diff --git a/bleach/__init__.py b/bleach/__init__.py
index bbcc2e03..8774e273 100644
--- a/bleach/__init__.py
+++ b/bleach/__init__.py
@@ -52,7 +52,7 @@ def clean(
 
     :arg str text: the text to clean
 
-    :arg list tags: allowed list of tags; defaults to
+    :arg set tags: set of allowed tags; defaults to
         ``bleach.sanitizer.ALLOWED_TAGS``
 
     :arg dict attributes: allowed attributes; can be a callable, list or dict;
diff --git a/bleach/html5lib_shim.py b/bleach/html5lib_shim.py
index d121953b..022fe19e 100644
--- a/bleach/html5lib_shim.py
+++ b/bleach/html5lib_shim.py
@@ -38,6 +38,9 @@
     allowed_protocols,
     allowed_css_properties,
     allowed_svg_properties,
+    attr_val_is_uri,
+    svg_attr_val_allows_ref,
+    svg_allow_local_href,
 )  # noqa: E402 module level import not at top of file
 from bleach._vendor.html5lib.filters.sanitizer import (
     Filter as SanitizerFilter,
diff --git a/bleach/sanitizer.py b/bleach/sanitizer.py
index da9637c1..35ccf71e 100644
--- a/bleach/sanitizer.py
+++ b/bleach/sanitizer.py
@@ -8,21 +8,23 @@
 from bleach import parse_shim
 
 
-#: List of allowed tags
-ALLOWED_TAGS = [
-    "a",
-    "abbr",
-    "acronym",
-    "b",
-    "blockquote",
-    "code",
-    "em",
-    "i",
-    "li",
-    "ol",
-    "strong",
-    "ul",
-]
+#: Set of allowed tags
+ALLOWED_TAGS = frozenset(
+    (
+        "a",
+        "abbr",
+        "acronym",
+        "b",
+        "blockquote",
+        "code",
+        "em",
+        "i",
+        "li",
+        "ol",
+        "strong",
+        "ul",
+    )
+)
 
 
 #: Map of allowed attributes by tag
@@ -93,7 +95,7 @@ def __init__(
     ):
         """Initializes a Cleaner
 
-        :arg list tags: allowed list of tags; defaults to
+        :arg set tags: set of allowed tags; defaults to
             ``bleach.sanitizer.ALLOWED_TAGS``
 
         :arg dict attributes: allowed attributes; can be a callable, list or dict;
@@ -190,13 +192,11 @@ def clean(self, text):
         dom = self.parser.parseFragment(text)
         filtered = BleachSanitizerFilter(
             source=self.walker(dom),
-            # Bleach-sanitizer-specific things
+            allowed_tags=self.tags,
             attributes=self.attributes,
-            strip_disallowed_elements=self.strip,
+            strip_disallowed_tags=self.strip,
             strip_html_comments=self.strip_comments,
             css_sanitizer=self.css_sanitizer,
-            # html5lib-sanitizer things
-            allowed_elements=self.tags,
             allowed_protocols=self.protocols,
         )
 
@@ -260,19 +260,21 @@ class BleachSanitizerFilter(html5lib_shim.SanitizerFilter):
     def __init__(
         self,
         source,
-        allowed_elements=ALLOWED_TAGS,
+        allowed_tags=ALLOWED_TAGS,
         attributes=ALLOWED_ATTRIBUTES,
         allowed_protocols=ALLOWED_PROTOCOLS,
-        strip_disallowed_elements=False,
+        attr_val_is_uri=html5lib_shim.attr_val_is_uri,
+        svg_attr_val_allows_ref=html5lib_shim.svg_attr_val_allows_ref,
+        svg_allow_local_href=html5lib_shim.svg_allow_local_href,
+        strip_disallowed_tags=False,
         strip_html_comments=True,
         css_sanitizer=None,
-        **kwargs,
     ):
         """Creates a BleachSanitizerFilter instance
 
         :arg source: html5lib TreeWalker stream as an html5lib TreeWalker
 
-        :arg list allowed_elements: allowed list of tags; defaults to
+        :arg set allowed_tags: set of allowed tags; defaults to
             ``bleach.sanitizer.ALLOWED_TAGS``
 
         :arg dict attributes: allowed attributes; can be a callable, list or dict;
@@ -281,8 +283,16 @@ def __init__(
         :arg list allowed_protocols: allowed list of protocols for links; defaults
             to ``bleach.sanitizer.ALLOWED_PROTOCOLS``
 
-        :arg bool strip_disallowed_elements: whether or not to strip disallowed
-            elements
+        :arg attr_val_is_uri: set of attributes that have URI values
+
+        :arg svg_attr_val_allows_ref: set of SVG attributes that can have
+            references
+
+        :arg svg_allow_local_href: set of SVG elements that can have local
+            hrefs
+
+        :arg bool strip_disallowed_tags: whether or not to strip disallowed
+            tags
 
         :arg bool strip_html_comments: whether or not to strip HTML comments
 
@@ -290,24 +300,24 @@ def __init__(
             sanitizing style attribute values and style text; defaults to None
 
         """
+        # NOTE(willkg): This is the superclass of
+        # html5lib.filters.sanitizer.Filter. We call this directly skipping the
+        # __init__ for html5lib.filters.sanitizer.Filter because that does
+        # things we don't need to do and kicks up the deprecation warning for
+        # using Sanitizer.
+        html5lib_shim.Filter.__init__(self, source)
+
+        self.allowed_tags = frozenset(allowed_tags)
+        self.allowed_protocols = allowed_protocols
+
         self.attr_filter = attribute_filter_factory(attributes)
-        self.strip_disallowed_elements = strip_disallowed_elements
+        self.strip_disallowed_tags = strip_disallowed_tags
         self.strip_html_comments = strip_html_comments
-        self.css_sanitizer = css_sanitizer
 
-        # filter out html5lib deprecation warnings to use bleach from BleachSanitizerFilter init
-        warnings.filterwarnings(
-            "ignore",
-            message="html5lib's sanitizer is deprecated",
-            category=DeprecationWarning,
-            module="bleach._vendor.html5lib",
-        )
-        return super().__init__(
-            source,
-            allowed_elements=allowed_elements,
-            allowed_protocols=allowed_protocols,
-            **kwargs,
-        )
+        self.attr_val_is_uri = attr_val_is_uri
+        self.svg_attr_val_allows_ref = svg_attr_val_allows_ref
+        self.css_sanitizer = css_sanitizer
+        self.svg_allow_local_href = svg_allow_local_href
 
     def sanitize_stream(self, token_iterator):
         for token in token_iterator:
@@ -377,10 +387,10 @@ def sanitize_token(self, token):
         """
         token_type = token["type"]
         if token_type in ["StartTag", "EndTag", "EmptyTag"]:
-            if token["name"] in self.allowed_elements:
+            if token["name"] in self.allowed_tags:
                 return self.allow_token(token)
 
-            elif self.strip_disallowed_elements:
+            elif self.strip_disallowed_tags:
                 return None
 
             else: