aforementioned works assume $r = 1$ for all data points. Notably, we also allow r to be negative, which means some data points prefer to be classified as label $-1$. This generalization is natural and very useful because it allows much richer agent preferences. For instance, it casts the adversarial/robust classification problem as a special case of our model as well (see discussions later). Intuitively, the set R captures the richness of agents' preferences. As we will prove, how rich it is will affect both the statistical learnability and computational tractability of the learning problem."}, {"name": "Setup: strategic manipulation and cost functions (Section 2)", "statement": "The Strategic Manipulation of Test Data. We consider strategic behaviors during the test phase and assume that the training data is unaltered/uncontaminated. An illustration of the setup can be found in Figure 1. A generic test data point is denoted as $(x,y,r)$ . The test data point is strategic and may shift its feature to vector z with cost $c(z;\\pmb{x})$ where $c : \\mathcal{X}\\times \\mathcal{X}\\to \\mathbb{R}_{\\ge 0}$ . In general, function c can be an arbitrary non-negative cost function. In our study of strategic linear classification, we assume the cost functions are induced by seminorms. We will consider the following two types of cost functions, with increasing generality.\n\n1. Instance-invariant cost functions: A cost function c is instance-invariant if there is a common function l such that $c(z;\\pmb{x})=l(\\pmb{z}-\\pmb{x})$ for any point $(x,y,r)$ .\n\n2. Instance-wise cost functions: With instance-wise cost functions, each data point is allowed to possess its own loss function. To capture this situation, we need to augment each data point's representation from $(x,y,r)$ to $({\\pmb x},y,r,l)$ , where l is a function depending on x such that $c ({\\boldsymbol{z}};{\\pmb x}) = l ({\\boldsymbol{z}}-{\\pmb x})$ for some semi-norm l. For notational convenience, we often refer to function l associated with this data point as $l_{x}$ though we emphasize that this is a slight abuse of notation because $l_{x}$ is determined not just by the data point's feature x but rather by the data point itself.4\n\nBoth cases have been considered in previous works. For instance, the separable cost function studied in (Hardt et al., 2016) is instance-wise, and the cost function induced by a seminorm as assumed by the main theorem of (Cullina et al., 2018) is instance-invariant. We shall prove later that the choice among these two types of cost functions will largely affect the efficient learnability of the problem."}, {"name": "Eq. (1): data point best response and tie-breaking", "statement": "Given a classifier h, the strategic test data point $(x,y,r)$ may shift its feature vector to z and would like to pick the best such z by solving the following optimization problem:\n\nData Point Best Response:\n\n$$\n\\Delta_{c}(\\pmb{x},r;h)=\\arg\\operatorname*{max}_{\\pmb{\\mathrm{z}}\\in\\mathcal{X}}\\big [\\mathbb{I}(h(\\pmb{z})=1)\\cdot r-c(\\pmb{z};\\pmb{x})\\big].\\tag{1}\n$$\n\nwhere $\\mathbb{I}(S)$ is the indicator function and $[\\mathbb{I}(h(z)=1)\\cdot r-c(z;\\pmb{x})]$ is the quasi-linear utility function of data point $(x,y,r)$ . We call $\\Delta_{c}({\\pmb x},r;h)$ the manipulated feature. When there are multiple best responses, we assume the data point may choose any best response and thus will adopt the standard worst-case analysis. Note that the test data's strategic behaviors do not change its true label. Such strategic gaming behaviors differ from strategic improvements (see (Miller et al., 2019) for more discussions on their differences). We also point out that the preference r is assumed to be known for any training data (though no need to be known for testing data). While this may appear a strong assumption at the first glance, we believe this parameter can be estimated in real applications (e.g., how much it matters to someone for being admitted to a university, for receiving a loan or for being tested COVID positive). An interesting future direction may be to understand how robust the strategic classifier is to the estimation error of r in the training data."}, {"name": "StraC problem and Eq. (2): strategic 0-1 loss (Section 2.1)", "statement": "A STRAC problem is described by a hypothesis class H, the set of preferences R and a manipulation cost function c. We thus denote it as $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ . Adopting the standard statistical learning framework, the input to our learning task is n uncontaminated training data points $({\\pmb x}_{1},y_{1},r_{1}),\\dots,({\\pmb x}_{n},y_{n},r_{n})$ drawn independently and identically (i.i.d.) from distribution D. Given these training data, we look to learn a classifier $h \\in \\mathcal H$ which minimizes the basic 0-1 loss, defined as follows:\n\nStrategic 0-1 Loss of classifier h :\n\n$$\nL_{c}(h;\\mathcal{D})=\\operatorname*{Pr}_{(\\mathbf{x},y,r)\\sim\\mathcal{D}}\\big [h(\\Delta_{c}(\\mathbf{x},r;h))\\neq y\\big].\\tag{2}\n$$\n\nNotably, the classifier h in the above loss definition takes the manipulated feature $\\textstyle\\Delta_{c}({\\pmb x},r;h)\\big)$ as input and, nevertheless, looks to correctly predict the true label y. For notational convenience, we sometimes omit c when it is clear from the context and simply write $\\Delta({\\pmb x},r;h)$ and $L(h;\\mathcal{D})$ ."}, {"name": "Definition 1 (Strategic Shattering Coefficients)", "statement": "In this section, we introduce the notion of strategic VC-dimension (SVC) and show that it properly captures the behaviors of a hypothesis class in the strategic setup introduced above. We then show the connection of SVC with previous studies on both strategic and adversarial learning. Before formally introducing SVC, we first define the shattering coefficients in strategic setups.\n\nDefinition 1 (Strategic Shattering Coefficients) The n-th shattering coefficient of any strategic classification problem $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ is defined as\n\n$$\n\\sigma_{n}(\\mathcal{H},R,c)=\\operatorname*{max}_{(\\boldsymbol{x},\\boldsymbol{r})\\in\\mathcal{X}^{n}\\times R^{n}}~|\\{(h(\\Delta_{c}(\\boldsymbol{x}_{1},\\boldsymbol{r}_{1};h)),\\cdots,h(\\Delta_{c}(\\boldsymbol{x}_{n},\\boldsymbol{r}_{n};h)):h\\in\\mathcal{H}\\}|,\n$$\n\nwhere $\\Delta_{c}(\\pmb{x}_{i},r_{i};h)$ defined in Eq. (1) is a best response of data point $({\\pmb x}_{i}, y_{i}, r_{i})$ to classifier h under cost function c.\n\nThat is, $\\sigma_{n}(\\mathcal{H},R,c)$ captures the maximum number of classification behaviors/outcomes (among all choices of data points) that classifiers in H can possibly induce by using manipulated features as input. Like classic definition of shattering coefficient, the $\\sigma_{n}(\\mathcal{H},R,c)$ here does not involve the labels of the data points at all. In contrast, in the shattering coefficient definition for adversarial VC-dimension of (Cullina et al., 2018), the \"max\" is allowed to be over data labels as well. This is an important difference compared to our setting. Given the definition of the strategic shattering coefficients, the definition of strategic VC-dimension is standard."}, {"name": "Definition 2 (Strategic VC-dimension), Eq. (3)", "statement": "Definition 2 The Strategic VC-dimension (SVC) for strategic classification problem $\\mathrm{STRAC}\\langle\\mathcal{H},R,c\\rangle$ is defined as\n\n$$\nS V C(\\mathcal{H},R,c)=\\operatorname*{sup}\\{n\\in\\mathbb{N}:\\sigma_{n}(\\mathcal{H},R,c)=2^{n}\\}.\\tag{3}\n$$"}, {"name": "Separable cost functions (Section 3.2)", "statement": "Not only restricting the set R of preference values can reduce the SVC. This subsection shows that restricting to special classes of cost functions can also lead to a small SVC. One special class of cost functions studied in many previous works is the separable cost functions (Hardt et al., 2016; Milli et al., 2019; Hu et al., 2019a). Formally, a cost function $c(z;\\pmb{x})$ is separable if there exists function $c_{1}, c_{2}: \\mathcal{X}\\to \\mathbb{R}$ such that $c(\\pmb{z};\\pmb{x})=\\operatorname*{max}\\left\\{c_{2}(\\pmb{z})-c_{1}(\\pmb{x}),0\\right\\}$\n\nThe following Proposition 9 shows that when the cost function is separable, SVC is at most 2 for any hypothesis class H and any class of preference set $R .^{5}$ Therefore, separable cost function essentially reduces any classification problem to a problem in lower dimension. Together with Theorem 4, Proposition 9 also recovers the PAC-learnability result of (Hardt et al., 2016) in their strategic-robust learning model (specifically, Theorem 1.8 of (2016)) and, moreover, generalizes their learnability from homogeneous agent preferences to the case with arbitrary agent preference values."}, {"name": "Remark after Proposition 9: role of 0 not in R", "statement": "The assumption $0 \\notin R$ implies that each agent must strictly prefer either label +1 or $- 1$ . This assumption is necessary since if $0\\in R,$ SVC will be at least the classic VC dimension of H and thus Proposition 9 cannot hold. We remark that the above SVC upper bound 2 holds for any hypothesis class H. This bound 2 is tight for some classes of hypothesis, e.g., linear classifiers."}], "nl_notes": "### Proposition 9\nREPAIRED STATEMENT (Proposition 9) — Proposition 9 as published is FALSE under the paper's own §2 convention that agents break ties in the worst case for the learner: there are instances in which three points are shattered, so SVC ≥ 3. The paper's Eq. (16) and Appendix A.4 instead break ties toward the agent's preferred label. This target is the repaired version: the bound must hold for every best-response selector Δ (`hΔ`) that breaks ties toward the agent's preferred label whenever such a best response exists (`htie`), with the strategic VC dimension defined through Δ.", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\ndef stratUtil {X : Type*} (c : X → X → ℝ) (h : X → Bool) (x : X) (r : ℝ) (z : X) : ℝ :=\n  (if h z then r else 0) - c z x\n\ndef IsBestResponse {X : Type*} (c : X → X → ℝ) (h : X → Bool) (x : X) (r : ℝ) (z : X) :\n    Prop :=\n  ∀ z' : X, stratUtil c h x r z' ≤ stratUtil c h x r z\n\ndef StrategicShatters {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) (n : ℕ) : Prop :=\n  ∃ x : Fin n → X, ∃ r : Fin n → ℝ, (∀ i, r i ∈ R) ∧\n    ∀ v : Fin n → Bool, ∃ h ∈ H, ∀ i, h (Δ h (x i) (r i)) = v i\n\nnoncomputable def SVC {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) : ℕ∞ :=\n  ⨆ (n : ℕ) (_ : StrategicShatters H R c Δ n), (n : ℕ∞)\n\nnoncomputable def strategicRisk {X : Type*} [MeasurableSpace X] (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) (h : X → Bool)\n    (D : MeasureTheory.Measure (X × Bool × ℝ)) : ℝ :=\n  (D {p : X × Bool × ℝ | h (Δ h p.1 p.2.2) ≠ p.2.1}).toReal\n\ndef empStrategicLoss {X : Type*} (c : X → X → ℝ) (Δ : (X → Bool) → X → ℝ → X)\n    (h : X → Bool) {n : ℕ} (s : Fin n → X × Bool × ℝ) : ℕ :=\n  (Finset.univ.filter (fun i => h (Δ h (s i).1 (s i).2.2) ≠ (s i).2.1)).card\n\ndef IsSERM {X : Type*} (H : Set (X → Bool)) (c : X → X → ℝ) (Δ : (X → Bool) → X → ℝ → X)\n    (hstar : X → Bool) {n : ℕ} (s : Fin n → X × Bool × ℝ) : Prop :=\n  hstar ∈ H ∧ ∀ h ∈ H, empStrategicLoss c Δ hstar s ≤ empStrategicLoss c Δ h s\n\ndef SeparableCost {X : Type*} (c : X → X → ℝ) : Prop :=\n  ∃ c₁ c₂ : X → ℝ, ∀ z x : X, c z x = max (c₂ z - c₁ x) 0\n\ndef dotp {d : ℕ} (w u : Fin d → ℝ) : ℝ := ∑ i, w i * u i\n\ntheorem strategic_agnostic_PAC :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (X : Type*) [MeasurableSpace X]\n        (H : Set (X → Bool)) (_hH : H.Nonempty) (_hH_ctble : H.Countable)\n        (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (_hc0 : ∀ x : X, c x x = 0)\n        (Δ : (X → Bool) → X → ℝ → X)\n        (_hΔ : ∀ h ∈ H, ∀ x, ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n        (_hmeas : ∀ h ∈ H, Measurable (fun p : X × Bool × ℝ => h (Δ h p.1 p.2.2)))\n        (d : ℕ) (_hd : SVC H R c Δ = (d : ℕ∞)),\n      ∃ m : ℝ → ℝ → ℕ,\n        (∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n            (m δ ε : ℝ) ≤ C * (ε ^ 2)⁻¹ * ((d : ℝ) + Real.log (1 / δ))) ∧\n        ∃ A : (N : ℕ) → (Fin N → X × Bool × ℝ) → (X → Bool),\n          (∀ N s, IsSERM H c Δ (A N s) s) ∧\n          ∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n          ∀ n : ℕ, m δ ε ≤ n →\n          ∀ D : MeasureTheory.Measure (X × Bool × ℝ), MeasureTheory.IsProbabilityMeasure D →\n\n          D {p : X × Bool × ℝ | p.2.2 ∉ R} = 0 →\n            1 - δ ≤\n              ((MeasureTheory.Measure.pi (fun _ : Fin n => D))\n                {s : Fin n → X × Bool × ℝ |\n                  strategicRisk c Δ (A n s) D - ⨅ h : ↥H, strategicRisk c Δ (h : X → Bool) D\n                    ≤ ε}).toReal := by\n  sorry\n\ntheorem separable_SVC_le_two\n    {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (Δ : (X → Bool) → X → ℝ → X)\n    (hΔ : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n\n    (htie : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R,\n      (∃ z, IsBestResponse c h x r z ∧ h z = decide (0 < r)) →\n      h (Δ h x r) = decide (0 < r))\n    (hR : (0 : ℝ) ∉ R) (hsep : SeparableCost c) (hc0 : ∀ x : X, c x x = 0) :\n    SVC H R c Δ ≤ 2 := by\n  sorry\n\ntheorem min_manipulation_cost_lower_bound\n    {d : ℕ} (l : (Fin d → ℝ) → ℝ) (w : Fin d → ℝ) (b : ℝ) (x : Fin d → ℝ)\n    (lstar : EReal) (hlstar : 0 < lstar)\n    (hnn  : ∀ u : Fin d → ℝ, 0 ≤ l u)\n    (hhom : ∀ (a : ℝ) (u : Fin d → ℝ), l (fun i => a * u i) = |a| * l u)\n    (htri : ∀ u v : Fin d → ℝ, l (fun i => u i + v i) ≤ l u + l v)\n    (hlub : IsLUB {t : EReal | ∃ z : Fin d → ℝ, l z ≤ 1 ∧ t = ((dotp w z : ℝ) : EReal)} lstar) :\n    IsLeast {t : EReal | ∃ x' : Fin d → ℝ, dotp w x' + b = 0 ∧\n        t = ((l (fun i => x' i - x i) : ℝ) : EReal)}\n      (((|dotp w x + b| : ℝ) : EReal) / lstar) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["strategic_agnostic_PAC", "separable_SVC_le_two", "min_manipulation_cost_lower_bound"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 2909, "end": 3342, "gold": "{X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (Δ : (X → Bool) → X → ℝ → X)\n    (hΔ : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n\n    (htie : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R,\n      (∃ z, IsBestResponse c h x r z ∧ h z = decide (0 < r)) →\n      h (Δ h x r) = decide (0 < r))\n    (hR : (0 : ℝ) ∉ R) (hsep : SeparableCost c) (hc0 : ∀ x : X, c x x = 0) :\n    SVC H R c Δ ≤ 2"}]}
{"item_id": "v24-21-1250::min_manipulation_cost_lower_bound::A-stmt", "paper_id": "v24-21-1250", "target_name": "min_manipulation_cost_lower_bound", "source": "Research-level", "nl_statement": "Lemma 12 For any seminorm $l : \\mathbb{R}^{d}\\to \\mathbb{R}_{\\ge 0}$ , and the cost function $c ({\\boldsymbol{z}};{\\pmb x}) = l ({\\boldsymbol{z}}-{\\pmb x})$ induced by l, the minimum manipulation cost for x to move to the hyperplane ${\\pmb w}\\cdot{\\pmb x}+ b = 0$ is given by the following:\n\n$$\n\\operatorname*{min}_{{\\pmb x}^{\\prime}}\\{c({\\pmb x}^{\\prime};{\\pmb x}):{\\pmb w}\\cdot{\\pmb x}^{\\prime}+b=0\\}=\\frac{|{\\pmb w}\\cdot{\\pmb x}+b|}{l^{*}({\\pmb w})}\n$$\n\nwhere $\\begin{array}{r}{l^{\\ast}(\\pmb{w})=\\operatorname*{sup}_{z\\in{\\cal B}}\\{\\pmb{w}\\cdot z\\}\\in\\mathbb{R}_{\\ge 0}\\cup\\{+\\infty\\}}\\end{array}$ , and $B=\\{z:l(z)\\leq 1\\}$ is the unit ball induced by l. Assume $l^{*}(\\pmb{w})>0$; when $l^{*}(\\pmb{w})= +\\infty$ the right-hand side is read as $0$.", "context": [{"name": "Setup: strategic manipulation and cost functions (Section 2)", "statement": "The Strategic Manipulation of Test Data. We consider strategic behaviors during the test phase and assume that the training data is unaltered/uncontaminated. An illustration of the setup can be found in Figure 1. A generic test data point is denoted as $(x,y,r)$ . The test data point is strategic and may shift its feature to vector z with cost $c(z;\\pmb{x})$ where $c : \\mathcal{X}\\times \\mathcal{X}\\to \\mathbb{R}_{\\ge 0}$ . In general, function c can be an arbitrary non-negative cost function. In our study of strategic linear classification, we assume the cost functions are induced by seminorms. We will consider the following two types of cost functions, with increasing generality.\n\n1. Instance-invariant cost functions: A cost function c is instance-invariant if there is a common function l such that $c(z;\\pmb{x})=l(\\pmb{z}-\\pmb{x})$ for any point $(x,y,r)$ .\n\n2. Instance-wise cost functions: With instance-wise cost functions, each data point is allowed to possess its own loss function. To capture this situation, we need to augment each data point's representation from $(x,y,r)$ to $({\\pmb x},y,r,l)$ , where l is a function depending on x such that $c ({\\boldsymbol{z}};{\\pmb x}) = l ({\\boldsymbol{z}}-{\\pmb x})$ for some semi-norm l. For notational convenience, we often refer to function l associated with this data point as $l_{x}$ though we emphasize that this is a slight abuse of notation because $l_{x}$ is determined not just by the data point's feature x but rather by the data point itself.4\n\nBoth cases have been considered in previous works. For instance, the separable cost function studied in (Hardt et al., 2016) is instance-wise, and the cost function induced by a seminorm as assumed by the main theorem of (Cullina et al., 2018) is instance-invariant. We shall prove later that the choice among these two types of cost functions will largely affect the efficient learnability of the problem."}, {"name": "Setup: strategic linear classification (Section 4)", "statement": "This section instantiates our previous general framework in one of the most fundamental special cases, i.e., linear classification. We will study both the statistical and computational efficiency in strategic linear classification. Naturally, we will restrict $\\mathcal{X}\\subseteq \\mathbb{R}^{d}$ in this section. Moreover, the cost functions are always assumed to be induced by seminorms.6 A linear classifier is defined by a hyperplane ${\\pmb w}\\cdot{\\pmb x}+ b = 0$; feature vector x is classified as +1 if and only if ${\\pmb w}\\cdot{\\pmb x}+ b \\ge 0$ With slight abuse of notation, we sometimes also call $({\\boldsymbol{\\mathbf{\\mathit{w}}}},{\\boldsymbol{b}})$ a hyperplane or a linear classifier. Let $\\mathcal{H}_{d}$ denote the hypothesis set of all d-dimensional linear classifiers. For linear classifier $({\\boldsymbol{\\mathbf{\\mathit{w}}}},{\\boldsymbol{b}})$ , the data point's best response can be more explicitly expressed as:\n\n$$\n\\Delta_{c}(x,r;w,b)=\\arg\\operatorname*{max}_{z}\\left[\\mathbb{I}(w\\cdot z+b\\ge 0)\\cdot r-c(z;\\pmb{x})\\right].\n$$"}, {"name": "Theorem 11 (statement; Lemma 12 is a tool in its proof)", "statement": "Theorem 11 Consider an instance of strategic linear classification $\\mathrm{STRAC}\\langle\\mathcal{H}_{d},R,c\\rangle$ . For any instance-invariant cost function $c ({\\boldsymbol{z}};{\\pmb x}) = l ({\\boldsymbol{z}}-{\\pmb x})$ where l is a seminorm, we have $S V C(\\mathcal{H}_{d},R,c)=d+1-\\mathrm{dim}(V_{l})$ for any bounded R, where $V_{l}$ is the largest linear space contained in the ball $B=\\{\\pmb{x}:l(\\pmb{x})\\leq 1\\}$\n\nIn particular, if l is a norm $(i . e .,l(\\pmb{x})=0\\operatorname{iff}\\pmb{x}=0)$ , then dim $(V_{l})=0$ and $SVC(\\mathcal{H},R,c)=$ $d + 1$"}, {"name": "Facts about the seminorm unit ball and l* (proof of Theorem 11)", "statement": "For any seminorm l, it is known the set $B=\\{x:l(x)\\leq 1\\}$ is nonempty, closed, convex, and origin-symmetric. Let $l^{*}(\\pmb{w})=\\operatorname*{sup}_{z\\in B}\\{\\pmb{w}\\cdot\\pmb{z}\\}$ . We have $l^{*}(w)>0$ for all $\\pmb{w} \\neq \\mathbf{0}$ since 0 is an interior point of B."}], "nl_notes": "", "lean": "import Mathlib\n\nnamespace ResearchLevel\n\ndef stratUtil {X : Type*} (c : X → X → ℝ) (h : X → Bool) (x : X) (r : ℝ) (z : X) : ℝ :=\n  (if h z then r else 0) - c z x\n\ndef IsBestResponse {X : Type*} (c : X → X → ℝ) (h : X → Bool) (x : X) (r : ℝ) (z : X) :\n    Prop :=\n  ∀ z' : X, stratUtil c h x r z' ≤ stratUtil c h x r z\n\ndef StrategicShatters {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) (n : ℕ) : Prop :=\n  ∃ x : Fin n → X, ∃ r : Fin n → ℝ, (∀ i, r i ∈ R) ∧\n    ∀ v : Fin n → Bool, ∃ h ∈ H, ∀ i, h (Δ h (x i) (r i)) = v i\n\nnoncomputable def SVC {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) : ℕ∞ :=\n  ⨆ (n : ℕ) (_ : StrategicShatters H R c Δ n), (n : ℕ∞)\n\nnoncomputable def strategicRisk {X : Type*} [MeasurableSpace X] (c : X → X → ℝ)\n    (Δ : (X → Bool) → X → ℝ → X) (h : X → Bool)\n    (D : MeasureTheory.Measure (X × Bool × ℝ)) : ℝ :=\n  (D {p : X × Bool × ℝ | h (Δ h p.1 p.2.2) ≠ p.2.1}).toReal\n\ndef empStrategicLoss {X : Type*} (c : X → X → ℝ) (Δ : (X → Bool) → X → ℝ → X)\n    (h : X → Bool) {n : ℕ} (s : Fin n → X × Bool × ℝ) : ℕ :=\n  (Finset.univ.filter (fun i => h (Δ h (s i).1 (s i).2.2) ≠ (s i).2.1)).card\n\ndef IsSERM {X : Type*} (H : Set (X → Bool)) (c : X → X → ℝ) (Δ : (X → Bool) → X → ℝ → X)\n    (hstar : X → Bool) {n : ℕ} (s : Fin n → X × Bool × ℝ) : Prop :=\n  hstar ∈ H ∧ ∀ h ∈ H, empStrategicLoss c Δ hstar s ≤ empStrategicLoss c Δ h s\n\ndef SeparableCost {X : Type*} (c : X → X → ℝ) : Prop :=\n  ∃ c₁ c₂ : X → ℝ, ∀ z x : X, c z x = max (c₂ z - c₁ x) 0\n\ndef dotp {d : ℕ} (w u : Fin d → ℝ) : ℝ := ∑ i, w i * u i\n\ntheorem strategic_agnostic_PAC :\n    ∃ C : ℝ, 0 < C ∧\n      ∀ (X : Type*) [MeasurableSpace X]\n        (H : Set (X → Bool)) (_hH : H.Nonempty) (_hH_ctble : H.Countable)\n        (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (_hc0 : ∀ x : X, c x x = 0)\n        (Δ : (X → Bool) → X → ℝ → X)\n        (_hΔ : ∀ h ∈ H, ∀ x, ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n        (_hmeas : ∀ h ∈ H, Measurable (fun p : X × Bool × ℝ => h (Δ h p.1 p.2.2)))\n        (d : ℕ) (_hd : SVC H R c Δ = (d : ℕ∞)),\n      ∃ m : ℝ → ℝ → ℕ,\n        (∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n            (m δ ε : ℝ) ≤ C * (ε ^ 2)⁻¹ * ((d : ℝ) + Real.log (1 / δ))) ∧\n        ∃ A : (N : ℕ) → (Fin N → X × Bool × ℝ) → (X → Bool),\n          (∀ N s, IsSERM H c Δ (A N s) s) ∧\n          ∀ δ ε : ℝ, 0 < δ → δ < 1 → 0 < ε → ε < 1 →\n          ∀ n : ℕ, m δ ε ≤ n →\n          ∀ D : MeasureTheory.Measure (X × Bool × ℝ), MeasureTheory.IsProbabilityMeasure D →\n\n          D {p : X × Bool × ℝ | p.2.2 ∉ R} = 0 →\n            1 - δ ≤\n              ((MeasureTheory.Measure.pi (fun _ : Fin n => D))\n                {s : Fin n → X × Bool × ℝ |\n                  strategicRisk c Δ (A n s) D - ⨅ h : ↥H, strategicRisk c Δ (h : X → Bool) D\n                    ≤ ε}).toReal := by\n  sorry\n\ntheorem separable_SVC_le_two\n    {X : Type*} (H : Set (X → Bool)) (R : Set ℝ) (c : X → X → ℝ) (_hc : ∀ x y : X, 0 ≤ c x y)\n    (Δ : (X → Bool) → X → ℝ → X)\n    (hΔ : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R, IsBestResponse c h x r (Δ h x r))\n\n    (htie : ∀ h ∈ H, ∀ (x : X), ∀ r ∈ R,\n      (∃ z, IsBestResponse c h x r z ∧ h z = decide (0 < r)) →\n      h (Δ h x r) = decide (0 < r))\n    (hR : (0 : ℝ) ∉ R) (hsep : SeparableCost c) (hc0 : ∀ x : X, c x x = 0) :\n    SVC H R c Δ ≤ 2 := by\n  sorry\n\ntheorem min_manipulation_cost_lower_bound\n    {d : ℕ} (l : (Fin d → ℝ) → ℝ) (w : Fin d → ℝ) (b : ℝ) (x : Fin d → ℝ)\n    (lstar : EReal) (hlstar : 0 < lstar)\n    (hnn  : ∀ u : Fin d → ℝ, 0 ≤ l u)\n    (hhom : ∀ (a : ℝ) (u : Fin d → ℝ), l (fun i => a * u i) = |a| * l u)\n    (htri : ∀ u v : Fin d → ℝ, l (fun i => u i + v i) ≤ l u + l v)\n    (hlub : IsLUB {t : EReal | ∃ z : Fin d → ℝ, l z ≤ 1 ∧ t = ((dotp w z : ℝ) : EReal)} lstar) :\n    IsLeast {t : EReal | ∃ x' : Fin d → ℝ, dotp w x' + b = 0 ∧\n        t = ((l (fun i => x' i - x i) : ℝ) : EReal)}\n      (((|dotp w x + b| : ℝ) : EReal) / lstar) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["strategic_agnostic_PAC", "separable_SVC_le_two", "min_manipulation_cost_lower_bound"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 3404, "end": 3952, "gold": "{d : ℕ} (l : (Fin d → ℝ) → ℝ) (w : Fin d → ℝ) (b : ℝ) (x : Fin d → ℝ)\n    (lstar : EReal) (hlstar : 0 < lstar)\n    (hnn  : ∀ u : Fin d → ℝ, 0 ≤ l u)\n    (hhom : ∀ (a : ℝ) (u : Fin d → ℝ), l (fun i => a * u i) = |a| * l u)\n    (htri : ∀ u v : Fin d → ℝ, l (fun i => u i + v i) ≤ l u + l v)\n    (hlub : IsLUB {t : EReal | ∃ z : Fin d → ℝ, l z ≤ 1 ∧ t = ((dotp w z : ℝ) : EReal)} lstar) :\n    IsLeast {t : EReal | ∃ x' : Fin d → ℝ, dotp w x' + b = 0 ∧\n        t = ((l (fun i => x' i - x i) : ℝ) : EReal)}\n      (((|dotp w x + b| : ℝ) : EReal) / lstar)"}]}
{"item_id": "v24-22-0522::lemma8::A-stmt", "paper_id": "v24-22-0522", "target_name": "lemma8", "source": "Research-level", "nl_statement": "Lemma 8 Suppose that Assumption 1 holds. Let $\\begin{array}{r}{\\eta \\le \\frac{1}{4 L}}\\end{array}$ and $0 \\leq \\theta \\leq 1$ . In each epoch of Algorithm 1 where the \"if condition\" triggers, when $\\begin{array}{r}{\\|\\nabla f(\\mathbf{y}^{\\mathcal{K}-1})\\|>\\frac{B}{\\eta}}\\end{array}$ , we have\n\n$$\nf(\\mathbf{x}^{\\mathcal{K}})-f(\\mathbf{x}^{0})\\leq-\\frac{B^{2}}{4\\eta}.\n$$", "context": [{"name": "Problem setup: problem (1) and approximate stationary point (Section 1)", "statement": "$$\n\\operatorname*{min}_{\\mathbf{x}\\in\\mathbb{R}^{d}}f(\\mathbf{x}),\\tag{1}\n$$\n\nwhere f (x) has Lipschitz continuous gradient and Hessian and it is bounded from below. Our goal is to find an $\\epsilon$-approximate first-order stationary point, defined as\n\n$$\n\\|\\nabla f(\\mathbf{x})\\|\\leq\\epsilon .\n$$"}, {"name": "Notation and Assumption 1 (Section 1.4)", "statement": "We use lowercase bold letters to represent vectors, uppercase bold letters for matrices, and non-bold (both lowercase and uppercase) letters for scalars. Denote $\\mathbf{x}_{j}$ and $\\nabla_{j}f(\\mathbf{x})$ as the jth element of x and $\\nabla f(\\mathbf{x})$ , respectively. For the vectors produced in the iterative algorithms, for example, x, denote $\\mathbf{x}^{k}$ to be the value at the kth iteration. We denote $\\| \\cdot \\|$ to be the $\\ell_{2}$ Euclidean norm for vectors, $\\| \\cdot \\|_{2}$ as the spectral norm and $\\| \\cdot \\|_{F}$ as the Frobenius norm for matrices. We make the following standard assumptions in this paper.\n\nAssumption 1 1. f (x) is L-gradient Lipschitz: $\\|\\nabla f(\\mathbf{x})-\\nabla f(\\mathbf{y})\\|\\leq L\\|\\mathbf{x}-\\mathbf{y}\\|,\\forall\\mathbf{x},\\mathbf{y}\\in$ $\\mathbb{R}^{d}$ ,\n\n2. f (x) is ρ-Hessian Lipschitz: $\\begin{array}{r}{\\|\\nabla^{2}f(\\mathbf{x})-\\nabla^{2}f(\\mathbf{y})\\|_{2}\\leq\\rho\\|\\mathbf{x}-\\mathbf{y}\\|,\\forall\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^{d},}\\end{array}$\n\nwhich yield the following two well-known inequalities:\n\n$$\n\\begin{array}{r}{|f(\\mathbf{y})-f(\\mathbf{x})-\\langle\\nabla f(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle|\\leq\\frac{L}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{r}{\\big|f(\\mathbf{y})-f(\\mathbf{x})-\\langle\\nabla f(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle-(\\mathbf{y}-\\mathbf{x})^{T}\\nabla^{2}f(\\mathbf{x})(\\mathbf{y}-\\mathbf{x})\\big|\\leq\\frac{\\rho}{6}\\|\\mathbf{y}-\\mathbf{x}\\|^{3}.}\\end{array}\\tag{3}\n$$\n\nWe also assume that the objective function is lower bounded, that is, $\\min_{\\mathbf{x}}f(\\mathbf{x})> -\\infty$"}, {"name": "Nesterov's AGD iteration (Section 2)", "statement": "Nesterov's classical AGD consists of the following iterations:\n\n$$\n\\begin{array}{r}{\\mathbf{y}^{k}=\\mathbf{x}^{k}+(1-\\theta)(\\mathbf{x}^{k}-\\mathbf{x}^{k-1}),\\quad\\mathbf{x}^{k+1}=\\mathbf{y}^{k}-\\eta\\nabla f(\\mathbf{y}^{k}),}\\end{array}\n$$"}, {"name": "Algorithm 1 (RAGD-NC) and its description", "statement": "Algorithm 1 Restarted AGD for Nonconvex Optimization (RAGD-NC)   \n1: Initialize ${\\bf x}^{- 1}={\\bf x}^{0}={\\bf x}_{i n t}, k = 0$   \n2: while $k < K$ do   \n3: $\\mathbf{y}^{k}=\\mathbf{x}^{k}+(1-\\theta)(\\mathbf{x}^{k}-\\mathbf{x}^{k-1})$   \n4: $\\mathbf{x}^{k+1}=\\mathbf{y}^{k}-\\eta\\nabla f(\\mathbf{y}^{k})$   \n5: $k = k + 1$   \n6: if $k \\textstyle \\sum_{t = 0}^{k - 1}\\|{\\bf x}^{t + 1}-{\\bf x}^{t}\\|^{2}> B^{2}$ then   \n7: $\\mathbf{x}^{- 1}= \\mathbf{x}^{0}= \\mathbf{x}^{k}, k = 0$   \n8: end if   \n9: end while   \n10: $K_{0}=\\mathrm{argmin}_{\\lfloor\\frac{K}{2}\\rfloor\\leq k\\leq K-1}\\|\\mathbf{x}^{k+1}-\\mathbf{x}^{k}\\|$   \n11: Output $\\begin{array}{r}{\\hat{\\mathbf{y}}= \\frac{1}{K_{0}+ 1}\\sum_{k = 0}^{K_{0}}\\mathbf{y}^{k}}\\end{array}$\n\nWe present our method in Algorithm 1. It runs Nesterov's classical AGD iterations until the \"if condition\" triggers. Then we reset $\\mathbf{x}^{0}$ and $\\mathbf{x}^{- 1}$ equal to $\\mathbf{x}^{k}$ and continue to the next round of AGD. The method terminates and outputs a specific average when the \"if condition\" does not trigger in K iterations. To simplify the description, we define one round of AGD between two successive restarts to be one \"epoch\". The restart trick, first proposed in (O'Donoghue and Candès, 2015), is motivated by (Fang et al., 2019), where a ball-mechanism is proposed as the stopping criteria to analyze SGD."}, {"name": "Epoch quantities: trigger index (7), (8a), (8b), (9), (10a), (10b) (Section 4.1)", "statement": "We prove the convergence rate of Algorithm 1 in this section. Denote $\\mathcal{K}$ to be the iteration number when the \"if condition\" on line 6 of Algorithm 1 triggers, that is,\n\n$$\n\\mathcal{K}=\\operatorname*{min}_{k}\\left\\{k\\bigg|k\\sum_{t=0}^{k-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}>B^{2}\\right\\}.\\tag{7}\n$$\n\nFor each epoch consisting of one round of AGD from iterations $k = 0$ to $k = \\mathcal{K}$ , we have\n\n$$\n1\\leq\\mathcal{K}\\leq K,\\qquad\\mathcal{K}\\sum_{t=0}^{\\mathcal{K}-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}>B^{2},\\qquad\\mathrm{and}\\tag{8a}\n$$\n\n$$\n\\|{\\bf x}^{k}-{\\bf x}^{0}\\|^{2}=\\left\\|\\sum_{t=0}^{k-1}{\\bf x}^{t+1}-{\\bf x}^{t}\\right\\|^{2}\\leq k\\sum_{t=0}^{k-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}\\leq B^{2},\\forall k<\\mathcal{K},\\tag{8b}\n$$\n\nwhere the last inequality comes from the definition of $\\mathcal{K}$. From the update of y on line 3 of Algorithm 1, we also have\n\n$$\n\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|\\leq\\|\\mathbf{x}^{k}-\\mathbf{x}^{0}\\|+\\|\\mathbf{x}^{k}-\\mathbf{x}^{k-1}\\|\\leq 2B,\\forall k<{\\mathcal{K}}.\\tag{9}\n$$\n\nOn the other hand, for the last epoch where the \"if condition\" does not trigger and the while loop breaks when k increases to $K$ ,we have\n\n$$\n\\|{\\mathbf{x}}^{k}-{\\mathbf{x}}^{0}\\|^{2}\\leq k\\sum_{t=0}^{k-1}\\|{\\mathbf{x}}^{t+1}-{\\mathbf{x}}^{t}\\|^{2}\\leq B^{2},\\forall k\\leq K,\\tag{10a}\n$$\n\n$$\n\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|\\leq 2B,\\forall k\\leq K.\\tag{10b}\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped NNReal BigOperators\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n\ntheorem lemma8 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (η θ B : ℝ) (hη : 0 < η) (hηL : η ≤ 1 / (4 * (L : ℝ)))\n    (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ)) (hθ0 : 0 ≤ θ) (hθ1 : θ ≤ 1) (hB : 0 ≤ B)\n    (K : ℕ) (hK : 1 ≤ K) (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (hbig : ‖f' (y (K - 1))‖ > B / η) :\n    f (x K) - f (x 0) ≤ -(B ^ 2 / (4 * η)) := by\n  sorry\n\ntheorem corollary12 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) ≤ 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2) :\n    f (x K) - f (x 0) ≤ -(7 * ε ^ ((3 : ℝ) / 2)) / (8 * Real.sqrt (ρ : ℝ)) := by\n  sorry\n\ntheorem lemma13 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) = 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h10a : ∀ k, k ≤ K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (K₀ : ℕ) (hK₀lo : K / 2 ≤ K₀) (hK₀hi : K₀ < K)\n    (hK₀min : ∀ k, K / 2 ≤ k → k < K → ‖x (K₀ + 1) - x K₀‖ ≤ ‖x (k + 1) - x k‖)\n    (yhat : E)\n    (hyhat : yhat = ((K₀ : ℝ) + 1)⁻¹ • ∑ k ∈ Finset.range (K₀ + 1), y k) :\n    ‖f' yhat‖ ≤ 82 * ε := by\n  sorry\ntheorem theorem1_complexity\n    (ε ρ L Δ : ℝ) (hε : 0 < ε) (hρ : 0 < ρ) (hL : 0 < L) (hΔ : 0 ≤ Δ)\n    (hK1 : (1 : ℝ) ≤ (1 : ℝ) / 2 * (L ^ 2 / (ε * ρ)) ^ ((1 : ℝ) / 4))\n    (N : ℕ) (F : ℕ → ℝ) (Fmin : ℝ)\n    (hdec : ∀ i, i < N →\n      F (i + 1) ≤ F i - 7 * ε ^ ((3 : ℝ) / 2) / (8 * Real.sqrt ρ))\n    (hlb : Fmin ≤ F N)\n    (hΔf : F 0 - Fmin ≤ Δ) :\n    (N : ℝ) * ((1 : ℝ) / 2 * (L ^ 2 / (ε * ρ)) ^ ((1 : ℝ) / 4))\n      ≤ Δ * Real.sqrt L * ρ ^ ((1 : ℝ) / 4) / ε ^ ((7 : ℝ) / 4) := by\n  set u : ℝ := ε ^ ((1:ℝ)/4) with hu\n  set v : ℝ := ρ ^ ((1:ℝ)/4) with hv\n  have hu0 : 0 < u := Real.rpow_pos_of_pos hε _\n  have hv0 : 0 < v := Real.rpow_pos_of_pos hρ _\n  have hsL0 : 0 < Real.sqrt L := Real.sqrt_pos.mpr hL\n  have hε32 : ε ^ ((3:ℝ)/2) = u^6 := by\n    rw [hu, ← Real.rpow_natCast (ε ^ ((1:ℝ)/4)) 6, ← Real.rpow_mul hε.le]\n    norm_num\n  have hε74 : ε ^ ((7:ℝ)/4) = u^7 := by\n    rw [hu, ← Real.rpow_natCast (ε ^ ((1:ℝ)/4)) 7, ← Real.rpow_mul hε.le]\n    norm_num\n  have hρs : Real.sqrt ρ = v^2 := by\n    rw [hv, ← Real.rpow_natCast (ρ ^ ((1:ℝ)/4)) 2, ← Real.rpow_mul hρ.le, Real.sqrt_eq_rpow]\n    norm_num\n  have hLrp : (L ^ 2 / (ε * ρ)) ^ ((1:ℝ)/4) = Real.sqrt L / (u * v) := by\n    rw [Real.div_rpow (by positivity) (by positivity), Real.mul_rpow hε.le hρ.le]\n    congr 1\n    rw [← Real.rpow_natCast L 2, ← Real.rpow_mul hL.le, Real.sqrt_eq_rpow]\n    norm_num\n  -- descent chain\n  set c : ℝ := 7 * ε ^ ((3:ℝ)/2) / (8 * Real.sqrt ρ) with hc\n  have hc0 : 0 < c := by\n    rw [hc]; positivity\n  have hstep : ∀ m, m ≤ N → F m ≤ F 0 - (m:ℝ) * c := by\n    intro m\n    induction m with\n    | zero => intro _; simp\n    | succ k ih =>\n        intro hk\n        have h1 := ih (by omega)\n        have h2 := hdec k (by omega)\n        push_cast\n        linarith\n  have hNc : (N:ℝ) * c ≤ Δ := by\n    have h := hstep N le_rfl\n    linarith\n  set R : ℝ := Real.sqrt L * v / u^7 with hR\n  have hR0 : 0 ≤ R := by rw [hR]; positivity\n  have hAeq : (1:ℝ)/2 * (L ^ 2 / (ε * ρ)) ^ ((1:ℝ)/4) = (4/7) * R * c := by\n    rw [hLrp, hR, hc, hε32, hρs]\n    field_simp\n    ring\n  have hgoal : Δ * Real.sqrt L * ρ ^ ((1:ℝ)/4) / ε ^ ((7:ℝ)/4) = Δ * R := by\n    rw [hR, ← hv, hε74]\n    ring\n  rw [hAeq, hgoal]\n  have : (N:ℝ) * ((4/7) * R * c) = (4/7) * R * ((N:ℝ) * c) := by ring\n  rw [this]\n  nlinarith [hR0, hNc, hΔ]\n\nend ResearchLevel\n", "target_names": ["lemma8", "corollary12", "lemma13"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 176, "end": 974, "gold": "[FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (η θ B : ℝ) (hη : 0 < η) (hηL : η ≤ 1 / (4 * (L : ℝ)))\n    (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ)) (hθ0 : 0 ≤ θ) (hθ1 : θ ≤ 1) (hB : 0 ≤ B)\n    (K : ℕ) (hK : 1 ≤ K) (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (hbig : ‖f' (y (K - 1))‖ > B / η) :\n    f (x K) - f (x 0) ≤ -(B ^ 2 / (4 * η))"}]}
{"item_id": "v24-22-0522::corollary12::A-stmt", "paper_id": "v24-22-0522", "target_name": "corollary12", "source": "Research-level", "nl_statement": "Corollary 12 Suppose that Assumption 1 holds. Use the parameter settings in Theorem 1. In each epoch of Algorithm 1 where the \"if condition\" triggers, we have\n\n$$\nf(\\mathbf{x}^{\\mathcal{K}})-f(\\mathbf{x}^{0})\\leq-\\frac{7\\epsilon^{3/2}}{8\\sqrt{\\rho}}.\\tag{20}\n$$", "context": [{"name": "Problem setup: problem (1) and approximate stationary point (Section 1)", "statement": "$$\n\\operatorname*{min}_{\\mathbf{x}\\in\\mathbb{R}^{d}}f(\\mathbf{x}),\\tag{1}\n$$\n\nwhere f (x) has Lipschitz continuous gradient and Hessian and it is bounded from below. Our goal is to find an $\\epsilon$-approximate first-order stationary point, defined as\n\n$$\n\\|\\nabla f(\\mathbf{x})\\|\\leq\\epsilon .\n$$"}, {"name": "Notation and Assumption 1 (Section 1.4)", "statement": "We use lowercase bold letters to represent vectors, uppercase bold letters for matrices, and non-bold (both lowercase and uppercase) letters for scalars. Denote $\\mathbf{x}_{j}$ and $\\nabla_{j}f(\\mathbf{x})$ as the jth element of x and $\\nabla f(\\mathbf{x})$ , respectively. For the vectors produced in the iterative algorithms, for example, x, denote $\\mathbf{x}^{k}$ to be the value at the kth iteration. We denote $\\| \\cdot \\|$ to be the $\\ell_{2}$ Euclidean norm for vectors, $\\| \\cdot \\|_{2}$ as the spectral norm and $\\| \\cdot \\|_{F}$ as the Frobenius norm for matrices. We make the following standard assumptions in this paper.\n\nAssumption 1 1. f (x) is L-gradient Lipschitz: $\\|\\nabla f(\\mathbf{x})-\\nabla f(\\mathbf{y})\\|\\leq L\\|\\mathbf{x}-\\mathbf{y}\\|,\\forall\\mathbf{x},\\mathbf{y}\\in$ $\\mathbb{R}^{d}$ ,\n\n2. f (x) is ρ-Hessian Lipschitz: $\\begin{array}{r}{\\|\\nabla^{2}f(\\mathbf{x})-\\nabla^{2}f(\\mathbf{y})\\|_{2}\\leq\\rho\\|\\mathbf{x}-\\mathbf{y}\\|,\\forall\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^{d},}\\end{array}$\n\nwhich yield the following two well-known inequalities:\n\n$$\n\\begin{array}{r}{|f(\\mathbf{y})-f(\\mathbf{x})-\\langle\\nabla f(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle|\\leq\\frac{L}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{r}{\\big|f(\\mathbf{y})-f(\\mathbf{x})-\\langle\\nabla f(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle-(\\mathbf{y}-\\mathbf{x})^{T}\\nabla^{2}f(\\mathbf{x})(\\mathbf{y}-\\mathbf{x})\\big|\\leq\\frac{\\rho}{6}\\|\\mathbf{y}-\\mathbf{x}\\|^{3}.}\\end{array}\\tag{3}\n$$\n\nWe also assume that the objective function is lower bounded, that is, $\\min_{\\mathbf{x}}f(\\mathbf{x})> -\\infty$"}, {"name": "Nesterov's AGD iteration (Section 2)", "statement": "Nesterov's classical AGD consists of the following iterations:\n\n$$\n\\begin{array}{r}{\\mathbf{y}^{k}=\\mathbf{x}^{k}+(1-\\theta)(\\mathbf{x}^{k}-\\mathbf{x}^{k-1}),\\quad\\mathbf{x}^{k+1}=\\mathbf{y}^{k}-\\eta\\nabla f(\\mathbf{y}^{k}),}\\end{array}\n$$"}, {"name": "Algorithm 1 (RAGD-NC) and its description", "statement": "Algorithm 1 Restarted AGD for Nonconvex Optimization (RAGD-NC)   \n1: Initialize ${\\bf x}^{- 1}={\\bf x}^{0}={\\bf x}_{i n t}, k = 0$   \n2: while $k < K$ do   \n3: $\\mathbf{y}^{k}=\\mathbf{x}^{k}+(1-\\theta)(\\mathbf{x}^{k}-\\mathbf{x}^{k-1})$   \n4: $\\mathbf{x}^{k+1}=\\mathbf{y}^{k}-\\eta\\nabla f(\\mathbf{y}^{k})$   \n5: $k = k + 1$   \n6: if $k \\textstyle \\sum_{t = 0}^{k - 1}\\|{\\bf x}^{t + 1}-{\\bf x}^{t}\\|^{2}> B^{2}$ then   \n7: $\\mathbf{x}^{- 1}= \\mathbf{x}^{0}= \\mathbf{x}^{k}, k = 0$   \n8: end if   \n9: end while   \n10: $K_{0}=\\mathrm{argmin}_{\\lfloor\\frac{K}{2}\\rfloor\\leq k\\leq K-1}\\|\\mathbf{x}^{k+1}-\\mathbf{x}^{k}\\|$   \n11: Output $\\begin{array}{r}{\\hat{\\mathbf{y}}= \\frac{1}{K_{0}+ 1}\\sum_{k = 0}^{K_{0}}\\mathbf{y}^{k}}\\end{array}$\n\nWe present our method in Algorithm 1. It runs Nesterov's classical AGD iterations until the \"if condition\" triggers. Then we reset $\\mathbf{x}^{0}$ and $\\mathbf{x}^{- 1}$ equal to $\\mathbf{x}^{k}$ and continue to the next round of AGD. The method terminates and outputs a specific average when the \"if condition\" does not trigger in K iterations. To simplify the description, we define one round of AGD between two successive restarts to be one \"epoch\". The restart trick, first proposed in (O'Donoghue and Candès, 2015), is motivated by (Fang et al., 2019), where a ball-mechanism is proposed as the stopping criteria to analyze SGD."}, {"name": "Theorem 1 (parameter settings)", "statement": "Theorem 1 Suppose that Assumption 1 holds. Let $\\begin{array}{r}{\\eta=\\frac{1}{4L},B=\\sqrt{\\frac{\\epsilon}{\\rho}},\\theta=4(\\epsilon\\rho\\eta^{2})^{1/4}\\in(0,1].}\\end{array}$ and $\\begin{array}{r}{K = \\frac{1}{\\theta}}\\end{array}$ . Then Algorithm 1 terminates in at most $\\frac{\\triangle_{f}L^{1 / 2}\\rho^{1 / 4}}{\\epsilon^{7 / 4}}$ gradient computations and the output satisfies $\\|\\nabla f(\\hat{\\mathbf{y}})\\|\\leq 82\\epsilon$ , where $\\begin{array}{r}{\\triangle_{f}=f(\\mathbf{x}_{i n t})-\\operatorname*{min}_{\\mathbf{x}}f(\\mathbf{x})}\\end{array}$"}, {"name": "Epoch quantities: trigger index (7), (8a), (8b), (9), (10a), (10b) (Section 4.1)", "statement": "We prove the convergence rate of Algorithm 1 in this section. Denote $\\mathcal{K}$ to be the iteration number when the \"if condition\" on line 6 of Algorithm 1 triggers, that is,\n\n$$\n\\mathcal{K}=\\operatorname*{min}_{k}\\left\\{k\\bigg|k\\sum_{t=0}^{k-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}>B^{2}\\right\\}.\\tag{7}\n$$\n\nFor each epoch consisting of one round of AGD from iterations $k = 0$ to $k = \\mathcal{K}$ , we have\n\n$$\n1\\leq\\mathcal{K}\\leq K,\\qquad\\mathcal{K}\\sum_{t=0}^{\\mathcal{K}-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}>B^{2},\\qquad\\mathrm{and}\\tag{8a}\n$$\n\n$$\n\\|{\\bf x}^{k}-{\\bf x}^{0}\\|^{2}=\\left\\|\\sum_{t=0}^{k-1}{\\bf x}^{t+1}-{\\bf x}^{t}\\right\\|^{2}\\leq k\\sum_{t=0}^{k-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}\\leq B^{2},\\forall k<\\mathcal{K},\\tag{8b}\n$$\n\nwhere the last inequality comes from the definition of $\\mathcal{K}$. From the update of y on line 3 of Algorithm 1, we also have\n\n$$\n\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|\\leq\\|\\mathbf{x}^{k}-\\mathbf{x}^{0}\\|+\\|\\mathbf{x}^{k}-\\mathbf{x}^{k-1}\\|\\leq 2B,\\forall k<{\\mathcal{K}}.\\tag{9}\n$$\n\nOn the other hand, for the last epoch where the \"if condition\" does not trigger and the while loop breaks when k increases to $K$ ,we have\n\n$$\n\\|{\\mathbf{x}}^{k}-{\\mathbf{x}}^{0}\\|^{2}\\leq k\\sum_{t=0}^{k-1}\\|{\\mathbf{x}}^{t+1}-{\\mathbf{x}}^{t}\\|^{2}\\leq B^{2},\\forall k\\leq K,\\tag{10a}\n$$\n\n$$\n\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|\\leq 2B,\\forall k\\leq K.\\tag{10b}\n$$"}, {"name": "Lemma 11", "statement": "Lemma 11 Suppose that Assumption 1 holds. Let $\\begin{array}{r}{\\eta \\le \\frac{1}{4 L}}\\end{array}$ and $0 < \\theta \\leq 1$ . In each epoch of Algorithm 1 where the \"if condition\" triggers, when $\\begin{array}{r}{\\|\\nabla f(\\mathbf{y}^{\\mathcal{K}-1})\\|\\le\\frac{B}{\\eta}}\\end{array}$ , we have\n\n$$\nf(\\mathbf{x}^{\\mathcal{K}})-f(\\mathbf{x}^{0})\\leq-\\frac{3\\theta B^{2}}{8\\eta K}+\\frac{10\\eta\\rho^{2}B^{4}K}{\\theta}+4.5\\rho B^{3}.\\tag{19}\n$$"}, {"name": "Lemma 8 (another result of the same paper)", "statement": "Lemma 8 Suppose that Assumption 1 holds. Let $\\begin{array}{r}{\\eta \\le \\frac{1}{4 L}}\\end{array}$ and $0 \\leq \\theta \\leq 1$ . In each epoch of Algorithm 1 where the \"if condition\" triggers, when $\\begin{array}{r}{\\|\\nabla f(\\mathbf{y}^{\\mathcal{K}-1})\\|>\\frac{B}{\\eta}}\\end{array}$ , we have\n\n$$\nf(\\mathbf{x}^{\\mathcal{K}})-f(\\mathbf{x}^{0})\\leq-\\frac{B^{2}}{4\\eta}.\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped NNReal BigOperators\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n\ntheorem lemma8 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (η θ B : ℝ) (hη : 0 < η) (hηL : η ≤ 1 / (4 * (L : ℝ)))\n    (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ)) (hθ0 : 0 ≤ θ) (hθ1 : θ ≤ 1) (hB : 0 ≤ B)\n    (K : ℕ) (hK : 1 ≤ K) (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (hbig : ‖f' (y (K - 1))‖ > B / η) :\n    f (x K) - f (x 0) ≤ -(B ^ 2 / (4 * η)) := by\n  sorry\n\ntheorem corollary12 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) ≤ 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2) :\n    f (x K) - f (x 0) ≤ -(7 * ε ^ ((3 : ℝ) / 2)) / (8 * Real.sqrt (ρ : ℝ)) := by\n  sorry\n\ntheorem lemma13 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) = 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h10a : ∀ k, k ≤ K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (K₀ : ℕ) (hK₀lo : K / 2 ≤ K₀) (hK₀hi : K₀ < K)\n    (hK₀min : ∀ k, K / 2 ≤ k → k < K → ‖x (K₀ + 1) - x K₀‖ ≤ ‖x (k + 1) - x k‖)\n    (yhat : E)\n    (hyhat : yhat = ((K₀ : ℝ) + 1)⁻¹ • ∑ k ∈ Finset.range (K₀ + 1), y k) :\n    ‖f' yhat‖ ≤ 82 * ε := by\n  sorry\ntheorem theorem1_complexity\n    (ε ρ L Δ : ℝ) (hε : 0 < ε) (hρ : 0 < ρ) (hL : 0 < L) (hΔ : 0 ≤ Δ)\n    (hK1 : (1 : ℝ) ≤ (1 : ℝ) / 2 * (L ^ 2 / (ε * ρ)) ^ ((1 : ℝ) / 4))\n    (N : ℕ) (F : ℕ → ℝ) (Fmin : ℝ)\n    (hdec : ∀ i, i < N →\n      F (i + 1) ≤ F i - 7 * ε ^ ((3 : ℝ) / 2) / (8 * Real.sqrt ρ))\n    (hlb : Fmin ≤ F N)\n    (hΔf : F 0 - Fmin ≤ Δ) :\n    (N : ℝ) * ((1 : ℝ) / 2 * (L ^ 2 / (ε * ρ)) ^ ((1 : ℝ) / 4))\n      ≤ Δ * Real.sqrt L * ρ ^ ((1 : ℝ) / 4) / ε ^ ((7 : ℝ) / 4) := by\n  set u : ℝ := ε ^ ((1:ℝ)/4) with hu\n  set v : ℝ := ρ ^ ((1:ℝ)/4) with hv\n  have hu0 : 0 < u := Real.rpow_pos_of_pos hε _\n  have hv0 : 0 < v := Real.rpow_pos_of_pos hρ _\n  have hsL0 : 0 < Real.sqrt L := Real.sqrt_pos.mpr hL\n  have hε32 : ε ^ ((3:ℝ)/2) = u^6 := by\n    rw [hu, ← Real.rpow_natCast (ε ^ ((1:ℝ)/4)) 6, ← Real.rpow_mul hε.le]\n    norm_num\n  have hε74 : ε ^ ((7:ℝ)/4) = u^7 := by\n    rw [hu, ← Real.rpow_natCast (ε ^ ((1:ℝ)/4)) 7, ← Real.rpow_mul hε.le]\n    norm_num\n  have hρs : Real.sqrt ρ = v^2 := by\n    rw [hv, ← Real.rpow_natCast (ρ ^ ((1:ℝ)/4)) 2, ← Real.rpow_mul hρ.le, Real.sqrt_eq_rpow]\n    norm_num\n  have hLrp : (L ^ 2 / (ε * ρ)) ^ ((1:ℝ)/4) = Real.sqrt L / (u * v) := by\n    rw [Real.div_rpow (by positivity) (by positivity), Real.mul_rpow hε.le hρ.le]\n    congr 1\n    rw [← Real.rpow_natCast L 2, ← Real.rpow_mul hL.le, Real.sqrt_eq_rpow]\n    norm_num\n  -- descent chain\n  set c : ℝ := 7 * ε ^ ((3:ℝ)/2) / (8 * Real.sqrt ρ) with hc\n  have hc0 : 0 < c := by\n    rw [hc]; positivity\n  have hstep : ∀ m, m ≤ N → F m ≤ F 0 - (m:ℝ) * c := by\n    intro m\n    induction m with\n    | zero => intro _; simp\n    | succ k ih =>\n        intro hk\n        have h1 := ih (by omega)\n        have h2 := hdec k (by omega)\n        push_cast\n        linarith\n  have hNc : (N:ℝ) * c ≤ Δ := by\n    have h := hstep N le_rfl\n    linarith\n  set R : ℝ := Real.sqrt L * v / u^7 with hR\n  have hR0 : 0 ≤ R := by rw [hR]; positivity\n  have hAeq : (1:ℝ)/2 * (L ^ 2 / (ε * ρ)) ^ ((1:ℝ)/4) = (4/7) * R * c := by\n    rw [hLrp, hR, hc, hε32, hρs]\n    field_simp\n    ring\n  have hgoal : Δ * Real.sqrt L * ρ ^ ((1:ℝ)/4) / ε ^ ((7:ℝ)/4) = Δ * R := by\n    rw [hR, ← hv, hε74]\n    ring\n  rw [hAeq, hgoal]\n  have : (N:ℝ) * ((4/7) * R * c) = (4/7) * R * ((N:ℝ) * c) := by ring\n  rw [this]\n  nlinarith [hR0, hNc, hΔ]\n\nend ResearchLevel\n", "target_names": ["lemma8", "corollary12", "lemma13"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 1010, "end": 1931, "gold": "[FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) ≤ 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2) :\n    f (x K) - f (x 0) ≤ -(7 * ε ^ ((3 : ℝ) / 2)) / (8 * Real.sqrt (ρ : ℝ))"}]}
{"item_id": "v24-22-0522::lemma13::A-stmt", "paper_id": "v24-22-0522", "target_name": "lemma13", "source": "Research-level", "nl_statement": "Lemma 13 Suppose that Assumption 1 holds. Use the parameter settings in Theorem 1. In the last epoch of Algorithm 1 where the \"if condition\" does not trigger, we have $\\|\\nabla f(\\hat{\\mathbf{y}})\\|\\leq 82\\epsilon$", "context": [{"name": "Problem setup: problem (1) and approximate stationary point (Section 1)", "statement": "$$\n\\operatorname*{min}_{\\mathbf{x}\\in\\mathbb{R}^{d}}f(\\mathbf{x}),\\tag{1}\n$$\n\nwhere f (x) has Lipschitz continuous gradient and Hessian and it is bounded from below. Our goal is to find an $\\epsilon$-approximate first-order stationary point, defined as\n\n$$\n\\|\\nabla f(\\mathbf{x})\\|\\leq\\epsilon .\n$$"}, {"name": "Notation and Assumption 1 (Section 1.4)", "statement": "We use lowercase bold letters to represent vectors, uppercase bold letters for matrices, and non-bold (both lowercase and uppercase) letters for scalars. Denote $\\mathbf{x}_{j}$ and $\\nabla_{j}f(\\mathbf{x})$ as the jth element of x and $\\nabla f(\\mathbf{x})$ , respectively. For the vectors produced in the iterative algorithms, for example, x, denote $\\mathbf{x}^{k}$ to be the value at the kth iteration. We denote $\\| \\cdot \\|$ to be the $\\ell_{2}$ Euclidean norm for vectors, $\\| \\cdot \\|_{2}$ as the spectral norm and $\\| \\cdot \\|_{F}$ as the Frobenius norm for matrices. We make the following standard assumptions in this paper.\n\nAssumption 1 1. f (x) is L-gradient Lipschitz: $\\|\\nabla f(\\mathbf{x})-\\nabla f(\\mathbf{y})\\|\\leq L\\|\\mathbf{x}-\\mathbf{y}\\|,\\forall\\mathbf{x},\\mathbf{y}\\in$ $\\mathbb{R}^{d}$ ,\n\n2. f (x) is ρ-Hessian Lipschitz: $\\begin{array}{r}{\\|\\nabla^{2}f(\\mathbf{x})-\\nabla^{2}f(\\mathbf{y})\\|_{2}\\leq\\rho\\|\\mathbf{x}-\\mathbf{y}\\|,\\forall\\mathbf{x},\\mathbf{y}\\in\\mathbb{R}^{d},}\\end{array}$\n\nwhich yield the following two well-known inequalities:\n\n$$\n\\begin{array}{r}{|f(\\mathbf{y})-f(\\mathbf{x})-\\langle\\nabla f(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle|\\leq\\frac{L}{2}\\|\\mathbf{y}-\\mathbf{x}\\|^{2},}\\end{array}\\tag{2}\n$$\n\n$$\n\\begin{array}{r}{\\big|f(\\mathbf{y})-f(\\mathbf{x})-\\langle\\nabla f(\\mathbf{x}),\\mathbf{y}-\\mathbf{x}\\rangle-(\\mathbf{y}-\\mathbf{x})^{T}\\nabla^{2}f(\\mathbf{x})(\\mathbf{y}-\\mathbf{x})\\big|\\leq\\frac{\\rho}{6}\\|\\mathbf{y}-\\mathbf{x}\\|^{3}.}\\end{array}\\tag{3}\n$$\n\nWe also assume that the objective function is lower bounded, that is, $\\min_{\\mathbf{x}}f(\\mathbf{x})> -\\infty$"}, {"name": "Nesterov's AGD iteration (Section 2)", "statement": "Nesterov's classical AGD consists of the following iterations:\n\n$$\n\\begin{array}{r}{\\mathbf{y}^{k}=\\mathbf{x}^{k}+(1-\\theta)(\\mathbf{x}^{k}-\\mathbf{x}^{k-1}),\\quad\\mathbf{x}^{k+1}=\\mathbf{y}^{k}-\\eta\\nabla f(\\mathbf{y}^{k}),}\\end{array}\n$$"}, {"name": "Algorithm 1 (RAGD-NC) and its description", "statement": "Algorithm 1 Restarted AGD for Nonconvex Optimization (RAGD-NC)   \n1: Initialize ${\\bf x}^{- 1}={\\bf x}^{0}={\\bf x}_{i n t}, k = 0$   \n2: while $k < K$ do   \n3: $\\mathbf{y}^{k}=\\mathbf{x}^{k}+(1-\\theta)(\\mathbf{x}^{k}-\\mathbf{x}^{k-1})$   \n4: $\\mathbf{x}^{k+1}=\\mathbf{y}^{k}-\\eta\\nabla f(\\mathbf{y}^{k})$   \n5: $k = k + 1$   \n6: if $k \\textstyle \\sum_{t = 0}^{k - 1}\\|{\\bf x}^{t + 1}-{\\bf x}^{t}\\|^{2}> B^{2}$ then   \n7: $\\mathbf{x}^{- 1}= \\mathbf{x}^{0}= \\mathbf{x}^{k}, k = 0$   \n8: end if   \n9: end while   \n10: $K_{0}=\\mathrm{argmin}_{\\lfloor\\frac{K}{2}\\rfloor\\leq k\\leq K-1}\\|\\mathbf{x}^{k+1}-\\mathbf{x}^{k}\\|$   \n11: Output $\\begin{array}{r}{\\hat{\\mathbf{y}}= \\frac{1}{K_{0}+ 1}\\sum_{k = 0}^{K_{0}}\\mathbf{y}^{k}}\\end{array}$\n\nWe present our method in Algorithm 1. It runs Nesterov's classical AGD iterations until the \"if condition\" triggers. Then we reset $\\mathbf{x}^{0}$ and $\\mathbf{x}^{- 1}$ equal to $\\mathbf{x}^{k}$ and continue to the next round of AGD. The method terminates and outputs a specific average when the \"if condition\" does not trigger in K iterations. To simplify the description, we define one round of AGD between two successive restarts to be one \"epoch\". The restart trick, first proposed in (O'Donoghue and Candès, 2015), is motivated by (Fang et al., 2019), where a ball-mechanism is proposed as the stopping criteria to analyze SGD."}, {"name": "Theorem 1 (parameter settings)", "statement": "Theorem 1 Suppose that Assumption 1 holds. Let $\\begin{array}{r}{\\eta=\\frac{1}{4L},B=\\sqrt{\\frac{\\epsilon}{\\rho}},\\theta=4(\\epsilon\\rho\\eta^{2})^{1/4}\\in(0,1].}\\end{array}$ and $\\begin{array}{r}{K = \\frac{1}{\\theta}}\\end{array}$ . Then Algorithm 1 terminates in at most $\\frac{\\triangle_{f}L^{1 / 2}\\rho^{1 / 4}}{\\epsilon^{7 / 4}}$ gradient computations and the output satisfies $\\|\\nabla f(\\hat{\\mathbf{y}})\\|\\leq 82\\epsilon$ , where $\\begin{array}{r}{\\triangle_{f}=f(\\mathbf{x}_{i n t})-\\operatorname*{min}_{\\mathbf{x}}f(\\mathbf{x})}\\end{array}$"}, {"name": "Epoch quantities: trigger index (7), (8a), (8b), (9), (10a), (10b) (Section 4.1)", "statement": "We prove the convergence rate of Algorithm 1 in this section. Denote $\\mathcal{K}$ to be the iteration number when the \"if condition\" on line 6 of Algorithm 1 triggers, that is,\n\n$$\n\\mathcal{K}=\\operatorname*{min}_{k}\\left\\{k\\bigg|k\\sum_{t=0}^{k-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}>B^{2}\\right\\}.\\tag{7}\n$$\n\nFor each epoch consisting of one round of AGD from iterations $k = 0$ to $k = \\mathcal{K}$ , we have\n\n$$\n1\\leq\\mathcal{K}\\leq K,\\qquad\\mathcal{K}\\sum_{t=0}^{\\mathcal{K}-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}>B^{2},\\qquad\\mathrm{and}\\tag{8a}\n$$\n\n$$\n\\|{\\bf x}^{k}-{\\bf x}^{0}\\|^{2}=\\left\\|\\sum_{t=0}^{k-1}{\\bf x}^{t+1}-{\\bf x}^{t}\\right\\|^{2}\\leq k\\sum_{t=0}^{k-1}\\|{\\bf x}^{t+1}-{\\bf x}^{t}\\|^{2}\\leq B^{2},\\forall k<\\mathcal{K},\\tag{8b}\n$$\n\nwhere the last inequality comes from the definition of $\\mathcal{K}$. From the update of y on line 3 of Algorithm 1, we also have\n\n$$\n\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|\\leq\\|\\mathbf{x}^{k}-\\mathbf{x}^{0}\\|+\\|\\mathbf{x}^{k}-\\mathbf{x}^{k-1}\\|\\leq 2B,\\forall k<{\\mathcal{K}}.\\tag{9}\n$$\n\nOn the other hand, for the last epoch where the \"if condition\" does not trigger and the while loop breaks when k increases to $K$ ,we have\n\n$$\n\\|{\\mathbf{x}}^{k}-{\\mathbf{x}}^{0}\\|^{2}\\leq k\\sum_{t=0}^{k-1}\\|{\\mathbf{x}}^{t+1}-{\\mathbf{x}}^{t}\\|^{2}\\leq B^{2},\\forall k\\leq K,\\tag{10a}\n$$\n\n$$\n\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|\\leq 2B,\\forall k\\leq K.\\tag{10b}\n$$"}, {"name": "Quadratic approximation setup: (12)-(15), S_1, S_2 (Section 4.1.2)", "statement": "If $\\|\\nabla f(\\mathbf{y}^{\\mathcal{K}-1})\\|\\leq\\frac{B}{\\eta}$ , then from the AGD iteration on line 4 and (9) we have\n\n$$\n\\begin{array}{r}{\\|{\\mathbf{x}}^{\\mathcal{K}}-{\\mathbf{x}}^{0}\\| \\leq \\|{\\mathbf{y}}^{\\mathcal{K} - 1}-{\\mathbf{x}}^{0}\\| + \\eta \\| \\nabla f ({\\mathbf{y}}^{\\mathcal{K} - 1}) \\| \\leq 3 B .}\\end{array}\n$$\n\nFor each epoch, denote $\\mathbf{H}=\\nabla^{2}f(\\mathbf{x}^{0})$ to be the Hessian matrix at the starting iterate and $\\mathbf{H}= \\mathbf{U}\\pmb{\\Lambda}\\mathbf{U}^{T}$ to be its eigenvalue decomposition with U, $\\pmb{\\Lambda}\\in \\mathbb{R}^{d \\times d}$ . Let $\\lambda_{j}$ be the jth eigenvalue. Define $\\widetilde{\\mathbf{x}}= \\mathbf{U}^{T}\\mathbf{x}, \\widetilde{\\mathbf{y}}= \\mathbf{U}^{T}\\mathbf{y}$ ,and $\\widetilde{\\nabla}f(\\mathbf{y})=\\mathbf{U}^{T}\\nabla f(\\mathbf{y})$ . As the Hessian is ρ-Lipschitz, we have\n\n$$\n\\begin{array}{l}{f({\\mathbf x}^{\\mathcal{K}})-f({\\mathbf x}^{0})\\le\\left\\langle\\nabla f({\\mathbf x}^{0}),{\\mathbf x}^{\\mathcal{K}}-{\\mathbf x}^{0}\\right\\rangle+\\displaystyle\\frac{1}{2}({\\mathbf x}^{\\mathcal{K}}-{\\mathbf x}^{0})^{T}{\\mathbf H}({\\mathbf x}^{\\mathcal{K}}-{\\mathbf x}^{0})+\\displaystyle\\frac{\\rho}{6}\\|{\\mathbf x}^{\\mathcal{K}}-{\\mathbf x}^{0}\\|^{3}}\\\\ {\\displaystyle=\\left\\langle\\widetilde\\nabla f({\\mathbf x}^{0}),\\widetilde{{\\mathbf x}}^{\\mathcal{K}}-\\widetilde{{\\mathbf x}}^{0}\\right\\rangle+\\displaystyle\\frac{1}{2}(\\widetilde{{\\mathbf x}}^{\\mathcal{K}}-\\widetilde{{\\mathbf x}}^{0})^{T}\\pmb{\\Lambda}(\\widetilde{{\\mathbf x}}^{\\mathcal{K}}-\\widetilde{{\\mathbf x}}^{0})+\\displaystyle\\frac{\\rho}{6}\\|{\\mathbf x}^{\\mathcal{K}}-{\\mathbf x}^{0}\\|^{3}}\\\\ {\\le g(\\widetilde{{\\mathbf x}}^{\\mathcal{K}})-g(\\widetilde{{\\mathbf x}}^{0})+4.5\\rho B^{3},}\\end{array}\\tag{12}\n$$\n\nwhere we denote\n\n$$\n\\begin{array}{l}{g({\\pmb x})=\\left\\langle\\widetilde{\\nabla}f({\\pmb x}^{0}),{\\pmb x}-\\widetilde{\\pmb x}^{0}\\right\\rangle+\\displaystyle\\frac{1}{2}({\\pmb x}-\\widetilde{\\pmb x}^{0})^{T}{\\pmb\\Lambda}({\\pmb x}-\\widetilde{\\pmb x}^{0})=\\sum_{j=1}^{d}g_{j}({\\pmb x}_{j}),}\\\\ {g_{j}(x)=\\left\\langle\\widetilde{\\nabla}_{j}f({\\pmb x}^{0}),x-\\widetilde{\\pmb x}_{j}^{0}\\right\\rangle+\\displaystyle\\frac{1}{2}\\lambda_{j}(x-\\widetilde{\\pmb x}_{j}^{0})^{2}.}\\end{array}\\tag{13}\n$$\n\nDenoting\n\n$$\n\\widetilde{\\delta}_{j}^{k}=\\widetilde{\\nabla}_{j}f(\\mathbf{y}^{k})-\\nabla g_{j}(\\widetilde{\\mathbf{y}}_{j}^{k}),\\qquad\\widetilde{\\delta}^{k}=\\widetilde{\\nabla}f(\\mathbf{y}^{k})-\\nabla g(\\widetilde{\\mathbf{y}}^{k}),\n$$\n\nthen the AGD iterations in Algorithm 1 can be rewritten as\n\n$$\n\\begin{array}{r l}&{\\widetilde{\\mathbf{y}}_{j}^{k}=\\widetilde{\\mathbf{x}}_{j}^{k}+(1-\\theta)(\\widetilde{\\mathbf{x}}_{j}^{k}-\\widetilde{\\mathbf{x}}_{j}^{k-1}),}\\\\ &{\\widetilde{\\mathbf{x}}_{j}^{k+1}=\\widetilde{\\mathbf{y}}_{j}^{k}-\\eta\\widetilde{\\nabla}_{j}f(\\mathbf{y}^{k})=\\widetilde{\\mathbf{y}}_{j}^{k}-\\eta\\nabla g_{j}(\\widetilde{\\mathbf{y}}_{j}^{k})-\\eta\\widetilde{\\delta}_{j}^{k},}\\end{array}\\tag{14a}\n$$\n\n(14b)\n\nand $\\lVert \\widetilde{\\delta}^{k}\\rVert$ can be bounded as\n\n$$\n\\begin{array}{r l r}{{\\|\\widetilde{\\delta}^{k}\\|=\\|\\widetilde{\\nabla}f(\\mathbf{y}^{k})-\\widetilde{\\nabla}f(\\mathbf{x}^{0})-\\pmb{\\Lambda}(\\widetilde{\\mathbf{y}}^{k}-\\widetilde{\\mathbf{x}}^{0})\\|}}\\\\ &{}&{=\\|\\nabla f(\\mathbf{y}^{k})-\\nabla f(\\mathbf{x}^{0})-\\mathbf{H}(\\mathbf{y}^{k}-\\mathbf{x}^{0})\\|}\\\\ &{}&{=\\|(\\int_{0}^{1}\\nabla^{2}f(\\mathbf{x}^{0}+t(\\mathbf{y}^{k}-\\mathbf{x}^{0}))-\\mathbf{H})(\\mathbf{y}^{k}-\\mathbf{x}^{0})dt\\|}\\\\ &{}&{\\leq\\frac{\\rho}{2}\\|\\mathbf{y}^{k}-\\mathbf{x}^{0}\\|^{2}\\leq 2\\rho B^{2}}\\end{array}\\tag{15}\n$$\n\nfor any $k < \\mathcal{K}$ , where we use the ρ-Lipschitz Hessian assumption and (9) in the last two inequalities, respectively.\n\nThanks to (12), to prove the decrease from $f(\\mathbf{x}^{0})$ to $f(\\mathbf{x}^{\\mathcal{K}})$ , we only need to study the decrease of g(x). Iterations (14a) and (14b) can be regarded as applying AGD to the quadratic approximation $g(\\mathbf{x})$ coordinately with the approximation error $\\widetilde{\\delta}^{k}$ , where the later can be controlled within $\\mathcal{O}(\\rho B^{2})$ . The quadratic approximation $g(\\mathbf{x})$ equals to the sum of d scalar functions $g_{j}(\\mathbf{x}_{j})$ . We decompose $g(\\mathbf{x})$ into $\\textstyle\\sum_{j\\in S_{1}}g_{j}(\\mathbf{x}_{j})$ and $\\textstyle\\sum_{j\\in S_{2}}g_{j}(\\mathbf{x}_{j})$ , where\n\n$$\n{\\mathcal{S}}_{1}=\\left\\{j:\\lambda_{j}\\geq-{\\frac{\\theta}{\\eta}}\\right\\}\\quad{\\mathrm{and}}\\quad{\\mathcal{S}}_{2}=\\left\\{j:\\lambda_{j}< -{\\frac{\\theta}{\\eta}}\\right\\}.\n$$\n\nWe see that $g_{j}(x)$ is approximate convex when $j \\in S_{1}$ , and strongly concave when $j \\in S_{2}$ We will prove the approximate decrease of $g_{j}(\\mathbf{x}_{j})$ in the above two cases. We first consider $\\textstyle\\sum_{j\\in S_{1}}g_{j}(\\mathbf{x}_{j})$ in the following lemma."}, {"name": "Remark 14 (role of the averaged output in (22))", "statement": "Remark 14 The purpose of using the specific average as the output and $k \\sum_{t = 0}^{k - 1}\\| \\mathbf{x}^{t + 1}- \\mathbf{x}^{t}\\|^{2}> B^{2}$ in the \"if condition\" in Algorithm 1, rather than $\\| \\mathbf{x}^{k}- \\mathbf{x}^{0}\\| \\geq B$ , is to establish (22)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped NNReal BigOperators\n\nnamespace ResearchLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n\ntheorem lemma8 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (η θ B : ℝ) (hη : 0 < η) (hηL : η ≤ 1 / (4 * (L : ℝ)))\n    (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ)) (hθ0 : 0 ≤ θ) (hθ1 : θ ≤ 1) (hB : 0 ≤ B)\n    (K : ℕ) (hK : 1 ≤ K) (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (hbig : ‖f' (y (K - 1))‖ > B / η) :\n    f (x K) - f (x 0) ≤ -(B ^ 2 / (4 * η)) := by\n  sorry\n\ntheorem corollary12 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) ≤ 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h8a : B ^ 2 < (K : ℝ) * ∑ t ∈ Finset.range K, ‖x (t + 1) - x t‖ ^ 2)\n    (h8b : ∀ k, k < K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2) :\n    f (x K) - f (x 0) ≤ -(7 * ε ^ ((3 : ℝ) / 2)) / (8 * Real.sqrt (ρ : ℝ)) := by\n  sorry\n\ntheorem lemma13 [FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) = 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h10a : ∀ k, k ≤ K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (K₀ : ℕ) (hK₀lo : K / 2 ≤ K₀) (hK₀hi : K₀ < K)\n    (hK₀min : ∀ k, K / 2 ≤ k → k < K → ‖x (K₀ + 1) - x K₀‖ ≤ ‖x (k + 1) - x k‖)\n    (yhat : E)\n    (hyhat : yhat = ((K₀ : ℝ) + 1)⁻¹ • ∑ k ∈ Finset.range (K₀ + 1), y k) :\n    ‖f' yhat‖ ≤ 82 * ε := by\n  sorry\ntheorem theorem1_complexity\n    (ε ρ L Δ : ℝ) (hε : 0 < ε) (hρ : 0 < ρ) (hL : 0 < L) (hΔ : 0 ≤ Δ)\n    (hK1 : (1 : ℝ) ≤ (1 : ℝ) / 2 * (L ^ 2 / (ε * ρ)) ^ ((1 : ℝ) / 4))\n    (N : ℕ) (F : ℕ → ℝ) (Fmin : ℝ)\n    (hdec : ∀ i, i < N →\n      F (i + 1) ≤ F i - 7 * ε ^ ((3 : ℝ) / 2) / (8 * Real.sqrt ρ))\n    (hlb : Fmin ≤ F N)\n    (hΔf : F 0 - Fmin ≤ Δ) :\n    (N : ℝ) * ((1 : ℝ) / 2 * (L ^ 2 / (ε * ρ)) ^ ((1 : ℝ) / 4))\n      ≤ Δ * Real.sqrt L * ρ ^ ((1 : ℝ) / 4) / ε ^ ((7 : ℝ) / 4) := by\n  set u : ℝ := ε ^ ((1:ℝ)/4) with hu\n  set v : ℝ := ρ ^ ((1:ℝ)/4) with hv\n  have hu0 : 0 < u := Real.rpow_pos_of_pos hε _\n  have hv0 : 0 < v := Real.rpow_pos_of_pos hρ _\n  have hsL0 : 0 < Real.sqrt L := Real.sqrt_pos.mpr hL\n  have hε32 : ε ^ ((3:ℝ)/2) = u^6 := by\n    rw [hu, ← Real.rpow_natCast (ε ^ ((1:ℝ)/4)) 6, ← Real.rpow_mul hε.le]\n    norm_num\n  have hε74 : ε ^ ((7:ℝ)/4) = u^7 := by\n    rw [hu, ← Real.rpow_natCast (ε ^ ((1:ℝ)/4)) 7, ← Real.rpow_mul hε.le]\n    norm_num\n  have hρs : Real.sqrt ρ = v^2 := by\n    rw [hv, ← Real.rpow_natCast (ρ ^ ((1:ℝ)/4)) 2, ← Real.rpow_mul hρ.le, Real.sqrt_eq_rpow]\n    norm_num\n  have hLrp : (L ^ 2 / (ε * ρ)) ^ ((1:ℝ)/4) = Real.sqrt L / (u * v) := by\n    rw [Real.div_rpow (by positivity) (by positivity), Real.mul_rpow hε.le hρ.le]\n    congr 1\n    rw [← Real.rpow_natCast L 2, ← Real.rpow_mul hL.le, Real.sqrt_eq_rpow]\n    norm_num\n  -- descent chain\n  set c : ℝ := 7 * ε ^ ((3:ℝ)/2) / (8 * Real.sqrt ρ) with hc\n  have hc0 : 0 < c := by\n    rw [hc]; positivity\n  have hstep : ∀ m, m ≤ N → F m ≤ F 0 - (m:ℝ) * c := by\n    intro m\n    induction m with\n    | zero => intro _; simp\n    | succ k ih =>\n        intro hk\n        have h1 := ih (by omega)\n        have h2 := hdec k (by omega)\n        push_cast\n        linarith\n  have hNc : (N:ℝ) * c ≤ Δ := by\n    have h := hstep N le_rfl\n    linarith\n  set R : ℝ := Real.sqrt L * v / u^7 with hR\n  have hR0 : 0 ≤ R := by rw [hR]; positivity\n  have hAeq : (1:ℝ)/2 * (L ^ 2 / (ε * ρ)) ^ ((1:ℝ)/4) = (4/7) * R * c := by\n    rw [hLrp, hR, hc, hε32, hρs]\n    field_simp\n    ring\n  have hgoal : Δ * Real.sqrt L * ρ ^ ((1:ℝ)/4) / ε ^ ((7:ℝ)/4) = Δ * R := by\n    rw [hR, ← hv, hε74]\n    ring\n  rw [hAeq, hgoal]\n  have : (N:ℝ) * ((4/7) * R * c) = (4/7) * R * ((N:ℝ) * c) := by ring\n  rw [this]\n  nlinarith [hR0, hNc, hΔ]\n\nend ResearchLevel\n", "target_names": ["lemma8", "corollary12", "lemma13"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 1963, "end": 2978, "gold": "[FiniteDimensional ℝ E]\n    {f : E → ℝ} {f' : E → E} {L ρ : ℝ≥0}\n    (hflb : BddBelow (Set.range f))\n    (hf' : ∀ x, HasGradientAt f (f' x) x) (hLip : LipschitzWith L f')\n    (Hf : E → (E →L[ℝ] E)) (hf'' : ∀ x, HasFDerivAt f' (Hf x) x)\n    (hHLip : LipschitzWith ρ Hf)\n    (ε : ℝ) (hε : 0 < ε) (hL0 : 0 < (L : ℝ)) (hρ0 : 0 < (ρ : ℝ))\n    (η θ B : ℝ)\n    (hηv : η = 1 / (4 * (L : ℝ)))\n    (hBv : B = Real.sqrt (ε / (ρ : ℝ)))\n    (hθv : θ = 4 * (ε * (ρ : ℝ) * η ^ 2) ^ ((1 : ℝ) / 4))\n    (hθ0 : 0 < θ) (hθ1 : θ ≤ 1)\n    (K : ℕ) (hK : 1 ≤ K) (hKv : (K : ℝ) = 1 / θ)\n    (x y : ℕ → E)\n    (hy : ∀ k, y k = x k + (1 - θ) • (x k - x (k - 1)))\n    (hx : ∀ k, x (k + 1) = y k - η • f' (y k))\n    (h10a : ∀ k, k ≤ K → (k : ℝ) * ∑ t ∈ Finset.range k, ‖x (t + 1) - x t‖ ^ 2 ≤ B ^ 2)\n    (K₀ : ℕ) (hK₀lo : K / 2 ≤ K₀) (hK₀hi : K₀ < K)\n    (hK₀min : ∀ k, K / 2 ≤ k → k < K → ‖x (K₀ + 1) - x K₀‖ ≤ ‖x (k + 1) - x k‖)\n    (yhat : E)\n    (hyhat : yhat = ((K₀ : ℝ) + 1)⁻¹ • ∑ k ∈ Finset.range (K₀ + 1), y k) :\n    ‖f' yhat‖ ≤ 82 * ε"}]}
{"item_id": "v3-brafman02a::simulation_lemma::A-stmt", "paper_id": "v3-brafman02a", "target_name": "simulation_lemma", "source": "Research-level", "nl_statement": "Lemma 4 Suppose $M$ and $\\bar{M}$ are SGs on $N$ states and that $\\bar{M}$ is an $\\frac{\\epsilon}{N T R_{m a x}}$ -approximation of $M$. Then, for any state $s$, any policy $\\pi$ of the agent and any policy $\\rho$ of the adversary,\n\n$$\n|U_{\\bar{M}}(s,\\pi,\\rho,T)-U_{M}(s,\\pi,\\rho,T)|\\le\\epsilon .\n$$", "context": [{"name": "Setup: stochastic games (Section 2.1)", "statement": "A game models how several agents interact. It involves a set of players; each player picks an action to carry out from a prescribed action set. The combination of the players' choices determines an outcome, which is expressed numerically as a payoff vector, namely a vector containing one value per player. Our focus is on fixed-sum games with two players (games where the entries of the payoff vector always add up to the same constant). The player we control is called the agent, and the opposing player is called the adversary.$^{2}$\n\nGames are commonly represented by a matrix; this representation is called the strategic form of the game. Rows of the matrix are indexed by the agent's actions and columns by the adversary's actions. When the agent plays its $i^{t h}$ action and the adversary plays its $j^{t h}$ action, the rewards received by the agent and by the adversary are those recorded in the entry at row i and column $j$ of the game matrix. For simplicity, we assume that the agent and the adversary have action sets of the same size; extending to action sets of different sizes is trivial.\n\nA stochastic game (SG) consists of the players playing, one after another, standard games drawn from a given set of games; this sequence may be infinite. Each time a game is played, the players obtain the payoff prescribed by that game's matrix and then move on to a new game. Which game comes next depends stochastically on the game just played and on the actions the players took in it. Formally:"}, {"name": "Definition 1 (stochastic game)", "statement": "Definition 1 A fixed-sum, two player, stochastic-game [SG] M with state set $S=\\{1,\\ldots,N\\}$ and action set $A=\\{a_{1},\\ldots,a_{k}\\}$ is made up of the following components:\n\n• Stage Games: to every state $s \\in S$ there is attached a two-player, fixed-sum game in strategic form in which each player's action set is A. The reward matrix of stage-game i is written $R^{i}$.\n\nProbabilistic Transition Function: $P_{M}(s,t,a,a^{\\prime})$ denotes the probability of moving from state s to state t when the first player (the agent) chooses a and the second player (the adversary) chooses a'."}, {"name": "SGs vs MDPs (Section 2.1)", "statement": "SGs resemble MDPs: in each model, actions cause the world to move between states. The key distinction is that an MDP's transition is governed by the action of one agent, while an SG's transition is governed by the joint action of the agent and the adversary. Moreover, in an SG the agent's reward for an action depends both on that action and on the adversary's action. To capture this, each state is assigned a game, and for this reason we use the words state and game interchangeably."}, {"name": "Rewards in [0, R_max], histories, policies, T-step average value (Section 2.1)", "statement": "To simplify the presentation, the payoffs of both players in every stage game are normalised to be nonnegative real numbers lying between 0 and some constant $R_{m a x}.$ The number of actions is also treated as a constant. A history of length t is an element of $(S\\times A^{2}\\times R)^{t}\\times S$ , that is, t state-action-reward triplets in sequence, followed by the state that is reached. The set H of possible histories is the union, over all $t \\geq 0$ , of the sets of histories of length t, where the histories of length 0 are the elements of S.\n\nFor a given SG, an agent policy maps H to the set of probability distributions over A. Thus, for every possible history, a policy specifies the probability with which each action is chosen.\n\nPolicies are evaluated by the average expected reward criterion, defined as follows. For an SG M and a natural number $T,$ we write $U_{M}(s,\\pi,\\rho,T)$ for the expected T-step undiscounted average reward of a policy π against an adversary using policy $\\rho,$ with both $\\pi$ and $\\rho$ run from a state $s \\in S$ (the subscript naming the SG is dropped when no confusion arises). The value that π can guarantee over T steps from s is denoted $U_{M}(s,\\pi,T)=\\operatorname*{min}_{\\rho \\text{ is a policy}}U_{M}(s,\\pi,\\rho,T)$. We then set $\\begin{array}{r}{U_{M}(s,\\pi)=\\operatorname*{lim}\\operatorname*{inf}_{T\\to\\infty}U_{M}(s,\\pi,T)}\\end{array}$ , and, lastly, $\\begin{array}{r}{U_{M}(\\pi)=\\operatorname*{min}_{s\\in S}U_{M}(s,\\pi).}\\end{array}$ $^{3}$"}, {"name": "Setup and notation for R-MAX (Section 3)", "statement": "As before, M is a stochastic game made up of a set $S=\\{G_{1},\\ldots,G_{N}\\}$ of stage-games, in each of which both the agent and the adversary choose from the action set $A=\\{a_{1},\\ldots,a_{k}\\}$. Each game has a reward matrix $R^{i}$ attached to it, and $R_{m,l}^{i}$ stands for the pair formed by the agent's and the adversary's rewards when actions $a_{m}$ and $a_{l}$ , respectively, are played in game $G_{i}.$ There is also a probabilistic transition function $P_{M}$ , where $P_{M}(s,t,a,a^{\\prime})$ gives the probability of moving from $G_{s}$ to $G_{t}$ when the agent plays a and the adversary plays $a^{\\prime}.$ It is helpful to regard $P_{M}(i,\\cdot,a,a^{\\prime})$ as attached to the entry $(a,a^{\\prime})$ of the stage-game $G_{i}$ ; in this way every model parameter, reward or transition alike, belongs to a joint action of a specific game. Fix $\\epsilon > 0$ . To keep the exposition simple, for most of the analysis we assume that T, the $\\epsilon$-return mixing time of the optimal policy, is known; how to relax this assumption is shown later."}, {"name": "Definition 3 (alpha-approximation)", "statement": "Definition 3 Suppose M and $\\bar{M}$ are SGs sharing the same state space and the same action space. $\\bar{M}$ is called an α-approximation of M if every state s satisfies the following:\n\n1. Writing $P_{M}(s,t,a,a^{\\prime})$ and $P_{\\bar{M}}(s,t,a,a^{\\prime})$ for the probability, in M and in $\\bar{M}$ respectively, of moving from state s to state t under the joint action $(a,a^{\\prime})$ of the agent and the adversary, we have $P_{M}(s,t,a,a^{\\prime})-\\alpha\\le P_{\\bar{M}}(s,t,a,a^{\\prime})\\le P_{M}(s,t,a,a^{\\prime})+\\alpha$\n\n2. For every state s, $\\bar{M}$ and M attach the same stage-game to s (so that the rewards coincide)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace ResearchLevel\n\nstructure SG (N k : ℕ) where\n  P : Fin N → Fin k → Fin k → Fin N → ℝ\n  R : Fin N → Fin k → Fin k → ℝ\n\nvariable {N k T : ℕ}\n\nabbrev Traj (N k T : ℕ) := (Fin (T + 1) → Fin N) × (Fin T → Fin k) × (Fin T → Fin k)\n\nabbrev Policy (N k T : ℕ) :=\n  (i : Fin T) → (Fin (i.val + 1) → Fin N) → (Fin i.val → Fin k) →\n    (Fin i.val → Fin k) → (Fin i.val → ℝ) → Fin k → ℝ\n\ndef stateHist (w : Traj N k T) (i : Fin T) : Fin (i.val + 1) → Fin N :=\n  fun j => w.1 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef agentHist (w : Traj N k T) (i : Fin T) : Fin i.val → Fin k :=\n  fun j => w.2.1 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef advHist (w : Traj N k T) (i : Fin T) : Fin i.val → Fin k :=\n  fun j => w.2.2 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef rewardHist (M : SG N k) (w : Traj N k T) (i : Fin T) : Fin i.val → ℝ :=\n  fun j => M.R (w.1 (Fin.castLE (by have := i.isLt; omega) j).castSucc)\n    (w.2.1 (Fin.castLE (by have := i.isLt; omega) j))\n    (w.2.2 (Fin.castLE (by have := i.isLt; omega) j))\n\nnoncomputable def pathProb (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) (w : Traj N k T) : ℝ :=\n  (if w.1 0 = s0 then (1 : ℝ) else 0) *\n    ∏ i : Fin T,\n      πa i (stateHist w i) (agentHist w i) (advHist w i) (rewardHist M w i) (w.2.1 i) *\n      πd i (stateHist w i) (agentHist w i) (advHist w i) (rewardHist M w i) (w.2.2 i) *\n      M.P (w.1 i.castSucc) (w.2.1 i) (w.2.2 i) (w.1 i.succ)\n\nnoncomputable def pathPayoff (M : SG N k) (w : Traj N k T) : ℝ :=\n  (1 / (T : ℝ)) * ∑ i : Fin T, M.R (w.1 i.castSucc) (w.2.1 i) (w.2.2 i)\n\nnoncomputable def U (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) : ℝ :=\n  ∑ w : Traj N k T, pathProb M πa πd s0 w * pathPayoff M w\n\ndef Approximates (Mbar M : SG N k) (α : ℝ) : Prop :=\n  (∀ s a a' t, |Mbar.P s a a' t - M.P s a a' t| ≤ α) ∧\n  (∀ s a a', Mbar.R s a a' = M.R s a a')\n\nabbrev UnknownSet (N k : ℕ) := Finset (Fin N × Fin k × Fin k)\n\ndef inducedSG (M : SG N k) (L : UnknownSet N k) (Rmax : ℝ) : SG (N + 1) k where\n  P := fun s a a' t =>\n    if hs : (s : ℕ) < N then\n      if (⟨s, hs⟩, a, a') ∈ L then (if t = Fin.last N then 1 else 0)\n      else if ht : (t : ℕ) < N then M.P ⟨s, hs⟩ a a' ⟨t, ht⟩ else 0\n    else if t = Fin.last N then 1 else 0\n  R := fun s a a' =>\n    if hs : (s : ℕ) < N then\n      if (⟨s, hs⟩, a, a') ∈ L then Rmax else M.R ⟨s, hs⟩ a a'\n    else Rmax\n\nnoncomputable def escapeProb (L : UnknownSet N k) (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) : ℝ :=\n  ∑ w : Traj N k T,\n    (if ∃ i : Fin T, (w.1 i.castSucc, w.2.1 i, w.2.2 i) ∈ L then\n        pathProb M πa πd s0 w else 0)\n\ntheorem simulation_lemma\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax ε : ℝ) (hRmax : 0 < Rmax) (hε : 0 < ε)\n    (M Mbar : SG N k)\n\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n\n    (happrox : Approximates Mbar M (ε / ((N : ℝ) * (T : ℝ) * Rmax))) :\n    |U Mbar πa πd s0 - U M πa πd s0| ≤ ε := by\n  sorry\n\ntheorem path_dist_l1_bound\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (α : ℝ) (hα : 0 ≤ α)\n    (M Mbar : SG N k)\n    (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n    (happrox : Approximates Mbar M α) :\n    ∑ w : Traj N k T,\n        |pathProb Mbar πa πd s0 w - pathProb M πa πd s0 w|\n      ≤ (N : ℝ) * (T : ℝ) * α := by\n  sorry\n\ntheorem value_diff_escape_bound\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (L : UnknownSet N k) (M : SG N k)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a') (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (πa' πd' : Policy (N + 1) k T)\n    (ha0' : ∀ i hs ha hd hr a, 0 ≤ πa' i hs ha hd hr a)\n    (hasum' : ∀ i hs ha hd hr, ∑ a, πa' i hs ha hd hr a = 1)\n    (hd0' : ∀ i hs ha hd hr a, 0 ≤ πd' i hs ha hd hr a)\n    (hdsum' : ∀ i hs ha hd hr, ∑ a, πd' i hs ha hd hr a = 1)\n    (hπa : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πa' i (fun j => (hs j).castSucc) ha hd hr a = πa i hs ha hd hr a)\n    (hπd : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πd' i (fun j => (hs j).castSucc) ha hd hr a = πd i hs ha hd hr a)\n    (s0 : Fin N) :\n    |U (inducedSG M L Rmax) πa' πd' (Fin.castSucc s0) - U M πa πd s0|\n      ≤ Rmax * escapeProb L M πa πd s0 := by\n  sorry\nend ResearchLevel\n", "target_names": ["simulation_lemma", "path_dist_l1_bound", "value_diff_escape_bound"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "holes": [{"start": 2654, "end": 3414, "gold": "(hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax ε : ℝ) (hRmax : 0 < Rmax) (hε : 0 < ε)\n    (M Mbar : SG N k)\n\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n\n    (happrox : Approximates Mbar M (ε / ((N : ℝ) * (T : ℝ) * Rmax))) :\n    |U Mbar πa πd s0 - U M πa πd s0| ≤ ε"}]}
{"item_id": "v3-brafman02a::path_dist_l1_bound::A-stmt", "paper_id": "v3-brafman02a", "target_name": "path_dist_l1_bound", "source": "Research-level", "nl_statement": "**Lemma 4, engine (path-distribution total-variation bound).**\n\nSame setting as Lemma 4: a two-player stochastic game on $N$ states, $k$ actions and horizon $T$, and $\\bar{M}$ approximating $M$ to within $\\alpha \\ge 0$ in the sense that rewards agree and $|\\bar{P}(s,a,a',t) - P(s,a,a',t)| \\le \\alpha$ for all $s, a, a', t$.\n\nClaim: the two induced distributions over length-$T$ trajectories (each trajectory being a sequence of $T+1$ states together with both players' $T$ actions) are close in total variation —\n\n$$\n\\sum_w |\\Pr_{\\bar{M}}(w) - \\Pr_M(w)| \\le N \\cdot T \\cdot \\alpha,\n$$\n\nthe sum being over all trajectories $w$ from $s_0$. This is the step that carries Lemma 4: a per-transition error of $\\alpha$ accumulates to at most $N \\cdot T \\cdot \\alpha$ over the whole path distribution, and bounded rewards then turn that into the value bound.", "context": [{"name": "Setup: stochastic games (Section 2.1)", "statement": "A game models how several agents interact. It involves a set of players; each player picks an action to carry out from a prescribed action set. The combination of the players' choices determines an outcome, which is expressed numerically as a payoff vector, namely a vector containing one value per player. Our focus is on fixed-sum games with two players (games where the entries of the payoff vector always add up to the same constant). The player we control is called the agent, and the opposing player is called the adversary.$^{2}$\n\nGames are commonly represented by a matrix; this representation is called the strategic form of the game. Rows of the matrix are indexed by the agent's actions and columns by the adversary's actions. When the agent plays its $i^{t h}$ action and the adversary plays its $j^{t h}$ action, the rewards received by the agent and by the adversary are those recorded in the entry at row i and column $j$ of the game matrix. For simplicity, we assume that the agent and the adversary have action sets of the same size; extending to action sets of different sizes is trivial.\n\nA stochastic game (SG) consists of the players playing, one after another, standard games drawn from a given set of games; this sequence may be infinite. Each time a game is played, the players obtain the payoff prescribed by that game's matrix and then move on to a new game. Which game comes next depends stochastically on the game just played and on the actions the players took in it. Formally:"}, {"name": "Definition 1 (stochastic game)", "statement": "Definition 1 A fixed-sum, two player, stochastic-game [SG] M with state set $S=\\{1,\\ldots,N\\}$ and action set $A=\\{a_{1},\\ldots,a_{k}\\}$ is made up of the following components:\n\n• Stage Games: to every state $s \\in S$ there is attached a two-player, fixed-sum game in strategic form in which each player's action set is A. The reward matrix of stage-game i is written $R^{i}$.\n\nProbabilistic Transition Function: $P_{M}(s,t,a,a^{\\prime})$ denotes the probability of moving from state s to state t when the first player (the agent) chooses a and the second player (the adversary) chooses a'."}, {"name": "SGs vs MDPs (Section 2.1)", "statement": "SGs resemble MDPs: in each model, actions cause the world to move between states. The key distinction is that an MDP's transition is governed by the action of one agent, while an SG's transition is governed by the joint action of the agent and the adversary. Moreover, in an SG the agent's reward for an action depends both on that action and on the adversary's action. To capture this, each state is assigned a game, and for this reason we use the words state and game interchangeably."}, {"name": "Rewards in [0, R_max], histories, policies, T-step average value (Section 2.1)", "statement": "To simplify the presentation, the payoffs of both players in every stage game are normalised to be nonnegative real numbers lying between 0 and some constant $R_{m a x}.$ The number of actions is also treated as a constant. A history of length t is an element of $(S\\times A^{2}\\times R)^{t}\\times S$ , that is, t state-action-reward triplets in sequence, followed by the state that is reached. The set H of possible histories is the union, over all $t \\geq 0$ , of the sets of histories of length t, where the histories of length 0 are the elements of S.\n\nFor a given SG, an agent policy maps H to the set of probability distributions over A. Thus, for every possible history, a policy specifies the probability with which each action is chosen.\n\nPolicies are evaluated by the average expected reward criterion, defined as follows. For an SG M and a natural number $T,$ we write $U_{M}(s,\\pi,\\rho,T)$ for the expected T-step undiscounted average reward of a policy π against an adversary using policy $\\rho,$ with both $\\pi$ and $\\rho$ run from a state $s \\in S$ (the subscript naming the SG is dropped when no confusion arises). The value that π can guarantee over T steps from s is denoted $U_{M}(s,\\pi,T)=\\operatorname*{min}_{\\rho \\text{ is a policy}}U_{M}(s,\\pi,\\rho,T)$. We then set $\\begin{array}{r}{U_{M}(s,\\pi)=\\operatorname*{lim}\\operatorname*{inf}_{T\\to\\infty}U_{M}(s,\\pi,T)}\\end{array}$ , and, lastly, $\\begin{array}{r}{U_{M}(\\pi)=\\operatorname*{min}_{s\\in S}U_{M}(s,\\pi).}\\end{array}$ $^{3}$"}, {"name": "Setup and notation for R-MAX (Section 3)", "statement": "As before, M is a stochastic game made up of a set $S=\\{G_{1},\\ldots,G_{N}\\}$ of stage-games, in each of which both the agent and the adversary choose from the action set $A=\\{a_{1},\\ldots,a_{k}\\}$. Each game has a reward matrix $R^{i}$ attached to it, and $R_{m,l}^{i}$ stands for the pair formed by the agent's and the adversary's rewards when actions $a_{m}$ and $a_{l}$ , respectively, are played in game $G_{i}.$ There is also a probabilistic transition function $P_{M}$ , where $P_{M}(s,t,a,a^{\\prime})$ gives the probability of moving from $G_{s}$ to $G_{t}$ when the agent plays a and the adversary plays $a^{\\prime}.$ It is helpful to regard $P_{M}(i,\\cdot,a,a^{\\prime})$ as attached to the entry $(a,a^{\\prime})$ of the stage-game $G_{i}$ ; in this way every model parameter, reward or transition alike, belongs to a joint action of a specific game. Fix $\\epsilon > 0$ . To keep the exposition simple, for most of the analysis we assume that T, the $\\epsilon$-return mixing time of the optimal policy, is known; how to relax this assumption is shown later."}, {"name": "Definition 3 (alpha-approximation)", "statement": "Definition 3 Suppose M and $\\bar{M}$ are SGs sharing the same state space and the same action space. $\\bar{M}$ is called an α-approximation of M if every state s satisfies the following:\n\n1. Writing $P_{M}(s,t,a,a^{\\prime})$ and $P_{\\bar{M}}(s,t,a,a^{\\prime})$ for the probability, in M and in $\\bar{M}$ respectively, of moving from state s to state t under the joint action $(a,a^{\\prime})$ of the agent and the adversary, we have $P_{M}(s,t,a,a^{\\prime})-\\alpha\\le P_{\\bar{M}}(s,t,a,a^{\\prime})\\le P_{M}(s,t,a,a^{\\prime})+\\alpha$\n\n2. For every state s, $\\bar{M}$ and M attach the same stage-game to s (so that the rewards coincide)."}, {"name": "Lemma 4 (Simulation Lemma) (another result of the same paper)", "statement": "Lemma 4 Suppose $M$ and $\\bar{M}$ are SGs on $N$ states and that $\\bar{M}$ is an $\\frac{\\epsilon}{N T R_{m a x}}$ -approximation of $M$. Then, for any state $s$, any policy $\\pi$ of the agent and any policy $\\rho$ of the adversary,\n\n$$\n|U_{\\bar{M}}(s,\\pi,\\rho,T)-U_{M}(s,\\pi,\\rho,T)|\\le\\epsilon .\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace ResearchLevel\n\nstructure SG (N k : ℕ) where\n  P : Fin N → Fin k → Fin k → Fin N → ℝ\n  R : Fin N → Fin k → Fin k → ℝ\n\nvariable {N k T : ℕ}\n\nabbrev Traj (N k T : ℕ) := (Fin (T + 1) → Fin N) × (Fin T → Fin k) × (Fin T → Fin k)\n\nabbrev Policy (N k T : ℕ) :=\n  (i : Fin T) → (Fin (i.val + 1) → Fin N) → (Fin i.val → Fin k) →\n    (Fin i.val → Fin k) → (Fin i.val → ℝ) → Fin k → ℝ\n\ndef stateHist (w : Traj N k T) (i : Fin T) : Fin (i.val + 1) → Fin N :=\n  fun j => w.1 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef agentHist (w : Traj N k T) (i : Fin T) : Fin i.val → Fin k :=\n  fun j => w.2.1 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef advHist (w : Traj N k T) (i : Fin T) : Fin i.val → Fin k :=\n  fun j => w.2.2 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef rewardHist (M : SG N k) (w : Traj N k T) (i : Fin T) : Fin i.val → ℝ :=\n  fun j => M.R (w.1 (Fin.castLE (by have := i.isLt; omega) j).castSucc)\n    (w.2.1 (Fin.castLE (by have := i.isLt; omega) j))\n    (w.2.2 (Fin.castLE (by have := i.isLt; omega) j))\n\nnoncomputable def pathProb (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) (w : Traj N k T) : ℝ :=\n  (if w.1 0 = s0 then (1 : ℝ) else 0) *\n    ∏ i : Fin T,\n      πa i (stateHist w i) (agentHist w i) (advHist w i) (rewardHist M w i) (w.2.1 i) *\n      πd i (stateHist w i) (agentHist w i) (advHist w i) (rewardHist M w i) (w.2.2 i) *\n      M.P (w.1 i.castSucc) (w.2.1 i) (w.2.2 i) (w.1 i.succ)\n\nnoncomputable def pathPayoff (M : SG N k) (w : Traj N k T) : ℝ :=\n  (1 / (T : ℝ)) * ∑ i : Fin T, M.R (w.1 i.castSucc) (w.2.1 i) (w.2.2 i)\n\nnoncomputable def U (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) : ℝ :=\n  ∑ w : Traj N k T, pathProb M πa πd s0 w * pathPayoff M w\n\ndef Approximates (Mbar M : SG N k) (α : ℝ) : Prop :=\n  (∀ s a a' t, |Mbar.P s a a' t - M.P s a a' t| ≤ α) ∧\n  (∀ s a a', Mbar.R s a a' = M.R s a a')\n\nabbrev UnknownSet (N k : ℕ) := Finset (Fin N × Fin k × Fin k)\n\ndef inducedSG (M : SG N k) (L : UnknownSet N k) (Rmax : ℝ) : SG (N + 1) k where\n  P := fun s a a' t =>\n    if hs : (s : ℕ) < N then\n      if (⟨s, hs⟩, a, a') ∈ L then (if t = Fin.last N then 1 else 0)\n      else if ht : (t : ℕ) < N then M.P ⟨s, hs⟩ a a' ⟨t, ht⟩ else 0\n    else if t = Fin.last N then 1 else 0\n  R := fun s a a' =>\n    if hs : (s : ℕ) < N then\n      if (⟨s, hs⟩, a, a') ∈ L then Rmax else M.R ⟨s, hs⟩ a a'\n    else Rmax\n\nnoncomputable def escapeProb (L : UnknownSet N k) (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) : ℝ :=\n  ∑ w : Traj N k T,\n    (if ∃ i : Fin T, (w.1 i.castSucc, w.2.1 i, w.2.2 i) ∈ L then\n        pathProb M πa πd s0 w else 0)\n\ntheorem simulation_lemma\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax ε : ℝ) (hRmax : 0 < Rmax) (hε : 0 < ε)\n    (M Mbar : SG N k)\n\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n\n    (happrox : Approximates Mbar M (ε / ((N : ℝ) * (T : ℝ) * Rmax))) :\n    |U Mbar πa πd s0 - U M πa πd s0| ≤ ε := by\n  sorry\n\ntheorem path_dist_l1_bound\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (α : ℝ) (hα : 0 ≤ α)\n    (M Mbar : SG N k)\n    (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n    (happrox : Approximates Mbar M α) :\n    ∑ w : Traj N k T,\n        |pathProb Mbar πa πd s0 w - pathProb M πa πd s0 w|\n      ≤ (N : ℝ) * (T : ℝ) * α := by\n  sorry\n\ntheorem value_diff_escape_bound\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (L : UnknownSet N k) (M : SG N k)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a') (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (πa' πd' : Policy (N + 1) k T)\n    (ha0' : ∀ i hs ha hd hr a, 0 ≤ πa' i hs ha hd hr a)\n    (hasum' : ∀ i hs ha hd hr, ∑ a, πa' i hs ha hd hr a = 1)\n    (hd0' : ∀ i hs ha hd hr a, 0 ≤ πd' i hs ha hd hr a)\n    (hdsum' : ∀ i hs ha hd hr, ∑ a, πd' i hs ha hd hr a = 1)\n    (hπa : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πa' i (fun j => (hs j).castSucc) ha hd hr a = πa i hs ha hd hr a)\n    (hπd : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πd' i (fun j => (hs j).castSucc) ha hd hr a = πd i hs ha hd hr a)\n    (s0 : Fin N) :\n    |U (inducedSG M L Rmax) πa' πd' (Fin.castSucc s0) - U M πa πd s0|\n      ≤ Rmax * escapeProb L M πa πd s0 := by\n  sorry\nend ResearchLevel\n", "target_names": ["simulation_lemma", "path_dist_l1_bound", "value_diff_escape_bound"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "holes": [{"start": 3461, "end": 4265, "gold": "(hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (α : ℝ) (hα : 0 ≤ α)\n    (M Mbar : SG N k)\n    (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n    (happrox : Approximates Mbar M α) :\n    ∑ w : Traj N k T,\n        |pathProb Mbar πa πd s0 w - pathProb M πa πd s0 w|\n      ≤ (N : ℝ) * (T : ℝ) * α"}]}
{"item_id": "v3-brafman02a::value_diff_escape_bound::A-stmt", "paper_id": "v3-brafman02a", "target_name": "value_diff_escape_bound", "source": "Research-level", "nl_statement": "**Lemma 6, core (implicit explore-or-exploit value bound).**\n\nSetting: a stochastic game $M$ on $N$ states as in Lemma 4, together with a set $L$ of (state, action, action) triples designated UNKNOWN. Build an auxiliary game on $N+1$ states by adding one absorbing state: from any triple in $L$ the auxiliary game moves to the new state with probability 1 and pays $R_{\\max}$; the new state is absorbing and also pays $R_{\\max}$; off $L$ the auxiliary game agrees with $M$ in both transitions and rewards. Lift a pair of policies for $M$ to the auxiliary game so that they behave identically on histories that never leave the original $N$ states.\n\nLet the escape probability be the probability, under $M$ and the original policies from $s_0$, that the trajectory uses at least one triple in $L$.\n\nClaim: the values of the two games differ by at most $R_{\\max}$ times that probability —\n\n$$\n|U_{\\mathrm{aux}}(\\pi_a', \\pi_d', s_0) - U_M(\\pi_a, \\pi_d, s_0)| \\le R_{\\max} \\cdot \\Pr[\\text{escape}].\n$$\n\nThis is the explore-or-exploit dichotomy in quantitative form: optimism about the unknown triples costs nothing unless they are actually reached, so either the auxiliary game's value is nearly the true value, or the unknown set gets visited with substantial probability.", "context": [{"name": "Setup: stochastic games (Section 2.1)", "statement": "A game models how several agents interact. It involves a set of players; each player picks an action to carry out from a prescribed action set. The combination of the players' choices determines an outcome, which is expressed numerically as a payoff vector, namely a vector containing one value per player. Our focus is on fixed-sum games with two players (games where the entries of the payoff vector always add up to the same constant). The player we control is called the agent, and the opposing player is called the adversary.$^{2}$\n\nGames are commonly represented by a matrix; this representation is called the strategic form of the game. Rows of the matrix are indexed by the agent's actions and columns by the adversary's actions. When the agent plays its $i^{t h}$ action and the adversary plays its $j^{t h}$ action, the rewards received by the agent and by the adversary are those recorded in the entry at row i and column $j$ of the game matrix. For simplicity, we assume that the agent and the adversary have action sets of the same size; extending to action sets of different sizes is trivial.\n\nA stochastic game (SG) consists of the players playing, one after another, standard games drawn from a given set of games; this sequence may be infinite. Each time a game is played, the players obtain the payoff prescribed by that game's matrix and then move on to a new game. Which game comes next depends stochastically on the game just played and on the actions the players took in it. Formally:"}, {"name": "Definition 1 (stochastic game)", "statement": "Definition 1 A fixed-sum, two player, stochastic-game [SG] M with state set $S=\\{1,\\ldots,N\\}$ and action set $A=\\{a_{1},\\ldots,a_{k}\\}$ is made up of the following components:\n\n• Stage Games: to every state $s \\in S$ there is attached a two-player, fixed-sum game in strategic form in which each player's action set is A. The reward matrix of stage-game i is written $R^{i}$.\n\nProbabilistic Transition Function: $P_{M}(s,t,a,a^{\\prime})$ denotes the probability of moving from state s to state t when the first player (the agent) chooses a and the second player (the adversary) chooses a'."}, {"name": "SGs vs MDPs (Section 2.1)", "statement": "SGs resemble MDPs: in each model, actions cause the world to move between states. The key distinction is that an MDP's transition is governed by the action of one agent, while an SG's transition is governed by the joint action of the agent and the adversary. Moreover, in an SG the agent's reward for an action depends both on that action and on the adversary's action. To capture this, each state is assigned a game, and for this reason we use the words state and game interchangeably."}, {"name": "Rewards in [0, R_max], histories, policies, T-step average value (Section 2.1)", "statement": "To simplify the presentation, the payoffs of both players in every stage game are normalised to be nonnegative real numbers lying between 0 and some constant $R_{m a x}.$ The number of actions is also treated as a constant. A history of length t is an element of $(S\\times A^{2}\\times R)^{t}\\times S$ , that is, t state-action-reward triplets in sequence, followed by the state that is reached. The set H of possible histories is the union, over all $t \\geq 0$ , of the sets of histories of length t, where the histories of length 0 are the elements of S.\n\nFor a given SG, an agent policy maps H to the set of probability distributions over A. Thus, for every possible history, a policy specifies the probability with which each action is chosen.\n\nPolicies are evaluated by the average expected reward criterion, defined as follows. For an SG M and a natural number $T,$ we write $U_{M}(s,\\pi,\\rho,T)$ for the expected T-step undiscounted average reward of a policy π against an adversary using policy $\\rho,$ with both $\\pi$ and $\\rho$ run from a state $s \\in S$ (the subscript naming the SG is dropped when no confusion arises). The value that π can guarantee over T steps from s is denoted $U_{M}(s,\\pi,T)=\\operatorname*{min}_{\\rho \\text{ is a policy}}U_{M}(s,\\pi,\\rho,T)$. We then set $\\begin{array}{r}{U_{M}(s,\\pi)=\\operatorname*{lim}\\operatorname*{inf}_{T\\to\\infty}U_{M}(s,\\pi,T)}\\end{array}$ , and, lastly, $\\begin{array}{r}{U_{M}(\\pi)=\\operatorname*{min}_{s\\in S}U_{M}(s,\\pi).}\\end{array}$ $^{3}$"}, {"name": "Setup and notation for R-MAX (Section 3)", "statement": "As before, M is a stochastic game made up of a set $S=\\{G_{1},\\ldots,G_{N}\\}$ of stage-games, in each of which both the agent and the adversary choose from the action set $A=\\{a_{1},\\ldots,a_{k}\\}$. Each game has a reward matrix $R^{i}$ attached to it, and $R_{m,l}^{i}$ stands for the pair formed by the agent's and the adversary's rewards when actions $a_{m}$ and $a_{l}$ , respectively, are played in game $G_{i}.$ There is also a probabilistic transition function $P_{M}$ , where $P_{M}(s,t,a,a^{\\prime})$ gives the probability of moving from $G_{s}$ to $G_{t}$ when the agent plays a and the adversary plays $a^{\\prime}.$ It is helpful to regard $P_{M}(i,\\cdot,a,a^{\\prime})$ as attached to the entry $(a,a^{\\prime})$ of the stage-game $G_{i}$ ; in this way every model parameter, reward or transition alike, belongs to a joint action of a specific game. Fix $\\epsilon > 0$ . To keep the exposition simple, for most of the analysis we assume that T, the $\\epsilon$-return mixing time of the optimal policy, is known; how to relax this assumption is shown later."}, {"name": "Algorithm R-MAX (Section 3)", "statement": "The R-MAX algorithm reads as follows.\n\nInput: N – how many stage games there are; k – how many actions each stage game has; $\\epsilon$ – the error bound; δ – the probability that the algorithm fails; $R_{m a x}$ – an upper bound on the reward function; T – the $\\epsilon$-return mixing time of an optimal policy.\n\nInitialize: Build a model $M'$ with $N{+ 1}$ stage-games $\\{G_{0},G_{1},\\dots,G_{N}\\}$ and k actions $\\{a_{1},\\ldots,a_{k}\\}$ . The games $G_{1},\\ldots,G_{N}$ stand for the real games and $\\{a_{1},\\ldots,a_{k}\\}$ for the real actions, while $G_{0}$ is an extra, fictitious game. Every entry of every game matrix is set to $(R_{max},0)$ initially.$^{4}$ Set $P_{M}(G_{i},G_{0},a,a^{\\prime})=1$ for every $i=0,\\ldots,N$ and every pair of actions $a,a^{\\prime}$\n\nAdditionally, for every entry of every game $G_{1},\\dots,G_{N}$ , keep track of: (1) a boolean flag known/unknown, whose initial value is unknown; (2) the states that have been reached by playing the joint action of this entry (together with the number of times each was reached); (3) the reward received (by both players) for playing the joint action of this entry. At the start, items 2 and 3 are empty.\n\n## Repeat:\n\nCompute and Act: For the current state, find a policy that is optimal over T steps, and follow it for T steps or until some new entry becomes known.\n\nObserve and update: After every joint action, proceed as follows. Let a be your action in $G_{i}$ and $a^{\\prime}$ the adversary's action.\n\nWhen the joint action $(a,a^{\\prime})$ is being performed in $G_{i}$ for the first time, set the reward associated with $(a,a^{\\prime})$ in $G_{i}$ to the observed one.\n\nUpdate the record of states that playing $(a,a^{\\prime})$ in $G_{i}$ has led to\n\nIf the record of states reached from this entry now holds $\\begin{array}{r}{K_{1}=\\operatorname*{max}((\\lceil\\frac{4 N T R_{m a x}}{\\epsilon})^{3}\\rceil,\\lceil-6 l n^{3}(\\frac{\\delta}{6 N k^{2}})\\rceil)+1}\\end{array}$ elements, flag this entry as known and set its transition probabilities to the observed frequencies.\n\nR-MAX is thus very simple. Its initial estimate of the model parameters assumes that every state and every joint action gives the maximal reward and moves with probability 1 to the fictitious stage-game $G_{0}$ . An optimal T-step policy is computed for the current model and then followed. After each joint action the agent lands in a new stage-game, and the transition is stored in the corresponding place. When enough information has been gathered about where a given joint action leads from a given stage-game, the model entries for that stage-game and joint action are updated. After every update of the model, an optimal policy is recomputed and the steps above are repeated."}, {"name": "Definition 5 (induced SG M_L) and R-max policy", "statement": "We now introduce induced SGs. Apart from the role played by $R_{m a x}$ , the definition mirrors the induced MDP of Kearns and Singh (1998). The agent chooses its policy on the basis of the induced SG.\n\nDefinition 5 Let M be an SG, and let L denote the set of entries $(G_{i},a,a^{\\prime})$ that are marked unknown; in other words, $(G_{i},a,a^{\\prime})\\in L$ means that the entry for the joint action $(a,a^{\\prime})$ in the stage-game $G_{i}$ is marked unknown. The SG $M_{L}$ is defined to coincide with M except that it has one more state, $G_{0}$ . Every entry of $M_{L}$ outside L has the same transitions and rewards as in M. Every entry that is in L or in $G_{0}$ moves to $G_{0}$ with probability 1 and gives reward $R_{m a x}$ to the agent and 0 to the adversary.$^{5}$\n\nFor an SG M with a set L of unknown states, $R^{M_{L}}$ -max stands for the optimal policy of the induced SG $M_{L}$ . If $M_{L}$ is clear from context, we just say R-MAX policy in place of $R^{M_{L}}$ -max policy."}, {"name": "Lemma 6 (implicit explore or exploit)", "statement": "Lemma 6 Let M be an SG and let L and $M_{L}$ be as above. Let $\\rho$ be any adversary policy, s any state, and $0 < \\alpha < 1$ . Then at least one of the following holds: (1) $V_{R-max}>Opt(\\Pi_{M}(\\epsilon,T))-\\alpha$ , where $V_{R - m a x}$ denotes the expected T-step average reward of the $R^{M_{L}}$ -max policy on M; or (2) when R-MAX is run on M for T steps, an unknown entry is played with probability at least $\\frac{\\alpha}{R_{m a x}}$\n\nThe adversary's policy $\\rho$ cannot be known ahead of time in practice, so we cannot tell in advance whether R-MAX will obtain near-optimal reward or will reach an unknown entry with sufficient probability. What matters is that one of the two will happen, whatever the adversary does."}, {"name": "Lemma 4 (Simulation Lemma) (another result of the same paper)", "statement": "Lemma 4 Suppose $M$ and $\\bar{M}$ are SGs on $N$ states and that $\\bar{M}$ is an $\\frac{\\epsilon}{N T R_{m a x}}$ -approximation of $M$. Then, for any state $s$, any policy $\\pi$ of the agent and any policy $\\rho$ of the adversary,\n\n$$\n|U_{\\bar{M}}(s,\\pi,\\rho,T)-U_{M}(s,\\pi,\\rho,T)|\\le\\epsilon .\n$$"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace ResearchLevel\n\nstructure SG (N k : ℕ) where\n  P : Fin N → Fin k → Fin k → Fin N → ℝ\n  R : Fin N → Fin k → Fin k → ℝ\n\nvariable {N k T : ℕ}\n\nabbrev Traj (N k T : ℕ) := (Fin (T + 1) → Fin N) × (Fin T → Fin k) × (Fin T → Fin k)\n\nabbrev Policy (N k T : ℕ) :=\n  (i : Fin T) → (Fin (i.val + 1) → Fin N) → (Fin i.val → Fin k) →\n    (Fin i.val → Fin k) → (Fin i.val → ℝ) → Fin k → ℝ\n\ndef stateHist (w : Traj N k T) (i : Fin T) : Fin (i.val + 1) → Fin N :=\n  fun j => w.1 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef agentHist (w : Traj N k T) (i : Fin T) : Fin i.val → Fin k :=\n  fun j => w.2.1 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef advHist (w : Traj N k T) (i : Fin T) : Fin i.val → Fin k :=\n  fun j => w.2.2 (Fin.castLE (by have := i.isLt; omega) j)\n\ndef rewardHist (M : SG N k) (w : Traj N k T) (i : Fin T) : Fin i.val → ℝ :=\n  fun j => M.R (w.1 (Fin.castLE (by have := i.isLt; omega) j).castSucc)\n    (w.2.1 (Fin.castLE (by have := i.isLt; omega) j))\n    (w.2.2 (Fin.castLE (by have := i.isLt; omega) j))\n\nnoncomputable def pathProb (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) (w : Traj N k T) : ℝ :=\n  (if w.1 0 = s0 then (1 : ℝ) else 0) *\n    ∏ i : Fin T,\n      πa i (stateHist w i) (agentHist w i) (advHist w i) (rewardHist M w i) (w.2.1 i) *\n      πd i (stateHist w i) (agentHist w i) (advHist w i) (rewardHist M w i) (w.2.2 i) *\n      M.P (w.1 i.castSucc) (w.2.1 i) (w.2.2 i) (w.1 i.succ)\n\nnoncomputable def pathPayoff (M : SG N k) (w : Traj N k T) : ℝ :=\n  (1 / (T : ℝ)) * ∑ i : Fin T, M.R (w.1 i.castSucc) (w.2.1 i) (w.2.2 i)\n\nnoncomputable def U (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) : ℝ :=\n  ∑ w : Traj N k T, pathProb M πa πd s0 w * pathPayoff M w\n\ndef Approximates (Mbar M : SG N k) (α : ℝ) : Prop :=\n  (∀ s a a' t, |Mbar.P s a a' t - M.P s a a' t| ≤ α) ∧\n  (∀ s a a', Mbar.R s a a' = M.R s a a')\n\nabbrev UnknownSet (N k : ℕ) := Finset (Fin N × Fin k × Fin k)\n\ndef inducedSG (M : SG N k) (L : UnknownSet N k) (Rmax : ℝ) : SG (N + 1) k where\n  P := fun s a a' t =>\n    if hs : (s : ℕ) < N then\n      if (⟨s, hs⟩, a, a') ∈ L then (if t = Fin.last N then 1 else 0)\n      else if ht : (t : ℕ) < N then M.P ⟨s, hs⟩ a a' ⟨t, ht⟩ else 0\n    else if t = Fin.last N then 1 else 0\n  R := fun s a a' =>\n    if hs : (s : ℕ) < N then\n      if (⟨s, hs⟩, a, a') ∈ L then Rmax else M.R ⟨s, hs⟩ a a'\n    else Rmax\n\nnoncomputable def escapeProb (L : UnknownSet N k) (M : SG N k)\n    (πa πd : Policy N k T) (s0 : Fin N) : ℝ :=\n  ∑ w : Traj N k T,\n    (if ∃ i : Fin T, (w.1 i.castSucc, w.2.1 i, w.2.2 i) ∈ L then\n        pathProb M πa πd s0 w else 0)\n\ntheorem simulation_lemma\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax ε : ℝ) (hRmax : 0 < Rmax) (hε : 0 < ε)\n    (M Mbar : SG N k)\n\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n\n    (happrox : Approximates Mbar M (ε / ((N : ℝ) * (T : ℝ) * Rmax))) :\n    |U Mbar πa πd s0 - U M πa πd s0| ≤ ε := by\n  sorry\n\ntheorem path_dist_l1_bound\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (α : ℝ) (hα : 0 ≤ α)\n    (M Mbar : SG N k)\n    (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a')\n    (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hMbarP0 : ∀ s a a' t, 0 ≤ Mbar.P s a a' t)\n    (hMbarPsum : ∀ s a a', ∑ t, Mbar.P s a a' t = 1)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (s0 : Fin N)\n    (happrox : Approximates Mbar M α) :\n    ∑ w : Traj N k T,\n        |pathProb Mbar πa πd s0 w - pathProb M πa πd s0 w|\n      ≤ (N : ℝ) * (T : ℝ) * α := by\n  sorry\n\ntheorem value_diff_escape_bound\n    (hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (L : UnknownSet N k) (M : SG N k)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a') (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (πa' πd' : Policy (N + 1) k T)\n    (ha0' : ∀ i hs ha hd hr a, 0 ≤ πa' i hs ha hd hr a)\n    (hasum' : ∀ i hs ha hd hr, ∑ a, πa' i hs ha hd hr a = 1)\n    (hd0' : ∀ i hs ha hd hr a, 0 ≤ πd' i hs ha hd hr a)\n    (hdsum' : ∀ i hs ha hd hr, ∑ a, πd' i hs ha hd hr a = 1)\n    (hπa : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πa' i (fun j => (hs j).castSucc) ha hd hr a = πa i hs ha hd hr a)\n    (hπd : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πd' i (fun j => (hs j).castSucc) ha hd hr a = πd i hs ha hd hr a)\n    (s0 : Fin N) :\n    |U (inducedSG M L Rmax) πa' πd' (Fin.castSucc s0) - U M πa πd s0|\n      ≤ Rmax * escapeProb L M πa πd s0 := by\n  sorry\nend ResearchLevel\n", "target_names": ["simulation_lemma", "path_dist_l1_bound", "value_diff_escape_bound"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Reinforcement Learning", "holes": [{"start": 4317, "end": 5516, "gold": "(hN : 0 < N) (hk : 0 < k) (hT : 0 < T) (Rmax : ℝ) (hRmax : 0 < Rmax)\n    (L : UnknownSet N k) (M : SG N k)\n    (hMP0 : ∀ s a a' t, 0 ≤ M.P s a a' t)\n    (hMPsum : ∀ s a a', ∑ t, M.P s a a' t = 1)\n    (hR0 : ∀ s a a', 0 ≤ M.R s a a') (hRle : ∀ s a a', M.R s a a' ≤ Rmax)\n    (πa πd : Policy N k T)\n    (ha0 : ∀ i hs ha hd hr a, 0 ≤ πa i hs ha hd hr a)\n    (hasum : ∀ i hs ha hd hr, ∑ a, πa i hs ha hd hr a = 1)\n    (hd0 : ∀ i hs ha hd hr a, 0 ≤ πd i hs ha hd hr a)\n    (hdsum : ∀ i hs ha hd hr, ∑ a, πd i hs ha hd hr a = 1)\n    (πa' πd' : Policy (N + 1) k T)\n    (ha0' : ∀ i hs ha hd hr a, 0 ≤ πa' i hs ha hd hr a)\n    (hasum' : ∀ i hs ha hd hr, ∑ a, πa' i hs ha hd hr a = 1)\n    (hd0' : ∀ i hs ha hd hr a, 0 ≤ πd' i hs ha hd hr a)\n    (hdsum' : ∀ i hs ha hd hr, ∑ a, πd' i hs ha hd hr a = 1)\n    (hπa : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πa' i (fun j => (hs j).castSucc) ha hd hr a = πa i hs ha hd hr a)\n    (hπd : ∀ (i : Fin T) (hs : Fin (i.val + 1) → Fin N) ha hd hr a,\n      πd' i (fun j => (hs j).castSucc) ha hd hr a = πd i hs ha hd hr a)\n    (s0 : Fin N) :\n    |U (inducedSG M L Rmax) πa' πd' (Fin.castSucc s0) - U M πa πd s0|\n      ≤ Rmax * escapeProb L M πa πd s0"}]}
{"item_id": "v24-22-0747::prop4::A-stmt", "paper_id": "v24-22-0747", "target_name": "prop4", "source": "Research-level", "nl_statement": "Proposition 4 For any distribution D, there exist a majority vote $B_{Q}$ such that\n\n$$\nR_{D}(B_{Q})\\leq\\mathcal{C}_{Q}^{D}=\\underset{x\\sim D_{\\mathcal{X}}}{\\mathbf{E}}\\left(\\underset{y\\sim D_{\\mathcal{Y}|x}}{\\mathbf{Var}}(y)\\right).\n$$\n\nMoreover, there is no majority vote which can produce a lower C-bound.", "context": [{"name": "Setup: binary classification, voters, weighted majority vote, Bayes and Gibbs risk (Section 2, eq. 1)", "statement": "Throughout this paper, we employ the same notation and definitions used by Germain et al. (2015) unless specified otherwise. That is, we consider a binary classification problem in which each example $(x,y)$ consists of an input x and output y belonging to the spaces $\\mathcal{X}$ and $\\mathcal{Y}=\\{-1,1\\}$ respectively. The examples are sampled i.i.d. according to a true unknown distribution D over $\\mathcal{X}\\times\\{-1,1\\}$ . We consider $\\mathcal{H}$, a finite set of functions $f:\\mathcal{X}\\to [-1,1]$. Each such $f$ defines a \"voter\" in the context of majority vote rules. One may see the output of a given voter $f$ as a measurement of the confidence of the voter toward a given label. For example, say we have two voters $f_{1}(x)=0.1$ and $f_{2}(x)=0.9$ , then both voters lean towards the label 1, but the former less confidently then the latter.\n\nA weighted majority vote is defined by a distribution $Q$ over $\\mathcal{H}.$ Such a distribution defines the weight given to each voter, denoted $Q(f)$ , which gives rise to the majority vote classifier $B_{Q}(x)=\\mathrm{sgn}\\left[\\mathbf{E}_{f\\sim Q}f(x)\\right]$ . This majority vote is also called the Bayes classifier$^{1}$ in the PAC-Bayesian literature, as opposed to the stochastic Gibbs Classifier $G_{Q}$; with $\\begin{array}{r}{\\operatorname*{Pr}(G_{Q}(x)=y)=\\frac{1}{2}(1+y\\mathbf{E}_{f\\sim Q}f(x))}\\end{array}$ for $y\\in\\{-1,1\\}$ . The probability that the majority vote will make an incorrect prediction of the output for a given example drawn from distribution D is called the risk of the majority vote, or Bayes risk$^{2}$, and is denoted $R_{D}(B_{Q})$ :\n\n$$\nR_{D}(B_{Q})=\\operatorname*{Pr}_{(x,y)\\sim D}(B_{Q}(x)\\neq y)=\\underset{(x,y)\\sim D}{\\mathbf{E}}I\\left(\\underset{f\\sim Q}{\\mathbf{E}}yf(x)\\leq 0\\right),\n$$\n\nwhere $I(\\cdot)$ is the indicator function taking value 0 if the input is false and 1 otherwise. Correspondingly, the probability that the Gibbs classifier makes an error is called the Gibbs risk and is denoted $R_{D}(G_{Q})$ :\n\n$$\nR_{D}(G_{Q})=\\operatorname*{Pr}_{(x,y)\\sim D}(G_{Q}(x)\\neq y)=\\underset{(x,y)\\sim D}{\\mathbf{E}}\\left(\\frac{1}{2}-\\frac{1}{2}\\underset{f\\sim Q}{\\mathbf{E}}yf(x)\\right).\n$$"}, {"name": "Margin, its moments, and the C-bound (Section 2; Lacasse et al. 2006 / Germain et al. 2015)", "statement": "The term $\\mathbf{E}_{f\\sim Q}yf(x)$ shared by both Bayes risk and Gibbs risk definitions is a commonly occurring expression in the context of majority votes. It is referred to as the margin of the majority vote and is denoted $M_{Q}(x,y)$ . Given that $(x,y){\\sim}D$ , Germain et al. (2015) study the margin as a random variable which they denote $M_{Q}^{D}$ . A longstanding intermediate result in PAC-Bayes analyses is the factor two bound between the Bayes Risk and the Gibbs risk (Langford and Shawe-Taylor, 2002; McAllester, 2003). That is, for fixed distributions D and $Q$, we have $R_{D}(B_{Q})\\leq 2R_{D}(G_{Q})$ . The C-bound (Lacasse et al., 2006; Germain et al., 2015) is a finer upper bound on the Bayes risk and is defined using the variance and first two moments of the margin $M_{Q}^{D}$ (denoted $\\mu_{1}(M_{Q}^{D})$ and $\\mu_{2}(M_{Q}^{D})$ respectively, with $\\mu_{i}(M_{Q}^{D})={\\bf E}_{(x,y)\\sim D}[M_{Q}(x,y)]^{i})$ . Equivalently, the C-bound can be expressed in terms of the Gibbs risk $R_{D}(G_{Q})$ and the expected disagreement (denoted $d_{Q}^{D}$ and defined below):\n\n$$\nR_{D}(B_{Q})\\leq{\\mathcal C}_{Q}^{D}=\\frac{\\mathbf{Var}(M_{Q}^{D})}{\\mu_{2}(M_{Q}^{D})}=1-\\frac{\\left(\\mu_{1}(M_{Q}^{D})\\right)^{2}}{\\mu_{2}(M_{Q}^{D})}=1-\\frac{(1-2 R_{D}(G_{Q}))^{2}}{1-2 d_{Q}^{D}},\n$$\n\nwhere\n\n$$\n\\begin{array}{c c}{{\\displaystyle\\mu_{1}\\big(M_{Q}^{D}\\big)=\\underset{(x,y)\\sim D}{\\mathbf{E}}\\,\\underset{f\\sim Q}{\\mathbf{E}}y f(x),}}&{{\\qquad\\mu_{2}\\big(M_{Q}^{D}\\big)=\\underset{x\\sim D_{\\mathcal{X}}}{\\mathbf{E}}\\left[\\underset{f\\sim Q}{\\mathbf{E}}f(x)\\right]^{2},}}\\\\ {{\\displaystyle{\\mathbf{Var}\\big(M_{Q}^{D}\\big)=\\mu_{2}\\big(M_{Q}^{D}\\big)-\\big(\\mu_{1}\\big(M_{Q}^{D}\\big)\\big)^{2},}}}&{{\\qquad d_{Q}^{D}=\\frac{1}{2}\\left(1-\\mu_{2}\\big(M_{Q}^{D}\\big)\\right).}}\\end{array}\n$$\n\nIn these definitions, $D_{\\mathcal{X}}$ denotes the marginal distribution on $\\mathcal{X}$ for a given distribution $D .^{1}$ We adopt this notation throughout the paper, as well as the use of $D_{\\mathcal{Y}| x}$ to denote the conditional distribution on the output space $\\mathcal{Y}$ given a fixed input $x \\in \\mathcal{X}$"}, {"name": "Sources of stochasticity: label noise, stochastic voters, stochastic majority vote (Section 2)", "statement": "An important note has to be made about the notions of stochasticity and determinism as they appear in the original paper. The generic use of these terms can be confusing to the reader as it is sometimes unclear what is meant to be stochastic or deterministic in some passages. To alleviate this ambiguity we differentiate three distinct sources of stochasticity that are of importance in the studied framework.\n\nLabel noise. The stochasticity of the labels arises when for a given value of $x \\sim D_{\\mathcal{X}}$ , there is a non-zero probability of observing more than one value of $y \\sim D_{\\mathcal{Y}| x}$.\n\nStochastic voters. Of note, Germain et al. (2015) introduced voters as functions of the form $f:\\mathcal{X}\\to [-1,1]$, implying they were deterministic predictors. Nevertheless, the property that a given voter is deterministic remains unused in their analysis of majority votes. We may wish to consider stochastic voters, such as voters for which the output follows some distribution. Say we consider a stochastic voter $f^{*}$ such that for any given input $x,$ the output of the voter is a random variable $f^{*}(x)\\sim W_{x}$ , where $W_{x}$ is a distribution over [−1, 1] that depends of the value $x .$ To apply the work of Germain et al. (2015), one needs only to consider $f(x)$ as the expected output of the stochastic voter $f^{*}$ given by $f(x)=\\mathbf{E}_{f^{*}(x)\\sim W_{x}}~f^{*}(x)$ . Doing so, the value $f(x)$ will be deterministic. Moreover, this is coherent with the idea that $f(x)$ is the confidence of the voter towards a given label. The discussion in this erratum henceforth encompasses both deterministic and stochastic voters and we use the general term voter to refer to voters of both natures.\n\nStochastic majority vote. We specified that the Bayes classifier $B_{Q}$ is deterministic while the Gibbs classifier $G_{Q}$ is stochastic. This distinction, made by Germain et al. (2015), is meant to distinguish between the stochasticity that arises from sampling a single voter (Gibbs) as opposed to averaging over the value of each voter $f \\in \\mathcal H$ (Bayes). It is not meant to refer to the nature of the individual voters as we remind that both Germain et al. (2015) and this erratum consider deterministic voters only.\n\nWe remind that this erratum is valid when considering the deterministic majority vote classifier that embed the expected value of its voters for the Bayes risk of Equation (1). The interested reader may note that the recent work of Zantedeschi et al. (2021) provide PAC-Bayes bounds for stochastic majority votes where the voters weights are random variables. While this is a related line of work, it differs from ours and Germain et al. (2015) as we only consider majority votes for which the weight of each voter is fixed."}, {"name": "Section 6 introduction (goal of Proposition 4)", "statement": "The counter-example given in Section 4 was based on the knowledge that the risk of the majority vote should never be inferior to minimal risk induced by the label noise in D. We have shown in Section 5 that the C-bound can converge to the minimal risk in a setting deprived of label noise by proving that in such a setting, using voters of sufficient quality renders Corrolary 2 true. In this section, we aim to generalise this result to problems which contain label noise. To do so, we introduce in Proposition 4 a theoretical lower-bound for the C-bound for a given distribution D as well as the majority vote which achieves this bound."}], "nl_notes": "### Proposition 4\nNOTE (Proposition 4) — The C-bound 1 − μ₁²/μ₂ is defined only when μ₂ > 0, so the Lean requires 0 < μ₂(S) in both parts: for the majority vote that attains E[Var(y|x)], and for every vote in the lower bound. When h = 0 almost everywhere (pure label noise), the paper's choice of vote S = h has μ₂ = 0 and cannot serve as the witness.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ResearchLevel\n\nnoncomputable def cMu1 {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  ∫ x, h x * S x ∂μ\n\nnoncomputable def cMu2 {X : Type*} [MeasurableSpace X] (μ : Measure X) (S : X → ℝ) : ℝ :=\n  ∫ x, (S x) ^ 2 ∂μ\n\nnoncomputable def cBound {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  1 - (cMu1 μ h S) ^ 2 / (cMu2 μ S)\n\nnoncomputable def labelVarMean {X : Type*} [MeasurableSpace X] (μ : Measure X) (h : X → ℝ) : ℝ :=\n  ∫ x, (1 - (h x) ^ 2) ∂μ\n\nnoncomputable def condErr (hx sx : ℝ) : ℝ :=\n  if 0 < sx then (1 - hx) / 2 else if sx < 0 then (1 + hx) / 2 else 1\n\nnoncomputable def riskD {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  ∫ x, condErr (h x) (S x) ∂μ\n\nnoncomputable def margin {Ω X ι : Type*} (Q : ι → ℝ) (f : ι → X → ℝ)\n    (xx : Ω → X) (yy : Ω → ℝ) : Ω → ℝ :=\n  fun ω => yy ω * ∑' i, Q i * f i (xx ω)\n\nnoncomputable def aggVote {Ω : Type*} (n : ℕ) (V : Fin n → Ω → ℝ) : Ω → ℝ :=\n  fun ω => (n : ℝ)⁻¹ * ∑ i, V i ω\n\ntheorem prop4 {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]\n    (h : X → ℝ) (hh_meas : Measurable h) (hh_bdd : ∀ x, |h x| ≤ 1) :\n    (∃ S : X → ℝ, Measurable S ∧ (∀ x, |S x| ≤ 1) ∧ 0 < cMu2 μ S ∧\n        riskD μ h S ≤ cBound μ h S ∧\n        cBound μ h S = labelVarMean μ h) ∧\n    (∀ S : X → ℝ, Measurable S → (∀ x, |S x| ≤ 1) → 0 < cMu2 μ S →\n        labelVarMean μ h ≤ cBound μ h S) := by\n  sorry\n\ntheorem prop3 {Ω X ι : Type*} [MeasurableSpace Ω] [MeasurableSpace X] [Countable ι]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (Q : ι → ℝ) (hQ_nonneg : ∀ i, 0 ≤ Q i) (hQ_summable : Summable Q) (hQ_sum : ∑' i, Q i = 1)\n    (f : ι → X → ℝ) (hf_bdd : ∀ i x, |f i x| ≤ 1)\n    (xx : Ω → X) (yy : Ω → ℝ) (hy : ∀ᵐ ω ∂P, yy ω = 1 ∨ yy ω = -1)\n    (g : X → ℝ) (hdet : ∀ᵐ ω ∂P, yy ω = g (xx ω))\n    (hacc : ∀ i, ∀ᵐ ω ∂P, 0 < yy ω * f i (xx ω))\n    (hmeas_x : Measurable xx) (hmeas_y : Measurable yy)\n    (hmeas_f : ∀ i, Measurable (f i)) :\n    Var[margin Q f xx yy; P]\n      ≤ (∑' i, (Q i) ^ 2)\n        + ∑' i, ∑' j : {j // j ≠ i},\n            Q i * Q (j : ι) * cov[fun ω => f i (xx ω), fun ω => f (j : ι) (xx ω); P] := by\n  sorry\n\ntheorem cbound_counterexample {Ω : Type*} [MeasurableSpace Ω]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (n : ℕ) (hn : 0 < n) (V : Fin n → Ω → ℝ) (Y : Ω → ℝ)\n    (hVint : ∀ i, Integrable (V i) P)\n    (hVcrossint : ∀ i j, Integrable (fun ω => V i ω * V j ω) P)\n    (hYint : Integrable Y P)\n    (hYSint : Integrable (fun ω => Y ω * aggVote n V ω) P)\n    (hV1 : ∀ i, ∫ ω, V i ω ∂P = 3 / 5)\n    (hV2 : ∀ i, ∫ ω, (V i ω) ^ 2 ∂P = 1)\n    (hVuncorr : ∀ i j, i ≠ j → ∫ ω, V i ω * V j ω ∂P = 9 / 25)\n    (hY1 : ∫ ω, Y ω ∂P = 3 / 5)\n    (hYsq : ∀ ω, (Y ω) ^ 2 = 1)\n    (hYindep : ∫ ω, Y ω * aggVote n V ω ∂P\n        = (∫ ω, Y ω ∂P) * (∫ ω, aggVote n V ω ∂P))\n    (X : Type*) [MeasurableSpace X] (xx : Ω → X) (v : Fin n → X → ℝ)\n    (hVroute : ∀ i ω, V i ω = v i (xx ω))\n    (hv_pm : ∀ i x, v i x = 1 ∨ v i x = -1)\n    (hv_meas : ∀ i, Measurable (v i))\n    (hVindep : ProbabilityTheory.iIndepFun V P)\n    (hYXindep : ProbabilityTheory.IndepFun Y xx P) :\n    cBound P Y (aggVote n V) = (400 + 144 * (n : ℝ)) / (400 + 225 * (n : ℝ))\n      ∧ (16 : ℝ) / 25 < cBound P Y (aggVote n V) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["prop4", "prop3", "cbound_counterexample"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1077, "end": 1473, "gold": "{X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]\n    (h : X → ℝ) (hh_meas : Measurable h) (hh_bdd : ∀ x, |h x| ≤ 1) :\n    (∃ S : X → ℝ, Measurable S ∧ (∀ x, |S x| ≤ 1) ∧ 0 < cMu2 μ S ∧\n        riskD μ h S ≤ cBound μ h S ∧\n        cBound μ h S = labelVarMean μ h) ∧\n    (∀ S : X → ℝ, Measurable S → (∀ x, |S x| ≤ 1) → 0 < cMu2 μ S →\n        labelVarMean μ h ≤ cBound μ h S)"}]}
{"item_id": "v24-22-0747::prop3::A-stmt", "paper_id": "v24-22-0747", "target_name": "prop3", "source": "Research-level", "nl_statement": "Proposition 3 (Proposition 15 in Germain et al. (2015) corrected) For any countable set of voters $\\mathcal{H}$, any distribution Q on ${\\mathcal{H}},$ and any distribution D on $\\mathcal{X}\\times\\{-1,1\\}$ such that there exists a function $g:\\mathcal{X}\\to\\{-1,1\\}$ for which $y=g(x)$ and $yf(x)>0$ for all $(x,y)\\sim D$ and $f \\in \\mathcal H$ , we have\n\n$$\n\\mathbf{Var}(M_{Q}^{D})\\leq\\sum_{f\\in\\mathcal{H}}Q^{2}(f)+\\sum_{f_{1}\\in\\mathcal{H}}\\sum_{f_{2}\\in\\mathcal{H}\\atop f_{2}\\neq f_{1}}Q(f_{1})Q(f_{2})\\mathbf{Cov}_{(x,y)\\sim D}(f_{1}(x),f_{2}(x)).\n$$", "context": [{"name": "Setup: binary classification, voters, weighted majority vote, Bayes and Gibbs risk (Section 2, eq. 1)", "statement": "Throughout this paper, we employ the same notation and definitions used by Germain et al. (2015) unless specified otherwise. That is, we consider a binary classification problem in which each example $(x,y)$ consists of an input x and output y belonging to the spaces $\\mathcal{X}$ and $\\mathcal{Y}=\\{-1,1\\}$ respectively. The examples are sampled i.i.d. according to a true unknown distribution D over $\\mathcal{X}\\times\\{-1,1\\}$ . We consider $\\mathcal{H}$, a finite set of functions $f:\\mathcal{X}\\to [-1,1]$. Each such $f$ defines a \"voter\" in the context of majority vote rules. One may see the output of a given voter $f$ as a measurement of the confidence of the voter toward a given label. For example, say we have two voters $f_{1}(x)=0.1$ and $f_{2}(x)=0.9$ , then both voters lean towards the label 1, but the former less confidently then the latter.\n\nA weighted majority vote is defined by a distribution $Q$ over $\\mathcal{H}.$ Such a distribution defines the weight given to each voter, denoted $Q(f)$ , which gives rise to the majority vote classifier $B_{Q}(x)=\\mathrm{sgn}\\left[\\mathbf{E}_{f\\sim Q}f(x)\\right]$ . This majority vote is also called the Bayes classifier$^{1}$ in the PAC-Bayesian literature, as opposed to the stochastic Gibbs Classifier $G_{Q}$; with $\\begin{array}{r}{\\operatorname*{Pr}(G_{Q}(x)=y)=\\frac{1}{2}(1+y\\mathbf{E}_{f\\sim Q}f(x))}\\end{array}$ for $y\\in\\{-1,1\\}$ . The probability that the majority vote will make an incorrect prediction of the output for a given example drawn from distribution D is called the risk of the majority vote, or Bayes risk$^{2}$, and is denoted $R_{D}(B_{Q})$ :\n\n$$\nR_{D}(B_{Q})=\\operatorname*{Pr}_{(x,y)\\sim D}(B_{Q}(x)\\neq y)=\\underset{(x,y)\\sim D}{\\mathbf{E}}I\\left(\\underset{f\\sim Q}{\\mathbf{E}}yf(x)\\leq 0\\right),\n$$\n\nwhere $I(\\cdot)$ is the indicator function taking value 0 if the input is false and 1 otherwise. Correspondingly, the probability that the Gibbs classifier makes an error is called the Gibbs risk and is denoted $R_{D}(G_{Q})$ :\n\n$$\nR_{D}(G_{Q})=\\operatorname*{Pr}_{(x,y)\\sim D}(G_{Q}(x)\\neq y)=\\underset{(x,y)\\sim D}{\\mathbf{E}}\\left(\\frac{1}{2}-\\frac{1}{2}\\underset{f\\sim Q}{\\mathbf{E}}yf(x)\\right).\n$$"}, {"name": "Margin, its moments, and the C-bound (Section 2; Lacasse et al. 2006 / Germain et al. 2015)", "statement": "The term $\\mathbf{E}_{f\\sim Q}yf(x)$ shared by both Bayes risk and Gibbs risk definitions is a commonly occurring expression in the context of majority votes. It is referred to as the margin of the majority vote and is denoted $M_{Q}(x,y)$ . Given that $(x,y){\\sim}D$ , Germain et al. (2015) study the margin as a random variable which they denote $M_{Q}^{D}$ . A longstanding intermediate result in PAC-Bayes analyses is the factor two bound between the Bayes Risk and the Gibbs risk (Langford and Shawe-Taylor, 2002; McAllester, 2003). That is, for fixed distributions D and $Q$, we have $R_{D}(B_{Q})\\leq 2R_{D}(G_{Q})$ . The C-bound (Lacasse et al., 2006; Germain et al., 2015) is a finer upper bound on the Bayes risk and is defined using the variance and first two moments of the margin $M_{Q}^{D}$ (denoted $\\mu_{1}(M_{Q}^{D})$ and $\\mu_{2}(M_{Q}^{D})$ respectively, with $\\mu_{i}(M_{Q}^{D})={\\bf E}_{(x,y)\\sim D}[M_{Q}(x,y)]^{i})$ . Equivalently, the C-bound can be expressed in terms of the Gibbs risk $R_{D}(G_{Q})$ and the expected disagreement (denoted $d_{Q}^{D}$ and defined below):\n\n$$\nR_{D}(B_{Q})\\leq{\\mathcal C}_{Q}^{D}=\\frac{\\mathbf{Var}(M_{Q}^{D})}{\\mu_{2}(M_{Q}^{D})}=1-\\frac{\\left(\\mu_{1}(M_{Q}^{D})\\right)^{2}}{\\mu_{2}(M_{Q}^{D})}=1-\\frac{(1-2 R_{D}(G_{Q}))^{2}}{1-2 d_{Q}^{D}},\n$$\n\nwhere\n\n$$\n\\begin{array}{c c}{{\\displaystyle\\mu_{1}\\big(M_{Q}^{D}\\big)=\\underset{(x,y)\\sim D}{\\mathbf{E}}\\,\\underset{f\\sim Q}{\\mathbf{E}}y f(x),}}&{{\\qquad\\mu_{2}\\big(M_{Q}^{D}\\big)=\\underset{x\\sim D_{\\mathcal{X}}}{\\mathbf{E}}\\left[\\underset{f\\sim Q}{\\mathbf{E}}f(x)\\right]^{2},}}\\\\ {{\\displaystyle{\\mathbf{Var}\\big(M_{Q}^{D}\\big)=\\mu_{2}\\big(M_{Q}^{D}\\big)-\\big(\\mu_{1}\\big(M_{Q}^{D}\\big)\\big)^{2},}}}&{{\\qquad d_{Q}^{D}=\\frac{1}{2}\\left(1-\\mu_{2}\\big(M_{Q}^{D}\\big)\\right).}}\\end{array}\n$$\n\nIn these definitions, $D_{\\mathcal{X}}$ denotes the marginal distribution on $\\mathcal{X}$ for a given distribution $D .^{1}$ We adopt this notation throughout the paper, as well as the use of $D_{\\mathcal{Y}| x}$ to denote the conditional distribution on the output space $\\mathcal{Y}$ given a fixed input $x \\in \\mathcal{X}$"}, {"name": "Sources of stochasticity: label noise, stochastic voters, stochastic majority vote (Section 2)", "statement": "An important note has to be made about the notions of stochasticity and determinism as they appear in the original paper. The generic use of these terms can be confusing to the reader as it is sometimes unclear what is meant to be stochastic or deterministic in some passages. To alleviate this ambiguity we differentiate three distinct sources of stochasticity that are of importance in the studied framework.\n\nLabel noise. The stochasticity of the labels arises when for a given value of $x \\sim D_{\\mathcal{X}}$ , there is a non-zero probability of observing more than one value of $y \\sim D_{\\mathcal{Y}| x}$.\n\nStochastic voters. Of note, Germain et al. (2015) introduced voters as functions of the form $f:\\mathcal{X}\\to [-1,1]$, implying they were deterministic predictors. Nevertheless, the property that a given voter is deterministic remains unused in their analysis of majority votes. We may wish to consider stochastic voters, such as voters for which the output follows some distribution. Say we consider a stochastic voter $f^{*}$ such that for any given input $x,$ the output of the voter is a random variable $f^{*}(x)\\sim W_{x}$ , where $W_{x}$ is a distribution over [−1, 1] that depends of the value $x .$ To apply the work of Germain et al. (2015), one needs only to consider $f(x)$ as the expected output of the stochastic voter $f^{*}$ given by $f(x)=\\mathbf{E}_{f^{*}(x)\\sim W_{x}}~f^{*}(x)$ . Doing so, the value $f(x)$ will be deterministic. Moreover, this is coherent with the idea that $f(x)$ is the confidence of the voter towards a given label. The discussion in this erratum henceforth encompasses both deterministic and stochastic voters and we use the general term voter to refer to voters of both natures.\n\nStochastic majority vote. We specified that the Bayes classifier $B_{Q}$ is deterministic while the Gibbs classifier $G_{Q}$ is stochastic. This distinction, made by Germain et al. (2015), is meant to distinguish between the stochasticity that arises from sampling a single voter (Gibbs) as opposed to averaging over the value of each voter $f \\in \\mathcal H$ (Bayes). It is not meant to refer to the nature of the individual voters as we remind that both Germain et al. (2015) and this erratum consider deterministic voters only.\n\nWe remind that this erratum is valid when considering the deterministic majority vote classifier that embed the expected value of its voters for the Bayes risk of Equation (1). The interested reader may note that the recent work of Zantedeschi et al. (2021) provide PAC-Bayes bounds for stochastic majority votes where the voters weights are random variables. While this is a related line of work, it differs from ours and Germain et al. (2015) as we only consider majority votes for which the weight of each voter is fixed."}, {"name": "Proposition 1 (Proposition 15 in Germain et al. 2015; erroneous)", "statement": "## Erroneous result\n\nProposition 1 (Proposition 15 in Germain et al. (2015)) For any countable set of voters $\\mathcal{H}$, any distribution Q on $\\mathcal{H}$, and any distribution D on $\\mathcal{X}\\times\\{-1,1\\}$ , we have\n\n$$\n\\mathbf{Var}(M_{Q}^{D})\\leq\\sum_{f\\in\\mathcal{H}}Q^{2}(f)+\\sum_{f_{1}\\in\\mathcal{H}}\\sum_{f_{2}\\in\\mathcal{H}\\atop f_{2}\\neq f_{1}}Q(f_{1})Q(f_{2})\\mathbf{Cov}_{(x,y)\\sim D}(f_{1}(x),f_{2}(x)).\n$$"}, {"name": "Original proof of Proposition 1 (Germain et al. 2015) and its flaw", "statement": "Proof We begin with the definition of the margin, which can be seen as a sum of random variables, and develop the variance of the sum into the sum of covariances:\n\n$$\n\\mathbf{Var}\\left(M_{Q}^{D}\\right)=\\mathbf{Var}_{(x,y)\\sim D}\\left(M_{Q}(x,y)\\right)\n$$\n\n$$\n=\\underset{(x,y)\\sim D}{\\mathbf{Var}}\\left(\\sum_{f\\in\\mathcal{H}}Q(f)y f(x)\\right)\n$$\n\n$$\n\\begin{array}{r l}&{=\\displaystyle\\sum_{f\\in\\mathcal{H}}Q^{2}(f)\\underset{(x,y)\\sim D}{\\mathbf{Var}}y f(x)+\\displaystyle\\sum_{f_{1}\\in\\mathcal{H}}\\displaystyle\\sum_{f_{2}\\in\\mathcal{H}\\atop f_{2}\\neq f_{1}}Q(f_{1})Q(f_{2})\\underset{(x,y)\\sim D}{\\mathbf{Cov}}(y f_{1}(x),y f_{2}(x))}\\\\ &{=\\displaystyle\\sum_{f\\in\\mathcal{H}}Q^{2}(f)\\underset{(x,y)\\sim D}{\\mathbf{Var}}y f(x)+\\displaystyle\\sum_{f_{1}\\in\\mathcal{H}}\\displaystyle\\sum_{f_{2}\\in\\mathcal{H}\\atop f_{2}\\neq f_{1}}Q(f_{1})Q(f_{2})\\underset{(x,y)\\sim D}{\\mathbf{Cov}}(f_{1}(x),f_{2}(x))}\\\\ &{\\leq\\displaystyle\\sum_{f\\in\\mathcal{H}}Q^{2}(f)+\\displaystyle\\sum_{f_{1}\\in\\mathcal{H}}\\displaystyle\\sum_{f_{2}\\in\\mathcal{H}\\atop f_{2}\\neq f_{1}}Q(f_{1})Q(f_{2})\\underset{(x,y)\\sim D}{\\mathbf{Cov}}(f_{1}(x),f_{2}(x)).}\\end{array}\n$$\n\nThe inequality is given by the fact that the definitions of $y$ and $f(x)$ implie $yf(x)\\in [-1,1]$, from which we infer that $\\underset{(x,y)\\sim D}{\\mathbf{Var}}{y f(x)}\\leq 1$ for all voters $f \\in{\\mathcal{H}}$.\n\nThe problem in this demonstration arises when we simplify $\\mathbf{Cov}_{(x,y)\\sim D}\\left(y f_{1}(x),y f_{2}(x)\\right)$ to $\\mathbf{Cov}_{(x,y)\\sim D}\\left(f_{1}(x),f_{2}(x)\\right)$ , which could also be written as $\\mathbf{Cov}_{x\\sim D_{\\mathcal{X}}}\\left(f_{1}(x),f_{2}(x)\\right)$ . As this final writing is independent of y, this equality would imply that the value y has no effect in the expression $\\mathbf{Cov}_{(x,y)\\sim D}\\left(y f_{1}(x),y f_{2}(x)\\right)$ . This simplification does not stand in the general case as it is possible to find various examples of functions $f_{i}$ and distributions D that do not satisfy this equality. As an example, consider $f_{1}(x)=1,f_{2}(x)= -1$ and $\\operatorname*{Pr}_{y\\sim D_{\\mathcal{Y}|x}}(y{=}1)=\\operatorname*{Pr}_{y\\sim D_{\\mathcal{Y}|x}}(y{=}-1)=\\textstyle\\frac{1}{2}$ for any given value $x \\in \\mathcal{X}$ . Using these values, we get the following, which disproves the equality assumed in the said simplification:\n\n$$\n\\begin{array}{r}{\\mathbf{Cov}(f_{1}(x),f_{2}(x))=\\mathbf{Cov}(1,-1)=0\\qquad}\\\\ {\\mathbf{Cov}(y f_{1}(x),y f_{2}(x))=\\mathbf{Cov}(y,-y)= -\\mathbf{Cov}(y,y)= -1 .}\\end{array}\n$$"}, {"name": "Assumptions 1-2: deterministic labels and sufficiently accurate voters (Section 5)", "statement": "We have now shown that Proposition 1 is incorrect as stated in the general case, namely when the labelling task includes label noise. The aim of the current section is to rectify Proposition 1 in a setting devoid of label noise, by introducing new assumptions on the data distribution and the voters. In the following section we build upon this knowledge a new, correct and improved result that stands in the general case.\n\nEven though the assumption that a given problem is devoid of label noise might not be realistic in most applications, we can see this case as an easy classification problem where an error of 0 should be achievable for some classifier. Given this assumption, we would thus desire a majority vote made with \"good enough\" voters to come arbitrarily close to null classification error. Moreover, we would expect a good bound on the risk of majority votes to reflect this fact by being potentially arbitrarily small in this setting.\n\nWe begin by noting that the assumption that distribution D is devoid of label noise implies that there exists a function that maps each input x to its correct label.\n\nAssumption 1: Deterministic labels. The data distribution D is such that there exists a function $g:\\mathcal{X}\\to\\{-1,1\\}$ for which $y=g(x)$ for all $(x,y)\\sim D$ , or equivalently $\\operatorname*{Pr}_{y\\sim D_{\\mathcal{Y}|x}}\\left(y{=}g(x)\\right)=1$ for all $x \\in \\mathcal{X}$\n\nThe counter-example of the previous section also illustrates that uncorrelated voters do not necessarily result in an accurate majority vote, as it does not imply that the input-output correlation is taken into account. Thus, we need a second assumption, capturing the fact that the majority vote must be made of \"good enough\" voters. Namely, we want each voter to classify this input correctly with a probability larger than $1 / 2$\n\nAssumption 2: Sufficiently accurate voters. For any given pair $(x,y)\\sim D$ , for any voter $f_{i}\\in\\mathcal{H},yf_{i}(x)>0;$ in other words, all voters are leaning towards the right label (although with scores, i.e., varying confidence levels) in a non ambivalent manner (i.e., $f_{i}(x)\\neq 0)$\n\nGiven the definition of a voter $f_{i}$, Assumption 2 implies that for any given input $x$ in $\\mathcal{X}$ and any given voter $f_{i}$ in $\\mathcal{H}$ , the expected output of $f_{i}(x)$ will agree with the true value of $y,$ meaning they will share the same sign. This assumption is very strong, as it will break when a single voter does not classify a given x with sufficient probability. It may seem trivial that under this assumption the majority vote will be correct every time and will thus have an actual risk of 0. Recall however that the majority vote risk $R_{D}(B_{Q})$ only considers the sign of the margin $[y\\,\\mathbf{E}_{f\\sim Q}f(x)]$ on $(x,y)\\sim D$ while the computation of the C-bound depends on the value of the margin itself. In other words, the risk $R_{D}(B_{Q})$ depends on which label the majority vote is most confident in but the C-bound further considers how confident the vote is. This implies that various majority votes which make the same prediction (same risk $R_{D}(B_{Q}))$ may have different C-bound as they may have different margins. Therefore, the C-bound is not trivially 0 under the above assumptions.\n\nWe now show that an updated Proposition 1 holds given these two new assumptions, namely that the labels are deterministic and the voters are sufficiently accurate."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ResearchLevel\n\nnoncomputable def cMu1 {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  ∫ x, h x * S x ∂μ\n\nnoncomputable def cMu2 {X : Type*} [MeasurableSpace X] (μ : Measure X) (S : X → ℝ) : ℝ :=\n  ∫ x, (S x) ^ 2 ∂μ\n\nnoncomputable def cBound {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  1 - (cMu1 μ h S) ^ 2 / (cMu2 μ S)\n\nnoncomputable def labelVarMean {X : Type*} [MeasurableSpace X] (μ : Measure X) (h : X → ℝ) : ℝ :=\n  ∫ x, (1 - (h x) ^ 2) ∂μ\n\nnoncomputable def condErr (hx sx : ℝ) : ℝ :=\n  if 0 < sx then (1 - hx) / 2 else if sx < 0 then (1 + hx) / 2 else 1\n\nnoncomputable def riskD {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  ∫ x, condErr (h x) (S x) ∂μ\n\nnoncomputable def margin {Ω X ι : Type*} (Q : ι → ℝ) (f : ι → X → ℝ)\n    (xx : Ω → X) (yy : Ω → ℝ) : Ω → ℝ :=\n  fun ω => yy ω * ∑' i, Q i * f i (xx ω)\n\nnoncomputable def aggVote {Ω : Type*} (n : ℕ) (V : Fin n → Ω → ℝ) : Ω → ℝ :=\n  fun ω => (n : ℝ)⁻¹ * ∑ i, V i ω\n\ntheorem prop4 {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]\n    (h : X → ℝ) (hh_meas : Measurable h) (hh_bdd : ∀ x, |h x| ≤ 1) :\n    (∃ S : X → ℝ, Measurable S ∧ (∀ x, |S x| ≤ 1) ∧ 0 < cMu2 μ S ∧\n        riskD μ h S ≤ cBound μ h S ∧\n        cBound μ h S = labelVarMean μ h) ∧\n    (∀ S : X → ℝ, Measurable S → (∀ x, |S x| ≤ 1) → 0 < cMu2 μ S →\n        labelVarMean μ h ≤ cBound μ h S) := by\n  sorry\n\ntheorem prop3 {Ω X ι : Type*} [MeasurableSpace Ω] [MeasurableSpace X] [Countable ι]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (Q : ι → ℝ) (hQ_nonneg : ∀ i, 0 ≤ Q i) (hQ_summable : Summable Q) (hQ_sum : ∑' i, Q i = 1)\n    (f : ι → X → ℝ) (hf_bdd : ∀ i x, |f i x| ≤ 1)\n    (xx : Ω → X) (yy : Ω → ℝ) (hy : ∀ᵐ ω ∂P, yy ω = 1 ∨ yy ω = -1)\n    (g : X → ℝ) (hdet : ∀ᵐ ω ∂P, yy ω = g (xx ω))\n    (hacc : ∀ i, ∀ᵐ ω ∂P, 0 < yy ω * f i (xx ω))\n    (hmeas_x : Measurable xx) (hmeas_y : Measurable yy)\n    (hmeas_f : ∀ i, Measurable (f i)) :\n    Var[margin Q f xx yy; P]\n      ≤ (∑' i, (Q i) ^ 2)\n        + ∑' i, ∑' j : {j // j ≠ i},\n            Q i * Q (j : ι) * cov[fun ω => f i (xx ω), fun ω => f (j : ι) (xx ω); P] := by\n  sorry\n\ntheorem cbound_counterexample {Ω : Type*} [MeasurableSpace Ω]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (n : ℕ) (hn : 0 < n) (V : Fin n → Ω → ℝ) (Y : Ω → ℝ)\n    (hVint : ∀ i, Integrable (V i) P)\n    (hVcrossint : ∀ i j, Integrable (fun ω => V i ω * V j ω) P)\n    (hYint : Integrable Y P)\n    (hYSint : Integrable (fun ω => Y ω * aggVote n V ω) P)\n    (hV1 : ∀ i, ∫ ω, V i ω ∂P = 3 / 5)\n    (hV2 : ∀ i, ∫ ω, (V i ω) ^ 2 ∂P = 1)\n    (hVuncorr : ∀ i j, i ≠ j → ∫ ω, V i ω * V j ω ∂P = 9 / 25)\n    (hY1 : ∫ ω, Y ω ∂P = 3 / 5)\n    (hYsq : ∀ ω, (Y ω) ^ 2 = 1)\n    (hYindep : ∫ ω, Y ω * aggVote n V ω ∂P\n        = (∫ ω, Y ω ∂P) * (∫ ω, aggVote n V ω ∂P))\n    (X : Type*) [MeasurableSpace X] (xx : Ω → X) (v : Fin n → X → ℝ)\n    (hVroute : ∀ i ω, V i ω = v i (xx ω))\n    (hv_pm : ∀ i x, v i x = 1 ∨ v i x = -1)\n    (hv_meas : ∀ i, Measurable (v i))\n    (hVindep : ProbabilityTheory.iIndepFun V P)\n    (hYXindep : ProbabilityTheory.IndepFun Y xx P) :\n    cBound P Y (aggVote n V) = (400 + 144 * (n : ℝ)) / (400 + 225 * (n : ℝ))\n      ∧ (16 : ℝ) / 25 < cBound P Y (aggVote n V) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["prop4", "prop3", "cbound_counterexample"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 1503, "end": 2201, "gold": "{Ω X ι : Type*} [MeasurableSpace Ω] [MeasurableSpace X] [Countable ι]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (Q : ι → ℝ) (hQ_nonneg : ∀ i, 0 ≤ Q i) (hQ_summable : Summable Q) (hQ_sum : ∑' i, Q i = 1)\n    (f : ι → X → ℝ) (hf_bdd : ∀ i x, |f i x| ≤ 1)\n    (xx : Ω → X) (yy : Ω → ℝ) (hy : ∀ᵐ ω ∂P, yy ω = 1 ∨ yy ω = -1)\n    (g : X → ℝ) (hdet : ∀ᵐ ω ∂P, yy ω = g (xx ω))\n    (hacc : ∀ i, ∀ᵐ ω ∂P, 0 < yy ω * f i (xx ω))\n    (hmeas_x : Measurable xx) (hmeas_y : Measurable yy)\n    (hmeas_f : ∀ i, Measurable (f i)) :\n    Var[margin Q f xx yy; P]\n      ≤ (∑' i, (Q i) ^ 2)\n        + ∑' i, ∑' j : {j // j ≠ i},\n            Q i * Q (j : ι) * cov[fun ω => f i (xx ω), fun ω => f (j : ι) (xx ω); P]"}]}
{"item_id": "v24-22-0747::cbound_counterexample::A-stmt", "paper_id": "v24-22-0747", "target_name": "cbound_counterexample", "source": "Research-level", "nl_statement": "Counter-example falsifying Corollary 2 (Corollary 16 in Germain et al. 2015). Corollary 2 claimed that for $n$ independent voters under a uniform distribution $Q$, the C-bound satisfies $R_D(B_Q) \\le C_Q^D \\le 1/(n(1-2 d_Q^D)) = 1/(n(1-2 R_D(G_Q)))$, and thus decreases asymptotically to 0 as $n$ grows. The erratum disproves this with a concrete construction (Section 4): the label $y$ is independent of the input $x$ with $\\Pr(y=1)=0.8$ (so $\\mathbb{E}[y]=3/5$), and there are $n$ mutually independent deterministic voters $f_1,\\dots,f_n$ each with $\\Pr(f_i=1)=0.8$ (so $\\mathbb{E}[f_i]=3/5$, $\\mathbb{E}[f_i^2]=1$, and $\\mathbb{E}[f_i f_j]=\\mathbb{E}[f_i]\\mathbb{E}[f_j]=9/25$ for $i \\neq j$), each given uniform weight $Q(f_i)=1/n$, giving the aggregated score $S(x)=(1/n) \\sum_i f_i(x)$. For this majority vote the first two margin moments are $\\mu_1 = \\mathbb{E}[y \\cdot S] = (3/5)(3/5) = 9/25$ and $\\mu_2 = \\mathbb{E}[S^2] = (16+9n)/(25n)$, so the C-bound equals $C_Q^D = 1 - \\mu_1^2/\\mu_2 = (400 + 144 n)/(400 + 225 n)$. In particular $C_Q^D > 0.64 = 16/25$ for every $n > 0$; hence the C-bound does not tend to 0 as the number of independent voters grows, contradicting Corollary 2 (and therefore Proposition 1).", "context": [{"name": "Setup: binary classification, voters, weighted majority vote, Bayes and Gibbs risk (Section 2, eq. 1)", "statement": "Throughout this paper, we employ the same notation and definitions used by Germain et al. (2015) unless specified otherwise. That is, we consider a binary classification problem in which each example $(x,y)$ consists of an input x and output y belonging to the spaces $\\mathcal{X}$ and $\\mathcal{Y}=\\{-1,1\\}$ respectively. The examples are sampled i.i.d. according to a true unknown distribution D over $\\mathcal{X}\\times\\{-1,1\\}$ . We consider $\\mathcal{H}$, a finite set of functions $f:\\mathcal{X}\\to [-1,1]$. Each such $f$ defines a \"voter\" in the context of majority vote rules. One may see the output of a given voter $f$ as a measurement of the confidence of the voter toward a given label. For example, say we have two voters $f_{1}(x)=0.1$ and $f_{2}(x)=0.9$ , then both voters lean towards the label 1, but the former less confidently then the latter.\n\nA weighted majority vote is defined by a distribution $Q$ over $\\mathcal{H}.$ Such a distribution defines the weight given to each voter, denoted $Q(f)$ , which gives rise to the majority vote classifier $B_{Q}(x)=\\mathrm{sgn}\\left[\\mathbf{E}_{f\\sim Q}f(x)\\right]$ . This majority vote is also called the Bayes classifier$^{1}$ in the PAC-Bayesian literature, as opposed to the stochastic Gibbs Classifier $G_{Q}$; with $\\begin{array}{r}{\\operatorname*{Pr}(G_{Q}(x)=y)=\\frac{1}{2}(1+y\\mathbf{E}_{f\\sim Q}f(x))}\\end{array}$ for $y\\in\\{-1,1\\}$ . The probability that the majority vote will make an incorrect prediction of the output for a given example drawn from distribution D is called the risk of the majority vote, or Bayes risk$^{2}$, and is denoted $R_{D}(B_{Q})$ :\n\n$$\nR_{D}(B_{Q})=\\operatorname*{Pr}_{(x,y)\\sim D}(B_{Q}(x)\\neq y)=\\underset{(x,y)\\sim D}{\\mathbf{E}}I\\left(\\underset{f\\sim Q}{\\mathbf{E}}yf(x)\\leq 0\\right),\n$$\n\nwhere $I(\\cdot)$ is the indicator function taking value 0 if the input is false and 1 otherwise. Correspondingly, the probability that the Gibbs classifier makes an error is called the Gibbs risk and is denoted $R_{D}(G_{Q})$ :\n\n$$\nR_{D}(G_{Q})=\\operatorname*{Pr}_{(x,y)\\sim D}(G_{Q}(x)\\neq y)=\\underset{(x,y)\\sim D}{\\mathbf{E}}\\left(\\frac{1}{2}-\\frac{1}{2}\\underset{f\\sim Q}{\\mathbf{E}}yf(x)\\right).\n$$"}, {"name": "Margin, its moments, and the C-bound (Section 2; Lacasse et al. 2006 / Germain et al. 2015)", "statement": "The term $\\mathbf{E}_{f\\sim Q}yf(x)$ shared by both Bayes risk and Gibbs risk definitions is a commonly occurring expression in the context of majority votes. It is referred to as the margin of the majority vote and is denoted $M_{Q}(x,y)$ . Given that $(x,y){\\sim}D$ , Germain et al. (2015) study the margin as a random variable which they denote $M_{Q}^{D}$ . A longstanding intermediate result in PAC-Bayes analyses is the factor two bound between the Bayes Risk and the Gibbs risk (Langford and Shawe-Taylor, 2002; McAllester, 2003). That is, for fixed distributions D and $Q$, we have $R_{D}(B_{Q})\\leq 2R_{D}(G_{Q})$ . The C-bound (Lacasse et al., 2006; Germain et al., 2015) is a finer upper bound on the Bayes risk and is defined using the variance and first two moments of the margin $M_{Q}^{D}$ (denoted $\\mu_{1}(M_{Q}^{D})$ and $\\mu_{2}(M_{Q}^{D})$ respectively, with $\\mu_{i}(M_{Q}^{D})={\\bf E}_{(x,y)\\sim D}[M_{Q}(x,y)]^{i})$ . Equivalently, the C-bound can be expressed in terms of the Gibbs risk $R_{D}(G_{Q})$ and the expected disagreement (denoted $d_{Q}^{D}$ and defined below):\n\n$$\nR_{D}(B_{Q})\\leq{\\mathcal C}_{Q}^{D}=\\frac{\\mathbf{Var}(M_{Q}^{D})}{\\mu_{2}(M_{Q}^{D})}=1-\\frac{\\left(\\mu_{1}(M_{Q}^{D})\\right)^{2}}{\\mu_{2}(M_{Q}^{D})}=1-\\frac{(1-2 R_{D}(G_{Q}))^{2}}{1-2 d_{Q}^{D}},\n$$\n\nwhere\n\n$$\n\\begin{array}{c c}{{\\displaystyle\\mu_{1}\\big(M_{Q}^{D}\\big)=\\underset{(x,y)\\sim D}{\\mathbf{E}}\\,\\underset{f\\sim Q}{\\mathbf{E}}y f(x),}}&{{\\qquad\\mu_{2}\\big(M_{Q}^{D}\\big)=\\underset{x\\sim D_{\\mathcal{X}}}{\\mathbf{E}}\\left[\\underset{f\\sim Q}{\\mathbf{E}}f(x)\\right]^{2},}}\\\\ {{\\displaystyle{\\mathbf{Var}\\big(M_{Q}^{D}\\big)=\\mu_{2}\\big(M_{Q}^{D}\\big)-\\big(\\mu_{1}\\big(M_{Q}^{D}\\big)\\big)^{2},}}}&{{\\qquad d_{Q}^{D}=\\frac{1}{2}\\left(1-\\mu_{2}\\big(M_{Q}^{D}\\big)\\right).}}\\end{array}\n$$\n\nIn these definitions, $D_{\\mathcal{X}}$ denotes the marginal distribution on $\\mathcal{X}$ for a given distribution $D .^{1}$ We adopt this notation throughout the paper, as well as the use of $D_{\\mathcal{Y}| x}$ to denote the conditional distribution on the output space $\\mathcal{Y}$ given a fixed input $x \\in \\mathcal{X}$"}, {"name": "Sources of stochasticity: label noise, stochastic voters, stochastic majority vote (Section 2)", "statement": "An important note has to be made about the notions of stochasticity and determinism as they appear in the original paper. The generic use of these terms can be confusing to the reader as it is sometimes unclear what is meant to be stochastic or deterministic in some passages. To alleviate this ambiguity we differentiate three distinct sources of stochasticity that are of importance in the studied framework.\n\nLabel noise. The stochasticity of the labels arises when for a given value of $x \\sim D_{\\mathcal{X}}$ , there is a non-zero probability of observing more than one value of $y \\sim D_{\\mathcal{Y}| x}$.\n\nStochastic voters. Of note, Germain et al. (2015) introduced voters as functions of the form $f:\\mathcal{X}\\to [-1,1]$, implying they were deterministic predictors. Nevertheless, the property that a given voter is deterministic remains unused in their analysis of majority votes. We may wish to consider stochastic voters, such as voters for which the output follows some distribution. Say we consider a stochastic voter $f^{*}$ such that for any given input $x,$ the output of the voter is a random variable $f^{*}(x)\\sim W_{x}$ , where $W_{x}$ is a distribution over [−1, 1] that depends of the value $x .$ To apply the work of Germain et al. (2015), one needs only to consider $f(x)$ as the expected output of the stochastic voter $f^{*}$ given by $f(x)=\\mathbf{E}_{f^{*}(x)\\sim W_{x}}~f^{*}(x)$ . Doing so, the value $f(x)$ will be deterministic. Moreover, this is coherent with the idea that $f(x)$ is the confidence of the voter towards a given label. The discussion in this erratum henceforth encompasses both deterministic and stochastic voters and we use the general term voter to refer to voters of both natures.\n\nStochastic majority vote. We specified that the Bayes classifier $B_{Q}$ is deterministic while the Gibbs classifier $G_{Q}$ is stochastic. This distinction, made by Germain et al. (2015), is meant to distinguish between the stochasticity that arises from sampling a single voter (Gibbs) as opposed to averaging over the value of each voter $f \\in \\mathcal H$ (Bayes). It is not meant to refer to the nature of the individual voters as we remind that both Germain et al. (2015) and this erratum consider deterministic voters only.\n\nWe remind that this erratum is valid when considering the deterministic majority vote classifier that embed the expected value of its voters for the Bayes risk of Equation (1). The interested reader may note that the recent work of Zantedeschi et al. (2021) provide PAC-Bayes bounds for stochastic majority votes where the voters weights are random variables. While this is a related line of work, it differs from ours and Germain et al. (2015) as we only consider majority votes for which the weight of each voter is fixed."}, {"name": "Proposition 1 (Proposition 15 in Germain et al. 2015; erroneous)", "statement": "## Erroneous result\n\nProposition 1 (Proposition 15 in Germain et al. (2015)) For any countable set of voters $\\mathcal{H}$, any distribution Q on $\\mathcal{H}$, and any distribution D on $\\mathcal{X}\\times\\{-1,1\\}$ , we have\n\n$$\n\\mathbf{Var}(M_{Q}^{D})\\leq\\sum_{f\\in\\mathcal{H}}Q^{2}(f)+\\sum_{f_{1}\\in\\mathcal{H}}\\sum_{f_{2}\\in\\mathcal{H}\\atop f_{2}\\neq f_{1}}Q(f_{1})Q(f_{2})\\mathbf{Cov}_{(x,y)\\sim D}(f_{1}(x),f_{2}(x)).\n$$"}, {"name": "Corollary 2 (Corollary 16 in Germain et al. 2015; erroneous) and its motivation", "statement": "Even though the demonstration of Proposition 1 is flawed, this does not necessarily imply that the proposition itself does not stand. In principle, the proposition might be valid due to a different argument. However, in order to show that Proposition 15 of Germain et al. (2015) is indeed false, we introduce the following corollary. If Proposition 1 were true, then so would Corollary 2 which follows directly from it.\n\n## Erroneous result\n\nCorollary 2 (Corollary 16 in Germain et al. (2015)) Given n independent voters under a uniform distribution $Q,$ we have\n\n$$\nR_{D}(B_{Q})\\leq\\mathcal{C}_{Q}^{D}\\leq\\frac{1}{n(1-2d_{Q}^{D})}=\\frac{1}{n(1-2R_{D}(G_{Q}))}.\n$$\n\nTo prove the above corollary, Germain et al. (2015) note that under the hypothesis that the n voters are independents under a uniform distribution, Proposition 1 implies that $\\mathbf{Var}(M_{Q}^{D})\\leq 1/n;$ the result then follows from the definition of the C-bound.\n\nAt first glance, the corollary seems intuitive. Increasing the number of voters should improve the quality of the classification. Therefore, we would expect the C-bound to decrease. The main problem with this result is that it implies that as the number of independent voters n goes to infinity, the right side goes to 0. This means that for any binary classification problem, the risk of the majority vote can be made arbitrarily small by using a sufficiently large set of independent voters. Though this may seem like a desirable property, it is not realistic in the presence of label noise. In such case, the noise associated with the output y given a certain x will induce a minimum amount of erroneous classification (i.e., irreducible error). Notice that the label noise is an inherent property of a given distribution D, and the irreducible error (also known as Bayes error in the literature) is achieved by an optimal prediction rule for D. This irreducible error is not 0 in the presence of label noise, which contradicts the statement of Corollary 2. To clearly illustrate this contradiction, consider the following example."}], "nl_notes": "### Counter-example (Section 4, falsifying Corollary 2)\nNOTE (Counter-example) — Do not construct the Section 4 example and do not state that one exists. State a conditional: for any probability space carrying a ±1 label Y and n voters with the means, second moments and pairwise products the text lists, the conclusion holds. Model each voter as V_i = v_i(x), a measurable ±1-valued function of an input x, with the voters mutually independent and the label independent of the input.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace ResearchLevel\n\nnoncomputable def cMu1 {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  ∫ x, h x * S x ∂μ\n\nnoncomputable def cMu2 {X : Type*} [MeasurableSpace X] (μ : Measure X) (S : X → ℝ) : ℝ :=\n  ∫ x, (S x) ^ 2 ∂μ\n\nnoncomputable def cBound {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  1 - (cMu1 μ h S) ^ 2 / (cMu2 μ S)\n\nnoncomputable def labelVarMean {X : Type*} [MeasurableSpace X] (μ : Measure X) (h : X → ℝ) : ℝ :=\n  ∫ x, (1 - (h x) ^ 2) ∂μ\n\nnoncomputable def condErr (hx sx : ℝ) : ℝ :=\n  if 0 < sx then (1 - hx) / 2 else if sx < 0 then (1 + hx) / 2 else 1\n\nnoncomputable def riskD {X : Type*} [MeasurableSpace X] (μ : Measure X) (h S : X → ℝ) : ℝ :=\n  ∫ x, condErr (h x) (S x) ∂μ\n\nnoncomputable def margin {Ω X ι : Type*} (Q : ι → ℝ) (f : ι → X → ℝ)\n    (xx : Ω → X) (yy : Ω → ℝ) : Ω → ℝ :=\n  fun ω => yy ω * ∑' i, Q i * f i (xx ω)\n\nnoncomputable def aggVote {Ω : Type*} (n : ℕ) (V : Fin n → Ω → ℝ) : Ω → ℝ :=\n  fun ω => (n : ℝ)⁻¹ * ∑ i, V i ω\n\ntheorem prop4 {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]\n    (h : X → ℝ) (hh_meas : Measurable h) (hh_bdd : ∀ x, |h x| ≤ 1) :\n    (∃ S : X → ℝ, Measurable S ∧ (∀ x, |S x| ≤ 1) ∧ 0 < cMu2 μ S ∧\n        riskD μ h S ≤ cBound μ h S ∧\n        cBound μ h S = labelVarMean μ h) ∧\n    (∀ S : X → ℝ, Measurable S → (∀ x, |S x| ≤ 1) → 0 < cMu2 μ S →\n        labelVarMean μ h ≤ cBound μ h S) := by\n  sorry\n\ntheorem prop3 {Ω X ι : Type*} [MeasurableSpace Ω] [MeasurableSpace X] [Countable ι]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (Q : ι → ℝ) (hQ_nonneg : ∀ i, 0 ≤ Q i) (hQ_summable : Summable Q) (hQ_sum : ∑' i, Q i = 1)\n    (f : ι → X → ℝ) (hf_bdd : ∀ i x, |f i x| ≤ 1)\n    (xx : Ω → X) (yy : Ω → ℝ) (hy : ∀ᵐ ω ∂P, yy ω = 1 ∨ yy ω = -1)\n    (g : X → ℝ) (hdet : ∀ᵐ ω ∂P, yy ω = g (xx ω))\n    (hacc : ∀ i, ∀ᵐ ω ∂P, 0 < yy ω * f i (xx ω))\n    (hmeas_x : Measurable xx) (hmeas_y : Measurable yy)\n    (hmeas_f : ∀ i, Measurable (f i)) :\n    Var[margin Q f xx yy; P]\n      ≤ (∑' i, (Q i) ^ 2)\n        + ∑' i, ∑' j : {j // j ≠ i},\n            Q i * Q (j : ι) * cov[fun ω => f i (xx ω), fun ω => f (j : ι) (xx ω); P] := by\n  sorry\n\ntheorem cbound_counterexample {Ω : Type*} [MeasurableSpace Ω]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (n : ℕ) (hn : 0 < n) (V : Fin n → Ω → ℝ) (Y : Ω → ℝ)\n    (hVint : ∀ i, Integrable (V i) P)\n    (hVcrossint : ∀ i j, Integrable (fun ω => V i ω * V j ω) P)\n    (hYint : Integrable Y P)\n    (hYSint : Integrable (fun ω => Y ω * aggVote n V ω) P)\n    (hV1 : ∀ i, ∫ ω, V i ω ∂P = 3 / 5)\n    (hV2 : ∀ i, ∫ ω, (V i ω) ^ 2 ∂P = 1)\n    (hVuncorr : ∀ i j, i ≠ j → ∫ ω, V i ω * V j ω ∂P = 9 / 25)\n    (hY1 : ∫ ω, Y ω ∂P = 3 / 5)\n    (hYsq : ∀ ω, (Y ω) ^ 2 = 1)\n    (hYindep : ∫ ω, Y ω * aggVote n V ω ∂P\n        = (∫ ω, Y ω ∂P) * (∫ ω, aggVote n V ω ∂P))\n    (X : Type*) [MeasurableSpace X] (xx : Ω → X) (v : Fin n → X → ℝ)\n    (hVroute : ∀ i ω, V i ω = v i (xx ω))\n    (hv_pm : ∀ i x, v i x = 1 ∨ v i x = -1)\n    (hv_meas : ∀ i, Measurable (v i))\n    (hVindep : ProbabilityTheory.iIndepFun V P)\n    (hYXindep : ProbabilityTheory.IndepFun Y xx P) :\n    cBound P Y (aggVote n V) = (400 + 144 * (n : ℝ)) / (400 + 225 * (n : ℝ))\n      ∧ (16 : ℝ) / 25 < cBound P Y (aggVote n V) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["prop4", "prop3", "cbound_counterexample"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 2247, "end": 3291, "gold": "{Ω : Type*} [MeasurableSpace Ω]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (n : ℕ) (hn : 0 < n) (V : Fin n → Ω → ℝ) (Y : Ω → ℝ)\n    (hVint : ∀ i, Integrable (V i) P)\n    (hVcrossint : ∀ i j, Integrable (fun ω => V i ω * V j ω) P)\n    (hYint : Integrable Y P)\n    (hYSint : Integrable (fun ω => Y ω * aggVote n V ω) P)\n    (hV1 : ∀ i, ∫ ω, V i ω ∂P = 3 / 5)\n    (hV2 : ∀ i, ∫ ω, (V i ω) ^ 2 ∂P = 1)\n    (hVuncorr : ∀ i j, i ≠ j → ∫ ω, V i ω * V j ω ∂P = 9 / 25)\n    (hY1 : ∫ ω, Y ω ∂P = 3 / 5)\n    (hYsq : ∀ ω, (Y ω) ^ 2 = 1)\n    (hYindep : ∫ ω, Y ω * aggVote n V ω ∂P\n        = (∫ ω, Y ω ∂P) * (∫ ω, aggVote n V ω ∂P))\n    (X : Type*) [MeasurableSpace X] (xx : Ω → X) (v : Fin n → X → ℝ)\n    (hVroute : ∀ i ω, V i ω = v i (xx ω))\n    (hv_pm : ∀ i x, v i x = 1 ∨ v i x = -1)\n    (hv_meas : ∀ i, Measurable (v i))\n    (hVindep : ProbabilityTheory.iIndepFun V P)\n    (hYXindep : ProbabilityTheory.IndepFun Y xx P) :\n    cBound P Y (aggVote n V) = (400 + 144 * (n : ℝ)) / (400 + 225 * (n : ℝ))\n      ∧ (16 : ℝ) / 25 < cBound P Y (aggVote n V)"}]}
{"item_id": "v25-22-1253::lemma6_lipschitz::A-stmt", "paper_id": "v25-22-1253", "target_name": "lemma6_lipschitz", "source": "Research-level", "nl_statement": "Lemma 6 (Lipschitz property of EU) $E_{\\beta}$ is Lipschitz continuous with respect to the supremum norm over ${\\mathcal{D}}_{M}$ with $L_{M}= |\\exp(\\beta M)-1|$ , Moreover, $L_{M}$ is tight with respect to both $\\beta$ and M, i.e., for every $\\beta \\neq 0$ and every $M > 0$ there exist $F,G\\in{\\mathcal{D}}_{M}$ with $\\| F - G \\|_{\\infty}> 0$ and $|E_{\\beta}(F)-E_{\\beta}(G)| =L_{M}\\|F-G\\|_{\\infty}$.", "context": [{"name": "Notations (Section 2.1)", "statement": "We write $[M:N]\\triangleq\\{M,M+1,\\ldots,N\\}$ and $[ N ] \\triangleq [ 1 : N ]$ for any positive integers $M \\leq N .$ We adopt the convention that $\\textstyle \\sum_{i = n}^{m}a_{i}\\triangleq 0{\\mathrm{~ i f ~}}n > m$ and $\\textstyle \\prod_{i = n}^{m}a_{i}\\triangleq 1{\\mathrm{~ i f ~}}n > m$ .We use $\\mathbb{I}\\{\\cdot \\}$ to denote the indicator function. For any $x \\in \\mathbb{R}$ , we define $[x]^{+}\\triangleq\\operatorname*{max}\\{x,0\\}$ . We denote by $\\delta_{c}$ the Dirac measure at $c .$ Similarly, $\\psi_{c}$ represents the unit step function at $c,$ corresponding to the CDF of $\\delta_{c}$ . We denote by $\\Delta(\\mathcal{S}),\\mathcal{D}(a,b),\\mathcal{D}_{M}$ and $\\mathcal{D}$ the set of distributions supported on a set $\\mathcal{S},[a,b],[0,M]$ and the set of all distributions respectively. We use $(x_{1},x_{2};p)$ to denote a binary r.v. taking values $x_{1}$ and $x_{2}$ with probability $1 - p$ and $p .$ For a discrete set $x=\\{x_{1},\\cdots,x_{n}\\}$ and a probability vector $p=(p_{1},\\cdot\\cdot\\cdot,p_{n})$ , the notation $(x,p)$ represents the discrete distribution with $\\mathbb{P}(X=x_{i})=p_{i}$ . For a discrete distribution $\\eta=\\left(x,p\\right)$ , we use $| \\eta | = | x |$ to denote the number of atoms of the distribution $\\eta$ . We use $\\tilde{\\mathcal{O}}(\\cdot)$ to denote $\\mathcal{O}(\\cdot)$ omitting logarithmic factors. A table of notation is provided in Appendix A."}, {"name": "Entropic risk measure EntRM (Section 2.3)", "statement": "We direct our attention to a specific distribution-invariant risk measure, EntRM, a prominent risk measure in domains requiring risk-sensitive decision-making, such as mathematical finance (Föllmer and Schied, 2016), Markovian decision processes (Bäuerle and Rieder, 2014). For a random variable $X \\sim F$ and a non-zero coefficient $\\beta,$ the EntRM is defined as:\n\n$$\nU_{\\beta}(X)\\triangleq{\\frac{1}{\\beta}}\\log\\left(\\mathbb{E}_{X\\sim F}\\left[e^{\\beta X}\\right]\\right)={\\frac{1}{\\beta}}\\log\\left(\\int_{\\mathbb{R}}e^{\\beta x}dF(x)\\right).\n$$"}, {"name": "Supremum distance, l1 distance, norm balls B_infty(F,c) and B_1(P,c)", "statement": "We begin by establishing additional notations. For two Cumulative Distribution Functions (CDFs) F and G, the supremum distance between them is defined as $\\| F - G \\|_{\\infty}\\triangleq$ $\\begin{array}{r}{\\operatorname*{sup}_{x}|F(x)-G(x)|}\\end{array}$ We define the $\\ell_{1}$ distance between two Probability Mass Functions $(\\mathrm{PMFs})$ with the same support P and $Q$ as $\\begin{array}{r}{\\| P - Q \\|_{1}\\triangleq \\sum_{i}| P_{i}- Q_{i}|}\\end{array}$ . Furthermore, the set $B_{\\infty}(F,c): =\\{G\\in\\mathcal{D}|\\|G-F\\|_{\\infty}\\leq c\\}$ denotes the supremum norm ball of CDFs centered at F with radius c. Analogously, $B_{1}(P,c)$ represents the $\\ell_{1}$ norm ball of PMFs centered at P with radius c."}, {"name": "Exponential utility E_beta (Section 4.1)", "statement": "Our analysis explores the relationship between EntRM and Exponential Utility (EU). The EU is defined as follows:\n\n$$\nE_{\\beta}(F)\\triangleq e^{\\beta U_{\\beta}(F)}=\\int_{\\mathbb{R}}e^{\\beta x}dF(x),\n$$\n\nwhere it serves as an exponential transformation of the EntRM. Notably, this transformation preserves the order in the sense that for any non-zero $\\beta,\\forall\\beta\\neq 0$\n\n$$\nU_{\\beta}(F)\\geq U_{\\beta}(G)\\Longleftrightarrow\\mathrm{sign}(\\beta)E_{\\beta}(F)\\geq\\mathrm{sign}(\\beta)E_{\\beta}(G).\n$$"}, {"name": "Lipschitz constant L_M of E_beta over D_M", "statement": "We further present two important properties of EU, instrumental in formulating the regret upper bounds: Lipschitz continuity and linearity. Denote by $L_{M}$ the Lipschitz constant of $E_{\\beta}: \\mathcal{D}_{M}\\to \\mathbb{R}$ with respect to the infinity norm $\\left\\| \\cdot \\right\\|_{\\infty}$ , which satisfies:\n\n$$\nE_{\\beta}(F)-E_{\\beta}(G)\\leq L_{M}\\|F-G\\|_{\\infty},\\forall F,G\\in\\mathcal{D}_{M}.\n$$"}, {"name": "Table of notation: CDF classes and the infinity ball (Appendix A)", "statement": "<table><tr><td>Symbol</td><td>Explanation</td></tr><tr><td> $\\mathcal{D}$ </td><td>The space of all CDFs</td></tr><tr><td> $\\mathcal{D}(a,b)$ </td><td>The space of all CDFs supported on  $[a,b]$ </td></tr><tr><td> ${\\mathcal{D}}_{M}$ </td><td>The space of all CDFs supported on  $[0,M]$ </td></tr><tr><td> $B_{\\infty}(F,c)$ </td><td>The  $\\left\\| \\cdot \\right\\|_{\\infty}$  norm ball centered at  $F$  with radius c</td></tr></table>"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def EU (β : ℝ) (μ : Measure ℝ) : ℝ := ∫ x, Real.exp (β * x) ∂μ\n\nnoncomputable def optimismInf (a b c : ℝ) (F : ℝ → ℝ) : ℝ → ℝ :=\n  fun x => max (F x - c * Set.indicator (Set.Ico a b) (fun _ => (1 : ℝ)) x) 0\n\ndef InDab (a b : ℝ) (F : ℝ → ℝ) : Prop :=\n  Monotone F ∧ (∀ x, F x ∈ Set.Icc (0 : ℝ) 1)\n    ∧ (∀ x, ContinuousWithinAt F (Set.Ici x) x)\n    ∧ (∀ x, x < a → F x = 0) ∧ (∀ x, b ≤ x → F x = 1)\n\ntheorem lemma6_lipschitz {M β : ℝ} (hM : 0 ≤ M) (hβ : β ≠ 0)\n    (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]\n    (hμ : μ (Set.Icc 0 M) = 1) (hν : ν (Set.Icc 0 M) = 1) :\n    (|EU β μ - EU β ν|\n      ≤ |Real.exp (β * M) - 1|\n          * ⨆ x : ℝ, |ProbabilityTheory.cdf μ x - ProbabilityTheory.cdf ν x|)\n    ∧ (0 < M → ∃ μ₀ ν₀ : Measure ℝ, IsProbabilityMeasure μ₀ ∧ IsProbabilityMeasure ν₀ ∧\n        μ₀ (Set.Icc 0 M) = 1 ∧ ν₀ (Set.Icc 0 M) = 1 ∧\n        0 < (⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) ∧\n        |EU β μ₀ - EU β ν₀|\n          = |Real.exp (β * M) - 1|\n              * ⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) := by\n  sorry\n\ntheorem lemma12_optimistic_bound {n : ℕ} (a b c : ℝ) (hab : a ≤ b) (hc0 : 0 < c)\n    (F : Fin n → ℝ → ℝ)\n    (hFmono : ∀ i, Monotone (F i))\n    (hFright : ∀ i x, ContinuousWithinAt (F i) (Set.Ici x) x)\n    (hFrange : ∀ i x, F i x ∈ Set.Icc (0 : ℝ) 1)\n    (hFlo : ∀ i x, x < a → F i x = 0)\n    (hFhi : ∀ i x, b ≤ x → F i x = 1)\n    (θ θ' : Fin n → ℝ)\n    (hθnn : ∀ i, 0 ≤ θ i) (hθ'nn : ∀ i, 0 ≤ θ' i)\n    (hθsum : ∑ i, θ i = 1) (hθ'sum : ∑ i, θ' i = 1)\n    (hc : ∑ i, |θ i - θ' i| ≤ c) :\n    ∀ x, optimismInf a b c (fun y => ∑ i, θ' i * F i y) x\n        ≤ (∑ i, θ i * F i x) := by\n  sorry\n\ntheorem lemma8_optimism_operator_argmax {a b c : ℝ} (hab : a ≤ b) (hc0 : 0 < c)\n    (F : ℝ → ℝ) (hF : InDab a b F)\n    (ρ : (ℝ → ℝ) → ℝ)\n    (hρM : ∀ P Q : ℝ → ℝ, InDab a b P → InDab a b Q → (∀ x, P x ≤ Q x) → ρ Q ≤ ρ P) :\n    (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n        ∀ x, optimismInf a b c F x ≤ G x)\n    ∧ ( InDab a b (optimismInf a b c F)\n        ∧ (∀ x, |optimismInf a b c F x - F x| ≤ c)\n        ∧ (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n            ρ G ≤ ρ (optimismInf a b c F)) ) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma6_lipschitz", "lemma12_optimistic_bound", "lemma8_optimism_operator_argmax"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 499, "end": 1170, "gold": "{M β : ℝ} (hM : 0 ≤ M) (hβ : β ≠ 0)\n    (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]\n    (hμ : μ (Set.Icc 0 M) = 1) (hν : ν (Set.Icc 0 M) = 1) :\n    (|EU β μ - EU β ν|\n      ≤ |Real.exp (β * M) - 1|\n          * ⨆ x : ℝ, |ProbabilityTheory.cdf μ x - ProbabilityTheory.cdf ν x|)\n    ∧ (0 < M → ∃ μ₀ ν₀ : Measure ℝ, IsProbabilityMeasure μ₀ ∧ IsProbabilityMeasure ν₀ ∧\n        μ₀ (Set.Icc 0 M) = 1 ∧ ν₀ (Set.Icc 0 M) = 1 ∧\n        0 < (⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) ∧\n        |EU β μ₀ - EU β ν₀|\n          = |Real.exp (β * M) - 1|\n              * ⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|)"}]}
{"item_id": "v25-22-1253::lemma12_optimistic_bound::A-stmt", "paper_id": "v25-22-1253", "target_name": "lemma12_optimistic_bound", "source": "Research-level", "nl_statement": "Lemma 12 (Bound on the optimistic constant) For any distributions $\\{F_{i}\\}_{i\\in [n]}\\subseteq\\mathcal{D}(a,b)$ and any $\\theta,\\theta^{\\prime}\\in\\Delta_{n}$ it holds that if $c \\geq \\| \\theta - \\theta^{\\prime}\\|_{1}$ , then\n\n$$\n\\sum_{i=1}^{n}\\theta_{i}F_{i}\\preceq\\mathrm{O}_{c}^{\\infty}\\left(\\sum_{i=1}^{n}\\theta_{i}^{\\prime}F_{i}\\right).\n$$", "context": [{"name": "Notations (Section 2.1)", "statement": "We write $[M:N]\\triangleq\\{M,M+1,\\ldots,N\\}$ and $[ N ] \\triangleq [ 1 : N ]$ for any positive integers $M \\leq N .$ We adopt the convention that $\\textstyle \\sum_{i = n}^{m}a_{i}\\triangleq 0{\\mathrm{~ i f ~}}n > m$ and $\\textstyle \\prod_{i = n}^{m}a_{i}\\triangleq 1{\\mathrm{~ i f ~}}n > m$ .We use $\\mathbb{I}\\{\\cdot \\}$ to denote the indicator function. For any $x \\in \\mathbb{R}$ , we define $[x]^{+}\\triangleq\\operatorname*{max}\\{x,0\\}$ . We denote by $\\delta_{c}$ the Dirac measure at $c .$ Similarly, $\\psi_{c}$ represents the unit step function at $c,$ corresponding to the CDF of $\\delta_{c}$ . We denote by $\\Delta(\\mathcal{S}),\\mathcal{D}(a,b),\\mathcal{D}_{M}$ and $\\mathcal{D}$ the set of distributions supported on a set $\\mathcal{S},[a,b],[0,M]$ and the set of all distributions respectively. We use $(x_{1},x_{2};p)$ to denote a binary r.v. taking values $x_{1}$ and $x_{2}$ with probability $1 - p$ and $p .$ For a discrete set $x=\\{x_{1},\\cdots,x_{n}\\}$ and a probability vector $p=(p_{1},\\cdot\\cdot\\cdot,p_{n})$ , the notation $(x,p)$ represents the discrete distribution with $\\mathbb{P}(X=x_{i})=p_{i}$ . For a discrete distribution $\\eta=\\left(x,p\\right)$ , we use $| \\eta | = | x |$ to denote the number of atoms of the distribution $\\eta$ . We use $\\tilde{\\mathcal{O}}(\\cdot)$ to denote $\\mathcal{O}(\\cdot)$ omitting logarithmic factors. A table of notation is provided in Appendix A."}, {"name": "Stochastic dominance and risk-measure properties (M), (TI), (DI) (Section 2.3)", "statement": "Consider two random variables $X \\sim F$ and $Y \\sim G$ . We assert that Y dominates $X,$ and correspondingly, G dominates $F,$ denoted as $Y \\succeq X$ and $G \\succeq F$ , if and only if for every real number x, the inequality $F(x)\\geq G(x)$ holds true. A risk measure, $\\rho,$ is a function mapping a set of random variables, denoted as $\\mathcal{X}$ , to the real numbers. This mapping adheres to several crucial properties:\n\n• Monotonicity (M): $X\\preceq Y\\Rightarrow\\rho(X)\\leq\\rho(Y),\\forall X,Y\\in{\\mathcal{X}}$\n\nTranslation-invariance $(\\mathbf{TI})\\colon\\rho(X+c)=\\rho(X)+c,\\forall X\\in{\\mathcal{X}},\\forall c\\in\\mathbb{R},$\n\nDistribution-invariance (DI): $F_{X_{1}}=F_{X_{2}}\\Rightarrow\\rho(X_{1})=\\rho(X_{2})$\n\nA mapping $\\rho : \\mathcal{X}\\to \\mathbb{R}$ qualifies as a risk measure if it satisfies both (M) and (TI). Additionally, a risk measure that also adheres to (DI) is termed a distribution-invariant risk measure. Distribution-invariant risk measure allows us to simplify notation by denoting $\\rho(F_{X})$ as $\\rho(X)$"}, {"name": "Supremum distance, l1 distance, norm balls B_infty(F,c) and B_1(P,c)", "statement": "We begin by establishing additional notations. For two Cumulative Distribution Functions (CDFs) F and G, the supremum distance between them is defined as $\\| F - G \\|_{\\infty}\\triangleq$ $\\begin{array}{r}{\\operatorname*{sup}_{x}|F(x)-G(x)|}\\end{array}$ We define the $\\ell_{1}$ distance between two Probability Mass Functions $(\\mathrm{PMFs})$ with the same support P and $Q$ as $\\begin{array}{r}{\\| P - Q \\|_{1}\\triangleq \\sum_{i}| P_{i}- Q_{i}|}\\end{array}$ . Furthermore, the set $B_{\\infty}(F,c): =\\{G\\in\\mathcal{D}|\\|G-F\\|_{\\infty}\\leq c\\}$ denotes the supremum norm ball of CDFs centered at F with radius c. Analogously, $B_{1}(P,c)$ represents the $\\ell_{1}$ norm ball of PMFs centered at P with radius c."}, {"name": "Definition 7 and the distributional optimism operator O_c^infty (Section 4.2)", "statement": "Definition 7 Given two CDFs F and G, we say that F is more optimistic than G if $F \\geq G$\n\nThis definition aligns with the intuitive notion that a more optimistic distribution should possess a larger EntRM value. Given that the exponential transformation preserves order, F is more optimistic than G if and only if $E_{\\beta}(F)\\ge E_{\\beta}(G)$ . Following the methodology of Keramati et al. (2020), we introduce the distributional optimism operator $\\mathrm{O}_{c}^{\\infty}:\\mathcal{D}(a,b)\\mapsto$ $\\mathcal{D}(a,b)$ for a level $c\\in(0,1)$ as\n\n$$\n(\\mathrm{O}_{c}^{\\infty}F)(x)\\triangleq [F(x)-c\\mathbb{I}_{[a,b)}(x)]^{+}.\n$$\n\nThis operator shifts the distribution F down by a maximum of c over $[a,b)$ , ensuring that the resulting function ${\\mathrm{O}}_{c}^{\\infty}F$ remains a valid CDF in $\\mathcal{D}(a,b)$ and optimistically dominates all other CDFs within the confidence ball $B_{\\infty}(F,c)^{3}$ . In particular, ${\\mathrm{O}}_{c}^{\\infty}F$ represents maximum permissible downward adjustment under the constraints of the infinity ball, ensuring that for any distribution $G\\in B_{\\infty}(F,c)$\n\n$$\n(\\mathrm{O}_{c}^{\\infty}F)(x)\\leq G(x),\\forall x\\in\\mathbb{R}.\n$$"}, {"name": "Lemma 11", "statement": "Lemma 11 For any $F_{i}\\in \\mathcal{D}$ and any $\\theta,\\theta^{\\prime}\\in\\Delta_{n}$ with any $n \\geq 2$ , it holds that\n\n$$\n\\left\\| \\sum_{i = 1}^{n}\\theta_{i}F_{i}- \\sum_{i = 1}^{n}\\theta_{i}^{\\prime}F_{i}\\right\\|_{\\infty}\\leq \\left\\| \\theta - \\theta^{\\prime}\\right\\|_{1}.\n$$"}, {"name": "Table of notation: CDF classes and the infinity ball (Appendix A)", "statement": "<table><tr><td>Symbol</td><td>Explanation</td></tr><tr><td> $\\mathcal{D}$ </td><td>The space of all CDFs</td></tr><tr><td> $\\mathcal{D}(a,b)$ </td><td>The space of all CDFs supported on  $[a,b]$ </td></tr><tr><td> ${\\mathcal{D}}_{M}$ </td><td>The space of all CDFs supported on  $[0,M]$ </td></tr><tr><td> $B_{\\infty}(F,c)$ </td><td>The  $\\left\\| \\cdot \\right\\|_{\\infty}$  norm ball centered at  $F$  with radius c</td></tr></table>"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def EU (β : ℝ) (μ : Measure ℝ) : ℝ := ∫ x, Real.exp (β * x) ∂μ\n\nnoncomputable def optimismInf (a b c : ℝ) (F : ℝ → ℝ) : ℝ → ℝ :=\n  fun x => max (F x - c * Set.indicator (Set.Ico a b) (fun _ => (1 : ℝ)) x) 0\n\ndef InDab (a b : ℝ) (F : ℝ → ℝ) : Prop :=\n  Monotone F ∧ (∀ x, F x ∈ Set.Icc (0 : ℝ) 1)\n    ∧ (∀ x, ContinuousWithinAt F (Set.Ici x) x)\n    ∧ (∀ x, x < a → F x = 0) ∧ (∀ x, b ≤ x → F x = 1)\n\ntheorem lemma6_lipschitz {M β : ℝ} (hM : 0 ≤ M) (hβ : β ≠ 0)\n    (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]\n    (hμ : μ (Set.Icc 0 M) = 1) (hν : ν (Set.Icc 0 M) = 1) :\n    (|EU β μ - EU β ν|\n      ≤ |Real.exp (β * M) - 1|\n          * ⨆ x : ℝ, |ProbabilityTheory.cdf μ x - ProbabilityTheory.cdf ν x|)\n    ∧ (0 < M → ∃ μ₀ ν₀ : Measure ℝ, IsProbabilityMeasure μ₀ ∧ IsProbabilityMeasure ν₀ ∧\n        μ₀ (Set.Icc 0 M) = 1 ∧ ν₀ (Set.Icc 0 M) = 1 ∧\n        0 < (⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) ∧\n        |EU β μ₀ - EU β ν₀|\n          = |Real.exp (β * M) - 1|\n              * ⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) := by\n  sorry\n\ntheorem lemma12_optimistic_bound {n : ℕ} (a b c : ℝ) (hab : a ≤ b) (hc0 : 0 < c)\n    (F : Fin n → ℝ → ℝ)\n    (hFmono : ∀ i, Monotone (F i))\n    (hFright : ∀ i x, ContinuousWithinAt (F i) (Set.Ici x) x)\n    (hFrange : ∀ i x, F i x ∈ Set.Icc (0 : ℝ) 1)\n    (hFlo : ∀ i x, x < a → F i x = 0)\n    (hFhi : ∀ i x, b ≤ x → F i x = 1)\n    (θ θ' : Fin n → ℝ)\n    (hθnn : ∀ i, 0 ≤ θ i) (hθ'nn : ∀ i, 0 ≤ θ' i)\n    (hθsum : ∑ i, θ i = 1) (hθ'sum : ∑ i, θ' i = 1)\n    (hc : ∑ i, |θ i - θ' i| ≤ c) :\n    ∀ x, optimismInf a b c (fun y => ∑ i, θ' i * F i y) x\n        ≤ (∑ i, θ i * F i x) := by\n  sorry\n\ntheorem lemma8_optimism_operator_argmax {a b c : ℝ} (hab : a ≤ b) (hc0 : 0 < c)\n    (F : ℝ → ℝ) (hF : InDab a b F)\n    (ρ : (ℝ → ℝ) → ℝ)\n    (hρM : ∀ P Q : ℝ → ℝ, InDab a b P → InDab a b Q → (∀ x, P x ≤ Q x) → ρ Q ≤ ρ P) :\n    (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n        ∀ x, optimismInf a b c F x ≤ G x)\n    ∧ ( InDab a b (optimismInf a b c F)\n        ∧ (∀ x, |optimismInf a b c F x - F x| ≤ c)\n        ∧ (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n            ρ G ≤ ρ (optimismInf a b c F)) ) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma6_lipschitz", "lemma12_optimistic_bound", "lemma8_optimism_operator_argmax"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 1219, "end": 1759, "gold": "{n : ℕ} (a b c : ℝ) (hab : a ≤ b) (hc0 : 0 < c)\n    (F : Fin n → ℝ → ℝ)\n    (hFmono : ∀ i, Monotone (F i))\n    (hFright : ∀ i x, ContinuousWithinAt (F i) (Set.Ici x) x)\n    (hFrange : ∀ i x, F i x ∈ Set.Icc (0 : ℝ) 1)\n    (hFlo : ∀ i x, x < a → F i x = 0)\n    (hFhi : ∀ i x, b ≤ x → F i x = 1)\n    (θ θ' : Fin n → ℝ)\n    (hθnn : ∀ i, 0 ≤ θ i) (hθ'nn : ∀ i, 0 ≤ θ' i)\n    (hθsum : ∑ i, θ i = 1) (hθ'sum : ∑ i, θ' i = 1)\n    (hc : ∑ i, |θ i - θ' i| ≤ c) :\n    ∀ x, optimismInf a b c (fun y => ∑ i, θ' i * F i y) x\n        ≤ (∑ i, θ i * F i x)"}]}
{"item_id": "v25-22-1253::lemma8_optimism_operator_argmax::A-stmt", "paper_id": "v25-22-1253", "target_name": "lemma8_optimism_operator_argmax", "source": "Research-level", "nl_statement": "Lemma 8 Let ρ be a functional (not necessarily a risk measure) satisfying (M). For any $G\\in{\\mathcal{D}}(a,b)$ , it holds that if $G\\in B_{\\infty}(F,c)$ , then $G \\preceq \\mathrm{O}_{c}^{\\infty}F$ . Moreover, it holds that\n\n$$\n\\mathrm{O}_{c}^{\\infty}F\\in\\arg\\operatorname*{max}_{G\\in B_{\\infty}(F,c)\\cap\\mathcal{D}(a,b)}\\rho(G).\n$$", "context": [{"name": "Notations (Section 2.1)", "statement": "We write $[M:N]\\triangleq\\{M,M+1,\\ldots,N\\}$ and $[ N ] \\triangleq [ 1 : N ]$ for any positive integers $M \\leq N .$ We adopt the convention that $\\textstyle \\sum_{i = n}^{m}a_{i}\\triangleq 0{\\mathrm{~ i f ~}}n > m$ and $\\textstyle \\prod_{i = n}^{m}a_{i}\\triangleq 1{\\mathrm{~ i f ~}}n > m$ .We use $\\mathbb{I}\\{\\cdot \\}$ to denote the indicator function. For any $x \\in \\mathbb{R}$ , we define $[x]^{+}\\triangleq\\operatorname*{max}\\{x,0\\}$ . We denote by $\\delta_{c}$ the Dirac measure at $c .$ Similarly, $\\psi_{c}$ represents the unit step function at $c,$ corresponding to the CDF of $\\delta_{c}$ . We denote by $\\Delta(\\mathcal{S}),\\mathcal{D}(a,b),\\mathcal{D}_{M}$ and $\\mathcal{D}$ the set of distributions supported on a set $\\mathcal{S},[a,b],[0,M]$ and the set of all distributions respectively. We use $(x_{1},x_{2};p)$ to denote a binary r.v. taking values $x_{1}$ and $x_{2}$ with probability $1 - p$ and $p .$ For a discrete set $x=\\{x_{1},\\cdots,x_{n}\\}$ and a probability vector $p=(p_{1},\\cdot\\cdot\\cdot,p_{n})$ , the notation $(x,p)$ represents the discrete distribution with $\\mathbb{P}(X=x_{i})=p_{i}$ . For a discrete distribution $\\eta=\\left(x,p\\right)$ , we use $| \\eta | = | x |$ to denote the number of atoms of the distribution $\\eta$ . We use $\\tilde{\\mathcal{O}}(\\cdot)$ to denote $\\mathcal{O}(\\cdot)$ omitting logarithmic factors. A table of notation is provided in Appendix A."}, {"name": "Stochastic dominance and risk-measure properties (M), (TI), (DI) (Section 2.3)", "statement": "Consider two random variables $X \\sim F$ and $Y \\sim G$ . We assert that Y dominates $X,$ and correspondingly, G dominates $F,$ denoted as $Y \\succeq X$ and $G \\succeq F$ , if and only if for every real number x, the inequality $F(x)\\geq G(x)$ holds true. A risk measure, $\\rho,$ is a function mapping a set of random variables, denoted as $\\mathcal{X}$ , to the real numbers. This mapping adheres to several crucial properties:\n\n• Monotonicity (M): $X\\preceq Y\\Rightarrow\\rho(X)\\leq\\rho(Y),\\forall X,Y\\in{\\mathcal{X}}$\n\nTranslation-invariance $(\\mathbf{TI})\\colon\\rho(X+c)=\\rho(X)+c,\\forall X\\in{\\mathcal{X}},\\forall c\\in\\mathbb{R},$\n\nDistribution-invariance (DI): $F_{X_{1}}=F_{X_{2}}\\Rightarrow\\rho(X_{1})=\\rho(X_{2})$\n\nA mapping $\\rho : \\mathcal{X}\\to \\mathbb{R}$ qualifies as a risk measure if it satisfies both (M) and (TI). Additionally, a risk measure that also adheres to (DI) is termed a distribution-invariant risk measure. Distribution-invariant risk measure allows us to simplify notation by denoting $\\rho(F_{X})$ as $\\rho(X)$"}, {"name": "Supremum distance, l1 distance, norm balls B_infty(F,c) and B_1(P,c)", "statement": "We begin by establishing additional notations. For two Cumulative Distribution Functions (CDFs) F and G, the supremum distance between them is defined as $\\| F - G \\|_{\\infty}\\triangleq$ $\\begin{array}{r}{\\operatorname*{sup}_{x}|F(x)-G(x)|}\\end{array}$ We define the $\\ell_{1}$ distance between two Probability Mass Functions $(\\mathrm{PMFs})$ with the same support P and $Q$ as $\\begin{array}{r}{\\| P - Q \\|_{1}\\triangleq \\sum_{i}| P_{i}- Q_{i}|}\\end{array}$ . Furthermore, the set $B_{\\infty}(F,c): =\\{G\\in\\mathcal{D}|\\|G-F\\|_{\\infty}\\leq c\\}$ denotes the supremum norm ball of CDFs centered at F with radius c. Analogously, $B_{1}(P,c)$ represents the $\\ell_{1}$ norm ball of PMFs centered at P with radius c."}, {"name": "Definition 7 and the distributional optimism operator O_c^infty (Section 4.2)", "statement": "Definition 7 Given two CDFs F and G, we say that F is more optimistic than G if $F \\geq G$\n\nThis definition aligns with the intuitive notion that a more optimistic distribution should possess a larger EntRM value. Given that the exponential transformation preserves order, F is more optimistic than G if and only if $E_{\\beta}(F)\\ge E_{\\beta}(G)$ . Following the methodology of Keramati et al. (2020), we introduce the distributional optimism operator $\\mathrm{O}_{c}^{\\infty}:\\mathcal{D}(a,b)\\mapsto$ $\\mathcal{D}(a,b)$ for a level $c\\in(0,1)$ as\n\n$$\n(\\mathrm{O}_{c}^{\\infty}F)(x)\\triangleq [F(x)-c\\mathbb{I}_{[a,b)}(x)]^{+}.\n$$\n\nThis operator shifts the distribution F down by a maximum of c over $[a,b)$ , ensuring that the resulting function ${\\mathrm{O}}_{c}^{\\infty}F$ remains a valid CDF in $\\mathcal{D}(a,b)$ and optimistically dominates all other CDFs within the confidence ball $B_{\\infty}(F,c)^{3}$ . In particular, ${\\mathrm{O}}_{c}^{\\infty}F$ represents maximum permissible downward adjustment under the constraints of the infinity ball, ensuring that for any distribution $G\\in B_{\\infty}(F,c)$\n\n$$\n(\\mathrm{O}_{c}^{\\infty}F)(x)\\leq G(x),\\forall x\\in\\mathbb{R}.\n$$"}, {"name": "Table of notation: CDF classes and the infinity ball (Appendix A)", "statement": "<table><tr><td>Symbol</td><td>Explanation</td></tr><tr><td> $\\mathcal{D}$ </td><td>The space of all CDFs</td></tr><tr><td> $\\mathcal{D}(a,b)$ </td><td>The space of all CDFs supported on  $[a,b]$ </td></tr><tr><td> ${\\mathcal{D}}_{M}$ </td><td>The space of all CDFs supported on  $[0,M]$ </td></tr><tr><td> $B_{\\infty}(F,c)$ </td><td>The  $\\left\\| \\cdot \\right\\|_{\\infty}$  norm ball centered at  $F$  with radius c</td></tr></table>"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace ResearchLevel\n\nnoncomputable def EU (β : ℝ) (μ : Measure ℝ) : ℝ := ∫ x, Real.exp (β * x) ∂μ\n\nnoncomputable def optimismInf (a b c : ℝ) (F : ℝ → ℝ) : ℝ → ℝ :=\n  fun x => max (F x - c * Set.indicator (Set.Ico a b) (fun _ => (1 : ℝ)) x) 0\n\ndef InDab (a b : ℝ) (F : ℝ → ℝ) : Prop :=\n  Monotone F ∧ (∀ x, F x ∈ Set.Icc (0 : ℝ) 1)\n    ∧ (∀ x, ContinuousWithinAt F (Set.Ici x) x)\n    ∧ (∀ x, x < a → F x = 0) ∧ (∀ x, b ≤ x → F x = 1)\n\ntheorem lemma6_lipschitz {M β : ℝ} (hM : 0 ≤ M) (hβ : β ≠ 0)\n    (μ ν : Measure ℝ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]\n    (hμ : μ (Set.Icc 0 M) = 1) (hν : ν (Set.Icc 0 M) = 1) :\n    (|EU β μ - EU β ν|\n      ≤ |Real.exp (β * M) - 1|\n          * ⨆ x : ℝ, |ProbabilityTheory.cdf μ x - ProbabilityTheory.cdf ν x|)\n    ∧ (0 < M → ∃ μ₀ ν₀ : Measure ℝ, IsProbabilityMeasure μ₀ ∧ IsProbabilityMeasure ν₀ ∧\n        μ₀ (Set.Icc 0 M) = 1 ∧ ν₀ (Set.Icc 0 M) = 1 ∧\n        0 < (⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) ∧\n        |EU β μ₀ - EU β ν₀|\n          = |Real.exp (β * M) - 1|\n              * ⨆ x : ℝ, |ProbabilityTheory.cdf μ₀ x - ProbabilityTheory.cdf ν₀ x|) := by\n  sorry\n\ntheorem lemma12_optimistic_bound {n : ℕ} (a b c : ℝ) (hab : a ≤ b) (hc0 : 0 < c)\n    (F : Fin n → ℝ → ℝ)\n    (hFmono : ∀ i, Monotone (F i))\n    (hFright : ∀ i x, ContinuousWithinAt (F i) (Set.Ici x) x)\n    (hFrange : ∀ i x, F i x ∈ Set.Icc (0 : ℝ) 1)\n    (hFlo : ∀ i x, x < a → F i x = 0)\n    (hFhi : ∀ i x, b ≤ x → F i x = 1)\n    (θ θ' : Fin n → ℝ)\n    (hθnn : ∀ i, 0 ≤ θ i) (hθ'nn : ∀ i, 0 ≤ θ' i)\n    (hθsum : ∑ i, θ i = 1) (hθ'sum : ∑ i, θ' i = 1)\n    (hc : ∑ i, |θ i - θ' i| ≤ c) :\n    ∀ x, optimismInf a b c (fun y => ∑ i, θ' i * F i y) x\n        ≤ (∑ i, θ i * F i x) := by\n  sorry\n\ntheorem lemma8_optimism_operator_argmax {a b c : ℝ} (hab : a ≤ b) (hc0 : 0 < c)\n    (F : ℝ → ℝ) (hF : InDab a b F)\n    (ρ : (ℝ → ℝ) → ℝ)\n    (hρM : ∀ P Q : ℝ → ℝ, InDab a b P → InDab a b Q → (∀ x, P x ≤ Q x) → ρ Q ≤ ρ P) :\n    (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n        ∀ x, optimismInf a b c F x ≤ G x)\n    ∧ ( InDab a b (optimismInf a b c F)\n        ∧ (∀ x, |optimismInf a b c F x - F x| ≤ c)\n        ∧ (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n            ρ G ≤ ρ (optimismInf a b c F)) ) := by\n  sorry\n\nend ResearchLevel\n", "target_names": ["lemma6_lipschitz", "lemma12_optimistic_bound", "lemma8_optimism_operator_argmax"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 1815, "end": 2295, "gold": "{a b c : ℝ} (hab : a ≤ b) (hc0 : 0 < c)\n    (F : ℝ → ℝ) (hF : InDab a b F)\n    (ρ : (ℝ → ℝ) → ℝ)\n    (hρM : ∀ P Q : ℝ → ℝ, InDab a b P → InDab a b Q → (∀ x, P x ≤ Q x) → ρ Q ≤ ρ P) :\n    (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n        ∀ x, optimismInf a b c F x ≤ G x)\n    ∧ ( InDab a b (optimismInf a b c F)\n        ∧ (∀ x, |optimismInf a b c F x - F x| ≤ c)\n        ∧ (∀ G : ℝ → ℝ, InDab a b G → (∀ x, |G x - F x| ≤ c) →\n            ρ G ≤ ρ (optimismInf a b c F)) )"}]}
{"item_id": "ch1_proposition_1_11::ch1_proposition_1_11::A-stmt", "paper_id": "ch1_proposition_1_11", "target_name": "ch1_proposition_1_11", "source": "Graduate-level", "nl_statement": "If \\(f\\) is twice continuously differentiable on its domain \\(\\Omega\\), then \\(f\\) is convex if and only if \\(\\nabla^2 f(x)\\succeq 0\\) for all \\(x\\in\\Omega\\).", "context": [{"name": "proposition 1.8", "statement": "Assume \\(f\\colon\\Omega\\to\\mathbb{R}\\) is differentiable. Then, \\(f\\) is convex if and only if for all \\(x,y\\in\\Omega\\) we have\n\\[\nf(y) \\geqslant f(x) + \\nabla f(x)^\\top (y-x)\\,.\n\\]"}, {"name": "proposition 1.7", "statement": "Assume \\(f\\colon\\Omega\\to\\mathbb{R}\\) is continuously differentiable along the line segment between two points \\(x\\) and \\(y\\). Then,\n\\[\nf(y) = f(x)\n+ \\nabla f(x)^\\top (y-x)\n+ \\int_0^1 (1-\\gamma)\\frac{\\partial^2 f(x+\\gamma(y-x))}{\\partial\\gamma^2}\\,\\mathrm{d}\\gamma\n\\]"}, {"name": "Convex function", "statement": "A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is convex if for all \\(x,y\\in\\Omega\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(f(\\gamma x+(1-\\gamma)y)\\leqslant \\gamma f(x)+(1-\\gamma)f(y)\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace\n\nnamespace GraduateLevel\n\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]\n\ntheorem ch1_proposition_1_11 {f : E → ℝ} {s : Set E}\n    (hs : Convex ℝ s) (hs_open : IsOpen s) (hf : ContDiffOn ℝ 2 f s) :\n    ConvexOn ℝ s f ↔ ∀ x ∈ s, ∀ u : E, 0 ≤ iteratedFDeriv ℝ 2 f x ![u, u] := sorry\n\nend GraduateLevel\n", "target_names": ["ch1_proposition_1_11"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 198, "end": 366, "gold": "{f : E → ℝ} {s : Set E}\n    (hs : Convex ℝ s) (hs_open : IsOpen s) (hf : ContDiffOn ℝ 2 f s) :\n    ConvexOn ℝ s f ↔ ∀ x ∈ s, ∀ u : E, 0 ≤ iteratedFDeriv ℝ 2 f x ![u, u]"}]}
{"item_id": "ch11_theorem_11_6::ch11_theorem_11_6::A-stmt", "paper_id": "ch11_theorem_11_6", "target_name": "ch11_theorem_11_6", "source": "Graduate-level", "nl_statement": "Assume \\(\\ell(w, z)\\) is \\(\\alpha\\)-strongly convex over the domain \\(\\Omega\\) and \\(L\\)-Lipschitz. Let \\(\\hat w_S = \\arg \\min_{w\\in \\Omega} \\frac{1}{m} \\sum_{i=1}^m \\ell(w, z_i)\\) denote the empirical risk minimizer (ERM). Then, ERM satisfies\n\\[\n\\Delta_{\\sup} (\\text{ERM}) \\leqslant \\frac{4L^2}{\\alpha m}\\,.\n\\]", "context": [{"name": "Strong convexity", "statement": "A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} if, for all \\(x,y\\in\\Omega\\), the following inequality holds for some \\(\\alpha>0\\): \\[ f(y) \\geqslant f(x) + g(x)^\\top(y-x) + \\frac{\\alpha}{2}\\left\\lVert x-y\\right\\rVert^2 \\]"}, {"name": "Uniform stability", "statement": "The uniform stability of an algorithm \\(A\\) is defined as\n\\[\n\\Delta_{\\sup}(A) = \\sup_{S, S' \\in (X\\times Y)^m } \\sup_{i \\in [m]} |\\ell(A(S), z_i') - \\ell(A(S^{(i)}, z_i')|\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\nopen Metric\n\nnamespace GraduateLevel\n\nnoncomputable def empRisk {E Z : Type*} {m : ℕ} (ℓ : E → Z → ℝ) (S : Fin m → Z) (w : E) : ℝ :=\n  (1 / (m : ℝ)) * ∑ i, ℓ w (S i)\n\ntheorem ch11_theorem_11_6\n    {E Z : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    {Ω : Set E} (hΩconv : Convex ℝ Ω) (hΩne : Ω.Nonempty)\n    (ℓ : E → Z → ℝ) (α L : ℝ) (hα : 0 < α) (hL : 0 ≤ L)\n    (m : ℕ) (hm : 0 < m)\n    (hsc : ∀ z : Z, StrongConvexOn Ω α (fun w => ℓ w z))\n    (hLip : ∀ z : Z, ∀ w ∈ Ω, ∀ w' ∈ Ω, |ℓ w z - ℓ w' z| ≤ L * ‖w - w'‖)\n    (erm : (Fin m → Z) → E)\n    (herm : ∀ S : Fin m → Z, IsMinOn (empRisk ℓ S) Ω (erm S) ∧ erm S ∈ Ω) :\n    ∀ (S S' : Fin m → Z) (i : Fin m),\n      |ℓ (erm (Function.update S i (S' i))) (S' i) - ℓ (erm S) (S' i)|\n        ≤ 4 * L ^ 2 / (α * m) := sorry\n\nend GraduateLevel\n", "target_names": ["ch11_theorem_11_6"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 238, "end": 810, "gold": "{E Z : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    {Ω : Set E} (hΩconv : Convex ℝ Ω) (hΩne : Ω.Nonempty)\n    (ℓ : E → Z → ℝ) (α L : ℝ) (hα : 0 < α) (hL : 0 ≤ L)\n    (m : ℕ) (hm : 0 < m)\n    (hsc : ∀ z : Z, StrongConvexOn Ω α (fun w => ℓ w z))\n    (hLip : ∀ z : Z, ∀ w ∈ Ω, ∀ w' ∈ Ω, |ℓ w z - ℓ w' z| ≤ L * ‖w - w'‖)\n    (erm : (Fin m → Z) → E)\n    (herm : ∀ S : Fin m → Z, IsMinOn (empRisk ℓ S) Ω (erm S) ∧ erm S ∈ Ω) :\n    ∀ (S S' : Fin m → Z) (i : Fin m),\n      |ℓ (erm (Function.update S i (S' i))) (S' i) - ℓ (erm S) (S' i)|\n        ≤ 4 * L ^ 2 / (α * m)"}]}
{"item_id": "ch13_proposition_13_2::ch13_proposition_13_2::A-stmt", "paper_id": "ch13_proposition_13_2", "target_name": "ch13_proposition_13_2", "source": "Graduate-level", "nl_statement": "Let \\(\\Omega \\subseteq \\mathbb{R}^d\\) be a closed convex set. Let \\(\\mathcal{A}\\) denote the affine set \\(\\{x : Ax = b\\}\\) for some \\(A \\in \\mathbb{R}^{n \\times d}\\) and \\(b \\in \\mathbb{R}^n\\). Suppose that the set \\(\\operatorname{ri}(\\Omega) \\cap \\mathcal{A}\\) is non-empty. Then for any \\(x \\in \\Omega \\cap \\mathcal{A}\\), \\[\\mathcal{N}_{\\Omega \\cap \\mathcal{A}}(x) = \\mathcal{N}_{\\Omega}(x) + \\{A^\\top \\lambda : \\lambda \\in \\mathbb{R}^n\\}\\,.\\]", "context": [{"name": "Convex set", "statement": "A set \\(K\\subseteq\\mathbb{R}^n\\) is convex if the line segment between any two points in \\(K\\) is also contained in \\(K\\). Formally, for all \\(x,y\\in K\\) and all scalars \\(\\gamma\\in[0,1]\\) we have \\(\\gamma x+(1-\\gamma)y\\in K\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped Pointwise RealInnerProductSpace\n\nnamespace GraduateLevel\n\ndef normalCone {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (S : Set E) (x : E) : Set E :=\n  {u | ∀ z ∈ S, inner ℝ (z - x) u ≤ 0}\n\ntheorem ch13_proposition_13_2\n    {d n : ℕ}\n    (Ω : Set (EuclideanSpace ℝ (Fin d)))\n    (hΩclosed : IsClosed Ω) (hΩconvex : Convex ℝ Ω)\n    (A : EuclideanSpace ℝ (Fin d) →ₗ[ℝ] EuclideanSpace ℝ (Fin n))\n    (b : EuclideanSpace ℝ (Fin n))\n    (𝒜 : Set (EuclideanSpace ℝ (Fin d)))\n    (h𝒜 : 𝒜 = {x | A x = b})\n    (hCQ : (intrinsicInterior ℝ Ω ∩ 𝒜).Nonempty)\n    (x : EuclideanSpace ℝ (Fin d)) (hx : x ∈ Ω ∩ 𝒜) :\n    normalCone (Ω ∩ 𝒜) x\n      = normalCone Ω x\n        + Set.range (fun μ : EuclideanSpace ℝ (Fin n) => (LinearMap.adjoint A) μ) := sorry\n\nend GraduateLevel\n", "target_names": ["ch13_proposition_13_2"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 269, "end": 775, "gold": "{d n : ℕ}\n    (Ω : Set (EuclideanSpace ℝ (Fin d)))\n    (hΩclosed : IsClosed Ω) (hΩconvex : Convex ℝ Ω)\n    (A : EuclideanSpace ℝ (Fin d) →ₗ[ℝ] EuclideanSpace ℝ (Fin n))\n    (b : EuclideanSpace ℝ (Fin n))\n    (𝒜 : Set (EuclideanSpace ℝ (Fin d)))\n    (h𝒜 : 𝒜 = {x | A x = b})\n    (hCQ : (intrinsicInterior ℝ Ω ∩ 𝒜).Nonempty)\n    (x : EuclideanSpace ℝ (Fin d)) (hx : x ∈ Ω ∩ 𝒜) :\n    normalCone (Ω ∩ 𝒜) x\n      = normalCone Ω x\n        + Set.range (fun μ : EuclideanSpace ℝ (Fin n) => (LinearMap.adjoint A) μ)"}]}
{"item_id": "ch17_proposition_17_3::ch17_proposition_17_3::A-stmt", "paper_id": "ch17_proposition_17_3", "target_name": "ch17_proposition_17_3", "source": "Graduate-level", "nl_statement": "Let \\(x^*\\) be an unconstrained local minimum of \\(f\\colon \\mathbb{R}^n \\to \\mathbb{R}\\) and assume \\(f\\) is continuously differentiable (\\(C^1\\)) in an open set containing \\(x^*\\). Then\n\\begin{enumerate}\n    \\item \\(\\nabla f(x^*) = 0\\) (First-Order Necessary Condition)\n    \\item If in addition \\(f\\) is twice continuously differentiable in an open set around \\(x^*\\), then \\(\\nabla^2 f(x^*) \\succeq 0\\). (Second Order Necessary Condition)\n\\end{enumerate}", "context": [{"name": "Local minimum", "statement": "A point \\(x^*\\) is an unconstrained \\textit{local minimum} if there exists \\(\\epsilon > 0\\) such that \\(f(x^*) \\leqslant f(x)\\) for all \\(x\\) with \\(\\|x - x^*\\| < \\epsilon\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped Gradient\n\nnamespace GraduateLevel\n\ntheorem ch17_proposition_17_3 {n : ℕ}\n    (f : EuclideanSpace ℝ (Fin n) → ℝ) (x : EuclideanSpace ℝ (Fin n))\n    (hmin : IsLocalMin f x) (hC1 : ContDiffAt ℝ 1 f x) :\n    gradient f x = 0\n      ∧ (ContDiffAt ℝ 2 f x →\n          ∀ d : EuclideanSpace ℝ (Fin n),\n            0 ≤ iteratedFDeriv ℝ 2 f x ![d, d]) := sorry\n\nend GraduateLevel\n", "target_names": ["ch17_proposition_17_3"], "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "holes": [{"start": 93, "end": 368, "gold": "{n : ℕ}\n    (f : EuclideanSpace ℝ (Fin n) → ℝ) (x : EuclideanSpace ℝ (Fin n))\n    (hmin : IsLocalMin f x) (hC1 : ContDiffAt ℝ 1 f x) :\n    gradient f x = 0\n      ∧ (ContDiffAt ℝ 2 f x →\n          ∀ d : EuclideanSpace ℝ (Fin n),\n            0 ≤ iteratedFDeriv ℝ 2 f x ![d, d])"}]}
{"item_id": "ch23_theorem_23_1::ch23_theorem_23_1::A-stmt", "paper_id": "ch23_theorem_23_1", "target_name": "ch23_theorem_23_1", "source": "Graduate-level", "nl_statement": "Newton's method uses the update rule \\(x_{t+1} = x_t - \\nabla^2 f(x_t)^{-1}\\nabla f(x_t)\\). Throughout, \\(\\alpha > 0\\) (not written in the lecture, which leaves it implicit by dividing by \\(\\alpha\\)). Given \\(f: \\mathbb{R}^n \\rightarrow \\mathbb{R}\\) and assuming that \\begin{enumerate} \\item \\(f\\) is twice continuously differentiable \\item \\(\\nabla^2 f(x)\\) is Lipschitz: \\(\\| \\nabla^2 f(x) - \\nabla^2 f(x') \\| \\leqslant \\|x-x'\\|\\) \\item \\(\\exists x^*\\) s.t. \\(\\nabla f(x^*) = 0\\) and \\(\\nabla^2 f(x^*) \\succeq \\alpha I\\) and \\(\\|x^0 - x^*\\| \\leqslant \\frac{\\alpha}{2}\\) \\end{enumerate} Then, \\(\\|x_{t+1} - x^*\\| \\leqslant \\frac{1}{\\alpha} \\|x_t - x^*\\|^2\\)", "context": null, "nl_notes": "", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace\nopen Real\n\nnamespace GraduateLevel\n\ntheorem ch23_theorem_23_1\n    {n : ℕ} (f : EuclideanSpace ℝ (Fin n) → ℝ) (α : ℝ) (hα : 0 < α)\n    (hf : ContDiff ℝ 2 f)\n    (hLip : LipschitzWith 1 (fun x => fderiv ℝ (gradient f) x))\n    (xstar : EuclideanSpace ℝ (Fin n))\n    (hstar : gradient f xstar = 0)\n    (hpd : ∀ v : EuclideanSpace ℝ (Fin n),\n        α * ‖v‖ ^ 2 ≤ ⟪(fderiv ℝ (gradient f) xstar) v, v⟫)\n    (x : ℕ → EuclideanSpace ℝ (Fin n))\n    (h0 : ‖x 0 - xstar‖ ≤ α / 2)\n    (hnewton : ∀ t, ∃ H : EuclideanSpace ℝ (Fin n) ≃L[ℝ] EuclideanSpace ℝ (Fin n),\n        (↑H = fderiv ℝ (gradient f) (x t)) ∧\n        x (t + 1) = x t - H.symm (gradient f (x t))) :\n    ∀ t, ‖x (t + 1) - xstar‖ ≤ (1 / α) * ‖x t - xstar‖ ^ 2 := sorry\n\nend GraduateLevel\n", "target_names": ["ch23_theorem_23_1"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 116, "end": 760, "gold": "{n : ℕ} (f : EuclideanSpace ℝ (Fin n) → ℝ) (α : ℝ) (hα : 0 < α)\n    (hf : ContDiff ℝ 2 f)\n    (hLip : LipschitzWith 1 (fun x => fderiv ℝ (gradient f) x))\n    (xstar : EuclideanSpace ℝ (Fin n))\n    (hstar : gradient f xstar = 0)\n    (hpd : ∀ v : EuclideanSpace ℝ (Fin n),\n        α * ‖v‖ ^ 2 ≤ ⟪(fderiv ℝ (gradient f) xstar) v, v⟫)\n    (x : ℕ → EuclideanSpace ℝ (Fin n))\n    (h0 : ‖x 0 - xstar‖ ≤ α / 2)\n    (hnewton : ∀ t, ∃ H : EuclideanSpace ℝ (Fin n) ≃L[ℝ] EuclideanSpace ℝ (Fin n),\n        (↑H = fderiv ℝ (gradient f) (x t)) ∧\n        x (t + 1) = x t - H.symm (gradient f (x t))) :\n    ∀ t, ‖x (t + 1) - xstar‖ ≤ (1 / α) * ‖x t - xstar‖ ^ 2"}]}
{"item_id": "ch14_theorem_14_1::ch14_theorem_14_1::A-stmt", "paper_id": "ch14_theorem_14_1", "target_name": "ch14_theorem_14_1", "source": "Graduate-level", "nl_statement": "\\[\\mathbb{P}\\left(\\forall f\\in F,R(f)\\leq\\frac{2^{9/2}}{\\sqrt{n}}\\int_{0}^{d(0,f)}\\log^{1/2}\\mathcal{D}(F,\\varepsilon,d)d\\varepsilon+2^{7/2}d(0,f)\\sqrt{\\frac{u}{n}}\\right)\\geq 1-e^{-u}\\]\nfor any \\(u > 0\\)", "context": [{"name": "definition 12.3", "statement": "The ε-packing number, \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) , is the marimal cardinality of an E-separated set.\nNote that \\(\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\) is decreasing in ε."}, {"name": "theorem 13.1", "statement": "Assume \\(\\mathcal{F}\\) is a VC-subgraph class and \\(V C(\\mathcal{F})=V\\) Suppose \\(-1\\leq f(x)\\leq 1\\) for all \\(f\\in{\\mathcal{F}}\\) and \\(x\\in\\mathcal{X}\\) .Let \\(x_{1},\\ldots,x_{n}\\in{\\mathcal{X}}\\) and define \\(\\begin{array}{ r }{d(f,g)=\\frac{1}{n}\\sum_{i=1}^{n}|f(x_{i})-g(x_{i})|}\\end{array}\\) . Then\n\\[\\mathcal{D}(\\mathcal{F},\\varepsilon,d)\\leq\\left(\\frac{8 e}{\\varepsilon}\\log\\frac{7}{\\varepsilon}\\right)^{V}.\\]\n(which \\(\\begin{array}{ r }{i s\\le\\left(\\frac{K}{\\varepsilon}\\right)^{V+\\delta}}\\end{array}\\) for some \\(\\delta . \\jmath\\)"}, {"name": "definition 12.2", "statement": "Given \\(\\varepsilon > 0\\) and \\(f_{1},\\ldots,f_{N}\\in\\mathcal{F}\\) ， we say that \\(f_{1},\\ldots,f_{N}\\) are E-separated if \\(d(f_{i},f_{j})>\\varepsilon\\) for any \\(i \\neq j\\)"}, {"name": "Lecture 14 setup (R, d, F) with the Lecture 7 Rademacher convention", "statement": "For \\(f \\in F \\subseteq [-1,1]^n\\), define \\(R(f) = \\frac{1}{n}\\sum_{i=1}^n \\varepsilon_i f_i\\), where \\(\\varepsilon_1,\\dots,\\varepsilon_n\\) are i.i.d. Rademacher random variables, \\(\\mathbb{P}(\\varepsilon_i = 1) = \\mathbb{P}(\\varepsilon_i = -1) = 1/2\\) (Lecture 7), and let \\(d(f,g) := \\left(\\frac{1}{n}\\sum_{i=1}^n (f_i - g_i)^2\\right)^{1/2}\\)."}], "nl_notes": "PROOF NOTE — The NL proof starts with 'Without loss of generality, assume 0 ∈ F'. That reduction is not free here: replacing F by F ∪ {0} can increase the packing number D(F, ε, d), while the statement and the Lean target use D(F, ε, d) for the given F, with no 0 ∈ F hypothesis. The statement is true as given; a proof has to handle 0 ∉ F without that reduction.", "lean": "import Mathlib\n\nopen MeasureTheory\nopen scoped BigOperators ENNReal\n\nnoncomputable section\n\nnamespace GraduateLevel\n\ndef EmpL2 (n : ℕ) : Type := Fin n → ℝ\n\ninstance (n : ℕ) : Zero (EmpL2 n) := inferInstanceAs (Zero (Fin n → ℝ))\n\ndef empEmbed (n : ℕ) : EmpL2 n → EuclideanSpace ℝ (Fin n) :=\n  fun f => (EuclideanSpace.equiv (Fin n) ℝ).symm (fun i => f i / Real.sqrt n)\n\ninstance (n : ℕ) : PseudoMetricSpace (EmpL2 n) :=\n  PseudoMetricSpace.induced (empEmbed n) inferInstance\n\ndef radProc (n : ℕ) (f : Fin n → ℝ) (ε : Fin n → ℝ) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i, ε i * f i\n\ndef IsPacking {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n  ↑t ⊆ s ∧ (↑t : Set A).Pairwise fun x y => eps < dist x y\n\nnoncomputable def packingNumber {A : Type*} [PseudoMetricSpace A] (eps : ℝ) (s : Set A) : WithTop ℕ :=\n  sSup {n | ∃ t, IsPacking t eps s ∧ ↑t.card = n}\n\ntheorem ch14_theorem_14_1\n    {n : ℕ} (hn : 0 < n)\n    (P : Measure (Fin n → ℝ)) [IsProbabilityMeasure P]\n    (hP : P = Measure.pi\n        (fun _ : Fin n => ((1 : ℝ≥0∞) / 2) • (Measure.dirac (-1 : ℝ) + Measure.dirac 1)))\n    (F : Set (EmpL2 n))\n    (hFbdd : ∀ f ∈ F, ∀ i, |(f : Fin n → ℝ) i| ≤ 1)\n    (hFin : ∀ ε : ℝ, 0 < ε → packingNumber ε F ≠ ⊤)\n    (hD : ∀ f ∈ F, ∀ ε ∈ Set.Ioo (0:ℝ) (dist (0 : EmpL2 n) f),\n        (1 : ℝ) ≤ (((packingNumber ε F).untopD 0 : ℕ) : ℝ))\n    (hInt : ∀ f ∈ F, IntervalIntegrable\n        (fun ε => Real.sqrt (Real.log (((packingNumber ε F).untopD 0 : ℕ) : ℝ)))\n        MeasureTheory.volume 0 (dist (0 : EmpL2 n) f))\n    (u : ℝ) (hu : 0 < u) :\n    (1 : ℝ) - Real.exp (-u) ≤\n      (P {ω | ∀ f ∈ F, radProc n (f : Fin n → ℝ) ω ≤\n          (2 ^ (9/2 : ℝ) / Real.sqrt n) *\n            (∫ ε in (0:ℝ)..(dist (0 : EmpL2 n) f),\n                Real.sqrt (Real.log (((packingNumber ε F).untopD 0 : ℕ) : ℝ)))\n          + 2 ^ (7/2 : ℝ) * dist (0 : EmpL2 n) f * Real.sqrt (u / n)}).toReal := sorry\n\nend GraduateLevel\n", "target_names": ["ch14_theorem_14_1"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 907, "end": 1884, "gold": "{n : ℕ} (hn : 0 < n)\n    (P : Measure (Fin n → ℝ)) [IsProbabilityMeasure P]\n    (hP : P = Measure.pi\n        (fun _ : Fin n => ((1 : ℝ≥0∞) / 2) • (Measure.dirac (-1 : ℝ) + Measure.dirac 1)))\n    (F : Set (EmpL2 n))\n    (hFbdd : ∀ f ∈ F, ∀ i, |(f : Fin n → ℝ) i| ≤ 1)\n    (hFin : ∀ ε : ℝ, 0 < ε → packingNumber ε F ≠ ⊤)\n    (hD : ∀ f ∈ F, ∀ ε ∈ Set.Ioo (0:ℝ) (dist (0 : EmpL2 n) f),\n        (1 : ℝ) ≤ (((packingNumber ε F).untopD 0 : ℕ) : ℝ))\n    (hInt : ∀ f ∈ F, IntervalIntegrable\n        (fun ε => Real.sqrt (Real.log (((packingNumber ε F).untopD 0 : ℕ) : ℝ)))\n        MeasureTheory.volume 0 (dist (0 : EmpL2 n) f))\n    (u : ℝ) (hu : 0 < u) :\n    (1 : ℝ) - Real.exp (-u) ≤\n      (P {ω | ∀ f ∈ F, radProc n (f : Fin n → ℝ) ω ≤\n          (2 ^ (9/2 : ℝ) / Real.sqrt n) *\n            (∫ ε in (0:ℝ)..(dist (0 : EmpL2 n) f),\n                Real.sqrt (Real.log (((packingNumber ε F).untopD 0 : ℕ) : ℝ)))\n          + 2 ^ (7/2 : ℝ) * dist (0 : EmpL2 n) f * Real.sqrt (u / n)}).toReal"}]}
{"item_id": "ch15_lemma_15_1::ch15_lemma_15_1::A-stmt", "paper_id": "ch15_lemma_15_1", "target_name": "ch15_lemma_15_1", "source": "Graduate-level", "nl_statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]", "context": null, "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch15_lemma_15_1\n    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]\n    {ξ ν : Ω → ℝ} (hξm : Measurable ξ) (hνm : Measurable ν)\n    {Γ γ : ℝ} (hΓ : 1 ≤ Γ) (hγ : 0 < γ)\n    (hint : ∀ a : ℝ, 0 < a →\n      Integrable (fun ω => max (ξ ω - a) 0) μ ∧ Integrable (fun ω => max (ν ω - a) 0) μ)\n    (htail : ∀ t : ℝ, 0 ≤ t → (μ {ω | t ≤ ν ω}).toReal ≤ Γ * Real.exp (-γ * t))\n    (hcmp : ∀ a : ℝ, 0 < a →\n      ∫ ω, max (ξ ω - a) 0 ∂μ ≤ ∫ ω, max (ν ω - a) 0 ∂μ)\n    {t : ℝ} (ht : 0 ≤ t) :\n    (μ {ω | t ≤ ξ ω}).toReal ≤ Γ * Real.exp 1 * Real.exp (-γ * t) := sorry\n\nend GraduateLevel\n", "target_names": ["ch15_lemma_15_1"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 94, "end": 643, "gold": "{Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]\n    {ξ ν : Ω → ℝ} (hξm : Measurable ξ) (hνm : Measurable ν)\n    {Γ γ : ℝ} (hΓ : 1 ≤ Γ) (hγ : 0 < γ)\n    (hint : ∀ a : ℝ, 0 < a →\n      Integrable (fun ω => max (ξ ω - a) 0) μ ∧ Integrable (fun ω => max (ν ω - a) 0) μ)\n    (htail : ∀ t : ℝ, 0 ≤ t → (μ {ω | t ≤ ν ω}).toReal ≤ Γ * Real.exp (-γ * t))\n    (hcmp : ∀ a : ℝ, 0 < a →\n      ∫ ω, max (ξ ω - a) 0 ∂μ ≤ ∫ ω, max (ν ω - a) 0 ∂μ)\n    {t : ℝ} (ht : 0 ≤ t) :\n    (μ {ω | t ≤ ξ ω}).toReal ≤ Γ * Real.exp 1 * Real.exp (-γ * t)"}]}
{"item_id": "ch26_lemma_26_1::ch26_lemma_26_1::A-stmt", "paper_id": "ch26_lemma_26_1", "target_name": "ch26_lemma_26_1", "source": "Graduate-level", "nl_statement": "\\[\\mathbb{E}\\operatorname*{sup}_{t\\in T}\\left|\\sum_{i=1}^{n}\\varepsilon_{i}\\varphi_{i}(t_{i})\\right|\\leq 2\\mathbb{E}\\operatorname*{sup}_{t\\in T}\\left|\\sum_{i=1}^{n}\\varepsilon_{i}t_{i}\\right|\\]", "context": [{"name": "theorem 26.1", "statement": "\\(I f - 1 \\leq f \\leq 1\\) ，then\n\\[\\mathbb{P}\\left(Z(x)\\leq 2\\mathbb{E}R(x)+2{\\sqrt{\\frac{2 t}{n}}}\\right)\\geq 1-e^{-t}.\\]\n\\(I f 0 \\leq f \\leq 1\\) ，then\n\\[\\mathbb{P}\\left(Z(x)\\le 2\\mathbb{E}R(x)+\\sqrt{\\frac{2 t}{n}}\\right)\\ge 1-e^{-t}.\\]\nConsider \\(\\begin{array}{ r }{\\mathbb E_{\\varepsilon}R(x)=\\mathbb E_{\\varepsilon}\\operatorname*{sup}_{f\\in\\mathcal{F}}\\frac{1}{n}\\sum_{i=1}^{n}\\varepsilon_{i}f(x_{i})}\\end{array}\\) . Since \\(x_{i}\\) are fixed, \\(f(x_{i})\\) are just vectors. Let \\(F\\subseteq\\mathbb{R}^{n},f\\in F\\) where \\(f=(f_{1},\\ldots,f_{n})\\)\nDefine contraction \\(\\varphi_{i}:\\mathbb{R}\\mapsto\\mathbb{R}\\) for \\(i=1,\\ldots,n\\) such that \\(\\varphi_{i}(0)=0\\) and \\(|\\varphi_{i}(s)-\\varphi_{i}(t)|\\leq|s-t|\\)\nLet \\(G:\\mathbb{R}\\mapsto\\mathbb{R}\\) be convex and non-decreasing.\nThe following theorem is called Comparison inequality for Rademacher process.\nTheorem 26.2.\n\\[\\mathbb{E}_{\\varepsilon}G\\left(\\operatorname*{sup}_{f\\in F}\\sum\\varepsilon_{i}\\varphi_{i}(f_{i})\\right)\\leq\\mathbb{E}_{\\varepsilon}G\\left(\\operatorname*{sup}_{f\\in F}\\sum\\varepsilon_{i}f_{i}\\right).\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ ({-1, 1} : Finset ℤ) }))\n\ntheorem ch26_lemma_26_1\n    {n : ℕ} {T : Type*} [Nonempty T]\n    (t : T → (Fin n → ℝ))\n    (φ : Fin n → ℝ → ℝ)\n    (hφ0 : ∀ i, φ i 0 = 0)\n    (hφL : ∀ i s s', |φ i s' - φ i s| ≤ |s' - s|)\n    (hbddφ : ∀ σ : RadSigns n,\n      BddAbove (Set.range (fun τ : T => |∑ i : Fin n, (σ i : ℝ) * φ i (t τ i)|)))\n    (hbdd : ∀ σ : RadSigns n,\n      BddAbove (Set.range (fun τ : T => |∑ i : Fin n, (σ i : ℝ) * t τ i|))) :\n    (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n        ∑ σ : RadSigns n,\n          ⨆ τ : T, |∑ i : Fin n, (σ i : ℝ) * φ i (t τ i)|\n      ≤ 2 * ((Fintype.card (RadSigns n) : ℝ)⁻¹ *\n        ∑ σ : RadSigns n,\n          ⨆ τ : T, |∑ i : Fin n, (σ i : ℝ) * t τ i|) := sorry\n\nend GraduateLevel\n", "target_names": ["ch26_lemma_26_1"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 285, "end": 917, "gold": "{n : ℕ} {T : Type*} [Nonempty T]\n    (t : T → (Fin n → ℝ))\n    (φ : Fin n → ℝ → ℝ)\n    (hφ0 : ∀ i, φ i 0 = 0)\n    (hφL : ∀ i s s', |φ i s' - φ i s| ≤ |s' - s|)\n    (hbddφ : ∀ σ : RadSigns n,\n      BddAbove (Set.range (fun τ : T => |∑ i : Fin n, (σ i : ℝ) * φ i (t τ i)|)))\n    (hbdd : ∀ σ : RadSigns n,\n      BddAbove (Set.range (fun τ : T => |∑ i : Fin n, (σ i : ℝ) * t τ i|))) :\n    (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n        ∑ σ : RadSigns n,\n          ⨆ τ : T, |∑ i : Fin n, (σ i : ℝ) * φ i (t τ i)|\n      ≤ 2 * ((Fintype.card (RadSigns n) : ℝ)⁻¹ *\n        ∑ σ : RadSigns n,\n          ⨆ τ : T, |∑ i : Fin n, (σ i : ℝ) * t τ i|)"}]}
{"item_id": "ch37_lemma_37_1::ch37_lemma_37_1::A-stmt", "paper_id": "ch37_lemma_37_1", "target_name": "ch37_lemma_37_1", "source": "Graduate-level", "nl_statement": "Let\n\\[V(x)=\\mathbb{E}_{x^{\\prime}}\\operatorname*{sup}_{f\\in\\mathcal{F}}\\sum_{i=1}^{n}(f(x_{i})-f(x_{i}^{\\prime}))^{2}\\]\nand \\(a \\leq f \\leq b\\) for all \\(f\\in{\\mathcal{F}}\\) . Then\n\\[\\begin{array}{ r }{\\operatorname{\\mathbb{P}}\\left(V\\leq 4\\operatorname{\\mathbb{E}}V+(b-a)^{2}t\\right)\\geq 1-4\\cdot 2^{-t}.}\\end{array}\\]", "context": [{"name": "theorem 33.1", "statement": "\\[\\mathbb{E}e^{{\\frac{1}{4}}d(A,x)}=\\int e^{{\\frac{1}{4}}d(A,x)}d P^{n}(x)\\leq{\\frac{1}{P^{n}(A)}}\\]\nand\n\\[P^{n}(d(A,x)\\geq t)\\leq{\\frac{1}{P^{n}(A)}}e^{-t/4}.\\]"}, {"name": "lemma 36.1", "statement": "Let \\(0\\leq g_{1},g_{2}\\leq 1,g_{i}:\\mathcal{X}\\mapsto [0,1]\\) . Then\n\\[\\int\\operatorname*{min}\\left(2,{\\frac{1}{g_{1}(x)}},{\\frac{1}{g_{2}(x)}}\\right)d P(x)\\cdot\\int g_{1}(x)d P(x)\\cdot\\int g_{2}(x)d P(x)\\leq 1\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace GraduateLevel\n\ntheorem ch37_lemma_37_1\n    {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]\n    (n : ℕ) (a b : ℝ) (hab : a ≤ b)\n    (F : Set (X → ℝ)) (hF : F.Nonempty)\n    (hbound : ∀ f ∈ F, ∀ x, a ≤ f x ∧ f x ≤ b)\n    (V : (Fin n → X) → ℝ)\n    (hVdef : V = fun x => ∫ x',\n      (⨆ f ∈ F, ∑ i : Fin n, (f (x i) - f (x' i))^2) ∂(Measure.pi (fun _ : Fin n => μ)))\n    (hVintegrand : ∀ x : Fin n → X,\n      Integrable (fun x' => ⨆ f ∈ F, ∑ i : Fin n, (f (x i) - f (x' i)) ^ 2)\n        (Measure.pi (fun _ : Fin n => μ)))\n    (hVmeas : Measurable V)\n    (hVint : Integrable V (Measure.pi (fun _ : Fin n => μ))) :\n    ∀ t : ℝ, 0 ≤ t →\n      ((Measure.pi (fun _ : Fin n => μ))\n        {x | V x ≤ 4 * (∫ x, V x ∂(Measure.pi (fun _ : Fin n => μ))) + (b - a)^2 * t}).toReal\n      ≥ 1 - 4 * (2 : ℝ) ^ (-t) := sorry\n\nend GraduateLevel\n", "target_names": ["ch37_lemma_37_1"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "holes": [{"start": 89, "end": 864, "gold": "{X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ]\n    (n : ℕ) (a b : ℝ) (hab : a ≤ b)\n    (F : Set (X → ℝ)) (hF : F.Nonempty)\n    (hbound : ∀ f ∈ F, ∀ x, a ≤ f x ∧ f x ≤ b)\n    (V : (Fin n → X) → ℝ)\n    (hVdef : V = fun x => ∫ x',\n      (⨆ f ∈ F, ∑ i : Fin n, (f (x i) - f (x' i))^2) ∂(Measure.pi (fun _ : Fin n => μ)))\n    (hVintegrand : ∀ x : Fin n → X,\n      Integrable (fun x' => ⨆ f ∈ F, ∑ i : Fin n, (f (x i) - f (x' i)) ^ 2)\n        (Measure.pi (fun _ : Fin n => μ)))\n    (hVmeas : Measurable V)\n    (hVint : Integrable V (Measure.pi (fun _ : Fin n => μ))) :\n    ∀ t : ℝ, 0 ≤ t →\n      ((Measure.pi (fun _ : Fin n => μ))\n        {x | V x ≤ 4 * (∫ x, V x ∂(Measure.pi (fun _ : Fin n => μ))) + (b - a)^2 * t}).toReal\n      ≥ 1 - 4 * (2 : ℝ) ^ (-t)"}]}
{"item_id": "ch4_theorem_4_17::ch4_theorem_4_17::A-stmt", "paper_id": "ch4_theorem_4_17", "target_name": "ch4_theorem_4_17", "source": "Graduate-level", "nl_statement": "If \\(\\mathcal{F}\\) is a function class from \\(Z \\mapsto \\mathbb{R}\\), then\n\\[\nR_S(\\mathcal{F}) \\leq 12\\int_{0}^{\\infty}\\sqrt{\\frac{\\log N(\\epsilon, \\mathcal{F}, L_2(P_n))}{n}}\\,d\\epsilon.\n\\]", "context": [{"name": "Empirical Rademacher complexity", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\)."}, {"name": "definition 4.14", "statement": "\\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\)."}, {"name": "definition 4.15", "statement": "The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]"}, {"name": "proposition 4.12", "statement": "Let \\(\\mathcal{F}\\) be a collection of functions mapping \\(Z \\mapsto \\mathbb{R}\\) and let \\(\\mathcal{Q}\\) be defined as in the referenced equation. Assume that \\(\\frac{1}{\\sqrt{n}} \\|v\\|_2 \\le M < \\infty\\) for all \\(v \\in \\mathcal{Q}\\). Then,\n\\[\nR_S(\\mathcal{F}) \\leq \\sqrt{\\frac{2 M^2 \\log |\\mathcal{Q}|}{n}}.\n\\]"}, {"name": "Output space \\(\\mathcal{Q}\\) and the \\(L_2(P_n)\\) metric", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), define the output space\n\\[\n\\mathcal{Q} \\triangleq \\left\\{ \\left(f(z_1), \\dots, f(z_n)\\right)^\\top : f \\in \\mathcal{F} \\right\\} \\subseteq \\mathbb{R}^n.\n\\]\nCorrespondingly, the empirical Rademacher complexity can be rewritten as a maximization over the output space \\(\\mathcal{Q}\\) instead of the function space \\(\\mathcal{F}\\):\n\\[\nR_S(\\mathcal{F}) = \\mathbb{E}_{\\sigma}\\left[ \\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle \\sigma, v \\rangle \\right].\n\\]\nWe use the metric \\(\\rho(v, v') = \\frac{1}{\\sqrt{n}} \\|v - v'\\|_2\\). In the literature, an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to \\(\\rho\\) is also referred to as an \\(\\epsilon\\)-cover of the function class \\(\\mathcal{F}\\) under the \\(L_2(P_n)\\) metric, where \\(P_n\\) denotes the empirical distribution, i.e. the uniform distribution over the observations \\(z_1, \\dots, z_n\\). In particular,\n\\[\nL_2(P_n)(f, f') = \\sqrt{\\frac{1}{n} \\sum_{i=1}^n (f(z_i) - f'(z_i))^2}.\n\\]\nWe can write a trivial correspondence between both the output and function class points of view as follows:\n\\[\nN(\\epsilon, \\mathcal{F}, L_2(P_n)) = N\\left(\\epsilon, \\mathcal{Q}, \\frac{1}{\\sqrt{n}} \\|\\cdot\\|_2\\right).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def empiricalNorm (n : ℕ) (f : Fin n → ℝ) : ℝ :=\n  Real.sqrt ((n : ℝ)⁻¹ * ∑ k : Fin n, f k ^ 2)\n\nlemma empiricalNorm_nonneg (n : ℕ) (f : Fin n → ℝ) : 0 ≤ empiricalNorm n f :=\n  Real.sqrt_nonneg _\n\nlemma empiricalNorm_zero (n : ℕ) : empiricalNorm n (0 : Fin n → ℝ) = 0 := by\n  unfold empiricalNorm; simp\n\nlemma empiricalNorm_neg (n : ℕ) (f : Fin n → ℝ) : empiricalNorm n (-f) = empiricalNorm n f := by\n  unfold empiricalNorm\n  congr 2\n  apply Finset.sum_congr rfl\n  intro i _\n  simp only [Pi.neg_apply, neg_sq]\n\ndef EmpiricalSpace (n : ℕ) := Fin n → ℝ\n\nnamespace EmpiricalSpace\n\ninstance (n : ℕ) : Inhabited (EmpiricalSpace n) := ⟨fun _ => 0⟩\ninstance (n : ℕ) : Zero (EmpiricalSpace n) := ⟨fun _ => 0⟩\ninstance (n : ℕ) : Add (EmpiricalSpace n) := ⟨fun a b i => a i + b i⟩\ninstance (n : ℕ) : Sub (EmpiricalSpace n) := ⟨fun a b i => a i - b i⟩\ninstance (n : ℕ) : Neg (EmpiricalSpace n) := ⟨fun a i => -a i⟩\n\nnoncomputable instance instDist (n : ℕ) : Dist (EmpiricalSpace n) where\n  dist a b := empiricalNorm n (a - b)\n\nnoncomputable instance instEDist (n : ℕ) : EDist (EmpiricalSpace n) where\n  edist a b := ENNReal.ofReal (dist a b)\n\nlemma dist_self (n : ℕ) (a : EmpiricalSpace n) : dist a a = 0 := by\n  show empiricalNorm n (a - a) = 0\n  have h : (a - a : EmpiricalSpace n) = 0 := by\n    funext i\n    show a i - a i = (0 : Fin n → ℝ) i\n    simp\n  rw [h]\n  exact empiricalNorm_zero n\n\nlemma dist_comm (n : ℕ) (a b : EmpiricalSpace n) : dist a b = dist b a := by\n  show empiricalNorm n (a - b) = empiricalNorm n (b - a)\n  have h : (a - b : EmpiricalSpace n) = -(b - a) := by\n    funext i\n    show a i - b i = -(b i - a i)\n    ring\n  rw [h]\n  exact empiricalNorm_neg n _\n\nlemma dist_triangle (n : ℕ) (a b c : EmpiricalSpace n) :\n    dist a c ≤ dist a b + dist b c := by\n  show empiricalNorm n (a - c) ≤ empiricalNorm n (a - b) + empiricalNorm n (b - c)\n  unfold empiricalNorm\n  have eq_ac : ∀ k, (a - c : EmpiricalSpace n) k = a k - c k := fun k => rfl\n  have eq_ab : ∀ k, (a - b : EmpiricalSpace n) k = a k - b k := fun k => rfl\n  have eq_bc : ∀ k, (b - c : EmpiricalSpace n) k = b k - c k := fun k => rfl\n  simp_rw [eq_ac, eq_ab, eq_bc]\n  by_cases hn : n = 0\n  · simp [hn]\n  have hn_pos : (0 : ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)\n  have hn_inv_nonneg : 0 ≤ (n : ℝ)⁻¹ := inv_nonneg.mpr (le_of_lt hn_pos)\n  have h_factor : ∀ f : Fin n → ℝ, Real.sqrt ((n : ℝ)⁻¹ * ∑ i, (f i)^2) =\n      Real.sqrt (n : ℝ)⁻¹ * Real.sqrt (∑ i, (f i)^2) := fun f => by\n    rw [Real.sqrt_mul hn_inv_nonneg]\n  rw [h_factor (fun i => a i - c i), h_factor (fun i => a i - b i),\n      h_factor (fun i => b i - c i), ← mul_add]\n  apply mul_le_mul_of_nonneg_left _ (Real.sqrt_nonneg _)\n  let e := EuclideanSpace.equiv (Fin n) ℝ\n  let a' : EuclideanSpace ℝ (Fin n) := e.symm a\n  let b' : EuclideanSpace ℝ (Fin n) := e.symm b\n  let c' : EuclideanSpace ℝ (Fin n) := e.symm c\n  have h_a' : ∀ i, a' i = a i := fun i => rfl\n  have h_b' : ∀ i, b' i = b i := fun i => rfl\n  have h_c' : ∀ i, c' i = c i := fun i => rfl\n  have h_norm : ∀ (x y : EuclideanSpace ℝ (Fin n)),\n      Real.sqrt (∑ i, (x i - y i)^2) = ‖x - y‖ := fun x y => by\n    rw [EuclideanSpace.norm_eq]\n    congr 1\n    apply Finset.sum_congr rfl\n    intro i _\n    rw [Real.norm_eq_abs, sq_abs]\n    rfl\n  simp only [← h_a', ← h_b', ← h_c']\n  rw [h_norm a' c', h_norm a' b', h_norm b' c']\n  have h_eq : a' - c' = (a' - b') + (b' - c') := (sub_add_sub_cancel a' b' c').symm\n  rw [h_eq]\n  exact norm_add_le (a' - b') (b' - c')\n\nlemma edist_dist (n : ℕ) (a b : EmpiricalSpace n) :\n    edist a b = ENNReal.ofReal (dist a b) := rfl\n\nnoncomputable instance instPseudoMetricSpace (n : ℕ) : PseudoMetricSpace (EmpiricalSpace n) where\n  dist_self := dist_self n\n  dist_comm := dist_comm n\n  dist_triangle := dist_triangle n\n  edist_dist := edist_dist n\n\nend EmpiricalSpace\n\ndef empiricalMetricImage {X : Type*} (n : ℕ) (x : Fin n → X) (g : X → ℝ) : EmpiricalSpace n :=\n  fun i => g (x i)\n\ndef IsENet {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n  s ⊆ ⋃ x ∈ t, Metric.closedBall x eps\n\nnoncomputable def coveringNumber {A : Type*} [PseudoMetricSpace A] (eps : ℝ) (s : Set A) : WithTop ℕ :=\n  sInf {n | ∃ t, IsENet t eps s ∧ (t.card : WithTop ℕ) = n}\n\ndef Signs (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (Signs n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empiricalRademacherComplexity_without_abs {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (Signs n) : ℝ)⁻¹ *\n    ∑ σ : Signs n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ntheorem ch4_theorem_4_17\n    {ι : Type*} [Nonempty ι] {𝒳 : Type*}\n    (n : ℕ) (hn : 0 < n)\n    (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳)\n    (s : Set (EmpiricalSpace n))\n    (hs : s = Set.range (fun i => empiricalMetricImage n S (f i)))\n    (hcov : ∀ ε ∈ Set.Ioi (0 : ℝ), coveringNumber ε s ≠ ⊤)\n    (hint : MeasureTheory.IntegrableOn\n        (fun ε : ℝ =>\n          Real.sqrt\n            (Real.log (((coveringNumber ε s).untopD 0 : ℕ) : ℝ)\n              / (n : ℝ)))\n        (Set.Ioi 0)) :\n    empiricalRademacherComplexity_without_abs n f S ≤\n      12 *\n        ∫ ε in Set.Ioi (0 : ℝ),\n          Real.sqrt\n            (Real.log (((coveringNumber ε s).untopD 0 : ℕ) : ℝ)\n              / (n : ℝ)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch4_theorem_4_17"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 4717, "end": 5373, "gold": "{ι : Type*} [Nonempty ι] {𝒳 : Type*}\n    (n : ℕ) (hn : 0 < n)\n    (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳)\n    (s : Set (EmpiricalSpace n))\n    (hs : s = Set.range (fun i => empiricalMetricImage n S (f i)))\n    (hcov : ∀ ε ∈ Set.Ioi (0 : ℝ), coveringNumber ε s ≠ ⊤)\n    (hint : MeasureTheory.IntegrableOn\n        (fun ε : ℝ =>\n          Real.sqrt\n            (Real.log (((coveringNumber ε s).untopD 0 : ℕ) : ℝ)\n              / (n : ℝ)))\n        (Set.Ioi 0)) :\n    empiricalRademacherComplexity_without_abs n f S ≤\n      12 *\n        ∫ ε in Set.Ioi (0 : ℝ),\n          Real.sqrt\n            (Real.log (((coveringNumber ε s).untopD 0 : ℕ) : ℝ)\n              / (n : ℝ))"}]}
{"item_id": "ch4_theorem_4_8::ch4_theorem_4_8::A-stmt", "paper_id": "ch4_theorem_4_8", "target_name": "ch4_theorem_4_8", "source": "Graduate-level", "nl_statement": "Assume that our infinite hypothesis class \\(\\mathcal{H}\\) can be parameterized by \\(\\mathcal{H} = \\{h_\\theta : \\theta \\in \\mathbb{R}^p, \\|\\theta\\|_2 \\leq B\\}\\). Suppose \\(\\ell((x,y), \\theta) \\in [0,1]\\), and \\(\\ell((x,y), \\theta)\\) is \\(\\kappa\\)-Lipschitz in \\(\\theta\\) with respect to the \\(\\ell_2\\)-norm for all \\((x, y)\\). Then, with probability at least \\(1-O(\\exp(-\\Omega(p)))\\), we have\n\\[\n\\forall \\theta, \\quad |\\widehat{L}(\\theta)- L(\\theta)| \\leq O\\!\\left(\\sqrt{\\frac{p \\max(\\ln(\\kappa Bn), 1)}{n}}\\right).\n\\]", "context": [{"name": "\\(\\kappa\\)-Lipschitz functions", "statement": "Let \\(\\kappa \\geq 0\\) and let \\(\\|\\cdot\\|\\) be a norm on the domain \\(D\\). A function \\(L : D \\to \\mathbb{R}\\) is said to be \\(\\kappa\\)-Lipschitz with respect to \\(\\|\\cdot\\|\\) if for all \\(\\theta, \\theta' \\in D\\),\n\\[\n|L(\\theta) - L(\\theta')| \\leq \\kappa\\,\\|\\theta - \\theta'\\|.\n\\]"}, {"name": "\\(\\epsilon\\)-cover of \\(\\ell_2\\) ball", "statement": "Let \\(B, \\epsilon > 0\\), and let \\(S = \\{x \\in \\mathbb{R}^p : \\|x\\|_2 \\leq B\\}\\). Then there exists an \\(\\epsilon\\)-cover of \\(S\\) with respect to the \\(\\ell_2\\)-norm with at most \\(\\max\\!\\left(\\left(\\dfrac{3B\\sqrt{p}}{\\epsilon}\\right)^p,\\, 1\\right)\\) elements."}, {"name": "Uniform convergence bound for finite hypothesis classes", "statement": "Suppose that the hypothesis class \\(\\mathcal{H}\\) is finite and the loss function \\(\\ell\\) is bounded in \\([0,1]\\), i.e. \\(0 \\leq \\ell((x,y),h) \\leq 1\\). Then for all \\(\\delta\\) with \\(0 < \\delta < \\tfrac{1}{2}\\), with probability at least \\(1 - \\delta\\),\n\\[\n|L(h) - \\widehat{L}(h)| \\leq \\sqrt{\\frac{\\ln|\\mathcal{H}| + \\ln(2/\\delta)}{2n}} \\qquad \\forall h \\in \\mathcal{H}.\n\\]\nAs a corollary,\n\\[\nL(\\hat{h}) - L(h^{*}) \\leq \\sqrt{\\frac{2(\\ln|\\mathcal{H}| + \\ln(2/\\delta))}{n}}.\n\\]"}, {"name": "Risk and empirical-risk minimizer setup for finite hypothesis classes", "statement": "Let \\(\\mathcal{H}\\) be a finite hypothesis class and let \\(\\ell\\) be a loss function with \\(0 \\leq \\ell((x,y),h) \\leq 1\\). For \\(h \\in \\mathcal{H}\\), define the population risk\n\\[\nL(h) = \\mathbb{E}_{(x,y) \\sim \\mathcal{D}}[\\ell((x,y),h)].\n\\]\nGiven \\(n\\) iid samples \\((x^{(1)},y^{(1)}), \\dots, (x^{(n)},y^{(n)})\\) from \\(\\mathcal{D}\\), define the empirical risk\n\\[\n\\widehat{L}(h) = \\frac{1}{n}\\sum_{i=1}^n \\ell((x^{(i)},y^{(i)}),h).\n\\]\nLet \\(\\hat{h} \\in \\arg\\min_{h \\in \\mathcal{H}} \\widehat{L}(h)\\) be an empirical-risk minimizer and let \\(h^* \\in \\arg\\min_{h \\in \\mathcal{H}} L(h)\\) be a population-risk minimizer."}, {"name": "Remark 4.6", "statement": "We can actually prove a stronger version of Lemma 4.5: there exists an \\(\\epsilon\\)-cover of \\(S\\) with at most \\(\\left(\\frac{3B}{\\epsilon}\\right)^p\\) elements."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch4_theorem_4_8 :\n    ∃ c₀ c₁ c₂ : ℝ, 0 < c₀ ∧ 0 < c₁ ∧ 0 < c₂ ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        {𝒳 𝒴 : Type*} [MeasurableSpace 𝒳] [MeasurableSpace 𝒴]\n        (p n : ℕ) [NeZero n] (κ B : ℝ)\n        (ℓ : (𝒳 × 𝒴) → EuclideanSpace ℝ (Fin p) → ℝ)\n        (Z : Fin n → Ω → (𝒳 × 𝒴)),\n        1 ≤ p → 1 ≤ n → 0 ≤ κ → 0 < B →\n        (∀ z θ, ‖θ‖ ≤ B → ℓ z θ ∈ Set.Icc (0 : ℝ) 1) →\n        (∀ z θ θ', ‖θ‖ ≤ B → ‖θ'‖ ≤ B → |ℓ z θ - ℓ z θ'| ≤ κ * dist θ θ') →\n        iIndepFun Z μ →\n        (∀ i, ProbabilityTheory.IdentDistrib (Z i) (Z 0) μ μ) →\n        (∀ θ, Measurable (fun z => ℓ z θ)) →\n        (∀ i θ, AEMeasurable (fun ω => ℓ (Z i ω) θ) μ) →\n        let L : EuclideanSpace ℝ (Fin p) → ℝ := fun θ => ∫ ω, ℓ (Z 0 ω) θ ∂μ\n        let Lhat : Ω → EuclideanSpace ℝ (Fin p) → ℝ := fun ω θ => (1 / (n : ℝ)) * ∑ i, ℓ (Z i ω) θ\n        1 - c₁ * Real.exp (- c₂ * p) ≤\n          μ.real {ω | ∀ θ : EuclideanSpace ℝ (Fin p), ‖θ‖ ≤ B →\n            |Lhat ω θ - L θ| ≤ c₀ * Real.sqrt (p * max (Real.log (κ * B * n)) 1 / n)} := sorry\n\nend GraduateLevel\n", "target_names": ["ch4_theorem_4_8"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 108, "end": 1152, "gold": ":\n    ∃ c₀ c₁ c₂ : ℝ, 0 < c₀ ∧ 0 < c₁ ∧ 0 < c₂ ∧\n      ∀ {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n        {𝒳 𝒴 : Type*} [MeasurableSpace 𝒳] [MeasurableSpace 𝒴]\n        (p n : ℕ) [NeZero n] (κ B : ℝ)\n        (ℓ : (𝒳 × 𝒴) → EuclideanSpace ℝ (Fin p) → ℝ)\n        (Z : Fin n → Ω → (𝒳 × 𝒴)),\n        1 ≤ p → 1 ≤ n → 0 ≤ κ → 0 < B →\n        (∀ z θ, ‖θ‖ ≤ B → ℓ z θ ∈ Set.Icc (0 : ℝ) 1) →\n        (∀ z θ θ', ‖θ‖ ≤ B → ‖θ'‖ ≤ B → |ℓ z θ - ℓ z θ'| ≤ κ * dist θ θ') →\n        iIndepFun Z μ →\n        (∀ i, ProbabilityTheory.IdentDistrib (Z i) (Z 0) μ μ) →\n        (∀ θ, Measurable (fun z => ℓ z θ)) →\n        (∀ i θ, AEMeasurable (fun ω => ℓ (Z i ω) θ) μ) →\n        let L : EuclideanSpace ℝ (Fin p) → ℝ := fun θ => ∫ ω, ℓ (Z 0 ω) θ ∂μ\n        let Lhat : Ω → EuclideanSpace ℝ (Fin p) → ℝ := fun ω θ => (1 / (n : ℝ)) * ∑ i, ℓ (Z i ω) θ\n        1 - c₁ * Real.exp (- c₂ * p) ≤\n          μ.real {ω | ∀ θ : EuclideanSpace ℝ (Fin p), ‖θ‖ ≤ B →\n            |Lhat ω θ - L θ| ≤ c₀ * Real.sqrt (p * max (Real.log (κ * B * n)) 1 / n)}"}]}
{"item_id": "ch5_theorem_5_10::ch5_theorem_5_10::A-stmt", "paper_id": "ch5_theorem_5_10", "target_name": "ch5_theorem_5_10", "source": "Graduate-level", "nl_statement": "Let \\(C(\\theta)=\\sum_{j=1}^{m}\\left|w_{j}\\right|\\left\\|u_{j}\\right\\|_{2}\\), and for some constant \\(B > 0\\) consider the hypothesis class\n\\[\n\\mathcal{H}=\\left\\{f_{\\theta} \\mid C(\\theta) \\leq B\\right\\}.\n\\]\nIf \\(\\left\\|x^{(i)}\\right\\|_{2} \\leq C\\) for all \\(i \\in\\{1, \\ldots, n\\}\\), then\n\\[\nR_{S}(\\mathcal{H}) \\leq \\frac{2 B C}{\\sqrt{n}}.\n\\]", "context": [{"name": "Removing the absolute value in Rademacher complexity", "statement": "Let \\(\\sigma = (\\sigma_1, \\dots, \\sigma_n)\\) and \\(f_{\\theta}(x) = \\left(f_{\\theta}\\left(x^{(1)}\\right), \\dots, f_{\\theta}\\left(x^{(n)}\\right)\\right)\\). Suppose that for any \\(\\sigma \\in \\{\\pm 1\\}^n\\), \\(\\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle \\geq 0\\). Then,\n\\[\n\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\left| \\langle \\sigma, f_{\\theta}(x) \\rangle \\right| \\right] \\leq 2\\, \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle \\right].\n\\]"}, {"name": "Talagrand's lemma", "statement": "Let \\(\\phi:\\mathbb{R}\\to\\mathbb{R}\\) be a \\(\\kappa\\)-Lipschitz function. Then\n\\[\nR_S(\\phi\\circ \\mathcal{H})\\leq \\kappa R_S(\\mathcal{H}),\n\\]\nwhere \\(\\phi\\circ\\mathcal{H} = \\{z\\mapsto \\phi(h(z))\\mid h\\in\\mathcal{H}\\}\\)."}, {"name": "Rademacher complexity of l2-bounded linear functions", "statement": "Let \\(\\mathcal{H} = \\{x \\mapsto \\langle w,x \\rangle \\mid w \\in \\mathbb{R}^d,\\, \\|w\\|_2 \\le B\\}\\) for some constant \\(B > 0\\). Moreover, assume \\(\\mathbb{E}_{x \\sim P}[\\|x\\|_2^2] \\leq C^2\\), where \\(P\\) is some distribution and \\(C > 0\\) is a constant. Then\n\\[\nR_S(\\mathcal{H}) \\le \\frac{B}{n} \\sqrt{\\sum_{i=1}^n \\|x^{(i)}\\|_2^2},\n\\]\nand\n\\[\nR_n(\\mathcal{H}) \\le \\frac{BC}{\\sqrt{n}}.\n\\]"}, {"name": "Empirical Rademacher complexity", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\)."}, {"name": "Two-layer ReLU network", "statement": "Given a width \\(m\\), let \\(w_1,\\dots,w_m \\in \\mathbb{R}\\), \\(u_1,\\dots,u_m \\in \\mathbb{R}^d\\), and let \\(\\phi\\) denote the ReLU activation. For \\(\\theta=(w_1,\\dots,w_m,u_1,\\dots,u_m)\\), define the two-layer ReLU network\n\\[\nf_{\\theta}(x)=\\sum_{j=1}^m w_j \\phi\\left(u_j^\\top x\\right).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped BigOperators\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ndef relu (x : ℝ) : ℝ := max 0 x\n\nnoncomputable def pathNorm {m d : ℕ} (w : Fin m → ℝ) (u : Fin m → EuclideanSpace ℝ (Fin d)) : ℝ :=\n  ∑ j : Fin m, |w j| * ‖u j‖\n\nnoncomputable def predict {m d : ℕ} (w : Fin m → ℝ) (u : Fin m → EuclideanSpace ℝ (Fin d))\n    (x : EuclideanSpace ℝ (Fin d)) : ℝ :=\n  ∑ j : Fin m, w j * relu (inner ℝ (u j) x)\n\ntheorem ch5_theorem_5_10\n    {m d n : ℕ} (B : ℝ) (Cdata : ℝ)\n    (hn : 0 < n) (hB : 0 < B) (hCdata : 0 ≤ Cdata)\n    (x : Fin n → EuclideanSpace ℝ (Fin d))\n    (hx : ∀ i, ‖x i‖ ≤ Cdata)\n    (F : {θ : (Fin m → ℝ) × (Fin m → EuclideanSpace ℝ (Fin d)) // pathNorm θ.1 θ.2 ≤ B}\n          → EuclideanSpace ℝ (Fin d) → ℝ)\n    (hF : ∀ θ, F θ = predict (θ : (Fin m → ℝ) × (Fin m → EuclideanSpace ℝ (Fin d))).1\n                          (θ : (Fin m → ℝ) × (Fin m → EuclideanSpace ℝ (Fin d))).2) :\n    empRadSigned n F x ≤ 2 * B * Cdata / Real.sqrt n := sorry\n\nend GraduateLevel\n", "target_names": ["ch5_theorem_5_10"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 815, "end": 1325, "gold": "{m d n : ℕ} (B : ℝ) (Cdata : ℝ)\n    (hn : 0 < n) (hB : 0 < B) (hCdata : 0 ≤ Cdata)\n    (x : Fin n → EuclideanSpace ℝ (Fin d))\n    (hx : ∀ i, ‖x i‖ ≤ Cdata)\n    (F : {θ : (Fin m → ℝ) × (Fin m → EuclideanSpace ℝ (Fin d)) // pathNorm θ.1 θ.2 ≤ B}\n          → EuclideanSpace ℝ (Fin d) → ℝ)\n    (hF : ∀ θ, F θ = predict (θ : (Fin m → ℝ) × (Fin m → EuclideanSpace ℝ (Fin d))).1\n                          (θ : (Fin m → ℝ) × (Fin m → EuclideanSpace ℝ (Fin d))).2) :\n    empRadSigned n F x ≤ 2 * B * Cdata / Real.sqrt n"}]}
{"item_id": "ch5_theorem_5_5::ch5_theorem_5_5::A-stmt", "paper_id": "ch5_theorem_5_5", "target_name": "ch5_theorem_5_5", "source": "Graduate-level", "nl_statement": "Let \\(\\mathcal{H} = \\{x \\mapsto \\langle w,x \\rangle \\mid w \\in \\mathbb{R}^d,\\, \\|w\\|_2 \\le B\\}\\) for some constant \\(B > 0\\). Moreover, assume \\(\\mathbb{E}_{x \\sim P}[\\|x\\|_2^2] \\leq C^2\\), where \\(P\\) is some distribution and \\(C > 0\\) is a constant. Then\n\\[\nR_S(\\mathcal{H}) \\le \\frac{B}{n} \\sqrt{\\sum_{i=1}^n \\|x^{(i)}\\|_2^2},\n\\]\nand\n\\[\nR_n(\\mathcal{H}) \\le \\frac{BC}{\\sqrt{n}}.\n\\]", "context": [{"name": "Empirical Rademacher complexity", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\)."}, {"name": "Rademacher complexity", "statement": "Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\nnoncomputable def popRadSigned {Ω : Type*} [MeasurableSpace Ω] {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (μ : Measure Ω) (X : Ω → 𝒳) : ℝ :=\n  ∫ ω : Fin n → Ω, empRadSigned n f (X ∘ ω) ∂(Measure.pi fun _ => μ)\n\nnoncomputable def l2LinearClass (d : ℕ) (B : ℝ) :\n    {w : EuclideanSpace ℝ (Fin d) // ‖w‖ ≤ B} → EuclideanSpace ℝ (Fin d) → ℝ :=\n  fun w x => @inner ℝ _ _ (w : EuclideanSpace ℝ (Fin d)) x\n\ntheorem ch5_theorem_5_5\n    {d n : ℕ} (hn : 0 < n) {B C : ℝ} (hB : 0 < B) (hC : 0 < C)\n    (S : Fin n → EuclideanSpace ℝ (Fin d)) :\n    empRadSigned n (l2LinearClass d B) S\n      ≤ (B / n) * Real.sqrt (∑ i : Fin n, ‖S i‖ ^ 2)\n    ∧ ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n        (X : Ω → EuclideanSpace ℝ (Fin d)),\n        Measurable X →\n        Integrable (fun ω => ‖X ω‖ ^ 2) P →\n        (∫ ω, ‖X ω‖ ^ 2 ∂P) ≤ C ^ 2 →\n        popRadSigned n (l2LinearClass d B) P X\n          ≤ B * C / Real.sqrt n := sorry\n\nend GraduateLevel\n", "target_names": ["ch5_theorem_5_5"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 901, "end": 1406, "gold": "{d n : ℕ} (hn : 0 < n) {B C : ℝ} (hB : 0 < B) (hC : 0 < C)\n    (S : Fin n → EuclideanSpace ℝ (Fin d)) :\n    empRadSigned n (l2LinearClass d B) S\n      ≤ (B / n) * Real.sqrt (∑ i : Fin n, ‖S i‖ ^ 2)\n    ∧ ∀ {Ω : Type} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n        (X : Ω → EuclideanSpace ℝ (Fin d)),\n        Measurable X →\n        Integrable (fun ω => ‖X ω‖ ^ 2) P →\n        (∫ ω, ‖X ω‖ ^ 2 ∂P) ≤ C ^ 2 →\n        popRadSigned n (l2LinearClass d B) P X\n          ≤ B * C / Real.sqrt n"}]}
{"item_id": "ch8_theorem_8_3::ch8_theorem_8_3::A-stmt", "paper_id": "ch8_theorem_8_3", "target_name": "ch8_theorem_8_3", "source": "Graduate-level", "nl_statement": "Suppose gradient descent on \\(\\widehat{L}(\\beta)\\) with initialization \\(\\beta^0 = 0\\) converges to a solution \\(\\hat{\\beta}\\) such that \\(\\widehat{L}(\\hat{\\beta}) = 0\\). Then \\(\\hat{\\beta} = \\beta^\\star\\).", "context": [{"name": "Minimum-norm least-squares solution", "statement": "From the referenced lemma, we can derive an explicit formula for the minimum norm interpolant \\(\\beta^\\star \\triangleq \\operatorname{argmin}_{\\beta:\\widehat{L}(\\beta)=0}\\|\\beta\\|_2\\). \\(\\beta^\\star = X^+\\vec{y}\\)."}, {"name": "Characterization of global minimizers of underdetermined least squares", "statement": "Let \\(X^+\\) denote the pseudoinverse of \\(X\\) (since \\(X\\) is full rank, \\(XX^\\top\\) is invertible and \\(X^+ = X^\\top(XX^\\top)^{-1}\\); note that \\(XX^+X = X\\)). Then \\(\\beta\\) is a global minimizer if and only if \\(\\beta = X^+\\vec{y} + \\zeta\\) for some \\(\\zeta\\) such that \\(\\zeta \\perp x_1, \\dots, x_n\\)."}, {"name": "Empirical least-squares loss", "statement": "In the underdetermined least-squares setting of Chapter 8, let \\(X \\in \\mathbb{R}^{n \\times d}\\) have rows \\(x^{(1)}, \\dots, x^{(n)}\\) and let \\(\\vec{y} \\in \\mathbb{R}^n\\) be the response vector, where \\(n < d\\). Assume \\(X\\) is full rank. Define the empirical least-squares loss by \\(\\widehat{L}(\\beta) = \\tfrac12\\|X\\beta - \\vec{y}\\|_2^2\\). Its gradient is \\(\\nabla \\widehat{L}(\\beta) = X^\\top(X\\beta - \\vec{y})\\), and \\(\\widehat{L}(\\beta) = 0\\) if and only if \\(X\\beta = \\vec{y}\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\nopen scoped InnerProductSpace\n\nnamespace GraduateLevel\n\nnoncomputable def designMatrixMul {n d : ℕ} (x : Fin n → EuclideanSpace ℝ (Fin d))\n    (θ : EuclideanSpace ℝ (Fin d)) : EuclideanSpace ℝ (Fin n) :=\n  (WithLp.equiv 2 (Fin n → ℝ)).symm (fun i => @inner ℝ _ _ θ (x i))\n\nnoncomputable def designMatrixLinearMap {n d : ℕ} (x : Fin n → EuclideanSpace ℝ (Fin d)) :\n    EuclideanSpace ℝ (Fin d) →ₗ[ℝ] EuclideanSpace ℝ (Fin n) where\n  toFun := designMatrixMul x\n  map_add' θ₁ θ₂ := by\n    ext i; simp only [designMatrixMul, inner_add_left]; rfl\n  map_smul' c θ := by\n    ext i; simp only [designMatrixMul, inner_smul_left, RCLike.conj_to_real]; rfl\n\nnoncomputable def gdStep {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    [CompleteSpace E] (f : E → ℝ) (α : ℝ) (x : E) : E :=\n  x - α • gradient f x\n\nnoncomputable def gdSeq {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    [CompleteSpace E] (f : E → ℝ) (α : ℕ → ℝ) (x₀ : E) : ℕ → E\n  | 0     => x₀\n  | k + 1 => gdStep f (α k) (gdSeq f α x₀ k)\n\ntheorem ch8_theorem_8_3\n    {n d : ℕ} (hnd : n < d)\n    (x : Fin n → EuclideanSpace ℝ (Fin d))\n    (y : EuclideanSpace ℝ (Fin n))\n    (η : ℝ)\n\n    (Lhat : EuclideanSpace ℝ (Fin d) → ℝ)\n    (hLhat : Lhat = fun β =>\n      (1 / 2 : ℝ) * ‖designMatrixMul x β - y‖ ^ 2)\n\n    (XXTinv : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (Fin n))\n    (hinvL : XXTinv ∘ₗ\n        (designMatrixLinearMap x ∘ₗ\n          LinearMap.adjoint (designMatrixLinearMap x))\n        = LinearMap.id)\n    (hinvR :\n        (designMatrixLinearMap x ∘ₗ\n          LinearMap.adjoint (designMatrixLinearMap x)) ∘ₗ XXTinv\n        = LinearMap.id)\n\n    (βhat : EuclideanSpace ℝ (Fin d))\n    (hconv : Filter.Tendsto\n        (gdSeq Lhat (fun _ => η) 0) Filter.atTop (nhds βhat))\n\n    (hfit : Lhat βhat = 0) :\n    βhat = LinearMap.adjoint (designMatrixLinearMap x) (XXTinv y) := sorry\n\nend GraduateLevel\n", "target_names": ["ch8_theorem_8_3"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 1088, "end": 1897, "gold": "{n d : ℕ} (hnd : n < d)\n    (x : Fin n → EuclideanSpace ℝ (Fin d))\n    (y : EuclideanSpace ℝ (Fin n))\n    (η : ℝ)\n\n    (Lhat : EuclideanSpace ℝ (Fin d) → ℝ)\n    (hLhat : Lhat = fun β =>\n      (1 / 2 : ℝ) * ‖designMatrixMul x β - y‖ ^ 2)\n\n    (XXTinv : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (Fin n))\n    (hinvL : XXTinv ∘ₗ\n        (designMatrixLinearMap x ∘ₗ\n          LinearMap.adjoint (designMatrixLinearMap x))\n        = LinearMap.id)\n    (hinvR :\n        (designMatrixLinearMap x ∘ₗ\n          LinearMap.adjoint (designMatrixLinearMap x)) ∘ₗ XXTinv\n        = LinearMap.id)\n\n    (βhat : EuclideanSpace ℝ (Fin d))\n    (hconv : Filter.Tendsto\n        (gdSeq Lhat (fun _ => η) 0) Filter.atTop (nhds βhat))\n\n    (hfit : Lhat βhat = 0) :\n    βhat = LinearMap.adjoint (designMatrixLinearMap x) (XXTinv y)"}]}
{"item_id": "ch9_lemma_9_1::ch9_lemma_9_1::A-stmt", "paper_id": "ch9_lemma_9_1", "target_name": "ch9_lemma_9_1", "source": "Graduate-level", "nl_statement": "Suppose \\(z \\sim \\mathcal{N}(v, I)\\). Then,\n\\[\n\\mathbb{E}[z \\otimes z \\otimes z] = v \\otimes v \\otimes v + \\sum_{l=1}^d \\mathbb{E}[z] \\otimes e_l \\otimes e_l + \\sum_{l=1}^d e_l \\otimes \\mathbb{E}[z] \\otimes e_l + \\sum_{l=1}^d e_l \\otimes e_l \\otimes \\mathbb{E}[z],\n\\]\nwhere \\(e_1,\\dots,e_d\\) denote the canonical basis vectors.", "context": null, "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace GraduateLevel\n\nnoncomputable def stdGaussianPi (n : ℕ) : Measure (Fin n → ℝ) :=\n  Measure.pi fun _ : Fin n => gaussianReal 0 1\n\ntheorem ch9_lemma_9_1 (d : ℕ) (v : Fin d → ℝ) (i j k : Fin d) :\n    ∫ ξ, (v i + ξ i) * (v j + ξ j) * (v k + ξ k) ∂(stdGaussianPi d)\n      = v i * v j * v k\n        + v i * (if j = k then 1 else 0)\n        + v j * (if i = k then 1 else 0)\n        + v k * (if i = j then 1 else 0) := sorry\n\nend GraduateLevel\n", "target_names": ["ch9_lemma_9_1"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 214, "end": 470, "gold": "(d : ℕ) (v : Fin d → ℝ) (i j k : Fin d) :\n    ∫ ξ, (v i + ξ i) * (v j + ξ j) * (v k + ξ k) ∂(stdGaussianPi d)\n      = v i * v j * v k\n        + v i * (if j = k then 1 else 0)\n        + v j * (if i = k then 1 else 0)\n        + v k * (if i = j then 1 else 0)"}]}
{"item_id": "ch2_corollary_2_18::ch2_corollary_2_18::A-stmt", "paper_id": "ch2_corollary_2_18", "target_name": "ch2_corollary_2_18", "source": "Graduate-level", "nl_statement": "Assume that \\(X_{1}\\in [0,1]\\).\nThen for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq(1+\\epsilon)\\mu\\right)\\leq\\exp\\left[\\frac{-n\\mu\\epsilon^{2}}{2+\\epsilon}\\right],\n\\]\n\\[\n\\Pr\\left(\\bar{X}_{n}\\leq(1-\\epsilon)\\mu\\right)\\leq\\exp\\left[\\frac{-n\\mu\\epsilon^{2}}{2}\\right].\n\\]\nMoreover, for \\( t > 0 \\), we have:\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\mu t}{n}}+\\frac{t}{3 n}\\right)\\leq e^{-t}.\n\\]", "context": [{"name": "theorem 2.17", "statement": "Assume that \\(X_{1}\\in [0,1]\\) . Then for all \\(\\epsilon >0\\) , we have\n\\[\n\\begin{array}{l}\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq e^{-n\\mathrm{KL}(\\mu+\\epsilon| |\\mu)},\\\\ \\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq e^{-n\\mathrm{KL}(\\mu-\\epsilon| |\\mu)},\\\\ \\end{array}\n\\]\nwhere \\(\\mathrm{KL}(z||\\mu)\\) is the Kullback-Leibler divergence (KL divergence) defined as\n\\[\n\\mathrm{KL}(z| |\\mu)=z\\ln{\\frac{z}{\\mu}}+(1-z)\\ln{\\frac{1-z}{1-\\mu}}.\n\\]"}, {"name": "theorem 2.10", "statement": "If \\( X_{1} \\) has a logarithmic moment generating function that satisfies (2.9) for \\( \\lambda > 0 \\), then all \\( \\epsilon > 0 \\):\n\\[\n\\operatorname*{Pr}(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq\\exp\\left[\\frac{-n\\epsilon^{2}}{2(\\alpha+\\beta\\epsilon)}\\right].\n\\]\nMoreover, for \\(t > 0\\), we have\n\\[\n\\operatorname*{Pr}\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\alpha t}{n}}+\\frac{\\beta t}{n}\\right)\\leq e^{-t}.\n\\]"}, {"name": "lemma 2.15", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]"}, {"name": "Lemma 2.9 (equation (2.9))", "statement": "Consider a random variable \\(X\\) so that \\(\\mathbb{E}[X] = \\mu\\). Assume that there exists \\(\\alpha > 0\\) and \\(\\beta \\geq 0\\) such that for \\(\\lambda \\in [0, \\beta^{-1})\\):\n\\[\n\\Lambda_{X}(\\lambda)\\leq\\lambda\\mu+\\frac{\\alpha\\lambda^{2}}{2(1-\\beta\\lambda)},\\tag{2.9}\n\\]"}, {"name": "Definition 2.4", "statement": "Given a random variable \\(X\\), we may define its logarithmic moment generating function as\n\\[\n\\Lambda_{X}(\\lambda)=\\ln\\mathbb{E}e^{\\lambda X}.\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch2_corollary_2_18\n    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {n : ℕ} [NeZero n] (X : Fin n → Ω → ℝ)\n    (hmeas : ∀ i, AEMeasurable (X i) P)\n    (hindep : iIndepFun X P)\n    (hbdd : ∀ i, ∀ᵐ ω ∂P, X i ω ∈ Set.Icc (0 : ℝ) 1)\n    (hident : ∀ i, Measure.map (X i) P = Measure.map (X 0) P)\n    (h_int : ∀ t, Integrable (fun ω => Real.exp (t * X 0 ω)) P)\n    (μ : ℝ) (hmu : μ = ∫ ω, X 0 ω ∂P) :\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | (1 + ε) * μ ≤ (∑ i, X i ω) / n}).toReal\n          ≤ Real.exp (-(n : ℝ) * μ * ε ^ 2 / (2 + ε))) ∧\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | (∑ i, X i ω) / n ≤ (1 - ε) * μ}).toReal\n          ≤ Real.exp (-(n : ℝ) * μ * ε ^ 2 / 2)) ∧\n    (∀ t : ℝ, 0 < t →\n        (P {ω | μ + Real.sqrt (2 * μ * t / n) + t / (3 * n) ≤ (∑ i, X i ω) / n}).toReal\n          ≤ Real.exp (-t)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch2_corollary_2_18"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 115, "end": 919, "gold": "{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {n : ℕ} [NeZero n] (X : Fin n → Ω → ℝ)\n    (hmeas : ∀ i, AEMeasurable (X i) P)\n    (hindep : iIndepFun X P)\n    (hbdd : ∀ i, ∀ᵐ ω ∂P, X i ω ∈ Set.Icc (0 : ℝ) 1)\n    (hident : ∀ i, Measure.map (X i) P = Measure.map (X 0) P)\n    (h_int : ∀ t, Integrable (fun ω => Real.exp (t * X 0 ω)) P)\n    (μ : ℝ) (hmu : μ = ∫ ω, X 0 ω ∂P) :\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | (1 + ε) * μ ≤ (∑ i, X i ω) / n}).toReal\n          ≤ Real.exp (-(n : ℝ) * μ * ε ^ 2 / (2 + ε))) ∧\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | (∑ i, X i ω) / n ≤ (1 - ε) * μ}).toReal\n          ≤ Real.exp (-(n : ℝ) * μ * ε ^ 2 / 2)) ∧\n    (∀ t : ℝ, 0 < t →\n        (P {ω | μ + Real.sqrt (2 * μ * t / n) + t / (3 * n) ≤ (∑ i, X i ω) / n}).toReal\n          ≤ Real.exp (-t))"}]}
{"item_id": "ch2_corollary_2_19::ch2_corollary_2_19::A-stmt", "paper_id": "ch2_corollary_2_19", "target_name": "ch2_corollary_2_19", "source": "Graduate-level", "nl_statement": "Assume that \\(X_{1}\\in \\{0,1\\}\\). Then for all \\(\\epsilon >0\\) that satisfies\n\\[\n\\epsilon^{\\prime}=\\epsilon+2\\sqrt{(\\mu+\\epsilon)(1-(\\mu+\\epsilon))/n}<1-\\mu,\n\\]\nand \\(n\\geq (1 - \\mu -\\epsilon) / (\\mu +\\epsilon)\\), we have\n\\[\n\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\geq 0.25\\exp\\left[-n\\mathrm{KL}(\\mu+\\epsilon^{\\prime}| |\\mu)-\\sqrt{n}\\Delta I\\right],\n\\]\nwhere\n\\[\n\\Delta I=2\\sqrt{(\\mu+\\epsilon)(1-\\mu-\\epsilon)}\\ln\\frac{(\\mu+\\epsilon^{\\prime})(1-\\mu)}{(1-(\\mu+\\epsilon^{\\prime}))\\mu}.\n\\]", "context": [{"name": "Chebyshev's Inequality", "statement": "We have\n\\[\n\\Pr\\left(\\left|\\bar{X}_{n}-\\mu\\right|\\geq\\epsilon\\right)\\leq\\frac{\\operatorname{Var}\\left(X_{1}\\right)}{n\\epsilon^{2}}.\\tag{2.2}\n\\]"}, {"name": "theorem 2.17", "statement": "Assume that \\(X_{1}\\in [0,1]\\) . Then for all \\(\\epsilon >0\\) , we have\n\\[\n\\begin{array}{l}\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq e^{-n\\mathrm{KL}(\\mu+\\epsilon| |\\mu)},\\\\ \\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq e^{-n\\mathrm{KL}(\\mu-\\epsilon| |\\mu)},\\\\ \\end{array}\n\\]\nwhere \\(\\mathrm{KL}(z||\\mu)\\) is the Kullback-Leibler divergence (KL divergence) defined as\n\\[\n\\mathrm{KL}(z| |\\mu)=z\\ln{\\frac{z}{\\mu}}+(1-z)\\ln{\\frac{1-z}{1-\\mu}}.\n\\]"}, {"name": "Theorem 2.7 (proof)", "statement": "Theorem 2.7. For all \\(\\epsilon' > \\epsilon > 0\\):\n\\[\n\\underline{\\lim}_{n\\rightarrow\\infty}\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\geq-I_{X_{1}}(\\mu+\\epsilon^{\\prime}).\n\\]\nSimilarly,\n\\[\n\\underline{\\lim}_{n\\rightarrow\\infty}\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\geq-I_{X_{1}}(\\mu-\\epsilon^{\\prime}).\n\\]\nProof. We only need to prove the first inequality. Consider \\(\\Pr(X_i \\leq x)\\) as a function of \\(x\\), and define a random variable \\(X_i'\\) with density at \\(x\\) as\n\\[\nd\\Pr(X_{i}^{\\prime}\\leq x)=e^{\\lambda x-\\Lambda_{X_{1}}(\\lambda)}d\\Pr(X_{i}\\leq x).\n\\]\nThis choice implies that\n\\[\n\\frac{d}{d\\lambda}\\Lambda_{X_{1}}(\\lambda)=\\frac{\\int x e^{\\lambda x}d\\Pr(X_{1}\\leq x)}{\\int e^{\\lambda x}d\\Pr(X_{1}\\leq x)}=\\mathbb{E}_{X_{1}^{\\prime}}X_{1}^{\\prime}.\n\\]\nWe now take \\(\\lambda\\) such that\n\\[\n\\lambda=\\arg\\max_{\\lambda^{\\prime}>0}\\left[\\lambda^{\\prime}(\\mu+\\epsilon^{\\prime})-\\Lambda_{X_{1}}(\\lambda^{\\prime})\\right].\n\\]\nBy setting the derivative to zero, we obtain\n\\[\n\\mathbb{E}_{X_{1}^{\\prime}}X_{1}^{\\prime}=\\frac{d}{d\\lambda}\\Lambda_{X_{1}}(\\lambda)=\\mu+\\epsilon^{\\prime},\\tag{2.3}\n\\]\n\\[\n-\\lambda(\\mu+\\epsilon^{\\prime})+\\Lambda(\\lambda)= -I(\\mu+\\epsilon^{\\prime}).\\tag{2.4}\n\\]\nLet \\(\\bar{X}_n' = n^{-1} \\sum_{i=1}^{n} X_i'\\). Then by the law of large numbers, we know that for \\(\\epsilon'' > \\epsilon'\\), we obtain from (2.3)\n\\[\n\\lim_{n\\rightarrow\\infty}\\Pr(\\bar{X}_{n}^{\\prime}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}])=1.\\tag{2.5}\n\\]\nSince the joint density of \\((X_1', \\ldots, X_n')\\) satisfies\n\\[\ne^{-\\lambda\\sum_{i=1}^{n}x_{i}+n\\Lambda_{X_{1}}(\\lambda)}\\prod_{i}d\\Pr(X_{i}^{\\prime}\\leq x_{i})=\\prod_{i}d\\Pr(X_{i}\\leq x_{i}),\\tag{2.6}\n\\]\nby using \\(\\mathbb{1}(\\cdot)\\) to denote the set indicator function, we obtain\n\\[\n\\begin{array}{l}\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\geq\\Pr(\\bar{X}_{n}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}])\\\\ =\\mathbb{E}_{X_{1},\\ldots,X_{n}}\\mathbb{1}(\\bar{X}_{n}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}])\\\\ =\\mathbb{E}_{X_{1}^{\\prime},\\ldots,X_{n}^{\\prime}}e^{-\\lambda n\\bar{X}_{n}^{\\prime}+n\\Lambda(\\lambda)}\\mathbb{1}(\\bar{X}_{n}^{\\prime}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}])\\\\ \\geq e^{-\\lambda n(\\mu+\\epsilon^{\\prime\\prime})+n\\Lambda(\\lambda)}\\Pr(\\bar{X}_{n}^{\\prime}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}]).\\\\ \\end{array}\n\\]\nThe first equality used the definition of \\(\\Pr(\\cdot)\\). The second equality used (2.6). The last inequality used Markov's inequality. Now by taking logarithm, and divide by \\(n\\), we obtain\n\\[\n\\begin{array}{l}\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\qquad(2.7)\\\\ \\geq-\\lambda(\\mu+\\epsilon^{\\prime\\prime})+\\Lambda(\\lambda)+\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}^{\\prime}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}])\\\\ = -I(\\mu+\\epsilon^{\\prime})-\\lambda(\\epsilon^{\\prime\\prime}-\\epsilon^{\\prime})+\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}^{\\prime}-\\mu\\in [\\epsilon,\\epsilon^{\\prime\\prime}]),\\qquad(2.8)\\\\ \\end{array}\n\\]\nThe equality used (2.4). Now we obtain the desired bound by letting \\(n \\to \\infty\\), applying (2.5), and letting \\(\\epsilon'' \\to \\epsilon'\\) so that \\(\\lambda(\\epsilon'' - \\epsilon') \\to 0\\) (this is true because \\(\\lambda\\) depends only on \\(\\epsilon'\\))."}, {"name": "Definition 2.4", "statement": "Definition 2.4. Given a random variable \\(X\\), we may define its logarithmic moment generating function as\n\\[\n\\Lambda_{X}(\\lambda)=\\ln\\mathbb{E}e^{\\lambda X}.\n\\]\nMoreover, given \\(z \\in \\mathbb{R}\\), the rate function \\(I_X(z)\\) is defined as\n\\[\nI_{X}(z)=\\left\\{\\begin{array}{l l}\\sup_{\\lambda>0}\\left[\\lambda z-\\Lambda_{X}(\\lambda)\\right]&z>\\mu\\\\ 0&z=\\mu\\\\ \\sup_{\\lambda<0}\\left[\\lambda z-\\Lambda_{X}(\\lambda)\\right]&z<\\mu,\\end{array}\\right.\n\\]\nwhere \\(\\mu = \\mathbb{E}[X]\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def bernoulliKL (z m : ℝ) : ℝ :=\n  z * Real.log (z / m) + (1 - z) * Real.log ((1 - z) / (1 - m))\n\ntheorem ch2_corollary_2_19\n    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]\n    {n : ℕ} (hn : 0 < n) (μ ε : ℝ)\n    (X : Fin n → Ω → ℝ)\n    (hmeas : ∀ i, Measurable (X i))\n    (hbin : ∀ i, ∀ ω, X i ω = 0 ∨ X i ω = 1)\n    (hindep : iIndepFun X P)\n    (hmean : ∀ i, ∫ ω, X i ω ∂P = μ)\n    (hμ0 : 0 < μ) (hμ1 : μ < 1) (hε : 0 < ε) (hμε : μ + ε < 1)\n    (hεp_lt : ε + 2 * Real.sqrt ((μ + ε) * (1 - (μ + ε)) / (n : ℝ)) < 1 - μ)\n    (hn_lb : (1 - μ - ε) / (μ + ε) ≤ (n : ℝ)) :\n    (let ε' := ε + 2 * Real.sqrt ((μ + ε) * (1 - (μ + ε)) / (n : ℝ));\n     let ΔI := 2 * Real.sqrt ((μ + ε) * (1 - μ - ε)) *\n       Real.log (((μ + ε') * (1 - μ)) / ((1 - (μ + ε')) * μ));\n     (P {ω | μ + ε ≤ (1 / (n : ℝ)) * ∑ i : Fin n, X i ω}).toReal ≥\n       (0.25 : ℝ) * Real.exp (-(n : ℝ) * bernoulliKL (μ + ε') μ - Real.sqrt (n : ℝ) * ΔI)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch2_corollary_2_19"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 227, "end": 1039, "gold": "{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]\n    {n : ℕ} (hn : 0 < n) (μ ε : ℝ)\n    (X : Fin n → Ω → ℝ)\n    (hmeas : ∀ i, Measurable (X i))\n    (hbin : ∀ i, ∀ ω, X i ω = 0 ∨ X i ω = 1)\n    (hindep : iIndepFun X P)\n    (hmean : ∀ i, ∫ ω, X i ω ∂P = μ)\n    (hμ0 : 0 < μ) (hμ1 : μ < 1) (hε : 0 < ε) (hμε : μ + ε < 1)\n    (hεp_lt : ε + 2 * Real.sqrt ((μ + ε) * (1 - (μ + ε)) / (n : ℝ)) < 1 - μ)\n    (hn_lb : (1 - μ - ε) / (μ + ε) ≤ (n : ℝ)) :\n    (let ε' := ε + 2 * Real.sqrt ((μ + ε) * (1 - (μ + ε)) / (n : ℝ));\n     let ΔI := 2 * Real.sqrt ((μ + ε) * (1 - μ - ε)) *\n       Real.log (((μ + ε') * (1 - μ)) / ((1 - (μ + ε')) * μ));\n     (P {ω | μ + ε ≤ (1 / (n : ℝ)) * ∑ i : Fin n, X i ω}).toReal ≥\n       (0.25 : ℝ) * Real.exp (-(n : ℝ) * bernoulliKL (μ + ε') μ - Real.sqrt (n : ℝ) * ΔI))"}]}
{"item_id": "ch2_lemma_2_15::ch2_lemma_2_15::A-stmt", "paper_id": "ch2_lemma_2_15", "target_name": "ch2_lemma_2_15", "source": "Graduate-level", "nl_statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]", "context": [{"name": "theorem 2.12", "statement": "If \\(X_{1}\\) is sub-Gaussian as in (2.10), then for all \\(t > 0\\):\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2 b t}{n}}\\right)\\leq e^{-t},\n\\]\n\\[\n\\Pr\\left(\\bar{X}_{n}\\leq\\mu-\\sqrt{\\frac{2 b t}{n}}\\right)\\leq e^{-t}.\n\\]"}, {"name": "definition 2.11", "statement": "A sub-Gaussian random variable \\(X\\) has quadratic logarithmic moment generating function for all \\(\\lambda \\in \\mathbb{R}\\):\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\lambda\\mu+\\frac{\\lambda^{2}}{2}b.\\tag{2.10}\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch2_lemma_2_15 {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)\n    [IsProbabilityMeasure P] {X : Ω → ℝ} (μ : ℝ)\n    (hX : AEMeasurable X P)\n    (hmem : ∀ᵐ ω ∂P, X ω ∈ Set.Icc (0 : ℝ) 1)\n    (hint : ∀ t : ℝ, Integrable (fun ω => Real.exp (t * X ω)) P)\n    (hmean : ∫ ω, X ω ∂P = μ) :\n    ∀ lam : ℝ,\n      cgf X P lam ≤ Real.log ((1 - μ) + μ * Real.exp lam) ∧\n      Real.log ((1 - μ) + μ * Real.exp lam) ≤ lam * μ + lam ^ 2 / 8 := sorry\n\nend GraduateLevel\n", "target_names": ["ch2_lemma_2_15"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 107, "end": 517, "gold": "{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω)\n    [IsProbabilityMeasure P] {X : Ω → ℝ} (μ : ℝ)\n    (hX : AEMeasurable X P)\n    (hmem : ∀ᵐ ω ∂P, X ω ∈ Set.Icc (0 : ℝ) 1)\n    (hint : ∀ t : ℝ, Integrable (fun ω => Real.exp (t * X ω)) P)\n    (hmean : ∫ ω, X ω ∂P = μ) :\n    ∀ lam : ℝ,\n      cgf X P lam ≤ Real.log ((1 - μ) + μ * Real.exp lam) ∧\n      Real.log ((1 - μ) + μ * Real.exp lam) ≤ lam * μ + lam ^ 2 / 8"}]}
{"item_id": "ch2_lemma_2_20::ch2_lemma_2_20::A-stmt", "paper_id": "ch2_lemma_2_20", "target_name": "ch2_lemma_2_20", "source": "Graduate-level", "nl_statement": "If \\(X - \\mathbb{E}X \\leq b\\), then \\(\\forall \\lambda \\geq 0\\):\n\\[\n\\ln \\mathbb{E} e^{\\lambda X} \\leq \\lambda \\mathbb{E}X + \\lambda^{2} \\phi(\\lambda b) \\operatorname{Var}(X),\n\\]\nwhere \\(\\phi(z) = (e^{z} - z - 1)/z^{2}\\).", "context": [{"name": "theorem 2.5", "statement": "For any \\(n\\) and \\(\\epsilon >0\\):\n\\[\n\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq-I_{X_{1}}(\\mu+\\epsilon)=\\inf_{\\lambda>0}\\left[-\\lambda(\\mu+\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}\\right],\n\\]\n\\[\n\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq-I_{X_{1}}(\\mu-\\epsilon)=\\inf_{\\lambda<0}\\left[-\\lambda(\\mu-\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}\\right].\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def phi (z : ℝ) : ℝ := if z = 0 then 1/2 else (Real.exp z - z - 1) / z ^ 2\n\ntheorem ch2_lemma_2_20 {Ω : Type*} [MeasurableSpace Ω]\n    (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ]\n    (X : Ω → ℝ) (b : ℝ)\n    (hX_int : MeasureTheory.Integrable X μ)\n    (hexpX_int : ∀ lam : ℝ, 0 ≤ lam →\n      MeasureTheory.Integrable (fun ω => Real.exp (lam * X ω)) μ)\n    (hX_L2 : MeasureTheory.MemLp X 2 μ)\n    (hb : ∀ᵐ ω ∂μ, X ω - (∫ x, X x ∂μ) ≤ b) :\n    ∀ lam : ℝ, 0 ≤ lam →\n      Real.log (∫ ω, Real.exp (lam * X ω) ∂μ)\n        ≤ lam * (∫ ω, X ω ∂μ)\n          + lam ^ 2 * phi (lam * b) * ProbabilityTheory.variance X μ := sorry\n\nend GraduateLevel\n", "target_names": ["ch2_lemma_2_20"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 197, "end": 729, "gold": "{Ω : Type*} [MeasurableSpace Ω]\n    (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ]\n    (X : Ω → ℝ) (b : ℝ)\n    (hX_int : MeasureTheory.Integrable X μ)\n    (hexpX_int : ∀ lam : ℝ, 0 ≤ lam →\n      MeasureTheory.Integrable (fun ω => Real.exp (lam * X ω)) μ)\n    (hX_L2 : MeasureTheory.MemLp X 2 μ)\n    (hb : ∀ᵐ ω ∂μ, X ω - (∫ x, X x ∂μ) ≤ b) :\n    ∀ lam : ℝ, 0 ≤ lam →\n      Real.log (∫ ω, Real.exp (lam * X ω) ∂μ)\n        ≤ lam * (∫ ω, X ω ∂μ)\n          + lam ^ 2 * phi (lam * b) * ProbabilityTheory.variance X μ"}]}
{"item_id": "ch2_theorem_2_16::ch2_theorem_2_16::A-stmt", "paper_id": "ch2_theorem_2_16", "target_name": "ch2_theorem_2_16", "source": "Graduate-level", "nl_statement": "Assume that \\( X_{1} \\in [0,1] \\). Then for all \\( \\epsilon > 0 \\):\n\\[\n\\begin{array}{l}\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq e^{-2n\\epsilon^{2}},\\\\ \\operatorname*{Pr}(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq e^{-2n\\epsilon^{2}}.\\\\ \\end{array}\n\\]", "context": [{"name": "theorem 2.12", "statement": "If \\(X_{1}\\) is sub-Gaussian as in (2.10), then for all \\(t > 0\\):\n\\[\n\\Pr\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2 b t}{n}}\\right)\\leq e^{-t},\n\\]\n\\[\n\\Pr\\left(\\bar{X}_{n}\\leq\\mu-\\sqrt{\\frac{2 b t}{n}}\\right)\\leq e^{-t}.\n\\]"}, {"name": "lemma 2.15", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]"}, {"name": "equation 2.1", "statement": "Let \\(X_1, \\ldots, X_n\\) be \\(n\\) real-valued independent and identically distributed (iid) random variables, with expectation \\(\\mu = \\mathbb{E} X_i\\). Let\n\\[\n\\bar{X}_n = \\frac{1}{n}\\sum_{i=1}^n X_i. \\tag{2.1}\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch2_theorem_2_16\n    {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {n : ℕ} (hn : 0 < n) (X : Fin n → Ω → ℝ)\n    (h_meas : ∀ i, AEMeasurable (X i) P)\n    (h_indep : iIndepFun X P)\n    (hident : ∀ i j, Measure.map (X i) P = Measure.map (X j) P)\n    (h_bound : ∀ i, ∀ᵐ ω ∂P, X i ω ∈ Set.Icc (0 : ℝ) 1)\n    (m : ℝ) (hm : ∀ i, ∫ ω, X i ω ∂P = m)\n    {ε : ℝ} (hε : 0 < ε) :\n    (P {ω | m + ε ≤ (n : ℝ)⁻¹ * ∑ i, X i ω}).toReal ≤ Real.exp (-2 * (n : ℝ) * ε ^ 2) ∧\n    (P {ω | (n : ℝ)⁻¹ * ∑ i, X i ω ≤ m - ε}).toReal ≤ Real.exp (-2 * (n : ℝ) * ε ^ 2) := sorry\n\nend GraduateLevel\n", "target_names": ["ch2_theorem_2_16"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 113, "end": 664, "gold": "{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]\n    {n : ℕ} (hn : 0 < n) (X : Fin n → Ω → ℝ)\n    (h_meas : ∀ i, AEMeasurable (X i) P)\n    (h_indep : iIndepFun X P)\n    (hident : ∀ i j, Measure.map (X i) P = Measure.map (X j) P)\n    (h_bound : ∀ i, ∀ᵐ ω ∂P, X i ω ∈ Set.Icc (0 : ℝ) 1)\n    (m : ℝ) (hm : ∀ i, ∫ ω, X i ω ∂P = m)\n    {ε : ℝ} (hε : 0 < ε) :\n    (P {ω | m + ε ≤ (n : ℝ)⁻¹ * ∑ i, X i ω}).toReal ≤ Real.exp (-2 * (n : ℝ) * ε ^ 2) ∧\n    (P {ω | (n : ℝ)⁻¹ * ∑ i, X i ω ≤ m - ε}).toReal ≤ Real.exp (-2 * (n : ℝ) * ε ^ 2)"}]}
{"item_id": "ch2_theorem_2_21::ch2_theorem_2_21::A-stmt", "paper_id": "ch2_theorem_2_21", "target_name": "ch2_theorem_2_21", "source": "Graduate-level", "nl_statement": "If \\( X_{1} \\leq \\mu + b \\), for some \\( b > 0 \\). Let \\( \\psi(z) = (1 + z)\\ln(1 + z) - z \\), then \\( \\forall \\epsilon > 0 \\):\n\\[\n\\Pr\\left[\\bar{X}_{n}\\geq\\mu+\\epsilon\\right]\\leq\\exp\\left[\\frac{-n\\operatorname{Var}\\left(X_{1}\\right)}{b^{2}}\\psi\\left(\\frac{\\epsilon b}{\\operatorname{Var}\\left(X_{1}\\right)}\\right)\\right],\n\\]\n\\[\n\\Pr [\\bar{X}_{n}\\geq\\mu+\\epsilon]\\leq\\exp\\left[\\frac{-n\\epsilon^{2}}{2\\operatorname{Var}(X_{1})+2\\epsilon b/3}\\right].\n\\]\nMoreover, for \\( t > 0 \\):\n\\[\n\\operatorname*{Pr}\\left[\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\operatorname{Var}(X_{1})t}{n}}+\\frac{bt}{3n}\\right]\\leq e^{-t}.\n\\]", "context": [{"name": "lemma 2.20", "statement": "If \\(X - \\mathbb{E}X \\leq b\\), then \\(\\forall \\lambda \\geq 0\\):\n\\[\n\\ln \\mathbb{E} e^{\\lambda X} \\leq \\lambda \\mathbb{E}X + \\lambda^{2} \\phi(\\lambda b) \\operatorname{Var}(X),\n\\]\nwhere \\(\\phi(z) = (e^{z} - z - 1)/z^{2}\\)."}, {"name": "theorem 2.5", "statement": "For any \\(n\\) and \\(\\epsilon >0\\):\n\\[\n\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq-I_{X_{1}}(\\mu+\\epsilon)=\\inf_{\\lambda>0}\\left[-\\lambda(\\mu+\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}\\right],\n\\]\n\\[\n\\frac{1}{n}\\ln\\Pr(\\bar{X}_{n}\\leq\\mu-\\epsilon)\\leq-I_{X_{1}}(\\mu-\\epsilon)=\\inf_{\\lambda<0}\\left[-\\lambda(\\mu-\\epsilon)+\\ln\\mathbb{E}e^{\\lambda X_{1}}\\right].\n\\]"}, {"name": "theorem 2.10", "statement": "If \\( X_{1} \\) has a logarithmic moment generating function that satisfies (2.9) for \\( \\lambda > 0 \\), then all \\( \\epsilon > 0 \\):\n\\[\n\\operatorname*{Pr}(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq\\exp\\left[\\frac{-n\\epsilon^{2}}{2(\\alpha+\\beta\\epsilon)}\\right].\n\\]\nMoreover, for \\(t > 0\\), we have\n\\[\n\\operatorname*{Pr}\\left(\\bar{X}_{n}\\geq\\mu+\\sqrt{\\frac{2\\alpha t}{n}}+\\frac{\\beta t}{n}\\right)\\leq e^{-t}.\n\\]"}, {"name": "Lemma 2.9 (assumption (2.9))", "statement": "Consider a random variable \\(X\\) so that \\(\\mathbb{E}[X] = \\mu\\). Assume that there exists \\(\\alpha > 0\\) and \\(\\beta \\geq 0\\) such that for \\(\\lambda \\in [0, \\beta^{-1})\\):\n\\[\n\\Lambda_X(\\lambda) \\leq \\lambda \\mu + \\frac{\\alpha \\lambda^{2}}{2(1 - \\beta \\lambda)}, \\tag{2.9}\n\\]"}, {"name": "Definition 2.4", "statement": "Given a random variable \\(X\\), we may define its logarithmic moment generating function as\n\\[\n\\Lambda_X(\\lambda) = \\ln \\mathbb{E} e^{\\lambda X}.\n\\]\nMoreover, given \\(z \\in \\mathbb{R}\\), the rate function \\(I_X(z)\\) is defined as\n\\[\nI_X(z) = \\begin{cases} \\sup_{\\lambda > 0} [\\lambda z - \\Lambda_X(\\lambda)] & z > \\mu \\\\ 0 & z = \\mu \\\\ \\sup_{\\lambda < 0} [\\lambda z - \\Lambda_X(\\lambda)] & z < \\mu, \\end{cases}\n\\]\nwhere \\(\\mu = \\mathbb{E}[X]\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch2_theorem_2_21\n    {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]\n    (X : ℕ → Ω → ℝ) (hX : ∀ i, Measurable (X i))\n    (hint_int : Integrable (X 0) P)\n    (hint_L2 : MemLp (X 0) 2 P)\n    (hvar : 0 < variance (X 0) P)\n    (hindep : iIndepFun X P)\n    (hident : ∀ i, IdentDistrib (X i) (X 0) P P)\n    (n : ℕ) (hn : 0 < n)\n    (μ : ℝ) (hμ : μ = ∫ ω, X 0 ω ∂P)\n    (b : ℝ) (hb_pos : 0 < b)\n    (hbound : ∀ᵐ ω ∂P, X 0 ω ≤ μ + b) :\n\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | μ + ε ≤ (∑ i ∈ Finset.range n, X i ω) / n}).toReal ≤\n          Real.exp ((-(n : ℝ) * variance (X 0) P / b ^ 2) *\n            (let z := ε * b / variance (X 0) P\n             (1 + z) * Real.log (1 + z) - z))) ∧\n\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | μ + ε ≤ (∑ i ∈ Finset.range n, X i ω) / n}).toReal ≤\n          Real.exp (-(n : ℝ) * ε ^ 2 / (2 * variance (X 0) P + 2 * ε * b / 3))) ∧\n\n    (∀ t : ℝ, 0 < t →\n        (P {ω | μ + (Real.sqrt (2 * variance (X 0) P * t / n) + b * t / (3 * n)) ≤\n              (∑ i ∈ Finset.range n, X i ω) / n}).toReal ≤ Real.exp (-t)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch2_theorem_2_21"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 113, "end": 1147, "gold": "{Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]\n    (X : ℕ → Ω → ℝ) (hX : ∀ i, Measurable (X i))\n    (hint_int : Integrable (X 0) P)\n    (hint_L2 : MemLp (X 0) 2 P)\n    (hvar : 0 < variance (X 0) P)\n    (hindep : iIndepFun X P)\n    (hident : ∀ i, IdentDistrib (X i) (X 0) P P)\n    (n : ℕ) (hn : 0 < n)\n    (μ : ℝ) (hμ : μ = ∫ ω, X 0 ω ∂P)\n    (b : ℝ) (hb_pos : 0 < b)\n    (hbound : ∀ᵐ ω ∂P, X 0 ω ≤ μ + b) :\n\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | μ + ε ≤ (∑ i ∈ Finset.range n, X i ω) / n}).toReal ≤\n          Real.exp ((-(n : ℝ) * variance (X 0) P / b ^ 2) *\n            (let z := ε * b / variance (X 0) P\n             (1 + z) * Real.log (1 + z) - z))) ∧\n\n    (∀ ε : ℝ, 0 < ε →\n        (P {ω | μ + ε ≤ (∑ i ∈ Finset.range n, X i ω) / n}).toReal ≤\n          Real.exp (-(n : ℝ) * ε ^ 2 / (2 * variance (X 0) P + 2 * ε * b / 3))) ∧\n\n    (∀ t : ℝ, 0 < t →\n        (P {ω | μ + (Real.sqrt (2 * variance (X 0) P * t / n) + b * t / (3 * n)) ≤\n              (∑ i ∈ Finset.range n, X i ω) / n}).toReal ≤ Real.exp (-t))"}]}
{"item_id": "ch4_lemma_4_16::ch4_lemma_4_16::A-stmt", "paper_id": "ch4_lemma_4_16", "target_name": "ch4_lemma_4_16", "source": "Graduate-level", "nl_statement": "If \\(\\operatorname{vc}(\\mathcal{G}) = d\\), then we have for all \\(n > 0\\) and empirical samples \\(\\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\} \\in \\mathcal{Z}^n\\):\n\\[\n|\\mathcal{G}(\\mathcal{S}_{n})|\\leq\\sum_{\\ell=0}^{d}\\binom{n}{\\ell}\\leq\\max(2,e n/d)^{d}.\n\\]", "context": [{"name": "VC-dimension", "statement": "Let \\(\\mathcal{G} = \\{\\phi(w,z) : w \\in \\Omega\\}\\) be a \\(\\{0,1\\}\\) valued binary function class of \\(z \\in \\mathcal{Z}\\) indexed by \\(w \\in \\Omega\\). Given an arbitrary set of \\(n\\) samples \\(\\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\} \\in \\mathcal{Z}^n\\), we are interested in the number of functions (uniform \\(L_\\infty\\) cover of the function class at \\(\\epsilon = 0\\)) that \\(\\mathcal{G}(\\mathcal{S}_n) = \\{[\\phi(w,Z_1), \\ldots, \\phi(w,Z_n)] : w \\in \\Omega\\}\\) can achieve.\nWe say that \\(\\mathcal{G}\\) shatters \\(S_{n}\\) if the number of elements \\(|\\mathcal{G}(S_n)|\\) is \\(2^{n}\\). That is, we can always find \\(w\\in \\Omega\\) so that \\(\\phi (w,z)\\) matches any arbitrary possible choice of \\(\\{0,1\\} ^n\\) values at the \\(n\\) points. The maximum \\(n\\) such that \\(\\mathcal{G}\\) shatters at least one instance of \\(S_{n}\\in \\mathcal{Z}^{n}\\), denoted by \\(\\mathrm{vc}(\\mathcal{G})\\), is called the VC-dimension of \\(\\mathcal{G}\\).\nNote that the maximum number of functions in \\(\\mathcal{G}(\\mathcal{S}_n)\\) is \\(2^n\\). If \\(n > d\\), then for any \\(n\\) samples \\(\\mathcal{S}_n\\), \\(\\mathcal{G}(\\mathcal{S}_n)\\) contains fewer than \\(2^n\\) elements. Surprisingly, if a binary-valued function class \\(\\mathcal{G}\\) has VC dimension \\(d\\), then when \\(n > d\\), the size of set \\(\\mathcal{G}(\\mathcal{S}_n)\\) can grow only polynomially in \\(n\\). This gives an \\(O(d\\ln n)\\) upper bound on the uniform entropy of the function class \\(\\mathcal{G}\\) with a finite VC-dimension (see Vapnik and Chervonenkis, 1968, 1971; Sauer, 1972)."}, {"name": "corollary 4.13", "statement": "If \\(\\phi(w, z) \\in [0,1]\\). Let \\(\\mathcal{G} = \\{\\phi(w, z) : w \\in \\Omega\\}\\). With probability at least \\(1 - \\delta\\), the approximate ERM method (3.3) satisfies the (additive) oracle inequality:\n\\[\n\\begin{array}{l}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(\\hat{w},Z)\\leq\\inf_{w\\in\\Omega}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z)+\\epsilon^{\\prime}\\\\ +\\inf_{\\epsilon>0}\\left[2\\epsilon+\\sqrt{\\frac{8\\ln(4 N_{1}(\\epsilon,\\mathcal{G},n)/\\delta)}{n}}\\right].\\\\ \\end{array}\n\\]\nMoreover, we have the following (multiplicative) oracle inequality for all \\(\\gamma \\in (0,1)\\): with probability at least \\(1 - \\delta\\),\n\\[\n\\begin{array}{l}(1-\\gamma)^{2}\\mathbb{E}_{(X,Y)\\sim\\mathcal{D}}\\phi(\\hat{w},Z)\\leq\\inf_{w\\in\\Omega}(1+\\gamma)\\mathbb{E}_{(X,Y)\\sim\\mathcal{D}}\\phi(w,Z)+\\epsilon^{\\prime}\\\\ +\\inf_{\\epsilon>0}\\left[2\\epsilon+\\frac{(6-3\\gamma)\\ln(4 N_{1}(\\epsilon,\\mathcal{G},n)/\\delta)}{2\\gamma n}\\right].\\\\ \\end{array}\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped ENNReal\n\nnamespace GraduateLevel\n\nstructure Predictor (𝒳 : Type*) (𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] where\n  f : 𝒳 → 𝒴'\n  hf : Measurable f\n\nabbrev HypothesisClass (𝒳 : Type*) (𝒴' : Type*) [MeasurableSpace 𝒳] [MeasurableSpace 𝒴'] : Type _ :=\n  Set (Predictor 𝒳 𝒴')\n\ndef Shatters {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool) (S : Finset 𝒳) : Prop :=\n  ∀ (b : ↥S → Bool), ∃ h ∈ H, ∀ (x : ↥S), h.f ↑x = b x\n\nnoncomputable def VCDim {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool) : ℕ∞ :=\n  ⨆ S, ⨆ (_ : Shatters H S), ↑S.card\n\ndef Restrictions {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool) (S : Finset 𝒳) : Set (↥S → Bool) :=\n  (fun h x => h.f ↑x) '' H\n\nnoncomputable def GrowthFunction {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool) (n : ℕ) : ℕ :=\n  sSup ((fun S => (Restrictions H S).ncard) '' {S | S.card = n})\n\ntheorem ch4_lemma_4_16\n    {𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool)\n    (d : ℕ) (hd : 0 < d) (hVC : VCDim H = (d : ℕ∞))\n    (n : ℕ) (hn : 0 < n) :\n    GrowthFunction H n ≤ ∑ ℓ ∈ Finset.range (d + 1), n.choose ℓ ∧\n    ((∑ ℓ ∈ Finset.range (d + 1), n.choose ℓ : ℕ) : ℝ)\n      ≤ max 2 (Real.exp 1 * n / d) ^ d := sorry\n\nend GraduateLevel\n", "target_names": ["ch4_lemma_4_16"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 943, "end": 1242, "gold": "{𝒳 : Type*} [MeasurableSpace 𝒳] (H : HypothesisClass 𝒳 Bool)\n    (d : ℕ) (hd : 0 < d) (hVC : VCDim H = (d : ℕ∞))\n    (n : ℕ) (hn : 0 < n) :\n    GrowthFunction H n ≤ ∑ ℓ ∈ Finset.range (d + 1), n.choose ℓ ∧\n    ((∑ ℓ ∈ Finset.range (d + 1), n.choose ℓ : ℕ) : ℝ)\n      ≤ max 2 (Real.exp 1 * n / d) ^ d"}]}
{"item_id": "ch4_lemma_4_8::symmetrization_4_8::A-stmt", "paper_id": "ch4_lemma_4_8", "target_name": "symmetrization_4_8", "source": "Graduate-level", "nl_statement": "(Symmetrization) Consider a real valued function family \\(F = \\{f : Z \\rightarrow \\mathbb{R}\\}\\). Assume there exists a function \\(\\psi : F \\times Z^{n} \\rightarrow \\mathbb{R}\\) and \\(\\epsilon_{n} : (0,1) \\rightarrow \\mathbb{R}\\) so that with probability at least \\(1-\\delta\\):\n\\[\n\\forall f \\in F, \\qquad f(\\sigma, S_{n}) \\leq \\psi(f, S_{n}) + \\epsilon_{n}(\\delta),\n\\]\nwhere the randomness is over both \\(S_{n} \\sim D^{n}\\) and \\(\\sigma\\). If there exists \\(\\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime})\\) so that the following superadditive inequality holds for all \\((S_{n}, S_{n}^{\\prime})\\)\n\\[\n\\psi(f, S_{n}) + \\psi(f, S_{n}^{\\prime}) \\leq \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}),\n\\]\nthen with probability at least \\(1-\\delta\\) over independent random data \\((S_{n}, S_{n}^{\\prime}) \\sim D^{2n}\\):\n\\[\n\\forall f \\in F, \\qquad f(S_{n}^{\\prime}) \\leq f(S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2).\n\\]", "context": [{"name": "corollary 2.27", "statement": "Let \\(\\sigma_{i} = \\{\\pm 1\\}\\) be independent Bernoulli random variables ( each takes value \\(\\pm 1\\) with equal probability). Let \\(a_{i}\\) be fixed numbers \\((i=1,\\dots,n)\\) Then for all \\(\\epsilon >0\\) ..\n\\[\n\\Pr\\left(n^{-1}\\sum_{i=1}^{n}\\sigma_{i}a_{i}\\geq\\epsilon\\right)\\leq\\exp\\left[-\\frac{n\\epsilon^{2}}{2 n^{-1}\\sum_{i=1}^{n}a_{i}^{2}}\\right].\n\\]"}, {"name": "definition 4.7", "statement": "Consider a real-valued function family \\(\\mathcal{F}:\\mathcal{Z}\\to \\mathbb{R}\\). Consider \\(n\\) iid Bernoulli random variables \\(\\sigma_{i}\\in \\{\\pm 1\\}\\), where \\(\\operatorname*{Pr}(\\sigma_i=1)=\\operatorname*{Pr}(\\sigma_i= -1)=0.5\\). The symmetrized empirical process is\n\\[\nf(\\sigma,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}f(Z_{i})\\qquad f\\in\\mathcal{F},\n\\]\nwhere the randomness is with respect to both \\(S_{n} = \\{Z_{i}\\}\\) and \\(\\sigma = \\{\\sigma_{i}\\}\\).\nNote that in order to bound the symmetrized empirical process, we only need a covering number result on \\( S_{n} \\). Therefore the symmetrization analysis allows us to work with empirical covering numbers. Next we will show that a bound on the symmetrized empirical process can be used to obtain uniform convergence of the empirical process \\( \\{f(S_{n}): f \\in \\mathcal{F}\\} \\) to the corresponding result on the validation data. Since one can relate the validation loss of a learning algorithm to its generalization error on the test data, we can use this result to obtain generalization bounds.\nThe following lemma shows that if we can obtain an upper bound on the symmetrized empirical process \\(\\{f(\\sigma, S_n) : f \\in \\mathcal{F}\\}\\), and the upper bound satisfies a superadditive property, then we can obtain the uniform convergence of the empirical process \\(\\{f(S_n) : f \\in \\mathcal{F}\\}\\) on the training data \\(S_n\\) to the corresponding result on the validation data \\(S_n'\\)."}, {"name": "Union Bound", "statement": "(Union Bound) Consider \\(m\\) events \\(E_1, \\ldots, E_m\\). The following probability inequality holds:\n\\[\n\\Pr (E_1 \\cup \\cdots \\cup E_m) \\leq \\sum_{j=1}^{m} \\Pr(E_j).\n\\]\n\nThe union bound has an alternative expression, which is often used in the learning theory analysis. If each \\(E_j\\) occurs with probability at least \\(1-\\delta_j\\) for \\(j=1, \\ldots, m\\), then with probability at least \\(1-\\sum_{j=1}^{m} \\delta_j\\), all of events \\(\\{E_j\\}\\) occur simultaneously for \\(j=1, \\ldots, m\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\nopen scoped ENNReal\n\nnamespace GraduateLevel\n\ndef Signs (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance instFintypeSigns {n} : Fintype (Signs n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\ninstance instMSSigns (n : ℕ) : MeasurableSpace (Signs n) := ⊤\n\ninstance instNESigns (n : ℕ) : Nonempty (Signs n) :=\n  ⟨fun _ => ⟨1, by decide⟩⟩\n\nnoncomputable def empMean {Z : Type*} {ι : Type*} {n : ℕ}\n    (f : ι → Z → ℝ) (i : ι) (S : Fin n → Z) : ℝ :=\n  (∑ k : Fin n, f i (S k)) / n\n\nnoncomputable def empSym {Z : Type*} {ι : Type*} {n : ℕ}\n    (f : ι → Z → ℝ) (i : ι) (σ : Signs n) (S : Fin n → Z) : ℝ :=\n  (∑ k : Fin n, ((σ k : ℤ) : ℝ) * f i (S k)) / n\n\nnoncomputable def μσ (n : ℕ) : Measure (Signs n) :=\n  (PMF.uniformOfFintype (Signs n)).toMeasure\n\ninstance (n : ℕ) : IsProbabilityMeasure (μσ n) := by\n  unfold μσ; infer_instance\n\ntheorem symmetrization_4_8\n    {Z : Type*} [MeasurableSpace Z] (D : Measure Z) [IsProbabilityMeasure D]\n    {ι : Type*} {n : ℕ} (hn : 0 < n)\n    (f : ι → Z → ℝ)\n    (ψ : ι → (Fin n → Z) → ℝ)\n    (ψtilde : ι → Multiset Z → ℝ)\n    (εn : ℝ → ℝ)\n    (δ : ℝ) (hδ : 0 < δ ∧ δ < 1)\n\n    (hmeas_hyp : ∀ δ' : ℝ,\n      MeasurableSet\n        {p : (Fin n → Z) × Signs n |\n          ∀ i, empSym f i p.2 p.1 ≤ ψ i p.1 + εn δ'})\n\n    (hmeas_concl :\n      MeasurableSet\n        {p : (Fin n → Z) × (Fin n → Z) |\n          ∀ i, empMean f i p.2 ≤ empMean f i p.1\n            + ψtilde i (((List.ofFn p.1 : List Z) : Multiset Z) + ((List.ofFn p.2 : List Z) : Multiset Z)) + 2 * εn (δ / 2)})\n\n    (h_hyp : ∀ δ' : ℝ, 0 < δ' → δ' < 1 →\n      ((Measure.pi fun _ => D).prod (μσ n))\n          {p : (Fin n → Z) × Signs n |\n            ∀ i, empSym f i p.2 p.1 ≤ ψ i p.1 + εn δ'}\n        ≥ ENNReal.ofReal (1 - δ'))\n\n    (h_super : ∀ i (S S' : Fin n → Z),\n      ψ i S + ψ i S' ≤ ψtilde i (((List.ofFn S : List Z) : Multiset Z) + ((List.ofFn S' : List Z) : Multiset Z))) :\n    ((Measure.pi fun _ => D).prod (Measure.pi fun _ => D))\n        {p : (Fin n → Z) × (Fin n → Z) |\n          ∀ i, empMean f i p.2 ≤ empMean f i p.1\n            + ψtilde i (((List.ofFn p.1 : List Z) : Multiset Z) + ((List.ofFn p.2 : List Z) : Multiset Z)) + 2 * εn (δ / 2)}\n      ≥ ENNReal.ofReal (1 - δ) := sorry\n\nend GraduateLevel\n", "target_names": ["symmetrization_4_8"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 944, "end": 2258, "gold": "{Z : Type*} [MeasurableSpace Z] (D : Measure Z) [IsProbabilityMeasure D]\n    {ι : Type*} {n : ℕ} (hn : 0 < n)\n    (f : ι → Z → ℝ)\n    (ψ : ι → (Fin n → Z) → ℝ)\n    (ψtilde : ι → Multiset Z → ℝ)\n    (εn : ℝ → ℝ)\n    (δ : ℝ) (hδ : 0 < δ ∧ δ < 1)\n\n    (hmeas_hyp : ∀ δ' : ℝ,\n      MeasurableSet\n        {p : (Fin n → Z) × Signs n |\n          ∀ i, empSym f i p.2 p.1 ≤ ψ i p.1 + εn δ'})\n\n    (hmeas_concl :\n      MeasurableSet\n        {p : (Fin n → Z) × (Fin n → Z) |\n          ∀ i, empMean f i p.2 ≤ empMean f i p.1\n            + ψtilde i (((List.ofFn p.1 : List Z) : Multiset Z) + ((List.ofFn p.2 : List Z) : Multiset Z)) + 2 * εn (δ / 2)})\n\n    (h_hyp : ∀ δ' : ℝ, 0 < δ' → δ' < 1 →\n      ((Measure.pi fun _ => D).prod (μσ n))\n          {p : (Fin n → Z) × Signs n |\n            ∀ i, empSym f i p.2 p.1 ≤ ψ i p.1 + εn δ'}\n        ≥ ENNReal.ofReal (1 - δ'))\n\n    (h_super : ∀ i (S S' : Fin n → Z),\n      ψ i S + ψ i S' ≤ ψtilde i (((List.ofFn S : List Z) : Multiset Z) + ((List.ofFn S' : List Z) : Multiset Z))) :\n    ((Measure.pi fun _ => D).prod (Measure.pi fun _ => D))\n        {p : (Fin n → Z) × (Fin n → Z) |\n          ∀ i, empMean f i p.2 ≤ empMean f i p.1\n            + ψtilde i (((List.ofFn p.1 : List Z) : Multiset Z) + ((List.ofFn p.2 : List Z) : Multiset Z)) + 2 * εn (δ / 2)}\n      ≥ ENNReal.ofReal (1 - δ)"}]}
{"item_id": "ch6_theorem_6_14::ch6_theorem_6_14::A-stmt", "paper_id": "ch6_theorem_6_14", "target_name": "ch6_theorem_6_14", "source": "Graduate-level", "nl_statement": "Consider offset function of (6.3). We have\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})\\leq 2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]\nConsequently, the approximate regularized ERM method of (6.1) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})]+\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]", "context": [{"name": "definition 6.6", "statement": "Consider any known data-dependent offset function \\( h(w, S_n) \\). Define the uniform convergence complexity of a function class \\( \\mathcal{G} \\) with offset \\( h \\) as\n\\[\n\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\sup_{w\\in\\Omega}[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})-h(w,\\mathcal{S}_{n})].\\tag{6.2}\n\\]\nThis quantity measures the one-sided expected uniform convergence of function class \\(\\mathcal{G}\\) with offset function \\(h(w, S_n)\\).\nWe note that\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}),\\qquad \\text{w i t h} h=0.\n\\]"}, {"name": "definition 6.11", "statement": "Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity."}, {"name": "equation (6.1)", "statement": "The approximate regularized ERM method is to find \\( \\hat{w} \\in \\Omega \\) such that, for some \\( \\epsilon' \\geq 0 \\),\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})+h(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}[\\phi(w,\\mathcal{S}_{n})+h(w,\\mathcal{S}_{n})]+\\epsilon'.\\tag{6.1}\n\\]\nWhen \\(h = 0\\), this reduces to the plain approximate ERM method."}, {"name": "theorem 6.7", "statement": "Let \\( S_{n} \\) be \\( n \\) iid samples from \\( \\mathcal{D} \\). Then the approximate ERM method of (6.1) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})\\right]+\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D}).\n\\]"}, {"name": "definition 3.8", "statement": "We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empOffsetRademacher {Ω Z : Type*} (n : ℕ)\n    (φ : Ω → Z → ℝ) (h₀ : Ω → ℝ) (h₁ : Ω → Z → ℝ) (S : Fin n → Z) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n,\n      ⨆ w : Ω,\n        ((n : ℝ)⁻¹ * ∑ i : Fin n,\n            (σ i : ℝ) * (φ w (S i) + (1/2) * h₁ w (S i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))\n\nnoncomputable def offsetRademacher {Ω Z : Type*} [MeasurableSpace Z] (n : ℕ)\n    (φ : Ω → Z → ℝ) (h₀ : Ω → ℝ) (h₁ : Ω → Z → ℝ) (D : Measure Z) : ℝ :=\n  (Measure.pi (fun _ : Fin n => D))[fun S => empOffsetRademacher n φ h₀ h₁ S]\n\nnoncomputable def offsetUniformConv {Ω Z : Type*} [MeasurableSpace Z] (n : ℕ)\n    (φ : Ω → Z → ℝ) (h₀ : Ω → ℝ) (h₁ : Ω → Z → ℝ) (D : Measure Z) : ℝ :=\n  (Measure.pi (fun _ : Fin n => D))[fun S =>\n    ⨆ w : Ω,\n      (D[fun z => φ w z]\n        - (n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i)\n        - (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))]\n\ntheorem ch6_theorem_6_14\n    {Ω Z : Type*} [Nonempty Ω] [MeasurableSpace Z]\n    (n : ℕ) (hn : 1 ≤ n)\n    (φ : Ω → Z → ℝ) (h₀ : Ω → ℝ) (h₁ : Ω → Z → ℝ)\n    (D : Measure Z) [IsProbabilityMeasure D]\n    (hIntφ : ∀ w, Integrable (fun z => φ w z) D)\n    (hInth₁ : ∀ w, Integrable (fun z => h₁ w z) D)\n    (hBddSup : ∀ S : Fin n → Z,\n      BddAbove (Set.range (fun w : Ω =>\n        D[fun z => φ w z]\n          - (n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i)\n          - (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))))\n    (hIntSup : Integrable (fun S : Fin n → Z =>\n        ⨆ w : Ω,\n          (D[fun z => φ w z]\n            - (n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i)\n            - (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))))\n      (Measure.pi (fun _ : Fin n => D)))\n    (hBddRad : ∀ (S : Fin n → Z) (σ : RadSigns n),\n      BddAbove (Set.range (fun w : Ω =>\n        ((n : ℝ)⁻¹ * ∑ i : Fin n,\n            (σ i : ℝ) * (φ w (S i) + (1/2) * h₁ w (S i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))))))\n    (hIntRad : Integrable (fun S : Fin n → Z => empOffsetRademacher n φ h₀ h₁ S)\n      (Measure.pi (fun _ : Fin n => D)))\n    (hBddSup2 : ∀ P : (Fin n → Z) × (Fin n → Z),\n      BddAbove (Set.range (fun w : Ω =>\n        ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.2 i) + (1/2) * h₁ w (P.2 i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.2 i)))\n          - ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.1 i) + (1/2) * h₁ w (P.1 i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.1 i))))))\n    (hIntSup2 : Integrable (fun P : (Fin n → Z) × (Fin n → Z) =>\n        ⨆ w : Ω,\n          ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.2 i) + (1/2) * h₁ w (P.2 i)))\n            - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.2 i)))\n            - ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.1 i) + (1/2) * h₁ w (P.1 i)))\n            - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.1 i))))\n      ((Measure.pi (fun _ : Fin n => D)).prod (Measure.pi (fun _ : Fin n => D)))) :\n    offsetUniformConv n φ h₀ h₁ D ≤ 2 * offsetRademacher n φ h₀ h₁ D\n    ∧\n    (∀ (ŵ : (Fin n → Z) → Ω) (ε' : ℝ), 0 ≤ ε' →\n      (∀ S : Fin n → Z,\n        ((n : ℝ)⁻¹ * ∑ i : Fin n, φ (ŵ S) (S i))\n            + ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ (ŵ S) + h₁ (ŵ S) (S i)))\n          ≤ (⨅ w : Ω,\n              ((n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i))\n                + ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))) + ε') →\n      Integrable (fun S : Fin n → Z => D[fun z => φ (ŵ S) z])\n        (Measure.pi (fun _ : Fin n => D)) →\n      BddBelow (Set.range (fun w : Ω =>\n        D[fun z => φ w z]\n          + (Measure.pi (fun _ : Fin n => D))[fun S =>\n              (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))])) →\n      (∀ S : Fin n → Z,\n        BddBelow (Set.range (fun w : Ω =>\n          ((n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i))\n            + ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))))) →\n      (Measure.pi (fun _ : Fin n => D))[fun S => D[fun z => φ (ŵ S) z]]\n        ≤ (⨅ w : Ω,\n            D[fun z => φ w z]\n              + (Measure.pi (fun _ : Fin n => D))[fun S =>\n                  (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))])\n          + ε' + 2 * offsetRademacher n φ h₀ h₁ D) := sorry\n\nend GraduateLevel\n", "target_names": ["ch6_theorem_6_14"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1202, "end": 4311, "gold": "{Ω Z : Type*} [Nonempty Ω] [MeasurableSpace Z]\n    (n : ℕ) (hn : 1 ≤ n)\n    (φ : Ω → Z → ℝ) (h₀ : Ω → ℝ) (h₁ : Ω → Z → ℝ)\n    (D : Measure Z) [IsProbabilityMeasure D]\n    (hIntφ : ∀ w, Integrable (fun z => φ w z) D)\n    (hInth₁ : ∀ w, Integrable (fun z => h₁ w z) D)\n    (hBddSup : ∀ S : Fin n → Z,\n      BddAbove (Set.range (fun w : Ω =>\n        D[fun z => φ w z]\n          - (n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i)\n          - (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))))\n    (hIntSup : Integrable (fun S : Fin n → Z =>\n        ⨆ w : Ω,\n          (D[fun z => φ w z]\n            - (n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i)\n            - (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))))\n      (Measure.pi (fun _ : Fin n => D)))\n    (hBddRad : ∀ (S : Fin n → Z) (σ : RadSigns n),\n      BddAbove (Set.range (fun w : Ω =>\n        ((n : ℝ)⁻¹ * ∑ i : Fin n,\n            (σ i : ℝ) * (φ w (S i) + (1/2) * h₁ w (S i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))))))\n    (hIntRad : Integrable (fun S : Fin n → Z => empOffsetRademacher n φ h₀ h₁ S)\n      (Measure.pi (fun _ : Fin n => D)))\n    (hBddSup2 : ∀ P : (Fin n → Z) × (Fin n → Z),\n      BddAbove (Set.range (fun w : Ω =>\n        ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.2 i) + (1/2) * h₁ w (P.2 i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.2 i)))\n          - ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.1 i) + (1/2) * h₁ w (P.1 i)))\n          - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.1 i))))))\n    (hIntSup2 : Integrable (fun P : (Fin n → Z) × (Fin n → Z) =>\n        ⨆ w : Ω,\n          ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.2 i) + (1/2) * h₁ w (P.2 i)))\n            - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.2 i)))\n            - ((n : ℝ)⁻¹ * ∑ i : Fin n, (φ w (P.1 i) + (1/2) * h₁ w (P.1 i)))\n            - (1/2) * ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (P.1 i))))\n      ((Measure.pi (fun _ : Fin n => D)).prod (Measure.pi (fun _ : Fin n => D)))) :\n    offsetUniformConv n φ h₀ h₁ D ≤ 2 * offsetRademacher n φ h₀ h₁ D\n    ∧\n    (∀ (ŵ : (Fin n → Z) → Ω) (ε' : ℝ), 0 ≤ ε' →\n      (∀ S : Fin n → Z,\n        ((n : ℝ)⁻¹ * ∑ i : Fin n, φ (ŵ S) (S i))\n            + ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ (ŵ S) + h₁ (ŵ S) (S i)))\n          ≤ (⨅ w : Ω,\n              ((n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i))\n                + ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))) + ε') →\n      Integrable (fun S : Fin n → Z => D[fun z => φ (ŵ S) z])\n        (Measure.pi (fun _ : Fin n => D)) →\n      BddBelow (Set.range (fun w : Ω =>\n        D[fun z => φ w z]\n          + (Measure.pi (fun _ : Fin n => D))[fun S =>\n              (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))])) →\n      (∀ S : Fin n → Z,\n        BddBelow (Set.range (fun w : Ω =>\n          ((n : ℝ)⁻¹ * ∑ i : Fin n, φ w (S i))\n            + ((n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i)))))) →\n      (Measure.pi (fun _ : Fin n => D))[fun S => D[fun z => φ (ŵ S) z]]\n        ≤ (⨅ w : Ω,\n            D[fun z => φ w z]\n              + (Measure.pi (fun _ : Fin n => D))[fun S =>\n                  (n : ℝ)⁻¹ * ∑ i : Fin n, (h₀ w + h₁ w (S i))])\n          + ε' + 2 * offsetRademacher n φ h₀ h₁ D)"}]}
{"item_id": "ch6_theorem_6_25::ch6_theorem_6_25::A-stmt", "paper_id": "ch6_theorem_6_25", "target_name": "ch6_theorem_6_25", "source": "Graduate-level", "nl_statement": "We have\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\inf_{\\epsilon\\geq 0}\\left[4\\epsilon+12\\int_{\\epsilon}^{\\infty}\\sqrt{\\frac{\\ln N(\\epsilon^{\\prime},\\mathcal{G},L_{2}(\\mathcal{S}_{n}))}{n}}d\\epsilon^{\\prime}\\right].\n\\]", "context": [{"name": "theorem 6.23", "statement": "If \\(\\mathcal{G}\\) is a finite function class with \\(|\\mathcal{G}| = N\\), then\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\sup_{g\\in\\mathcal{G}}\\|g\\|_{L_{2}(\\mathcal{S}_{n})}\\cdot\\sqrt{\\frac{2\\ln N}{n}}.\n\\]\nIf moreover, for all \\( g \\in \\mathcal{G} \\), \\( g(z) \\in [0,1] \\). Consider the offset decomposition (6.3), and let \\( h_0(g) = 0 \\), and \\( h_1(g, S_n) = (\\gamma / n)\\sum_{i=1}^{n} g(Z_i) \\). Then we have\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})\\leq\\frac{(1+0.5\\gamma)^{2}\\ln N}{\\gamma n}.\n\\]"}, {"name": "Empirical and Uniform Covering Number", "statement": "Given an empirical distribution \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), we define the pseudometric \\( d = L_{p}(S_{n}) \\) as\n\\[\nd(\\phi,\\phi^{\\prime})=\\left[\\frac{1}{n}\\sum_{i=1}^{n}|\\phi(Z_{i})-\\phi^{\\prime}(Z_{i})|^{p}\\right]^{1/p}.\n\\]\nThe corresponding metric covering number \\(N(\\epsilon, \\mathcal{G}, L_p(\\mathcal{S}_n))\\) is referred to as the\nempirical \\( L_{p} \\) covering number. Given \\( n \\), the largest \\( L_{p} \\) covering number over empirical distribution \\( S_{n} \\) is referred to as the uniform \\( L_{p} \\) covering number\n\\[\nN_{p}(\\epsilon,\\mathcal{G},n)=\\sup_{\\mathcal{S}_{n}}N(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{S}_{n})).\n\\]"}, {"name": "definition 6.3", "statement": "Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ndef isENet {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n  s ⊆ ⋃ x ∈ t, Metric.closedBall x eps\n\nnoncomputable def coveringNum {A : Type*} [PseudoMetricSpace A] (eps : ℝ) (s : Set A) : WithTop ℕ :=\n  sInf {n | ∃ t, isENet t eps s ∧ (t.card : WithTop ℕ) = n}\n\nnoncomputable def dudleyIntegrand6_25 {ι : Type*} [PseudoMetricSpace ι]\n    (n : ℕ) (eps' : ℝ) : ℝ :=\n  Real.sqrt\n    (Real.log (((coveringNum eps' (Set.univ : Set ι)).untopD 0 : ℕ) : ℝ)\n      / (n : ℝ))\n\ntheorem ch6_theorem_6_25\n    {ι : Type*} {𝒳 : Type*} [PseudoMetricSpace ι]\n    (n : ℕ) (hn : 1 ≤ n)\n    (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳)\n    (hdist : ∀ i j : ι,\n      dist i j = Real.sqrt ((n : ℝ)⁻¹ * ∑ k : Fin n, (f i (S k) - f j (S k)) ^ 2))\n    (hTB : TotallyBounded (Set.univ : Set ι))\n    (hBdd : BddAbove (Set.range (fun i : ι =>\n      Real.sqrt ((n : ℝ)⁻¹ * ∑ k : Fin n, (f i (S k)) ^ 2))))\n    (hInt : ∀ eps ∈ Set.Ici (0 : ℝ),\n      IntegrableOn (fun eps' => dudleyIntegrand6_25 (ι := ι) n eps') (Set.Ioi eps)) :\n    empRadSigned n f S\n      ≤ ⨅ eps : Set.Ici (0 : ℝ),\n          (4 * (eps : ℝ)\n            + 12 * ∫ eps' in Set.Ioi (eps : ℝ), dudleyIntegrand6_25 (ι := ι) n eps') := sorry\n\nend GraduateLevel\n", "target_names": ["ch6_theorem_6_25"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 973, "end": 1629, "gold": "{ι : Type*} {𝒳 : Type*} [PseudoMetricSpace ι]\n    (n : ℕ) (hn : 1 ≤ n)\n    (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳)\n    (hdist : ∀ i j : ι,\n      dist i j = Real.sqrt ((n : ℝ)⁻¹ * ∑ k : Fin n, (f i (S k) - f j (S k)) ^ 2))\n    (hTB : TotallyBounded (Set.univ : Set ι))\n    (hBdd : BddAbove (Set.range (fun i : ι =>\n      Real.sqrt ((n : ℝ)⁻¹ * ∑ k : Fin n, (f i (S k)) ^ 2))))\n    (hInt : ∀ eps ∈ Set.Ici (0 : ℝ),\n      IntegrableOn (fun eps' => dudleyIntegrand6_25 (ι := ι) n eps') (Set.Ioi eps)) :\n    empRadSigned n f S\n      ≤ ⨅ eps : Set.Ici (0 : ℝ),\n          (4 * (eps : ℝ)\n            + 12 * ∫ eps' in Set.Ioi (eps : ℝ), dudleyIntegrand6_25 (ι := ι) n eps')"}]}
{"item_id": "ch8_theorem_8_9::ch8_theorem_8_9::A-stmt", "paper_id": "ch8_theorem_8_9", "target_name": "ch8_theorem_8_9", "source": "Graduate-level", "nl_statement": "Consider \\( h \\) with decomposition 6.3 as \\( h(f,z) = h_0(f) + h_1(f,z) \\). Assume that for some \\( M \\geq 0 \\):\n\\[\n\\sup_{f\\in\\mathcal{F}}\\sup_{z,z^{\\prime}}[|\\phi(f,z)-\\phi(f,z^{\\prime})| + |h_{1}(f,z)-h_{1}(f,z^{\\prime})|]\\leq M.\n\\]\nWe have with probability at least \\(1 - \\delta\\)\n\\[\nR_{n}^{h}(\\mathcal{F},\\mathcal{D})\\leq R^{h}(\\mathcal{F},\\mathcal{S}_{n})+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]\nSimilarly, with probability at least \\(1 - \\delta\\), we have\n\\[\nR_{n}^{h}(\\mathcal{F},\\mathcal{D})\\geq R^{h}(\\mathcal{F},\\mathcal{S}_{n})-M\\sqrt{\\frac{\\ln(1/\\delta)}{2 n}}.\n\\]", "context": [{"name": "McDiarmid's Inequality", "statement": "Consider \\( n \\) independent random variables \\( X_{1},\\ldots,X_{n} \\), and a real-valued function \\( f(X_{1},\\ldots,X_{n}) \\) that satisfies the following inequality\n\\[\n\\sup_{x_{1},\\dots,x_{n},x_{i}^{\\prime}}|f(x_{1},\\dots,x_{n})-f(x_{1},\\dots,x_{i-1},x_{i}^{\\prime},x_{i+1},\\dots,x_{n})|\\leq c_{i}\n\\]\nfor all \\(1 \\leq i \\leq n\\). Then for all \\(\\epsilon > 0\\):\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\geq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)+\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]\nSimilarly:\n\\[\n\\Pr\\left[f\\left(X_{1},\\dots,X_{n}\\right)\\leq\\mathbb{E}f\\left(X_{1},\\dots,X_{n}\\right)-\\epsilon\\right]\\leq\\exp\\left(\\frac{-2\\epsilon^{2}}{\\sum_{i=1}^{n}c_{i}^{2}}\\right).\n\\]"}, {"name": "definition 6.11", "statement": "Consider a function class \\(\\mathcal{G} = \\{\\phi(w, Z) : w \\in \\Omega\\}\\), and let \\(h\\) be an offset function of the following form\n\\[\nh(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}h(w,Z_{i}),\\quad h(w,z)=h_{0}(w)+h_{1}(w,z).\\tag{6.3}\n\\]\nGiven \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), the (one-sided) empirical Rademacher complexity of \\( \\mathcal{G} \\) with offset \\( h \\) decomposition (6.3) is defined as\n\\[\nR^{h}(\\mathcal{G},\\mathcal{S}_{n})=\\mathbb{E}_{\\sigma}\\sup_{w\\in\\Omega}\\left[\\frac{1}{n}\\sum_{i=1}^{n}\\sigma_{i}[\\phi(w,Z_{i})+0. 5h_{1}(w,Z_{i})]-0. 5h(w,\\mathcal{S}_{n})\\right],\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}^{h}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R^{h}(\\mathcal{G},\\mathcal{S}_{n}).\n\\]\nWe note that the standard Rademacher complexity can be regarded as a special case of the offset Rademacher complexity with \\( h(\\cdot) = h_0(\\cdot) = h_1(\\cdot) = 0 \\):\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})=R^{0}(\\mathcal{G},\\mathcal{S}_{n}),\\quad R_{n}(\\mathcal{G},\\mathcal{D})=R_{n}^{0}(\\mathcal{G},\\mathcal{D}).\n\\]\nIt should be pointed out that the decomposition of \\( h \\) in (6.3) may not be unique, and the offset Rademacher complexity relies on the specific decomposition used.\nAs shown in Example 6.10, we allow distribution dependent offset in the definition of uniform convergence complexity. We also allow distribution dependent offset in the definition of offset Rademacher complexity."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\nopen scoped BigOperators\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def offsetH {F : Type*} {𝒵 : Type*} {n : ℕ}\n    (h0 : F → ℝ) (h1 : F → 𝒵 → ℝ) (f : F) (s : Fin n → 𝒵) : ℝ :=\n  h0 f + (n : ℝ)⁻¹ * ∑ i : Fin n, h1 f (s i)\n\nnoncomputable def offsetEmpRC {F : Type*} {𝒵 : Type*} (n : ℕ)\n    (φ : F → 𝒵 → ℝ) (h0 : F → ℝ) (h1 : F → 𝒵 → ℝ) (s : Fin n → 𝒵) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n,\n      ⨆ f : F,\n        ( (n : ℝ)⁻¹ * ∑ i : Fin n, (σ i : ℝ) * (φ f (s i) + (1/2) * h1 f (s i))\n          - (1/2) * offsetH h0 h1 f s )\n\ntheorem ch8_theorem_8_9\n    {F : Type*} [Nonempty F]\n    {𝒵 : Type*} [MeasurableSpace 𝒵] [Nonempty 𝒵]\n    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]\n    {n : ℕ} (hn : 0 < n)\n    (Z : Fin n → Ω → 𝒵)\n    (hZmeas : ∀ i, Measurable (Z i))\n    (hZindep : ProbabilityTheory.iIndepFun Z μ)\n    (hZident : ∀ i j, Measure.map (Z i) μ = Measure.map (Z j) μ)\n    (φ : F → 𝒵 → ℝ) (h0 : F → ℝ) (h1 : F → 𝒵 → ℝ)\n\n    (hBdd : ∀ (s : Fin n → 𝒵) (σ : RadSigns n),\n        BddAbove (Set.range (fun f : F =>\n          (n : ℝ)⁻¹ * ∑ i : Fin n, (σ i : ℝ) * (φ f (s i) + (1/2) * h1 f (s i))\n            - (1/2) * offsetH h0 h1 f s)))\n\n    (hRmeas : Measurable (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s))\n\n    (hRint : Integrable (fun x => offsetEmpRC n φ h0 h1 (fun i => Z i x)) μ)\n    (M : ℝ) (hM : 0 ≤ M)\n\n    (hSens : ∀ (f : F) (z z' : 𝒵),\n        |φ f z - φ f z'| + |h1 f z - h1 f z'| ≤ M)\n    {δ : ℝ} (hδ0 : 0 < δ) (hδ1 : δ ≤ 1) :\n\n    ( 1 - δ ≤\n        (μ {ω |\n          (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i ω)\n            ≥ (∫ x, (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i x) ∂μ)\n                - M * Real.sqrt (Real.log (1/δ) / (2 * n)) }).toReal )\n    ∧\n    ( 1 - δ ≤\n        (μ {ω |\n          (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i ω)\n            ≤ (∫ x, (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i x) ∂μ)\n                + M * Real.sqrt (Real.log (1/δ) / (2 * n)) }).toReal ) := sorry\n\nend GraduateLevel\n", "target_names": ["ch8_theorem_8_9"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 797, "end": 2249, "gold": "{F : Type*} [Nonempty F]\n    {𝒵 : Type*} [MeasurableSpace 𝒵] [Nonempty 𝒵]\n    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]\n    {n : ℕ} (hn : 0 < n)\n    (Z : Fin n → Ω → 𝒵)\n    (hZmeas : ∀ i, Measurable (Z i))\n    (hZindep : ProbabilityTheory.iIndepFun Z μ)\n    (hZident : ∀ i j, Measure.map (Z i) μ = Measure.map (Z j) μ)\n    (φ : F → 𝒵 → ℝ) (h0 : F → ℝ) (h1 : F → 𝒵 → ℝ)\n\n    (hBdd : ∀ (s : Fin n → 𝒵) (σ : RadSigns n),\n        BddAbove (Set.range (fun f : F =>\n          (n : ℝ)⁻¹ * ∑ i : Fin n, (σ i : ℝ) * (φ f (s i) + (1/2) * h1 f (s i))\n            - (1/2) * offsetH h0 h1 f s)))\n\n    (hRmeas : Measurable (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s))\n\n    (hRint : Integrable (fun x => offsetEmpRC n φ h0 h1 (fun i => Z i x)) μ)\n    (M : ℝ) (hM : 0 ≤ M)\n\n    (hSens : ∀ (f : F) (z z' : 𝒵),\n        |φ f z - φ f z'| + |h1 f z - h1 f z'| ≤ M)\n    {δ : ℝ} (hδ0 : 0 < δ) (hδ1 : δ ≤ 1) :\n\n    ( 1 - δ ≤\n        (μ {ω |\n          (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i ω)\n            ≥ (∫ x, (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i x) ∂μ)\n                - M * Real.sqrt (Real.log (1/δ) / (2 * n)) }).toReal )\n    ∧\n    ( 1 - δ ≤\n        (μ {ω |\n          (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i ω)\n            ≤ (∫ x, (fun s : Fin n → 𝒵 => offsetEmpRC n φ h0 h1 s) (fun i => Z i x) ∂μ)\n                + M * Real.sqrt (Real.log (1/δ) / (2 * n)) }).toReal )"}]}
{"item_id": "ch10_corollary_10_17::ch10_corollary_10_17::A-stmt", "paper_id": "ch10_corollary_10_17", "target_name": "ch10_corollary_10_17", "source": "Graduate-level", "nl_statement": "Let \\(\\mathcal{F}_A = \\{f(q,x):\\mathrm{KL}(q||q_0)\\leq A^2\\}\\) be entropy regularized functions of (10.11). Then\n\\[\nR\\left(\\mathcal{F}_{A},\\mathcal{S}_{n}\\right)\\leq\\sqrt{\\frac{2}{n}}A\\sup_{\\theta}\\sqrt{\\frac{1}{n}\\sum_{i=1}^{n}\\psi\\left(\\theta,X_{i}\\right)^{2}}.\n\\]", "context": [{"name": "theorem 10.16", "statement": "Consider \\( h(q) = h_0(q) = \\lambda \\mathrm{KL}(q||q_0) \\), and \\( \\mathcal{F} = \\{f(q,x)\\} \\) in (10.11). The offset Rademacher complexity can be bounded as\n\\[\nR^{h}(\\mathcal{F},\\mathcal{S}_{n})\\leq\\frac{\\lambda}{2}\\ln\\mathbb{E}_{\\theta\\sim q_{0}}\\exp\\left[\\frac{2}{\\lambda^{2}n^{2}}\\sum_{i=1}^{n}\\psi(\\theta,X_{i})^{2}\\right]\\leq\\frac{1}{\\lambda n^{2}}\\sup_{\\theta}\\sum_{i=1}^{n}\\psi(\\theta,X_{i})^{2}.\n\\]"}, {"name": "equation (10.11)", "statement": "Let \\(\\Theta\\) be a parameter space, let \\(q\\) and the reference distribution \\(q_0\\) be probability distributions on \\(\\Theta\\), and let \\(\\psi(\\theta,x)\\) be a base function indexed by \\(\\theta\\in\\Theta\\). Define\n\\[\nf(q,x)=\\mathbb{E}_{\\theta\\sim q}\\psi(\\theta,x).\n\\tag{10.11}\n\\]\nFor \\(A\\geq 0\\), the entropy-regularized function class is\n\\[\n\\mathcal{F}_A=\\{f(q,x):\\mathrm{KL}(q||q_0)\\leq A^2\\}.\n\\]"}, {"name": "definition 6.3", "statement": "Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\nopen scoped ENNReal\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ntheorem ch10_corollary_10_17\n    {Θ : Type*} [MeasurableSpace Θ] {𝒳 : Type*}\n    {n : ℕ} (hn : 0 < n)\n    (A : ℝ) (hA : 0 ≤ A)\n    (q_0 : Measure Θ) [IsProbabilityMeasure q_0]\n    (ψ : Θ → 𝒳 → ℝ)\n    (S : Fin n → 𝒳)\n    (hint : ∀ (q : Measure Θ), IsProbabilityMeasure q →\n        InformationTheory.klDiv q q_0 ≤ ENNReal.ofReal (A^2) →\n        ∀ x : 𝒳, Integrable (fun θ => ψ θ x) q)\n    (hbdd : BddAbove (Set.range (fun θ : Θ => (1 / (n : ℝ)) * ∑ i, (ψ θ (S i))^2))) :\n    empRadSigned\n        (ι := {q : Measure Θ //\n            IsProbabilityMeasure q ∧\n              InformationTheory.klDiv q q_0 ≤ ENNReal.ofReal (A^2)})\n        n\n        (fun q x => ∫ θ, ψ θ x ∂(q : Measure Θ))\n        S\n      ≤ Real.sqrt (2 / n) * A *\n          ⨆ θ : Θ, Real.sqrt ((1 / (n : ℝ)) * ∑ i, (ψ θ (S i))^2) := sorry\n\nend GraduateLevel\n", "target_names": ["ch10_corollary_10_17"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 516, "end": 1273, "gold": "{Θ : Type*} [MeasurableSpace Θ] {𝒳 : Type*}\n    {n : ℕ} (hn : 0 < n)\n    (A : ℝ) (hA : 0 ≤ A)\n    (q_0 : Measure Θ) [IsProbabilityMeasure q_0]\n    (ψ : Θ → 𝒳 → ℝ)\n    (S : Fin n → 𝒳)\n    (hint : ∀ (q : Measure Θ), IsProbabilityMeasure q →\n        InformationTheory.klDiv q q_0 ≤ ENNReal.ofReal (A^2) →\n        ∀ x : 𝒳, Integrable (fun θ => ψ θ x) q)\n    (hbdd : BddAbove (Set.range (fun θ : Θ => (1 / (n : ℝ)) * ∑ i, (ψ θ (S i))^2))) :\n    empRadSigned\n        (ι := {q : Measure Θ //\n            IsProbabilityMeasure q ∧\n              InformationTheory.klDiv q q_0 ≤ ENNReal.ofReal (A^2)})\n        n\n        (fun q x => ∫ θ, ψ θ x ∂(q : Measure Θ))\n        S\n      ≤ Real.sqrt (2 / n) * A *\n          ⨆ θ : Θ, Real.sqrt ((1 / (n : ℝ)) * ∑ i, (ψ θ (S i))^2)"}]}
{"item_id": "ch10_proposition_10_1::ch10_proposition_10_1::A-stmt", "paper_id": "ch10_proposition_10_1", "target_name": "ch10_proposition_10_1", "source": "Graduate-level", "nl_statement": "Assume that the \\(m\\) feature functions \\(\\{\\psi_j(X)\\}\\) are orthonormal when \\(X\\sim \\mathcal{D}\\). Then there exists an absolute constant \\(c > 0\\) such that for sufficiently large \\(n\\),\n\\[\nR_{n}(\\mathcal{F}^{\\prime},\\mathcal{D})\\geq c\\sqrt{\\frac{m}{n}}A,\n\\]\nwhere \\(\\mathcal{F}'\\) is given by (10.2).", "context": [{"name": "theorem 5.3", "statement": "Let \\(\\| \\cdot \\|\\) be a seminorm on \\(\\mathbb{R}^k\\). Let \\(B(r) = \\{z \\in \\mathbb{R}^k : \\| z \\| \\leq r\\}\\) be the \\(\\| \\cdot \\|\\)-ball with radius \\(r\\). Then\n\\[\nM(\\epsilon,B(r),\\|\\cdot\\|)\\leq\\left(1+2 r/\\epsilon\\right)^{k}.\n\\]\nMoreover, if \\(\\| \\cdot \\|\\) is a norm, then\n\\[\nN(\\epsilon,B(r),\\|\\cdot\\|)\\geq(r/\\epsilon)^{k}.\n\\]"}, {"name": "equation 10.2", "statement": "\\[\n\\mathcal{F}^{\\prime}=\\{f(w,x)=w^\\top\\psi(x):\\|w\\|_2\\leq A\\}.\n\\]"}, {"name": "definition 6.3", "statement": "Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\nnoncomputable def popRadSigned {Ω : Type*} [MeasurableSpace Ω] {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (μ : Measure Ω) (X : Ω → 𝒳) : ℝ :=\n  ∫ ω : Fin n → Ω, empRadSigned n f (X ∘ ω) ∂(Measure.pi fun _ => μ)\n\nabbrev L2BallIndex (m : ℕ) (A : ℝ) : Type :=\n  {w : EuclideanSpace ℝ (Fin m) // ‖w‖ ≤ A}\n\ndef linPred (m : ℕ) (A : ℝ) {𝒳 : Type*} (ψ : 𝒳 → Fin m → ℝ)\n    (w : L2BallIndex m A) (x : 𝒳) : ℝ :=\n  ∑ j : Fin m, w.1 j * ψ x j\n\ntheorem ch10_proposition_10_1 {𝒳 : Type*} [MeasurableSpace 𝒳] :\n    ∃ c : ℝ, 0 < c ∧\n      ∀ (m : ℕ) (A : ℝ) (ψ : 𝒳 → Fin m → ℝ) (D : Measure 𝒳)\n        [IsProbabilityMeasure D],\n        0 ≤ A →\n        (∀ j, Measurable (fun x => ψ x j)) →\n        (∀ j k, Integrable (fun x => ψ x j * ψ x k) D) →\n        (∀ j k, ∫ x, ψ x j * ψ x k ∂D = if j = k then 1 else 0) →\n        ∃ N₀ : ℕ, ∀ n : ℕ, N₀ ≤ n →\n          c * Real.sqrt ((m : ℝ) / (n : ℝ)) * A ≤\n            popRadSigned n (linPred m A ψ) D (id : 𝒳 → 𝒳) := sorry\n\nend GraduateLevel\n", "target_names": ["ch10_proposition_10_1"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 916, "end": 1392, "gold": "{𝒳 : Type*} [MeasurableSpace 𝒳] :\n    ∃ c : ℝ, 0 < c ∧\n      ∀ (m : ℕ) (A : ℝ) (ψ : 𝒳 → Fin m → ℝ) (D : Measure 𝒳)\n        [IsProbabilityMeasure D],\n        0 ≤ A →\n        (∀ j, Measurable (fun x => ψ x j)) →\n        (∀ j k, Integrable (fun x => ψ x j * ψ x k) D) →\n        (∀ j k, ∫ x, ψ x j * ψ x k ∂D = if j = k then 1 else 0) →\n        ∃ N₀ : ℕ, ∀ n : ℕ, N₀ ≤ n →\n          c * Real.sqrt ((m : ℝ) / (n : ℝ)) * A ≤\n            popRadSigned n (linPred m A ψ) D (id : 𝒳 → 𝒳)"}]}
{"item_id": "ch10_proposition_10_35::ch10_proposition_10_35::A-stmt", "paper_id": "ch10_proposition_10_35", "target_name": "ch10_proposition_10_35", "source": "Graduate-level", "nl_statement": "Let \\(\\hat{F} = \\mathrm{supp}(\\hat{w})\\). If \\(X_{\\hat{F}}^{\\top}X_{\\hat{F}}\\) is positive definite, and\n\\[\nX_{\\hat{F}}^{\\top}\\left(X_{\\hat{F}}\\hat{w}_{\\hat{F}}-Y\\right)+\\lambda\\operatorname{sign}(\\hat{w})_{\\hat{F}}=0,\n\\]\n\\[\nX_{j}^{\\top}(X_{\\hat{F}}\\hat{w}_{\\hat{F}}-Y)\\in(-\\lambda,\\lambda)\\qquad(j\\not\\in\\hat{F}),\n\\]\nthen \\(\\hat{w}\\) is the unique solution of (10.20).", "context": [{"name": "definition 10.2", "statement": "The sparsity pattern, or support of a weight vector \\( w \\in \\mathbb{R}^m \\) is defined as\n\\[\n\\operatorname{supp}(w)=\\{j:w_{j}\\neq 0\\},\n\\]\nand the \\(L_0\\) norm of \\(w\\) is defined as\n\\[\n\\left\\|w\\right\\|_{0}=\\left|\\operatorname{supp}(w)\\right|.\n\\]"}, {"name": "equation 10.20", "statement": "\\[\\hat{w} = \\arg\\min_w Q_{L_1}(w); \\quad Q_{L_1}(w) = \\frac{1}{2}\\left\\|Xw - Y\\right\\|_2^2 + \\lambda \\left\\|w\\right\\|_1, \\tag{10.20}\\] where \\(\\lambda > 0\\) is an appropriately chosen regularization parameter."}, {"name": "proposition 10.34", "statement": "Let \\(\\hat{F} = \\mathrm{supp}(\\hat{w})\\), then \\(\\hat{w}\\) is a solution of (10.20) if and\nonly if\n\\[\nX_{\\hat{F}}^{\\top}(X_{\\hat{F}}\\hat{w}_{\\hat{F}}-Y)+\\lambda\\mathrm{sign}(\\hat{w})_{\\hat{F}}=0,\n\\]\n\\[\nX_{j}^{\\top}(X_{\\hat{F}}\\hat{w}_{\\hat{F}}-Y)\\in [-\\lambda,\\lambda]\\qquad(j\\not\\in\\hat{F}),\n\\]\nwhere \\( X_{\\hat{F}} \\) contains columns of \\( X \\) with columns in \\( \\hat{F} \\), and \\( X_{j} \\) is the \\( j \\)-th column of \\( X \\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Matrix Real\n\nnamespace GraduateLevel\n\nclass PosDefAsymm {α : Type*} [Fintype α] (A : Matrix α α ℝ) : Prop where\n  pd : ∀ x, x ≠ 0 → 0 < x ⬝ᵥ (A *ᵥ x)\n\nnoncomputable def lassoObjective {n : ℕ} {ι : Type*} [Fintype ι]\n    (X : Matrix (Fin n) ι ℝ) (Y : Fin n → ℝ) (lam : ℝ) (w : ι → ℝ) : ℝ :=\n  (1 / 2) * (∑ i : Fin n, (X.mulVec w i - Y i) ^ 2) + lam * (∑ j : ι, |w j|)\n\ntheorem ch10_proposition_10_35\n    {n : ℕ} {ι : Type*} [Fintype ι] [DecidableEq ι]\n    (X : Matrix (Fin n) ι ℝ) (Y : Fin n → ℝ) (lam : ℝ) (ŵ : ι → ℝ)\n    (hlam : 0 ≤ lam)\n    (hPD : PosDefAsymm\n      ((X.submatrix (id : Fin n → Fin n) (Subtype.val : {j // ŵ j ≠ 0} → ι))ᵀ *\n       (X.submatrix (id : Fin n → Fin n) (Subtype.val : {j // ŵ j ≠ 0} → ι))))\n    (hStat : ∀ j : ι, ŵ j ≠ 0 →\n      (∑ i : Fin n, X i j * (X.mulVec ŵ i - Y i)) + lam * Real.sign (ŵ j) = 0)\n    (hOff : ∀ j : ι, ŵ j = 0 →\n      |∑ i : Fin n, X i j * (X.mulVec ŵ i - Y i)| < lam) :\n    IsMinOn (lassoObjective X Y lam) Set.univ ŵ ∧\n      (∀ w, IsMinOn (lassoObjective X Y lam) Set.univ w → w = ŵ) := sorry\n\nend GraduateLevel\n", "target_names": ["ch10_proposition_10_35"], "topic_domain": "Optimization", "topic_subtopic": "Convex Analysis & Duality", "holes": [{"start": 438, "end": 1071, "gold": "{n : ℕ} {ι : Type*} [Fintype ι] [DecidableEq ι]\n    (X : Matrix (Fin n) ι ℝ) (Y : Fin n → ℝ) (lam : ℝ) (ŵ : ι → ℝ)\n    (hlam : 0 ≤ lam)\n    (hPD : PosDefAsymm\n      ((X.submatrix (id : Fin n → Fin n) (Subtype.val : {j // ŵ j ≠ 0} → ι))ᵀ *\n       (X.submatrix (id : Fin n → Fin n) (Subtype.val : {j // ŵ j ≠ 0} → ι))))\n    (hStat : ∀ j : ι, ŵ j ≠ 0 →\n      (∑ i : Fin n, X i j * (X.mulVec ŵ i - Y i)) + lam * Real.sign (ŵ j) = 0)\n    (hOff : ∀ j : ι, ŵ j = 0 →\n      |∑ i : Fin n, X i j * (X.mulVec ŵ i - Y i)| < lam) :\n    IsMinOn (lassoObjective X Y lam) Set.univ ŵ ∧\n      (∀ w, IsMinOn (lassoObjective X Y lam) Set.univ w → w = ŵ)"}]}
{"item_id": "ch11_proposition_11_9::ch11_proposition_11_9::A-stmt", "paper_id": "ch11_proposition_11_9", "target_name": "ch11_proposition_11_9", "source": "Graduate-level", "nl_statement": "Let \\(\\mathcal{F} = \\{h(\\theta^\\top x):\\theta \\in \\mathbb{R}^d\\}\\) and \\(\\mathcal{F}_{\\pm} = \\mathcal{F}\\cup -\\mathcal{F}\\). Let\n\\[\n\\mathcal{F}_{A}^{1}=\\{\\mathbb{E}_{\\theta\\sim\\mu}u(\\theta)h(\\theta^{\\top}x):\\mathbb{E}_{\\theta\\sim\\mu}|u(\\theta)|\\leq A\\}.\n\\]\nThen for all monotone function \\( h(\\cdot) \\) with \\( h(\\cdot) \\in [0, M] \\):\n\\[\nR\\left(\\mathcal{F}_{\\pm},\\mathcal{S}_{n}\\right)\\leq M\\frac{32\\sqrt{d+1}}{\\sqrt{n}}\n\\]\n\\[\nR\\left(\\mathcal{F}_{A}^{1},\\mathcal{S}_{n}\\right)\\leq A M\\frac{32\\sqrt{d+1}}{\\sqrt{n}}.\n\\]", "context": [{"name": "theorem 5.11", "statement": "Assume that \\(\\mathcal{G}\\) is a VC subgraph class, with VC dimension \\(d\\), and all \\(\\phi \\in \\mathcal{G}\\) are bounded: \\(\\phi(Z) \\in [0,1]\\). Then for any distribution \\(\\mathcal{D}\\) over \\(Z\\), \\(\\epsilon \\in (0,1]\\) and \\(p \\in [1,\\infty)\\), we have\n\\[\n\\ln M(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq 1+\\ln(d+1)+d\\ln(2 e/\\epsilon^{p}).\n\\]\nMoreover,\n\\[\n\\ln N_{\\infty}(\\epsilon,\\mathcal{G},n)\\leq d\\ln\\max  [2,e n/(d\\epsilon)].\n\\]"}, {"name": "example 6.26", "statement": "From Corollary 5.7, we know that if a binary-valued function class \\(\\mathcal{G}\\) (or a VC-subgraph class with values in \\([0,1]\\)) has VC-dimension \\(d\\), then\n\\[\n\\ln N_{2}(\\epsilon,\\mathcal{G},n)\\leq 1+\\ln(d+1)+d\\ln(2 e/\\epsilon^{2}).\n\\]\nSince \\(N_{2}(0.5,\\mathcal{G},n) = 1\\), we have\n\\[\n12\\int_{0}^{\\infty}\\sqrt{\\ln N_{2}(\\epsilon,\\mathcal{G},n)}d\\epsilon\\leq 12\\int_{0}^{0. 5}\\sqrt{1+\\ln(d+1)+d\\ln(2e/\\epsilon^{2})}d\\epsilon\\leq 16\\sqrt{d}.\n\\]\nIt follows that\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n})\\leq\\frac{16\\sqrt{d}}{\\sqrt{n}}.\n\\]\nThe constant isn't optimal, and in fact a better constant can be obtained using packing number (see Example 6.4). The result implies the following uniform convergence result: with probability at least \\(1 - \\delta\\), for all \\(w \\in \\Omega\\),\n\\[\n\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\frac{32\\sqrt{d}}{\\sqrt{n}}+\\sqrt{\\frac{\\ln(1/\\delta)}{2n}}.\n\\]\nThis bound removes a \\( \\ln n \\) factor from the additive uniform convergence bound in Theorem 4.17."}, {"name": "theorem 10.8", "statement": "We have\n\\[\nR(\\mathrm{CONV}(\\Psi),\\mathcal{S}_{n})=R(\\Psi,\\mathcal{S}_{n}).\n\\]\nIf either \\(\\Psi = -\\Psi\\) or \\(0 \\in \\Psi\\), then the following equality holds:\n\\[\nR\\left(\\mathcal{F}_{A,L_{1}}^{+}(\\Psi),\\mathcal{S}_{n}\\right)=A\\cdot R(\\Psi,\\mathcal{S}_{n}).\n\\]\nIf \\(\\Psi = -\\Psi\\), then the following equality holds:\n\\[\nR\\left(\\mathcal{F}_{A,L_{1}}(\\Psi),\\mathcal{S}_{n}\\right)=A\\cdot R(\\Psi,\\mathcal{S}_{n}).\n\\]"}, {"name": "definition 6.3", "statement": "Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ndef dotProd {d : ℕ} (θ x : Fin d → ℝ) : ℝ := ∑ i, θ i * x i\n\nabbrev IdxPM (d : ℕ) : Type := (Fin d → ℝ) × Bool\n\ndef famPM {d : ℕ} (h : ℝ → ℝ) : IdxPM d → (Fin d → ℝ) → ℝ :=\n  fun p x => (if p.2 then (1 : ℝ) else -1) * h (dotProd p.1 x)\n\nabbrev IdxA {d : ℕ} (μ : MeasureTheory.Measure (Fin d → ℝ)) (A : ℝ) : Type :=\n  {u : (Fin d → ℝ) → ℝ // Integrable u μ ∧ (∫ θ, |u θ| ∂μ) ≤ A}\n\nnoncomputable def famA {d : ℕ} (μ : MeasureTheory.Measure (Fin d → ℝ)) (A : ℝ) (h : ℝ → ℝ) :\n    IdxA μ A → (Fin d → ℝ) → ℝ :=\n  fun u x => ∫ θ, u.1 θ * h (dotProd θ x) ∂μ\n\ntheorem ch11_proposition_11_9\n    {d n : ℕ} {M A : ℝ}\n    (μ : MeasureTheory.Measure (Fin d → ℝ)) [IsProbabilityMeasure μ]\n    (S : Fin n → (Fin d → ℝ))\n    (h : ℝ → ℝ)\n    (hmono : Monotone h ∨ Antitone h)\n    (hrange : ∀ t, h t ∈ Set.Icc (0 : ℝ) M)\n    (hM : 0 ≤ M) (hA : 0 ≤ A) (hn : 0 < n) :\n    empRadSigned n (famPM (d := d) h) S\n        ≤ M * 32 * Real.sqrt (d + 1) / Real.sqrt n\n      ∧\n    empRadSigned n (famA μ A h) S\n        ≤ A * M * 32 * Real.sqrt (d + 1) / Real.sqrt n := sorry\n\nend GraduateLevel\n", "target_names": ["ch11_proposition_11_9"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 1050, "end": 1499, "gold": "{d n : ℕ} {M A : ℝ}\n    (μ : MeasureTheory.Measure (Fin d → ℝ)) [IsProbabilityMeasure μ]\n    (S : Fin n → (Fin d → ℝ))\n    (h : ℝ → ℝ)\n    (hmono : Monotone h ∨ Antitone h)\n    (hrange : ∀ t, h t ∈ Set.Icc (0 : ℝ) M)\n    (hM : 0 ≤ M) (hA : 0 ≤ A) (hn : 0 < n) :\n    empRadSigned n (famPM (d := d) h) S\n        ≤ M * 32 * Real.sqrt (d + 1) / Real.sqrt n\n      ∧\n    empRadSigned n (famA μ A h) S\n        ≤ A * M * 32 * Real.sqrt (d + 1) / Real.sqrt n"}]}
{"item_id": "ch12_lemma_12_27::ch12_lemma_12_27::A-stmt", "paper_id": "ch12_lemma_12_27", "target_name": "ch12_lemma_12_27", "source": "Graduate-level", "nl_statement": "Consider a finite family of distributions \\(\\mathcal{P}\\). Let \\(d \\geq 1\\) be an integer, and \\(Q\\) can be decomposed as\n\\[\nQ(\\theta,\\mathcal{D})=\\sum_{j=1}^{d}Q_{j}(\\theta,\\mathcal{D}),\n\\]\nwhere \\( Q_{j} \\geq 0 \\) are all non-negative. Assume for all \\( j \\), there exists a partition \\( M_{j} \\) of \\( \\mathcal{P} \\). We use notation \\( \\mathcal{D}' \\sim_{j} \\mathcal{D} \\) to indicate that \\( \\mathcal{D}' \\) and \\( \\mathcal{D} \\) belong to the same partition in \\( M_{j} \\). Let \\( m_{j}(\\mathcal{D}) \\) be the number of elements in the partition containing \\( \\mathcal{D} \\). Assume there exist \\( \\epsilon, \\beta \\geq 0 \\) such that\n\\[\n\\begin{array}{l}\\forall\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D},\\mathcal{D}^{\\prime}\\neq\\mathcal{D}:\\quad\\inf_{\\theta}[Q_{j}(\\theta,\\mathcal{D}^{\\prime})+Q_{j}(\\theta,\\mathcal{D})]\\geq\\epsilon,\\\\ \\forall\\mathcal{D}\\in\\mathcal{P}:\\frac{1}{d(\\mathcal{P})}\\sum_{j=1}^{d}\\sum_{\\mathcal{D}\\in\\mathcal{P}_{j}}\\frac{1}{m_{j}(\\mathcal{D})-1}\\sum_{\\mathcal{D}^{\\prime}\\sim_{j}\\mathcal{D}}\\|\\mathcal{D}^{\\prime}-\\mathcal{D}\\|_{\\mathrm{TV}}\\leq\\beta,\\\\ \\end{array}\n\\]\nwhere \\(\\mathcal{P}_j = \\{\\mathcal{D}\\in \\mathcal{P}:m_j(\\mathcal{D}) > 1\\}\\) and \\(d(\\mathcal{P}) = \\sum_{j = 1}^{d}|\\mathcal{P}_j|\\). Let \\(\\mathcal{A}(Z)\\) be any estimator, we have\n\\[\n\\frac{1}{|\\mathcal{P}|}\\sum_{\\mathcal{D}\\in\\mathcal{P}}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\mathbb{E}_{\\mathcal{A}}Q(\\mathcal{A}(Z),\\mathcal{D})\\geq\\frac{\\epsilon d(\\mathcal{P})}{2|\\mathcal{P}|}[1-\\beta],\n\\]\nwhere \\(\\mathbb{E}_A\\) is with respect to the internal randomization in \\(\\mathcal{A}\\).", "context": null, "nl_notes": "NOTE — Condition (b) is printed in the source with two notational slips, which the NL statement reproduces: the leading 'for all D ∈ P' binds nothing (it is re-bound by the inner sum over D ∈ P_j), and the innermost sum over D' ∼_j D should exclude D' = D (the omitted term is ‖D − D‖_TV = 0). The Lean hypothesis `hb` is the single scalar inequality this describes, with the innermost sum over i' related to i and i' ≠ i.", "lean": "import Mathlib\n\nopen MeasureTheory Finset\n\nnamespace GraduateLevel\n\nnoncomputable def tvDist {Ω : Type*} [MeasurableSpace Ω]\n    (μ ν : Measure Ω) [IsFiniteMeasure μ] [IsFiniteMeasure ν] : ℝ :=\n  (((μ.toSignedMeasure - ν.toSignedMeasure).totalVariation Set.univ).toReal) / 2\n\nnoncomputable def blockSize {ι : Type*} [Fintype ι] (M : Setoid ι)\n    [DecidableRel M.r] (i : ι) : ℕ :=\n  (Finset.univ.filter (fun i' => M.r i i')).card\n\ntheorem ch12_lemma_12_27\n    {Ω : Type*} [MeasurableSpace Ω]\n    {R : Type*} [MeasurableSpace R] (μR : Measure R) [IsProbabilityMeasure μR]\n    {ι : Type*} [Fintype ι] [Nonempty ι] [DecidableEq ι]\n    {κ : Type*} [Fintype κ] [Nonempty κ]\n    {Θ : Type*}\n    (D : ι → Measure Ω) [hP : ∀ i, IsProbabilityMeasure (D i)]\n    (Qcomp : κ → Θ → ι → ℝ)\n    (hQnonneg : ∀ j θ i, 0 ≤ Qcomp j θ i)\n    (M : κ → Setoid ι) [hM : ∀ j, DecidableRel (M j).r]\n    (ε β : ℝ) (hε : 0 ≤ ε) (hβ : 0 ≤ β)\n    (A : Ω → R → Θ)\n    (q : Ω → ι → ℝ)\n    (hQIntR : ∀ (j : κ) (ω : Ω) (i : ι), Integrable (fun r => Qcomp j (A ω r) i) μR)\n    (hQIntΩ : ∀ (j : κ) (i : ι), Integrable (fun ω => ∫ r, Qcomp j (A ω r) i ∂μR) (D i))\n    (hqdef : ∀ ω i, q ω i = ∫ r, (∑ j : κ, Qcomp j (A ω r) i) ∂μR)\n    (hqInt : ∀ i, Integrable (fun ω => q ω i) (D i))\n    (Pj : κ → Finset ι)\n    (hPj : ∀ j, Pj j = Finset.univ.filter (fun i => 1 < blockSize (M j) i))\n    (dP : ℕ) (hdP : dP = ∑ j : κ, (Pj j).card) (hdPpos : 0 < dP)\n    (ha : ∀ (j : κ) (i i' : ι) (θ : Θ), (M j).r i i' → i' ≠ i →\n        ε ≤ Qcomp j θ i' + Qcomp j θ i)\n    (hb : (1 / (dP : ℝ)) *\n        ∑ j : κ, ∑ i ∈ Pj j,\n          (1 / ((blockSize (M j) i : ℝ) - 1)) *\n            ∑ i' ∈ Finset.univ.filter (fun i' => (M j).r i i' ∧ i' ≠ i),\n              tvDist (D i') (D i) ≤ β) :\n    (ε * (dP : ℝ) / (2 * (Fintype.card ι : ℝ))) * (1 - β) ≤\n    (1 / (Fintype.card ι : ℝ)) *\n      ∑ i : ι, ∫ ω, q ω i ∂(D i) := sorry\n\nend GraduateLevel\n", "target_names": ["ch12_lemma_12_27"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "holes": [{"start": 460, "end": 1860, "gold": "{Ω : Type*} [MeasurableSpace Ω]\n    {R : Type*} [MeasurableSpace R] (μR : Measure R) [IsProbabilityMeasure μR]\n    {ι : Type*} [Fintype ι] [Nonempty ι] [DecidableEq ι]\n    {κ : Type*} [Fintype κ] [Nonempty κ]\n    {Θ : Type*}\n    (D : ι → Measure Ω) [hP : ∀ i, IsProbabilityMeasure (D i)]\n    (Qcomp : κ → Θ → ι → ℝ)\n    (hQnonneg : ∀ j θ i, 0 ≤ Qcomp j θ i)\n    (M : κ → Setoid ι) [hM : ∀ j, DecidableRel (M j).r]\n    (ε β : ℝ) (hε : 0 ≤ ε) (hβ : 0 ≤ β)\n    (A : Ω → R → Θ)\n    (q : Ω → ι → ℝ)\n    (hQIntR : ∀ (j : κ) (ω : Ω) (i : ι), Integrable (fun r => Qcomp j (A ω r) i) μR)\n    (hQIntΩ : ∀ (j : κ) (i : ι), Integrable (fun ω => ∫ r, Qcomp j (A ω r) i ∂μR) (D i))\n    (hqdef : ∀ ω i, q ω i = ∫ r, (∑ j : κ, Qcomp j (A ω r) i) ∂μR)\n    (hqInt : ∀ i, Integrable (fun ω => q ω i) (D i))\n    (Pj : κ → Finset ι)\n    (hPj : ∀ j, Pj j = Finset.univ.filter (fun i => 1 < blockSize (M j) i))\n    (dP : ℕ) (hdP : dP = ∑ j : κ, (Pj j).card) (hdPpos : 0 < dP)\n    (ha : ∀ (j : κ) (i i' : ι) (θ : Θ), (M j).r i i' → i' ≠ i →\n        ε ≤ Qcomp j θ i' + Qcomp j θ i)\n    (hb : (1 / (dP : ℝ)) *\n        ∑ j : κ, ∑ i ∈ Pj j,\n          (1 / ((blockSize (M j) i : ℝ) - 1)) *\n            ∑ i' ∈ Finset.univ.filter (fun i' => (M j).r i i' ∧ i' ≠ i),\n              tvDist (D i') (D i) ≤ β) :\n    (ε * (dP : ℝ) / (2 * (Fintype.card ι : ℝ))) * (1 - β) ≤\n    (1 / (Fintype.card ι : ℝ)) *\n      ∑ i : ι, ∫ ω, q ω i ∂(D i)"}]}
{"item_id": "ch12_theorem_12_11::ch12_theorem_12_11::A-stmt", "paper_id": "ch12_theorem_12_11", "target_name": "ch12_theorem_12_11", "source": "Graduate-level", "nl_statement": "Consider a finite family of distributions \\(\\mathcal{P} = \\{\\mathcal{D}_1, \\ldots, \\mathcal{D}_N\\}\\). Given a loss function \\(Q\\) on \\(\\Theta \\times \\mathcal{P}\\), let\n\\[\nm=\\sup_{\\theta\\in\\Theta}\\left|\\left\\{k:Q(\\theta,\\mathcal{D}_{k})<\\epsilon\\right\\}\\right|.\n\\]\nAssume that \\(j\\) is a random variable that is uniformly distributed in \\(\\{1, \\dots, N\\}\\), and conditioned on \\(j\\), \\(Z \\sim \\mathcal{D}_j\\). Given any (possibly random) estimator \\(\\mathcal{A}(Z)\\). Then\n\\[\n\\frac{1}{N}\\sum_{j=1}^{N}\\operatorname*{Pr}_{Z\\sim\\mathcal{D}_{j}}(Q(\\mathcal{A}(Z),\\mathcal{D}_{j})<\\epsilon)\\leq\\max\\left(\\frac{m}{N},\\frac{I(j,Z)+\\ln 2}{\\ln(N/m)}\\right),\n\\]\nwhere \\( I(j,Z) \\) is the mutual information of \\( j \\) and \\( Z \\). The probability includes possible randomization in \\( \\mathcal{A} \\).", "context": [{"name": "definition 12.9", "statement": "Consider a distribution family \\(\\mathcal{P}\\) on sample space \\(\\mathcal{Z}\\), a parameter space \\(\\Theta\\). A learning algorithm \\(\\mathcal{A}:\\mathcal{Z}^n\\to \\Theta\\), a loss function \\(Q:\\Theta \\times \\mathcal{P}\\rightarrow \\mathbb{R}\\). Then the worst case expected risk of a learning algorithm (i.e., a statistical estimator) \\(\\mathcal{A}\\) with respect to \\(\\mathcal{P}\\) is given by\n\\[\nr_{n}(\\mathcal{A},\\mathcal{P},Q)=\\sup_{\\mathcal{D}\\in\\mathcal{P}}\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\mathbb{E}_{\\mathcal{A}}Q(\\mathcal{A}(\\mathcal{S}_{n}),\\mathcal{D}),\n\\]\nwhere \\(\\mathbb{E}_{\\mathcal{A}}\\) is the expectation over any internal randomization of \\(\\mathcal{A}\\). Moreover, the minimax risk is defined as:\n\\[\nr_{n}(\\mathcal{P},Q)=\\inf_{\\mathcal{A}}r_{n}(\\mathcal{A},\\mathcal{P},Q).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\nopen scoped ENNReal\n\nnamespace GraduateLevel\n\nnoncomputable def lowLossCount {N : ℕ} {Θ : Type*} (Q : Θ → Fin N → ℝ) (ε : ℝ)\n    (θ : Θ) : ℕ :=\n  (Finset.univ.filter (fun k : Fin N => Q θ k < ε)).card\n\nnoncomputable def jointLaw {N : ℕ} [NeZero N] {Z : Type*} [MeasurableSpace Z]\n    (D : Fin N → Measure Z) : Measure (Fin N × Z) :=\n  Measure.bind ((PMF.uniformOfFintype (Fin N)).toMeasure)\n    (fun j => (D j).map (fun z => (j, z)))\n\nnoncomputable def prodMarginals {N : ℕ} [NeZero N] {Z : Type*} [MeasurableSpace Z]\n    (D : Fin N → Measure Z) : Measure (Fin N × Z) :=\n  ((jointLaw D).map Prod.fst).prod ((jointLaw D).map Prod.snd)\n\ntheorem ch12_theorem_12_11\n    {N : ℕ} [NeZero N]\n    {Z : Type*} [MeasurableSpace Z]\n    {Θ : Type*} [MeasurableSpace Θ]\n    (D : Fin N → Measure Z) (hD : ∀ j, IsProbabilityMeasure (D j))\n    (Q : Θ → Fin N → ℝ)\n    (ε : ℝ)\n    (m : ℕ)\n    (hm_def : m = ⨆ θ : Θ, lowLossCount Q ε θ)\n    (A : Z → Measure Θ) (hA : ∀ z, IsProbabilityMeasure (A z))\n    (hAmeas : Measurable A)\n    (hQmeas : ∀ j : Fin N, MeasurableSet {θ : Θ | Q θ j < ε})\n    (I : ℝ)\n    (hI : I = (InformationTheory.klDiv (jointLaw D) (prodMarginals D)).toReal)\n    (hI_fin : InformationTheory.klDiv (jointLaw D) (prodMarginals D) ≠ ⊤) :\n    (1 / (N : ℝ)) * ∑ j : Fin N, ((D j).bind A {θ : Θ | Q θ j < ε}).toReal\n      ≤ max ((m : ℝ) / (N : ℝ)) ((I + Real.log 2) / Real.log ((N : ℝ) / (m : ℝ))) := sorry\n\nend GraduateLevel\n", "target_names": ["ch12_theorem_12_11"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "holes": [{"start": 724, "end": 1453, "gold": "{N : ℕ} [NeZero N]\n    {Z : Type*} [MeasurableSpace Z]\n    {Θ : Type*} [MeasurableSpace Θ]\n    (D : Fin N → Measure Z) (hD : ∀ j, IsProbabilityMeasure (D j))\n    (Q : Θ → Fin N → ℝ)\n    (ε : ℝ)\n    (m : ℕ)\n    (hm_def : m = ⨆ θ : Θ, lowLossCount Q ε θ)\n    (A : Z → Measure Θ) (hA : ∀ z, IsProbabilityMeasure (A z))\n    (hAmeas : Measurable A)\n    (hQmeas : ∀ j : Fin N, MeasurableSet {θ : Θ | Q θ j < ε})\n    (I : ℝ)\n    (hI : I = (InformationTheory.klDiv (jointLaw D) (prodMarginals D)).toReal)\n    (hI_fin : InformationTheory.klDiv (jointLaw D) (prodMarginals D) ≠ ⊤) :\n    (1 / (N : ℝ)) * ∑ j : Fin N, ((D j).bind A {θ : Θ | Q θ j < ε}).toReal\n      ≤ max ((m : ℝ) / (N : ℝ)) ((I + Real.log 2) / Real.log ((N : ℝ) / (m : ℝ)))"}]}
{"item_id": "ch13_theorem_13_5::ch13_theorem_13_5::A-stmt", "paper_id": "ch13_theorem_13_5", "target_name": "ch13_theorem_13_5", "source": "Graduate-level", "nl_statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) such that \\(\\xi_{i}\\in [0,1]\\) for all \\(i\\). We have for \\(\\lambda >0\\), with probability at least \\(1 - \\delta\\):\n\\[\n\\forall n>0:\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\frac{\\lambda}{1-e^{-\\lambda}}\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}+\\frac{\\ln(1/\\delta)}{\\left(1-e^{-\\lambda}\\right)n}.\n\\]\nSimilarly, for \\(\\lambda > 0\\), with probability at least \\(1 - \\delta\\):\n\\[\n\\forall n>0:\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}>\\frac{\\lambda}{e^{\\lambda}-1}\\frac{1}{n}\\sum_{i=1}^{n}\\xi_{i}-\\frac{\\ln(1/\\delta)}{(e^{\\lambda}-1)n}.\n\\]", "context": [{"name": "lemma 2.15", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]"}, {"name": "theorem 13.2", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) with filtration \\(\\{\\mathcal{F}_t\\}\\). We have for any \\(\\delta \\in (0,1)\\) and \\(\\lambda >0\\):\n\\[\n\\operatorname*{Pr}\\left[\\exists n>0: -\\sum_{i=1}^{n}\\xi_{i}\\geq\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right]\\leq\\delta .\n\\]\nMoreover, consider a sequence of \\(\\{z_t\\in \\mathbb{R}\\}\\) adapted to \\(\\{\\mathcal{F}_t\\}\\), and events \\(A_{t}\\) on \\(\\mathcal{F}_t\\):\n\\[\n\\ln\\Pr\\left[\\exists n>0:\\sum_{i=1}^{n}\\xi_{i}\\leq z_{n}\\&\\mathcal{S}_{n}\\in A_{n}\\right]\\leq\\inf_{\\lambda>0}\\sup_{n>0}\\sup_{\\mathcal{S}_{n}\\in A_{n}}\\left[\\lambda z_{n}+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right].\n\\]"}, {"name": "section 13.1 sequential observation setup", "statement": "In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch13_theorem_13_5\n    {Ω : Type*} {m : MeasurableSpace Ω} (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (ℱ : MeasureTheory.Filtration ℕ m)\n    (ξ : ℕ → Ω → ℝ)\n    (lam δ : ℝ) (hlam : 0 < lam) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (hmeas : ∀ i, Measurable (ξ i))\n    (hadapt : MeasureTheory.Adapted ℱ ξ)\n    (hbound : ∀ i ω, ξ i ω ∈ Set.Icc (0 : ℝ) 1) :\n    (1 - δ ≤ (μ {ω : Ω | ∀ n : ℕ, 0 < n →\n        (n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, (μ[ξ i | ℱ (i - 1)]) ω\n          < (lam / (1 - Real.exp (-lam))) * ((n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, ξ i ω)\n              + Real.log (1 / δ) / ((1 - Real.exp (-lam)) * n)}).toReal)\n    ∧\n    (1 - δ ≤ (μ {ω : Ω | ∀ n : ℕ, 0 < n →\n        (n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, (μ[ξ i | ℱ (i - 1)]) ω\n          > (lam / (Real.exp lam - 1)) * ((n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, ξ i ω)\n              - Real.log (1 / δ) / ((Real.exp lam - 1) * n)}).toReal) := sorry\n\nend GraduateLevel\n", "target_names": ["ch13_theorem_13_5"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Martingales & Dependent Data", "holes": [{"start": 114, "end": 969, "gold": "{Ω : Type*} {m : MeasurableSpace Ω} (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (ℱ : MeasureTheory.Filtration ℕ m)\n    (ξ : ℕ → Ω → ℝ)\n    (lam δ : ℝ) (hlam : 0 < lam) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (hmeas : ∀ i, Measurable (ξ i))\n    (hadapt : MeasureTheory.Adapted ℱ ξ)\n    (hbound : ∀ i ω, ξ i ω ∈ Set.Icc (0 : ℝ) 1) :\n    (1 - δ ≤ (μ {ω : Ω | ∀ n : ℕ, 0 < n →\n        (n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, (μ[ξ i | ℱ (i - 1)]) ω\n          < (lam / (1 - Real.exp (-lam))) * ((n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, ξ i ω)\n              + Real.log (1 / δ) / ((1 - Real.exp (-lam)) * n)}).toReal)\n    ∧\n    (1 - δ ≤ (μ {ω : Ω | ∀ n : ℕ, 0 < n →\n        (n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, (μ[ξ i | ℱ (i - 1)]) ω\n          > (lam / (Real.exp lam - 1)) * ((n : ℝ)⁻¹ * ∑ i ∈ Finset.Icc 1 n, ξ i ω)\n              - Real.log (1 / δ) / ((Real.exp lam - 1) * n)}).toReal)"}]}
{"item_id": "ch15_corollary_15_7::ch15_corollary_15_7::A-stmt", "paper_id": "ch15_corollary_15_7", "target_name": "ch15_corollary_15_7", "source": "Graduate-level", "nl_statement": "Assume that \\( Y_{t} \\in [0, M] \\) for \\( t \\geq 1 \\). Consider the ridge regression method of (15.3), with \\(\\Lambda_0\\) symmetric positive definite, and let\n\\[\n\\hat{Y}_{t}=\\max(0,\\min(M,\\hat{w}(\\mathcal{S}_{t-1})^{\\top}\\psi(X_{t}))).\n\\]\nWe have\n\\[\n\\sum_{t=1}^{T}(Y_{t}-\\hat{Y}_{t})^{2}\\leq\\inf_{w}\\left[\\sum_{t=1}^{T}(Y_{t}-w^{\\top}\\psi(X_{t}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+M^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\n\\]\nwhere\n\\[\n\\Lambda_{T}=\\Lambda_{0}+\\sum_{s=1}^{T}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]", "context": [{"name": "theorem 15.6", "statement": "Consider the ridge regression method of (15.3). We have the following result for any \\(\\sigma \\geq 0\\) and for all observed sequence \\(S_T\\):\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{b_{t}}+\\sigma^{2}\\ln b_{t}\\right]\\\\ =\\inf_{w}\\left[\\sum_{t=1}^{T}\\left(Y_{t}-w^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+\\sigma^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\\\\ \\end{array}\n\\]\nwhere\n\\[\n\\Lambda_{t}=\\Lambda_{0}+\\sum_{s=1}^{t}\\psi(X_{s})\\psi(X_{s})^{\\top},\n\\]\nand \\(b_{t} = 1 + \\psi (X_{t})^{\\top}\\Lambda_{t - 1}^{-1}\\psi (X_{t}).\\)"}, {"name": "equation (15.3)", "statement": "The ridge regression method defines, for each \\(n\\),\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\arg\\min_{w}\\left[\\sum_{i=1}^{n}(Y_{i}-w^{\\top}\\psi(X_{i}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right].\n\\]\nEquivalently,\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\Lambda_{n}^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i},\\quad\\Lambda_{n}=\\Lambda_{0}+\\sum_{s=1}^{n}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]"}, {"name": "equation (15.3)", "statement": "Let \\(\\Lambda_0\\) be a symmetric positive definite regularizer and let \\(\\psi\\) be a feature map. Given \\(\\mathcal{S}_n = \\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\), define\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}\n\\]\nand the ridge regression estimator\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i}.\\tag{15.3}\n\\]\nEquivalently, \\(\\hat{w}(\\mathcal{S}_{n})\\) minimizes\n\\[\n\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}.\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen Matrix\n\nnamespace GraduateLevel\n\ntheorem ch15_corollary_15_7\n    {𝒳 : Type*} {d T : ℕ}\n    (Λ₀ : Matrix (Fin d) (Fin d) ℝ) (hΛ0 : Λ₀.PosDef)\n    (ψ : 𝒳 → (Fin d → ℝ)) (X : ℕ → 𝒳) (Y : ℕ → ℝ)\n    (M : ℝ) (hM : 0 ≤ M) (hYrange : ∀ t, 1 ≤ t → Y t ∈ Set.Icc 0 M)\n    (Λ : ℕ → Matrix (Fin d) (Fin d) ℝ)\n    (hΛ : ∀ n, Λ n = Λ₀ + ∑ s ∈ Finset.Icc 1 n, Matrix.vecMulVec (ψ (X s)) (ψ (X s)))\n    (ŵ : ℕ → (Fin d → ℝ))\n    (hŵ : ∀ n, ŵ n = (Λ n)⁻¹.mulVec (∑ i ∈ Finset.Icc 1 n, Y i • ψ (X i)))\n    (Ŷ : ℕ → ℝ)\n    (hŶ : ∀ t, Ŷ t = max 0 (min M ((ŵ (t - 1)) ⬝ᵥ (ψ (X t))))) :\n    (∑ t ∈ Finset.Icc 1 T, (Y t - Ŷ t) ^ 2)\n      ≤ (⨅ w : Fin d → ℝ,\n            ((∑ t ∈ Finset.Icc 1 T, (Y t - (w ⬝ᵥ (ψ (X t)))) ^ 2)\n              + (Λ₀.mulVec w) ⬝ᵥ w))\n          + M ^ 2 * Real.log |(Λ₀⁻¹ * Λ T).det| := sorry\n\nend GraduateLevel\n", "target_names": ["ch15_corollary_15_7"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 86, "end": 807, "gold": "{𝒳 : Type*} {d T : ℕ}\n    (Λ₀ : Matrix (Fin d) (Fin d) ℝ) (hΛ0 : Λ₀.PosDef)\n    (ψ : 𝒳 → (Fin d → ℝ)) (X : ℕ → 𝒳) (Y : ℕ → ℝ)\n    (M : ℝ) (hM : 0 ≤ M) (hYrange : ∀ t, 1 ≤ t → Y t ∈ Set.Icc 0 M)\n    (Λ : ℕ → Matrix (Fin d) (Fin d) ℝ)\n    (hΛ : ∀ n, Λ n = Λ₀ + ∑ s ∈ Finset.Icc 1 n, Matrix.vecMulVec (ψ (X s)) (ψ (X s)))\n    (ŵ : ℕ → (Fin d → ℝ))\n    (hŵ : ∀ n, ŵ n = (Λ n)⁻¹.mulVec (∑ i ∈ Finset.Icc 1 n, Y i • ψ (X i)))\n    (Ŷ : ℕ → ℝ)\n    (hŶ : ∀ t, Ŷ t = max 0 (min M ((ŵ (t - 1)) ⬝ᵥ (ψ (X t))))) :\n    (∑ t ∈ Finset.Icc 1 T, (Y t - Ŷ t) ^ 2)\n      ≤ (⨅ w : Fin d → ℝ,\n            ((∑ t ∈ Finset.Icc 1 T, (Y t - (w ⬝ᵥ (ψ (X t)))) ^ 2)\n              + (Λ₀.mulVec w) ⬝ᵥ w))\n          + M ^ 2 * Real.log |(Λ₀⁻¹ * Λ T).det|"}]}
{"item_id": "ch16_theorem_16_17::ch16_theorem_16_17::A-stmt", "paper_id": "ch16_theorem_16_17", "target_name": "ch16_theorem_16_17", "source": "Graduate-level", "nl_statement": "Consider Algorithm 16.5. Let \\( G_{*} = \\max_{a}\\sum_{t = 1}^{T}r_{t}(a) \\). We have the following bound for the adversarial regret (16.1):\n\\[\n\\mathrm{REG}_{T}\\leq(e-1)\\gamma G_{*}+\\frac{K\\ln K}{\\gamma}.\n\\]", "context": [{"name": "EXP3", "statement": "Algorithm 16.5: EXP3\nInput: \\(K, T, \\gamma \\in (0,1]\\)  \nfor \\(a = 1, \\ldots, K\\) do  \n\\(\\begin{array}{r}\\lfloor \\mathrm{Let}w_0(a) = 1 \\end{array}\\)  \nfor \\(t = 1,2,\\dots,T\\) do  \n\\(\\begin{array}{rl}&{\\mathrm{Let}w_{t-1}=\\sum_{a=1}^{K}w_{t-1}(a)}\\\\ &{\\mathrm{for}a=1,\\ldots,K\\mathrm{do}}\\\\ &{\\mathrm{Let}\\hat{\\pi}_t(a)=(1-\\gamma)w_{t-1}(a)/w_{t-1}+\\gamma/K}\\\\ &{\\mathrm{Sample}a_t\\mathrm{according to}\\hat{\\pi}_t(\\cdot)}\\\\ &{\\mathrm{Pull}\\mathrm{arm}a_t\\mathrm{and observe reward}r_t(a_t)\\in [0,1]}\\\\ &{\\mathrm{for}a=1,\\ldots,K\\mathrm{do}}\\\\ &{\\mathrm{Let}\\hat{r}_t(a,a_t)=r_t(a_t)\\mathbb{1}(a=a_t)/\\hat{\\pi}_t(a_t)}\\\\ &{\\mathrm{Let}w_t(a)=w_{t-1}(a)\\exp(\\gamma\\hat{r}_t(a,a_t)/K)}\\end{array}\\)"}, {"name": "definition 16.1", "statement": "In the multi-armed bandit (or MAB) problem, we consider \\(K\\) arms. The environment generates a sequence of reward vectors for time steps \\(t \\geq 1\\) as \\(r_t = [r_t(1), \\ldots, r_t(K)]\\). Each \\(r_t(a)\\) is associated with an arm \\(a \\in \\{1, \\ldots, K\\}\\). In the literature, an arm \\(a\\) is also referred to as an action. At each time step \\(t = 1, 2, \\ldots, T\\),\n- The player pulls one of the arms \\( a_{t} \\in \\{1, \\dots, K\\} \\).\n- The environment returns the reward \\( r_t(a_t) \\), but does not reveal information on any other arm \\( a \\neq a_t \\).\nAt each time \\( t \\), a (randomized) bandit algorithm takes the historic observations observed so far, and maps it to a distribution \\( \\hat{\\pi}_{t-1} \\) over actions \\( a \\in \\{1, \\ldots, K\\} \\). We then draw a random action \\( a_t \\) (arm) from \\( \\hat{\\pi}_{t-1} \\). Here \\( \\hat{\\pi}_{t-1} \\) is referred to as policy.\nIn the adversarial bandit problem, we are given an arbitrary reward sequence \\(\\{[r_1(1),\\ldots,r_t(K)]:t\\geq 1\\}\\) before hand. The adversary in this setting is called oblivious, who decides the rewards of all the rounds before the game starts (but may take advantage of the bandit algorithm's weakness). For this reward se\nquence, the expected cumulative reward of a randomized bandit algorithm is\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t}),\n\\]\nwhere the randomization is over the internal randomization of the bandit algorithm. The expected regret is defined as\n\\[\n\\operatorname{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\mathbb{E}\\sum_{t=1}^{T}r_{t}\\left(a_{t}\\right).\\tag{16.1}\n\\]\nIf the bandit algorithm is deterministic: i.e. pulling a single arm \\(a_{t}\\) only at any time step, then the regret is\n\\[\n\\mathrm{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\sum_{t=1}^{T}r_{t}(a_{t}).\n\\]\nIn the adversarial bandit problem, our goal is to minimize the expected regret \\(\\mathrm{REG}_T\\) (and the expectation is with respect to the internal randomization of the algorithm instead of the data).\nAnother form of bandit problem, called stochastic bandit, assumes that the reward \\( r_t(a) \\) is drawn independently from a distribution \\( \\mathcal{D}_a \\), with mean \\( \\mu(a) = \\mathbb{E}_{r_t(a) \\sim \\mathcal{D}_a} [r_t(a)] \\). In this setting, the goal of bandit is to find \\( a \\) that maximizes the expected reward \\( \\mu(a) \\). We are interested in the expected regret defined as:\n\\[\n\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu\\left(a_{t}\\right).\\tag{16.2}\n\\]\nAgain, this regret is defined for each realization of the stochastic rewards, and the expectation is with respect to the randomness of the algorithm. If algorithm is deterministic and outputs \\( a_{t} \\) at each time \\( t \\), then\n\\[\n\\mathrm{REG}_{T}=T\\max_{a}\\mu(a)-\\sum_{t=1}^{T}\\mu(a_{t}).\n\\]\nWe can further include randomization over data into the regret of stochastic bandit, by considering expected regret as follows:\n\\[\n\\mathbb{E}\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu(a_{t}),\n\\]\nwhere the expectation is with respect to both the data and the internal randomization of the learning algorithm."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory\nopen scoped MeasureTheory\n\nnamespace GraduateLevel\n\ndef bestArmReward {K T : ℕ} (hK : 1 ≤ K) (r : Fin T → Fin K → ℝ) : ℝ :=\n  haveI : Nonempty (Fin K) := ⟨⟨0, by omega⟩⟩\n  (Finset.univ : Finset (Fin K)).sup' Finset.univ_nonempty (fun a => ∑ t : Fin T, r t a)\n\ntheorem ch16_theorem_16_17\n    {K T : ℕ} (hK : 1 ≤ K) (γ : ℝ) (hγ : 0 < γ) (hγ1 : γ ≤ 1)\n    (r : Fin T → Fin K → ℝ) (hr : ∀ t a, r t a ∈ Set.Icc (0 : ℝ) 1)\n    {Ω : Type*} {m0 : MeasurableSpace Ω} (μ : @Measure Ω m0) [IsProbabilityMeasure μ]\n    (w : Fin (T + 1) → Ω → Fin K → ℝ) (pol : Fin T → Ω → Fin K → ℝ)\n    (a' : Fin T → Ω → Fin K) (rhat : Fin T → Ω → Fin K → ℝ)\n    (ha'meas : ∀ t, Measurable (a' t)) (hw0 : ∀ ω a, w 0 ω a = 1)\n    (hwnonneg : ∀ (t : Fin (T + 1)) ω a, 0 ≤ w t ω a)\n    (hwpos : ∀ (t : Fin T) ω, 0 < ∑ b : Fin K, w t.castSucc ω b)\n    (hpol : ∀ (t : Fin T) ω a, pol t ω a = (1 - γ) * (w t.castSucc ω a / ∑ b : Fin K, w t.castSucc ω b) + γ / K)\n    (hpolpos : ∀ (t : Fin T) ω a, 0 < pol t ω a)\n    (hrhat : ∀ (t : Fin T) ω a, rhat t ω a = if a = a' t ω then r t (a' t ω) / pol t ω (a' t ω) else 0)\n    (hwupd : ∀ (t : Fin T) ω a, w t.succ ω a = w t.castSucc ω a * Real.exp (γ * rhat t ω a / K))\n    (ℱ : Fin T → MeasurableSpace Ω)\n    (hℱ : ∀ t, ℱ t ≤ m0)\n    (ha'adapted : ∀ (s t : Fin T), s < t → Measurable[ℱ t] (a' s))\n    (hpol_meas : ∀ (t : Fin T) (a : Fin K), Measurable[ℱ t] (fun ω => pol t ω a))\n    (hsample : ∀ (t : Fin T) (a : Fin K),\n        μ[(fun ω => if a' t ω = a then (1 : ℝ) else 0) | ℱ t]\n          =ᵐ[μ] fun ω => pol t ω a)\n    (hint_rhat : ∀ (t : Fin T) (a : Fin K), Integrable (fun ω => rhat t ω a) μ)\n    (hint_rew : Integrable (fun ω => ∑ t : Fin T, r t (a' t ω)) μ)\n    (REG : ℝ) (hREG : REG = bestArmReward hK r - ∫ ω, ∑ t : Fin T, r t (a' t ω) ∂μ) :\n    REG ≤ (Real.exp 1 - 1) * γ * bestArmReward hK r + (K * Real.log K) / γ := sorry\n\nend GraduateLevel\n", "target_names": ["ch16_theorem_16_17"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 326, "end": 1871, "gold": "{K T : ℕ} (hK : 1 ≤ K) (γ : ℝ) (hγ : 0 < γ) (hγ1 : γ ≤ 1)\n    (r : Fin T → Fin K → ℝ) (hr : ∀ t a, r t a ∈ Set.Icc (0 : ℝ) 1)\n    {Ω : Type*} {m0 : MeasurableSpace Ω} (μ : @Measure Ω m0) [IsProbabilityMeasure μ]\n    (w : Fin (T + 1) → Ω → Fin K → ℝ) (pol : Fin T → Ω → Fin K → ℝ)\n    (a' : Fin T → Ω → Fin K) (rhat : Fin T → Ω → Fin K → ℝ)\n    (ha'meas : ∀ t, Measurable (a' t)) (hw0 : ∀ ω a, w 0 ω a = 1)\n    (hwnonneg : ∀ (t : Fin (T + 1)) ω a, 0 ≤ w t ω a)\n    (hwpos : ∀ (t : Fin T) ω, 0 < ∑ b : Fin K, w t.castSucc ω b)\n    (hpol : ∀ (t : Fin T) ω a, pol t ω a = (1 - γ) * (w t.castSucc ω a / ∑ b : Fin K, w t.castSucc ω b) + γ / K)\n    (hpolpos : ∀ (t : Fin T) ω a, 0 < pol t ω a)\n    (hrhat : ∀ (t : Fin T) ω a, rhat t ω a = if a = a' t ω then r t (a' t ω) / pol t ω (a' t ω) else 0)\n    (hwupd : ∀ (t : Fin T) ω a, w t.succ ω a = w t.castSucc ω a * Real.exp (γ * rhat t ω a / K))\n    (ℱ : Fin T → MeasurableSpace Ω)\n    (hℱ : ∀ t, ℱ t ≤ m0)\n    (ha'adapted : ∀ (s t : Fin T), s < t → Measurable[ℱ t] (a' s))\n    (hpol_meas : ∀ (t : Fin T) (a : Fin K), Measurable[ℱ t] (fun ω => pol t ω a))\n    (hsample : ∀ (t : Fin T) (a : Fin K),\n        μ[(fun ω => if a' t ω = a then (1 : ℝ) else 0) | ℱ t]\n          =ᵐ[μ] fun ω => pol t ω a)\n    (hint_rhat : ∀ (t : Fin T) (a : Fin K), Integrable (fun ω => rhat t ω a) μ)\n    (hint_rew : Integrable (fun ω => ∑ t : Fin T, r t (a' t ω)) μ)\n    (REG : ℝ) (hREG : REG = bestArmReward hK r - ∫ ω, ∑ t : Fin T, r t (a' t ω) ∂μ) :\n    REG ≤ (Real.exp 1 - 1) * γ * bestArmReward hK r + (K * Real.log K) / γ"}]}
{"item_id": "ch7_theorem_7_7::ch7_theorem_7_7::A-stmt", "paper_id": "ch7_theorem_7_7", "target_name": "ch7_theorem_7_7", "source": "Graduate-level", "nl_statement": "All local minima of \\(g\\) are global minima (even though \\(g\\) is non-convex).", "context": [{"name": "Local minimum of a function", "statement": "We say that \\(x\\) is a local minimum of a function \\(f\\) if there exists an open neighborhood \\(N\\) around \\(x\\) such that in \\(N\\), the function values are at least \\(f(x)\\)."}, {"name": "PCA rank-1 objective and assumptions", "statement": "Let \\(M \\in \\mathbb{R}^{d \\times d}\\) be symmetric positive semidefinite. For \\(x \\in \\mathbb{R}^d\\), define \\(g(x) := \\frac{1}{2}\\|M - xx^\\top\\|_F^2\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen scoped Matrix\n\nnamespace GraduateLevel\n\nnoncomputable def pcaObjective {d : ℕ} (M : Matrix (Fin d) (Fin d) ℝ) (x : Fin d → ℝ) : ℝ :=\n  (1 / 2) * ∑ i : Fin d, ∑ j : Fin d, (M i j - Matrix.vecMulVec x x i j) ^ 2\n\ntheorem ch7_theorem_7_7\n    {d : ℕ} (M : Matrix (Fin d) (Fin d) ℝ)\n    (hSymm : M.IsSymm) (hPSD : M.PosSemidef)\n    (x : Fin d → ℝ)\n    (hLocal : IsLocalMin (pcaObjective M) x) :\n    ∀ y : Fin d → ℝ, pcaObjective M x ≤ pcaObjective M y := sorry\n\nend GraduateLevel\n", "target_names": ["ch7_theorem_7_7"], "topic_domain": "Optimization", "topic_subtopic": "Non-convex Geometry", "holes": [{"start": 260, "end": 467, "gold": "{d : ℕ} (M : Matrix (Fin d) (Fin d) ℝ)\n    (hSymm : M.IsSymm) (hPSD : M.PosSemidef)\n    (x : Fin d → ℝ)\n    (hLocal : IsLocalMin (pcaObjective M) x) :\n    ∀ y : Fin d → ℝ, pcaObjective M x ≤ pcaObjective M y"}]}
{"item_id": "ch4_theorem_4_18::ch4_theorem_4_18::A-stmt", "paper_id": "ch4_theorem_4_18", "target_name": "ch4_theorem_4_18", "source": "Graduate-level", "nl_statement": "Suppose for all \\(f \\in \\mathcal{F}\\), \\(0 \\leq f(z) \\leq 1\\). Then, with probability at least \\(1-\\delta\\),\n\\[\n\\sup_{f\\in \\mathcal{F}} \\left[ \\frac{1}{n} \\sum^n_{i=1} f(z_i) - \\mathbb{E}[f(z)] \\right] \\leq 2 R_S(\\mathcal{F}) + 3\\sqrt{\\frac{\\log{(2/\\delta)}}{2n}}.\n\\]", "context": [{"name": "Empirical Rademacher complexity", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\).\nAs the name suggests, the expectation of the empirical Rademacher complexity is the Rademacher complexity:\n\\[\nR_n(\\mathcal{F}) = \\mathbb{E}_{\\substack{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P \\\\ S = \\{z_1, \\dots, z_n\\}}}\\left[ R_S(\\mathcal{F}) \\right].\n\\]"}, {"name": "Symmetrization bound by Rademacher complexity", "statement": "\\[\n\\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\sup_{f\\in F} \\left[ \\frac{1}{n} \\sum^n_{i=1} f(z_i) - \\mathbb{E}_{z\\sim P} [f(z)] \\right]\\right] \\leq 2 R_n(\\mathcal{F}).\n\\]"}, {"name": "McDiarmid's inequality", "statement": "Suppose \\(f : \\mathbb{R}^n \\to \\mathbb{R}\\) satisfies the bounded difference condition: there exist constants \\(c_1, \\ldots, c_n \\in \\mathbb{R}\\) such that for all real numbers \\(x_1, \\ldots, x_n\\) and \\(x_i'\\),\n\\[\n|f(x_1, \\ldots, x_n) - f(x_1, \\ldots, x_{i-1}, x_i', x_{i+1}, \\ldots, x_n)| \\leq c_i.\n\\]\nThen, for any independent random variables \\(X_1, \\ldots, X_n\\),\n\\[\n\\mathbb{P}\\left[f(X_1, \\ldots, X_n) - \\mathbb{E}[f(X_1, \\ldots, X_n)] \\geq t\\right] \\leq \\exp\\left(-\\frac{2t^2}{\\sum_{i=1}^n c_i^2}\\right).\n\\]\nMoreover, \\(f(X_1, \\ldots, X_n)\\) is \\(O\\!\\left(\\sqrt{\\sum_{i=1}^n c_i^2}\\right)\\)-sub-Gaussian."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ntheorem ch4_theorem_4_18\n    {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    {ι : Type v} [Nonempty ι]\n    {𝒳 : Type w} [MeasurableSpace 𝒳]\n    (n : ℕ) (hn : 0 < n)\n    (f : ι → 𝒳 → ℝ)\n    (X : Ω → 𝒳) (hX : Measurable X)\n    (hf01 : ∀ i, ∀ z : 𝒳, 0 ≤ f i z ∧ f i z ≤ 1)\n    (hfmeas : ∀ i, Measurable (fun z => f i z))\n    (hgmeas : Measurable (fun ω : Fin n → Ω =>\n        ⨆ i, ((n : ℝ)⁻¹ * ∑ k : Fin n, f i (X (ω k)) - ∫ x, f i (X x) ∂μ)))\n    (hRmeas : Measurable (fun ω : Fin n → Ω =>\n        empRadSigned n f (fun k => X (ω k))))\n    (hint : ∀ i, Integrable (fun ω => f i (X ω)) μ)\n    (hsym : (∫ ω, (⨆ i, ((n : ℝ)⁻¹ * ∑ k : Fin n, f i (X (ω k)) - ∫ x, f i (X x) ∂μ))\n                ∂(Measure.pi (fun _ : Fin n => μ)))\n              ≤ 2 * ∫ ω, empRadSigned n f (fun k => X (ω k))\n                        ∂(Measure.pi (fun _ : Fin n => μ)))\n    (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ ≤ 1) :\n    (Measure.pi (fun _ : Fin n => μ))\n        {ω : Fin n → Ω |\n          (⨆ i, ((n : ℝ)⁻¹ * ∑ k : Fin n, f i (X (ω k)) - ∫ x, f i (X x) ∂μ))\n            ≤ 2 * empRadSigned n f\n                  (fun k => X (ω k))\n              + 3 * Real.sqrt (Real.log (2 / δ) / (2 * n))}\n      ≥ ENNReal.ofReal (1 - δ) := sorry\n\nend GraduateLevel\n", "target_names": ["ch4_theorem_4_18"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 492, "end": 1683, "gold": "{Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    {ι : Type v} [Nonempty ι]\n    {𝒳 : Type w} [MeasurableSpace 𝒳]\n    (n : ℕ) (hn : 0 < n)\n    (f : ι → 𝒳 → ℝ)\n    (X : Ω → 𝒳) (hX : Measurable X)\n    (hf01 : ∀ i, ∀ z : 𝒳, 0 ≤ f i z ∧ f i z ≤ 1)\n    (hfmeas : ∀ i, Measurable (fun z => f i z))\n    (hgmeas : Measurable (fun ω : Fin n → Ω =>\n        ⨆ i, ((n : ℝ)⁻¹ * ∑ k : Fin n, f i (X (ω k)) - ∫ x, f i (X x) ∂μ)))\n    (hRmeas : Measurable (fun ω : Fin n → Ω =>\n        empRadSigned n f (fun k => X (ω k))))\n    (hint : ∀ i, Integrable (fun ω => f i (X ω)) μ)\n    (hsym : (∫ ω, (⨆ i, ((n : ℝ)⁻¹ * ∑ k : Fin n, f i (X (ω k)) - ∫ x, f i (X x) ∂μ))\n                ∂(Measure.pi (fun _ : Fin n => μ)))\n              ≤ 2 * ∫ ω, empRadSigned n f (fun k => X (ω k))\n                        ∂(Measure.pi (fun _ : Fin n => μ)))\n    (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ ≤ 1) :\n    (Measure.pi (fun _ : Fin n => μ))\n        {ω : Fin n → Ω |\n          (⨆ i, ((n : ℝ)⁻¹ * ∑ k : Fin n, f i (X (ω k)) - ∫ x, f i (X x) ∂μ))\n            ≤ 2 * empRadSigned n f\n                  (fun k => X (ω k))\n              + 3 * Real.sqrt (Real.log (2 / δ) / (2 * n))}\n      ≥ ENNReal.ofReal (1 - δ)"}]}
{"item_id": "ch4_theorem_4_16::ch4_theorem_4_16::A-stmt", "paper_id": "ch4_theorem_4_16", "target_name": "ch4_theorem_4_16", "source": "Graduate-level", "nl_statement": "Let \\(\\mathcal{F}\\) be a family of functions \\(Z \\mapsto [-1,1]\\). Then\n\\[\nR_S(\\mathcal{F})\\le\\inf_{\\epsilon>0}\\left(\\epsilon+\\sqrt{\\frac{2\\log N(\\epsilon,\\mathcal{F},L_2(P_n))}{n}}\\right).\n\\]", "context": [{"name": "Empirical Rademacher complexity", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), the empirical Rademacher complexity is defined as\n\\[\nR_S(\\mathcal{F}) \\stackrel{\\Delta}{=} \\mathbb{E}_{\\sigma_1,\\dots, \\sigma_n} \\left[ \\sup_{f\\in \\mathcal{F}} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right].\n\\]\n\\(R_S(\\mathcal{F})\\) is a function of both the function class \\(\\mathcal{F}\\) and the dataset \\(S\\)."}, {"name": "definition 4.14", "statement": "\\(\\mathcal{C}\\) is an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to metric \\(\\rho\\) if for all \\(v' \\in \\mathcal{Q}\\), there exists \\(v \\in \\mathcal{C}\\) such that \\(\\rho(v,v')\\le \\epsilon\\)."}, {"name": "definition 4.15", "statement": "The covering number is defined as the minimum size of an \\(\\epsilon\\)-cover, or explicitly:\n\\[\nN(\\epsilon, \\mathcal{Q}, \\rho) \\stackrel{\\Delta}{=} (\\text{min size of } \\epsilon\\text{-cover of } \\mathcal{Q} \\text{ w.r.t. metric } \\rho).\n\\]"}, {"name": "proposition 4.12", "statement": "Let \\(\\mathcal{F}\\) be a collection of functions mapping \\(Z \\mapsto \\mathbb{R}\\) and let \\(\\mathcal{Q}\\) be defined as in the referenced equation. Assume that \\(\\frac{1}{\\sqrt{n}} \\|v\\|_2 \\le M < \\infty\\) for all \\(v \\in \\mathcal{Q}\\). Then,\n\\[\nR_S(\\mathcal{F}) \\leq \\sqrt{\\frac{2 M^2 \\log |\\mathcal{Q}|}{n}}.\n\\]"}, {"name": "Output space \\(\\mathcal{Q}\\) and the \\(L_2(P_n)\\) metric", "statement": "Given a dataset \\(S = \\{z_1, \\dots, z_n\\}\\), define the output space\n\\[\n\\mathcal{Q} \\triangleq \\left\\{ \\left(f(z_1), \\dots, f(z_n)\\right)^\\top : f \\in \\mathcal{F} \\right\\} \\subseteq \\mathbb{R}^n.\n\\]\nCorrespondingly, the empirical Rademacher complexity can be rewritten as a maximization over the output space \\(\\mathcal{Q}\\) instead of the function space \\(\\mathcal{F}\\):\n\\[\nR_S(\\mathcal{F}) = \\mathbb{E}_{\\sigma}\\left[ \\sup_{v \\in \\mathcal{Q}} \\frac{1}{n} \\langle \\sigma, v \\rangle \\right].\n\\]\nWe use the metric \\(\\rho(v, v') = \\frac{1}{\\sqrt{n}} \\|v - v'\\|_2\\). In the literature, an \\(\\epsilon\\)-cover of \\(\\mathcal{Q}\\) with respect to \\(\\rho\\) is also referred to as an \\(\\epsilon\\)-cover of the function class \\(\\mathcal{F}\\) under the \\(L_2(P_n)\\) metric, where \\(P_n\\) denotes the empirical distribution, i.e. the uniform distribution over the observations \\(z_1, \\dots, z_n\\). In particular,\n\\[\nL_2(P_n)(f, f') = \\sqrt{\\frac{1}{n} \\sum_{i=1}^n (f(z_i) - f'(z_i))^2}.\n\\]\nWe can write a trivial correspondence between both the output and function class points of view as follows:\n\\[\nN(\\epsilon, \\mathcal{F}, L_2(P_n)) = N\\left(\\epsilon, \\mathcal{Q}, \\frac{1}{\\sqrt{n}} \\|\\cdot\\|_2\\right).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def empiricalNorm (n : ℕ) (f : Fin n → ℝ) : ℝ :=\n  Real.sqrt ((n : ℝ)⁻¹ * ∑ k : Fin n, f k ^ 2)\n\nlemma empiricalNorm_nonneg (n : ℕ) (f : Fin n → ℝ) : 0 ≤ empiricalNorm n f :=\n  Real.sqrt_nonneg _\n\nlemma empiricalNorm_zero (n : ℕ) : empiricalNorm n (0 : Fin n → ℝ) = 0 := by\n  unfold empiricalNorm\n  simp\n\nlemma empiricalNorm_neg (n : ℕ) (f : Fin n → ℝ) : empiricalNorm n (-f) = empiricalNorm n f := by\n  unfold empiricalNorm\n  congr 2\n  apply Finset.sum_congr rfl\n  intro i _\n  simp only [Pi.neg_apply, neg_sq]\n\ndef EmpiricalSpace (n : ℕ) := Fin n → ℝ\n\nnamespace EmpiricalSpace\n\ninstance (n : ℕ) : Inhabited (EmpiricalSpace n) := ⟨fun _ => 0⟩\ninstance (n : ℕ) : Zero (EmpiricalSpace n) := ⟨fun _ => 0⟩\ninstance (n : ℕ) : Add (EmpiricalSpace n) := ⟨fun a b i => a i + b i⟩\ninstance (n : ℕ) : Sub (EmpiricalSpace n) := ⟨fun a b i => a i - b i⟩\ninstance (n : ℕ) : Neg (EmpiricalSpace n) := ⟨fun a i => -a i⟩\n\nnoncomputable instance instDist (n : ℕ) : Dist (EmpiricalSpace n) where\n  dist a b := empiricalNorm n (a - b)\n\nnoncomputable instance instEDist (n : ℕ) : EDist (EmpiricalSpace n) where\n  edist a b := ENNReal.ofReal (dist a b)\n\nlemma dist_self (n : ℕ) (a : EmpiricalSpace n) : dist a a = 0 := by\n  show empiricalNorm n (a - a) = 0\n  have h : (a - a : EmpiricalSpace n) = 0 := by\n    funext i\n    show a i - a i = (0 : Fin n → ℝ) i\n    simp\n  rw [h]\n  exact empiricalNorm_zero n\n\nlemma dist_comm (n : ℕ) (a b : EmpiricalSpace n) : dist a b = dist b a := by\n  show empiricalNorm n (a - b) = empiricalNorm n (b - a)\n  have h : (a - b : EmpiricalSpace n) = -(b - a) := by\n    funext i\n    show a i - b i = -(b i - a i)\n    ring\n  rw [h]\n  exact empiricalNorm_neg n _\n\nlemma dist_triangle (n : ℕ) (a b c : EmpiricalSpace n) :\n    dist a c ≤ dist a b + dist b c := by\n  show empiricalNorm n (a - c) ≤ empiricalNorm n (a - b) + empiricalNorm n (b - c)\n  unfold empiricalNorm\n  have eq_ac : ∀ k, (a - c : EmpiricalSpace n) k = a k - c k := fun k => rfl\n  have eq_ab : ∀ k, (a - b : EmpiricalSpace n) k = a k - b k := fun k => rfl\n  have eq_bc : ∀ k, (b - c : EmpiricalSpace n) k = b k - c k := fun k => rfl\n  simp_rw [eq_ac, eq_ab, eq_bc]\n  by_cases hn : n = 0\n  · simp [hn]\n  have hn_pos : (0 : ℝ) < n := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)\n  have hn_inv_nonneg : 0 ≤ (n : ℝ)⁻¹ := inv_nonneg.mpr (le_of_lt hn_pos)\n  have h_factor : ∀ f : Fin n → ℝ, Real.sqrt ((n : ℝ)⁻¹ * ∑ i, (f i)^2) =\n      Real.sqrt (n : ℝ)⁻¹ * Real.sqrt (∑ i, (f i)^2) := fun f => by\n    rw [Real.sqrt_mul hn_inv_nonneg]\n  rw [h_factor (fun i => a i - c i), h_factor (fun i => a i - b i),\n      h_factor (fun i => b i - c i), ← mul_add]\n  apply mul_le_mul_of_nonneg_left _ (Real.sqrt_nonneg _)\n  let e := EuclideanSpace.equiv (Fin n) ℝ\n  let a' : EuclideanSpace ℝ (Fin n) := e.symm a\n  let b' : EuclideanSpace ℝ (Fin n) := e.symm b\n  let c' : EuclideanSpace ℝ (Fin n) := e.symm c\n  have h_a' : ∀ i, a' i = a i := fun i => rfl\n  have h_b' : ∀ i, b' i = b i := fun i => rfl\n  have h_c' : ∀ i, c' i = c i := fun i => rfl\n  have h_norm : ∀ (x y : EuclideanSpace ℝ (Fin n)),\n      Real.sqrt (∑ i, (x i - y i)^2) = ‖x - y‖ := fun x y => by\n    rw [EuclideanSpace.norm_eq]\n    congr 1\n    apply Finset.sum_congr rfl\n    intro i _\n    rw [Real.norm_eq_abs, sq_abs]\n    rfl\n  simp only [← h_a', ← h_b', ← h_c']\n  rw [h_norm a' c', h_norm a' b', h_norm b' c']\n  have h_eq : a' - c' = (a' - b') + (b' - c') := (sub_add_sub_cancel a' b' c').symm\n  rw [h_eq]\n  exact norm_add_le (a' - b') (b' - c')\n\nlemma edist_dist (n : ℕ) (a b : EmpiricalSpace n) :\n    edist a b = ENNReal.ofReal (dist a b) := rfl\n\nnoncomputable instance instPseudoMetricSpace (n : ℕ) : PseudoMetricSpace (EmpiricalSpace n) where\n  dist_self := dist_self n\n  dist_comm := dist_comm n\n  dist_triangle := dist_triangle n\n  edist_dist := edist_dist n\n\nend EmpiricalSpace\n\ndef empiricalMetricImage {X : Type*} (n : ℕ) (x : Fin n → X) (g : X → ℝ) : EmpiricalSpace n :=\n  fun i => g (x i)\n\ndef IsENet {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n  s ⊆ ⋃ x ∈ t, Metric.closedBall x eps\n\nnoncomputable def coveringNumber {A : Type*} [PseudoMetricSpace A] (eps : ℝ) (s : Set A) : WithTop Nat :=\n  sInf {n : WithTop Nat | ∃ t : Finset A, IsENet t eps s ∧ (t.card : WithTop Nat) = n}\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\ntheorem ch4_theorem_4_16\n    {ι : Type*} [Nonempty ι] {𝒳 : Type*}\n    (n : ℕ) (hn : 0 < n)\n    (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳)\n    (hf : ∀ i, ∀ z, f i z ∈ Set.Icc (-1 : ℝ) 1)\n    (s : Set (EmpiricalSpace n))\n    (hs : s = Set.range (fun i => empiricalMetricImage n S (f i)))\n    (hcov : ∀ ε ∈ Set.Ioi (0 : ℝ), coveringNumber ε s ≠ ⊤) :\n    empRadSigned n f S ≤\n      ⨅ ε : Set.Ioi (0 : ℝ),\n        ((ε : ℝ) +\n          Real.sqrt\n            (2 * Real.log (((coveringNumber (ε : ℝ) s).untopD 0 : ℕ) : ℝ)\n              / (n : ℝ))) := sorry\n\nend GraduateLevel\n", "target_names": ["ch4_theorem_4_16"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 4730, "end": 5229, "gold": "{ι : Type*} [Nonempty ι] {𝒳 : Type*}\n    (n : ℕ) (hn : 0 < n)\n    (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳)\n    (hf : ∀ i, ∀ z, f i z ∈ Set.Icc (-1 : ℝ) 1)\n    (s : Set (EmpiricalSpace n))\n    (hs : s = Set.range (fun i => empiricalMetricImage n S (f i)))\n    (hcov : ∀ ε ∈ Set.Ioi (0 : ℝ), coveringNumber ε s ≠ ⊤) :\n    empRadSigned n f S ≤\n      ⨅ ε : Set.Ioi (0 : ℝ),\n        ((ε : ℝ) +\n          Real.sqrt\n            (2 * Real.log (((coveringNumber (ε : ℝ) s).untopD 0 : ℕ) : ℝ)\n              / (n : ℝ)))"}]}
{"item_id": "ch5_theorem_5_9::ch5_theorem_5_9::A-stmt", "paper_id": "ch5_theorem_5_9", "target_name": "ch5_theorem_5_9", "source": "Graduate-level", "nl_statement": "For some constants \\(B_w > 0\\) and \\(B_u > 0\\), let\n\\[\n\\mathcal{H} = \\left\\{ f_\\theta \\mid \\left\\|w\\right\\|_2 \\leq B_w,\\ \\left\\|u_i\\right\\|_2 \\leq B_u,\\ \\forall i \\in \\{1,2,\\dots,m\\} \\right\\},\n\\]\nand suppose \\(\\mathbb{E}\\left[\\left\\|x\\right\\|_2^2\\right] \\leq C^2\\). Then\n\\[\nR_n(\\mathcal{H}) \\le 2 B_w B_u C\\sqrt{\\frac{m}{n}}.\n\\]", "context": [{"name": "Removing the absolute value in Rademacher complexity", "statement": "Let \\(\\sigma = (\\sigma_1, \\dots, \\sigma_n)\\) and \\(f_{\\theta}(x) = \\left(f_{\\theta}\\left(x^{(1)}\\right), \\dots, f_{\\theta}\\left(x^{(n)}\\right)\\right)\\). Suppose that for any \\(\\sigma \\in \\{\\pm 1\\}^n\\), \\(\\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle \\geq 0\\). Then,\n\\[\n\\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\left| \\langle \\sigma, f_{\\theta}(x) \\rangle \\right| \\right] \\leq 2\\, \\mathbb{E}_{\\sigma}\\left[ \\sup_{\\theta} \\langle \\sigma, f_{\\theta}(x) \\rangle \\right].\n\\]"}, {"name": "Talagrand's lemma", "statement": "Let \\(\\phi:\\mathbb{R}\\to\\mathbb{R}\\) be a \\(\\kappa\\)-Lipschitz function. Then\n\\[\nR_S(\\phi\\circ \\mathcal{H})\\leq \\kappa R_S(\\mathcal{H}),\n\\]\nwhere \\(\\phi\\circ\\mathcal{H} = \\{z\\mapsto \\phi(h(z))\\mid h\\in\\mathcal{H}\\}\\)."}, {"name": "Rademacher complexity of l2-bounded linear functions", "statement": "Let \\(\\mathcal{H} = \\{x \\mapsto \\langle w,x \\rangle \\mid w \\in \\mathbb{R}^d,\\, \\|w\\|_2 \\le B\\}\\) for some constant \\(B > 0\\). Moreover, assume \\(\\mathbb{E}_{x \\sim P}[\\|x\\|_2^2] \\leq C^2\\), where \\(P\\) is some distribution and \\(C > 0\\) is a constant. Then\n\\[\nR_S(\\mathcal{H}) \\le \\frac{B}{n} \\sqrt{\\sum_{i=1}^n \\|x^{(i)}\\|_2^2},\n\\]\nand\n\\[\nR_n(\\mathcal{H}) \\le \\frac{BC}{\\sqrt{n}}.\n\\]"}, {"name": "Rademacher complexity", "statement": "Let \\(\\mathcal{F}\\) be a family of functions mapping \\(Z \\mapsto \\mathbb{R}\\), and let \\(P\\) be a distribution over \\(Z\\). The (average) Rademacher complexity of \\(\\mathcal{F}\\) is defined as\n\\[\nR_n(F) \\triangleq \\mathbb{E}_{z_1, \\dots, z_n \\overset{\\text{iid}}{\\sim} P} \\left[ \\mathbb{E}_{\\sigma_1, \\dots, \\sigma_n \\overset{\\text{iid}}{\\sim}\\{ \\pm 1 \\}} \\left[ \\sup_{f\\in F} \\frac{1}{n} \\sum^n_{i=1} \\sigma_i f(z_i) \\right] \\right],\n\\]\nwhere \\(\\sigma_1, \\dots, \\sigma_n\\) are independent Rademacher random variables, i.e. each taking on the value of \\(1\\) or \\(-1\\) with probability \\(1/2\\)."}, {"name": "Two-layer ReLU network", "statement": "Given a width \\(m\\), let \\(w_1,\\dots,w_m \\in \\mathbb{R}\\), \\(u_1,\\dots,u_m \\in \\mathbb{R}^d\\), and let \\(\\phi\\) denote the ReLU activation. For \\(\\theta=(w_1,\\dots,w_m,u_1,\\dots,u_m)\\), define the two-layer ReLU network\n\\[\nf_{\\theta}(x)=\\sum_{j=1}^m w_j \\phi\\left(u_j^\\top x\\right).\n\\]\nThroughout this section, we use the following notation: \\(\\theta = (w, U)\\) are the parameters of the model with \\(w \\in \\mathbb{R}^m\\) and \\(U \\in \\mathbb{R}^{m\\times d}\\), where \\(m\\) denotes the number of hidden units. We use \\(u_i \\in \\mathbb{R}^d\\) to denote the \\(i\\)-th row of \\(U\\) (written as a column vector). \\(\\phi(z) = \\max(z,0)\\) is the ReLU activation function applied element-wise. \\(f_\\theta(x) = \\langle w, \\phi(Ux)\\rangle = w^\\top \\phi(Ux)\\) is the model. \\(\\{(x^{(i)}, y^{(i)})\\}_{i=1}^n\\) is the training set, with \\(x^{(i)} \\in \\mathbb{R}^d\\) and \\(y^{(i)} \\in \\mathbb{R}\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\nnoncomputable def popRadSigned {Ω : Type*} [MeasurableSpace Ω] {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (μ : Measure Ω) (X : Ω → 𝒳) : ℝ :=\n  ∫ ω : Fin n → Ω, empRadSigned n f (X ∘ ω) ∂(Measure.pi fun _ => μ)\n\ndef reluAct : ℝ → ℝ := fun x => max 0 x\n\ndef NetParam (d m : ℕ) (Bw Bu : ℝ) : Type := {p : EuclideanSpace ℝ (Fin m) × (Fin m → EuclideanSpace ℝ (Fin d)) // ‖p.1‖ ≤ Bw ∧ ∀ i, ‖p.2 i‖ ≤ Bu}\n\nnoncomputable def netFun {d m : ℕ} {Bw Bu : ℝ} (p : NetParam d m Bw Bu) (x : EuclideanSpace ℝ (Fin d)) : ℝ := ∑ j : Fin m, (p.1.1 j) * reluAct (inner ℝ (p.1.2 j) x)\n\ntheorem ch5_theorem_5_9 {d m n : ℕ} (Bw Bu C : ℝ) (hBw : 0 < Bw) (hBu : 0 < Bu) (hC : 0 ≤ C) (hn : 0 < n) {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → EuclideanSpace ℝ (Fin d)) (hXmeas : AEMeasurable X μ) (hXsq_int : Integrable (fun ω => ‖X ω‖ ^ 2) μ) (hmom : ∫ ω, ‖X ω‖ ^ 2 ∂μ ≤ C ^ 2) : popRadSigned n (fun (p : NetParam d m Bw Bu) => netFun p) μ X ≤ 2 * Bw * Bu * C * Real.sqrt (m / n) := sorry\n\nend GraduateLevel\n", "target_names": ["ch5_theorem_5_9"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 1062, "end": 1463, "gold": "{d m n : ℕ} (Bw Bu C : ℝ) (hBw : 0 < Bw) (hBu : 0 < Bu) (hC : 0 ≤ C) (hn : 0 < n) {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : Ω → EuclideanSpace ℝ (Fin d)) (hXmeas : AEMeasurable X μ) (hXsq_int : Integrable (fun ω => ‖X ω‖ ^ 2) μ) (hmom : ∫ ω, ‖X ω‖ ^ 2 ∂μ ≤ C ^ 2) : popRadSigned n (fun (p : NetParam d m Bw Bu) => netFun p) μ X ≤ 2 * Bw * Bu * C * Real.sqrt (m / n)"}]}
{"item_id": "ch9_theorem_9_2::ch9_theorem_9_2::A-stmt", "paper_id": "ch9_theorem_9_2", "target_name": "ch9_theorem_9_2", "source": "Graduate-level", "nl_statement": "Let \\(t \\leqslant n\\), \\(L, R > 0\\). There exists a convex \\(L\\)-Lipschitz function \\(f\\) such that any black-box procedure started at \\(x_0 = 0\\) satisfies \\[ \\min_{1 \\leqslant s \\leqslant t} f(x_s) - \\min_{x \\in B_2(R)} f(x) \\geqslant \\frac{RL}{2(1+\\sqrt{t})}. \\] Furthermore, there is an \\(\\alpha\\)-strongly convex, \\(L\\)-Lipschitz function \\(f\\) such that \\[ \\min_{1 \\leqslant s \\leqslant t} f(x_s) - \\min_{x \\in B_2(\\frac{L}{2\\alpha})} f(x) \\geqslant \\frac{L^2}{8\\alpha t}. \\]", "context": [{"name": "Black-Box Procedure", "statement": "A \\textit{black-box procedure} generates a sequence of points \\(\\{x_t\\}\\) such that \\[ x_{t+1} \\in x_0 + \\Span\\{g_1, \\dots, g_t\\}, \\] and \\(g_s \\in \\partial f (x_s)\\)."}], "nl_notes": "SOURCE NOTE — the second claim of the source's Theorem 9.2 (eq. (9)) asserts an 'alpha-strongly convex, L-Lipschitz function f'. Read literally (Lipschitz on all of R^n), no such f exists for n >= 1: for g in the subdifferential at 0, f(r e_1) - f(0) >= -||g|| r + (alpha/2) r^2, which exceeds L r once r is large. So the printed claim (9) is false as written.\nThe formal statement follows the proof. The second conjunct requires `LipschitzOnWith L.toNNReal f (Metric.closedBall 0 (L / (2 * alpha)))`, and the first conjunct likewise uses `Metric.closedBall 0 R`. The subgradient oracle is chosen existentially (`exists G`, a fixed selector with `G z in SubderivAt f z`), matching the proof's 'the gradient oracle returns g_i = alpha x + gamma e_i'; quantifying over every subgradient choice makes the claim false for t >= 2.", "lean": "import Mathlib\n\nopen MeasureTheory RealInnerProductSpace\n\nnamespace GraduateLevel\n\ndef HasSubgradientAt {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (f : E → ℝ) (g x : E) : Prop :=\n  ∀ y, f y ≥ f x + ⟪g, y - x⟫\n\ndef SubderivAt {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    (f : E → ℝ) (x : E) : Set E :=\n  {g : E | HasSubgradientAt f g x}\n\ndef OracleProcedure {n : ℕ} (f : EuclideanSpace ℝ (Fin n) → ℝ)\n    (G : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n))\n    (x : ℕ → EuclideanSpace ℝ (Fin n)) : Prop :=\n  x 0 = 0 ∧\n  (∀ s, x (s + 1) - x 0 ∈\n    Submodule.span ℝ ((fun i => G (x i)) '' Set.Icc 1 s))\n\ntheorem ch9_theorem_9_2 (n t : ℕ) (L R α : ℝ)\n    (ht : 1 ≤ t) (htn : t ≤ n) (hL : 0 < L) (hR : 0 < R) (hα : 0 < α) :\n    (∃ f : EuclideanSpace ℝ (Fin n) → ℝ,\n      ConvexOn ℝ Set.univ f ∧\n      LipschitzOnWith L.toNNReal f (Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) R) ∧\n      ∃ G : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n),\n        (∀ z, G z ∈ SubderivAt f z) ∧\n        ∀ x : ℕ → EuclideanSpace ℝ (Fin n), OracleProcedure f G x →\n          ((Finset.Icc 1 t).image (fun s => f (x s))).min'\n            (by simpa [Finset.image_nonempty, Finset.nonempty_Icc] using ht)\n            - sInf (f '' Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) R)\n            ≥ R * L / (2 * (1 + Real.sqrt t))) ∧\n    (∃ f : EuclideanSpace ℝ (Fin n) → ℝ,\n      StrongConvexOn Set.univ α f ∧\n      LipschitzOnWith L.toNNReal f\n        (Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) (L / (2 * α))) ∧\n      ∃ G : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n),\n        (∀ z, G z ∈ SubderivAt f z) ∧\n        ∀ x : ℕ → EuclideanSpace ℝ (Fin n), OracleProcedure f G x →\n          ((Finset.Icc 1 t).image (fun s => f (x s))).min'\n            (by simpa [Finset.image_nonempty, Finset.nonempty_Icc] using ht)\n            - sInf (f '' Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) (L / (2 * α)))\n            ≥ L ^ 2 / (8 * α * t)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch9_theorem_9_2"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 670, "end": 1975, "gold": "(n t : ℕ) (L R α : ℝ)\n    (ht : 1 ≤ t) (htn : t ≤ n) (hL : 0 < L) (hR : 0 < R) (hα : 0 < α) :\n    (∃ f : EuclideanSpace ℝ (Fin n) → ℝ,\n      ConvexOn ℝ Set.univ f ∧\n      LipschitzOnWith L.toNNReal f (Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) R) ∧\n      ∃ G : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n),\n        (∀ z, G z ∈ SubderivAt f z) ∧\n        ∀ x : ℕ → EuclideanSpace ℝ (Fin n), OracleProcedure f G x →\n          ((Finset.Icc 1 t).image (fun s => f (x s))).min'\n            (by simpa [Finset.image_nonempty, Finset.nonempty_Icc] using ht)\n            - sInf (f '' Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) R)\n            ≥ R * L / (2 * (1 + Real.sqrt t))) ∧\n    (∃ f : EuclideanSpace ℝ (Fin n) → ℝ,\n      StrongConvexOn Set.univ α f ∧\n      LipschitzOnWith L.toNNReal f\n        (Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) (L / (2 * α))) ∧\n      ∃ G : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n),\n        (∀ z, G z ∈ SubderivAt f z) ∧\n        ∀ x : ℕ → EuclideanSpace ℝ (Fin n), OracleProcedure f G x →\n          ((Finset.Icc 1 t).image (fun s => f (x s))).min'\n            (by simpa [Finset.image_nonempty, Finset.nonempty_Icc] using ht)\n            - sInf (f '' Metric.closedBall (0 : EuclideanSpace ℝ (Fin n)) (L / (2 * α)))\n            ≥ L ^ 2 / (8 * α * t))"}]}
{"item_id": "ch3_theorem_3_3::ch3_theorem_3_3::A-stmt", "paper_id": "ch3_theorem_3_3", "target_name": "ch3_theorem_3_3", "source": "Graduate-level", "nl_statement": "Assume \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} and \\emph{\\(L\\)-Lipschitz}. Let \\(x^{*}\\) be an optimizer of \\(f\\), and let \\(x_{s}\\) be the updated point at step \\(s\\) using projected gradient descent. Let the max number of iterations be \\(t\\) with an adaptive step size \\(\\eta_{s} = \\frac{2}{\\alpha(s+1)}\\), then \\[ f\\left(\\sum_{s=1}^{t}\\frac{2s}{t(t+1)}x_{s}\\right) - f(x^{*})\\leqslant\\frac{2 L^{2}}{\\alpha(t+1)} \\]", "context": [{"name": "lemma 2.3", "statement": "\\[\n\\| \\Pi_{\\Omega}(y) - x \\|^2 \\leqslant \\| y - x \\|^2 - \\| y - \\Pi_{\\Omega}(y) \\|^2\n\\]"}, {"name": "Jensen's inequality", "statement": "Assume \\(f\\colon\\Omega\\to\\mathbb{R}\\) is a convex function and \\(x_{1}, x_{2}, \\ldots, x_{n}, \\sum_{i=1}^{n}\\gamma_{i}x_{i}/\\sum_{i=1}^{n}\\gamma_{i}\\in\\Omega\\) with weights \\(\\gamma_{i} > 0\\), then \\[ f\\left(\\frac{\\sum_{i=1}^{n}\\gamma_{i}x_{i}}{\\sum_{i=1}^{n}\\gamma_{i}}\\right) \\leqslant \\frac{\\sum_{i=1}^{n}\\gamma_{i}f(x_{i})}{\\sum_{i=1}^{n}\\gamma_{i}} \\]"}, {"name": "Strong convexity", "statement": "A function \\(f\\colon\\Omega\\to\\mathbb{R}\\) is \\emph{\\(\\alpha\\)-strongly convex} if, for all \\(x,y\\in\\Omega\\), the following inequality holds for some \\(\\alpha>0\\): \\[ f(y) \\geqslant f(x) + g(x)^\\top(y-x) + \\frac{\\alpha}{2}\\left\\lVert x-y\\right\\rVert^2 \\]"}], "nl_notes": "NOTE — Model ∇f(x_s) as the g(x) of Definition 3.1: a vector satisfying the subgradient inequality of f at x_s on Ω. Do not assume f is differentiable. Assume this subgradient has norm at most L at every iterate x_s, 1 ≤ s ≤ t.", "lean": "import Mathlib\n\nopen scoped RealInnerProductSpace\nopen Finset\n\nnamespace GraduateLevel\n\ntheorem ch3_theorem_3_3\n    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n    (Ω : Set E) (f : E → ℝ) (f' : E → E)\n    (proj : E → E)\n    (α : ℝ) (L : NNReal) (t : ℕ)\n    (x : ℕ → E)\n    (xopt : E)\n\n    (hsc : StrongConvexOn Ω α f)\n\n    (hLip : LipschitzOnWith L f Ω)\n\n    (hgrad : ∀ s, 1 ≤ s → s ≤ t →\n      (∀ y ∈ Ω, f y ≥ f (x s) + inner ℝ (f' (x s)) (y - x s)) ∧ ‖f' (x s)‖ ≤ (L : ℝ))\n\n    (hα : 0 < α)\n\n    (ht : 1 ≤ t)\n\n    (hΩ : Convex ℝ Ω) (hΩc : IsClosed Ω)\n    (hprojmem : ∀ y, proj y ∈ Ω)\n    (hproj : ∀ y, ∀ z ∈ Ω, ‖proj y - z‖ ^ 2 ≤ ‖y - z‖ ^ 2 - ‖y - proj y‖ ^ 2)\n\n    (hx1 : x 1 ∈ Ω)\n    (hrec : ∀ s, 1 ≤ s → x (s + 1) = proj (x s - (2 / (α * (s + 1))) • f' (x s)))\n\n    (hxopt_mem : xopt ∈ Ω)\n    (hxopt_min : ∀ z ∈ Ω, f xopt ≤ f z) :\n    f (∑ s ∈ Finset.Icc 1 t, (2 * (s : ℝ) / ((t : ℝ) * ((t : ℝ) + 1))) • x s) - f xopt\n      ≤ 2 * (L : ℝ) ^ 2 / (α * ((t : ℝ) + 1)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch3_theorem_3_3"], "topic_domain": "Optimization", "topic_subtopic": "Algorithms & Convergence", "holes": [{"start": 116, "end": 1002, "gold": "{E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n    (Ω : Set E) (f : E → ℝ) (f' : E → E)\n    (proj : E → E)\n    (α : ℝ) (L : NNReal) (t : ℕ)\n    (x : ℕ → E)\n    (xopt : E)\n\n    (hsc : StrongConvexOn Ω α f)\n\n    (hLip : LipschitzOnWith L f Ω)\n\n    (hgrad : ∀ s, 1 ≤ s → s ≤ t →\n      (∀ y ∈ Ω, f y ≥ f (x s) + inner ℝ (f' (x s)) (y - x s)) ∧ ‖f' (x s)‖ ≤ (L : ℝ))\n\n    (hα : 0 < α)\n\n    (ht : 1 ≤ t)\n\n    (hΩ : Convex ℝ Ω) (hΩc : IsClosed Ω)\n    (hprojmem : ∀ y, proj y ∈ Ω)\n    (hproj : ∀ y, ∀ z ∈ Ω, ‖proj y - z‖ ^ 2 ≤ ‖y - z‖ ^ 2 - ‖y - proj y‖ ^ 2)\n\n    (hx1 : x 1 ∈ Ω)\n    (hrec : ∀ s, 1 ≤ s → x (s + 1) = proj (x s - (2 / (α * (s + 1))) • f' (x s)))\n\n    (hxopt_mem : xopt ∈ Ω)\n    (hxopt_min : ∀ z ∈ Ω, f xopt ≤ f z) :\n    f (∑ s ∈ Finset.Icc 1 t, (2 * (s : ℝ) / ((t : ℝ) * ((t : ℝ) + 1))) • x s) - f xopt\n      ≤ 2 * (L : ℝ) ^ 2 / (α * ((t : ℝ) + 1))"}]}
{"item_id": "ch41_theorem_41_2::ch41_theorem_41_2::A-stmt", "paper_id": "ch41_theorem_41_2", "target_name": "ch41_theorem_41_2", "source": "Graduate-level", "nl_statement": "Let \\(x_1, \\cdots, x_n\\) be independent random variables (Lecture 41, opening paragraph). [Editorial, not in the source: each \\(f_i\\) is a function of \\(x_i\\), so that \\(Z = \\sup_{f\\in\\mathcal F}\\sum_{i=1}^n f_i(x_i)\\).] Let \\(\\mathcal{F}\\) be a finite set of functions \\(|{\\mathcal{F}}| <\\infty .\\forall f=(f_{1},\\cdots,f_{n})\\in{\\mathcal{F}},0\\leq f_{i}\\leq 1,Z=\\) \\(\\textstyle\\operatorname*{sup}_{f}\\sum_{i=1}^{n}f_{i}\\) ， and define h as \\(h(u)=(1+u)\\log(1+u)-u\\) where \\(u \\geq 0\\) Then \\(\\mathbb{P}(Z\\geq\\mathbb{E}Z+x)\\leq e^{-\\mathbb{E}Z\\cdot h(x/\\mathbb{E}Z)}\\)", "context": [{"name": "Tensorization of entropy lemma (recalled, Lecture 41 opening paragraph)", "statement": "Recall the tensorization of entropy lemma we proved previously. Let \\(x_1, \\cdots, x_n\\) be independent random variables, \\(x'_1, \\cdots, x'_n\\) be their independent copies, \\(Z = Z(x_1, \\cdots, x_n)\\), \\(Z^i = Z(x_1, \\cdots, x_{i-1}, x'_i, x_{i+1}, \\cdots, x_n)\\), and \\(\\phi(x) = e^x - x - 1\\). We have \\(\\mathbb{E}\\lambda Z e^{\\lambda Z} - \\mathbb{E}e^{\\lambda Z}\\log\\mathbb{E}e^{\\lambda Z} \\le \\mathbb{E}e^{\\lambda Z}\\sum_{i=1}^n \\phi(-\\lambda(Z - Z^i))\\)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def bennettH (u : ℝ) : ℝ := (1 + u) * Real.log (1 + u) - u\n\ntheorem ch41_theorem_41_2\n    {n : ℕ}\n    {α : Fin n → Type*} [∀ i, MeasurableSpace (α i)]\n    (μ : ∀ i, Measure (α i)) [∀ i, IsProbabilityMeasure (μ i)]\n    (F : Finset (∀ i, α i → ℝ)) (hF : F.Nonempty)\n    (hmeas : ∀ f ∈ F, ∀ i, Measurable (f i))\n    (hbound : ∀ f ∈ F, ∀ i (a : α i), 0 ≤ f i a ∧ f i a ≤ 1)\n    (Z : (∀ i, α i) → ℝ)\n    (hZ : ∀ ω, Z ω = F.sup' hF (fun f => ∑ i : Fin n, f i (ω i)))\n    (hZint : Integrable Z (Measure.pi μ))\n    (hEZ : 0 < ∫ ω, Z ω ∂(Measure.pi μ))\n    {x : ℝ} (hx : 0 ≤ x) :\n    ((Measure.pi μ) {ω | Z ω ≥ (∫ ω, Z ω ∂(Measure.pi μ)) + x}).toReal\n      ≤ Real.exp (-(∫ ω, Z ω ∂(Measure.pi μ)) *\n          bennettH (x / (∫ ω, Z ω ∂(Measure.pi μ)))) := sorry\n\nend GraduateLevel\n", "target_names": ["ch41_theorem_41_2"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "holes": [{"start": 170, "end": 822, "gold": "{n : ℕ}\n    {α : Fin n → Type*} [∀ i, MeasurableSpace (α i)]\n    (μ : ∀ i, Measure (α i)) [∀ i, IsProbabilityMeasure (μ i)]\n    (F : Finset (∀ i, α i → ℝ)) (hF : F.Nonempty)\n    (hmeas : ∀ f ∈ F, ∀ i, Measurable (f i))\n    (hbound : ∀ f ∈ F, ∀ i (a : α i), 0 ≤ f i a ∧ f i a ≤ 1)\n    (Z : (∀ i, α i) → ℝ)\n    (hZ : ∀ ω, Z ω = F.sup' hF (fun f => ∑ i : Fin n, f i (ω i)))\n    (hZint : Integrable Z (Measure.pi μ))\n    (hEZ : 0 < ∫ ω, Z ω ∂(Measure.pi μ))\n    {x : ℝ} (hx : 0 ≤ x) :\n    ((Measure.pi μ) {ω | Z ω ≥ (∫ ω, Z ω ∂(Measure.pi μ)) + x}).toReal\n      ≤ Real.exp (-(∫ ω, Z ω ∂(Measure.pi μ)) *\n          bennettH (x / (∫ ω, Z ω ∂(Measure.pi μ))))"}]}
{"item_id": "ch15_lemma_15_2::ch15_lemma_15_2::A-stmt", "paper_id": "ch15_lemma_15_2", "target_name": "ch15_lemma_15_2", "source": "Graduate-level", "nl_statement": "Let \\(x=(x_{1},\\ldots,x_{n}),x^{\\prime}=(x_{1}^{\\prime},\\ldots,x_{n}^{\\prime})\\) . If for functions \\(\\varphi_{1}(x,x^{\\prime}),\\varphi_{2}(x,x^{\\prime}),\\varphi_{3}(x,x^{\\prime})\\)\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\varphi_{1}(x,x^{\\prime})\\geq\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma e^{-\\gamma t}}\\end{array}\\]\nthen\n\\[\\begin{array}{ r }{\\mathbb{P}\\left(\\mathbb{E}_{x^{\\prime}}\\varphi_{1}(x,x^{\\prime})\\geq\\mathbb{E}_{x^{\\prime}}\\varphi_{2}(x,x^{\\prime})+\\sqrt{\\mathbb{E}_{x^{\\prime}}\\varphi_{3}(x,x^{\\prime})\\cdot t}\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.}\\end{array}\\]\n(i.e. if the inequality holds, then it holds with averaging over one of the copies)", "context": [{"name": "lemma 15.1", "statement": "Let \\(\\xi,\\nu\\) - random variables.Assume that\n\\[\\mathbb{P}\\left(\\nu\\geq t\\right)\\leq\\Gamma e^{-\\gamma t}\\]\nwhere \\(\\Gamma\\geq 1,t\\geq 0\\) ，and \\(\\gamma > 0\\) . Furthermore, for all \\(a > 0\\) assume that\n\\[\\mathbb{E}\\phi(\\xi)\\le\\mathbb{E}\\phi(\\nu)\\]\nwhere \\(\\phi(x)=(x-a)_{+}\\) .Then\n\\[\\mathbb{P}\\left(\\xi\\geq t\\right)\\leq\\Gamma\\cdot e\\cdot e^{-\\gamma t}.\\]"}, {"name": "theorem 14.1", "statement": "\\[\\mathbb{P}\\left(\\forall f\\in F,R(f)\\leq\\frac{2^{9/2}}{\\sqrt{n}}\\int_{0}^{d(0,f)}\\log^{1/2}\\mathcal{D}(F,\\varepsilon,d)d\\varepsilon+2^{7/2}d(0,f)\\sqrt{\\frac{u}{n}}\\right)\\geq 1-e^{-u}\\]\nfor any \\(u > 0\\)"}], "nl_notes": "NOTE — The NL states the tail hypothesis at a single t. The Lean reads it as the source's proof uses it (through Lemma 15.1): `hyp` requires P(φ₁ ≥ φ₂ + √(φ₃·s)) ≤ Γe^{−γs} for every s ≥ 0, with Γ ≥ 1, γ > 0 and t ≥ 0 (`hΓ`, `hγ`, `ht`); the conclusion is at the given t.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch15_lemma_15_2\n    {Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']\n    (μ : Measure Ω) (μ' : Measure Ω')\n    [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']\n    (phi1 phi2 phi3 : Ω → Ω' → ℝ)\n    (hmeas1 : Measurable (Function.uncurry phi1))\n    (hmeas2 : Measurable (Function.uncurry phi2))\n    (hmeas3 : Measurable (Function.uncurry phi3))\n    (hphi3_nonneg : ∀ x x', 0 ≤ phi3 x x')\n    (hint1 : ∀ x, Integrable (fun x' => phi1 x x') μ')\n    (hint2 : ∀ x, Integrable (fun x' => phi2 x x') μ')\n    (hint3 : ∀ x, Integrable (fun x' => phi3 x x') μ')\n    (hmavg1 : Measurable (fun x => ∫ x', phi1 x x' ∂μ'))\n    (hmavg2 : Measurable (fun x => ∫ x', phi2 x x' ∂μ'))\n    (hmavg3 : Measurable (fun x => ∫ x', phi3 x x' ∂μ'))\n    (Γ γ t : ℝ) (hΓ : 1 ≤ Γ) (hγ : 0 < γ) (ht : 0 ≤ t)\n    (hyp : ∀ s : ℝ, 0 ≤ s →\n      (μ.prod μ') {p : Ω × Ω' | phi1 p.1 p.2 ≥ phi2 p.1 p.2 + Real.sqrt (phi3 p.1 p.2 * s)}\n        ≤ ENNReal.ofReal (Γ * Real.exp (-γ * s))) :\n    μ {x : Ω | (∫ x', phi1 x x' ∂μ') ≥ (∫ x', phi2 x x' ∂μ')\n        + Real.sqrt ((∫ x', phi3 x x' ∂μ') * t)}\n      ≤ ENNReal.ofReal (Γ * Real.exp 1 * Real.exp (-γ * t)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch15_lemma_15_2"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "holes": [{"start": 112, "end": 1220, "gold": "{Ω Ω' : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω']\n    (μ : Measure Ω) (μ' : Measure Ω')\n    [IsProbabilityMeasure μ] [IsProbabilityMeasure μ']\n    (phi1 phi2 phi3 : Ω → Ω' → ℝ)\n    (hmeas1 : Measurable (Function.uncurry phi1))\n    (hmeas2 : Measurable (Function.uncurry phi2))\n    (hmeas3 : Measurable (Function.uncurry phi3))\n    (hphi3_nonneg : ∀ x x', 0 ≤ phi3 x x')\n    (hint1 : ∀ x, Integrable (fun x' => phi1 x x') μ')\n    (hint2 : ∀ x, Integrable (fun x' => phi2 x x') μ')\n    (hint3 : ∀ x, Integrable (fun x' => phi3 x x') μ')\n    (hmavg1 : Measurable (fun x => ∫ x', phi1 x x' ∂μ'))\n    (hmavg2 : Measurable (fun x => ∫ x', phi2 x x' ∂μ'))\n    (hmavg3 : Measurable (fun x => ∫ x', phi3 x x' ∂μ'))\n    (Γ γ t : ℝ) (hΓ : 1 ≤ Γ) (hγ : 0 < γ) (ht : 0 ≤ t)\n    (hyp : ∀ s : ℝ, 0 ≤ s →\n      (μ.prod μ') {p : Ω × Ω' | phi1 p.1 p.2 ≥ phi2 p.1 p.2 + Real.sqrt (phi3 p.1 p.2 * s)}\n        ≤ ENNReal.ofReal (Γ * Real.exp (-γ * s))) :\n    μ {x : Ω | (∫ x', phi1 x x' ∂μ') ≥ (∫ x', phi2 x x' ∂μ')\n        + Real.sqrt ((∫ x', phi3 x x' ∂μ') * t)}\n      ≤ ENNReal.ofReal (Γ * Real.exp 1 * Real.exp (-γ * t))"}]}
{"item_id": "ch7_theorem_7_3::ch7_theorem_7_3::A-stmt", "paper_id": "ch7_theorem_7_3", "target_name": "ch7_theorem_7_3", "source": "Graduate-level", "nl_statement": "Let \\(a_{1},\\cdots,a_{n}\\in\\mathbb{R},\\epsilon_{i},\\cdots,\\epsilon_{n}\\) be i.i.d. Rademacher random variables: \\(\\mathbb{P}(\\epsilon_{i}=1)=\\mathbb{P}(\\epsilon_{i}= -1)=0.5\\) and \\(0 < p < \\infty\\) . Then\n\\[A_{p}\\cdot\\left(\\sum_{i=1}^{n}\\left|a_{i}\\right|^{2}\\right)^{1/2}\\leq\\left(\\mathbb{E}\\left|\\sum_{i=1}^{n}a_{i}\\epsilon_{i}\\right|^{p}\\right)^{1/p}\\leq B_{p}\\cdot\\left(\\sum_{i=1}^{n}\\left|a_{i}\\right|^{2}\\right)^{1/2}\\]\nfor some constants \\(A_{p}\\) and \\(B_{p}\\) depending on p.", "context": [{"name": "theorem 7.1", "statement": "For \\(t \\geq 0\\)\n\\[\\mathbb{P}\\left(\\sum_{i=1}^{n}\\varepsilon_{i}a_{i}\\geq t\\right)\\leq\\exp\\left(-\\frac{t^{2}}{2\\sum_{i=1}^{n}a_{i}^{2}}\\right).\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ntheorem ch7_theorem_7_3 :\n    ∀ p : ℝ, 0 < p →\n      ∃ Ap Bp : ℝ, 0 < Ap ∧ 0 < Bp ∧\n        ∀ (n : ℕ) {Ω : Type} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n          (ε : Fin n → Ω → ℝ)\n          (_hindep : iIndepFun ε μ)\n          (_hmeas : ∀ i, AEMeasurable (ε i) μ)\n          (_hrad : ∀ i, μ {ω | ε i ω = 1} = 1/2 ∧ μ {ω | ε i ω = -1} = 1/2)\n          (a : Fin n → ℝ),\n          Ap * (∑ i, (a i)^2) ^ ((1:ℝ)/2)\n              ≤ (∫ ω, |∑ i, a i * ε i ω| ^ p ∂μ) ^ (1/p)\n            ∧ (∫ ω, |∑ i, a i * ε i ω| ^ p ∂μ) ^ (1/p)\n              ≤ Bp * (∑ i, (a i)^2) ^ ((1:ℝ)/2) := sorry\n\nend GraduateLevel\n", "target_names": ["ch7_theorem_7_3"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 108, "end": 675, "gold": ":\n    ∀ p : ℝ, 0 < p →\n      ∃ Ap Bp : ℝ, 0 < Ap ∧ 0 < Bp ∧\n        ∀ (n : ℕ) {Ω : Type} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n          (ε : Fin n → Ω → ℝ)\n          (_hindep : iIndepFun ε μ)\n          (_hmeas : ∀ i, AEMeasurable (ε i) μ)\n          (_hrad : ∀ i, μ {ω | ε i ω = 1} = 1/2 ∧ μ {ω | ε i ω = -1} = 1/2)\n          (a : Fin n → ℝ),\n          Ap * (∑ i, (a i)^2) ^ ((1:ℝ)/2)\n              ≤ (∫ ω, |∑ i, a i * ε i ω| ^ p ∂μ) ^ (1/p)\n            ∧ (∫ ω, |∑ i, a i * ε i ω| ^ p ∂μ) ^ (1/p)\n              ≤ Bp * (∑ i, (a i)^2) ^ ((1:ℝ)/2)"}]}
{"item_id": "ch10_lemma_10_1::ch10_lemma_10_1::A-stmt", "paper_id": "ch10_lemma_10_1", "target_name": "ch10_lemma_10_1", "source": "Graduate-level", "nl_statement": "If \\(t\\geq{\\sqrt{\\frac{2}{n}}}.\\) then\n\\[\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\mathbb{P}\\left(C\\right)\\right|\\geq t\\right)\\leq 2\\mathbb{P}\\left(\\operatorname*{sup}_{C\\in\\mathcal{C}}\\left|\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}\\in C)-\\frac{1}{n}\\sum_{i=1}^{n}I(X_{i}^{\\prime}\\in C)\\right|\\geq t/2\\right).\\]", "context": null, "nl_notes": "NOTE — The lemma as printed states no sampling model. The Lean uses the course's standard one: X_1, …, X_n have common law μ𝒳 and are independent (`hXlaw`, `hXindep`), X'_1, …, X'_n is an independent copy (`hX'law`, `hX'indep`, with X read on the first and X' on the second factor of μ.prod μ), and the sets in the class are measurable (`hC`). Read literally, with X' unconstrained (e.g. X' = X), the lemma is false.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real Filter\nopen scoped ENNReal NNReal\n\nnamespace GraduateLevel\n\nnoncomputable def empFreq {Ω 𝒳 ι : Type*} (n : ℕ) (C : ι → Set 𝒳) (X : Fin n → Ω → 𝒳) (i' : ι) (ω : Ω) : ℝ :=\n  (n : ℝ)⁻¹ * ∑ k : Fin n, (C i').indicator (fun _ => (1 : ℝ)) (X k ω)\n\nnoncomputable def supDevTruth {Ω 𝒳 ι : Type*} [MeasurableSpace 𝒳] (n : ℕ) (C : ι → Set 𝒳) (μ𝒳 : Measure 𝒳) (X : Fin n → Ω → 𝒳) (ω : Ω) : ℝ :=\n  ⨆ i' : ι, |empFreq n C X i' ω - (μ𝒳 (C i')).toReal|\n\nnoncomputable def supDevGhost {Ω 𝒳 ι : Type*} (n : ℕ) (C : ι → Set 𝒳) (X X' : Fin n → Ω → 𝒳) (ω : Ω) : ℝ :=\n  ⨆ i' : ι, |empFreq n C X i' ω - empFreq n C X' i' ω|\n\ntheorem ch10_lemma_10_1\n    {Ω 𝒳 ι : Type*} [MeasurableSpace Ω] [MeasurableSpace 𝒳]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (μ𝒳 : Measure 𝒳)\n    (n : ℕ) [NeZero n] (hn : 0 < n)\n    (C : ι → Set 𝒳) (hC : ∀ i' : ι, MeasurableSet (C i'))\n    (X X' : Fin n → Ω → 𝒳)\n    (hXmeas : ∀ k, Measurable (X k)) (hX'meas : ∀ k, Measurable (X' k))\n    (hXlaw : ∀ k, Measure.map (X k) μ = μ𝒳)\n    (hX'law : ∀ k, Measure.map (X' k) μ = μ𝒳)\n    (hXindep : iIndepFun X μ) (hX'indep : iIndepFun X' μ)\n    (hmTruth : Measurable\n      (fun p : Ω × Ω => supDevTruth n C μ𝒳 (fun (k : Fin n) (q : Ω × Ω) => X k q.1) p))\n    (hmGhost : Measurable\n      (fun p : Ω × Ω =>\n        supDevGhost n C (fun (k : Fin n) (q : Ω × Ω) => X k q.1)\n          (fun (k : Fin n) (q : Ω × Ω) => X' k q.2) p))\n    (t : ℝ) (ht : t ≥ Real.sqrt (2 / (n : ℝ))) :\n    ((μ.prod μ)\n        {p : Ω × Ω | supDevTruth n C μ𝒳 (fun (k : Fin n) (q : Ω × Ω) => X k q.1) p ≥ t}).toReal\n      ≤ 2 * ((μ.prod μ)\n        {p : Ω × Ω |\n          supDevGhost n C (fun (k : Fin n) (q : Ω × Ω) => X k q.1)\n            (fun (k : Fin n) (q : Ω × Ω) => X' k q.2) p ≥ t / 2}).toReal := sorry\n\nend GraduateLevel\n", "target_names": ["ch10_lemma_10_1"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 689, "end": 1782, "gold": "{Ω 𝒳 ι : Type*} [MeasurableSpace Ω] [MeasurableSpace 𝒳]\n    (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (μ𝒳 : Measure 𝒳)\n    (n : ℕ) [NeZero n] (hn : 0 < n)\n    (C : ι → Set 𝒳) (hC : ∀ i' : ι, MeasurableSet (C i'))\n    (X X' : Fin n → Ω → 𝒳)\n    (hXmeas : ∀ k, Measurable (X k)) (hX'meas : ∀ k, Measurable (X' k))\n    (hXlaw : ∀ k, Measure.map (X k) μ = μ𝒳)\n    (hX'law : ∀ k, Measure.map (X' k) μ = μ𝒳)\n    (hXindep : iIndepFun X μ) (hX'indep : iIndepFun X' μ)\n    (hmTruth : Measurable\n      (fun p : Ω × Ω => supDevTruth n C μ𝒳 (fun (k : Fin n) (q : Ω × Ω) => X k q.1) p))\n    (hmGhost : Measurable\n      (fun p : Ω × Ω =>\n        supDevGhost n C (fun (k : Fin n) (q : Ω × Ω) => X k q.1)\n          (fun (k : Fin n) (q : Ω × Ω) => X' k q.2) p))\n    (t : ℝ) (ht : t ≥ Real.sqrt (2 / (n : ℝ))) :\n    ((μ.prod μ)\n        {p : Ω × Ω | supDevTruth n C μ𝒳 (fun (k : Fin n) (q : Ω × Ω) => X k q.1) p ≥ t}).toReal\n      ≤ 2 * ((μ.prod μ)\n        {p : Ω × Ω |\n          supDevGhost n C (fun (k : Fin n) (q : Ω × Ω) => X k q.1)\n            (fun (k : Fin n) (q : Ω × Ω) => X' k q.2) p ≥ t / 2}).toReal"}]}
{"item_id": "ch4_theorem_4_12::ch4_theorem_4_12::A-stmt", "paper_id": "ch4_theorem_4_12", "target_name": "ch4_theorem_4_12", "source": "Graduate-level", "nl_statement": "Assume that \\(\\phi(w, z) \\in [0,1]\\) for all \\(w\\) and \\(z\\). Then given \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\), the following inequality holds:\n\\[\n\\forall w\\in\\Omega:\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\epsilon_{n}(\\delta),\n\\]\nwhere\n\\[\n\\epsilon_{n}(\\delta)=\\inf_{\\epsilon>0}\\left[2\\epsilon+3\\sqrt{\\frac{\\ln(3 N_{1}(\\epsilon,\\mathcal{G},2 n)/\\delta)}{2 n}}\\right].\n\\]\nMoreover, for any \\(\\gamma \\in (0,1)\\), with probability at least \\(1 - \\delta\\), the following inequality holds:\n\\[\n\\forall w\\in\\Omega:(1-\\gamma)^{2}\\phi(w,\\mathcal{D})\\leq\\phi(w,\\mathcal{S}_{n})+\\epsilon_{n}(\\delta),\n\\]\nwhere\n\\[\n\\epsilon_{n}(\\delta)=\\inf_{\\epsilon>0}\\left[2\\epsilon+\\frac{(5-4\\gamma)\\ln(3 N_{1}(\\epsilon,\\mathcal{G},n)/\\delta)}{2\\gamma n}\\right].\n\\]", "context": [{"name": "corollary 2.27", "statement": "Let \\(\\sigma_{i} = \\{\\pm 1\\}\\) be independent Bernoulli random variables ( each takes value \\(\\pm 1\\) with equal probability). Let \\(a_{i}\\) be fixed numbers \\((i=1,\\dots,n)\\) Then for all \\(\\epsilon >0\\) ..\n\\[\n\\Pr\\left(n^{-1}\\sum_{i=1}^{n}\\sigma_{i}a_{i}\\geq\\epsilon\\right)\\leq\\exp\\left[-\\frac{n\\epsilon^{2}}{2 n^{-1}\\sum_{i=1}^{n}a_{i}^{2}}\\right].\n\\]"}, {"name": "Lemma 4.8", "statement": "(Symmetrization) Consider a real valued function family \\(F = \\{f : Z \\rightarrow \\mathbb{R}\\}\\). Assume there exists a function \\(\\psi : F \\times Z^{n} \\rightarrow \\mathbb{R}\\) and \\(\\epsilon_{n} : (0,1) \\rightarrow \\mathbb{R}\\) so that with probability at least \\(1-\\delta\\):\n\\[\n\\forall f \\in F, \\qquad f(\\sigma, S_{n}) \\leq \\psi(f, S_{n}) + \\epsilon_{n}(\\delta),\n\\]\nwhere the randomness is over both \\(S_{n} \\sim D^{n}\\) and \\(\\sigma\\). If there exists \\(\\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime})\\) so that the following superadditive inequality holds for all \\((S_{n}, S_{n}^{\\prime})\\)\n\\[\n\\psi(f, S_{n}) + \\psi(f, S_{n}^{\\prime}) \\leq \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}),\n\\]\nthen with probability at least \\(1-\\delta\\) over independent random data \\((S_{n}, S_{n}^{\\prime}) \\sim D^{2n}\\):\n\\[\n\\forall f \\in F, \\qquad f(S_{n}^{\\prime}) \\leq f(S_{n}) + \\tilde{\\psi}(f, S_{n} \\cup S_{n}^{\\prime}) + 2\\epsilon_{n}(\\delta/2).\n\\]"}, {"name": "lemma 4.11", "statement": "Let \\(\\psi_{\\mathrm{trn}}: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R}\\), \\(\\psi_{\\mathrm{val}}: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R}\\), \\(\\psi_{\\mathrm{tst}}: \\mathcal{F} \\times \\mathcal{D} \\to \\mathbb{R}\\) be appropriately training (where \\(\\mathcal{D}\\) denotes probability distributions on \\(\\mathcal{Z}\\)), validation, and test statistics. Assume that for any \\(\\delta_1 \\in (0,1)\\), the following uniform convergence result holds. With probability at least \\(1 - \\delta_1\\) over randomly drawn training and validation sets \\((S_n, S_n') \\sim \\mathcal{D}^{2n}\\):\n\\[\n\\forall f\\in\\mathcal{F}:\\psi_{\\mathrm{val}}(f,\\mathcal{S}_{n}^{\\prime})\\leq\\psi_{\\mathrm{trn}}(f,\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1}).\n\\]\nMoreover, assume \\(\\forall f\\in \\mathcal{F}\\), we have with probability \\(1 - \\delta_{2}\\) over randomly drawn \\(\\mathcal{S}_n^\\prime \\sim \\mathcal{D}\\):\n\\[\n\\psi_{\\mathrm{tst}}(f,\\mathcal{D})\\leq\\psi_{\\mathrm{val}}(f,\\mathcal{S}_{n}^{\\prime})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]\nThen the following uniform convergence statement holds. With probability at least \\(1 - \\delta_1 - \\delta_2\\),\n\\[\n\\forall f\\in\\mathcal{F}:\\psi_{\\mathrm{tst}}(f,\\mathcal{D})\\leq\\psi_{\\mathrm{trn}}(f,\\mathcal{S}_{n})+\\epsilon_{n}^{1}(\\delta_{1})+\\epsilon_{n}^{2}(\\delta_{2}).\n\\]"}, {"name": "Union Bound", "statement": "(Union Bound) Consider \\(m\\) events \\(E_1, \\ldots, E_m\\). The following probability inequality holds:\n\\[\n\\Pr (E_1 \\cup \\cdots \\cup E_m) \\leq \\sum_{j=1}^{m} \\Pr(E_j).\n\\]\n\nThe union bound has an alternative expression, which is often used in the learning theory analysis. If each \\(E_j\\) occurs with probability at least \\(1-\\delta_j\\) for \\(j=1, \\ldots, m\\), then with probability at least \\(1-\\sum_{j=1}^{m} \\delta_j\\), all of events \\(\\{E_j\\}\\) occur simultaneously for \\(j=1, \\ldots, m\\)."}, {"name": "definition 4.1", "statement": "Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]"}, {"name": "definition 3.8", "statement": "We define\n\\[\n\\phi(w,z)=L\\left(f(w,x),y\\right)-L_{*}(x,y),\\tag{3.4}\n\\]\nfor \\( w \\in \\Omega \\) and \\( z = (x, y) \\in \\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y} \\), and a pre-chosen \\( L_{*}(x, y) \\) of \\( z = (x, y) \\) that does not depend on \\( w \\).\nFor training data \\( S_{n}=\\{Z_{i}=(X_{i},Y_{i}):i=1,\\ldots,n\\} \\), we define the training loss for \\( w\\in \\Omega \\) as\n\\[\n\\phi(w,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i}).\\tag{3.5}\n\\]\nMoreover, for a distribution \\(\\mathcal{D}\\) on \\(\\mathcal{Z}\\), we define the test loss for \\(w \\in \\Omega\\) as:\n\\[\n\\phi(w,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(w,Z).\\tag{3.6}\n\\]\nSince \\( L_{*}(x,y) \\) does not depend on \\( w \\), the ERM solution with respect to the loss \\( L(\\cdot) \\) is equivalent to the ERM solution with respect to \\( \\phi (w,\\cdot) \\). Therefore with the simplified notations, the approximate ERM method (3.3) is a special case of the following method:\n\\[\n\\phi(\\hat{w},\\mathcal{S}_{n})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{S}_{n})+\\epsilon^{\\prime}.\\tag{3.7}\n\\]\nIn general, we may simply take \\( L_{*}(x,y) = 0 \\) in (3.4). However, for some applications, we may choose a non-zero \\( L_{*}(x,y) \\) so that\n\\[\nL(f(w,x),y)-L_{*}(x,y)\n\\]\nhas a small variance. For least squares loss, this can be achieved with \\( L_{*}(x,y) = L(f_{*}(x),y) \\) where \\( f_{*}(x) \\) is the optimal prediction function that minimizes the test loss as shown in Example 3.9. The smaller variance, combined with Bernstein's inequality, implies better generalization bound (see Section 3.6 for more details)."}, {"name": "Empirical and Uniform Covering Number", "statement": "Given an empirical distribution \\( S_{n}=\\{Z_{1},\\ldots,Z_{n}\\} \\), we define the pseudometric \\( d = L_{p}(S_{n}) \\) as\n\\[\nd(\\phi,\\phi^{\\prime})=\\left[\\frac{1}{n}\\sum_{i=1}^{n}|\\phi(Z_{i})-\\phi^{\\prime}(Z_{i})|^{p}\\right]^{1/p}.\n\\]\nThe corresponding metric covering number \\(N(\\epsilon, \\mathcal{G}, L_p(\\mathcal{S}_n))\\) is referred to as the\nempirical \\( L_{p} \\) covering number. Given \\( n \\), the largest \\( L_{p} \\) covering number over empirical distribution \\( S_{n} \\) is referred to as the uniform \\( L_{p} \\) covering number\n\\[\nN_{p}(\\epsilon,\\mathcal{G},n)=\\sup_{\\mathcal{S}_{n}}N(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{S}_{n})).\n\\]"}, {"name": "Bracketing Number", "statement": "Let \\(\\mathcal{G} = \\{\\phi(w, \\cdot) : w \\in \\Omega\\}\\) be a real-valued function class, equipped with a pseudometric \\(d\\). We say\n\\[\n\\mathcal{G}(\\epsilon)=\\left\\{\\left[\\phi_{1}^{L}(z),\\phi_{1}^{U}(z)\\right],\\dots,\\left[\\phi_{N}^{L}(z),\\phi_{N}^{U}(z)\\right]\\right\\}\n\\]\nis an \\(\\epsilon\\)-bracket of \\(\\mathcal{G}\\) under metric \\(d\\) if for all \\(w \\in \\Omega\\), there exists \\(j = j(w)\\) such that \\(\\forall z\\):\n\\[\n\\phi_{j}^{L}(z)\\leq\\phi(w,z)\\leq\\phi_{j}^{U}(z),\\quad d(\\phi_{j}^{L},\\phi_{j}^{U})\\leq\\epsilon .\n\\]\nThe \\(\\epsilon\\)-bracketing number is the smallest cardinality \\(N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The quantity \\(\\ln N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) is called \\(\\epsilon\\) bracketing entropy.\nIn particular, given a distribution \\(\\mathcal{D}\\) and \\(p \\geq 1\\), we define \\(L_{p}\\)-seminorm in function space as\n\\[\n\\left\\|f-f^{\\prime}\\right\\|_{L_{p}(\\mathcal{D})}=\\left[\\mathbb{E}_{Z\\sim\\mathcal{D}}|f(Z)-f^{\\prime}(Z)|^{p}\\right]^{1/p}.\\tag{3.19}\n\\]\nIt induces a pseudometric, denoted as \\( d = L_{p}(\\mathcal{D}) \\), and the corresponding bracketing number is \\( N_{\\parallel}(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D})) \\).\nFor \\(p = \\infty\\), the \\(L_{\\infty}(\\mathcal{D})\\)-seminorm is defined as the essential supremum seminorm, which leads to the pseudometric\n\\[\nd(f,f^{\\prime})=\\inf\\left\\{\\omega:\\Pr_{\\mathcal{D}}[|f(Z)-f^{\\prime}(Z)|\\leq\\omega]=1\\right\\}.\n\\]\nHowever, when \\( p = \\infty \\), instead of using the \\( L_{\\infty} \\) bracketing number, it is more conventional to use the equivalent notion of \\( L_{\\infty} \\) covering number (see Proposition 4.3)."}], "nl_notes": "NOTE — The Lean assumes the parameter set is countable (`hW_ctble`), which the NL statement does not state. Without it the statement is false: on the first uncountable ordinal with the countable/co-countable σ-algebra and φ(w, z) = 0 if z ≤ w else 1, every w has test loss 1 while any finite sample admits a w with empirical loss 0. This is the usual countability / permissibility convention for uniform bounds over a function class.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def empLoss {𝒵 : Type*} {Ωp : Type*} {n : ℕ}\n    (φ : Ωp → 𝒵 → ℝ) (Z : Fin n → 𝒵) (w : Ωp) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i : Fin n, φ w (Z i)\n\nnoncomputable def testLoss {𝒵 : Type*} [MeasurableSpace 𝒵] {Ωp : Type*}\n    (φ : Ωp → 𝒵 → ℝ) (D : Measure 𝒵) (w : Ωp) : ℝ :=\n  ∫ z, φ w z ∂D\n\ndef IsUnifL1Cover {𝒵 : Type*} [MeasurableSpace 𝒵] {Ωp : Type*}\n    (D : Measure 𝒵)\n    (C : Finset (𝒵 → ℝ)) (φ : Ωp → 𝒵 → ℝ) (𝒲 : Set Ωp) (ε : ℝ) {m : ℕ}\n    (S : Fin m → 𝒵) : Prop :=\n  (∀ g ∈ C, (∀ z, g z ∈ Set.Icc (0 : ℝ) 1) ∧ AEMeasurable g D) ∧\n  (∀ w ∈ 𝒲, ∃ g ∈ C,\n    (1 / (m : ℝ)) * ∑ j : Fin m, |φ w (S j) - g (S j)| ≤ ε)\n\ndef IsUnifL1CoveringNumber {𝒵 : Type*} [MeasurableSpace 𝒵] {Ωp : Type*}\n    (D : Measure 𝒵)\n    (N : ℕ) (φ : Ωp → 𝒵 → ℝ) (𝒲 : Set Ωp) (ε : ℝ) (m : ℕ) : Prop :=\n  (∀ S : Fin m → 𝒵, ∃ C : Finset (𝒵 → ℝ),\n      C.card ≤ N ∧ IsUnifL1Cover D C φ 𝒲 ε S) ∧\n  (∀ M : ℕ,\n    (∀ S : Fin m → 𝒵, ∃ C : Finset (𝒵 → ℝ),\n        C.card ≤ M ∧ IsUnifL1Cover D C φ 𝒲 ε S) →\n      N ≤ M)\n\ntheorem ch4_theorem_4_12\n    {𝒵 : Type*} [MeasurableSpace 𝒵]\n    {Ωp : Type*}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (D : Measure 𝒵) [IsProbabilityMeasure D]\n    (𝒲 : Set Ωp) (hW : 𝒲.Nonempty) (hW_ctble : 𝒲.Countable)\n    (φ : Ωp → 𝒵 → ℝ)\n    (hφ_bdd : ∀ w ∈ 𝒲, ∀ z, φ w z ∈ Set.Icc (0 : ℝ) 1)\n    (hφ_meas : ∀ w ∈ 𝒲, AEMeasurable (fun z => φ w z) D)\n    (n : ℕ) (hn : 0 < n)\n    (Z : Ω → Fin n → 𝒵)\n    (hZ_meas : ∀ i, AEMeasurable (fun ω => Z ω i) μ)\n    (hZ_iid : (iIndepFun (fun i ω => Z ω i) μ) ∧\n      (∀ i, Measure.map (fun ω => Z ω i) μ = D))\n    (N₁ : ℝ → ℕ → ℕ)\n    (hN₁ : ∀ ε > (0 : ℝ),\n      IsUnifL1CoveringNumber D (N₁ ε n) φ 𝒲 ε n ∧\n      IsUnifL1CoveringNumber D (N₁ ε (2 * n)) φ 𝒲 ε (2 * n))\n    (δ : ℝ) (hδ : δ ∈ Set.Ioo (0 : ℝ) 1) :\n    (1 - δ ≤ μ.real {ω |\n        ∀ w ∈ 𝒲,\n          testLoss φ D w ≤ empLoss φ (Z ω) w\n            + ⨅ ε : {e : ℝ // 0 < e},\n                (2 * (ε : ℝ)\n                  + 3 * Real.sqrt\n                      (Real.log (3 * (N₁ (ε : ℝ) (2 * n) : ℝ) / δ) / (2 * n)))})\n    ∧\n    (∀ γ ∈ Set.Ioo (0 : ℝ) 1,\n      1 - δ ≤ μ.real {ω |\n        ∀ w ∈ 𝒲,\n          (1 - γ) ^ 2 * testLoss φ D w ≤ empLoss φ (Z ω) w\n            + ⨅ ε : {e : ℝ // 0 < e},\n                (2 * (ε : ℝ)\n                  + (5 - 4 * γ) * Real.log (3 * (N₁ (ε : ℝ) n : ℝ) / δ)\n                      / (2 * γ * n))}) := sorry\n\nend GraduateLevel\n", "target_names": ["ch4_theorem_4_12"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1110, "end": 2459, "gold": "{𝒵 : Type*} [MeasurableSpace 𝒵]\n    {Ωp : Type*}\n    {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]\n    (D : Measure 𝒵) [IsProbabilityMeasure D]\n    (𝒲 : Set Ωp) (hW : 𝒲.Nonempty) (hW_ctble : 𝒲.Countable)\n    (φ : Ωp → 𝒵 → ℝ)\n    (hφ_bdd : ∀ w ∈ 𝒲, ∀ z, φ w z ∈ Set.Icc (0 : ℝ) 1)\n    (hφ_meas : ∀ w ∈ 𝒲, AEMeasurable (fun z => φ w z) D)\n    (n : ℕ) (hn : 0 < n)\n    (Z : Ω → Fin n → 𝒵)\n    (hZ_meas : ∀ i, AEMeasurable (fun ω => Z ω i) μ)\n    (hZ_iid : (iIndepFun (fun i ω => Z ω i) μ) ∧\n      (∀ i, Measure.map (fun ω => Z ω i) μ = D))\n    (N₁ : ℝ → ℕ → ℕ)\n    (hN₁ : ∀ ε > (0 : ℝ),\n      IsUnifL1CoveringNumber D (N₁ ε n) φ 𝒲 ε n ∧\n      IsUnifL1CoveringNumber D (N₁ ε (2 * n)) φ 𝒲 ε (2 * n))\n    (δ : ℝ) (hδ : δ ∈ Set.Ioo (0 : ℝ) 1) :\n    (1 - δ ≤ μ.real {ω |\n        ∀ w ∈ 𝒲,\n          testLoss φ D w ≤ empLoss φ (Z ω) w\n            + ⨅ ε : {e : ℝ // 0 < e},\n                (2 * (ε : ℝ)\n                  + 3 * Real.sqrt\n                      (Real.log (3 * (N₁ (ε : ℝ) (2 * n) : ℝ) / δ) / (2 * n)))})\n    ∧\n    (∀ γ ∈ Set.Ioo (0 : ℝ) 1,\n      1 - δ ≤ μ.real {ω |\n        ∀ w ∈ 𝒲,\n          (1 - γ) ^ 2 * testLoss φ D w ≤ empLoss φ (Z ω) w\n            + ⨅ ε : {e : ℝ // 0 < e},\n                (2 * (ε : ℝ)\n                  + (5 - 4 * γ) * Real.log (3 * (N₁ (ε : ℝ) n : ℝ) / δ)\n                      / (2 * γ * n))})"}]}
{"item_id": "ch16_theorem_16_3::ch16_theorem_16_3::A-stmt", "paper_id": "ch16_theorem_16_3", "target_name": "ch16_theorem_16_3", "source": "Graduate-level", "nl_statement": "Assume that rewards \\(r_t(a) \\in [0,1]\\). Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). With a choice of\n\\[\n\\hat{c}_{t}(a)=\\sqrt{\\frac{\\ln(2(\\hat{n}_{t}(a)+1)^{2}/\\delta)}{2\\hat{n}_{t}(a)}},\n\\]\nwe have with probability at least \\(1 - \\delta\\):\n\\[\n\\mathrm{REG}_{T}\\leq(K-1)+\\sqrt{8\\ln(2 K T^{2}/\\delta)(T-K)K}.\n\\]", "context": [{"name": "Martingale Sub-Gaussian Inequality", "statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) Assume each \\(\\xi_{i}\\) is sub-Gaussian with respect to \\(Z_{i}^{(y)}\\):\n\\[\n\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\lambda\\xi_{i}}\\leq\\lambda\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}+\\frac{\\lambda^{2}\\sigma_{i}^{2}}{2}\n\\]\nfor some \\(\\sigma_{i}\\) that may depend on \\(S_{i - 1}\\) and \\(Z_{i}^{(x)}\\). Then for all \\(\\sigma >0\\), with probability at least \\(1 - \\delta\\)\n\\[\n\\forall n>0:\\sum_{i=1}^{n}\\mathbb{E}_{Z_{i}^{(y)}}\\xi_{i}<\\sum_{i=1}^{n}\\xi_{i}+\\left(\\sigma+\\frac{\\sum_{i=1}^{n}\\sigma_{i}^{2}}{\\sigma}\\right)\\sqrt{\\frac{\\ln(1/\\delta)}{2}}.\n\\]"}, {"name": "lemma 16.2", "statement": "Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). Let\n\\[\n\\delta_{1}=\\Pr [\\exists t>K:\\mu(a_{*})>\\hat{\\mu}_{t-1}(a_{*})+\\hat{c}_{t-1}(a_{*})]\n\\]\nbe the probability that the upper confident bound fails on \\( a_* \\). Let\n\\[\n\\delta_{2}=\\Pr\\left[\\exists t>K\\&a\\in\\{1,\\dots,K\\}\\backslash\\left\\{a_{*}\\right\\}:\\mu(a)<\\hat{\\mu}_{t-1}(a)-\\hat{c}_{t-1}^{\\prime}(a)\\right]\n\\]\nbe the probability that the lower confident bound fails on \\( a \\neq a_* \\). Then for Algorithm 16.1, we have with probability at least \\( 1 - \\delta_1 - \\delta_2 \\):\n\\[\n\\operatorname{REG}_{T}\\leq\\sum_{a=1}^{K}[\\mu(a_{*})-\\mu(a)]+\\sum_{t=K+1}^{T}[\\hat{c}_{t-1}(a_{t})+\\hat{c}_{t-1}^{\\prime}(a_{t})]\\mathbb{1}(a_{t}\\neq a_{*}).\n\\]"}, {"name": "definition 16.1", "statement": "In the multi-armed bandit (or MAB) problem, we consider \\(K\\) arms. The environment generates a sequence of reward vectors for time steps \\(t \\geq 1\\) as \\(r_t = [r_t(1), \\ldots, r_t(K)]\\). Each \\(r_t(a)\\) is associated with an arm \\(a \\in \\{1, \\ldots, K\\}\\). In the literature, an arm \\(a\\) is also referred to as an action. At each time step \\(t = 1, 2, \\ldots, T\\),\n- The player pulls one of the arms \\( a_{t} \\in \\{1, \\dots, K\\} \\).\n- The environment returns the reward \\( r_t(a_t) \\), but does not reveal information on any other arm \\( a \\neq a_t \\).\nAt each time \\( t \\), a (randomized) bandit algorithm takes the historic observations observed so far, and maps it to a distribution \\( \\hat{\\pi}_{t-1} \\) over actions \\( a \\in \\{1, \\ldots, K\\} \\). We then draw a random action \\( a_t \\) (arm) from \\( \\hat{\\pi}_{t-1} \\). Here \\( \\hat{\\pi}_{t-1} \\) is referred to as policy.\nIn the adversarial bandit problem, we are given an arbitrary reward sequence \\(\\{[r_1(1),\\ldots,r_t(K)]:t\\geq 1\\}\\) before hand. The adversary in this setting is called oblivious, who decides the rewards of all the rounds before the game starts (but may take advantage of the bandit algorithm's weakness). For this reward se\nquence, the expected cumulative reward of a randomized bandit algorithm is\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t}),\n\\]\nwhere the randomization is over the internal randomization of the bandit algorithm. The expected regret is defined as\n\\[\n\\operatorname{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\mathbb{E}\\sum_{t=1}^{T}r_{t}\\left(a_{t}\\right).\\tag{16.1}\n\\]\nIf the bandit algorithm is deterministic: i.e. pulling a single arm \\(a_{t}\\) only at any time step, then the regret is\n\\[\n\\mathrm{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\sum_{t=1}^{T}r_{t}(a_{t}).\n\\]\nIn the adversarial bandit problem, our goal is to minimize the expected regret \\(\\mathrm{REG}_T\\) (and the expectation is with respect to the internal randomization of the algorithm instead of the data).\nAnother form of bandit problem, called stochastic bandit, assumes that the reward \\( r_t(a) \\) is drawn independently from a distribution \\( \\mathcal{D}_a \\), with mean \\( \\mu(a) = \\mathbb{E}_{r_t(a) \\sim \\mathcal{D}_a} [r_t(a)] \\). In this setting, the goal of bandit is to find \\( a \\) that maximizes the expected reward \\( \\mu(a) \\). We are interested in the expected regret defined as:\n\\[\n\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu\\left(a_{t}\\right).\\tag{16.2}\n\\]\nAgain, this regret is defined for each realization of the stochastic rewards, and the expectation is with respect to the randomness of the algorithm. If algorithm is deterministic and outputs \\( a_{t} \\) at each time \\( t \\), then\n\\[\n\\mathrm{REG}_{T}=T\\max_{a}\\mu(a)-\\sum_{t=1}^{T}\\mu(a_{t}).\n\\]\nWe can further include randomization over data into the regret of stochastic bandit, by considering expected regret as follows:\n\\[\n\\mathbb{E}\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu(a_{t}),\n\\]\nwhere the expectation is with respect to both the data and the internal randomization of the learning algorithm."}, {"name": "UCB Algorithm", "statement": "Algorithm 16.1: UCB Algorithm\nInput: \\(K\\) and \\(T\\geq K\\)   \n1 for \\(a=1,\\dots,K\\) do   \n2 Let \\(\\hat{n}_0(a) = 0\\)   \n3 Let \\(\\hat{\\mu}_0(a) = 0\\)   \n4 for \\(t=1,2,\\ldots,T\\) do   \n5 if \\(t\\leq K\\) then   \n6 Let \\(a_{t} = t\\)   \n7 else   \n8 Let \\(a_{t}\\in \\arg \\max_{a}[\\hat{\\mu}_{t - 1}(a) + \\hat{c}_{t - 1}(a)]\\) according to (16.3)   \n9 Pull arm \\(a_{t}\\) and observe reward \\(r_t(a_t)\\)"}, {"name": "equation 16.3", "statement": "For each arm \\(a\\), define the number of pulls up to time \\(t\\) and the corresponding empirical mean by\n\\[\n\\hat n_t(a)=\\sum_{s=1}^{t}\\mathbb{1}(a_s=a), \\qquad\n\\hat\\mu_t(a)=\\frac{1}{\\hat n_t(a)}\\sum_{s=1}^{t} r_s(a_s)\\mathbb{1}(a_s=a),\n\\]\nwith the initialization \\(\\hat n_0(a)=0\\) and \\(\\hat\\mu_0(a)=0\\). In the UCB algorithm, after the initialization rounds, the arm is selected according to\n\\[\na_t\\in\\arg\\max_a\\left[\\hat\\mu_{t-1}(a)+\\hat c_{t-1}(a)\\right]. \\tag{16.3}\n\\]\nHere \\(\\hat c_t(a)\\) denotes the confidence width used by the algorithm."}], "nl_notes": "NOTE — Do not use `ucbAlgorithmNL`. Model the run of Algorithm 16.1 by an arbitrary measurable, deterministic arm-selection rule that plays the arms in round-robin order during the warm-up and afterwards picks some maximiser of μ̂ + ĉ, with ties broken arbitrarily.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real Finset\nopen scoped ENNReal NNReal\n\nnamespace GraduateLevel\n\nlemma measurable_encode {α : Type*} {_ : MeasurableSpace α} [Encodable α]\n    [MeasurableSingletonClass α] :\n    Measurable (Encodable.encode (α := α)) := by\n  refine measurable_to_nat fun a ↦ ?_\n  have : Encodable.encode ⁻¹' {Encodable.encode a} = {a} := by ext; simp\n  rw [this]\n  exact measurableSet_singleton _\n\nlemma measurableEmbedding_encode (α : Type*) {_ : MeasurableSpace α} [Encodable α]\n    [MeasurableSingletonClass α] :\n    MeasurableEmbedding (Encodable.encode (α := α)) where\n  injective := Encodable.encode_injective\n  measurable := measurable_encode\n  measurableSet_image' _ _ := .of_discrete\n\nsection Finite\n\nvariable {𝓧 𝓨 α : Type*} {m𝓧 : MeasurableSpace 𝓧} {m𝓨 : MeasurableSpace 𝓨}\n  {mα : MeasurableSpace α} [TopologicalSpace α] [LinearOrder α]\n  [OpensMeasurableSpace α] [OrderClosedTopology α] [SecondCountableTopology α]\n\nlemma measurableSet_isMax [Countable 𝓨]\n    {f : 𝓧 → 𝓨 → α} (hf : ∀ y, Measurable (fun x ↦ f x y)) (y : 𝓨) :\n    MeasurableSet {x | ∀ z, f x z ≤ f x y} := by\n  rw [show {x | ∀ y', f x y' ≤ f x y} = ⋂ y', {x | f x y' ≤ f x y} by ext; simp]\n  exact MeasurableSet.iInter fun z ↦ measurableSet_le (by fun_prop) (by fun_prop)\n\nlemma exists_isMaxOn' {α : Type*} [LinearOrder α]\n    [Nonempty 𝓨] [Finite 𝓨] [Encodable 𝓨] (f : 𝓧 → 𝓨 → α) (x : 𝓧) :\n    ∃ n : ℕ, ∃ y, n = Encodable.encode y ∧ ∀ z, f x z ≤ f x y := by\n  obtain ⟨y, h⟩ := Finite.exists_max (f x)\n  exact ⟨Encodable.encode y, y, rfl, h⟩\n\nnoncomputable\ndef measurableArgmax [Nonempty 𝓨] [Finite 𝓨] [Encodable 𝓨] [MeasurableSingletonClass 𝓨]\n    (f : 𝓧 → 𝓨 → α)\n    [∀ x, DecidablePred fun n ↦ ∃ y, n = Encodable.encode y ∧ ∀ (z : 𝓨), f x z ≤ f x y]\n    (x : 𝓧) :\n    𝓨 :=\n  (measurableEmbedding_encode 𝓨).invFun (Nat.find (exists_isMaxOn' f x))\n\nlemma measurable_measurableArgmax [Nonempty 𝓨] [Finite 𝓨] [Encodable 𝓨]\n    [MeasurableSingletonClass 𝓨]\n    {f : 𝓧 → 𝓨 → α}\n    [∀ x, DecidablePred fun n ↦ ∃ y, n = Encodable.encode y ∧ ∀ (z : 𝓨), f x z ≤ f x y]\n    (hf : ∀ y, Measurable (fun x ↦ f x y)) :\n    Measurable (measurableArgmax f) := by\n  refine (MeasurableEmbedding.measurable_invFun (measurableEmbedding_encode 𝓨)).comp ?_\n  refine measurable_find _ fun n ↦ ?_\n  have : {x | ∃ y, n = Encodable.encode y ∧ ∀ (z : 𝓨), f x z ≤ f x y}\n      = ⋃ y, ({x | n = Encodable.encode y} ∩ {x | ∀ z, f x z ≤ f x y}) := by ext; simp\n  rw [this]\n  refine MeasurableSet.iUnion fun y ↦ (MeasurableSet.inter (by simp) ?_)\n  exact measurableSet_isMax (by fun_prop) y\n\nend Finite\n\nstructure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where\n\n  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐\n  [h_policy : ∀ n, IsMarkovKernel (policy n)]\n\n  p0 : Measure 𝓐\n  [hp0 : IsProbabilityMeasure p0]\n\nattribute [instance] Algorithm.h_policy Algorithm.hp0\n\nstructure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where\n\n  feedback : (n : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨\n  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]\n\n  ν0 : Kernel 𝓐 𝓨\n  [hp0 : IsMarkovKernel ν0]\n\nattribute [instance] Environment.h_feedback Environment.hp0\n\nsection Traj\n\nvariable {𝓐 𝓨 : Type*} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]\n\nnoncomputable\ndef stepKernel (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :\n    Kernel (Iic n → 𝓐 × 𝓨) (𝓐 × 𝓨) :=\n  alg.policy n ⊗ₖ env.feedback n\n\ninstance (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :\n    IsMarkovKernel (stepKernel alg env n) := by\n  rw [stepKernel]; infer_instance\n\nnoncomputable\ndef trajMeasure (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :\n    Measure (ℕ → 𝓐 × 𝓨) :=\n  Kernel.trajMeasure (alg.p0 ⊗ₘ env.ν0) (stepKernel alg env)\n\ndef action (n : ℕ) (h : ℕ → 𝓐 × 𝓨) : 𝓐 := (h n).1\n\nend Traj\n\nnoncomputable\ndef regret {𝓐 Ω : Type*} [MeasurableSpace 𝓐] (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) :\n    ℝ :=\n  (t : ℝ) * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]\n\nsection History\n\nvariable {𝓐 R : Type*} [DecidableEq 𝓐] {n : ℕ}\n\nnoncomputable def pullCount' (n : ℕ) (h : Iic n → 𝓐 × R) (a : 𝓐) : ℕ :=\n  #(univ.filter (fun s ↦ (h s).1 = a))\n\nlemma pullCount'_eq_sum (n : ℕ) (h : Iic n → 𝓐 × R) (a : 𝓐) :\n    pullCount' n h a = ∑ s : Iic n, if (h s).1 = a then 1 else 0 := by\n  rw [pullCount', Finset.card_filter]\n\nnoncomputable def sumRewards' (n : ℕ) (h : Iic n → 𝓐 × ℝ) (a : 𝓐) : ℝ :=\n  ∑ s, if (h s).1 = a then (h s).2 else 0\n\nnoncomputable def empMean' (n : ℕ) (h : Iic n → 𝓐 × ℝ) (a : 𝓐) : ℝ :=\n  sumRewards' n h a / pullCount' n h a\n\nvariable [MeasurableSpace 𝓐] [MeasurableSingletonClass 𝓐] [MeasurableSpace R]\n\n@[fun_prop]\nlemma measurable_pullCount' (n : ℕ) (a : 𝓐) :\n    Measurable (fun h : Iic n → 𝓐 × R ↦ pullCount' n h a) := by\n  simp_rw [pullCount'_eq_sum]\n  have h_meas s : Measurable (fun (h : Iic n → 𝓐 × R) ↦ if (h s).1 = a then 1 else 0) := by\n    refine Measurable.ite ?_ (by fun_prop) (by fun_prop)\n    exact (measurableSet_singleton _).preimage (by fun_prop)\n  fun_prop\n\n@[fun_prop]\nlemma measurable_sumRewards' (n : ℕ) (a : 𝓐) :\n    Measurable (fun h : Iic n → 𝓐 × ℝ ↦ sumRewards' n h a) := by\n  simp_rw [sumRewards']\n  have h_meas s : Measurable (fun (h : Iic n → 𝓐 × ℝ) ↦ if (h s).1 = a then (h s).2 else 0) := by\n    refine Measurable.ite ?_ (by fun_prop) (by fun_prop)\n    exact (measurableSet_singleton _).preimage (by fun_prop)\n  fun_prop\n\n@[fun_prop]\nlemma measurable_empMean' (n : ℕ) (a : 𝓐) :\n    Measurable (fun h : Iic n → 𝓐 × ℝ ↦ empMean' n h a) := by\n  unfold empMean'; fun_prop\n\nend History\n\ndef RoundRobin.nextAction {K : ℕ} (hK : 0 < K) (n : ℕ) : Fin K := ⟨(n + 1) % K, Nat.mod_lt _ hK⟩\n\nsection Interaction\n\nvariable {𝓐 𝓨 : Type*} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]\n\nnoncomputable def detAlgorithm (nextA : (n : ℕ) → (Iic n → 𝓐 × 𝓨) → 𝓐)\n    (h_next : ∀ n, Measurable (nextA n)) (action0 : 𝓐) : Algorithm 𝓐 𝓨 where\n  policy n := Kernel.deterministic (nextA n) (h_next n)\n  p0 := Measure.dirac action0\n\nnoncomputable def stationaryEnv (ν : Kernel 𝓐 𝓨) [IsMarkovKernel ν] : Environment 𝓐 𝓨 where\n  feedback _ := ν.comap Prod.snd measurable_snd\n  ν0 := ν\n\nend Interaction\n\nnoncomputable def ucbWidthNL {K : ℕ} (δ : ℝ) (n : ℕ) (h : Iic n → Fin K × ℝ) (a : Fin K) : ℝ :=\n  Real.sqrt (Real.log (2 * ((pullCount' n h a : ℝ) + 1) ^ 2 / δ) / (2 * pullCount' n h a))\n\n@[fun_prop]\nlemma measurable_ucbWidthNL {K : ℕ} (δ : ℝ) (n : ℕ) (a : Fin K) :\n    Measurable (fun h : Iic n → Fin K × ℝ ↦ ucbWidthNL δ n h a) := by\n  unfold ucbWidthNL; fun_prop\n\nopen Classical in\n\nnoncomputable def nextArmNL {K : ℕ} (hK : 0 < K) (δ : ℝ) (n : ℕ) (h : Iic n → Fin K × ℝ) : Fin K :=\n  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK\n  if n < K - 1 then RoundRobin.nextAction hK n else\n  measurableArgmax (fun h a ↦ empMean' n h a + ucbWidthNL δ n h a) h\n\nopen Classical in\n@[fun_prop]\nlemma measurable_nextArmNL {K : ℕ} (hK : 0 < K) (δ : ℝ) (n : ℕ) :\n    Measurable (nextArmNL hK δ n) := by\n  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK\n  unfold nextArmNL\n  split_ifs with hc\n  · exact measurable_const\n  · refine measurable_measurableArgmax fun a ↦ ?_\n    fun_prop\n\nnoncomputable def ucbAlgorithmNL {K : ℕ} (hK : 0 < K) (δ : ℝ) : Algorithm (Fin K) ℝ :=\n  detAlgorithm (nextArmNL hK δ) (measurable_nextArmNL hK δ) ⟨0, hK⟩\n\ntheorem ch16_theorem_16_3 {K T : ℕ} (hK : 0 < K) (hKT : K ≤ T)\n    {δ : ℝ} (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (ν : Kernel (Fin K) ℝ) [IsMarkovKernel ν]\n    (hrew : ∀ a, (ν a) (Set.Icc 0 1) = 1)\n    (nextA : (n : ℕ) → (Iic n → Fin K × ℝ) → Fin K)\n    (hmeasA : ∀ n, Measurable (nextA n))\n    (hwarm : ∀ n h, n < K - 1 → nextA n h = RoundRobin.nextAction hK n)\n    (hargmax : ∀ n h a, K - 1 ≤ n →\n      empMean' n h a + ucbWidthNL δ n h a ≤\n        empMean' n h (nextA n h) + ucbWidthNL δ n h (nextA n h)) :\n    (trajMeasure (detAlgorithm nextA hmeasA ⟨0, hK⟩) (stationaryEnv ν))\n        {ω | regret ν (action (𝓐 := Fin K) (𝓨 := ℝ)) T ω ≤\n          ((K:ℝ) - 1) + Real.sqrt (8 * Real.log (2 * K * T ^ 2 / δ) * ((T:ℝ) - K) * K)}\n      ≥ ENNReal.ofReal (1 - δ) := sorry\n\nend GraduateLevel\n", "target_names": ["ch16_theorem_16_3"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 7277, "end": 8003, "gold": "{K T : ℕ} (hK : 0 < K) (hKT : K ≤ T)\n    {δ : ℝ} (hδ0 : 0 < δ) (hδ1 : δ < 1)\n    (ν : Kernel (Fin K) ℝ) [IsMarkovKernel ν]\n    (hrew : ∀ a, (ν a) (Set.Icc 0 1) = 1)\n    (nextA : (n : ℕ) → (Iic n → Fin K × ℝ) → Fin K)\n    (hmeasA : ∀ n, Measurable (nextA n))\n    (hwarm : ∀ n h, n < K - 1 → nextA n h = RoundRobin.nextAction hK n)\n    (hargmax : ∀ n h a, K - 1 ≤ n →\n      empMean' n h a + ucbWidthNL δ n h a ≤\n        empMean' n h (nextA n h) + ucbWidthNL δ n h (nextA n h)) :\n    (trajMeasure (detAlgorithm nextA hmeasA ⟨0, hK⟩) (stationaryEnv ν))\n        {ω | regret ν (action (𝓐 := Fin K) (𝓨 := ℝ)) T ω ≤\n          ((K:ℝ) - 1) + Real.sqrt (8 * Real.log (2 * K * T ^ 2 / δ) * ((T:ℝ) - K) * K)}\n      ≥ ENNReal.ofReal (1 - δ)"}]}
{"item_id": "ch6_proposition_6_46::ch6_proposition_6_46::A-stmt", "paper_id": "ch6_proposition_6_46", "target_name": "ch6_proposition_6_46", "source": "Graduate-level", "nl_statement": "Consider function class \\(\\mathcal{G} = \\{\\phi(w, z) : w \\in \\Omega\\}\\) with \\(h(w) = \\phi(w, \\mathcal{D})\\) and \\(\\inf_{w \\in \\Omega} h(w) = 0\\). Assume that \\(|\\phi(\\cdot)| \\leq M\\) and the variance condition (3.14) holds with \\(c_0 = 0\\). Assume that for any \\(b > 0\\), we have\n\\[\n\\sup_{\\mathcal{S}_{n}}R\\left(\\left\\{\\phi(w,\\cdot):\\frac{1}{n}\\sum_{i=1}^{n}\\phi(w,Z_{i})^{2}\\leq b\\right\\},\\mathcal{S}_{n}\\right)\\leq r_{n}(b),\n\\]\nwhere \\(r_n(b)\\) is a continuous concave function of \\(b\\). Let \\(\\alpha \\leq 0.5c_1 / M\\) and\n\\[\nb_{0}=\\sup\\{b>0:b\\leq(4 c_{1}/\\alpha)r_{n}(b)\\},\n\\]\nthen \\(\\bar{r}_n^h(\\alpha,\\mathcal{G},\\mathcal{D})\\leq 0.5\\alpha b_0/c_1\\)", "context": [{"name": "theorem 6.28", "statement": "Let \\(\\{\\phi_i\\}_{i=1}^n\\) be functions with Lipschitz constants \\(\\{\\gamma_i\\}_{i=1}^n\\) respectively. That is, \\(\\forall i \\in [n]\\):\n\\[\n\\left|\\phi_{i}(\\theta)-\\phi_{i}\\left(\\theta^{\\prime}\\right)\\right|\\leq\\gamma_{i}\\left|\\theta-\\theta^{\\prime}\\right|.\n\\]\nThen for any real valued function \\( h: \\mathcal{F} \\times \\mathcal{Z}^n \\to \\mathbb{R} \\), and \\( S_n = \\{Z_1, \\ldots, Z_n\\} \\subset \\mathcal{Z}^n \\), we have\n\\[\n\\mathbb{E}_{\\sigma}\\sup_{f\\in\\mathcal{F}}\\left[\\sum_{i=1}^{n}\\sigma_{i}\\phi_{i}(f(Z_{i}))-h(f,\\mathcal{S}_{n})\\right]\\leq\\mathbb{E}_{\\sigma}\\sup_{f\\in\\mathcal{F}}\\left[\\sum_{i=1}^{n}\\sigma_{i}\\gamma_{i}f(Z_{i})-h(f,\\mathcal{S}_{n})\\right].\n\\]"}, {"name": "theorem 6.4", "statement": "We have\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})\\leq 2 R_{n}(\\mathcal{G},\\mathcal{D}).\n\\]\nConsequently, the approximate ERM method of (3.7) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\phi(w,\\mathcal{D})+\\epsilon^{\\prime}+2 R_{n}(\\mathcal{G},\\mathcal{D}).\n\\]"}, {"name": "definition 6.35", "statement": "Given \\(\\mathcal{D}\\) and \\(\\mathcal{F}\\), and consider a localization function \\(h: \\mathcal{F} \\to \\mathbb{R}\\) such that \\(b_0 = \\inf_{f \\in \\mathcal{F}} h(f) > -\\infty\\). Define localized function class \\(\\mathcal{F}^h(b) = \\{f \\in \\mathcal{F}: h(f) \\leq b\\}\\) for all \\(b > b_0\\). For any \\(\\alpha > 0\\), the rate function with respect to localization \\(h\\) is defined as\n\\[\n\\bar{r}_{n}^{h}(\\alpha,\\mathcal{F},\\mathcal{D})=\\sup\\left\\{r:r\\leq\\inf_{r^{\\prime}>\\max(r,\\alpha b_{0})}\\epsilon_{n}\\left(\\mathcal{F}^{h}\\big(r^{\\prime}/\\alpha\\big),\\mathcal{D}\\right)\\right\\}.\n\\]\nWe note that the requirement of \\( r' > \\alpha b_0 \\) in Definition 6.35 is only to make sure that \\( \\mathcal{F}^h(r'/\\alpha) \\) is always non-empty, and thus \\( \\epsilon_n(\\mathcal{F}^h(r'/\\alpha), \\mathcal{D}) \\) is well-defined."}, {"name": "definition 6.1", "statement": "Given an empirical process \\(\\{\\phi(w, S_n) : w \\in \\Omega\\}\\), with \\(S_n \\sim \\mathcal{D}^n\\). Define the expected supremum of this empirical process as\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}}\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})\\right],\n\\]\nwhich will be referred to as the uniform convergence complexity of the function class \\(\\mathcal{G}\\).\nThe smaller this quantity is, the closer the gap between the training error and the test error is, which implies that we have less overfitting. In Chapter 3, we\nobtained large probability uniform convergence results for empirical processes, and then derived oracle inequalities in large probabilities. In the following, we show that if average convergence can be obtained, then we can derive oracle inequalities in expectation directly."}, {"name": "definition 6.3", "statement": "Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]"}, {"name": "Variance Condition", "statement": "Given a function class \\(\\mathcal{G}\\). We say it satisfies the variance condition if there exists \\(c_0, c_1 > 0\\) such that for all \\(\\phi(z) \\in \\mathcal{G}\\):\n\\[\n\\operatorname{Var}_{Z\\sim\\mathcal{D}}(\\phi(Z))\\leq c_{0}^{2}+c_{1}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(Z),\\tag{3.13}\n\\]\nwhere we require that \\(\\mathbb{E}_{Z\\sim \\mathcal{D}}\\phi (Z)\\geq -c_0^2 /c_1\\) for all \\(\\phi \\in \\mathcal{G}\\)\nIn applications, the following modification of the variance condition is often more convenient to employ\n\\[\n\\mathbb{E}_{Z\\sim\\mathcal{D}}[\\phi(Z)^{2}]\\leq c_{0}^{2}+c_{1}\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(Z).\\tag{3.14}\n\\]\nIt is easy to see that (3.14) implies (3.13). If \\(\\phi(Z)\\) is bounded, then the two conditions are equivalent.\nIn general, if the variance condition (3.13) only holds for \\( c_{1} = 0 \\), then we can only obtain a convergence rate that is \\( O(1 / \\sqrt{n}) \\) at the best. When the variance condition holds for \\( c_{1} > 0 \\) and \\( c_{0} = 0 \\), then we may be able to obtain a convergence rate faster than \\( O(1 / \\sqrt{n}) \\) by using Bernstein's inequality. The ability to achieve faster convergence is the main reason to study this condition."}], "nl_notes": "NOTE — The Lean assumes the index set Ω is countable (`[Countable Ω]`), which the NL statement does not state. The conclusion is built from integrals of suprema over the class {φ w : w ∈ Ω}, and without countability the statement is false (on the first uncountable ordinal with the countable/co-countable 0–1 measure, φ(w, z) = −1 if z ≤ w else 0 makes the conclusion read 1 ≤ 3/4). This is the usual countability / permissibility convention for such suprema.", "lean": "import Mathlib\n\nopen MeasureTheory Real Set\nopen scoped BigOperators ENNReal NNReal\n\nnamespace GraduateLevel\n\nnoncomputable def popVal {𝒵 Ω : Type*} [MeasurableSpace 𝒵]\n    (D : Measure 𝒵) (φ : Ω → 𝒵 → ℝ) (w : Ω) : ℝ :=\n  ∫ z, φ w z ∂D\n\nnoncomputable def empMean {𝒵 : Type*} (n : ℕ) (g : 𝒵 → ℝ) (S : Fin n → 𝒵) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i, g (S i)\n\nnoncomputable def empRad {𝒵 Ω : Type*} (n : ℕ) (φ : Ω → 𝒵 → ℝ)\n    (P : Set Ω) (S : Fin n → 𝒵) : ℝ :=\n  (1 / (2 ^ n : ℝ)) * ∑ σ : Fin n → Bool,\n    ⨆ w : P, (1 / (n : ℝ)) * ∑ i, (if σ i then (1 : ℝ) else -1) * φ w.1 (S i)\n\nnoncomputable def unifComplexity {𝒵 Ω : Type*} [MeasurableSpace 𝒵]\n    (n : ℕ) (D : Measure 𝒵) (φ : Ω → 𝒵 → ℝ) (P : Set Ω) : ℝ :=\n  ∫ S, (⨆ w : P, (popVal D φ w.1 - empMean n (φ w.1) S))\n    ∂(Measure.pi (fun _ : Fin n => D))\n\nnoncomputable def bZero (c₁ α : ℝ) (r : ℝ → ℝ) : ℝ :=\n  sSup {b : ℝ | 0 < b ∧ b ≤ (4 * c₁ / α) * r b}\n\ntheorem ch6_proposition_6_46\n    {𝒵 Ω : Type*} [MeasurableSpace 𝒵] [Nonempty Ω] [Countable Ω]\n    (D : Measure 𝒵) [IsProbabilityMeasure D]\n    (n : ℕ) (hn : 1 ≤ n)\n    (φ : Ω → 𝒵 → ℝ)\n    (M c₁ α : ℝ)\n    (hM : 0 < M) (hc₁ : 0 < c₁) (hα : 0 < α)\n    (hInf : ⨅ w, popVal D φ w = 0)\n    (hMeas : ∀ w, AEStronglyMeasurable (fun z => φ w z) D)\n    (hInt : ∀ w, Integrable (fun z => φ w z) D)\n    (hInt2 : ∀ w, Integrable (fun z => (φ w z) ^ 2) D)\n    (hBound : ∀ w z, |φ w z| ≤ M)\n    (hVar : ∀ w, (∫ z, (φ w z) ^ 2 ∂D) ≤ c₁ * ∫ z, φ w z ∂D)\n    (r : ℝ → ℝ)\n    (hrCont : ContinuousOn r (Ici 0))\n    (hrConcave : ConcaveOn ℝ (Ici 0) r)\n    (hRad : ∀ b > (0 : ℝ),\n      (⨆ S : Fin n → 𝒵, empRad n φ {w | empMean n (fun z => (φ w z) ^ 2) S ≤ b} S) ≤ r b)\n    (hαM : α ≤ (1 / 2) * c₁ / M)\n    (hbNE : {b : ℝ | 0 < b ∧ b ≤ (4 * c₁ / α) * r b}.Nonempty)\n    (hbBdd : BddAbove {b : ℝ | 0 < b ∧ b ≤ (4 * c₁ / α) * r b})\n    (rbar : ℝ)\n    (hrbarDef : rbar = sSup {ρ : ℝ | ρ ≤\n      sInf ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0)))})\n    (hrbarNE : {ρ : ℝ | ρ ≤\n      sInf ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0)))}.Nonempty)\n    (hrbarBdd : BddAbove {ρ : ℝ | ρ ≤\n      sInf ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0)))})\n    (hInnerBdd : ∀ ρ : ℝ,\n      BddBelow ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0))))\n    (hUnifInt : ∀ r' : ℝ,\n      Integrable (fun S : Fin n → 𝒵 =>\n        ⨆ w : {w | popVal D φ w ≤ r' / α}, (popVal D φ w.1 - empMean n (φ w.1) S))\n        (Measure.pi (fun _ : Fin n => D)))\n\n    (hUnifInt2 : ∀ r' : ℝ,\n      Integrable (fun S : Fin n → 𝒵 =>\n        ⨆ w ∈ {w | popVal D φ w ≤ r' / α}, empMean n (fun z => (φ w z) ^ 2) S)\n        (Measure.pi (fun _ : Fin n => D))) :\n    rbar ≤ (1 / 2) * α * bZero c₁ α r / c₁ := sorry\n\nend GraduateLevel\n", "target_names": ["ch6_proposition_6_46"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 936, "end": 2792, "gold": "{𝒵 Ω : Type*} [MeasurableSpace 𝒵] [Nonempty Ω] [Countable Ω]\n    (D : Measure 𝒵) [IsProbabilityMeasure D]\n    (n : ℕ) (hn : 1 ≤ n)\n    (φ : Ω → 𝒵 → ℝ)\n    (M c₁ α : ℝ)\n    (hM : 0 < M) (hc₁ : 0 < c₁) (hα : 0 < α)\n    (hInf : ⨅ w, popVal D φ w = 0)\n    (hMeas : ∀ w, AEStronglyMeasurable (fun z => φ w z) D)\n    (hInt : ∀ w, Integrable (fun z => φ w z) D)\n    (hInt2 : ∀ w, Integrable (fun z => (φ w z) ^ 2) D)\n    (hBound : ∀ w z, |φ w z| ≤ M)\n    (hVar : ∀ w, (∫ z, (φ w z) ^ 2 ∂D) ≤ c₁ * ∫ z, φ w z ∂D)\n    (r : ℝ → ℝ)\n    (hrCont : ContinuousOn r (Ici 0))\n    (hrConcave : ConcaveOn ℝ (Ici 0) r)\n    (hRad : ∀ b > (0 : ℝ),\n      (⨆ S : Fin n → 𝒵, empRad n φ {w | empMean n (fun z => (φ w z) ^ 2) S ≤ b} S) ≤ r b)\n    (hαM : α ≤ (1 / 2) * c₁ / M)\n    (hbNE : {b : ℝ | 0 < b ∧ b ≤ (4 * c₁ / α) * r b}.Nonempty)\n    (hbBdd : BddAbove {b : ℝ | 0 < b ∧ b ≤ (4 * c₁ / α) * r b})\n    (rbar : ℝ)\n    (hrbarDef : rbar = sSup {ρ : ℝ | ρ ≤\n      sInf ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0)))})\n    (hrbarNE : {ρ : ℝ | ρ ≤\n      sInf ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0)))}.Nonempty)\n    (hrbarBdd : BddAbove {ρ : ℝ | ρ ≤\n      sInf ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0)))})\n    (hInnerBdd : ∀ ρ : ℝ,\n      BddBelow ((fun r' => unifComplexity n D φ {w | popVal D φ w ≤ r' / α}) '' Ioi (max ρ (α * 0))))\n    (hUnifInt : ∀ r' : ℝ,\n      Integrable (fun S : Fin n → 𝒵 =>\n        ⨆ w : {w | popVal D φ w ≤ r' / α}, (popVal D φ w.1 - empMean n (φ w.1) S))\n        (Measure.pi (fun _ : Fin n => D)))\n\n    (hUnifInt2 : ∀ r' : ℝ,\n      Integrable (fun S : Fin n → 𝒵 =>\n        ⨆ w ∈ {w | popVal D φ w ≤ r' / α}, empMean n (fun z => (φ w z) ^ 2) S)\n        (Measure.pi (fun _ : Fin n => D))) :\n    rbar ≤ (1 / 2) * α * bZero c₁ α r / c₁"}]}
{"item_id": "ch13_theorem_13_2::ch13_theorem_13_2::A-stmt", "paper_id": "ch13_theorem_13_2", "target_name": "ch13_theorem_13_2", "source": "Graduate-level", "nl_statement": "Consider a sequence of random functions \\(\\xi_1(S_1),\\ldots,\\xi_t(S_t),\\ldots\\) with filtration \\(\\{\\mathcal{F}_t\\}\\). We have for any \\(\\delta \\in (0,1)\\) and \\(\\lambda >0\\):\n\\[\n\\operatorname*{Pr}\\left[\\exists n>0: -\\sum_{i=1}^{n}\\xi_{i}\\geq\\frac{\\ln(1/\\delta)}{\\lambda}+\\frac{1}{\\lambda}\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right]\\leq\\delta .\n\\]\nMoreover, consider a sequence of \\(\\{z_t\\in \\mathbb{R}\\}\\) adapted to \\(\\{\\mathcal{F}_t\\}\\), and events \\(A_{t}\\) on \\(\\mathcal{F}_t\\):\n\\[\n\\ln\\Pr\\left[\\exists n>0:\\sum_{i=1}^{n}\\xi_{i}\\leq z_{n}\\&\\mathcal{S}_{n}\\in A_{n}\\right]\\leq\\inf_{\\lambda>0}\\sup_{n>0}\\sup_{\\mathcal{S}_{n}\\in A_{n}}\\left[\\lambda z_{n}+\\sum_{i=1}^{n}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{-\\lambda\\xi_{i}}\\right].\n\\]", "context": [{"name": "lemma 13.1", "statement": "Consider a sequence of real-valued random (measurable) functions \\(\\xi_1(S_1),\\ldots,\\xi_T(S_T)\\). Let \\(\\tau \\leq T\\) be a stopping time so that \\(\\mathbb{1}(t\\leq \\tau)\\) is measurable in \\(S_{t}\\). We have\n\\[\n\\mathbb{E}_{\\mathcal{S}_{T}}\\exp\\left(\\sum_{i=1}^{\\tau}\\xi_{i}-\\sum_{i=1}^{\\tau}\\ln\\mathbb{E}_{Z_{i}^{(y)}}e^{\\xi_{i}}\\right)=1.\n\\]"}, {"name": "section 13.1 sequential observation setup", "statement": "In Section 13.1, data are observed sequentially. At each time \\(t\\), after the history \\(S_{t-1}\\) has been observed, the next observation \\(Z_t\\) is drawn according to a distribution that may depend on that history. The observation is decomposed as \\(Z_t=(Z_t^{(x)},Z_t^{(y)})\\), where \\(Z_t^{(x)}\\) is the component determined by the past history or the chosen action at time \\(t\\), and \\(Z_t^{(y)}\\) is the random response component. The history after time \\(t\\) is \\(S_t=(Z_1,\\ldots,Z_t)\\), and \\(\\mathcal{S}_T\\) denotes the full sequential experiment/history through time \\(T\\). For a function depending on \\(S_t\\), the notation \\(\\mathbb{E}_{Z_t^{(y)}}[\\cdot]\\) means conditional expectation over the response component \\(Z_t^{(y)}\\), conditional on \\(S_{t-1}\\) and on \\(Z_t^{(x)}\\). Stopping times are adapted to the filtration generated by the histories \\(S_t\\)."}], "nl_notes": "NOTE — The conditional MGF c(λ, i) is specified by its set-integral characterisation, not through Mathlib's `condExp`: for every λ > 0 and every F_{i−1}-measurable set s, ∫_s exp(−λ ξ_i) dμ = ∫_s c(λ, i) dμ, both as lintegrals of nonnegative functions (`hc_mgf`). No integrability of exp(−λ ξ_i) is assumed; where integrability is needed (e.g. for the likelihood ratio) it has to be derived. c is real-valued and pointwise positive.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory\n\nnamespace GraduateLevel\n\ndef partialSum {Ω : Type*} (ξ : ℕ → Ω → ℝ) (n : ℕ) (ω : Ω) : ℝ :=\n  ∑ i ∈ Finset.Icc 1 n, ξ i ω\n\ntheorem ch13_theorem_13_2\n    {Ω : Type*} [m₀ : MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]\n    (ℱ : Filtration ℕ m₀) (ξ : ℕ → Ω → ℝ) (hξ_adapted : Adapted ℱ ξ)\n    (z : ℕ → Ω → ℝ) (hz_adapted : Adapted ℱ z)\n    (A : ℕ → Set Ω) (hA : ∀ n, MeasurableSet[ℱ n] (A n))\n    (c : ℝ → ℕ → Ω → ℝ)\n    (hc_meas : ∀ lam i, StronglyMeasurable[ℱ (i - 1)] (c lam i))\n    (hc_mgf : ∀ lam i, 0 < lam → ∀ s, MeasurableSet[ℱ (i - 1)] s →\n        ∫⁻ ω in s, ENNReal.ofReal (Real.exp (-lam * ξ i ω)) ∂μ\n          = ∫⁻ ω in s, ENNReal.ofReal (c lam i ω) ∂μ)\n    (hc_pos : ∀ lam i ω, 0 < lam → 0 < c lam i ω) :\n    (∀ δ : ℝ, 0 < δ → δ < 1 → ∀ lam : ℝ, 0 < lam →\n        (μ {ω | ∃ n, 0 < n ∧\n            Real.log (1 / δ) / lam + (1 / lam) * (∑ i ∈ Finset.Icc 1 n, Real.log (c lam i ω))\n              ≤ - partialSum ξ n ω}).toReal ≤ δ)\n    ∧ (ENNReal.log (μ {ω | ∃ n, 0 < n ∧ partialSum ξ n ω ≤ z n ω ∧ ω ∈ A n})\n          ≤ ⨅ lam : {l : ℝ // 0 < l}, ⨆ n : {k : ℕ // 0 < k}, ⨆ ω ∈ A (n : ℕ),\n              (((lam : ℝ) * z (n : ℕ) ω\n                  + ∑ i ∈ Finset.Icc 1 (n : ℕ), Real.log (c (lam : ℝ) i ω) : ℝ) : EReal)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch13_theorem_13_2"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Martingales & Dependent Data", "holes": [{"start": 206, "end": 1290, "gold": "{Ω : Type*} [m₀ : MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]\n    (ℱ : Filtration ℕ m₀) (ξ : ℕ → Ω → ℝ) (hξ_adapted : Adapted ℱ ξ)\n    (z : ℕ → Ω → ℝ) (hz_adapted : Adapted ℱ z)\n    (A : ℕ → Set Ω) (hA : ∀ n, MeasurableSet[ℱ n] (A n))\n    (c : ℝ → ℕ → Ω → ℝ)\n    (hc_meas : ∀ lam i, StronglyMeasurable[ℱ (i - 1)] (c lam i))\n    (hc_mgf : ∀ lam i, 0 < lam → ∀ s, MeasurableSet[ℱ (i - 1)] s →\n        ∫⁻ ω in s, ENNReal.ofReal (Real.exp (-lam * ξ i ω)) ∂μ\n          = ∫⁻ ω in s, ENNReal.ofReal (c lam i ω) ∂μ)\n    (hc_pos : ∀ lam i ω, 0 < lam → 0 < c lam i ω) :\n    (∀ δ : ℝ, 0 < δ → δ < 1 → ∀ lam : ℝ, 0 < lam →\n        (μ {ω | ∃ n, 0 < n ∧\n            Real.log (1 / δ) / lam + (1 / lam) * (∑ i ∈ Finset.Icc 1 n, Real.log (c lam i ω))\n              ≤ - partialSum ξ n ω}).toReal ≤ δ)\n    ∧ (ENNReal.log (μ {ω | ∃ n, 0 < n ∧ partialSum ξ n ω ≤ z n ω ∧ ω ∈ A n})\n          ≤ ⨅ lam : {l : ℝ // 0 < l}, ⨆ n : {k : ℕ // 0 < k}, ⨆ ω ∈ A (n : ℕ),\n              (((lam : ℝ) * z (n : ℕ) ω\n                  + ∑ i ∈ Finset.Icc 1 (n : ℕ), Real.log (c (lam : ℝ) i ω) : ℝ) : EReal))"}]}
{"item_id": "ch8_theorem_8_2::ch8_theorem_8_2::A-stmt", "paper_id": "ch8_theorem_8_2", "target_name": "ch8_theorem_8_2", "source": "Graduate-level", "nl_statement": "Assume \\(\\sup_{Z,Z'}[\\phi(f,Z) - \\phi(f,Z')] \\leq M\\) for all \\(f\\), and the validation data \\(\\bar{S}_m\\) contains \\(m\\) iid samples from \\(\\mathcal{D}\\). Given training data \\(S_n\\), consider (8.2) with\n\\[\nr_{m}(q)=M\\sqrt{\\frac{\\ln(1/q)}{2 m}}.\n\\]\nThen with probability at least \\(1 - \\delta\\) over the random selection of \\(S_{m}\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta},\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta}Q(\\theta,\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\bar{\\mathcal{S}}_{m})+\\tilde{\\epsilon}+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 m}}.\n\\]\nThis implies the following oracle inequality. With probability at least \\(1 - \\delta\\) over the random sampling of \\(\\bar{S}_m\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta},\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta}[\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+r_{m}(q(\\theta))]+\\tilde{\\epsilon}+M\\sqrt{\\frac{2\\ln(2/\\delta)}{m}},\n\\]\nwhere \\(q(\\theta)\\) satisfies (8.1).", "context": [{"name": "Union Bound", "statement": "(Union Bound) Consider \\(m\\) events \\(E_1, \\ldots, E_m\\). The following probability inequality holds:\n\\[\n\\Pr (E_1 \\cup \\cdots \\cup E_m) \\leq \\sum_{j=1}^{m} \\Pr(E_j).\n\\]\n\nThe union bound has an alternative expression, which is often used in the learning theory analysis. If each \\(E_j\\) occurs with probability at least \\(1-\\delta_j\\) for \\(j=1, \\ldots, m\\), then with probability at least \\(1-\\sum_{j=1}^{m} \\delta_j\\), all of events \\(\\{E_j\\}\\) occur simultaneously for \\(j=1, \\ldots, m\\)."}, {"name": "definition 8.1", "statement": "Consider a loss function \\(\\phi(f, z): \\mathcal{F} \\times \\mathcal{Z} \\to \\mathbb{R}\\), and a model family \\(\\{\\mathcal{A}(\\theta, S_n): \\Theta \\times \\mathcal{Z}^n \\to \\mathcal{F}, n \\geq 0\\}\\). Consider \\(N \\geq n \\geq 0\\), and iid dataset \\(S_n \\subset S_N \\sim \\mathcal{D}^N\\). A model selection algorithm \\(\\bar{\\mathcal{A}}\\) maps \\(S_N\\) to \\(\\hat{\\theta} = \\hat{\\theta}(S_N) \\in \\Theta\\),\nand then train a model \\(\\hat{f} = \\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_N), \\mathcal{S}_n) = \\bar{\\mathcal{A}}(\\mathcal{S}_N)\\). It satisfies an \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) oracle inequality if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(\\mathcal{S}_N\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_{N}),\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)].\n\\]\nMore generally, a learning algorithm \\(\\bar{\\mathcal{A}}: S_N \\to \\mathcal{F}\\) is \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) adaptive to the model family \\(\\{\\mathcal{A}(\\theta, \\cdot): \\theta \\in \\Theta\\}\\) if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(S_N\\):\n\\[\n\\phi(\\bar{\\mathcal{A}}(\\mathcal{S}_{N}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}\\left[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)\\right].\n\\]\nNote that we have used a slightly different notation \\(\\phi(f, Z)\\), instead of \\(\\phi(w, Z)\\) as in earlier chapters, to emphasize that in model selection, the prediction function \\(f(w, \\cdot)\\) from different models can be compared, but the parameter \\(w\\) of the functions may not be comparable when different models have different functional forms such as decision trees versus neural networks. Without causing confusion, one may consider \\(f\\) to be the function represented by \\(w\\), so that \\(\\phi(f, Z) = L(f(X), Y)\\) and \\(\\phi(w, Z) = L(f(w, X), Y)\\). Similar to Definition 3.8, we use the simplified notation\n\\[\n\\phi(f,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(f,Z),\\quad\\phi(f,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}\\phi(f,Z).\n\\]\nThe concept adaptivity is more general than model selection, as the algorithm does not need to choose a specific model \\(\\mathcal{A}(\\theta,\\cdot)\\) indexed by \\(\\theta\\). For example, a commonly used method to achieve adaptivity is model averaging, where we take the average of several different model outputs \\(\\hat{f}_{\\theta_j} = \\mathcal{A}(\\theta_j,\\mathcal{S}_n)\\) as\n\\[\n\\hat{f}_{\\mathrm{avg}}(x)=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{f}_{\\theta_{j}}(x).\n\\]\nIf different models have the same functional form, with a vector model parameter \\( w \\in \\mathbb{R}^d \\): \\( \\hat{f}_{\\theta_j}(x) = f(\\hat{w}_j, x) \\). Then one may also average the model parameters as follows:\n\\[\n\\hat{w}_{\\mathrm{avg}}=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{w}_{j}.\n\\]\nIn general, the training data \\( S_{n} \\) is a subset of \\( S_{N} \\), and we allow \\( n < N \\) in order to analyze validation methods which are commonly used in practice. In addition to the large probability bounds in Definition 8.1, one may also derive bounds that hold for expectation over \\( S_{N} \\)."}, {"name": "equation (8.2)", "statement": "Given training data \\(S_n\\) and validation data \\(\\bar{S}_m\\), define the validation objective\n\\[\nQ(\\theta, f, \\bar{\\mathcal{S}}_m) = \\phi(f, \\bar{\\mathcal{S}}_m) + r_m(q(\\theta)).\n\\]\nWe select \\(\\hat{\\theta}\\) by approximate minimization:\n\\[\nQ(\\hat{\\theta}, \\mathcal{A}(\\hat{\\theta}, \\mathcal{S}_n), \\bar{\\mathcal{S}}_m) \\leq \\inf_{\\theta} Q(\\theta, \\mathcal{A}(\\theta, \\mathcal{S}_n), \\bar{\\mathcal{S}}_m) + \\tilde{\\epsilon}. \\tag{8.2}\n\\]\nHere \\(\\tilde{\\epsilon}\\) is the additive slack in the validation-based model-selection rule."}, {"name": "equation (8.1)", "statement": "Let \\(q(\\theta)\\geq 0\\) for \\(\\theta = 1,2,\\ldots\\) satisfy\n\\[\n\\sum_{\\theta=1}^{\\infty} q(\\theta) \\leq 1. \\tag{8.1}\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def rPenalty (M : ℝ) (m : ℕ) (q : ℝ) : ℝ :=\n  M * Real.sqrt (Real.log (1 / q) / (2 * m))\n\nnoncomputable def popLoss {F Z : Type*} [MeasurableSpace Z] (φ : F → Z → ℝ) (D : MeasureTheory.Measure Z) (f : F) : ℝ :=\n  ∫ z, φ f z ∂D\n\nnoncomputable def empLoss {F Z : Type*} (φ : F → Z → ℝ) {m : ℕ} (S : Fin m → Z) (f : F) : ℝ :=\n  (1 / (m : ℝ)) * ∑ i, φ f (S i)\n\nnoncomputable def Qobj {F Z Θ : Type*} (φ : F → Z → ℝ) {m : ℕ} (S : Fin m → Z)\n    (M : ℝ) (q : Θ → ℝ) (θ : Θ) (f : F) : ℝ :=\n  empLoss φ S f + rPenalty M m (q θ)\n\ntheorem ch8_theorem_8_2\n    {F Z Θ : Type*} [MeasurableSpace Z] [Countable Θ] [Nonempty Θ]\n\n    (φ : F → Z → ℝ)\n\n    (D : MeasureTheory.Measure Z) [MeasureTheory.IsProbabilityMeasure D]\n\n    (M : ℝ) (hM : 0 ≤ M)\n    (hrange : ∀ (f : F) (z z' : Z), φ f z - φ f z' ≤ M)\n\n    {m : ℕ} (hm : 0 < m)\n\n    {Ω : Type*} [MeasurableSpace Ω]\n    (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P]\n\n    (S : Fin m → Ω → Z)\n    (hSmeas : ∀ i, Measurable (S i))\n    (hSindep : ProbabilityTheory.iIndepFun S P)\n    (hSlaw : ∀ i, MeasureTheory.Measure.map (S i) P = D)\n\n    (fhat : Θ → F)\n\n    (hintg : ∀ θ, MeasureTheory.Integrable (fun z => φ (fhat θ) z) D)\n\n    (hφmeas : ∀ θ, AEMeasurable (fun z => φ (fhat θ) z) D)\n\n    (hφbdd : ∀ θ, ∃ a : ℝ, ∀ᵐ z ∂D, φ (fhat θ) z ∈ Set.Icc a (a + M))\n\n    (hφSmeas : ∀ θ i, AEMeasurable (fun ω => φ (fhat θ) (S i ω)) P)\n\n    (hφSbdd : ∀ θ, ∃ a : ℝ, ∀ i, ∀ᵐ ω ∂P, φ (fhat θ) (S i ω) ∈ Set.Icc a (a + M))\n\n    (q : Θ → ℝ) (hqpos : ∀ θ, 0 < q θ) (hsummable : Summable q) (hqsum : ∑' θ, q θ ≤ 1)\n\n    (εt : ℝ) (hεt : 0 ≤ εt)\n\n    (thetahat : Ω → Θ)\n    (hsel : ∀ ω, Qobj φ (fun i => S i ω) M q (thetahat ω) (fhat (thetahat ω))\n              ≤ (⨅ θ, Qobj φ (fun i => S i ω) M q θ (fhat θ)) + εt)\n\n    (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n\n    (hbddQ : ∀ ω, BddBelow (Set.range (fun θ => Qobj φ (fun i => S i ω) M q θ (fhat θ))))\n    (hbddOr : BddBelow (Set.range (fun θ => popLoss φ D (fhat θ) + rPenalty M m (q θ)))) :\n\n    (1 - δ ≤ P.real {ω |\n        popLoss φ D (fhat (thetahat ω))\n          ≤ (⨅ θ, Qobj φ (fun i => S i ω) M q θ (fhat θ)) + εt\n            + M * Real.sqrt (Real.log (1 / δ) / (2 * m))})\n    ∧\n\n    (1 - δ ≤ P.real {ω |\n        popLoss φ D (fhat (thetahat ω))\n          ≤ (⨅ θ, popLoss φ D (fhat θ) + rPenalty M m (q θ)) + εt\n            + M * Real.sqrt (2 * Real.log (2 / δ) / m)}) := sorry\n\nend GraduateLevel\n", "target_names": ["ch8_theorem_8_2"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 647, "end": 2457, "gold": "{F Z Θ : Type*} [MeasurableSpace Z] [Countable Θ] [Nonempty Θ]\n\n    (φ : F → Z → ℝ)\n\n    (D : MeasureTheory.Measure Z) [MeasureTheory.IsProbabilityMeasure D]\n\n    (M : ℝ) (hM : 0 ≤ M)\n    (hrange : ∀ (f : F) (z z' : Z), φ f z - φ f z' ≤ M)\n\n    {m : ℕ} (hm : 0 < m)\n\n    {Ω : Type*} [MeasurableSpace Ω]\n    (P : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure P]\n\n    (S : Fin m → Ω → Z)\n    (hSmeas : ∀ i, Measurable (S i))\n    (hSindep : ProbabilityTheory.iIndepFun S P)\n    (hSlaw : ∀ i, MeasureTheory.Measure.map (S i) P = D)\n\n    (fhat : Θ → F)\n\n    (hintg : ∀ θ, MeasureTheory.Integrable (fun z => φ (fhat θ) z) D)\n\n    (hφmeas : ∀ θ, AEMeasurable (fun z => φ (fhat θ) z) D)\n\n    (hφbdd : ∀ θ, ∃ a : ℝ, ∀ᵐ z ∂D, φ (fhat θ) z ∈ Set.Icc a (a + M))\n\n    (hφSmeas : ∀ θ i, AEMeasurable (fun ω => φ (fhat θ) (S i ω)) P)\n\n    (hφSbdd : ∀ θ, ∃ a : ℝ, ∀ i, ∀ᵐ ω ∂P, φ (fhat θ) (S i ω) ∈ Set.Icc a (a + M))\n\n    (q : Θ → ℝ) (hqpos : ∀ θ, 0 < q θ) (hsummable : Summable q) (hqsum : ∑' θ, q θ ≤ 1)\n\n    (εt : ℝ) (hεt : 0 ≤ εt)\n\n    (thetahat : Ω → Θ)\n    (hsel : ∀ ω, Qobj φ (fun i => S i ω) M q (thetahat ω) (fhat (thetahat ω))\n              ≤ (⨅ θ, Qobj φ (fun i => S i ω) M q θ (fhat θ)) + εt)\n\n    (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1)\n\n    (hbddQ : ∀ ω, BddBelow (Set.range (fun θ => Qobj φ (fun i => S i ω) M q θ (fhat θ))))\n    (hbddOr : BddBelow (Set.range (fun θ => popLoss φ D (fhat θ) + rPenalty M m (q θ)))) :\n\n    (1 - δ ≤ P.real {ω |\n        popLoss φ D (fhat (thetahat ω))\n          ≤ (⨅ θ, Qobj φ (fun i => S i ω) M q θ (fhat θ)) + εt\n            + M * Real.sqrt (Real.log (1 / δ) / (2 * m))})\n    ∧\n\n    (1 - δ ≤ P.real {ω |\n        popLoss φ D (fhat (thetahat ω))\n          ≤ (⨅ θ, popLoss φ D (fhat θ) + rPenalty M m (q θ)) + εt\n            + M * Real.sqrt (2 * Real.log (2 / δ) / m)})"}]}
{"item_id": "ch14_theorem_14_15::ch14_theorem_14_15::A-stmt", "paper_id": "ch14_theorem_14_15", "target_name": "ch14_theorem_14_15", "source": "Graduate-level", "nl_statement": "Assume that for all \\(t\\):\n\\[\n\\sup_{w\\in\\Omega}\\ell_{t}(w)-\\inf_{w\\in\\Omega}\\ell_{t}(w)\\leq M,\n\\]\nthen Algorithm 14.4 has regret\n\\[\n\\sum_{t=1}^{T}\\mathbb{E}_{w_{t-1}\\sim p_{t-1}(\\cdot)}\\ell_{t}(w_{t-1})\\leq\\inf_{p\\in\\Delta(\\Omega)}\\left[\\mathbb{E}_{w\\sim p}\\sum_{t=1}^{T}\\ell_{t}(w)+\\frac{1}{\\eta}\\mathrm{KL}(p| |p_{0})\\right]+\\frac{\\eta TM^{2}}{8},\n\\]\nwhere \\(\\Delta (\\Omega)\\) denotes the set of probability distributions on \\(\\Omega\\)", "context": [{"name": "Hedge Algorithm", "statement": "Hedge Algorithm\nInput: \\(T\\), prior \\(p_{0}(w)\\) on \\(\\Omega\\), learning rate \\(\\eta > 0\\)\n1 Randomly draw \\(w_{0} \\sim p_{0}(w)\\)\n2 for \\(t = 1, 2, \\ldots, T\\) do\n3 \\quad Observe loss \\(\\ell_{t}(w_{t-1})\\)\n4 \\quad Randomly draw \\(w_{t} \\sim p_{t}(w)\\) according to \\((14.9)\\)"}, {"name": "lemma 2.15", "statement": "Consider a random variable \\(X \\in [0,1]\\) and \\(\\mathbb{E}X = \\mu\\). We have the following inequality:\n\\[\n\\ln\\mathbb{E}e^{\\lambda X}\\leq\\ln [(1-\\mu)e^{0}+\\mu e^{\\lambda}]\\leq\\lambda\\mu+\\lambda^{2}/8.\n\\]"}, {"name": "proposition 7.16", "statement": "Given any function \\(U(w)\\), we have\n\\[\n\\min_{p\\in\\Delta(\\Omega)}\\left[\\mathbb{E}_{w\\sim p}U(w)+\\mathrm{KL}(p| |p_{0})\\right]= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\exp(-U(w)),\n\\]\nand the solution is achieved by the Gibbs distribution \\( q(w) \\propto p_0(w) \\exp(-U(w)) \\). Here \\( \\Delta(\\Omega) \\) denotes the set of probability distributions on \\( \\Omega \\)."}, {"name": "proposition 14.14", "statement": "Consider \\(\\Omega = \\{w_0, w_1, \\ldots, w_T\\}\\) that contains \\(T\\) members. Given any deterministic online learning algorithm \\(\\mathcal{A}\\) that returns a model parameter \\(w_{t-1}\\) based on \\(\\{\\ell_s : s = 1, \\ldots, t-1\\}\\), there exists a loss sequence \\(\\ell_t(w) \\in \\{0, 1\\}\\) so that\n\\[\n\\inf_{w}\\sum_{t=1}^{T}\\ell_{t}(w)=0,\\qquad\\sum_{t=1}^{T}\\ell_{t}(w_{t-1})=T.\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\nopen scoped ENNReal\n\nnamespace GraduateLevel\n\nnoncomputable abbrev klDiv {α : Type*} [MeasurableSpace α] (μ ν : Measure α) : ℝ≥0∞ :=\n  _root_.InformationTheory.klDiv μ ν\n\n\ntheorem ch14_theorem_14_15\n    {Ω : Type*} [MeasurableSpace Ω] [Nonempty Ω]\n    (T : ℕ) (η M : ℝ) (hη : 0 < η)\n    (ℓ : ℕ → Ω → ℝ)\n    (p₀ : Measure Ω) [IsProbabilityMeasure p₀]\n    (p : ℕ → Measure Ω) (hp₀ : p 0 = p₀)\n    (hp : ∀ t < T, IsProbabilityMeasure (p t))\n    (hpt_ac : ∀ t < T, p t ≪ p₀)\n    (hℓmeas : ∀ t < T, Measurable (ℓ (t + 1)))\n    (hℓint : ∀ t < T, Integrable (ℓ (t + 1)) (p t))\n    (hℓintq : ∀ t < T, ∀ q : Measure Ω, IsProbabilityMeasure q → q ≪ p₀ →\n      Integrable (ℓ (t + 1)) q)\n    (hbdd : ∀ t < T, BddAbove (Set.range (ℓ (t + 1))) ∧ BddBelow (Set.range (ℓ (t + 1))))\n    (hgibbs : ∀ t < T, ∃ Z : ℝ, 0 < Z ∧\n      (p t).rnDeriv p₀ =ᵐ[p₀]\n        fun w => ENNReal.ofReal (Real.exp (-η * (∑ s ∈ Finset.range t, ℓ (s + 1) w)) / Z))\n    (hosc : ∀ t < T, (⨆ w, ℓ (t + 1) w) - (⨅ w, ℓ (t + 1) w) ≤ M) :\n    ∀ q : Measure Ω, IsProbabilityMeasure q → q ≪ p₀ →\n        GraduateLevel.klDiv q p₀ ≠ ⊤ →\n      (∑ t ∈ Finset.range T, ∫ w, ℓ (t + 1) w ∂(p t))\n        ≤ (∫ w, (∑ t ∈ Finset.range T, ℓ (t + 1) w) ∂q)\n          + (1 / η) * (GraduateLevel.klDiv q p₀).toReal\n          + η * (T : ℝ) * M ^ 2 / 8 := sorry\n\nend GraduateLevel\n", "target_names": ["ch14_theorem_14_15"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 261, "end": 1348, "gold": "{Ω : Type*} [MeasurableSpace Ω] [Nonempty Ω]\n    (T : ℕ) (η M : ℝ) (hη : 0 < η)\n    (ℓ : ℕ → Ω → ℝ)\n    (p₀ : Measure Ω) [IsProbabilityMeasure p₀]\n    (p : ℕ → Measure Ω) (hp₀ : p 0 = p₀)\n    (hp : ∀ t < T, IsProbabilityMeasure (p t))\n    (hpt_ac : ∀ t < T, p t ≪ p₀)\n    (hℓmeas : ∀ t < T, Measurable (ℓ (t + 1)))\n    (hℓint : ∀ t < T, Integrable (ℓ (t + 1)) (p t))\n    (hℓintq : ∀ t < T, ∀ q : Measure Ω, IsProbabilityMeasure q → q ≪ p₀ →\n      Integrable (ℓ (t + 1)) q)\n    (hbdd : ∀ t < T, BddAbove (Set.range (ℓ (t + 1))) ∧ BddBelow (Set.range (ℓ (t + 1))))\n    (hgibbs : ∀ t < T, ∃ Z : ℝ, 0 < Z ∧\n      (p t).rnDeriv p₀ =ᵐ[p₀]\n        fun w => ENNReal.ofReal (Real.exp (-η * (∑ s ∈ Finset.range t, ℓ (s + 1) w)) / Z))\n    (hosc : ∀ t < T, (⨆ w, ℓ (t + 1) w) - (⨅ w, ℓ (t + 1) w) ≤ M) :\n    ∀ q : Measure Ω, IsProbabilityMeasure q → q ≪ p₀ →\n        GraduateLevel.klDiv q p₀ ≠ ⊤ →\n      (∑ t ∈ Finset.range T, ∫ w, ℓ (t + 1) w ∂(p t))\n        ≤ (∫ w, (∑ t ∈ Finset.range T, ℓ (t + 1) w) ∂q)\n          + (1 / η) * (GraduateLevel.klDiv q p₀).toReal\n          + η * (T : ℝ) * M ^ 2 / 8"}]}
{"item_id": "ch9_theorem_9_35::ch9_ridge_excess_risk_bound::A-stmt", "paper_id": "ch9_theorem_9_35", "target_name": "ch9_ridge_excess_risk_bound", "source": "Graduate-level", "nl_statement": "Assume that \\( \\mathcal{S}_n = \\{(X_i, Y_i)\\}_{1}^{n} \\sim \\mathcal{D}^n \\), and \\( \\operatorname{Var}[\\epsilon_i | X_i] \\leq \\sigma^2 \\) for all \\( X_i \\). If we choose \\( \\lambda \\) such that \\( \\lambda n > B^2 \\geq \\sup_x \\psi(x)^\\top \\psi(x) \\), then (9.18) satisfies\n\\[\n\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}\\mathbb{E}_{X\\sim\\mathcal{D}}(\\hat{w}^{\\top}\\psi(X)-w_{*}^{\\top}\\psi(X))^{2}\\leq\\frac{d_{1,\\lambda^{\\prime}}}{n}(\\sigma^{2}+\\lambda n\\|w_{*}\\|_{2}^{2}),\n\\]\nwhere\n\\[\nd_{1,\\lambda^{\\prime}}=\\mathrm{trace}\\left((\\Sigma_{\\mathcal{D}}+\\lambda^{\\prime}I)^{-1}\\Sigma_{\\mathcal{D}}\\right),\\quad\\Sigma_{\\mathcal{D}}=\\mathbb{E}_{\\mathcal{D}}\\psi(X)\\psi(X)^{\\top},\n\\]\nand \\(\\lambda' = (\\lambda n - B^2) / (n + 1)\\).", "context": [{"name": "lemma 9.34", "statement": "Let \\(\\{Y_i\\}_1^n\\) be independent samples conditioned on \\(\\{X_i\\}_1^n\\) according to (9.19). Given any vector \\(X\\) that is independent of \\(\\{Y_i\\}_1^n\\). If \\(\\mathbb{E}[\\epsilon_i|X_i] \\leq \\sigma^2\\) for all \\(i\\), then we have\n\\[\n\\mathbb{E}_{\\left\\{Y_{i}\\right\\}_{1}^{n}\\mid\\left\\{X_{i}\\right\\}_{1}^{n}}\\left[\\left(\\hat{w}^{\\top}\\psi(X)-w_{*}^{\\top}\\psi(X)\\right)^{2}\\right]\\leq\\left(\\left(\\sigma^{2}/n\\right)+\\lambda\\|w_{*}\\|_{2}^{2}\\right)\\|\\psi(X)\\|_{\\hat{\\Sigma}_{\\lambda}^{-1}}^{2},\n\\]\nwhere\n\\[\n\\hat{\\Sigma}_{\\lambda}=\\frac{1}{n}\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}+\\lambda I.\n\\]\nMoreover if \\(\\epsilon\\) is sub-Gaussian:\n\\[\n\\forall i:\\ln\\mathbb{E}\\exp [\\mu\\epsilon_{i}|X_{i}]\\leq\\frac{\\mu^{2}\\sigma^{2}}{2},\n\\]\nthen the following inequalities hold for conditional probability of \\(\\{Y_i\\}_1^n|\\{X_i\\}_1^n\\):\n\\[\n\\Pr\\left[\\hat{w}^{\\top}\\psi(X)-w_{*}^{\\top}\\psi(X)\\leq\\left(\\sqrt{\\lambda}\\|w_{*}\\|_{2}+\\sigma\\sqrt{\\frac{2\\ln(1/\\delta)}{n}}\\right)\\|\\psi(X)\\|_{\\hat{\\Sigma}_{\\lambda}^{-1}}\\right]\\leq\\delta,\n\\]\n\\[\n\\Pr\\left[w_{*}^{\\top}\\psi(X)-\\hat{w}^{\\top}\\psi(X)\\leq\\left(\\sqrt{\\lambda}\\|w_{*}\\|_{2}+\\sigma\\sqrt{\\frac{2\\ln(1/\\delta)}{n}}\\right)\\|\\psi(X)\\|_{\\hat{\\Sigma}_{\\lambda}^{-1}}\\right]\\leq\\delta .\n\\]"}, {"name": "equation 9.18", "statement": "The ridge-regression estimator \\(\\hat w\\) on training data \\(\\mathcal S_n = \\{(X_i,Y_i)\\}_{i=1}^n\\) is the regularized least-squares solution\n\\[\n\\hat{w}=\\hat{\\Sigma}_{\\lambda}^{-1}\\frac{1}{n}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i},\\qquad\\hat{\\Sigma}_{\\lambda}=\\frac{1}{n}\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}+\\lambda I.\n\\]"}, {"name": "equation 9.19", "statement": "The response model is\n\\[\nY_i=w_{*}^{\\top}\\psi(X_i)+\\epsilon_i,\n\\]\nwhere \\(w_*\\) is the target parameter and the noise variables satisfy \\(\\mathbb E[\\epsilon_i \\mid X_i] = 0\\) and \\(\\operatorname{Var}[\\epsilon_i \\mid X_i] \\leq \\sigma^2\\)."}], "nl_notes": "NOTE — Model one data point as an outcome ω of a probability space with law D, and the training sample as n independent outcomes (ω_1, …, ω_n) ~ Dⁿ. The covariate X and the feature vector ψ = feat ∘ X are functions of ω. Give each sample position i its own response and noise functions on that space, Y_i = w_*ᵀψ + ε_i, evaluated at ω_i. Take the noise conditions conditionally on σ(X), and express \"identically distributed\" as: (X, Y_i) has the same law under D for every i.", "lean": "import Mathlib\n\nopen MeasureTheory\n\nnamespace GraduateLevel\n\nnoncomputable def featureCovariance {Ω : Type*} [MeasurableSpace Ω] {d : ℕ}\n    (D : Measure Ω) (ψ : Ω → (Fin d → ℝ)) : Matrix (Fin d) (Fin d) ℝ :=\n  Matrix.of fun j k => ∫ x, ψ x j * ψ x k ∂D\n\nnoncomputable def linPred {Ω : Type*} {d : ℕ}\n    (w : Fin d → ℝ) (ψ : Ω → (Fin d → ℝ)) (x : Ω) : ℝ :=\n  ∑ j, w j * ψ x j\n\ntheorem ch9_ridge_excess_risk_bound\n    {Ω : Type*} [MeasurableSpace Ω] {𝒳 : Type*} [MeasurableSpace 𝒳] {d n : ℕ}\n    (D : Measure Ω) [IsProbabilityMeasure D]\n    (X : Ω → 𝒳) (feat : 𝒳 → (Fin d → ℝ))\n    (ψ : Ω → (Fin d → ℝ)) (wstar : Fin d → ℝ)\n    (ε : Fin n → Ω → ℝ) (Y : Fin n → Ω → ℝ)\n    (what : (Fin n → Ω) → (Fin d → ℝ))\n    (lam B σ : ℝ)\n    (hn : 1 ≤ n)\n    (hXmeas : Measurable X)\n    (hfeatmeas : Measurable feat)\n    (hψfac : ψ = fun ω => feat (X ω))\n    (hψmeas : Measurable ψ)\n    (hψ : ∀ j, AEMeasurable (fun x => ψ x j) D)\n    (hψint : ∀ j k, Integrable (fun x => ψ x j * ψ x k) D)\n    (hY : ∀ i x, Y i x = linPred wstar ψ x + ε i x)\n    (hεint : ∀ i, Integrable (ε i) D)\n    (hεsqint : ∀ i, Integrable (fun ω => (ε i ω)^2) D)\n    (hεmean : ∀ i, MeasureTheory.condExp (MeasurableSpace.comap X inferInstance) D (ε i) =ᵐ[D] 0)\n    (hεvar : ∀ i, MeasureTheory.condExp (MeasurableSpace.comap X inferInstance) D (fun ω => (ε i ω)^2) ≤ᵐ[D] fun _ => σ^2)\n    (hiid : ∀ i j, D.map (fun ω => (X ω, Y i ω)) = D.map (fun ω => (X ω, Y j ω)))\n    (hB : ∀ x, (∑ j, ψ x j ^ 2) ≤ B^2)\n    (hlamB : B^2 < lam * n)\n    (hExcInt : ∀ S, Integrable (fun x => (linPred (what S) ψ x - linPred wstar ψ x)^2) D)\n    (hOuterInt : Integrable (fun S => ∫ x, (linPred (what S) ψ x - linPred wstar ψ x)^2 ∂D) (Measure.pi fun _ => D))\n    (hwhat : ∀ S : Fin n → Ω, what S = ((((n : ℝ)⁻¹) • Matrix.of (fun j k => ∑ i, ψ (S i) j * ψ (S i) k) + lam • (1 : Matrix (Fin d) (Fin d) ℝ))⁻¹).mulVec (fun j => ((n : ℝ)⁻¹) * ∑ i, ψ (S i) j * Y i (S i))) :\n    ∫ S, (∫ x, (linPred (what S) ψ x - linPred wstar ψ x)^2 ∂D) ∂(Measure.pi fun _ => D)\n      ≤ ((Matrix.trace ((featureCovariance D ψ + ((lam * n - B^2) / (n + 1)) • (1 : Matrix (Fin d) (Fin d) ℝ))⁻¹ * featureCovariance D ψ)) / n) * (σ^2 + lam * n * (∑ j, wstar j ^ 2)) := sorry\n\nend GraduateLevel\n", "target_names": ["ch9_ridge_excess_risk_bound"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "holes": [{"start": 418, "end": 2180, "gold": "{Ω : Type*} [MeasurableSpace Ω] {𝒳 : Type*} [MeasurableSpace 𝒳] {d n : ℕ}\n    (D : Measure Ω) [IsProbabilityMeasure D]\n    (X : Ω → 𝒳) (feat : 𝒳 → (Fin d → ℝ))\n    (ψ : Ω → (Fin d → ℝ)) (wstar : Fin d → ℝ)\n    (ε : Fin n → Ω → ℝ) (Y : Fin n → Ω → ℝ)\n    (what : (Fin n → Ω) → (Fin d → ℝ))\n    (lam B σ : ℝ)\n    (hn : 1 ≤ n)\n    (hXmeas : Measurable X)\n    (hfeatmeas : Measurable feat)\n    (hψfac : ψ = fun ω => feat (X ω))\n    (hψmeas : Measurable ψ)\n    (hψ : ∀ j, AEMeasurable (fun x => ψ x j) D)\n    (hψint : ∀ j k, Integrable (fun x => ψ x j * ψ x k) D)\n    (hY : ∀ i x, Y i x = linPred wstar ψ x + ε i x)\n    (hεint : ∀ i, Integrable (ε i) D)\n    (hεsqint : ∀ i, Integrable (fun ω => (ε i ω)^2) D)\n    (hεmean : ∀ i, MeasureTheory.condExp (MeasurableSpace.comap X inferInstance) D (ε i) =ᵐ[D] 0)\n    (hεvar : ∀ i, MeasureTheory.condExp (MeasurableSpace.comap X inferInstance) D (fun ω => (ε i ω)^2) ≤ᵐ[D] fun _ => σ^2)\n    (hiid : ∀ i j, D.map (fun ω => (X ω, Y i ω)) = D.map (fun ω => (X ω, Y j ω)))\n    (hB : ∀ x, (∑ j, ψ x j ^ 2) ≤ B^2)\n    (hlamB : B^2 < lam * n)\n    (hExcInt : ∀ S, Integrable (fun x => (linPred (what S) ψ x - linPred wstar ψ x)^2) D)\n    (hOuterInt : Integrable (fun S => ∫ x, (linPred (what S) ψ x - linPred wstar ψ x)^2 ∂D) (Measure.pi fun _ => D))\n    (hwhat : ∀ S : Fin n → Ω, what S = ((((n : ℝ)⁻¹) • Matrix.of (fun j k => ∑ i, ψ (S i) j * ψ (S i) k) + lam • (1 : Matrix (Fin d) (Fin d) ℝ))⁻¹).mulVec (fun j => ((n : ℝ)⁻¹) * ∑ i, ψ (S i) j * Y i (S i))) :\n    ∫ S, (∫ x, (linPred (what S) ψ x - linPred wstar ψ x)^2 ∂D) ∂(Measure.pi fun _ => D)\n      ≤ ((Matrix.trace ((featureCovariance D ψ + ((lam * n - B^2) / (n + 1)) • (1 : Matrix (Fin d) (Fin d) ℝ))⁻¹ * featureCovariance D ψ)) / n) * (σ^2 + lam * n * (∑ j, wstar j ^ 2))"}]}
{"item_id": "ch5_theorem_5_14::ch5_theorem_5_14::A-stmt", "paper_id": "ch5_theorem_5_14", "target_name": "ch5_theorem_5_14", "source": "Graduate-level", "nl_statement": "Consider any class \\(\\mathcal{F} = \\{f_1, \\ldots, f_d\\}\\), and assume that for a distribution \\(\\mathcal{D}\\), \\(\\sup_{f \\in \\mathcal{F}} \\|f\\|_{L_2(\\mathcal{D})} \\leq A\\). Then\n\\[\n\\ln N(\\epsilon,\\operatorname{CONV}(\\mathcal{F}),L_{2}(\\mathcal{D}))\\leq\\lceil A^{2}/\\epsilon^{2}\\rceil\\ln [e+e d\\epsilon^{2}/A^{2}].\n\\]", "context": [{"name": "definition 5.12", "statement": "The convex hull of a function class \\(\\mathcal{F} = \\{f(\\theta, x) : \\theta \\in \\Theta\\}\\) is defined as\n\\[\n\\operatorname{CONV}(\\mathcal{F})=\\left\\{\\sum_{j=1}^{m}w_{j}f\\left(\\theta_{j},x\\right):m>0,\\|w\\|_{1}=1,w_{j}\\geq 0,\\theta_{j}\\in\\Theta\\right\\}.\n\\]\nWe also include the closure of the finite sum functions above with respect to an appropriate topology in the convex hull."}, {"name": "definition 4.1", "statement": "Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite\nset \\(\\mathcal{G}(\\epsilon) \\subset \\mathcal{V}\\) is an \\(\\epsilon\\)-cover (or \\(\\epsilon\\)-net) of \\(\\mathcal{G} \\subset \\mathcal{V}\\) if, for all \\(\\phi \\in \\mathcal{G}\\), there exists \\(\\phi' \\in \\mathcal{G}(\\epsilon)\\) so that \\(d(\\phi', \\phi) \\leq \\epsilon\\). The \\(\\epsilon\\)-covering number of \\(\\mathcal{G}\\) with metric \\(d\\) is the smallest cardinality \\(N(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The number \\(\\ln N(\\epsilon, \\mathcal{G}, d)\\) is called the \\(\\epsilon\\)-entropy.\nFor a function class \\(\\mathcal{G}\\) with seminorm \\(L_{p}(\\mathcal{D})\\) in (3.19), we denote the corresponding \\(L_{p}(\\mathcal{D})\\)-covering number as \\(N(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D}))\\). When \\(1 \\leq p \\leq q\\), we have\n\\[\nN(\\epsilon,\\mathcal{G},L_{p}(\\mathcal{D}))\\leq N(\\epsilon,\\mathcal{G},L_{q}(\\mathcal{D})).\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Set\nopen scoped ENNReal\n\nnamespace GraduateLevel\n\ndef IsENet {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n  s ⊆ ⋃ x ∈ t, Metric.closedBall x eps\n\nnoncomputable def coveringNumber {A : Type*} [PseudoMetricSpace A] (eps : ℝ) (s : Set A) :\n    WithTop ℕ :=\n  sInf {n | ∃ t, IsENet t eps s ∧ ↑t.card = n}\n\ntheorem ch5_theorem_5_14\n    {α : Type*} [MeasurableSpace α] (D : Measure α) [IsProbabilityMeasure D]\n    [Fact (1 ≤ (2 : ℝ≥0∞))]\n    {d : ℕ} (hd : 0 < d) (f : Fin d → (α → ℝ))\n    (hf : ∀ j, MemLp (f j) 2 D)\n    (ε A : ℝ) (hε : 0 < ε) (hA : 0 < A)\n    (hbound : ∀ j, eLpNorm (f j) 2 D ≤ ENNReal.ofReal A) :\n    let S : Set (Lp ℝ 2 D) := Set.range (fun j : Fin d => (hf j).toLp (f j))\n    coveringNumber ε (convexHull ℝ S) ≠ ⊤ ∧\n    Real.log (ENNReal.toReal (ENat.toENNReal (coveringNumber ε (convexHull ℝ S))))\n      ≤ (⌈A ^ 2 / ε ^ 2⌉₊ : ℝ)\n        * Real.log (Real.exp 1 + Real.exp 1 * (d : ℝ) * ε ^ 2 / A ^ 2) := sorry\n\nend GraduateLevel\n", "target_names": ["ch5_theorem_5_14"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 402, "end": 986, "gold": "{α : Type*} [MeasurableSpace α] (D : Measure α) [IsProbabilityMeasure D]\n    [Fact (1 ≤ (2 : ℝ≥0∞))]\n    {d : ℕ} (hd : 0 < d) (f : Fin d → (α → ℝ))\n    (hf : ∀ j, MemLp (f j) 2 D)\n    (ε A : ℝ) (hε : 0 < ε) (hA : 0 < A)\n    (hbound : ∀ j, eLpNorm (f j) 2 D ≤ ENNReal.ofReal A) :\n    let S : Set (Lp ℝ 2 D) := Set.range (fun j : Fin d => (hf j).toLp (f j))\n    coveringNumber ε (convexHull ℝ S) ≠ ⊤ ∧\n    Real.log (ENNReal.toReal (ENat.toENNReal (coveringNumber ε (convexHull ℝ S))))\n      ≤ (⌈A ^ 2 / ε ^ 2⌉₊ : ℝ)\n        * Real.log (Real.exp 1 + Real.exp 1 * (d : ℝ) * ε ^ 2 / A ^ 2)"}]}
{"item_id": "ch6_corollary_6_34::corollary_6_34::A-stmt", "paper_id": "ch6_corollary_6_34", "target_name": "corollary_6_34", "source": "Graduate-level", "nl_statement": "Consider a real valued function class \\(\\mathcal{F} = \\{f(z):\\mathcal{Z}\\to \\mathbb{R}\\}\\). Let \\(\\mathcal{D}\\) be a distribution on \\(\\mathcal{Z}\\). Assume that there exists \\(M,\\sigma >0\\) so that \\(\\forall f\\in \\mathcal{F}\\),\n\\(\\sigma^2 \\geq \\operatorname{Var}_{Z \\sim \\mathcal{D}}[f(Z)]\\), and \\(\\sup_{z' \\in \\mathcal{Z}} [\\mathbb{E}_{Z \\sim \\mathcal{D}} f(Z) - f(z')] \\leq M\\). Let \\(\\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\}\\) be \\(n\\) independent random variables from \\(\\mathcal{D}\\). Then with probability at least \\(1 - \\delta\\) over \\(\\mathcal{S}_n\\), for all \\(f \\in \\mathcal{F}\\),\n\\[\n\\begin{array}{l}\\mathbb{E}_{Z\\sim\\mathcal{D}}f(Z)-\\frac{1}{n}\\sum_{i=1}^{n}f\\left(Z_{i}\\right)\\\\ \\leq\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+\\sqrt{\\frac{(4 M\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+2\\sigma^{2})\\ln(1/\\delta)}{n}}+\\frac{M\\ln(1/\\delta)}{3 n}\\\\ \\leq 2\\epsilon_{n}(\\mathcal{F},\\mathcal{D})+\\sqrt{\\frac{2\\sigma^{2}\\ln(1/\\delta)}{n}}+\\frac{4 M\\ln(1/\\delta)}{3 n},\\\\ \\end{array}\n\\]\nwhere \\(\\epsilon_{n}(\\mathcal{F},\\mathcal{D})\\) is Definition 6.1.", "context": [{"name": "definition 6.1", "statement": "Given an empirical process \\(\\{\\phi(w, S_n) : w \\in \\Omega\\}\\), with \\(S_n \\sim \\mathcal{D}^n\\). Define the expected supremum of this empirical process as\n\\[\n\\epsilon_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}}\\sup_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})-\\phi(w,\\mathcal{S}_{n})\\right],\n\\]\nwhich will be referred to as the uniform convergence complexity of the function class \\(\\mathcal{G}\\).\nThe smaller this quantity is, the closer the gap between the training error and the test error is, which implies that we have less overfitting. In Chapter 3, we\nobtained large probability uniform convergence results for empirical processes, and then derived oracle inequalities in large probabilities. In the following, we show that if average convergence can be obtained, then we can derive oracle inequalities in expectation directly."}, {"name": "Bousquet, 2002", "statement": "Consider iid random variables \\((Z_{1},\\ldots,Z_{n})\\sim\\) \\(\\mathcal{D}^n\\) . Let \\(\\zeta\\) be a real-valued function of \\((Z_{1},\\dots,Z_{n})\\) . Moreover, for each \\(k\\in [n]\\) let \\(\\zeta_{k}\\) be a real-valued function of \\((Z_{1},\\ldots,Z_{k-1},Z_{k+1},\\ldots,Z_{n})\\) so that\n\\[\n\\sum_{k=1}^{n}\\left[\\zeta-\\zeta_{k}\\right]\\leq\\zeta .\n\\]\nAssume that for each \\(k\\), there exists a function \\(\\zeta_k^\\prime\\) of \\((Z_{1},\\ldots,Z_{n})\\) such that\n\\[\n\\zeta_{k}^{\\prime}\\leq\\zeta-\\zeta_{k}\\leq M,\\quad\\mathbb{E}_{Z_{k}}\\zeta_{k}^{\\prime}\\geq 0,\\quad\\zeta_{k}^{\\prime}\\leq u M.\n\\]\nWe have for all \\(t \\geq 0\\):\n\\[\n\\operatorname*{Pr}\\left[\\zeta\\geq\\mathbb{E}\\zeta+\\sqrt{2((1+u)M\\mathbb{E}\\zeta+n\\sigma^{2})t}+\\frac{tM}{3}\\right]\\leq e^{-t},\n\\]\nwhere \\(\\sigma^2\\geq n^{-1}\\sum_{k = 1}^{n}\\mathbb{E}_{Z_k}(\\zeta_k')^2\\)"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def epsN {𝒵 : Type*} [MeasurableSpace 𝒵] (n : ℕ)\n    (F : Set (𝒵 → ℝ)) (D : Measure 𝒵) : ℝ :=\n  ∫ ω, (⨆ f : F, ((∫ z, f.1 z ∂D) - (1 / (n : ℝ)) * ∑ i, f.1 (ω i)))\n    ∂(Measure.pi (fun _ : Fin n => D))\n\ntheorem corollary_6_34\n    {𝒵 : Type*} [MeasurableSpace 𝒵]\n    (n : ℕ) (hn : 0 < n)\n    (F : Set (𝒵 → ℝ)) (hF : F.Nonempty)\n    (D : Measure 𝒵) [IsProbabilityMeasure D]\n    (hMemLp : ∀ f ∈ F, MemLp f 2 D)\n    (hInt : Integrable (fun ω => ⨆ f : F,\n        ((∫ z, f.1 z ∂D) - (1 / (n : ℝ)) * ∑ i, f.1 (ω i)))\n      (Measure.pi (fun _ : Fin n => D)))\n    (M σ : ℝ) (hM : 0 < M) (hσ : 0 < σ)\n    (hVar : ∀ f ∈ F, ProbabilityTheory.variance f D ≤ σ ^ 2)\n    (hSup : ∀ f ∈ F, ∀ z' : 𝒵, (∫ z, f z ∂D) - f z' ≤ M)\n    (hBousquet : ∀ (ζ : (Fin n → 𝒵) → ℝ) (ζk ζ' : Fin n → (Fin n → 𝒵) → ℝ)\n        (u Mb sb t : ℝ), 0 ≤ u → 0 ≤ t → 0 < Mb →\n        (∀ k ω z, ζk k (Function.update ω k z) = ζk k ω) →\n        (∀ ω, ∑ k, (ζ ω - ζk k ω) ≤ ζ ω) →\n        (∀ k ω, ζ' k ω ≤ ζ ω - ζk k ω) →\n        (∀ k ω, ζ ω - ζk k ω ≤ Mb) →\n        (∀ k ω, ζ' k ω ≤ u * Mb) →\n        Integrable ζ (Measure.pi (fun _ : Fin n => D)) →\n        (∀ k ω, Integrable (fun z => ζ' k (Function.update ω k z)) D) →\n        (∀ k ω, Integrable (fun z => (ζ' k (Function.update ω k z)) ^ 2) D) →\n        (∀ k ω, 0 ≤ ∫ z, ζ' k (Function.update ω k z) ∂D) →\n        (∀ k ω, ∫ z, (ζ' k (Function.update ω k z)) ^ 2 ∂D ≤ sb ^ 2) →\n        (Measure.pi (fun _ : Fin n => D))\n            {ω : Fin n → 𝒵 |\n              (∫ x, ζ x ∂(Measure.pi (fun _ : Fin n => D)))\n                + Real.sqrt (2 * ((1 + u) * Mb\n                      * (∫ x, ζ x ∂(Measure.pi (fun _ : Fin n => D))) + (n : ℝ) * sb ^ 2) * t)\n                + t * Mb / 3 ≤ ζ ω}\n          ≤ ENNReal.ofReal (Real.exp (-t)))\n    (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1) :\n    (1 - δ ≤ ((Measure.pi (fun _ : Fin n => D))\n        {ω : Fin n → 𝒵 | ∀ f ∈ F,\n          ((∫ z, f z ∂D) - (1 / (n : ℝ)) * ∑ i, f (ω i) ≤\n            epsN n F D + Real.sqrt (((4 * M * epsN n F D + 2 * σ ^ 2) * Real.log (1 / δ)) / (n : ℝ))\n            + (M * Real.log (1 / δ)) / (3 * (n : ℝ)))\n          ∧ ((∫ z, f z ∂D) - (1 / (n : ℝ)) * ∑ i, f (ω i) ≤\n            2 * epsN n F D + Real.sqrt ((2 * σ ^ 2 * Real.log (1 / δ)) / (n : ℝ))\n            + (4 * M * Real.log (1 / δ)) / (3 * (n : ℝ)))}).toReal)\n    ∧ (epsN n F D + Real.sqrt (((4 * M * epsN n F D + 2 * σ ^ 2) * Real.log (1 / δ)) / (n : ℝ))\n        + (M * Real.log (1 / δ)) / (3 * (n : ℝ))\n      ≤ 2 * epsN n F D + Real.sqrt ((2 * σ ^ 2 * Real.log (1 / δ)) / (n : ℝ))\n        + (4 * M * Real.log (1 / δ)) / (3 * (n : ℝ))) := sorry\n\nend GraduateLevel\n", "target_names": ["corollary_6_34"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Functional & Supremum Concentration", "holes": [{"start": 328, "end": 2661, "gold": "{𝒵 : Type*} [MeasurableSpace 𝒵]\n    (n : ℕ) (hn : 0 < n)\n    (F : Set (𝒵 → ℝ)) (hF : F.Nonempty)\n    (D : Measure 𝒵) [IsProbabilityMeasure D]\n    (hMemLp : ∀ f ∈ F, MemLp f 2 D)\n    (hInt : Integrable (fun ω => ⨆ f : F,\n        ((∫ z, f.1 z ∂D) - (1 / (n : ℝ)) * ∑ i, f.1 (ω i)))\n      (Measure.pi (fun _ : Fin n => D)))\n    (M σ : ℝ) (hM : 0 < M) (hσ : 0 < σ)\n    (hVar : ∀ f ∈ F, ProbabilityTheory.variance f D ≤ σ ^ 2)\n    (hSup : ∀ f ∈ F, ∀ z' : 𝒵, (∫ z, f z ∂D) - f z' ≤ M)\n    (hBousquet : ∀ (ζ : (Fin n → 𝒵) → ℝ) (ζk ζ' : Fin n → (Fin n → 𝒵) → ℝ)\n        (u Mb sb t : ℝ), 0 ≤ u → 0 ≤ t → 0 < Mb →\n        (∀ k ω z, ζk k (Function.update ω k z) = ζk k ω) →\n        (∀ ω, ∑ k, (ζ ω - ζk k ω) ≤ ζ ω) →\n        (∀ k ω, ζ' k ω ≤ ζ ω - ζk k ω) →\n        (∀ k ω, ζ ω - ζk k ω ≤ Mb) →\n        (∀ k ω, ζ' k ω ≤ u * Mb) →\n        Integrable ζ (Measure.pi (fun _ : Fin n => D)) →\n        (∀ k ω, Integrable (fun z => ζ' k (Function.update ω k z)) D) →\n        (∀ k ω, Integrable (fun z => (ζ' k (Function.update ω k z)) ^ 2) D) →\n        (∀ k ω, 0 ≤ ∫ z, ζ' k (Function.update ω k z) ∂D) →\n        (∀ k ω, ∫ z, (ζ' k (Function.update ω k z)) ^ 2 ∂D ≤ sb ^ 2) →\n        (Measure.pi (fun _ : Fin n => D))\n            {ω : Fin n → 𝒵 |\n              (∫ x, ζ x ∂(Measure.pi (fun _ : Fin n => D)))\n                + Real.sqrt (2 * ((1 + u) * Mb\n                      * (∫ x, ζ x ∂(Measure.pi (fun _ : Fin n => D))) + (n : ℝ) * sb ^ 2) * t)\n                + t * Mb / 3 ≤ ζ ω}\n          ≤ ENNReal.ofReal (Real.exp (-t)))\n    (δ : ℝ) (hδ0 : 0 < δ) (hδ1 : δ < 1) :\n    (1 - δ ≤ ((Measure.pi (fun _ : Fin n => D))\n        {ω : Fin n → 𝒵 | ∀ f ∈ F,\n          ((∫ z, f z ∂D) - (1 / (n : ℝ)) * ∑ i, f (ω i) ≤\n            epsN n F D + Real.sqrt (((4 * M * epsN n F D + 2 * σ ^ 2) * Real.log (1 / δ)) / (n : ℝ))\n            + (M * Real.log (1 / δ)) / (3 * (n : ℝ)))\n          ∧ ((∫ z, f z ∂D) - (1 / (n : ℝ)) * ∑ i, f (ω i) ≤\n            2 * epsN n F D + Real.sqrt ((2 * σ ^ 2 * Real.log (1 / δ)) / (n : ℝ))\n            + (4 * M * Real.log (1 / δ)) / (3 * (n : ℝ)))}).toReal)\n    ∧ (epsN n F D + Real.sqrt (((4 * M * epsN n F D + 2 * σ ^ 2) * Real.log (1 / δ)) / (n : ℝ))\n        + (M * Real.log (1 / δ)) / (3 * (n : ℝ))\n      ≤ 2 * epsN n F D + Real.sqrt ((2 * σ ^ 2 * Real.log (1 / δ)) / (n : ℝ))\n        + (4 * M * Real.log (1 / δ)) / (3 * (n : ℝ)))"}]}
{"item_id": "ch15_theorem_15_6::ch15_theorem_15_6::A-stmt", "paper_id": "ch15_theorem_15_6", "target_name": "ch15_theorem_15_6", "source": "Graduate-level", "nl_statement": "Consider the ridge regression method of (15.3), with \\(\\Lambda_0\\) symmetric positive definite. We have the following result for any \\(\\sigma \\geq 0\\) and for all observed sequence \\(S_T\\):\n\\[\n\\begin{array}{l}\\sum_{t=1}^{T}\\left[\\frac{\\left(Y_{t}-\\hat{w}\\left(\\mathcal{S}_{t-1}\\right)^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}}{b_{t}}+\\sigma^{2}\\ln b_{t}\\right]\\\\ =\\inf_{w}\\left[\\sum_{t=1}^{T}\\left(Y_{t}-w^{\\top}\\psi\\left(X_{t}\\right)\\right)^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right]+\\sigma^{2}\\ln\\left|\\Lambda_{0}^{-1}\\Lambda_{T}\\right|,\\\\ \\end{array}\n\\]\nwhere\n\\[\n\\Lambda_{t}=\\Lambda_{0}+\\sum_{s=1}^{t}\\psi(X_{s})\\psi(X_{s})^{\\top},\n\\]\nand \\(b_{t} = 1 + \\psi (X_{t})^{\\top}\\Lambda_{t - 1}^{-1}\\psi (X_{t}).\\)", "context": [{"name": "proposition 15.5", "statement": "Consider probability model\n\\[\np(y|w,x)=N(w^{\\top}\\psi(x),\\sigma^{2}),\n\\]\nwith prior\n\\[\np_{0}(w)=N(0,\\sigma^{2}\\Lambda_{0}^{-1}).\n\\]\nThen given \\(\\mathcal{S}_n=\\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\) , we have\n\\[\np(w|\\mathcal{S}_{n})=N(\\hat{w}(\\mathcal{S}_{n}),\\sigma^{2}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}),\n\\]\nwhere \\(\\hat{w} (S_n)\\) is given by (15.3) and\n\\[\n\\hat{\\Lambda}\\left(\\mathcal{S}_{n}\\right)=\\sum_{i=1}^{n}\\psi\\left(X_{i}\\right)\\psi\\left(X_{i}\\right)^{\\top}+\\Lambda_{0}.\n\\]\nGiven \\( x \\), the posterior distribution of \\( y \\) is\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=N\\left(\\hat{w}(\\mathcal{S}_{n})^{\\top}\\psi(x),\\sigma^{2}+\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\psi(x)\\right).\n\\]"}, {"name": "theorem 15.3", "statement": "We have\n\\[\n\\begin{array}{l}-\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid X_{t},\\mathcal{S}_{t-1}\\right)= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\prod_{t=1}^{T}p\\left(Y_{t}\\mid w,X_{t}\\right)\\\\ =\\inf_{q\\in\\Delta(\\Omega)}\\left[-\\mathbb{E}_{w\\sim q}\\sum_{t=1}^{T}\\ln p(Y_{t}|w,X_{t})+\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\Delta (\\Omega)\\) is the set of probability distributions over \\(\\Omega\\)"}, {"name": "equation (15.3)", "statement": "Let \\(\\Lambda_0\\) be a symmetric positive definite regularizer and let \\(\\psi\\) be a feature map. Given \\(\\mathcal{S}_n = \\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\), define\n\\[\n\\hat{\\Lambda}(\\mathcal{S}_{n})=\\Lambda_{0}+\\sum_{i=1}^{n}\\psi(X_{i})\\psi(X_{i})^{\\top}\n\\]\nand the ridge regression estimator\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i}.\\tag{15.3}\n\\]\nEquivalently, \\(\\hat{w}(\\mathcal{S}_{n})\\) minimizes\n\\[\n\\sum_{i=1}^{n}(w^{\\top}\\psi(X_{i})-Y_{i})^{2}+\\|w\\|_{\\Lambda_{0}}^{2}.\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen Matrix Finset\n\nnamespace GraduateLevel\n\ndef lambdaHat {d : ℕ} (Λ₀ : Matrix (Fin d) (Fin d) ℝ)\n    (ψ : ℕ → (Fin d → ℝ)) (n : ℕ) : Matrix (Fin d) (Fin d) ℝ :=\n  Λ₀ + ∑ i ∈ Finset.range n, Matrix.vecMulVec (ψ i) (ψ i)\n\nnoncomputable def ridgeEst {d : ℕ} (Λ₀ : Matrix (Fin d) (Fin d) ℝ)\n    (ψ : ℕ → (Fin d → ℝ)) (Y : ℕ → ℝ) (n : ℕ) : Fin d → ℝ :=\n  (lambdaHat Λ₀ ψ n)⁻¹ *ᵥ (∑ i ∈ Finset.range n, Y i • ψ i)\n\nnoncomputable def bStep {d : ℕ} (Λ₀ : Matrix (Fin d) (Fin d) ℝ)\n    (ψ : ℕ → (Fin d → ℝ)) (t : ℕ) : ℝ :=\n  1 + (ψ t) ⬝ᵥ ((lambdaHat Λ₀ ψ t)⁻¹ *ᵥ (ψ t))\n\ndef batchObjective {d : ℕ} (Λ₀ : Matrix (Fin d) (Fin d) ℝ)\n    (ψ : ℕ → (Fin d → ℝ)) (Y : ℕ → ℝ) (T : ℕ) (w : Fin d → ℝ) : ℝ :=\n  (∑ t ∈ Finset.range T, (Y t - w ⬝ᵥ (ψ t)) ^ 2) + w ⬝ᵥ (Λ₀ *ᵥ w)\n\ntheorem ch15_theorem_15_6 {d : ℕ}\n    (Λ₀ : Matrix (Fin d) (Fin d) ℝ) (hΛ₀ : Λ₀.PosDef)\n    (ψ : ℕ → (Fin d → ℝ)) (Y : ℕ → ℝ)\n    (σ : ℝ) (hσ : 0 ≤ σ) (T : ℕ) :\n    (∑ t ∈ Finset.range T,\n        ((Y t - (ridgeEst Λ₀ ψ Y t) ⬝ᵥ (ψ t)) ^ 2 / bStep Λ₀ ψ t\n          + σ ^ 2 * Real.log (bStep Λ₀ ψ t)))\n      = (⨅ w : Fin d → ℝ, batchObjective Λ₀ ψ Y T w)\n        + σ ^ 2 * Real.log ((Λ₀⁻¹ * lambdaHat Λ₀ ψ T).det) := sorry\n\nend GraduateLevel\n", "target_names": ["ch15_theorem_15_6"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Online Learning & Regret", "holes": [{"start": 801, "end": 1185, "gold": "{d : ℕ}\n    (Λ₀ : Matrix (Fin d) (Fin d) ℝ) (hΛ₀ : Λ₀.PosDef)\n    (ψ : ℕ → (Fin d → ℝ)) (Y : ℕ → ℝ)\n    (σ : ℝ) (hσ : 0 ≤ σ) (T : ℕ) :\n    (∑ t ∈ Finset.range T,\n        ((Y t - (ridgeEst Λ₀ ψ Y t) ⬝ᵥ (ψ t)) ^ 2 / bStep Λ₀ ψ t\n          + σ ^ 2 * Real.log (bStep Λ₀ ψ t)))\n      = (⨅ w : Fin d → ℝ, batchObjective Λ₀ ψ Y T w)\n        + σ ^ 2 * Real.log ((Λ₀⁻¹ * lambdaHat Λ₀ ψ T).det)"}]}
{"item_id": "ch15_proposition_15_5::ch15_proposition_15_5::A-stmt", "paper_id": "ch15_proposition_15_5", "target_name": "ch15_proposition_15_5", "source": "Graduate-level", "nl_statement": "Consider probability model\n\\[\np(y|w,x)=N(w^{\\top}\\psi(x),\\sigma^{2}),\n\\]\nwith prior\n\\[\np_{0}(w)=N(0,\\sigma^{2}\\Lambda_{0}^{-1}).\n\\]\nThen given \\(\\mathcal{S}_n=\\{(X_1,Y_1),\\ldots,(X_n,Y_n)\\}\\) , we have\n\\[\np(w|\\mathcal{S}_{n})=N(\\hat{w}(\\mathcal{S}_{n}),\\sigma^{2}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}),\n\\]\nwhere \\(\\hat{w} (S_n)\\) is given by (15.3) and\n\\[\n\\hat{\\Lambda}\\left(\\mathcal{S}_{n}\\right)=\\sum_{i=1}^{n}\\psi\\left(X_{i}\\right)\\psi\\left(X_{i}\\right)^{\\top}+\\Lambda_{0}.\n\\]\nGiven \\( x \\), the posterior distribution of \\( y \\) is\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=N\\left(\\hat{w}(\\mathcal{S}_{n})^{\\top}\\psi(x),\\sigma^{2}+\\sigma^{2}\\psi(x)^{\\top}\\hat{\\Lambda}(\\mathcal{S}_{n})^{-1}\\psi(x)\\right).\n\\]", "context": [{"name": "equation (15.3)", "statement": "The ridge regression method defines, for each \\(n\\),\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\arg\\min_{w}\\left[\\sum_{i=1}^{n}(Y_{i}-w^{\\top}\\psi(X_{i}))^{2}+\\|w\\|_{\\Lambda_{0}}^{2}\\right].\n\\]\nEquivalently,\n\\[\n\\hat{w}(\\mathcal{S}_{n})=\\Lambda_{n}^{-1}\\sum_{i=1}^{n}\\psi(X_{i})Y_{i},\\quad\\Lambda_{n}=\\Lambda_{0}+\\sum_{s=1}^{n}\\psi(X_{s})\\psi(X_{s})^{\\top}.\n\\]"}, {"name": "theorem 15.3", "statement": "We have\n\\[\n\\begin{array}{l}-\\sum_{t=1}^{T}\\ln\\hat{p}\\left(Y_{t}\\mid X_{t},\\mathcal{S}_{t-1}\\right)= -\\ln\\mathbb{E}_{w\\sim p_{0}}\\prod_{t=1}^{T}p\\left(Y_{t}\\mid w,X_{t}\\right)\\\\ =\\inf_{q\\in\\Delta(\\Omega)}\\left[-\\mathbb{E}_{w\\sim q}\\sum_{t=1}^{T}\\ln p(Y_{t}|w,X_{t})+\\mathbb{E}_{w\\sim q}\\ln\\frac{q(w)}{p_{0}(w)}\\right],\\\\ \\end{array}\n\\]\nwhere \\(\\Delta (\\Omega)\\) is the set of probability distributions over \\(\\Omega\\)"}, {"name": "Equations (15.1)-(15.2) (posterior and Bayesian posterior averaging estimator)", "statement": "In the Bayesian setting, the optimal statistical estimation of posterior density for log-loss is the Bayesian posterior averaging estimator. Here we consider a prior \\(p_{0}(w)\\) on \\(\\Omega\\). Given the training data \\(\\mathcal{S}_n = \\{Z_1, \\ldots, Z_n\\}\\), the posterior distribution is\n\\[\np(w|\\mathcal{S}_{n})=\\frac{\\prod_{i=1}^{n}p(Y_{i}|w,X_{i})p_{0}(w)}{\\int_{\\Omega}\\prod_{i=1}^{n}p(Y_{i}|w^{\\prime},X_{i})p_{0}(w^{\\prime})d w^{\\prime}}.\\tag{15.1}\n\\]\nUnlike ERM, the Bayesian posterior averaging method does not use \\(p(y|\\hat{w}, x)\\) for a specific \\(\\hat{w} \\in \\Omega\\) that is estimated from the training data, but instead use the averaged probability estimate over the posterior as follows:\n\\[\n\\hat{p}(y|x,\\mathcal{S}_{n})=\\int_{\\Omega}p(y|w,x)p(w|\\mathcal{S}_{n})d w.\\tag{15.2}\n\\]"}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real Matrix\n\nnamespace GraduateLevel\n\nnoncomputable def postPrec {d n : ℕ} (Λ₀ : Matrix (Fin d) (Fin d) ℝ)\n    (ψ : (Fin n) → (Fin d → ℝ)) : Matrix (Fin d) (Fin d) ℝ :=\n  Λ₀ + ∑ i : Fin n, vecMulVec (ψ i) (ψ i)\n\nnoncomputable def postMean {d n : ℕ} (Λ₀ : Matrix (Fin d) (Fin d) ℝ)\n    (ψ : (Fin n) → (Fin d → ℝ)) (Y : (Fin n) → ℝ) : Fin d → ℝ :=\n  (postPrec Λ₀ ψ)⁻¹.mulVec (∑ i : Fin n, (Y i) • (ψ i))\n\nnoncomputable def postUnnorm {d n : ℕ} (σ : ℝ)\n    (ψ : (Fin n) → (Fin d → ℝ)) (Y : (Fin n) → ℝ)\n    (prior : Measure (Fin d → ℝ)) : Measure (Fin d → ℝ) :=\n  prior.withDensity (fun w => ENNReal.ofReal\n    (∏ i : Fin n, gaussianPDFReal (w ⬝ᵥ ψ i) ((σ^2).toNNReal) (Y i)))\n\nnoncomputable def posteriorMeasure {d n : ℕ} (σ : ℝ)\n    (ψ : (Fin n) → (Fin d → ℝ)) (Y : (Fin n) → ℝ)\n    (prior : Measure (Fin d → ℝ)) : Measure (Fin d → ℝ) :=\n  ((postUnnorm σ ψ Y prior) Set.univ)⁻¹ • postUnnorm σ ψ Y prior\n\nnoncomputable def predictiveMeasure {d n : ℕ} (σ : ℝ)\n    (ψ : (Fin n) → (Fin d → ℝ)) (Y : (Fin n) → ℝ) (φ : Fin d → ℝ)\n    (prior : Measure (Fin d → ℝ)) : Measure ℝ :=\n  (posteriorMeasure σ ψ Y prior).bind (fun w => gaussianReal (w ⬝ᵥ φ) ((σ^2).toNNReal))\n\ntheorem ch15_proposition_15_5\n    {d n : ℕ} (σ : ℝ) (hσ : 0 < σ)\n    (Λ₀ : Matrix (Fin d) (Fin d) ℝ) (hΛ₀ : Λ₀.PosDef)\n    (ψ : (Fin n) → (Fin d → ℝ)) (Y : (Fin n) → ℝ)\n    (φ : Fin d → ℝ)\n    (prior : Measure (Fin d → ℝ))\n    (hprior : ∀ c : Fin d → ℝ,\n        prior.map (fun w => c ⬝ᵥ w)\n          = gaussianReal 0 ((σ^2 * (c ⬝ᵥ Λ₀⁻¹.mulVec c)).toNNReal)) :\n    IsProbabilityMeasure (posteriorMeasure σ ψ Y prior)\n    ∧ (∀ c : Fin d → ℝ,\n       (posteriorMeasure σ ψ Y prior).map (fun w => c ⬝ᵥ w)\n         = gaussianReal (c ⬝ᵥ postMean Λ₀ ψ Y)\n             ((σ^2 * (c ⬝ᵥ (postPrec Λ₀ ψ)⁻¹.mulVec c)).toNNReal))\n    ∧ predictiveMeasure σ ψ Y φ prior\n        = gaussianReal (postMean Λ₀ ψ Y ⬝ᵥ φ)\n            ((σ^2 + σ^2 * (φ ⬝ᵥ (postPrec Λ₀ ψ)⁻¹.mulVec φ)).toNNReal) := sorry\n\nend GraduateLevel\n", "target_names": ["ch15_proposition_15_5"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Statistical Estimation & Minimax Theory", "holes": [{"start": 1248, "end": 1982, "gold": "{d n : ℕ} (σ : ℝ) (hσ : 0 < σ)\n    (Λ₀ : Matrix (Fin d) (Fin d) ℝ) (hΛ₀ : Λ₀.PosDef)\n    (ψ : (Fin n) → (Fin d → ℝ)) (Y : (Fin n) → ℝ)\n    (φ : Fin d → ℝ)\n    (prior : Measure (Fin d → ℝ))\n    (hprior : ∀ c : Fin d → ℝ,\n        prior.map (fun w => c ⬝ᵥ w)\n          = gaussianReal 0 ((σ^2 * (c ⬝ᵥ Λ₀⁻¹.mulVec c)).toNNReal)) :\n    IsProbabilityMeasure (posteriorMeasure σ ψ Y prior)\n    ∧ (∀ c : Fin d → ℝ,\n       (posteriorMeasure σ ψ Y prior).map (fun w => c ⬝ᵥ w)\n         = gaussianReal (c ⬝ᵥ postMean Λ₀ ψ Y)\n             ((σ^2 * (c ⬝ᵥ (postPrec Λ₀ ψ)⁻¹.mulVec c)).toNNReal))\n    ∧ predictiveMeasure σ ψ Y φ prior\n        = gaussianReal (postMean Λ₀ ψ Y ⬝ᵥ φ)\n            ((σ^2 + σ^2 * (φ ⬝ᵥ (postPrec Λ₀ ψ)⁻¹.mulVec φ)).toNNReal)"}]}
{"item_id": "ch2_theorem_2_1::theorem_2_1::A-stmt", "paper_id": "ch2_theorem_2_1", "target_name": "theorem_2_1", "source": "Graduate-level", "nl_statement": "Let \\(X_{1},\\ldots,X_{n}\\) be \\(n\\) iid Gaussian random variables \\(X_{i}\\sim N(\\mu,\\sigma^{2})\\) and let \\(\\bar{X}_n = n^{-1}\\sum_{i = 1}^n X_i\\). Then given any \\(\\epsilon >0\\):\n\\[\n0.5e^{-n(\\epsilon+\\sigma/\\sqrt{n})^{2}/2\\sigma^{2}}\\leq\\operatorname*{Pr}(\\bar{X}_{n}\\geq\\mu+\\epsilon)\\leq 0.5e^{-n\\epsilon^{2}/2\\sigma^{2}}.\n\\]", "context": null, "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\nnoncomputable def sampleMean {Ω : Type*} {n : ℕ} (X : Fin n → Ω → ℝ) (ω : Ω) : ℝ :=\n  (1 / (n : ℝ)) * ∑ i, X i ω\n\ntheorem theorem_2_1 {Ω : Type*} [MeasurableSpace Ω]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (n : ℕ) (μ σ : ℝ) (X : Fin n → Ω → ℝ)\n    (hσ : 0 < σ) (hn : 0 < n)\n    (hX_meas : ∀ i, Measurable (X i))\n    (hX_indep : ProbabilityTheory.iIndepFun X P)\n    (hX_gaussian :\n      ∀ i, MeasureTheory.Measure.map (X i) P =\n        ProbabilityTheory.gaussianReal μ (Real.toNNReal (σ ^ 2))) :\n    ∀ ε : ℝ, 0 < ε →\n      0.5 * Real.exp (-(n : ℝ) * (ε + σ / Real.sqrt (n : ℝ)) ^ 2 / (2 * σ ^ 2)) ≤\n          (P {ω : Ω | μ + ε ≤ sampleMean X ω}).toReal ∧\n        (P {ω : Ω | μ + ε ≤ sampleMean X ω}).toReal ≤\n          0.5 * Real.exp (-(n : ℝ) * ε ^ 2 / (2 * σ ^ 2)) := sorry\n\nend GraduateLevel\n", "target_names": ["theorem_2_1"], "topic_domain": "Probability & Concentration", "topic_subtopic": "Distributions, Tails & Moments", "holes": [{"start": 218, "end": 858, "gold": "{Ω : Type*} [MeasurableSpace Ω]\n    (P : Measure Ω) [IsProbabilityMeasure P]\n    (n : ℕ) (μ σ : ℝ) (X : Fin n → Ω → ℝ)\n    (hσ : 0 < σ) (hn : 0 < n)\n    (hX_meas : ∀ i, Measurable (X i))\n    (hX_indep : ProbabilityTheory.iIndepFun X P)\n    (hX_gaussian :\n      ∀ i, MeasureTheory.Measure.map (X i) P =\n        ProbabilityTheory.gaussianReal μ (Real.toNNReal (σ ^ 2))) :\n    ∀ ε : ℝ, 0 < ε →\n      0.5 * Real.exp (-(n : ℝ) * (ε + σ / Real.sqrt (n : ℝ)) ^ 2 / (2 * σ ^ 2)) ≤\n          (P {ω : Ω | μ + ε ≤ sampleMean X ω}).toReal ∧\n        (P {ω : Ω | μ + ε ≤ sampleMean X ω}).toReal ≤\n          0.5 * Real.exp (-(n : ℝ) * ε ^ 2 / (2 * σ ^ 2))"}]}
{"item_id": "ch8_corollary_8_3::ch8_corollary_8_3::A-stmt", "paper_id": "ch8_corollary_8_3", "target_name": "ch8_corollary_8_3", "source": "Graduate-level", "nl_statement": "Consider a countable family of approximate ERM algorithms \\(\\{\\mathcal{A}(\\theta, \\cdot): \\theta = 1, 2, \\ldots\\}\\), each characterized by its model space \\(\\mathcal{F}(\\theta)\\), and returns a function \\(\\hat{f}_{\\theta} \\in \\mathcal{F}(\\theta)\\) such that\n\\[\n\\phi(\\hat{f},\\mathcal{S}_{n})\\leq\\inf_{f\\in\\mathcal{F}(\\theta)}\\phi(f,\\mathcal{S}_{n})+\\epsilon^{\\prime},\n\\]\nwhere we use the notation of Definition 8.1.\nAssume further that \\(\\sup_{Z,Z'}[\\phi(f,Z) - \\phi(f,Z')]\\leq M\\) for all \\(f\\), and we use (8.2) to select the best model \\(\\hat{\\theta}\\) on the validation data \\(\\bar{S}_m\\), with the choice\n\\[\nr_{m}(q)=M\\sqrt{\\frac{\\ln(1/q)}{2 m}}.\n\\]\nThen the following result holds with probability at least \\(1 - \\delta\\) over random selection\nof \\(S_{n}\\) and \\(\\bar{S}_m\\) ..\n\\[\n\\begin{array}{l}\\phi(\\mathcal{A}(\\hat{\\theta},\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta}\\left[\\inf_{f\\in\\mathcal{F}(\\theta)}\\phi(f,\\mathcal{D})+2 R_{n}(\\mathcal{G}(\\theta),\\mathcal{D})+r_{m}(q(\\theta))\\right]\\\\ +\\tilde{\\epsilon}+\\epsilon^{\\prime}+M\\sqrt{\\frac{2\\ln(4/\\delta)}{n}}+M\\sqrt{\\frac{2\\ln(4/\\delta)}{m}},\\\\ \\end{array}\n\\]\nwhere \\( R_{n}(\\mathcal{G}(\\theta),\\mathcal{D}) \\) is the Rademacher complexity of \\( \\mathcal{G}(\\theta) = \\{\\phi (f,\\cdot):f\\in \\mathcal{F}(\\theta)\\} \\) and \\( q(\\theta) \\) satisfies (8.1).", "context": [{"name": "definition 8.1", "statement": "Consider a loss function \\(\\phi(f, z): \\mathcal{F} \\times \\mathcal{Z} \\to \\mathbb{R}\\), and a model family \\(\\{\\mathcal{A}(\\theta, S_n): \\Theta \\times \\mathcal{Z}^n \\to \\mathcal{F}, n \\geq 0\\}\\). Consider \\(N \\geq n \\geq 0\\), and iid dataset \\(S_n \\subset S_N \\sim \\mathcal{D}^N\\). A model selection algorithm \\(\\bar{\\mathcal{A}}\\) maps \\(S_N\\) to \\(\\hat{\\theta} = \\hat{\\theta}(S_N) \\in \\Theta\\),\nand then train a model \\(\\hat{f} = \\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_N), \\mathcal{S}_n) = \\bar{\\mathcal{A}}(\\mathcal{S}_N)\\). It satisfies an \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) oracle inequality if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(\\mathcal{S}_N\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta}(\\mathcal{S}_{N}),\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)].\n\\]\nMore generally, a learning algorithm \\(\\bar{\\mathcal{A}}: S_N \\to \\mathcal{F}\\) is \\(\\epsilon_{n,N}(\\cdot, \\cdot)\\) adaptive to the model family \\(\\{\\mathcal{A}(\\theta, \\cdot): \\theta \\in \\Theta\\}\\) if there exists \\(\\epsilon_{n,N}(\\theta, \\delta)\\), such that for all \\(\\delta \\in (0,1)\\), with probability at least \\(1 - \\delta\\) over \\(S_N\\):\n\\[\n\\phi(\\bar{\\mathcal{A}}(\\mathcal{S}_{N}),\\mathcal{D})\\leq\\inf_{\\theta\\in\\Theta}\\left[\\mathbb{E}_{\\mathcal{S}_{n}}\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+\\epsilon_{n,N}(\\theta,\\delta)\\right].\n\\]\nNote that we have used a slightly different notation \\(\\phi(f, Z)\\), instead of \\(\\phi(w, Z)\\) as in earlier chapters, to emphasize that in model selection, the prediction function \\(f(w, \\cdot)\\) from different models can be compared, but the parameter \\(w\\) of the functions may not be comparable when different models have different functional forms such as decision trees versus neural networks. Without causing confusion, one may consider \\(f\\) to be the function represented by \\(w\\), so that \\(\\phi(f, Z) = L(f(X), Y)\\) and \\(\\phi(w, Z) = L(f(w, X), Y)\\). Similar to Definition 3.8, we use the simplified notation\n\\[\n\\phi(f,\\mathcal{D})=\\mathbb{E}_{Z\\sim\\mathcal{D}}\\phi(f,Z),\\quad\\phi(f,\\mathcal{S}_{n})=\\frac{1}{n}\\sum_{Z\\in\\mathcal{S}_{n}}\\phi(f,Z).\n\\]\nThe concept adaptivity is more general than model selection, as the algorithm does not need to choose a specific model \\(\\mathcal{A}(\\theta,\\cdot)\\) indexed by \\(\\theta\\). For example, a commonly used method to achieve adaptivity is model averaging, where we take the average of several different model outputs \\(\\hat{f}_{\\theta_j} = \\mathcal{A}(\\theta_j,\\mathcal{S}_n)\\) as\n\\[\n\\hat{f}_{\\mathrm{avg}}(x)=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{f}_{\\theta_{j}}(x).\n\\]\nIf different models have the same functional form, with a vector model parameter \\( w \\in \\mathbb{R}^d \\): \\( \\hat{f}_{\\theta_j}(x) = f(\\hat{w}_j, x) \\). Then one may also average the model parameters as follows:\n\\[\n\\hat{w}_{\\mathrm{avg}}=\\frac{1}{k}\\sum_{j=1}^{k}\\hat{w}_{j}.\n\\]\nIn general, the training data \\( S_{n} \\) is a subset of \\( S_{N} \\), and we allow \\( n < N \\) in order to analyze validation methods which are commonly used in practice. In addition to the large probability bounds in Definition 8.1, one may also derive bounds that hold for expectation over \\( S_{N} \\)."}, {"name": "Model Selction on Validation Data", "statement": "Assume \\(\\sup_{Z,Z'}[\\phi(f,Z) - \\phi(f,Z')] \\leq M\\) for all \\(f\\), and the validation data \\(\\bar{S}_m\\) contains \\(m\\) iid samples from \\(\\mathcal{D}\\). Given training data \\(S_n\\), consider (8.2) with\n\\[\nr_{m}(q)=M\\sqrt{\\frac{\\ln(1/q)}{2 m}}.\n\\]\nThen with probability at least \\(1 - \\delta\\) over the random selection of \\(S_{m}\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta},\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta}Q(\\theta,\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\bar{\\mathcal{S}}_{m})+\\tilde{\\epsilon}+M\\sqrt{\\frac{\\ln(1/\\delta)}{2 m}}.\n\\]\nThis implies the following oracle inequality. With probability at least \\(1 - \\delta\\) over the random sampling of \\(\\bar{S}_m\\):\n\\[\n\\phi(\\mathcal{A}(\\hat{\\theta},\\mathcal{S}_{n}),\\mathcal{D})\\leq\\inf_{\\theta}[\\phi(\\mathcal{A}(\\theta,\\mathcal{S}_{n}),\\mathcal{D})+r_{m}(q(\\theta))]+\\tilde{\\epsilon}+M\\sqrt{\\frac{2\\ln(2/\\delta)}{m}},\n\\]\nwhere \\(q(\\theta)\\) satisfies (8.1)."}, {"name": "corollary 6.21", "statement": "Assume that for some \\(M \\geq 0\\):\n\\[\n\\sup_{w\\in\\Omega}\\left[\\sup_{z,z^{\\prime}}[\\phi(w,z)-\\phi(w,z^{\\prime})]+\\Delta_{n}h(w)\\right]\\leq M.\n\\]\nThen the approximate ERM method (6.1) satisfies the following oracle inequality. With probability at least \\( 1 - \\delta - \\delta' \\):\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln\\left(1/\\delta^{\\prime}\\right)}{2 n}}\\right]\\\\ +\\epsilon^{\\prime}+\\epsilon_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]\nIf \\( h(\\cdot) \\) has the decomposition (6.3), then\n\\[\n\\begin{array}{l}\\phi(\\hat{w},\\mathcal{D})\\leq\\inf_{w\\in\\Omega}\\left[\\phi(w,\\mathcal{D})+\\mathbb{E}_{\\mathcal{S}_{n}}h(w,\\mathcal{S}_{n})+\\Delta_{n}h(w)\\sqrt{\\frac{\\ln(1/\\delta^{\\prime})}{2 n}}\\right]\\\\ +\\epsilon^{\\prime}+2 R_{n}^{h}(\\mathcal{G},\\mathcal{D})+2 M\\sqrt{\\frac{\\ln(2/\\delta)}{2 n}}.\\\\ \\end{array}\n\\]"}, {"name": "Union Bound", "statement": "(Union Bound) Consider \\(m\\) events \\(E_1, \\ldots, E_m\\). The following probability inequality holds:\n\\[\n\\Pr (E_1 \\cup \\cdots \\cup E_m) \\leq \\sum_{j=1}^{m} \\Pr(E_j).\n\\]\n\nThe union bound has an alternative expression, which is often used in the learning theory analysis. If each \\(E_j\\) occurs with probability at least \\(1-\\delta_j\\) for \\(j=1, \\ldots, m\\), then with probability at least \\(1-\\sum_{j=1}^{m} \\delta_j\\), all of events \\(\\{E_j\\}\\) occur simultaneously for \\(j=1, \\ldots, m\\)."}, {"name": "definition 6.3", "statement": "Given \\(S_{n}=\\{Z_{1},\\ldots,Z_{n}\\}\\), the (one-sided) empirical Rademacher complexity of \\(\\mathcal{G}\\) is defined as\n\\[\nR(\\mathcal{G},\\mathcal{S}_{n}) = \\mathbb{E}_{\\sigma}\\sup_{w\\in \\Omega}\\frac{1}{n}\\sum_{i = 1}^{n}\\sigma_{i}\\phi (w,Z_{i}),\n\\]\nwhere \\(\\sigma_1, \\ldots, \\sigma_n\\) are independent uniform \\(\\{\\pm 1\\}\\)-valued Bernoulli random variables. Moreover, the expected Rademacher complexity is\n\\[\nR_{n}(\\mathcal{G},\\mathcal{D})=\\mathbb{E}_{\\mathcal{S}_{n}\\sim\\mathcal{D}^{n}}R(\\mathcal{G},\\mathcal{S}_{n}).\n\\]"}, {"name": "equation (8.1)", "statement": "Let \\(q(\\theta)\\geq 0\\) for \\(\\theta = 1,2,\\ldots\\) satisfy\n\\[\n\\sum_{\\theta=1}^{\\infty} q(\\theta) \\leq 1. \\tag{8.1}\n\\]"}, {"name": "equation (8.2)", "statement": "Given training data \\(S_n\\) and validation data \\(\\bar{S}_m\\), define the validation objective\n\\[\nQ(\\theta, f, \\bar{\\mathcal{S}}_m) = \\phi(f, \\bar{\\mathcal{S}}_m) + r_m(q(\\theta)).\n\\]\nWe select \\(\\hat{\\theta}\\) by approximate minimization:\n\\[\nQ(\\hat{\\theta}, \\mathcal{A}(\\hat{\\theta}, \\mathcal{S}_n), \\bar{\\mathcal{S}}_m) \\leq \\inf_{\\theta} Q(\\theta, \\mathcal{A}(\\theta, \\mathcal{S}_n), \\bar{\\mathcal{S}}_m) + \\tilde{\\epsilon}. \\tag{8.2}\n\\]\nHere \\(\\tilde{\\epsilon}\\) is the additive slack in the validation-based model-selection rule."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real\n\nnamespace GraduateLevel\n\ndef RadSigns (n : ℕ) : Type := Fin n → ({-1, 1} : Finset ℤ)\n\ninstance {n} : Fintype (RadSigns n) :=\n  inferInstanceAs (Fintype (Fin n → { x // x ∈ {-1, 1} }))\n\nnoncomputable def empRadSigned {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (S : Fin n → 𝒳) : ℝ :=\n  (Fintype.card (RadSigns n) : ℝ)⁻¹ *\n    ∑ σ : RadSigns n, ⨆ i, (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * f i (S k)\n\nnoncomputable def popRadSigned {Ω : Type*} [MeasurableSpace Ω] {ι : Type*} {𝒳 : Type*}\n    (n : ℕ) (f : ι → 𝒳 → ℝ) (μ : Measure Ω) (X : Ω → 𝒳) : ℝ :=\n  ∫ ω : Fin n → Ω, empRadSigned n f (X ∘ ω) ∂(Measure.pi fun _ => μ)\n\nnoncomputable def rmPenalty (M : ℝ) (m : ℕ) (q : ℝ) : ℝ :=\n  M * Real.sqrt (Real.log (1 / q) / (2 * m))\n\nnoncomputable def radCx {Z Pred : Type*} [MeasurableSpace Z]\n    (n : ℕ) (φ : Pred → Z → ℝ) (F : Set Pred) (D : Measure Z) : ℝ :=\n  popRadSigned n (fun f : F => fun z => φ (f : Pred) z) D id\n\n\ntheorem ch8_corollary_8_3\n    {Z Pred : Type*} [MeasurableSpace Z] [Nonempty Z] [MeasurableSpace Pred]\n    (D : Measure Z) [IsProbabilityMeasure D]\n    (φ : Pred → Z → ℝ)\n    (F : ℕ+ → Set Pred)\n    (hFne : ∀ θ, (F θ).Nonempty)\n    (hint : ∀ f, Integrable (fun z => φ f z) D)\n    (M ε' εt δ : ℝ) (n m : ℕ)\n    (hn : 0 < n) (hm : 0 < m)\n    (hδ : 0 < δ ∧ δ < 1)\n    (hM : ∀ f, ∀ z z' : Z, φ f z - φ f z' ≤ M)\n    (hBddAbove : ∀ (θ : ℕ+) (s : Fin n → Z) (σ : RadSigns n),\n        BddAbove (Set.range (fun f : F θ =>\n          (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * φ (f : Pred) (s k))))\n    (hRadInt : ∀ θ : ℕ+, Integrable\n        (fun s : Fin n → Z => empRadSigned n\n          (fun f : F θ => fun z => φ (f : Pred) z) s)\n        (Measure.pi (fun _ : Fin n => D)))\n    (q : ℕ+ → ℝ) (hq_pos : ∀ θ, 0 < q θ)\n    (hq_summable : Summable q)\n    (hq_sum : ∑' θ, q θ ≤ 1)\n    {Ωs : Type*} [MeasurableSpace Ωs] (P : Measure Ωs) [IsProbabilityMeasure P]\n    (Strain : Ωs → Fin n → Z) (Sval : Ωs → Fin m → Z)\n    (hStrain_meas : Measurable Strain) (hSval_meas : Measurable Sval)\n    (hStrain_law : ∀ i, Measure.map (fun ω => Strain ω i) P = D)\n    (hSval_law : ∀ i, Measure.map (fun ω => Sval ω i) P = D)\n    (hStrain_indep : iIndepFun (fun i ω => Strain ω i) P)\n    (hSval_indep : iIndepFun (fun i ω => Sval ω i) P)\n    (hSplit_indep : IndepFun Strain Sval P)\n    (A : ℕ+ → (Fin n → Z) → Pred) (hA_mem : ∀ θ s, A θ s ∈ F θ)\n    (hA_meas : ∀ θ, Measurable (A θ))\n\n    (hφ_meas : Measurable (fun p : Pred × Z => φ p.1 p.2))\n    (hERM : ∀ θ s, (n:ℝ)⁻¹ * ∑ i, φ (A θ s) (s i) ≤ (⨅ f : F θ, (n:ℝ)⁻¹ * ∑ i, φ (f:Pred) (s i)) + ε')\n    (thetaHat : (Fin n → Z) → (Fin m → Z) → ℕ+)\n\n    (hSel : ∀ s v, ((m:ℝ)⁻¹ * ∑ i, φ (A (thetaHat s v) s) (v i) + rmPenalty M m (q (thetaHat s v))) ≤ (⨅ θ:ℕ+, ((m:ℝ)⁻¹ * ∑ i, φ (A θ s) (v i) + rmPenalty M m (q θ))) + εt)\n    (hbddSel : ∀ (s : Fin n → Z) (v : Fin m → Z), BddBelow (Set.range (fun θ : ℕ+ =>\n      (m:ℝ)⁻¹ * ∑ i, φ (A θ s) (v i) + rmPenalty M m (q θ))))\n    (hbddOracle : BddBelow (Set.range (fun θ : ℕ+ =>\n      (⨅ f : F θ, ∫ z, φ (f:Pred) z ∂D) + 2 * radCx n φ (F θ) D + rmPenalty M m (q θ)))) :\n    1 - δ ≤ (P {ω : Ωs | ∫ z, φ (A (thetaHat (Strain ω) (Sval ω)) (Strain ω)) z ∂D ≤ (⨅ θ:ℕ+, ((⨅ f : F θ, ∫ z, φ (f:Pred) z ∂D) + 2 * radCx n φ (F θ) D + rmPenalty M m (q θ))) + εt + ε' + M * Real.sqrt (2 * Real.log (4/δ) / n) + M * Real.sqrt (2 * Real.log (4/δ) / m)}).toReal := sorry\n\nend GraduateLevel\n", "target_names": ["ch8_corollary_8_3"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Generalization & Learnability", "holes": [{"start": 1011, "end": 3384, "gold": "{Z Pred : Type*} [MeasurableSpace Z] [Nonempty Z] [MeasurableSpace Pred]\n    (D : Measure Z) [IsProbabilityMeasure D]\n    (φ : Pred → Z → ℝ)\n    (F : ℕ+ → Set Pred)\n    (hFne : ∀ θ, (F θ).Nonempty)\n    (hint : ∀ f, Integrable (fun z => φ f z) D)\n    (M ε' εt δ : ℝ) (n m : ℕ)\n    (hn : 0 < n) (hm : 0 < m)\n    (hδ : 0 < δ ∧ δ < 1)\n    (hM : ∀ f, ∀ z z' : Z, φ f z - φ f z' ≤ M)\n    (hBddAbove : ∀ (θ : ℕ+) (s : Fin n → Z) (σ : RadSigns n),\n        BddAbove (Set.range (fun f : F θ =>\n          (n : ℝ)⁻¹ * ∑ k : Fin n, (σ k : ℝ) * φ (f : Pred) (s k))))\n    (hRadInt : ∀ θ : ℕ+, Integrable\n        (fun s : Fin n → Z => empRadSigned n\n          (fun f : F θ => fun z => φ (f : Pred) z) s)\n        (Measure.pi (fun _ : Fin n => D)))\n    (q : ℕ+ → ℝ) (hq_pos : ∀ θ, 0 < q θ)\n    (hq_summable : Summable q)\n    (hq_sum : ∑' θ, q θ ≤ 1)\n    {Ωs : Type*} [MeasurableSpace Ωs] (P : Measure Ωs) [IsProbabilityMeasure P]\n    (Strain : Ωs → Fin n → Z) (Sval : Ωs → Fin m → Z)\n    (hStrain_meas : Measurable Strain) (hSval_meas : Measurable Sval)\n    (hStrain_law : ∀ i, Measure.map (fun ω => Strain ω i) P = D)\n    (hSval_law : ∀ i, Measure.map (fun ω => Sval ω i) P = D)\n    (hStrain_indep : iIndepFun (fun i ω => Strain ω i) P)\n    (hSval_indep : iIndepFun (fun i ω => Sval ω i) P)\n    (hSplit_indep : IndepFun Strain Sval P)\n    (A : ℕ+ → (Fin n → Z) → Pred) (hA_mem : ∀ θ s, A θ s ∈ F θ)\n    (hA_meas : ∀ θ, Measurable (A θ))\n\n    (hφ_meas : Measurable (fun p : Pred × Z => φ p.1 p.2))\n    (hERM : ∀ θ s, (n:ℝ)⁻¹ * ∑ i, φ (A θ s) (s i) ≤ (⨅ f : F θ, (n:ℝ)⁻¹ * ∑ i, φ (f:Pred) (s i)) + ε')\n    (thetaHat : (Fin n → Z) → (Fin m → Z) → ℕ+)\n\n    (hSel : ∀ s v, ((m:ℝ)⁻¹ * ∑ i, φ (A (thetaHat s v) s) (v i) + rmPenalty M m (q (thetaHat s v))) ≤ (⨅ θ:ℕ+, ((m:ℝ)⁻¹ * ∑ i, φ (A θ s) (v i) + rmPenalty M m (q θ))) + εt)\n    (hbddSel : ∀ (s : Fin n → Z) (v : Fin m → Z), BddBelow (Set.range (fun θ : ℕ+ =>\n      (m:ℝ)⁻¹ * ∑ i, φ (A θ s) (v i) + rmPenalty M m (q θ))))\n    (hbddOracle : BddBelow (Set.range (fun θ : ℕ+ =>\n      (⨅ f : F θ, ∫ z, φ (f:Pred) z ∂D) + 2 * radCx n φ (F θ) D + rmPenalty M m (q θ)))) :\n    1 - δ ≤ (P {ω : Ωs | ∫ z, φ (A (thetaHat (Strain ω) (Sval ω)) (Strain ω)) z ∂D ≤ (⨅ θ:ℕ+, ((⨅ f : F θ, ∫ z, φ (f:Pred) z ∂D) + 2 * radCx n φ (F θ) D + rmPenalty M m (q θ))) + εt + ε' + M * Real.sqrt (2 * Real.log (4/δ) / n) + M * Real.sqrt (2 * Real.log (4/δ) / m)}).toReal"}]}
{"item_id": "ch16_theorem_16_7::ch16_theorem_16_7::A-stmt", "paper_id": "ch16_theorem_16_7", "target_name": "ch16_theorem_16_7", "source": "Graduate-level", "nl_statement": "Assume that the reward \\(r_t(a) \\in [0,1]\\). Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). With a choice of\n\\[\n\\hat{c}_{t}(a)=\\sqrt{\\frac{\\ln(2(\\hat{n}_{t}(a)+1)^{2}/\\delta)}{2\\hat{n}_{t}(a)}},\n\\]\nwe have\n\\[\n\\mathrm{REG}_{T}\\leq\\inf_{\\Delta_{0}>0}\\left[(T-K)\\Delta_{0}+\\sum_{a=1}^{K}\\frac{2\\ln(2 K T^{2}/\\delta)}{\\Delta(a)}\\mathbb{1}(\\Delta(a)>\\Delta_{0})\\right]+\\sum_{a=1}^{K}\\Delta(a).\n\\]", "context": [{"name": "theorem 16.3", "statement": "Assume that rewards \\(r_t(a) \\in [0,1]\\). Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). With a choice of\n\\[\n\\hat{c}_{t}(a)=\\sqrt{\\frac{\\ln(2(\\hat{n}_{t}(a)+1)^{2}/\\delta)}{2\\hat{n}_{t}(a)}},\n\\]\nwe have with probability at least \\(1 - \\delta\\):\n\\[\n\\mathrm{REG}_{T}\\leq(K-1)+\\sqrt{8\\ln(2 K T^{2}/\\delta)(T-K)K}.\n\\]"}, {"name": "lemma 16.6", "statement": "Let \\(a_* \\in \\arg \\max_a \\mu(a)\\). Let\n\\[\n\\delta_{1}=\\Pr\\left[\\exists t>K:\\mu\\left(a_{*}\\right)>\\hat{\\mu}_{t-1}\\left(a_{*}\\right)+\\hat{c}_{t-1}\\left(a_{*}\\right)\\right],\n\\]\n\\[\n\\delta_{2}=\\Pr\\left[\\exists t>K,a\\in\\{1,\\dots,K\\}\\backslash\\{a_{*}\\}:\\mu(a)<\\hat{\\mu}_{t-1}(a)-\\hat{c}_{t-1}^{\\prime}(a)\\right].\n\\]\nDefine \\(\\Delta(a) = \\mu(a_{*}) - \\mu(a)\\), and\n\\[\nT(a)=\\max\\left\\{m:\\Delta(a)\\leq\\hat{c}_{t-1}(a)+\\hat{c}_{t-1}^{\\prime}(a),\\hat{n}_{t-1}(a)=m,K<t\\leq T\\right\\}\\cup\\{0\\}.\n\\]\nThen for Algorithm 16.1, we have with probability at least \\(1 - \\delta_{1} - \\delta_{2}\\):\n\\[\n\\operatorname{REG}_{T}\\leq\\inf_{\\Delta_{0}>0}\\left[\\sum_{a=1}^{K}T(a)\\Delta(a)\\mathbb{1}(\\Delta(a)>\\Delta_{0})+(T-K)\\Delta_{0}\\right]+\\sum_{a=1}^{K}\\Delta(a).\n\\]"}, {"name": "definition 16.1", "statement": "In the multi-armed bandit (or MAB) problem, we consider \\(K\\) arms. The environment generates a sequence of reward vectors for time steps \\(t \\geq 1\\) as \\(r_t = [r_t(1), \\ldots, r_t(K)]\\). Each \\(r_t(a)\\) is associated with an arm \\(a \\in \\{1, \\ldots, K\\}\\). In the literature, an arm \\(a\\) is also referred to as an action. At each time step \\(t = 1, 2, \\ldots, T\\),\n- The player pulls one of the arms \\( a_{t} \\in \\{1, \\dots, K\\} \\).\n- The environment returns the reward \\( r_t(a_t) \\), but does not reveal information on any other arm \\( a \\neq a_t \\).\nAt each time \\( t \\), a (randomized) bandit algorithm takes the historic observations observed so far, and maps it to a distribution \\( \\hat{\\pi}_{t-1} \\) over actions \\( a \\in \\{1, \\ldots, K\\} \\). We then draw a random action \\( a_t \\) (arm) from \\( \\hat{\\pi}_{t-1} \\). Here \\( \\hat{\\pi}_{t-1} \\) is referred to as policy.\nIn the adversarial bandit problem, we are given an arbitrary reward sequence \\(\\{[r_1(1),\\ldots,r_t(K)]:t\\geq 1\\}\\) before hand. The adversary in this setting is called oblivious, who decides the rewards of all the rounds before the game starts (but may take advantage of the bandit algorithm's weakness). For this reward se\nquence, the expected cumulative reward of a randomized bandit algorithm is\n\\[\n\\mathbb{E}\\sum_{t=1}^{T}r_{t}(a_{t}),\n\\]\nwhere the randomization is over the internal randomization of the bandit algorithm. The expected regret is defined as\n\\[\n\\operatorname{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\mathbb{E}\\sum_{t=1}^{T}r_{t}\\left(a_{t}\\right).\\tag{16.1}\n\\]\nIf the bandit algorithm is deterministic: i.e. pulling a single arm \\(a_{t}\\) only at any time step, then the regret is\n\\[\n\\mathrm{REG}_{T}=\\max_{a}\\sum_{t=1}^{T}r_{t}(a)-\\sum_{t=1}^{T}r_{t}(a_{t}).\n\\]\nIn the adversarial bandit problem, our goal is to minimize the expected regret \\(\\mathrm{REG}_T\\) (and the expectation is with respect to the internal randomization of the algorithm instead of the data).\nAnother form of bandit problem, called stochastic bandit, assumes that the reward \\( r_t(a) \\) is drawn independently from a distribution \\( \\mathcal{D}_a \\), with mean \\( \\mu(a) = \\mathbb{E}_{r_t(a) \\sim \\mathcal{D}_a} [r_t(a)] \\). In this setting, the goal of bandit is to find \\( a \\) that maximizes the expected reward \\( \\mu(a) \\). We are interested in the expected regret defined as:\n\\[\n\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu\\left(a_{t}\\right).\\tag{16.2}\n\\]\nAgain, this regret is defined for each realization of the stochastic rewards, and the expectation is with respect to the randomness of the algorithm. If algorithm is deterministic and outputs \\( a_{t} \\) at each time \\( t \\), then\n\\[\n\\mathrm{REG}_{T}=T\\max_{a}\\mu(a)-\\sum_{t=1}^{T}\\mu(a_{t}).\n\\]\nWe can further include randomization over data into the regret of stochastic bandit, by considering expected regret as follows:\n\\[\n\\mathbb{E}\\operatorname{REG}_{T}=T\\max_{a}\\mu(a)-\\mathbb{E}\\sum_{t=1}^{T}\\mu(a_{t}),\n\\]\nwhere the expectation is with respect to both the data and the internal randomization of the learning algorithm."}, {"name": "UCB Algorithm", "statement": "Algorithm 16.1: UCB Algorithm\nInput: \\(K\\) and \\(T\\geq K\\)   \n1 for \\(a=1,\\dots,K\\) do   \n2 Let \\(\\hat{n}_0(a) = 0\\)   \n3 Let \\(\\hat{\\mu}_0(a) = 0\\)   \n4 for \\(t=1,2,\\ldots,T\\) do   \n5 if \\(t\\leq K\\) then   \n6 Let \\(a_{t} = t\\)   \n7 else   \n8 Let \\(a_{t}\\in \\arg \\max_{a}[\\hat{\\mu}_{t - 1}(a) + \\hat{c}_{t - 1}(a)]\\) according to (16.3)   \n9 Pull arm \\(a_{t}\\) and observe reward \\(r_t(a_t)\\)"}, {"name": "equation 16.3", "statement": "For each arm \\(a\\), define the number of pulls up to time \\(t\\) and the corresponding empirical mean by\n\\[\n\\hat n_t(a)=\\sum_{s=1}^{t}\\mathbb{1}(a_s=a), \\qquad\n\\hat\\mu_t(a)=\\frac{1}{\\hat n_t(a)}\\sum_{s=1}^{t} r_s(a_s)\\mathbb{1}(a_s=a),\n\\]\nwith the initialization \\(\\hat n_0(a)=0\\) and \\(\\hat\\mu_0(a)=0\\). In the UCB algorithm, after the initialization rounds, the arm is selected according to\n\\[\na_t\\in\\arg\\max_a\\left[\\hat\\mu_{t-1}(a)+\\hat c_{t-1}(a)\\right]. \\tag{16.3}\n\\]\nHere \\(\\hat c_t(a)\\) denotes the confidence width used by the algorithm."}], "nl_notes": "NOTE — Do not use `nlAlgo`. Model the run of Algorithm 16.1 by an arbitrary measurable, deterministic arm-selection rule that plays the arms in round-robin order during the warm-up and afterwards picks some maximiser of μ̂ + ĉ, with ties broken arbitrarily.", "lean": "import Mathlib\n\nopen MeasureTheory ProbabilityTheory Real Finset\n\nnamespace GraduateLevel\n\nlemma measurable_encode {α : Type*} {_ : MeasurableSpace α} [Encodable α]\n    [MeasurableSingletonClass α] :\n    Measurable (Encodable.encode (α := α)) := by\n  refine measurable_to_nat fun a ↦ ?_\n  have : Encodable.encode ⁻¹' {Encodable.encode a} = {a} := by ext; simp\n  rw [this]\n  exact measurableSet_singleton _\n\nlemma measurableEmbedding_encode (α : Type*) {_ : MeasurableSpace α} [Encodable α]\n    [MeasurableSingletonClass α] :\n    MeasurableEmbedding (Encodable.encode (α := α)) where\n  injective := Encodable.encode_injective\n  measurable := measurable_encode\n  measurableSet_image' _ _ := .of_discrete\n\nsection Finite\n\nvariable {𝓧 𝓨 α : Type*} {m𝓧 : MeasurableSpace 𝓧} {m𝓨 : MeasurableSpace 𝓨}\n  {mα : MeasurableSpace α} [TopologicalSpace α] [LinearOrder α]\n  [OpensMeasurableSpace α] [OrderClosedTopology α] [SecondCountableTopology α]\n\nlemma measurableSet_isMax [Countable 𝓨]\n    {f : 𝓧 → 𝓨 → α} (hf : ∀ y, Measurable (fun x ↦ f x y)) (y : 𝓨) :\n    MeasurableSet {x | ∀ z, f x z ≤ f x y} := by\n  rw [show {x | ∀ y', f x y' ≤ f x y} = ⋂ y', {x | f x y' ≤ f x y} by ext; simp]\n  exact MeasurableSet.iInter fun z ↦ measurableSet_le (by fun_prop) (by fun_prop)\n\nlemma exists_isMaxOn' {α : Type*} [LinearOrder α]\n    [Nonempty 𝓨] [Finite 𝓨] [Encodable 𝓨] (f : 𝓧 → 𝓨 → α) (x : 𝓧) :\n    ∃ n : ℕ, ∃ y, n = Encodable.encode y ∧ ∀ z, f x z ≤ f x y := by\n  obtain ⟨y, h⟩ := Finite.exists_max (f x)\n  exact ⟨Encodable.encode y, y, rfl, h⟩\n\nnoncomputable\ndef measurableArgmax [Nonempty 𝓨] [Finite 𝓨] [Encodable 𝓨] [MeasurableSingletonClass 𝓨]\n    (f : 𝓧 → 𝓨 → α)\n    [∀ x, DecidablePred fun n ↦ ∃ y, n = Encodable.encode y ∧ ∀ (z : 𝓨), f x z ≤ f x y]\n    (x : 𝓧) :\n    𝓨 :=\n  (measurableEmbedding_encode 𝓨).invFun (Nat.find (exists_isMaxOn' f x))\n\nlemma measurable_measurableArgmax [Nonempty 𝓨] [Finite 𝓨] [Encodable 𝓨]\n    [MeasurableSingletonClass 𝓨]\n    {f : 𝓧 → 𝓨 → α}\n    [∀ x, DecidablePred fun n ↦ ∃ y, n = Encodable.encode y ∧ ∀ (z : 𝓨), f x z ≤ f x y]\n    (hf : ∀ y, Measurable (fun x ↦ f x y)) :\n    Measurable (measurableArgmax f) := by\n  refine (MeasurableEmbedding.measurable_invFun (measurableEmbedding_encode 𝓨)).comp ?_\n  refine measurable_find _ fun n ↦ ?_\n  have : {x | ∃ y, n = Encodable.encode y ∧ ∀ (z : 𝓨), f x z ≤ f x y}\n      = ⋃ y, ({x | n = Encodable.encode y} ∩ {x | ∀ z, f x z ≤ f x y}) := by ext; simp\n  rw [this]\n  refine MeasurableSet.iUnion fun y ↦ (MeasurableSet.inter (by simp) ?_)\n  exact measurableSet_isMax (by fun_prop) y\n\nend Finite\n\nvariable {𝓐 𝓨 : Type*} [MeasurableSpace 𝓐] [MeasurableSpace 𝓨]\n\nstructure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where\n  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐\n  [h_policy : ∀ n, IsMarkovKernel (policy n)]\n  p0 : Measure 𝓐\n  [hp0 : IsProbabilityMeasure p0]\n\nattribute [instance] Algorithm.h_policy Algorithm.hp0\n\nstructure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where\n  feedback : (n : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨\n  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]\n  ν0 : Kernel 𝓐 𝓨\n  [hp0 : IsMarkovKernel ν0]\n\nattribute [instance] Environment.h_feedback Environment.hp0\n\nnoncomputable\ndef stepKernel (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :\n    Kernel (Iic n → 𝓐 × 𝓨) (𝓐 × 𝓨) :=\n  alg.policy n ⊗ₖ env.feedback n\n\ninstance (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) (n : ℕ) :\n    IsMarkovKernel (stepKernel alg env n) := by\n  rw [stepKernel]; infer_instance\n\nnoncomputable\ndef trajMeasure (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨) :\n    Measure (ℕ → 𝓐 × 𝓨) :=\n  Kernel.trajMeasure (alg.p0 ⊗ₘ env.ν0) (stepKernel alg env)\n\nnoncomputable\ndef gap {𝓐 : Type*} [MeasurableSpace 𝓐] (ν : Kernel 𝓐 ℝ) (a : 𝓐) : ℝ :=\n  (⨆ i, (ν i)[id]) - (ν a)[id]\n\nnoncomputable\ndef regret {𝓐 Ω : Type*} [MeasurableSpace 𝓐] (ν : Kernel 𝓐 ℝ) (A : ℕ → Ω → 𝓐) (t : ℕ) (ω : Ω) : ℝ :=\n  t * (⨆ a, (ν a)[id]) - ∑ s ∈ range t, (ν (A s ω))[id]\n\nsection History\n\nvariable {𝓐 R : Type*} [DecidableEq 𝓐] {n : ℕ}\n\nnoncomputable def pullCount' (n : ℕ) (h : Iic n → 𝓐 × R) (a : 𝓐) : ℕ :=\n  #(univ.filter (fun s ↦ (h s).1 = a))\n\nlemma pullCount'_eq_sum (n : ℕ) (h : Iic n → 𝓐 × R) (a : 𝓐) :\n    pullCount' n h a = ∑ s : Iic n, if (h s).1 = a then 1 else 0 := by\n  rw [pullCount', Finset.card_filter]\n\nnoncomputable def sumRewards' (n : ℕ) (h : Iic n → 𝓐 × ℝ) (a : 𝓐) : ℝ :=\n  ∑ s, if (h s).1 = a then (h s).2 else 0\n\nnoncomputable def empMean' (n : ℕ) (h : Iic n → 𝓐 × ℝ) (a : 𝓐) : ℝ :=\n  sumRewards' n h a / pullCount' n h a\n\nvariable [MeasurableSpace 𝓐] [MeasurableSingletonClass 𝓐] [MeasurableSpace R]\n\n@[fun_prop]\nlemma measurable_pullCount' (n : ℕ) (a : 𝓐) :\n    Measurable (fun h : Iic n → 𝓐 × R ↦ pullCount' n h a) := by\n  simp_rw [pullCount'_eq_sum]\n  have h_meas s : Measurable (fun (h : Iic n → 𝓐 × R) ↦ if (h s).1 = a then 1 else 0) := by\n    refine Measurable.ite ?_ (by fun_prop) (by fun_prop)\n    exact (measurableSet_singleton _).preimage (by fun_prop)\n  fun_prop\n\n@[fun_prop]\nlemma measurable_sumRewards' (n : ℕ) (a : 𝓐) :\n    Measurable (fun h : Iic n → 𝓐 × ℝ ↦ sumRewards' n h a) := by\n  simp_rw [sumRewards']\n  have h_meas s : Measurable (fun (h : Iic n → 𝓐 × ℝ) ↦ if (h s).1 = a then (h s).2 else 0) := by\n    refine Measurable.ite ?_ (by fun_prop) (by fun_prop)\n    exact (measurableSet_singleton _).preimage (by fun_prop)\n  fun_prop\n\n@[fun_prop]\nlemma measurable_empMean' (n : ℕ) (a : 𝓐) :\n    Measurable (fun h : Iic n → 𝓐 × ℝ ↦ empMean' n h a) := by\n  unfold empMean'; fun_prop\n\nend History\n\ndef RoundRobin.nextAction {K : ℕ} (hK : 0 < K) (n : ℕ) : Fin K := ⟨(n + 1) % K, Nat.mod_lt _ hK⟩\n\nnoncomputable def detAlgorithm (nextA : (n : ℕ) → (Iic n → 𝓐 × 𝓨) → 𝓐)\n    (h_next : ∀ n, Measurable (nextA n)) (action0 : 𝓐) : Algorithm 𝓐 𝓨 where\n  policy n := Kernel.deterministic (nextA n) (h_next n)\n  p0 := Measure.dirac action0\n\nnoncomputable def stationaryEnv (ν : Kernel 𝓐 𝓨) [IsMarkovKernel ν] : Environment 𝓐 𝓨 where\n  feedback _ := ν.comap Prod.snd measurable_snd\n  ν0 := ν\n\nnoncomputable def cHat {K : ℕ} (δ : ℝ) (n : ℕ) (h : Iic n → Fin K × ℝ) (a : Fin K) : ℝ :=\n  Real.sqrt (Real.log (2 * (pullCount' n h a + 1) ^ 2 / δ) / (2 * pullCount' n h a))\n\n@[fun_prop]\nlemma measurable_cHat {K : ℕ} (δ : ℝ) (n : ℕ) (a : Fin K) :\n    Measurable (fun h : Iic n → Fin K × ℝ ↦ cHat δ n h a) := by\n  unfold cHat; fun_prop\n\nopen Classical in\n\nnoncomputable def nlNextArm {K : ℕ} (hK : 0 < K) (δ : ℝ) (n : ℕ) (h : Iic n → Fin K × ℝ) : Fin K :=\n  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK\n  if n < K - 1 then RoundRobin.nextAction hK n else\n  measurableArgmax (fun h a ↦ empMean' n h a + cHat δ n h a) h\n\nopen Classical in\n@[fun_prop]\nlemma measurable_nlNextArm {K : ℕ} (hK : 0 < K) (δ : ℝ) (n : ℕ) :\n    Measurable (nlNextArm hK δ n) := by\n  have : Nonempty (Fin K) := Fin.pos_iff_nonempty.mp hK\n  unfold nlNextArm\n  split_ifs with hc\n  · exact measurable_const\n  · refine measurable_measurableArgmax fun a ↦ ?_\n    fun_prop\n\nnoncomputable def nlAlgo {K : ℕ} (hK : 0 < K) (δ : ℝ) : Algorithm (Fin K) ℝ :=\n  detAlgorithm (nlNextArm hK δ) (measurable_nlNextArm hK δ) ⟨0, hK⟩\n\ntheorem ch16_theorem_16_7 {K : ℕ} (hK : 0 < K) (ν : Kernel (Fin K) ℝ) [IsMarkovKernel ν]\n    (T : ℕ) (hKT : K ≤ T) (hT : 0 < T) (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)\n    (hrewards : ∀ a, ∀ᵐ x ∂(ν a), x ∈ Set.Icc (0 : ℝ) 1)\n    (nextA : (n : ℕ) → (Iic n → Fin K × ℝ) → Fin K)\n    (hmeasA : ∀ n, Measurable (nextA n))\n    (hwarm : ∀ n h, n < K - 1 → nextA n h = RoundRobin.nextAction hK n)\n    (hargmax : ∀ n h a, K - 1 ≤ n →\n      empMean' n h a + cHat δ n h a ≤ empMean' n h (nextA n h) + cHat δ n h (nextA n h)) :\n    (trajMeasure (detAlgorithm nextA hmeasA ⟨0, hK⟩) (stationaryEnv ν))\n        {ω : ℕ → Fin K × ℝ |\n          regret ν (fun s ω => (ω s).1) T ω ≤\n            (⨅ Δ₀ : {x : ℝ // 0 < x},\n              ((T : ℝ) - K) * (Δ₀ : ℝ) +\n                ∑ a : Fin K,\n                  Set.indicator {a' : Fin K | (Δ₀ : ℝ) < gap ν a'}\n                    (fun a' => 2 * Real.log (2 * K * T ^ 2 / δ) / gap ν a') a) +\n            ∑ a : Fin K, gap ν a}\n      ≥ ENNReal.ofReal (1 - δ) := sorry\n\nend GraduateLevel\n", "target_names": ["ch16_theorem_16_7"], "topic_domain": "Online Learning & RL", "topic_subtopic": "Bandits", "holes": [{"start": 7133, "end": 8086, "gold": "{K : ℕ} (hK : 0 < K) (ν : Kernel (Fin K) ℝ) [IsMarkovKernel ν]\n    (T : ℕ) (hKT : K ≤ T) (hT : 0 < T) (δ : ℝ) (hδ : 0 < δ) (hδ1 : δ ≤ 1)\n    (hrewards : ∀ a, ∀ᵐ x ∂(ν a), x ∈ Set.Icc (0 : ℝ) 1)\n    (nextA : (n : ℕ) → (Iic n → Fin K × ℝ) → Fin K)\n    (hmeasA : ∀ n, Measurable (nextA n))\n    (hwarm : ∀ n h, n < K - 1 → nextA n h = RoundRobin.nextAction hK n)\n    (hargmax : ∀ n h a, K - 1 ≤ n →\n      empMean' n h a + cHat δ n h a ≤ empMean' n h (nextA n h) + cHat δ n h (nextA n h)) :\n    (trajMeasure (detAlgorithm nextA hmeasA ⟨0, hK⟩) (stationaryEnv ν))\n        {ω : ℕ → Fin K × ℝ |\n          regret ν (fun s ω => (ω s).1) T ω ≤\n            (⨅ Δ₀ : {x : ℝ // 0 < x},\n              ((T : ℝ) - K) * (Δ₀ : ℝ) +\n                ∑ a : Fin K,\n                  Set.indicator {a' : Fin K | (Δ₀ : ℝ) < gap ν a'}\n                    (fun a' => 2 * Real.log (2 * K * T ^ 2 / δ) / gap ν a') a) +\n            ∑ a : Fin K, gap ν a}\n      ≥ ENNReal.ofReal (1 - δ)"}]}
{"item_id": "ch5_theorem_5_13::ch5_theorem_5_13::A-stmt", "paper_id": "ch5_theorem_5_13", "target_name": "ch5_theorem_5_13", "source": "Graduate-level", "nl_statement": "Consider a finite function class \\(\\mathcal{F} = \\{f_1, \\ldots, f_d\\}\\), and assume that for a distribution \\(\\mathcal{D}\\) and \\(p \\in [1, p]\\), \\(\\sup_{f \\in F} \\|f\\|_{L_p(\\mathcal{D})} \\leq A\\). Then for \\(\\epsilon \\leq A\\):\n\\[\n\\ln M(\\epsilon,\\operatorname{CONV}(\\mathcal{F}),L_{p}(\\mathcal{D}))\\leq d\\ln(3 A/\\epsilon).\n\\]", "context": [{"name": "theorem 5.3", "statement": "Let \\(\\| \\cdot \\|\\) be a seminorm on \\(\\mathbb{R}^k\\). Let \\(B(r) = \\{z \\in \\mathbb{R}^k : \\| z \\| \\leq r\\}\\) be the \\(\\| \\cdot \\|\\)-ball with radius \\(r\\). Then\n\\[\nM(\\epsilon,B(r),\\|\\cdot\\|)\\leq\\left(1+2 r/\\epsilon\\right)^{k}.\n\\]\nMoreover, if \\(\\| \\cdot \\|\\) is a norm, then\n\\[\nN(\\epsilon,B(r),\\|\\cdot\\|)\\geq(r/\\epsilon)^{k}.\n\\]"}, {"name": "definition 5.12", "statement": "The convex hull of a function class \\(\\mathcal{F} = \\{f(\\theta, x) : \\theta \\in \\Theta\\}\\) is defined as\n\\[\n\\operatorname{CONV}(\\mathcal{F})=\\left\\{\\sum_{j=1}^{m}w_{j}f\\left(\\theta_{j},x\\right):m>0,\\|w\\|_{1}=1,w_{j}\\geq 0,\\theta_{j}\\in\\Theta\\right\\}.\n\\]\nWe also include the closure of the finite sum functions above with respect to an appropriate topology in the convex hull."}, {"name": "Packing Number", "statement": "Let \\((\\mathcal{V},d)\\) be a pseudometric space with metric \\(d(\\cdot,\\cdot)\\). A finite subset \\(\\mathcal{G}(\\epsilon)\\subset \\mathcal{G}\\) is an \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\) if \\(d(\\phi,\\phi^{\\prime})>\\epsilon\\) for all \\(\\phi,\\phi^{\\prime}\\in\\mathcal{G}(\\epsilon)\\). The \\(\\epsilon\\)-packing number of \\(\\mathcal{G}\\), denoted by \\(M(\\epsilon,\\mathcal{G},d)\\), is the largest cardinality of \\(\\epsilon\\)-packing of \\(\\mathcal{G}\\).\nThe following results illustrate the equivalence between covering number and packing number. One advantage of using an \\(\\epsilon\\) packing of \\(\\mathcal{G}\\) instead of an \\(\\epsilon\\) cover of \\(\\mathcal{G}\\) is that all members in the \\(\\epsilon\\) packing also belong to \\(\\mathcal{G}\\). There if members in \\(\\mathcal{G}\\) satisfy certain assumptions such as the variance condition, then members of its \\(\\epsilon\\) packing also satisfy such assumptions. For this reason, we will use packing numbers instead of covering numbers in some of the theoretical analysis in later chapters."}, {"name": "Bracketing Number", "statement": "Let \\(\\mathcal{G} = \\{\\phi(w, \\cdot) : w \\in \\Omega\\}\\) be a real-valued function class, equipped with a pseudometric \\(d\\). We say\n\\[\n\\mathcal{G}(\\epsilon)=\\left\\{\\left[\\phi_{1}^{L}(z),\\phi_{1}^{U}(z)\\right],\\dots,\\left[\\phi_{N}^{L}(z),\\phi_{N}^{U}(z)\\right]\\right\\}\n\\]\nis an \\(\\epsilon\\)-bracket of \\(\\mathcal{G}\\) under metric \\(d\\) if for all \\(w \\in \\Omega\\), there exists \\(j = j(w)\\) such that \\(\\forall z\\):\n\\[\n\\phi_{j}^{L}(z)\\leq\\phi(w,z)\\leq\\phi_{j}^{U}(z),\\quad d(\\phi_{j}^{L},\\phi_{j}^{U})\\leq\\epsilon .\n\\]\nThe \\(\\epsilon\\)-bracketing number is the smallest cardinality \\(N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) of such \\(\\mathcal{G}(\\epsilon)\\). The quantity \\(\\ln N_{\\mathbb{I}}(\\epsilon, \\mathcal{G}, d)\\) is called \\(\\epsilon\\) bracketing entropy.\nIn particular, given a distribution \\(\\mathcal{D}\\) and \\(p \\geq 1\\), we define \\(L_{p}\\)-seminorm in function space as\n\\[\n\\left\\|f-f^{\\prime}\\right\\|_{L_{p}(\\mathcal{D})}=\\left[\\mathbb{E}_{Z\\sim\\mathcal{D}}|f(Z)-f^{\\prime}(Z)|^{p}\\right]^{1/p}.\\tag{3.19}\n\\]\nIt induces a pseudometric, denoted as \\( d = L_{p}(\\mathcal{D}) \\), and the corresponding bracketing number is \\( N_{\\parallel}(\\epsilon, \\mathcal{G}, L_{p}(\\mathcal{D})) \\).\nFor \\(p = \\infty\\), the \\(L_{\\infty}(\\mathcal{D})\\)-seminorm is defined as the essential supremum seminorm, which leads to the pseudometric\n\\[\nd(f,f^{\\prime})=\\inf\\left\\{\\omega:\\Pr_{\\mathcal{D}}[|f(Z)-f^{\\prime}(Z)|\\leq\\omega]=1\\right\\}.\n\\]\nHowever, when \\( p = \\infty \\), instead of using the \\( L_{\\infty} \\) bracketing number, it is more conventional to use the equivalent notion of \\( L_{\\infty} \\) covering number (see Proposition 4.3)."}], "nl_notes": "", "lean": "import Mathlib\n\nopen MeasureTheory Real ENNReal\n\nnamespace GraduateLevel\n\ndef IsPacking {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n  ↑t ⊆ s ∧ (↑t : Set A).Pairwise fun x y => eps < dist x y\n\ntheorem ch5_theorem_5_13\n    {Ω : Type*} [MeasurableSpace Ω]\n    (D : Measure Ω) [IsProbabilityMeasure D]\n    (d : ℕ)\n    (p : ℝ≥0∞) [Fact (1 ≤ p)] (hp_top : p ≠ ⊤)\n    (f : Fin d → Lp ℝ p D)\n    (A : ℝ) (hA : 0 < A)\n    (hbound : ∀ j, ‖f j‖ ≤ A)\n    (ε : ℝ) (hε : 0 < ε) (hεA : ε ≤ A)\n    (Q : Finset (Lp ℝ p D))\n    (hQpack : IsPacking Q ε (convexHull ℝ (Set.range f))) :\n    Real.log (Q.card : ℝ) ≤ (d : ℝ) * Real.log (3 * A / ε) := sorry\n\nend GraduateLevel\n", "target_names": ["ch5_theorem_5_13"], "topic_domain": "Statistical Learning Theory", "topic_subtopic": "Function Class Complexity", "holes": [{"start": 258, "end": 661, "gold": "{Ω : Type*} [MeasurableSpace Ω]\n    (D : Measure Ω) [IsProbabilityMeasure D]\n    (d : ℕ)\n    (p : ℝ≥0∞) [Fact (1 ≤ p)] (hp_top : p ≠ ⊤)\n    (f : Fin d → Lp ℝ p D)\n    (A : ℝ) (hA : 0 < A)\n    (hbound : ∀ j, ‖f j‖ ≤ A)\n    (ε : ℝ) (hε : 0 < ε) (hεA : ε ≤ A)\n    (Q : Finset (Lp ℝ p D))\n    (hQpack : IsPacking Q ε (convexHull ℝ (Set.range f))) :\n    Real.log (Q.card : ℝ) ≤ (d : ℝ) * Real.log (3 * A / ε)"}]}